On the Observability of Individual Population III Stars and Their Stellar-mass Black Hole Accretion Disks through Cluster Caustic Transits

In Figure 1, the green unfilled squares at 2–3 μm indicate the Kelsall et al. (1998) COBE DIRBE sky-SB from zodiacal light, which is scattered sunlight. At 3–200 μm, these COBE DIRBE points are dominated by the ∼200 K thermal dust component in the zodiacal belt. Most of this dust is piled up in the asteroid belt and is clearly a limiting factor for near–mid-IR observations, including for JWST observations at λ ≳ 3.5 μm. Figure 1 plots with solid green points the zodiacal foreground as measured from low-Earth orbit using the panchromatic Hubble Space Telescope (HST) Wide Field Camera 3 (WFC3) Early Release Science (ERS) observations of Windhorst et al. (2011, hereafter W11) and its precursor data from the Great Orbiting Observatories Deep Survey (GOODS) Advanced Camera for Surveys (ACS) data (Giavalisco et al. 2004). This includes the zodiacal sky measurements in the Hubble Ultra Deep Field (HUDF) by Hathi et al. (2008). The green dotted line is the solar energy spectrum (Kurucz 2005) normalized to these HST data.

All units in Figure 1 have been converted to $\nu {I}_{\nu }$ in units of nW m⁻² sr⁻¹. For reference, 1.00 nW m⁻² sr⁻¹ corresponds to 28.41 mag arcsec⁻² at 2.00 μm, which is indicated by the orange K-band SB-scale in AB-mag arcsec⁻² on the right vertical axis of Figure 1. At other near-IR wavelengths, one can derive the SB-scale corresponding to the $\nu {I}_{\nu }$ scale on the left by adding −2.5 log (λ/2.0 μm) to the K-band scale on the right.

An important comment on the WFC3 ERS data of W11 is in order here. Figure 1 shows that 8 of the 10 ERS filters have sky-background measurements in line with the zodiacal foreground at those wavelengths. However, their bluest and reddest filters (WFC3/UVIS F225W and WFC3/IR F160W) have a sky level significantly in excess of the zodiacal foreground for this ecliptic latitude. This was expected for the F225W filter, as this bluest WFC3 filter was intentionally scheduled at the end of each available HST orbit, so that any Earthshine would add some sky level to the highly readnoise-limited UV images, since the zodiacal sky is darkest at the shortest HST wavelengths. Indeed, the resulting F225W background level was significantly higher than that expected from the zodiacal sky alone. In all other ERS filters, every possible effort was made to avoid the Earth's limb, but this was not fully successful for the WFC3/IR filter F160W, and its resulting sky background was ∼0.3 dex higher than expected, despite our scheduling attempts to avoid this. In the remaining eight ERS filters, the root mean square (rms) variation from the best-fit normalized solar energy spectrum is 10%–20%, illustrating that even in the case of requesting HST "LOW-SKY" observations—and going to great lengths in the Astronomer's Proposal Tool (APT) scheduling requests to make sure that the sky background remains close to the theoretical zodiacal minimum—some Earthshine may have nonetheless leaked into the low-Earth orbit observations.

The red dots in Figure 1 indicate the integrated EBL measurements derived from the panchromatic (0.1–500 μm) discrete galaxy counts from GALEX, HST, ground-based, Spitzer, WISE, and Herschel surveys, as summarized in Driver et al. (2016, hereafter D16), which incorporated the panchromatic HST galaxy counts at $\lambda \simeq 0.2\mbox{--}2$ μm to AB ≲ 29–30 mag discussed in W11. From 0.1–500 μm, the discrete galaxy counts converge well at almost all wavelengths, except for the less deep Spitzer/WISE galaxy counts at 8–12 μm, where the galaxy count extrapolation that yields the iEBL integral is ∼40% uncertain. Typically, the normalized differential galaxy counts in D16 reach a peak at AB ≃ 19–25 mag, where most of their iEBL energy is contained. At all wavelengths except 8–12 μm, the normalized differential counts converge—with a slope flatter than 0.4 dex/mag—to a finite sky integral that results in a well-determined iEBL value for discrete objects to within 10%–20%, including random errors, count extrapolation errors, and cosmic variance that were determined through Monte Carlo simulations. For clarity, error bars are omitted from Figure 1, but these can be found in D16. The iEBL from discrete objects is thus well-determined to within ≲20% in general, as indicated by the small scatter in the red dots in Figure 1 compared to the iEBL models of Andrews et al. (2017b).

The red, green, and purple dashed lines indicate the contributions that spheroids, disks, and unobscured active galactic nuclei (AGNs) at z ≲ 6 may contribute to the EBL energy, following Andrews et al. (2017a, 2017b). Obscured AGNs in these models are incorporated into the spheroidal galaxies and not plotted separately. The contribution from unobscured AGNs to the discrete iEBL is uncertain, but at their median redshift of ${z}_{\mathrm{med}}$ ≃ 2, AGNs may produce enough total rest-frame UV radiation (at ${\lambda }_{\mathrm{rest}}$ ≳ 912–1216 Å) to contribute significantly to the observed near-UV background (${\lambda }_{\mathrm{obs}}\,$≲ 0.4 μm). Even below ${\lambda }_{\mathrm{rest}}\simeq 912$ Å, AGNs at z ≃ 2–3 may produce non-negligible LyC radiation (possibly made visible through outflows) to the reionizing budget at these redshifts (e.g., Madau & Haardt 2015; Smith et al. 2018). As we will discuss below, these discrete object iEBL measurements are directly relevant to the possible sky-SB contributed from unresolved objects, such as Pop III stars and their stellar-mass BH accretion disks at z ≳ 7.

The light gray unfilled downwards triangles in Figure 1 indicate the direct measurements or limits to the EBL, which are in general absolute measurements, and are summarized in detail in Dwek & Krennrich (2013) and D16. Most of these direct EBL estimates are a factor of 3–5× higher than the integrated and extrapolated discrete-objects counts (the iEBL), and about  ≳ 2× higher in the far-IR, although the latter is in general agreement within the errors. Given that non-zodiacal foreground light may enter into the low-Earth orbit observations at the ≳10% level as discussed above, it is therefore possible that the true level of foreground (zodiacal+Earthshine and other straylight components) may have been undersubtracted in some of the direct EBL measurements.

Here we summarize arguments that the diffuse EBL is likely smaller than the iEBL that comes from discrete objects, especially in the near-IR. This will help us derive our first constraints to the diffuse EBL that may be caused by Pop III stars and their stellar-mass BH accretion disks. Any real diffuse EBL could be due to faint Inter-galaxy Halo Light (IHL; Cooray et al. 2012), IntraCluster Light (ICL), or IntraGroup Light (IGL) not measured by Source Extractor-type algorithms (Bertin & Arnouts 1996) in discrete object surveys, or to truly diffuse, unresolved populations, such as Pop III stars and their BH accretion disks. Our reasoning that there may not be a large amount of near-IR diffuse light hidden is as follows:

(1) Independent diffuse EBL estimates at 0.3–20 μm come from γ-ray blazar spectra and how much these are distorted from their original power-law shape. When a γ-ray from the blazar hits an intervening EBL photon, this can result in pair production and energy loss in the power-law spectrum. This constrains the total EBL level that each of the low-redshift blazar γ-ray photons are exposed to (Dwek & Krennrich 2013; Lorentz et al. 2015). Figure 1 indicates the resulting EBL constraints as a gray shaded region+blue line and a green shaded region+dark-green line from the blazar surveys with the High Energy Stereoscopic System (H.E.S.S.; H.E.S.S. Collaboration et al. 2013, 2017) and the Major Atmospheric Gamma Imaging Cherenkov telescope (MAGIC; Ahnen et al. 2016), respectively. The MAGIC shaded region in Figure 1 is smaller than that of HESS, and closer to the red iEBL points of D16. The extent to which the γ-ray blazar spectra deviate from their intrinsic power laws constrains the amplitude and shape of the foreground component of the EBL spectrum directly (Biteau & Williams 2015), which is completely independent from having to subtract the zodiacal foreground. Biteau & Williams (2015) found that the amount of diffuse EBL at $\lambda \simeq 1\mbox{--}5$ μm is ≲1–2 nW m⁻² sr⁻¹. For a detailed discussion of these blazar data and their constraints on the EBL, we refer the reader to Dwek & Krennrich (2013) and D16. In short, the allowed amount of total 1–5 μm EBL from the γ-ray blazar spectral constraints is generally quite consistent with the integrated and extrapolated discrete galaxy counts (red dots in Figure 1) summarized in D16.

At 0.45–0.65 μm, the diffuse blazar EBL and the discrete iEBL measurements are—to within their errors—also consistent with the direct Pioneer spacecraft measurements (Matsuoka et al. 2011), which were made at a distance of 4.6 au from the Sun, i.e., well away from most of the zodiacal foreground brightness (blue unfilled circles in Figure 1). The direct R-band Pioneer EBL measurement was confirmed through the first measurement in a broader R-band with the Long Range Reconnaissance Imager instrument on board the New Horizons spacecraft on its way to Pluto at ∼7–16 au from the Sun (Zemcov et al. 2017), albeit with a larger error bar, which will improve as further New Horizons data are taken. At these very large distances from the Sun, the uncertainties due to the zodiacal foreground are much smaller than those from low-Earth orbit. Ground-based optical spectroscopy of dark clouds was done by Mattila et al. (2017) to remove the Diffuse Galactic Light (DGL), suggesting a diffuse EBL component at $\lambda \simeq 0.4\mbox{--}0.6$ μm possibly as high as ∼4–6 nW m⁻² sr⁻¹. The good correspondence at λ ≲ 1 μm among the iEBL from the discrete extrapolated galaxy counts (D16), the direct Pioneer and New Horizons B+R-band observations at 4.6–16 au, and the independent constraints from the H.E.S.S./MAGIC blazar γ-ray spectra in Figure 1, suggests that a low-redshift, truly diffuse EBL component at $\lambda \simeq 0.4\mbox{--}1$ μm may not exceed the iEBL component itself, which is ∼4–10 nW m⁻² sr⁻¹.

Despite uncertainties in the optical diffuse EBL, the 1–4 μm iEBL results are consistent with the blazar constraints on the diffuse EBL to within their errors. Below, we will therefore adopt an upper limit to the diffuse 1–4 μm EBL based on the difference between the γ-ray blazar constraints from H.E.S.S. + MAGIC and the integrated plus extrapolated galaxy counts of D16. If any diffuse 1–4 μm EBL were truly 3×–5× higher than what the red dots in Figure 1 indicate, such a high EBL level would have distorted the blazar spectra more than is observed in Figure 1. Comparing the H.E.S.S. and MAGIC blazar constraints to the EBL from the discrete galaxy counts in Figure 1 suggests that a diffuse 1–4 μm EBL component (Biteau & Williams 2015) may add ∼20% to the iEBL from discrete objects (D16).

(2) The deepest ground-based surveys with large telescopes do not detect an excessive amount of light in the outskirts of galaxies that have total fluxes of AB ≃ 20–23 mag. It is precisely in this flux range where most of the iEBL is generated in the observed blue wavelength regime (see D16). For instance, Ashcraft et al. (2017) present 32 hr LBT U-band images sorted as a function of image FWHM value. The best 10% of their 320 images with the highest resolution (FWHM ≲ 07) reach AB ≲ 27.0 mag for point-source detection, while their best-depth 32 hr image has FWHM ≲ 18, reaches AB ≲ 28.0 mag for point sources, and has a 1σ SB sensitivity of AB ≲ 32 mag arcsec⁻². Ashcraft et al. (2017) then compare the light profiles of 220 galaxies with total fluxes of AB ≃ 20–23 mag in both their highest-resolution images and in their best-depth LBT U-band image, and find that no more than 5%–10% of the total galaxy flux is missing in the high-resolution images compared to the deeper low-resolution images. That is, at least in the U-band for galaxies AB ≃ 20–23 mag—over which most of the iEBL is generated (see Section 2.1)—no more than 10% of the light appears to be hidden in the outskirts of these galaxies down to AB ≲ 32 mag arcsec⁻². The integrated and extrapolated U-band galaxy counts of W11, D16 and Ashcraft et al. (2017) are consistent with the H.E.S.S. and MAGIC blazar constraints at 0.36 μm, with little room to hide more than 10%–20% in diffuse EBL at 0.36 μm. Longer-wavelength studies of this depth have been done with the 10 meter Grand Canary Telescope (Trujillo & Fliri 2016), with similar results in the r-band.

(3) Combining the constraints from the previous two arguments, we derive the following limit to the diffuse 1–4 μm EBL: (1) a diffuse 1–4 μm EBL component can add ∼20% to the iEBL from discrete objects, and (2) no more than 10%–20% in diffuse EBL seems to be hidden in galaxy outskirts to AB ≲32 mag arcsec⁻². More could come from low-redshift ICL, but cluster galaxies comprise a small fraction of the total galaxy population. Some could come from IGL at low redshifts, since most galaxies reside in galaxy groups (Robotham et al. 2011). Where the IGL has been measured, it does not appear to dominate the total stellar light in galaxy groups (e.g., Robotham et al. 2011) or in galaxy clusters (e.g., Morishita et al. 2017; Griffiths et al. 2018 and references therein). In all, Figure 1 suggests that diffuse 1–4 μm EBL may well be as low as 20% of the discrete iEBL, or ≲1–2 nW m⁻² sr⁻¹ at 2 μm. We will use this level as a conservative or "hard" upper limit for any Pop III contribution to the near-IR EBL, as indicated by the orange circle with its dotted 1–4 μm range in Figure 1.

If the diffuse 1–4 μm EBL from Pop III stars or accretion disks at z ≳ 7 was much larger than our hard upper limit of ≲1–2 nW m⁻² sr⁻¹ at 2 μm, it would exceed the known components from unobscured AGNs and even galaxy disks (blue and green dashed lines in Figure 1) at z ≲ 6, which would be unheard of at any other wavelength in the electromagnetic spectrum. That is, the diffuse 1–4 μm EBL from Pop III stars and/or their accretion disks is likely well below the level indicated by our hard upper limit at 1–2 nW m⁻² sr⁻¹ in Figure 1. In Section 4.4, we will estimate the Pop III caustic transit rate for a range of possible diffuse 1–4 μm EBL values, and estimate which SB levels may result in observable numbers of Pop III caustic transits during JWST's lifetime.

Next, we adopt tighter constraints to the sky-SB from Pop III stars from recent theoretical and observational constraints, and from Pop III stellar-mass BH accretion disks using recent near-IR–X-ray power-spectrum results. We need both sky-SB constraints to estimate their cluster caustic transits in Sections 4.4 and 6.2, respectively.

The thermal brightness of the zodiacal belt rapidly increases at wavelengths λ ≳ 4 μm (Figure 1), and so in the calculations below we do not anticipate easily detecting Pop III caustic transits with JWST at wavelengths longer than 4 μm. For Pop III objects at z ≳ 7, the wavelength range of interest is therefore $\lambda \simeq 1\mbox{--}4$ μm. The geometric average of this wavelength range is λ = 2.0 μm, which is also equal to the JWST diffraction limit (Rieke et al. 2005). JWST NIRCam will be most sensitive over the wavelength range of 2–3.5 μm, where the zodiacal sky from L2 is darkest (Figure 1 and W11).

2.3.1. Diffuse EBL Limits Adopted for Pop III Stars

Based on metallicity arguments, Madau & Silk (2005) provided a constraint suggesting that Pop III stars must contribute less than a few nW m⁻² sr⁻¹ to the (1–4 μm) InfraRed Background (IRB). This is consistent with our hard diffuse-EBL upper limit in Section 2.2. Cooray et al. (2012) provide a detailed Pop III model for reionization and estimate the Pop III flux to be ≲0.04 nW m⁻² sr⁻¹ (see their Figure 4), which we confirm below. Bovill (2016) suggests a Pop III star density of 0.1–10³ stars per arcsec² between z ≃ 10–30, which we consider in more detail in Section 3.5. For their expected range in luminosities, Pop III stars could have an observed flux of AB ≃ 35–41.5 mag over the redshift range of z ≃ 7–17 (see Section 3). As an example, if there existed ∼1000 Pop III stars of 100 M_⊙ each per arcsec², then their integrated 2.0 μm sky-SB would be  ≳33 mag arcsec⁻² or ≲0.016 nW m⁻² sr⁻¹, which is comparable to the Cooray et al. (2012) limit.

To confirm these numbers, we will estimate the average sky-SB from star-forming objects at z ≃ 7–8 from the actual HUDF data corrected for incompleteness. For our caustic transit calculations, we need to estimate the maximum possible SB from Pop III stars at z ≳ 7 to use as the most conservative upper limit. This needs to take into account that the steep faint end of the galaxy luminosity function (LF) at z ≳ 7 will contribute additional flux from unseen Pop III objects beyond the detection limit of the deepest HST and JWST images, and an estimate of the maximum additional sky-SB from z ≃ 9 to z ≃ 17. We proceed with this calculation in three steps:

(a) The average sky-SB from star-forming objects at z ≃ 7–8 from the actual HUDF data corrected for incompleteness: we use the actual HUDF data at z ≃ 7–8 (Table A1 of Bouwens et al. 2015) to estimate the observed surface densities of star-forming objects at z ≃ 7 and z ≃ 8 to an average sky-SB. In the 4.7 arcmin² effective area of the WFC3/IR data, there are 56 dropout candidates detected at z ≃ 7 to the HUDF limit of AB ≃ 30.0 mag, while there are 28 dropout candidates at z ≃ 8 to AB ≲ 30 mag. These can be directly converted to a total sky-SB, in this case from the objects detected to AB ≲ 30 mag. We need to correct these observed surface densities by about a factor of 1.8, since in the deepest HUDF WFC3/IR images, at least ∼45% of the detector pixels are covered by the wings of the foreground objects (Koekemoer et al. 2013). Our own insertion of artificial objects into the HUDF WFC3/IR images confirms this correction factor.

(b) Maximum contribution from the steep faint end of the galaxy luminosity function down to the luminosity of single Pop III stars: next, we correct this upper limit to the 2.0 μm sky-SB that comes from z ≃ 7–8 for the flux of objects that will have been missed below the current HUDF object detection limit of AB ≃ 30 mag. At z ≃ 7 to z ≃ 10, the AB ≃ 30 mag HUDF detection limits correspond to absolute magnitudes of ${M}_{\mathrm{AB}}$ ≃ –17.5 mag. According to the fits to the available galaxy LF data in Figures 5–6 of Finkelstein (2016), the faint-end slope of the galaxy LF at z ≳ 7 may become as steep as α ≲ –2.0 to −2.3, while the characteristic Schechter luminosities (L^∗ or M^∗) and space densities (${{\rm{\Phi }}}^{* }$) may well continue to get fainter and decline at z ≳ 7, respectively. High-resolution hierarchical simulations of the faint-end galaxy LF-slope evolution with redshift (e.g., Morgan et al. 2015) suggested values of $\alpha \simeq -2.1$ from z ≃ 11 to z ≃ 4. This is about as steep as the Initial Mass Function (IMF) slope for more massive stars (Coulter et al. 2017), and would occur if the luminosity density is dominated by Pop III stars with M ≳ 100 M_⊙, for which $L\propto {M}$ approximately holds. We discuss this further in Sections 3.1 and 3.4.

We will adopt for simplicity in our extrapolation $\alpha \simeq -2.0$, so that each additional luminosity bin with objects at z ≳ 7 that are currently beyond the HST detection limit would contribute roughly equal amounts of energy to the sky-SB. Since the M* values at z ≃ 7–8 in the best fits of Finkelstein (2016) are about ${M}^{* }\simeq -20.5$ mag, the sky-SB in the HUDF from objects that are currently resolved into galaxies comes effectively from a ∼3 mag range in the observed LF. If we extrapolate this LF with a faint-end Schechter slope α = −2.0 from ${M}_{\mathrm{AB}}$ ≃ −17.5 mag to ${M}_{\mathrm{AB}}$ ≃ −7 mag (i.e., the luminosity of a 20 M_⊙-star; see Section 3), then the integrated 2.0 μm sky-SB will be ∼3× brighter than the estimate from (a) alone.

Integrating the maximum SB that can come from Pop III stars at z ≳ 7 to ${M}_{\mathrm{AB}}$ ≃ −7 mag is meaningful and necessary, since at this luminosity a faint star-forming "object" would simply consist of a single unresolved Pop III star with M ≳ 20 M_⊙ and ${M}_{\mathrm{AB}}$ ≃ −7 mag, which is the faintest JWST could detect at z ≳ 7 during a favorable caustic transit (see Section 3). Given the homology relations in Section 3.1, the M–L relation for such massive stars becomes approximately L ∝ M, so that the very faint-end slope of the object luminosity function may reflect the bright-end slope of the stellar-mass function at z ≳ 8.

If we integrate down to the limit of a 1.5 M_⊙ Pop III star luminosity of ${M}_{\mathrm{AB}}$ ≃ +2 mag at z ≳ 7 (see Section 3), then the maximum 2.0 μm sky-SB will be ∼5× brighter than the estimate from (a). Since these are the coolest stars that can contribute to reionization (Section 3), we will use this multiplier to derive the most conservative upper limit to the 2.0 μm sky-SB that may come from z ≳ 7. The maximum 2.0 μm sky-SB we then obtain from the entire object LF to ${M}_{\mathrm{AB}}$ = +2 mag is 32.2 mag arcsec⁻² at z ≃ 7 and 32.8 mag arcsec⁻² at z ≃ 8.

(c) Maximum contribution from the cosmic star formation history at z ≳ 8: lastly, we need to correct these limits for the maximum contribution from the LF of star-forming objects at z ≃ 9–17 that is not yet accounted for. For this, we use a best fit of the cosmic star-formation history (SFH) data summarized by Madau & Dickinson (2014) and Finkelstein (2016). Equation (15) of Madau & Dickinson (2014) gives a best fit to the cosmic SFR data over the entire redshift range 0 ≲ z ≲ 8:

$$\begin{eqnarray}&&\psi (z)=0.015\,\displaystyle \frac{{(1+z)}^{2.7}}{1+{[(1+z)/2.9]}^{5.6}}\,{M}_{\odot }\,{\mathrm{yr}}^{-1}\,{\mathrm{Mpc}}^{-3}.\end{eqnarray} \tag{ 1 }$$

Their best fit has its peak in the cosmic SFR at z ≃ 1.9. The best-fit power-law slope for $z\gg 2$ is approximately 2.7–5.6 ≃ −2.9, so that at z ≃ 7 the SFR is ∼1.0 dex or 2.5 mag lower than at z ≃ 1.9. This decline is also seen in the more recent HST WFC3 data reviewed by Finkelstein (2015, 2016) and Madau & Fragos (2017), who find a slightly steeper decline of ∝(1+z)^−3.6 to (1+z)^−4.2 when only fitting the data for z ≳ 2. The difference in slope could be due to a truly steeper decline in the cosmic SFR at z ≳ 8, the smaller fitted redshift range used in these more recent papers, or larger incompleteness corrections for dropout samples at z ≳ 7, as discussed in (a).

To obtain the most conservative upper limit to the integrated sky-SB from z = 7 to z = 17, we will use the highest predicted SFR at z ≳ 8. Hence, we will use the extrapolation of Madau & Dickinson (2014) in Equation (1), since it is ∼0.3 dex above the fits of Finkelstein (2016) and Madau & Fragos (2017) to the most recent WFC3 data at z ≃ 8–10. The extrapolation of Equation (1) is also consistent with the hierarchical model predictions of Sarmento et al. (2018) at 7 ≲ z ≲ 20, which approximately match the Madau & Dickinson (2014) results at z ≃ 7–8. The extrapolation of Equation (1) thus yields the highest observed sky-SB that may come from star-forming objects at z ≳ 8, which is used in Section 4.4 to predict the highest level of caustic transits that could be seen. That is, if the true Pop III star sky-SB is lower than what we predict from Equation (1) here, then the caustic transit rates will be correspondingly smaller, as indicated in Figure 1 and discussed in Section 4.4.

With the Madau & Dickinson (2014) extrapolation of Equation (1), more than half of the sky-SB that comes from 7 ≲ z ≲ 17 is already obtained from the redshift shell at z ≃ 7, while about 75% comes from the two redshift shells at z ≃ 7 and z ≃ 8 combined. Each redshift shell here is assumed to have a width of ${\rm{\Delta }}z\simeq 1$. The contributions from the redshift shells at z ≳ 12 are negligibly small. Integrating the sky-SB produced by each redshift shell by Equation (1) from z ≃ 7–17 thus produces approximately 1.33× the flux than that from the z ≃ 7–8 redshift shells alone, where we directly summed the observed sky-SB in (a).

This then results in a most conservative upper limit to the 2.0 μm sky-SB from star-forming objects at 7 ≲ z ≲ 17 down to the luminosity of a single Pop III star. This upper limit to the 2.0 μm Pop III star sky-SB is ≳31.4 ± 0.6 mag arcsec⁻² or ≲0.06 nW m⁻² sr⁻¹, which is indicated by the dark orange unfilled triangle and its error and wavelength range in Figure 1.

2.3.2. Diffuse EBL Limits Adopted for Pop III Stellar-mass Black Hole Accretion Disks

We first need to outline the plausible physical parameter ranges for Pop III stars. Figure 2 shows the zero-age main sequence (ZAMS) in an HR diagram for stellar evolution models with Z = 0.00 Z_⊙, and the inset shows their corresponding mass–radius relation. These non-rotating, zero-metallicity, zero mass-loss, single-star 1–1000 M_⊙ models were calculated using the MESA software instrument (Paxton et al. 2011, 2013, 2015) with the same physical and numerical parameters as those in Farmer et al. (2015, 2016) and Fields et al. (2016). We also calculated MESA models for Z = 10⁻⁸ Z_⊙, and their results were very similar to Z = 0.00 Z_⊙. This is because stars more massive than ∼2 M_⊙ make enough of their own carbon in their cores to run the CNO cycle appropriate for their mass. In other words, there is a floor metallicity, which—if not provided by the star's birth composition—will be made by the star itself, and convective episodes may bring part of these self-made metals to the stellar photosphere. Here, we adopt the set of Z = 0.00 Z_⊙ MESA models for Pop III stars and will discuss the possible effects of metallicity in more detail below and in Section 5.2.

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There may be model-dependent variations in the MS ages, depending on the age definitions and on the chemical mixing algorithms used (e.g., convection, overshooting, etc). For details, we refer to Paxton et al. (2011, 2013, 2015). In our MESA models, the ZAMS by definition starts when the nuclear luminosity reaches 90% of the total stellar luminosity. The Terminal-Age Main Sequence (TAMS) is defined when the central hydrogen mass fraction drops to below 10⁻⁶ of the star's core mass, which is when the ZAMS ends. At this point, shell hydrogen burning dominates the energy production, and can be taken as the "beginning" of the "Giant Branch." Core helium depletion is defined as the stage in the star's evolution when the fraction of ⁴He drops below 10⁻⁶ of the core mass of the star. These definitions are more precise than the common use of "Red Giant Branch" or "Asymptotic Giant Branch," but for the sake of brevity, we will henceforth refer to these latter stages as the "RGB" and "AGB," respectively. The MS age adopted here is defined as the time between the start of the ZAMS and the start of the RGB (core-hydrogen depletion), while the "Giant Branch" (GB) age is defined between the start of the RGB and the end of the AGB, when the star has run out of ⁴He.

Stars more massive than ≳100 M_⊙ are radiation-pressure dominated. For CNO burning, constant electron scattering, and radiative transport, the ZAMS "homology" relations for massive stars (Hoyle & Lyttleton 1942; Faulkner 1967; Pagel & Portinari 1998; Bromm et al. 2001; Portinari et al. 2010) are

$$\begin{eqnarray}\left(\displaystyle \frac{R}{10\ {R}_{\odot }}\right) & \simeq & {\left(\displaystyle \frac{Z}{{10}^{-7}{Z}_{\odot }}\right)}^{1/11}{\left(\displaystyle \frac{M}{400{M}_{\odot }}\right)}^{5/11},\\ \left(\displaystyle \frac{{T}_{\mathrm{eff}}}{{10}^{5}\ {\rm{K}}}\right) & \simeq & {\left(\displaystyle \frac{Z}{{10}^{-7}{Z}_{\odot }}\right)}^{-1/20}{\left(\displaystyle \frac{M}{100{M}_{\odot }}\right)}^{1/40},\\ \left(\displaystyle \frac{L}{{L}_{\odot }}\right) & \simeq & \left(\displaystyle \frac{{L}_{\mathrm{edd}}}{{L}_{\odot }}\right)\simeq {10}^{5}\left(\displaystyle \frac{M}{{M}_{\odot }}\right),\end{eqnarray} \tag{ 2 }$$

and they approximate the trends shown by our detailed MESA models for Z ≲ 10⁻⁸ Z_⊙ (Figure 2 and Table 1).

Table 1.  Adopted Pop III Star Physical Parameters from MESA Models^a

Mass	Age	Teff	$\mathrm{log}R$	$\mathrm{log}{L}_{\mathrm{bol}}$	Teff	$\mathrm{log}R$	$\mathrm{log}{L}_{\mathrm{bol}}$	Age	${T}_{\mathrm{eff}}$	$\mathrm{log}R$	$\mathrm{log}{L}_{\mathrm{bol}}$	Age	Timeb
	Pre-MS	at ZAMS			at Hydrogen Depletion				at Helium Depletion				AGB-MS
(${M}_{\odot })$	(Myr)	(K)	(R⊙)	(L⊙)	(K)	(R⊙)	(L⊙)	Myr	(K)	(R⊙)	(L⊙)	Myr	(Myr)
1.0	9.28	7.266e3	−0.0581	0.2825	6.999e3	0.5119	1.3576	5882	⋯c	⋯	⋯	6420	538
1.5	6.11	1.065e4	−0.0203	1.0227	1.181e4	0.3292	1.9015	1501	8.149e3	0.7913	2.1804	1670	169
2.0	3.02	1.367e4	0.0108	1.5177	1.611e4	0.2498	2.2815	642	1.145e4	0.6685	2.5249	702	60
3.0	1.38	1.899e4	0.0487	2.1654	2.311e4	0.1843	2.7770	201	1.736e4	0.5510	3.0138	228	27
5.0	0.56	2.805e4	0.0911	2.9274	3.206e4	0.1903	3.3581	53	2.658e4	0.4608	3.5732	70	17
10	0.23	4.508e4	0.1462	3.8618	4.174e4	0.3807	4.1972	17	3.938e4	0.4811	4.2968	19	1.6
15	0.13	5.789e4	0.1803	4.3647	4.624e4	0.5401	4.6937	10	4.215e4	0.6581	4.7691	11	0.8
20	0.09	6.754e4	0.2183	4.7082	4.864e4	0.6612	5.0240	7.8	4.386e4	0.7879	5.0975	8.4	0.6
30	0.05	7.737e4	0.3270	5.1619	5.180e4	0.8120	5.4347	5.6	4.006e4	1.0688	5.5016	6.0	0.5
50	0.03	8.713e4	0.4570	5.6283	5.490e4	0.9722	5.8562	3.7	3.536e4	1.3862	5.9200	4.3	0.5
100	0.02	9.796e4	0.6147	6.1470	5.173e4	1.2610	6.3303	2.8	3.392e4	1.6437	6.3627	3.1	0.3
300	0.02	1.074e5	0.8697	6.8172	4.882e4	1.6111	6.9301	2.1	3.165e4	2.0041	6.9631	2.4	0.3
1000	0.02	1.080e5	1.1090	7.3047	4.807e4	1.8740	7.4288	2.1	3.122e4	2.2119	7.3549	2.4	0.3

Notes.

^aAll physical Pop III star parameters were calculated using MESA models with zero initial metallicity (Z = 0.00 Z_⊙), zero mass loss, zero rotation, and no stellar duplicity (i.e., no binaries/multiple stars). Pop III star parameters are listed with a sufficient number of significant digits to be able to integrate them assuming blackbody spectra, which is needed in Section 3.3. ^bAges are listed for the pre-main-sequence collapse (pre-MS), the core+shell hydrogen-burning phase (H-depletion), the core+shell helium-burning phase (He-depletion), and the total giant branch lifetime (i.e., the AGB--MS age difference). The latter provides an upper limit to the BH feeding times due to Roche-lobe overflow in non-zero-metallicity massive-star binaries, as discussed in Section 5. ^cThe 1.0 M_⊙, Z = 0.00 Z_⊙ model did not ignite helium and may thus turn directly into a helium white dwarf, so no AGB parameters are listed here. (Its AGB ages and K-corrections in Section 3.3 are those of the Z = 10⁻⁸ Z_⊙ model, which did end in a white dwarf).

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Metallicity affects the evolution of single stars in four distinct ways: it sets their initial abundance, and it impacts their energy generation, opacity, and their mass-loss mechanism. For low-metallicity stars, these homology relations approximate the metallicity dependence of their radii and luminosities. For metallicities Z ≲ 10⁻⁴ Z_⊙, Equation (2) suggest that the ZAMS tracks in Figure 2 will shift systematically by a factor of ∼0.7 toward lower effective temperatures almost independently of mass, while the ZAMS luminosities would be nearly independent of the metallicity. The ZAMS radii will correspondingly shift by a factor of ∼2 toward larger values. Given the other much larger uncertainties in our Pop III star caustic transit calculations in Section 4.4, we will adopt the physical parameter values of the zero-metallicity Pop III stars (or Z = 0.00 Z_⊙) from our MESA modeling runs.

Equation (2) suggests that the bolometric luminosities of zero-metallicity Pop III stars—as modeled in our MESA runs—are to first order directly proportional to their ZAMS mass, while the mass–radius and mass–T_eff relations have much shallower slopes. All three parameters in Equation (2) need to be carefully traced as a function of ZAMS mass for our caustic transit calculations in Section 4.4.

A closer inspection of Figure 2 suggests that the third line of Equation (2) is only approximately correct for stars with M ≳ 100 M_⊙. Over the mass range of 1 ≲ M ≲ 1000 M_⊙, the bolometric ZAMS luminosities of Pop III stars in Figure 2 scale to a better approximation with ZAMS mass M as

$$\begin{eqnarray}L & \simeq & {L}_{100}\ {(M/100{M}_{\odot })}^{1.16},(100\lesssim M\lesssim 1000\,{M}_{\odot }),\\ & \simeq & {L}_{100}\ {(M/100{M}_{\odot })}^{2.06},\ \ \ \ \ (20\lesssim M\lesssim 100\ {M}_{\odot }),\\ & \simeq & {L}_{20}\ {(M/20{M}_{\odot })}^{3.20},\ \ \ \ \ \ (1\lesssim M\lesssim 20\ {M}_{\odot }),\end{eqnarray} \tag{ 3 }$$

where L₁₀₀ and L₂₀ are the luminosities of a 100 M_⊙ and a 20 M_⊙ star, respectively. The first two segments in this equation are rescaled to the parameters of a 100 M_⊙ star, and the third to a 20 M_⊙ star. The first segment is the nearly linear mass–luminosity relation for the most massive (M ≳ 100 M_⊙) Pop III stars in Equation (2), the second segment is a good approximation for Pop III stars in the intermediate-mass range (20 ≲ M ≲ 100 M_⊙), and the third segment is the M/L relation for Pop III stars with M ≃ 1–20 M_⊙, which has a slope of ∼3.2, similar to the slope of the M/L relation for lower-mass stars in our own Galaxy. Our caustic transit calculations in Section 4.4 are dependent on stellar luminosity, and so we will propagate the segmented M/L relation of Equation (3) into the relevant equations (Equations (19)–(30)) in Section 4.4.

Here we discuss in more detail the MESA Pop III star physical parameters that are needed to estimate their resulting caustic transit rates at z ≳ 7:

Masses: The mass range for Pop III stars that have luminosities bright enough for caustic transit detection by JWST at z ≳ 7 is 30–1000 M_⊙ (see Table 1 and Section 4.4). This corresponds to a logarithmic mass range of M ≃ 175 M_⊙ ± 0.75 dex, with a corresponding bolometric absolute magnitude range of ${M}_{\mathrm{AB}}$ ≃ −10.8 ± 2.5 mag. As discussed in Section 5.1, LIGO has detected several BHs at the lower end of this mass range, some of which may be the remnants of Pop III stars. BH accretion disks in stellar binaries are therefore considered in Sections 5–6 as possible additional sources of caustic transits at z ≳ 7. When computing physical quantities below, we use a solar mass of 1.989 × 10³⁰ kg (Mamajek et al. 2015).

Temperatures: Effective temperatures were determined by integrating a finely zoned MESA photospheric model inward to an optical depth of τ = 1. This radial location becomes the effective radius ${R}_{\mathrm{eff}}$ at which the effective temperature is ${T}_{\mathrm{eff}}$. Zero-metallicity Pop III stars have ZAMS photospheric temperatures ranging from ${T}_{\mathrm{eff}}$ ≃ 7300–108,000 K for masses in the range of M ≃ 1–1000 M_⊙, as summarized in Table 1. During the RGB stage, Pop III stars with M ≃ 1–1000 M_⊙ have lower temperatures ranging from 7000–55,000 K, while during the AGB stage, their temperatures are even lower, ranging from 6300–44,000 K (see Figure 2). For both post-MS stages, the photospheric temperatures change nonmonotonically with mass. This will affect their bolometric, IGM, and K-corrections in a nonlinear way as a function of mass and stellar evolution stage (see Section 3.3). When calculating their bolometric corrections using blackbody curves below, we use as reference a solar photospheric temperature of 5772 K (Mamajek et al. 2015; Prša et al. 2016).

Radii: Figure 2 and Table 1 show that zero-metallicity Pop III stars have ZAMS effective radii ranging from ${R}_{\mathrm{eff}}$ ≃ 0.9–13 R_⊙. These are in line with previous predictions (e.g., Woosley et al. 2002; Hirschi 2007; Ohkubo et al. 2009; Yusof et al. 2013). In 2015, the IAU adopted—for stellar normalization purposes—a value of the solar radius of 1.00 R_⊙ ≡ 695,700 km (Mamajek et al. 2015; Prša et al. 2016), which was guided by recent space-based measurements (e.g., Emilio et al. 2012). Hence, the Pop III star ZAMS radii in Table 1 range from ${R}_{\mathrm{Pop}\mathrm{III}}=6.05\times {10}^{8}-8.97\times {10}^{9}$ m, which are the values we use for Pop III star caustic transit rate predictions in Section 4.4. The Pop III star radii are at most between 1.3×–5.8× larger during their RGB phase, and 2.3×–14× larger during their AGB phase (see Table 1).

Luminosities: The MESA models shown in Figure 2 yield the bolometric absolute magnitudes of Pop III stars as a function of their mass and for different stellar evolution ages. Zero-metallicity Pop III stars have ZAMS luminosities ranging from ${L}_{\mathrm{bol}}$ ≃ 1.9 L_⊙−2.0 × 10⁷ L_⊙ for masses M ≃ 1–1000 M_⊙, respectively. During the RGB stage, their luminosities range from ${L}_{\mathrm{bol}}$ ≃ 23 L_⊙−2.7 × 10⁷ L_⊙, while during the AGB stage, they are ${L}_{\mathrm{bol}}$ ≃ 40 L_⊙−2.4 × 10⁷ L_⊙. These are the full bolometric stellar luminosities as predicted by the MESA code.

Several stellar atmosphere calculations have suggested that zero-metallicity Pop III stars in the ZAMS, RGB, and AGB stages can be approximated as blackbody emitters (e.g., Bromm et al. 2001) due to the lack of atomic absorption features or line blanketing in their spectra. For our calculations below, we therefore approximate the Pop III stars with blackbody curves of the same photospheric temperatures and radii from the MESA models summarized in Tables 2–4. We integrated these blackbody curves in ${I}_{\nu }$ from hard X-ray to radio wavelengths, and use their listed stellar radii (in kilometers using R_⊙ above) to predict the theoretical luminosities integrated under the full Planck curve for stars of that size. We use these results to convert their bolometric luminosities to observed apparent magnitudes in JWST's near-IR filters, which requires the distance modulus (DM) as a function of redshift, their bolometric corrections, corrections for IGM transmission, and their K-corrections (see Section 3.3).

Table 2.  Implied ZAMS Pop III Star Observational Parameters Relevant to Caustic Transit Calculations

Massa	${T}_{\mathrm{eff}}$ b	Radiusc	${L}_{\mathrm{bol}}$ d	${M}_{\mathrm{bol}}$ e	Bolo+IGM+K-corrf			ZAMS ${m}_{\mathrm{UV}}$ g			${t}_{\mathrm{rise}}$ h	Transiti
ZAMS	at ZAMS				z = 7	z = 12	z = 17	z = 7	z = 12	z = 17	caust	Rate
(M⊙)	(K)	(R⊙)	(L⊙)	(AB)	(AB-mag)			(AB-mag)			(hr)	(/cl/yr)
1.0	7.266e3	0.87	1.92	+4.03	+4.44	+3.13	+2.61	57.71	57.74	58.07	0.17	8 × 105
1.5	1.065e4	0.95	10.5	+2.18	+1.45	+0.42	−0.06	52.87	53.18	53.55	0.18	1.1 × 104
2.0	1.367e4	1.03	32.9	+0.95	+0.30	−0.59	−1.06	50.49	50.93	51.31	0.20	1.5 × 103
3.0	1.899e4	1.12	146.	−0.67	−0.51	−1.26	−1.72	48.06	48.64	49.03	0.22	182
5.0	2.805e4	1.23	846.	−2.58	−0.70	−1.35	−1.80	45.96	46.65	47.04	0.24	29.1
10	4.508e4	1.40	7.28e3	−4.91	−0.22	−0.79	−1.23	44.10	44.88	45.27	0.27	5.70
15	5.789e4	1.51	2.32e4	−6.17	+0.23	−0.30	−0.75	43.30	44.10	44.50	0.29	2.78
20	6.754e4	1.65	5.11e4	−7.03	+0.56	+0.04	−0.40	42.77	43.59	43.99	0.32	1.74
30	7.737e4	2.12	1.45e5	−8.16	+0.88	+0.36	−0.08	41.95	42.78	43.17	0.41?	0.82?
50	8.713e4	2.86	4.25e5	−9.33	+1.17	+0.66	+0.22	41.08	41.91	42.31	0.55*	0.37*
100	9.796e4	4.12	1.40e6	−10.63	+1.47	+0.96	+0.52	40.08	40.91	41.31	0.80*	0.15*
300	1.074e5	7.41	6.56e6	−12.30	+1.71	+1.21	+0.77	38.64	39.48	39.88	1.43*	0.039*
1000	1.080e5	12.9	2.02e7	−13.52	+1.72	+1.22	+0.78	37.44	38.28	38.68	2.48*	0.013*

Notes.

^aStellar mass in M_⊙. The physical parameters listed are for Pop III ZAMS stars in Table 1, as modeled with the MESA code for zero initial metallicity, zero mass loss, no rotation, and no stellar duplicity. ^bPop III star photospheric temperature ${T}_{\mathrm{eff}}$ in Kelvin. ^cPop III star radius ${R}_{\mathrm{eff}}$ at ${T}_{\mathrm{eff}}$ in R_⊙. ^dPop III star bolometric luminosity ${L}_{\mathrm{bol}}$ in L_⊙. ^ePop III star bolometric absolute magnitude ${M}_{\mathrm{bol}}$ in AB-mag. ^fCombined bolometric + IGM + K-corrections to Pop III star ${L}_{\mathrm{bol}}$ at z = 7, z = 12, and z = 17, respectively, from Section 3.3. ^gPop III star apparent rest-frame-UV AB-magnitudes at z = 7, z = 12, and z = 17 in 2016 Planck cosmology (Planck Collaboration et al. 2016b) using the NIRCam filters that sample the rest-frame UV 1500 Å, assuming K-corrections as in columns 6–8 and no dust (for a discussion of dust, see Section 6.1). Distance moduli used are DM = 49.24, 50.58, and 51.42 mag at z = 7, z = 12, and z = 17, respectively. ^hUpper limits to the caustic transit rise time t_rise (in hours) as estimated in Section 4.4. Asterisks indicate Pop III star ZAMS masses M ≳ 50 M_⊙, for which cluster caustic transit events are possibly observable to the detection limits of JWST medium-deep to deep survey epochs reaching AB ≲ 28.5–29 mag, assuming that caustic transit magnifications of $\mu \simeq {10}^{3}\mbox{--}{10}^{5}$ can elevate Pop III stars with AB ≲ 35–41.5 mag temporarily above these JWST detection limits. Stars with M ≲ 30 M_⊙ (labeled "?") likely remain undetectable even through caustic transits. ⁱThe cluster caustic transit rate for Pop III stars (number of events per cluster per year) at z = 12 as estimated in Section 4.4 by directly using Equation (27). Appendices C–D summarize the uncertainties relevant to caustic transits.

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Table 3.  Implied Red Giant Branch Pop III Star Observational Parameters Relevant to Caustic Transit Calculations

Massa	${T}_{\mathrm{eff}}$ b	Radiusc	${L}_{\mathrm{bol}}$ d	${M}_{\mathrm{bol}}$ e	Bolo+IGM+K-corrf			Giant Branch ${m}_{\mathrm{UV}}$ g			${t}_{\mathrm{rise}}$ h	Transiti
GB	at Hydrogen Depletion				z = 7	z = 12	z = 17	z = 7	z = 12	z = 17	Caust	Rate
(M⊙)	(K)	(R⊙)	(L⊙)	(AB)	(AB-mag)			(AB-mag)			(hr)	(/cl/yr)
1.0	6.999e3	3.25	22.8	+1.35	+4.83	+3.48	+2.96	55.42	55.41	55.73	0.63	9 × 104
1.5	1.181e4	2.13	79.7	−0.01	+0.91	−0.06	−0.53	50.13	50.51	50.88	0.41	1.0 × 103
2.0	1.611e4	1.78	191.	−0.96	−0.19	−1.01	−1.47	48.08	48.60	48.99	0.34	175
3.0	2.311e4	1.53	598.	−2.20	−0.69	−1.39	−1.84	46.35	46.99	47.38	0.30	39.8
5.0	3.206e4	1.55	2.28e3	−3.66	−0.63	−1.25	−1.70	44.95	45.67	46.07	0.30	11.8
10	4.174e4	2.40	1.57e4	−5.75	−0.34	−0.92	−1.36	43.15	43.91	44.31	0.46	2.33
15	4.624e4	3.47	4.94e4	−6.99	−0.18	−0.74	−1.19	42.06	42.84	43.24	0.67?	0.87?
20	4.864e4	4.58	1.06e5	−7.82	−0.10	−0.65	−1.09	41.32	42.11	42.51	0.88*	0.44*
30	5.180e4	6.49	2.72e5	−8.85	+0.02	−0.53	−0.97	40.41	41.20	41.60	1.25*	0.19*
50	5.490e4	9.38	7.18e5	−9.90	+0.13	−0.42	−0.86	39.47	40.26	40.66	1.81*	0.081*
100	5.173e4	18.2	2.14e6	−11.09	+0.02	−0.53	−0.98	38.17	38.96	39.36	3.52*	0.024*
300	4.882e4	40.8	8.51e6	−12.59	−0.09	−0.65	−1.09	36.57	37.35	37.75	7.88*	0.006*
1000	4.807e4	74.8	2.68e7	−13.83	−0.12	−0.67	−1.12	35.29	36.07	36.47	14.44*	0.002*

Note.

^a–iare as in Table 2. All parameters in this table are for Pop III stars at hydrogen depletion (RGB).

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Table 4.  Implied AGB Pop III Star Observational Parameters Relevant to Caustic Transit Calculations

Massa	${T}_{\mathrm{eff}}$ b	Radiusc	${L}_{\mathrm{bol}}$ d	${M}_{\mathrm{bol}}$ e	Bolo+IGM+K-corrf			AGB ${m}_{\mathrm{UV}}$ g			${t}_{\mathrm{rise}}$ h	Transiti
AGB	at Helium Depletion				z = 7	z = 12	z = 17	z = 7	z = 12	z = 17	Caust	Rate
(M⊙)	(K)	(R⊙)	(L⊙)	(AB)	(AB-mag)			(AB-mag)			(hr)	(/cl/yr)
1.0	6.312e3j	5.23j	39.8j	+0.74	+6.01	+4.57	+4.03	55.99	55.89	56.19	1.01	1.4 × 105
1.5	8.149e3	6.18	151	−0.71	+3.36	+2.14	+1.64	51.89	52.01	52.35	1.19	4.0 × 103
2.0	1.145e4	4.66	335	−1.57	+1.06	+0.07	−0.40	48.73	49.08	49.45	0.90	273
3.0	1.736e4	3.56	1.03e3	−2.79	−0.36	−1.15	−1.60	46.09	46.64	47.03	0.69	28.9
5.0	2.658e4	2.89	3.74e3	−4.19	−0.72	−1.38	−1.82	44.33	45.01	45.41	0.56	6.43
10	3.938e4	3.03	1.98e4	−6.00	−0.42	−1.00	−1.45	42.82	43.57	43.97	0.58	1.71
15	4.215e4	4.55	5.88e4	−7.18	−0.33	−0.90	−1.34	41.73	42.50	42.89	0.88?	0.64?
20	4.386e4	6.14	1.25e5	−8.00	−0.27	−0.84	−1.28	40.97	41.74	42.14	1.19*	0.32*
30	4.006e4	11.7	3.17e5	−9.01	−0.40	−0.98	−1.42	39.83	40.59	40.98	2.26*	0.11*
50	3.536e4	24.3	8.32e5	−10.06	−0.55	−1.15	−1.59	38.63	39.37	39.77	4.70*	0.036*
100	3.392e4	44.0	2.31e6	−11.17	−0.59	−1.19	−1.64	37.49	38.22	38.61	8.50*	0.012*
300	3.165e4	101.	9.19e6	−12.67	−0.64	−1.26	−1.71	35.93	36.65	37.04	19.49*	0.003*
1000	3.122e4	163.	2.26e7	−13.65	−0.65	−1.28	−1.72	34.94	35.66	36.05	31.45*	0.001*

Notes.

^a–iare as in Table 2. All parameters in this table are for Pop III stars at helium depletion (AGB). ^jThe 1.0 M_⊙, Z = 0.00 Z_⊙ MESA model did not ignite helium and may thus become a helium white dwarf. To complete our calculations, its AGB parameters listed are for the Z = 10⁻⁸ Z_⊙ model, which did end in a white dwarf.

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To normalize these calculations, we use the bolometric luminosity of the Sun, L_⊙ = 3.828 × 10²⁶ W, which by definition is produced by a blackbody with the effective temperature (T_eff = 5772 K) and radius (R_⊙ = 6.957 × 10⁸ m) adopted for the Sun (Mamajek et al. 2015). This corresponds to an absolute bolometric AB-magnitude of the Sun, which is by definition ${M}_{\mathrm{bol}}(\,\odot \,)\equiv +4.74$ mag (Bessell et al. 1998; Casagrande et al. 2006). Hence, all our Pop III star absolute magnitudes in Tables 2–4 are normalized to this ${M}_{\mathrm{bol}}$ value of the Sun.

This worked well for all Pop III stars in our MESA runs, except for the 1.0 M_⊙ zero-metallicity AGB model, whose MESA predictions were 2% lower than a blackbody curve at its specified T_eff. Its MESA model did not ignite helium, so the star may turn directly into a helium white dwarf. Hence, no AGB parameters are listed in Table 1 for a 1.0 M_⊙ zero-metallicity AGB star. To permit caustic calculations for a 1.0 M_⊙ AGB star in any case, Table 4 lists the parameters for the Z = 10⁻⁸ Z_⊙ MESA model, which did result in a white dwarf.

Ages: Pre-MS ages were estimated from the collapse time of a gas cloud of the specified mass—using atomic and molecular H cooling—to the onset of ZAMS stage, and are not added to the other ages below. Table 1 lists the MESA ages for the (estimated) pre-main sequence (pre-MS) collapse time, the core hydrogen-burning phase (H depletion), the shell hydrogen- and core+shell helium-burning phases (He depletion), and for the total time spent on the giant branches. That is, (${\tau }_{\mathrm{AGB}}$–${\tau }_{\mathrm{MS}}$) estimates the lifetime of the Red Giant Branch, Hot Horizontal Branch, and Asymptotic Giant Branch together (RGB+HHB+AGB). Note that for the most massive Pop III stars, HHB (core He burning) is of very short duration, and the stars essentially quickly transit from shell H burning to shell He burning in one smooth, nearly horizontal giant branch toward cooler ${T}_{\mathrm{eff}}$ values. We will refer to the combined RGB+HHB+AGB phases as the giant branch ("GB"). For all Pop III stars in Table 1, the time between core H depletion and core He depletion is about 8%–14% of the time between ZAMS and core H -depletion, with an average of ∼12%. For our caustic transit calculations in Section 4.4, we will take the approximate duration of the Pop III RGB and AGB stages each to be about 6% of their ZAMS lifetime.

For Pop III stars, we will use the post-MS lifetimes in Section 5 to estimate the maximum time that a lower-mass He-burning star may be feeding the accretion disk around a BH that was left over from a more massive Pop III companion star at z ≳ 7. This assumes no major mass exchange during the prior stellar evolution stages, i.e., we assume that stars in multiple systems evolve in isolation during the ZAMS stage following the MESA models.

For the most massive Pop III stars, their MS lifetime τ scales roughly as mass/luminosity. Since for the highest masses (M ≳ 100 M_⊙) luminosities are directly proportional to their ZAMS mass (see Equation (2)), the MESA models yield MS ages of 5.6–2.1 Myr that are only weakly dependent on the ZAMS mass for the mass range of 30–1000 M_⊙, respectively (see Table 1). The shortest MS lifetime possible is ∼2.07 Myr, which happens when the star is radiating at the Eddington luminosity, and so its age becomes nearly independent of mass and only a function of fundamental constants. The MESA models in Table 1 indeed approach this MS age limit for M ≃ 300–1000 M_⊙ to within the modeling uncertainties.

In summary, Tables 1–4 show that Pop III stars in the mass range of M ≃ 30–1000 M_⊙ have ZAMS photospheric temperatures of 77,000–108,000 K, bolometric luminosities of ${L}_{\mathrm{bol}}$ ≃ 10^5.2–10^7.3 L_⊙, stellar radii of ${R}_{\mathrm{MS}}$ ≃ 2–13 R_⊙, and main-sequence (MS) lifetimes of ${\tau }_{\mathrm{MS}}$ ≃ 2.1–5.6 Myr. They may therefore be bright enough for occasional caustic transit detections by JWST, which is summarized in columns 9–13 of Tables 2–4, as calculated in Section 4.4.

As discussed in Sections 2.3 and 4.4, we only use upper limits to the integrated 1–4 μm sky-SB to estimate the maximum Pop III object caustic transit rate. Hence, the actual Pop III star lifetimes do not directly enter these calculations. However, plausible differences in Pop III star GB lifetimes as a function of ZAMS mass are relevant when estimating limits to the caustic transit rates from Pop III stellar-mass BH accretion disks compared to those from Pop III stars. This is discussed in Sections 5–6, in which we consider the case where Pop III BH feeding may be done by lower-mass companion stars as soon as these can form in slightly polluted environments.

The effect of binaries and stars of higher multiplicities is a complex subject that can have a significant effect on population synthesis models of galaxies (e.g., Zhang et al. 2010; Conroy 2013; Stanway et al. 2016). For our present purposes, we must address the fact that the multiplicity factor MF is nearly unity for O-stars, at least in the local universe (e.g., Duchêne & Kraus 2013):

$$\begin{eqnarray}&&\mathrm{MF}=\displaystyle \frac{(B+T+Q)}{(S+B+T+Q)}\simeq 1.\end{eqnarray} \tag{ 4 }$$

Here, S is the number of single stars in a coeval stellar population, B the number of binary stars, T the number of triples, and Q the number of quads, etc., implying that one gets essentially a factor ≳2 increase in the luminosity from binary, triple, and quad stars together.

3.2.1. Multiplicity—Low-mass End

3.2.2. Multiplicity—High-mass End

3.2.3. Massive Star Multiplicity at Low Redshifts

O stars in nearby surveys show significant multiplicity (≳80%) and have a rather flat mass-ratio distribution:

$$\begin{eqnarray}&&q\equiv {M}_{\sec }/{M}_{\mathrm{pri}},\end{eqnarray} \tag{ 5 }$$

where M_pri is the more massive star (Duchêne & Kraus 2013). In theory, the q-value distribution can be as steep as the Salpeter (1955) mass function slope, i.e.,

$$\begin{eqnarray}&&N(q)\propto {q}^{-2.35}.\end{eqnarray} \tag{ 6 }$$

The observed q-distribution of nearby O-stars seems to have a slope much flatter than the IMF slope (Duchêne & Kraus 2013). Since nearby surveys of double/multiple stars may suffer from flux bias, very faint low-mass stellar companions around more massive stars are harder to find. In Sections 5–6, we will therefore assume that slightly polluted early massive stars have a mass ratio no steeper than the IMF slope, if they already occur in binaries.

Figure 2 of Duchêne & Kraus (2013) suggests that the majority of nearby binary OB stars have typical orbital periods in the range of ∼10–130 days and typical orbital separations between 0.067–0.51 au or 14 ≲ D ≲ 110 R_⊙. Larger and smaller separations can occur as well, some as small as a few R_⊙. Each of their Roche lobes will be about half that in effective radius, or 7 ≲ R ≲ 55 R_⊙. Following the MESA models of Tables 1–4, it is therefore possible that if Pop III or II.5 stars exist in binaries at z ≳ 7 as in OB binary stars today, their lower-mass companion stars with M ≳ 2 M_⊙ will fill their Roche lobes in the GB stage at z ≳ 7 with sizes 2 ≲ R ≲ 160 R_⊙ (see column 3 of Tables 3–4). This would feed the BH remnant from the more massive Pop III star during their AGB–MS lifetime, which could last up to ≲60 Myr (Section 3.2.1). Their BH UV accretion disk half-light radii (${r}_{\mathrm{hl}}$) are estimated in Section 5.5 to be in the range of 1 ≲ ${r}_{\mathrm{hl}}$ ≲ 30 R_⊙, and so in general will fit inside these Roche lobes when the companion star in the binary reaches the AGB stage.

Since we do not include Pop III star multiplicity in the MESA models, this will render the Pop III caustic transit rates of Section 4.4 more conservative. This is because caustic transit detections may be ≳0.75 mag brighter for binary stars—and possibly multipeaked in their detailed time sequence—than we estimate for single Pop III stars in Section 4.4. One example of a multiple caustic transit event has been suggested for a possible massive binary star at z ≃ 1.5 (Kelly et al. 2018). Future work that includes the evolution of massive binaries with possible mass exchange will need to model the evolutionary tracks of zero- or very low-metallicity stars, and how this may affect the BH feeding timescales.

The luminosities and absolute magnitudes of Pop III stars summarized in Section 3.1 and Tables 1–4 were calculated by the MESA code in bolometric solar units without making bolometric corrections, corrections for IGM transmission, or K-corrections. We thus need to correct the theoretical Pop III star luminosities for these effects to predict the ${m}_{\mathrm{AB}}$ values observed in the JWST filters at a given redshift. The exact bolometric and K-corrections cannot be computed until the actual object redshifts have been estimated from the eight-band JWST NIRCam photometry and/or measured with JWST NIRSpec or NIRISS spectra. For our current photometric predictions, we will therefore proceed as outlined below.

3.3.1. Pop III Star Bolometric+IGM Corrections

3.3.2. Pop III Star K-corrections

For each fixed JWST filter, we need to apply the K-correction to the Pop III star flux at λ ≳ 1216 Å that makes it through the IGM. Hogg (1999) and Hogg et al. (2002 and references therein) define the K-correction in ${I}_{\nu }$ units as follows:

$$\begin{eqnarray}&&K=-2.5\ \mathrm{log}\ [(1+z){L}_{(1+z)\nu }/{L}_{{\nu }_{e}}]\ \ (\mathrm{mag}).\end{eqnarray} \tag{ 7 }$$

This includes the effects of bandpass shifting due to the object's redshift and the change of the object's rest-frame SED with frequency or wavelength. The factor of (1+z) accounts for the fact that the flux and luminosity are not bolometric, but are flux densities per unit frequency (Hogg et al. 2002). Due to the complete IGM absorption for λ ≲ 1216 Å at z ≃ 7–17, our specific K-term needs to correct the flux predicted in the bluest available JWST filter that is completely longwards of redshifted Lyα for these effects. Hence, the luminosity ratio in Equation (7) only needs to account for the brighter flux shortward of this filter that still makes it through the IGM.

At z = 7, the bluest available JWST filter that is above the Lyman-break (F115W or short J-band) samples rest-frame ${\lambda }_{\mathrm{rest}}\simeq 1440\,\mathring{\rm A}$, while at z = 12 the F200W filter (short K-band) samples ${\lambda }_{\mathrm{rest}}\simeq 1645$ Å, and at z = 17 the F277W filter samples ${\lambda }_{\mathrm{rest}}\simeq 1540\,\mathring{\rm A}$. Since the most massive Pop III ZAMS stars have nearly uniform temperatures of T ≃ 10⁵ K (Table 1), all 1–5 μm JWST filters sample the Rayleigh–Jeans tail of their SEDs at z ≃ 7–17, so that ${I}_{\nu }\propto {\nu }^{2}$. The peak in their rest-frame (${I}_{\nu }$) SEDs thus occurs only a factor of 2–3.5 below the central rest-frame wavelength sampled by these filters. We then compute this K-correction from the flux bluewards of each filter down to the 1216 Å IGM transmission cutoff. Using Equation (7), the K-correction follows from these wavelength ratios as

$$\begin{eqnarray}&&K\simeq -2.5\ \mathrm{log}\ [(1+z){(1216/{\lambda }_{c})}^{2}]\ (\mathrm{mag}).\end{eqnarray} \tag{ 8 }$$

Here, ${\lambda }_{c}$ is the central rest-frame wavelength of the bluest JWST filter used for object detection at each redshift listed in Tables 2–4. The wavelength ratio (${\lambda }_{\mathrm{Ly}\alpha }$/${\lambda }_{c}$)² reflects the fact that we can only sample the ${I}_{\nu }\propto {\nu }^{2}$ tail of the very blue SEDs longwards of Lyα at z ≳ 7. For the extremely blue Pop III stars, this total K-term amounts to about −1.9 mag brighter at z = 7, −2.5 mag brighter at z = 12, and −2.9 mag brighter at z = 17. The K-correction gets somewhat brighter at the higher redshifts, because the (1+z) factor dominates the term from the wavelength or frequency ratio. For all longer wavelength JWST filters, the K-corrections can be computed similarly.

3.3.3. The Combined Bolometric+IGM+K-corrections

The combined bolometric+IGM+K-corrections (hereafter "BIK"-corrections) were calculated in two independent ways. First, all integrations were done in the observed frame while folding their blackbody SEDs with the JWST NIRCam filters and integrating the bolometric+IGM corrections as described in Section 3.3.1, and the K-corrections as in Section 3.3.2. Second, the calculations were done in the rest frame after folding the blackbody SEDs with the appropriately blueshifted JWST NIRCam filter curves, and integrating between each filters' rest-frame FWHM wavelength cutoffs. The NIRCam interference filters were designed to resemble block functions (Rieke et al. 2005), so this approximation is valid. Both methods gave similar ${m}_{\mathrm{AB}}$ flux results to within ≲0.05–0.20 mag, with the second method producing ${m}_{\mathrm{AB}}$ fluxes that are on average only 0.09 mag brighter than the first method. The ${m}_{\mathrm{AB}}$ values listed in Tables 2–4 are therefore the slightly brighter values from the second method, since our caustic transit calculations use in all cases the upper limits in predicted fluxes.

These combined BIK-corrections need to be added to the ${M}_{\mathrm{AB}}$ values from Tables 2–4 to yield the predicted ${m}_{\mathrm{AB}}$ values that JWST would observe:

$$\begin{eqnarray}&&{m}_{\mathrm{AB}}(z)={M}_{\mathrm{AB}}+\mathrm{DM}(z)+(\mathrm{BC}+\mathrm{IGM}+{\rm{K}})(z),\end{eqnarray} \tag{ 9 }$$

where the distance moduli DM at redshift z are listed in the footnotes of Table 2. To first order, the bolometric+IGM and K-corrections are comparable in magnitude, but are opposite in sign for the Pop III stars in Table 1. The combined BIK-corrections are, therefore, in general, modest, but can, on average, result in making objects ∼1–2 mag fainter than what the intrinsic bolometric luminosities from the MESA models would yield.

We list the combined corrections for z = 7, z = 12, and z = 17 in columns 6–8 of Tables 2–4 for the ZAMS, RGB, and AGB, respectively. Columns 9–11 list the resulting apparent rest-frame UV AB-magnitudes for these redshifts that result from the absolute bolometric magnitudes in column 5 and these combined corrections, respectively. These are directly used in the calculation of our caustic transit rise times and rates in Section 4.4, which are listed in columns 12–13, respectively.

For the lowest-mass stars that have ${T}_{\mathrm{eff}}$ ≲ 10⁴ K in Tables 2–4, the combined BIK-corrections are significantly positive, since for such cool stars most of their blackbody-like SED is redshifted to well longwards of the bluest JWST filter that samples above the 1216 Å break at the anticipated object redshift. These cool, low-mass Pop III stars are thus predicted to be always very faint (AB ≳ 50 mag), and permanently out of reach for JWST caustic transits. At the intermediate temperatures of Pop III RGB and AGB stars of ${T}_{\mathrm{eff}}$ ≃ 50,000−30,000 K, respectively (see Figure 2 and Section 3.1), the combined BIK-corrections are in general negative but no brighter than −1 to −2 mag (Tables 3–4), because the peak of their blackbody SED falls between the Lyα 1216 Å IGM cutoff and the rest-frame wavelength range covered by the bluest NIRCam filter that JWST will use to detect the Pop III object at z ≃ 7–17, so that the full benefit of the K-correction for a very blue SED is achieved. For much hotter ZAMS Pop III star temperatures of ${T}_{\mathrm{eff}}\simeq {10}^{5}$ K, the BIK-corrections are generally positive but no dimmer than +1 to +2 mag (Table 2), because most of the energy in their blackbody SED now falls well below the Lyα 1216 Å IGM cutoff. Hence, the combined BIK-corrections are more advantageous for detecting the cooler Pop III RGB and AGB phases than for the hotter Pop III ZAMS stars. This can be seen in columns 6–8 of Tables 2–4 and will be folded into the caustic transit rate calculations in Section 4.4.

The ZAMS Pop III mass–luminosity relation discussed in Section 3.1 and Table 2 has important implications for the mass range that dominates the luminosity density of a faint star-forming object at z ≳ 7. This is indicated in Figure 3, where the ZAMS M/L relation from Table 2 is indicated by the solid black line:

$$\begin{eqnarray}&&{L}_{\mathrm{bol}}\propto {M}^{\lambda },\end{eqnarray} \tag{ 10 }$$

with the mass-dependent slope λ from Table 2, as approximated by the segmented power laws in Equation (3).

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In our caustic transit calculations of Section 4.4, we assume that small early star-forming objects exist at z ≳ 7. These will be mostly fainter than the HST or JWST detection limits, and contribute a total 1–4 μm sky-SB whose upper limits were discussed in Section 2.3.1. In this context, it is necessary to consider which stars will dominate the luminosity density of these faint star-forming objects, which is defined as the number of stars per unit area on the sky. We consider their IMF to be a power law,

$$\begin{eqnarray}&&{dN}/{dM}\propto {M}^{-\alpha },\end{eqnarray} \tag{ 11 }$$

with three different IMF slopes in Figure 3 (dotted curves), ranging from "top heavy" (α = 1.5; blue), "intermediate" (α = 2.0; green), and "steep" (α = 2.5; orange), that bracket a range of plausible IMFs (dotted curves; e.g., Scalo 1986; Bastian et al. 2010; Coulter et al. 2017). The ZAMS Pop III M/L relation is folded with these three IMF slopes in Figure 3 to yield the luminosity density:

$$\begin{eqnarray}&&{L}_{\mathrm{bol}}({dN}/{dM})\propto {M}^{\lambda }{M}^{-\alpha }\propto {M}^{\delta },\end{eqnarray} \tag{ 12 }$$

where δ = (λ–α) is the slope of the luminosity density versus mass relation. For an IMF slope $\alpha \simeq 2.0$ and the mass-dependent slope of the M/L relation in Equation (3), we infer a strongly positive slope of the luminosity density versus mass relation (dashed green line): $\delta \simeq +1.2$ for M ≃ 1–20 M_⊙, a nearly zero slope ($\delta \simeq 0.1$) for M ≃ 20–100 M_⊙, and a negative slope ($\delta \simeq -0.8$) for M ≃ 100–1000 M_⊙.

For an IMF slope of $\alpha \simeq 2.0$, most of the Pop III ZAMS bolometric energy from faint star-forming objects at z ≳ 7 is thus produced by stars with masses between 10–100 M_⊙, with a somewhat smaller contribution from stars with M ≃ 100–1000 M_⊙, and a much smaller contribution from M ≃ 1–10 M_⊙, which is compounded by the significant K-correction for the lowest-mass stars (Section 3.3). For an IMF slope of $\alpha \simeq 2.0$, the Pop III luminosity density peaks at around 30 M_⊙ with a broad plateau (green dashed curve in Figure 3). These are the Pop III stars with the most advantageous bolometric+IGM+K-correction values (Tables 2–4) and are within reach for JWST, assuming caustic transits can occur as described in Section 4.4.

For a top-heavy IMF, most of the luminosity density would be produced by stars with M ≃ 100 M_⊙ (blue dashed lines in Figure 3), while for a very steep Salpeter-like IMF, most energy is still produced by stars as massive as M ≃ 20 M_⊙ (orange dashed lines). Hence, irrespective of any reasonable IMF slope at z ≳ 7, the Pop III star mass–luminosity relation implies that the highest near-IR sky-SB will be produced by stars with 20 ≲ M ≲ 100 M_⊙. These are precisely the stars that are most likely to become visible during caustic transits, as discussed in Section 4.4.

Mas-Ribas et al. (2016) give the number of Lyman–Werner (LW) photons produced by ZAMS Pop III stars as a function of their mass. A 300 M_⊙ Pop III star emits $\dot{N}=3.1\times {10}^{49}$ photons s⁻¹ in the LW band, which spans the energy range of E = 11.2–13.6 eV. From these numbers, we find a flux of ${m}_{\mathrm{LW}}\simeq 38.6$ mag at z ≃ 12. The ZAMS Pop III stars from the MESA runs in Tables 1–2 are about 0.3 mag brighter in their bolometric ${M}_{\mathrm{AB}}$ magnitude.

We estimate the surface density of Pop III stars using the models of Sarmento et al. (2018) in which the Pop III star formation rate density reaches $\sim {10}^{-3.5}$ M_⊙ yr⁻¹ Mpc⁻³ at z = 12 (see their Figure 2). If we assume that these consist of M = 300 M_⊙ stars that each live $t\simeq 2\times {10}^{6}\,\mathrm{yr}$ (Schaerer 2002, see Table 1 here), then the total number of Pop III stars per arcsec² is

$$\begin{eqnarray}&&\approx \ 0.03\left(\displaystyle \frac{{\mathrm{SFR}}_{\mathrm{Pop}\mathrm{III}}}{{10}^{-3.5}{M}_{\odot }/\mathrm{yr}}\right){\left(\displaystyle \frac{{M}_{* }}{300{M}_{\odot }}\right)}^{-1}\displaystyle \frac{\mathrm{stars}}{{(^{\prime\prime} )}^{2}{\rm{\Delta }}z}.\end{eqnarray} \tag{ 13 }$$

If each of these stars has ${m}_{\mathrm{AB}}$ ≳ 38.6 mag (see Table 2), then the total surface brightness in Pop III stars could be fainter than ∼36.0 mag arcsec⁻².

Theoretically, it appears unlikely that Pop III stars alone can fully account for an IR background with SB ≃ 31 mag arcsec⁻². In order to reach SB ≃ 31 mag arcsec⁻², we need ∼10³ massive Pop III stars per arcsec², each with AB ≃ 38.5 mag. Most of these must be at lower redshift (z ≲12), because of the strong redshift dependence of dS/dz, as discussed in Section 2.3.1. This would require ≳100 massive Pop III stars per arcsec² at z ≃ 12, with a weak z-dependence. This is much larger than the numbers we calculate above, although close to the "no LW" case of Trenti & Stiavelli (2009).

Strong LW radiation from massive Pop III stars can significantly suppress the formation of subsequent lower-mass stars in their immediate environment, resulting in a possibly much dimmer total Pop III sky-SB of ∼36 mag arcsec⁻². This SB-level is also indicated in orange in Figure 1. In Sections 4.5 and 7–8, we will therefore consider a range of Pop III near-IR SB-levels and the scope of observing programs needed for JWST to detect their resulting caustic transits in each case.

To address the caustic transit rate and duration for Pop III stars, we first need to evaluate the plausible limits on the transverse velocities of massive lensing clusters, their typical caustic lengths, and the possible effects from microlensing. From this, we will in Section 4.4 estimate the cluster caustic transit rates for the Pop III stars of Section 3.

A Pop III caustic transit observing program with JWST would likely select the best lensing clusters that also have matching deep HST images in previous epochs—including WFC3 IR data—such as the Hubble Frontier Field clusters (HFF; e.g., Kawamata et al. 2016; Acebron et al. 2017; Lagattuta et al. 2017; Lotz et al. 2017; Mahler et al. 2018) or the CLASH clusters (e.g., Postman et al. 2012; Rydberg et al. 2015).

The HFF clusters were chosen to have the capability for significant lensing magnification. Many are highly elongated and could well have significant internal velocities between cluster subcomponents and/or a significant space velocity compared to the nearby cosmic web, as discussed in Section 4.2 and Appendix A. Indeed, in two of these clusters (MACS J0416–2403 and MACS J1149.5+2223) to date, possible caustic transits have been identified at lower redshifts (z ≃ 1.0–1.5; Diego et al. 2017; Kelly et al. 2018; Rodney et al. 2017).

A JWST lensing cluster program should select the best lensing clusters with redshifts 0.3 ≲ z ≲ 0.5. This is because of a combination of the following two factors. First, the SED of the 5–8 Gyr old stellar population in these clusters at z ≃ 0.4 peaks at $\lambda \simeq 1.6$ μm in the rest frame (e.g., Kim et al. 2017). This includes the SED of the substantial ICL that is present in massive virialized clusters. Also, the zodiacal foreground in JWST's second Lagrange point (L2) orbit strongly declines between 1–3.5 μm (see Figure 1), so the best JWST sensitivity per unit time is obtained in the observed wavelength range of 2–3.5 μm. This is the critical wavelength range for detecting First Light objects at z ≲ 17. Hence, ideally, the redshift of lensing clusters to be observed with JWST should be kept at z ≲ 0.5. This is so that the rest-frame peak SED of the cluster galaxies and the ICL does not redshift as much into the most sensitive 2.5–3.5 μm NIRCam filters and thus compromise the ability to make First Light object detections, including Pop III caustic transits.

Higher-redshift clusters will of course have lower ICL and cluster galaxy brightness, because of the stronger cosmological (1+z)⁴ SB dimming. They are also less massive by selection, and may therefore not always be the most optimal gravitational lenses. Because of their younger ages, they may also be less virialized. There exist exceptional clusters at z ≳ 0.5, of course, that could be used for lensing studies with JWST, such as, e.g., El Gordo at z ≃ 0.87 (Zitrin et al. 2013).

In Section 4.5, we suggest that 3–30 clusters need to be monitored by JWST for several years for Pop III caustic transit studies. In the end, practical arguments, such as available HST images—especially at shorter wavelengths (λ ≲ 0.6 μm) than those that JWST can observe—the quality of available lensing models, ancillary data such as available ground-based spectra and X-ray images, and the ability to schedule JWST observations for at least half a year during each JWST cycle will likely determine which cluster sample is best suited for Pop III caustic transit observations with JWST.

In this section, we consider the possible maximum transverse (or tangential) velocity, v_T, of a massive cluster, which has visible substructure in both its measured redshift/velocity distribution as well as in its spatial extent on the sky (see Figure 4 and Appendix A).

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4.2.1. General Limits to Transverse Velocities of Nearby Clusters

4.2.2. Maximum Transverse Velocities Adopted for Lensing Clusters at 0.3 ≲ z ≲ 0.5

Throughout, we take (${V}_{T},s$) to mean the net transverse velocity, accounting for the transverse motion of both the observer, the lens, and the source planes. The effective transverse velocity ${V}_{T},s$ when observing a source at ${z}_{s}\simeq 7\mbox{--}17$ lensed by a cluster at ${z}_{d}\simeq 0.4$ is computed as follows, starting with the sum of the relevant velocity vectors scaled with the appropriate angular diameter distances, using Equation B9 of Kayser et al. (1986):

$$\begin{eqnarray}{{\boldsymbol{V}}}_{T},s\ & = & \ {{\boldsymbol{v}}}_{s}/(1+{z}_{s})\ +\ {{\boldsymbol{v}}}_{\mathrm{obs}}({D}_{{ds}}/{D}_{d})/(1+{z}_{d})\\ & & -\ {{\boldsymbol{v}}}_{T}({D}_{s}/{D}_{d})/(1+{z}_{d}).\end{eqnarray} \tag{ 14 }$$

Here, the first term due to the source motion at z ≃ 7–17 is negligible at v_s ≲ 30/(1 + z_s) ≲ 2–4 km s⁻¹(Barkana & Loeb 2002), using the expected small velocity dispersion of the low-mass halos they likely reside in at high redshifts. The unknown bulk motion of the halo at z ≳ 7 could increase this to several 100/(1+z) km s⁻¹, or ≲40 km s⁻¹. In either case, this first term is much smaller than the last two terms. The second term is due to the velocity of ${v}_{\mathrm{obs}}\simeq 370$ km s⁻¹ of the solar system (moving in the Galaxy and the Local Group, and with the Virgo cluster) toward the CMB from Section 4.2.1—modulated by the Earth's motion around the Sun of ∼30 km s⁻¹–and scales with the ratio of the angular diameter distance between the deflector to source, D_ds, and the angular diameter distance to the cluster deflector, D_d. The third term v_T is due to the transverse cluster motion itself, and scales with the angular diameter distance ratio of the source to the deflector, ${D}_{s}/{D}_{d}$. We ignore here the intrinsic velocities of any microlenses in the lens plane, since these are demagnified in the source plane by large factors, as shown by Kayser et al. (1986) in the high-magnification regime of interest.

To assess the transverse velocity v_T for lensing clusters at 0.3 ≲ z ≲ 0.5, we perturbed the observed redshift distribution of three promising HFF clusters with a random space velocity and determine how much of the projected space velocity can be added to the transverse direction, before the other projected component added to the line-of-sight velocities disturbs the observed cluster redshift distribution too much. Details of this simulation are given in Appendix A and its figure. These show that adding space velocities with projected transverse components much larger than v_T ≃ 1000 km s⁻¹ imply projected components of this space velocity added along the line of sight that are not consistent with the available redshift data in the cluster core, although somewhat smaller values are certainly allowed. This results in an upper limit of v_T ≲ 1000 km s⁻¹ for the maximum transverse velocity of these clusters at 0.3 ≲ z ≲ 0.5 in the plane of the sky. For some substructures in each cluster, the v_T values may well be as high as 1000 km s⁻¹. Since we cannot currently distinguish if the whole cluster or only several of its subclumps are moving transversely at v_T ≲ 1000 km s⁻¹, the integral caustic length that we defined in Section 4.3 and use in Section 4.4 to calculate the caustic transit rate is therefore also an upper limit.

For the HFF cluster subsample discussed in Appendix A at least, it seems appropriate to use v_T ≲ 1000 km s⁻¹ for the maximum transverse velocity of these clusters in the plane of the sky, as projected from their space velocity. Kelly et al. (2018) find that for the cluster MACS J0416–2403—behind which their caustic-transiting star at z ≃ 1.5 was identified—the transverse velocity is about 1000 km s⁻¹. From large N-body simulations, Watson et al. (2014) find that most pairwise halo velocities are ≲3000 km s⁻¹ with a median of ∼1000 km s⁻¹.

The maximum effect for caustic transits in Equation (14) is obtained when the solar system velocity vector, ${{\boldsymbol{v}}}_{\mathrm{obs}}$, toward the CMB as projected on the sky at the cluster location is exactly anti-aligned with the transverse cluster velocity vector, v_T, which is captured by the minus sign in Equation (14). If both lens and observer are going in the same direction, one would obtain the smallest value for this velocity difference. The actual transverse vector sum will be different for each cluster by an unknown amount, depending on how the unknown direction of the transverse vector of each cluster, ${{\boldsymbol{v}}}_{T}$, aligns with the velocity vector, ${{\boldsymbol{v}}}_{\mathrm{obs}}$, of the solar system toward the CMB. Because both projected transverse velocities are vectors, the typical expected value of the velocities is the rms, not the velocity difference. Hence, we add both in quadrature, so that for ${z}_{d}\simeq 0.4$ and ${z}_{s}\simeq 12$ we obtain

$$\begin{eqnarray}&&| {V}_{T},s| \simeq \sqrt{{[370\sqrt{2/3}\times 0.40]}^{2}\ +\ {[1000\times 0.48]}^{2}}\end{eqnarray} \tag{ 15 }$$

using the velocities discussed above and the appropriate angular diameter distance ratios for our adopted cosmology. This amounts to ${V}_{T},s$ ≃ 495 km s⁻¹. When exactly anti-aligned, the two velocity components would just add without the transverse projection factor $\sqrt{2/3}$ of the solar system velocity at the location of the lensing cluster, so that ${V}_{T},s$ ≲ 502 km s⁻¹. Given the significant differences in the allowed v_T values between the three HFF clusters discussed in Appendix A, we will adopt for our caustic transit calculations in Section 4.4 an upper limit of ${V}_{T},s$ ≲ 1000 km s⁻¹.

Gravitational lensing modeling will result in lensing maps in the plane of the cluster. The clusters selected for JWST are assumed to be in the redshift range 0.3 ≲ z ≲ 0.5, following the arguments of Section 4.1. An example of a lensing map is shown in Figure 4(a) for sources at z = 10 behind the HFF cluster MACS J1149.5+2223 at z ≃ 0.4 (e.g., Lotz et al. 2017). Detailed lensing maps of the HFF clusters have been made by, e.g., Jauzac et al. (2014, 2015), Lam et al. (2014), Diego et al. (2015a, 2015b, 2016a, 2016b), and Caminha et al. (2017). An example of a caustic map for a source at z = 10 behind MACS J1149.5+2223 is given in Figure 4(b). These caustic maps are similar for Pop III sources at 7 ≲ z ≲ 17, which may be observable to JWST via caustic transits.

Making lensing maps is a complex process that introduces its own uncertainties, which depend, e.g., on the detailed input cluster mass distribution, the number and redshift distribution of available sources with multiple images through which the lensing model gets refined, the point spread function (PSF), quality, and the depth of the images at HST resolution used to reconstruct the lensed sources, and on a number of other factors such as the actual amount of cluster substructure present. For details, we refer to, e.g., Meneghetti et al. (2017), where errors in the reconstruction of HFF-like clusters are discussed based on simulated data.

In essence, the more numerous the input redshifts and the better the input imaging data are, the more reliable the lensing model will become. Ideally, the entire gravitational field of the cluster with all its substructure, dwarf galaxies, detailed ICL distribution, and stellar microlenses would have been modeled. For exact caustic transit modeling of a known source at lower redshift (e.g., Diego et al. 2017; Kelly et al. 2018), detailed lensing modeling is necessary, since a caustic transit may have been observed at one location, where the local gravitational lensing model then exactly matters for the correct interpretation of the observed data. This is, e.g., the case when a known background galaxy provides the stellar object that transits the cluster caustic at a specific location.

For the present work, detailed lensing models are not required, since Pop III stars at z ≳ 7 may be present everywhere at average sky-SB levels no brighter than the upper limits adopted in Section 2.3, where we calculated that most of their diffuse flux will come from objects that are well beyond the HST and JWST point-source detection limits. For the current caustic transit calculations, we will assume that the integrated near-IR sky-SB of these "unresolved" objects at z ≳ 7 is rather uniform (see Section 2.3.1). That is, we do not need to know the exact lensing model at each location along the caustic, since Pop III caustic transits at z ≳ 7 can happen anywhere at unpredictable locations along the caustic lines in the cluster.

Instead, we need the general properties of the caustics to estimate the rate of transits and an estimate of the maximum magnification around the caustic as a function of source angular size. We are interested in the statistics/probability of seeing these rare events. Hence, the typical global properties of the lenses, such as the area in the background plane with magnifications above a given threshold or the statistical presence of microlenses that can modify these properties in the high-magnification regime, are the only quantities that are relevant here.

For the present goal of order-of-magnitude estimates of Pop III object caustic transits at z ≳ 7, we will thus assume average caustic lengths L and geometry. Line integration of the lensing models in clusters like Figure 4(b) shows that their typical total caustic length is ${L}_{\mathrm{caust}}$ ≲ 100'', which we will use as an upper limit in the caustic transit calculations in Sections 4.4 and 6.2.

4.3.1. Details of Lensing Magnification Near the Cluster Caustics

Although usually represented in the source plane for convenience, caustics actually form in the observer plane, and it is the relative motion with respect to these caustics that produce the peaks in the observed light curves (see Section 4.2.2). For most clusters, caustics tend to adopt a diamond-like shape aligned in the same direction as the main symmetry axis of the ellipsoid that encloses the cluster (see Figures 4(a) and (b)). Since these three planes are uncorrelated, the vector of the relative transverse velocity can point in any direction with respect to the caustic pattern. As discussed in Section 4.2.2, we should consider all possible directions when estimating caustic transit rates. In detail, the velocity of the caustics is complex, because the shape of the network changes in addition to the transverse movement. Given the other larger uncertainties in cluster geometry, v_T, and ${L}_{\mathrm{caust}}$, we will henceforth ignore the projection effects from the angle, i, between the cluster's unknown main velocity vector and the main direction of the caustic at each location, which will average out to $\langle \sin ({\rm{i}})\rangle \simeq 1/2$.

When a background star crosses a cluster caustic, it can be magnified by a factor of up to $\mu \simeq {10}^{5}$–10⁶ for a short period of time (few weeks to months; see Section 4.4), depending on the strength of the caustic and the stellar radius. This magnification can thus boost the apparent brightness of the star by ∼12.5–15 mag. Possible modifications from microlensing are discussed in Section 4.3.2 and Appendices B–B.2. Fainter Pop III stars with AB ≃ 41–43 mag could then be observed with JWST to AB ≲ 28.5–29 mag during one of these caustic crossing events. At larger distances from the caustic ($\simeq 1$ pc), the magnification is more moderate ($\mu \simeq {10}^{3}$), and only the brightest stars with AB ≲ 36 mag could be observed via a caustic transit, but they could remain visible for many years because they remain visible farther away from the caustic. Microlensing can reduce these magnifications and spread the microlensed events over a larger area still, which lengthens their visibility in time. This is discussed further in Appendices B–B.2.

For the more ubiquitous fold caustics, the magnification near a caustic varies with the distance to the caustic, d, as

$$\begin{eqnarray}&&\mu ={B}_{o}/\sqrt{d},\end{eqnarray} \tag{ 16 }$$

where B_o is a constant that depends on the derivatives of the gravitational potential. For clusters like the HFFs, B_o is normally in the range 10–20, while d is expressed in arcseconds (see, e.g., Miralda-Escude 1991; Diego et al. 2017, for a detailed discussion). Hence, for a background Pop III star at z ≳ 7, magnifications of order $\mu \simeq {10}^{3}$ can be attained once the background star is  ≃1 pc away from the caustic (or $d\simeq 0\buildrel{\prime\prime}\over{.} 001$). For an HFF-like cluster with ${L}_{\mathrm{caust}}\simeq 100$'' at z ≃ 12, this implies that an area of ∼0.1 arcsec² in the source plane can magnify background stars by more than a factor of $\mu \simeq {10}^{3}$, so that any star brighter than AB ≃ 36 mag can be lensed to above the detection limit of JWST and produce a double image separated by less than 05 around the critical curve. When we get closer to the critical curve, the double-lensed image that would appear on each side of the critical curve will be unresolved at JWST's near-IR resolution of ∼008 FWHM if the separation between the two images is smaller than ∼25 milliarcsec. At these separations, the total magnification would be $\mu \simeq {10}^{4}$, and any star brighter than AB ≃ 38 mag could be lensed to above the detection limit of JWST. This corresponds to an area of  ≃3 × 10⁻⁴ arcsec² in the source plane.

At even smaller distances to the true caustic, fainter and smaller stars would become visible, but the probability of magnifying a star in this narrower region would be smaller. Clearly, there is a trade-off between the luminosity function slope of the background stars and the probability of being magnified above a certain value, which is given by the area in the source plane, $A(\gt \mu )$, that has a magnification larger than μ. This magnification area seems to follow a power law for the more ubiquitous fold caustics, $A(\gt \mu )={B}_{o}/{\mu }^{2}$, where B_o will vary somewhat from cluster to cluster.

Owing to this scaling of the area with ${\mu }^{2}$, it is easy to see (Kelly et al. 2018) that the optimal trade-off between the luminosity function and $A(\gt \mu )$ happens when the stellar LF slope is close to $\alpha \simeq -2$, where ${dN}/{dL}\propto {L}^{\alpha }$ is the luminosity function of the background stars. One possible complication when observing a lensed bright star at moderate magnifications (i.e., a star brighter than AB ≃ 36 mag and with μ ≲ 10³) is that the timescale for the flux variation could be very long (≳ hundreds of years). This makes it a challenging task to distinguish between a lensed Pop III star and a larger unlensed substructure, such as a globular cluster in the background galaxy. Spectroscopy of brighter caustic transits will be necessary to help reveal their nature, as discussed in Section 7. Microlensing fluctuations will likely make the light curve of a caustic transit more variable and spread over a longer period of time, as discussed in Section 4.3.2 and Appendices B–B.2.

4.3.2. Possible Role of Microlenses during Caustic Transits

The question that we will address in this section is: if a fraction of the diffuse near-IR background is generated by Pop III stars—with a conservative upper limit to their near-IR sky-SB of ≳31 mag arcsec⁻² (Section 2.3)—then what is the probability that JWST will catch one of these Pop III stars being lensed by a cluster caustic transit?

For our calculations, we start with the premise that this maximum 1–4 μm sky-SB is made up of ZAMS Pop III stars with AB ≳ 37.5 mag at z ≳ 7 (Table 2). During their RGB and AGB stages, these Pop III stars may be as "bright" as AB ≳ 35 mag at z ≳ 7 (Tables 3–4). Pop III stars in the mass range of 30 ≲ M ≲ 1000 M_⊙ are the most likely to be detected by JWST at z ≳ 7 at AB ≲ 28.5–29 mag if the caustic magnifications reach μ ≳ 10⁴–10⁵. We will assume that the geometrical optics approximation still holds in this very small source regime. To reference our calculations following Equation (3), we define the apparent magnitude of a 100 M_⊙ star with luminosity L₁₀₀ at z = 12 as m₁₀₀, and that of a 20 M_⊙ star with luminosity L₂₀ at z = 12 as m₂₀. The AB-magnitudes at other redshifts scale with the DM and BIK-corrections in Tables 2–4.

Our caustic transit calculations depend on stellar luminosity, which depends on the ZAMS mass following the Pop III star mass–luminosity relation of Equation (3). To generalize our caustic transit calculations in the relevant equations below, we will propagate the three different mass-dependent power-law slopes in Equation (3) over the entire Pop III ZAMS mass range of 1 ≲ M ≲ 1000 M_⊙. For 10 ≲ M ≲ 1000 M_⊙, the ZAMS radii scale as

$$\begin{eqnarray}&&R={R}_{100}{(M/100{M}_{\odot })}^{0.45},\end{eqnarray} \tag{ 17 }$$

following the first line of Equation 2 in Section 3.1.

Given a population of Pop III stars with luminosity L, the number density N(L) required to make up a surface brightness of AB ≃ 31 mag arcsec⁻² follows from

$$\begin{eqnarray}({m}_{100}-31) & = & 2.5\ {\mathrm{log}}_{10}(N(L)\times {L}_{100}),\mathrm{or}:\\ & & N(L)\times {L}_{100}={10}^{({m}_{100}-31)/2.5}.\end{eqnarray} \tag{ 18 }$$

Given the segmented M/L relation in Equation (3), we can generalize this as a function of mass M as follows:

$$\begin{eqnarray}N(M) & = & {10}^{\displaystyle \frac{({m}_{100}-2.5\mathrm{log}(L/{L}_{100})-31)}{2.5}}\\ & = & {\left(\displaystyle \frac{L}{{L}_{100}}\right)}^{-1}{10}^{\displaystyle \frac{({m}_{100}-31)}{2.5}}\\ & \simeq & {\left(\displaystyle \frac{M}{100}\right)}^{-1.16}{10}^{\displaystyle \frac{({m}_{100}-31)}{2.5}},\ \mathrm{for}\ M\gtrsim 100\ {M}_{\odot },\\ & \simeq & {\left(\displaystyle \frac{M}{100}\right)}^{-2.06}{10}^{\displaystyle \frac{({m}_{100}-31)}{2.5}},\ \mathrm{for}\ 20\lesssim M\lesssim 100\ {M}_{\odot },\\ & \simeq & 2.33{\left(\displaystyle \frac{M}{20}\right)}^{-3.20}{10}^{\displaystyle \frac{({m}_{20}-31)}{2.5}},\ \mathrm{for}\ M\lesssim 20\ {M}_{\odot },\end{eqnarray} \tag{ 19 }$$

in units of ${{\rm{arcsec}}}^{-2}$, while all masses are in M_⊙. The extra constant in the last line of Equations (19)–(30) reflects the change in normalization at 100 M_⊙ in the first two mass ranges to 20 M_⊙ in the last mass range.

To be observed with JWST in a single epoch at a flux limit of AB ≲  28.5 mag, a star of mass M would need a lensing magnification of

$$\begin{eqnarray}\mu (M) & = & {10}^{\displaystyle \frac{({m}_{100}-2.5\mathrm{log}(L/{L}_{100})-28.5)}{2.5}}\\ & = & {\left(\displaystyle \frac{L}{{L}_{100}}\right)}^{-1}{10}^{\displaystyle \frac{({m}_{100}-28.5)}{2.5}}\\ & \simeq & {\left(\displaystyle \frac{M}{100}\right)}^{-1.16}{10}^{\displaystyle \frac{({m}_{100}-28.5)}{2.5}},\ \mathrm{for}\ M\gtrsim 100\ {M}_{\odot },\\ & \simeq & {\left(\displaystyle \frac{M}{100}\right)}^{-2.06}{10}^{\displaystyle \frac{({m}_{100}-28.5)}{2.5}},\ \mathrm{for}\ 20\lesssim M\lesssim 100\ {M}_{\odot },\\ & \simeq & 2.33\ {\left(\displaystyle \frac{M}{20}\right)}^{-3.20}{10}^{\displaystyle \frac{({m}_{20}-28.5)}{2.5}},\ \mathrm{for}\ M\lesssim 20\ {M}_{\odot }.\end{eqnarray} \tag{ 20 }$$

Now the typical magnification at a distance ${d}_{\mu }\simeq 1$'' from a caustic is μ(1'') = 10 (see Section 4.3 and Figure 4(b)), using the conservative lower value of ${B}_{o}\simeq 10$ in Section 4.3.1, so that

$$\begin{eqnarray}&&\mu \ \simeq 10\displaystyle \frac{1}{\sqrt{{d}_{\mu }}}.\end{eqnarray} \tag{ 21 }$$

The angular distance (in arcsec) from the true caustic maximum corresponding to a magnification μ is then

$$\begin{eqnarray}{d}_{\mu }(M) & = & 100\ {\mu }^{-2}\\ & = & 100{\left(\displaystyle \frac{L}{{L}_{100}}\right)}^{2}{10}^{\displaystyle \frac{-({m}_{100}-28.5)}{1.25}}{\rm{arcsec}}\\ & \simeq & 100{\left(\displaystyle \frac{M}{100}\right)}^{2.32}{10}^{\displaystyle \frac{-({m}_{100}-28.5)}{1.25}},\ \mathrm{for}\ M\gtrsim 100\ {M}_{\odot },\\ & \simeq & 100{\left(\displaystyle \frac{M}{100}\right)}^{4.12}{10}^{\displaystyle \frac{-({m}_{100}-28.5)}{1.25}},20\lesssim M\lesssim 100\ {M}_{\odot },\\ & \simeq & 18.4{\left(\displaystyle \frac{M}{20}\right)}^{6.4}{10}^{\displaystyle \frac{-({m}_{20}-28.5)}{1.25}},1\lesssim M\lesssim 20\ {M}_{\odot }.\end{eqnarray} \tag{ 22 }$$

To estimate the relevant timescales, we need to know the crossing time for a caustic passing over a distance ${d}_{\mu }$, and the crossing time for a typical radius of a ZAMS Pop III star with M ≃ 30–1000 M_⊙. Our adopted cosmology yields 3740 pc per arcsec at z ≃ 12. Using the 2012 IAU value for one astronomical unit (au) of 149.6  × 10⁹ m (Prša et al. 2016), 1.0 R_⊙ (or 695,700 km; Section 3.1) then corresponds to ∼6.03 × 10⁻¹² arcsec at z ≃ 12. Hence, the ZAMS Pop III stars in Table 1 are ∼5.2 × 10⁻¹²–7.78 × 10⁻¹¹ arcsec across at z ≃ 12, and at most between ∼1.3×–14× larger during their RGB–AGB phases, which together last ∼8× shorter than the ZAMS (Section 3.1). Pop III RGB–AGB star caustic transits will thus be more rare, although according to Tables 3–4 also ∼1.5–2.5 mag brighter than those of ZAMS Pop III stars (Table 2). To obtain lower limits to the caustic transit rise times and upper limits to their caustic transit rates, we will therefore use the Pop III star ZAMS parameters in Tables 1–2.

The upper limit of ${V}_{T},s$ ≲ 1000 km s⁻¹ (Section 4.2 and Appendix A) corresponds to an angular speed of $d\theta /{dt}\simeq 1.83\times {10}^{-7}$ arcsec/yr for a galaxy cluster at z ≃ 0.4. At this redshift, there are ∼5590 pc/arcsec in Planck cosmology. Using Equation (17), the crossing times for the Pop III star radius R(M) across the $\mu \gt {10}^{4}$ magnification region ${\theta }_{\mu }(M)$—needed to make stars of mass M detectable to JWST—leads then to a mass-dependent Pop III star caustic transit timescale of

$$\begin{eqnarray}&&{t}_{R}(M)=\displaystyle \frac{R}{{v}_{{\rm{T}},{\rm{s}}}}=\displaystyle \frac{R}{{R}_{100}}\displaystyle \frac{{R}_{100}}{{v}_{{\rm{T}},{\rm{s}}}}\simeq \displaystyle \frac{{R}_{100}}{{v}_{{\rm{T}},{\rm{s}}}}{\left(\displaystyle \frac{M}{100}\right)}^{0.45},\end{eqnarray} \tag{ 23 }$$

and:

$$\begin{eqnarray}{\theta }_{\mu }(M) & = & {\left(\displaystyle \frac{d\theta }{{dt}}\right)}^{-1}\displaystyle \frac{100^{\prime\prime} }{{\mu }^{2}(M)}\\ & \simeq & \displaystyle \frac{100}{d\theta /{dt}}{\left(\displaystyle \frac{M}{100}\right)}^{2.32}{10}^{\displaystyle \frac{-({m}_{100}-28.5)}{1.25}},\mathrm{for}\ M\gtrsim 100\ {M}_{\odot },\\ & \simeq & \displaystyle \frac{100}{d\theta /{dt}}{\left(\displaystyle \frac{M}{100}\right)}^{4.12}{10}^{\displaystyle \frac{-({m}_{100}-28.5)}{1.25}},\ 20\lesssim M\lesssim 100\ {M}_{\odot },\\ & \simeq & \displaystyle \frac{18.4}{d\theta /{dt}}{\left(\displaystyle \frac{M}{20}\right)}^{6.4}{10}^{\displaystyle \frac{-({m}_{20}-28.5)}{1.25}},{for}\ M\lesssim 20\ {M}_{\odot }.\end{eqnarray} \tag{ 24 }$$

This implies that the brightening time—defined as the time for the magnification to go from zero to its maximum value—for a Pop III star is very short (∼0.5–3 hr) when it transits the caustic starting at the "highest-magnification edge." The star would then stay bright for several months to a year, with brightness decaying as 1/$\sqrt{t-{t}_{o}}$, where ($t-{t}_{o}$) is the time since the stellar disk started the caustic crossing at time t_o.

Also, this entire process is reversible, so one could witness a very slow rise of an object's flux as 1/$\sqrt{{t}_{f}-t}$ when it moves toward the caustic starting from the low-magnification end, followed by an abrupt disappearance once it crosses the caustic at some future time t_f. We will discuss below and in Section 7 how JWST may detect each of these cases.

To calculate how often we expect such a brightening, we assume that the cluster has a length ${L}_{\mathrm{caust}}\simeq 100$'' of caustic (see Section 4.3). To calculate the rate at which Pop III lensing occurs, we need the area crossed by the caustics per unit time. This change in area is given by

$$\begin{eqnarray}\displaystyle \frac{{dA}}{{dt}} & = & {L}_{\mathrm{caust}}\times {v}_{{\rm{T}}}=100\times 2.4\times {10}^{-14}\ {{\rm{arcsec}}}^{2}/{\rm{s}}\\ & \simeq & 1.6\times {10}^{-5}\left(\displaystyle \frac{{L}_{\mathrm{caust}}}{100^{\prime\prime} }\right)\left(\displaystyle \frac{{v}_{{\rm{T}}}}{1000\ \mathrm{km}\,{{\rm{s}}}^{-1}}\right){({\prime\prime} )}^{2}\,{\mathrm{yr}}^{-1}.\end{eqnarray} \tag{ 25 }$$

The surface areas referred to here are all in the source plane, and so there is no depletion correction for magnification. The number of events therefore follows from the surface density of Pop III stars N, yielding

$$\begin{eqnarray}\displaystyle \frac{{{dN}}_{\mathrm{lens}}}{{dt}} & = & N(M)\times \displaystyle \frac{{dA}}{{dt}}\\ & = & \displaystyle \frac{{dA}}{{dt}}{\left(\displaystyle \frac{L}{{L}_{100}}\right)}^{-1}{10}^{\displaystyle \frac{({m}_{100}-31)}{2.5}}\,{\mathrm{yr}}^{-1}.\end{eqnarray} \tag{ 26 }$$

To quantify the values for N, t_μ, and $\tfrac{{{dN}}_{\mathrm{lens}}}{{dt}}$, we base our numbers on the discussion of the physical parameters of Pop III stars in Section 3.1. From Table 2, the luminosity of a 100 M_⊙ star is ∼1.40 × 10⁶ L_⊙, giving an absolute magnitude of −10.63 AB-mag, using ${\rm{M}}\equiv +4.74$ mag for the absolute magnitude of the Sun (see Section 3.1). Including the bolometric+IGM+K-corrections of Table 2, the corresponding apparent magnitude m₁₀₀ at z = 12 is then m₁₀₀ = 40.91 mag, assuming no extinction. For a discussion of dust, see Section 6.1. From Table 1, the radius of a 100 M_⊙ star is R₁₀₀ = 4.12 R_⊙. We then find the following caustic crossing rate for lensed Pop III stars:

$$\begin{eqnarray}&&\displaystyle \frac{{{dN}}_{\mathrm{lens}}}{{dt}}=N(M)\times \displaystyle \frac{{dA}}{{dt}}\\ &&=\,0.064\ \displaystyle \frac{{L}_{\mathrm{caust}}}{100^{\prime\prime} }\ \displaystyle \frac{{v}_{{\rm{T}}}}{1000}{\left(\displaystyle \frac{M}{100}\right)}^{-1.16},\mathrm{for}\ M\gtrsim 100\ {M}_{\odot },\\ &&=\,0.064\ \displaystyle \frac{{L}_{\mathrm{caust}}}{100^{\prime\prime} }\ \displaystyle \frac{{v}_{{\rm{T}}}}{1000}{\left(\displaystyle \frac{M}{100}\right)}^{-2.06},20\lesssim M\lesssim 100\ {M}_{\odot },\\ &&=\,1.76\ \displaystyle \frac{{L}_{\mathrm{caust}}}{100^{\prime\prime} }\ \displaystyle \frac{{v}_{{\rm{T}}}}{1000}{\left(\displaystyle \frac{M}{20}\right)}^{-3.20},1\lesssim M\lesssim 20\ {M}_{\odot }\end{eqnarray} \tag{ 27 }$$

per year. The duration of a brightening time is then

$$\begin{eqnarray}&&{t}_{R}(M)=9.1\times {10}^{-5}{\left(\displaystyle \frac{M}{100}\right)}^{0.45}{\rm{yr}},\end{eqnarray} \tag{ 28 }$$

which for a 100 M_⊙ ZAMS star is about 0.80 hr for v_T ≲ 1000 km s⁻¹. The range in rise times in Table 2 is ∼0.4–2.5 hr for M ≃ 30–1000 M_⊙ Pop III stars.

The time spent above the detection limit is

$$\begin{eqnarray}{t}_{\mu }(M) & \simeq & 0.4{\left(\displaystyle \frac{{v}_{{\rm{T}}}}{1000}\right)}^{-1}{\left(\displaystyle \frac{M}{100}\right)}^{2.32},\mathrm{for}\ M\gtrsim 100\ {M}_{\odot },\\ & \simeq & 0.4{\left(\displaystyle \frac{{v}_{{\rm{T}}}}{1000}\right)}^{-1}{\left(\displaystyle \frac{M}{100}\right)}^{4.12},\ \mathrm{for}\ 20\lesssim M\lesssim 100\ {M}_{\odot },\\ & \simeq & 5.3\times {10}^{-4}{\left(\displaystyle \frac{{v}_{{\rm{T}}}}{1000}\right)}^{-1}{\left(\displaystyle \frac{M}{20}\right)}^{6.4},\mathrm{for}\ M\lesssim 20\ {M}_{\odot }\end{eqnarray} \tag{ 29 }$$

in units of years. This assumes that after the flux has peaked upon caustic crossing, the flux declines as 1/$\sqrt{t}$ following Equation (21), assuming constant velocity v_T.

The number of lensed Pop III stars visible at a given time is then

$$\begin{eqnarray}{N}_{\mathrm{lens}} & = & {t}_{\mu }(M)\displaystyle \frac{{{dN}}_{\mathrm{lens}}}{{dt}}\\ & \simeq & 0.026\ \displaystyle \frac{{L}_{\mathrm{caust}}}{100^{\prime\prime} }{\left(\displaystyle \frac{M}{100}\right)}^{1.16},\mathrm{for}\ M\gtrsim 100\ {M}_{\odot },\\ & \simeq & 0.026\ \displaystyle \frac{{L}_{\mathrm{caust}}}{100^{\prime\prime} }{\left(\displaystyle \frac{M}{100}\right)}^{2.06},\mathrm{for}\ 20\lesssim M\lesssim 100\ {M}_{\odot },\\ & \simeq & 9.4\times {10}^{-4}\displaystyle \frac{{L}_{\mathrm{caust}}}{100^{\prime\prime} }{\left(\displaystyle \frac{M}{20}\right)}^{3.20},1\lesssim M\lesssim 20\ {M}_{\odot }.\end{eqnarray} \tag{ 30 }$$

Note that while ${t}_{\mu }(M)$ and $\tfrac{{{dN}}_{\mathrm{lens}}}{{dt}}$ are sensitive to the transverse velocity v_T, the visible number of events is not. Equations (27)–(30) contain the key relations of this paper for calculating Pop III star caustic transits.

For an IR background of ≳31 mag arcsec⁻² (Section 2.3) made up of AB ≃ 41 mag Pop III stars with M ≃ 100 M_⊙, we estimate that one lensing event can be observed above a flux limit of AB ≃ 28.5 mag per cluster per ∼2.7 years, or one event when monitoring ∼3 clusters during a year. Because these events should stay detectable at $\mu \gt \mu (M=100\ {M}_{\odot })$ for ${t}_{\mu }\simeq 0.4$ years, this implies that ∼0.15 such lensed Pop III sources per cluster would be observed above the flux limit at any given time.

These results are sensitive to the luminosity of the Pop III stars. For example, let us instead try the extreme case where the Pop III stars are 1000 M_⊙, and so have AB ≃ 38.3 mag at z ≃ 12 rather than AB ≃ 40.9 mag. This implies that the source needs to be magnified less, and so it can be observed while farther from the caustic, with a visible time of ${t}_{\mu }=40\,\mathrm{yr}$. However, the rate is lower with ${{dN}}_{\mathrm{lens}}/{dt}\simeq 0.013$ per year per cluster, giving one event per cluster per 75 years. In this case of brighter, more massive stars, we find that ∼0.5 events per cluster would be visible at any given time.

Thus, for 100 M_⊙ Pop III stars, about six clusters observed twice about six months apart would make the likelihood of observing a lensed Pop III star of order unity, while for more massive stars, detecting a new lensing event (with a time baseline limited to 1 year) would require the observation of a larger number of clusters in proportion to the mass M. For lower-mass stars, fewer clusters would need to be observed, as long as they can appear magnified above the detection thresholds of Tables 2–4.

The observed rate of events will thus also depend on the mass function of Pop III stars. A mass-function weighted average over Equation (27) is

$$\begin{eqnarray}\left\langle \displaystyle \frac{{{dN}}_{\mathrm{lens}}}{{dt}}\right\rangle & = & 0.064\left(\displaystyle \frac{{L}_{\mathrm{caust}}}{100^{\prime\prime} }\right)\left(\displaystyle \frac{{v}_{{\rm{T}}}}{1000\ \mathrm{km}\,{{\rm{s}}}^{-1}}\right)\\ & & \times {\displaystyle \int }_{{M}_{\min }}^{{M}_{\max }}{dM}{\left(\displaystyle \frac{M}{100}\right)}^{-1}\displaystyle \frac{{dP}}{{dM}}{dM}\,{\mathrm{yr}}^{-1}.\end{eqnarray} \tag{ 31 }$$

Here, dP/dM is the normalized instantaneous mass function of Pop III stars, i.e., the population of stars available to be lensed, not the entire IMF. We assume a power-law mass function ${dP}/{dM}\propto {M}^{-\alpha }$ with slope $| \alpha | \gt 1$ in the range ${M}_{\min }\lt M\lt {M}_{\max }$, leading to

$$\begin{eqnarray}\left\langle \displaystyle \frac{{{dN}}_{\mathrm{lens}}}{{dt}}\right\rangle & = & 0.064\left(\displaystyle \frac{{L}_{\mathrm{caust}}}{100^{\prime\prime} }\right)\left(\displaystyle \frac{{v}_{{\rm{T}}}}{1000\,\mathrm{km}\,{{\rm{s}}}^{-1}}\right){\left(\displaystyle \frac{{M}_{\min }}{100}\right)}^{-1}\\ & & \times \left(\displaystyle \frac{\alpha -1}{\alpha }\right)\left(\displaystyle \frac{1-{({M}_{\min }/{M}_{\max })}^{\alpha }}{1-{({M}_{\min }/{M}_{\max })}^{\alpha -1}}\right)\,{\mathrm{yr}}^{-1}.\end{eqnarray} \tag{ 32 }$$

For ${M}_{\max }\gg {M}_{\min }$, the last term is close to unity, while for steep mass functions with $\alpha \gg 1$ the integral converges to the value of $\langle {\textstyle \tfrac{{dN}_{{\rm{l}}{\rm{e}}{\rm{n}}{\rm{s}}}}{dt}} \rangle \rightarrow {\textstyle \tfrac{{dN}_{{\rm{l}}{\rm{e}}{\rm{n}}{\rm{s}}}}{dt}}({M}_{min})$. The choice of the mass function slope $\alpha \simeq 2.0$ was discussed in Sections 2.3.1 and 3.4, and is used below.

We chose the lower-mass boundary here at 30 M_⊙ since such stars may be visible through caustic transits to JWST (Tables 2–4), and such stars may produce BH leftovers in the mass range already observed by LIGO at M ≳ 14 M_⊙ (e.g., Abbott et al. 2016e, see also Section 5.1). Lower Pop III stellar masses would render the stars too faint to be reasonably observed at AB ≲ 28.5 mag through caustic transits in a single JWST epoch (Table 2), except for perhaps RGB–AGB Pop III stars with M ≳ 15–20 M_⊙. The latter may be visible because their K-corrections are more advantageous (Tables 3–4) than for the much hotter ZAMS Pop III stars (Section 3.3.3).

For the lowest Pop III star mass considered here (${M}_{\min }\,=30$ M_⊙), its physical parameters given in Section 3.1, and adopting an IMF slope of $\alpha \simeq 2.0$, we get the following upper limits for L ≃ 100'' and ${V}_{T},s$ ≲ 1000 km s⁻¹:

$$\begin{eqnarray}&&\left\langle \displaystyle \frac{{{dN}}_{\mathrm{lens}}}{{dt}}\right\rangle \leqslant 0.064{\left(\displaystyle \frac{30}{100}\right)}^{-1}\left(\displaystyle \frac{1.0}{2.0}\right)\,{\mathrm{yr}}^{-1}.\end{eqnarray} \tag{ 33 }$$

For a ZAMS Pop III star mass function slope of $\alpha \simeq 2$, the weights for each mass bin in Table 2 are very similar at 0.23–0.17 following Equation (32).

The resulting total transit rates for stars with M ≳ 30 M_⊙ that are in principle observable with JWST across the caustics are then predicted to be $d{N}_{{\rm{l}}{\rm{e}}{\rm{n}}{\rm{s}}}/dt$ ≲ 0.30 events per cluster per year. These caustic transits that may be visible to JWST are marked with an asterisk in columns 12–13 of Tables 2–4.

To this we need to add the caustic transit rates expected for the RGB from Table 3 and the AGB from Table 4. These must be weighted with their approximate lifetimes compared to the ZAMS, which are ∼6% of the ZAMS lifetime (Section 3.1) for each of the RGB and AGB phases detectable by JWST. This amounts to an additional 0.01 transits per cluster for each of the RGB and AGB phases. Hence, the weighted total number of caustic transits for ZAMS, RGB and AGB Pop III stars in Tables 2–4 are ∼0.32 per cluster per year.

From Section 4.4, it follows that in order to see one Pop III star caustic transit per year at the top of the Pop III star mass function (M ≳  15–30 M_⊙), one would need to observe about three clusters at least two times per year about six months apart in one to two successive JWST cycles. Observing more often when scheduling allows for clusters at higher zodiacal latitude would, of course, be preferred. The first exposure pair would be needed to identify a potential Pop III star caustic transit event, and the last pair is needed to monitor its expected decay on a timescale of less than one year. Imaging in all eight broadband NIRCam filters is essential to identify the high-redshift dropout nature of a potential caustic transit event and to identify foreground interloping events, which will be more numerous but interesting in their own right. For the brighter caustic transit events, follow-up spectroscopy should be attempted to confirm the nature of the transit, as described in Section 7.

The caustic transit rate of 0.32 per cluster per year is indicated by the orange upper limit in Figure 1. If the actual 2.0 μm SB of Pop III stars is dimmer than ∼31 mag arcsec⁻², then their caustic transit rate would be correspondingly lower. This is indicated in Figure 1 for SB levels (in light orange) that are 10×, 100×, and 1000× dimmer than ∼31.0 mag arcsec⁻², with the corresponding caustic transit rates indicated in dark orange. The minimum number of caustic transits JWST could reasonably see—in a large monitoring program spread over many years—is a Pop III SB of ∼36 mag arcsec⁻², which would require monitoring 30 clusters at least twice every year over 10 years. Such a large JWST observing program could reach the level of ∼10 Pop III objects per arcsec² and would need to be a dedicated multiyear community effort.

To reach levels of only a few Pop III objects per arcsec² (SB ≳ 37 mag arcsec⁻² in Figure 1) through JWST caustic transits would either require observing either ∼100 clusters per year for 10 years—prohibitive in terms of JWST time—or the existence of stellar-mass BH accretion disks that are feeding much longer than massive Pop III stars live on average. This is discussed in Section 5.3.

Appendix C discusses the uncertainty estimates in the main parameters that determine the caustic transit rates and rise times of Pop III stars at z ≳ 7. The combined uncertainty in their caustic transit rates follows from the multiplicative sources of error in Equations (19), (25), and (26). These are the adopted effective caustic length L_caust (with ∼0.3 dex uncertainty), the cluster transverse velocity v_T (∼0.3 dex), the Pop III stellar luminosity L at z ≳ 7 (∼0.2 dex), the uncertainty from the presence of microlensing in the ICL (≳ 0.5 dex), and the uncertainty in the 1–4 μm sky-SB from Pop III stars (≳0.5 dex). Further details are given in Appendix C.

These five main parameters that determine the Pop III star caustic transit rates are independent. Therefore, the combined uncertainty in the Pop III star caustic transit rates follows from taking these factors in quadrature and is estimated to be at least 0.7 dex, which is indicated by the vertical (dark orange) error range in Figure 1. For this reason, a JWST survey to find caustic transits at z ≳ 7 will need to be prepared to cover at least this factor of 5 uncertainty in Pop III star caustic transit rates. Since these uncertainty factors can be larger, JWST may need to observe at least 3–30 clusters per year during the first couple years of its lifetime. Such a survey would need to be maintained until a sufficient number of Pop III star caustic transits have been detected, at which point the actual Pop III star caustic transit rate can be better estimated, and the survey strategy updated accordingly.

If Pop III stars at z ≳ 7 are weakly clustered, their SB may be fairly uniform compared to the size of the caustics (see Section 4.3 and Figure 4(b)). Therefore, one could instead monitor fewer clusters for a correspondingly longer period of time. That is, for the anticipated 5–10 year lifetime of JWST (see Section 7), one could instead monitor a number of well-understood lensing clusters at high zodiacal latitude every few months during JWST's lifetime. Any of these possibilities would constitute a minimum observing program to potentially identify Pop III star caustic transits during the lifetime of JWST. The program could then be adjusted after the number of caustic transits at z ≳ 7 is known after the first couple of years when monitoring a number of clusters. The presence of microlensing will likely also require observing the clusters more frequently to catch caustic transits at shorter timescales, as discussed in Section 4.3.2 and Appendix B.

The mass of the final Pop III star end product is more nuanced than just the BH mass. For example, theoretical models predict that stars in the general mass range of 100 M_⊙ $\lesssim \,M\lesssim 200$ M_⊙ do not lose much mass and that they may undergo an ${e}^{+}-{e}^{-}$ pair-creation instability (Barkat et al. 1967; Fraley 1968; Wheeler 1977; Sugimoto & Nomoto 1980; Bond et al. 1984; Fryer et al. 2001; Woosley et al. 2002; Kozyreva et al. 2017; Woosley 2017). Such stars may undergo thermonuclear explosions that completely disrupt the star without forming a stellar-mass BH and eject a large amount of iron-group elements, especially ⁵⁶Ni (e.g., Smith et al. 2007; Kozyreva & Blinnikov 2015). Theoretical models predict that stars with 260 M_⊙ $\lesssim \,M\lesssim 5$ × 10⁵ M_⊙ enter the pair-instability region but are too massive to be disrupted. They undergo standard core collapse and form intermediate-mass black holes (IMBHs; Fryer et al. 2001; Chatzopoulos et al. 2013; Belczynski et al. 2016). Figure 12 of Woosley et al. (2002) offers a map of the Pop III initial–final mass relation for massive stars from stellar evolution theory. For the Pop III ZAMS mass range in our MESA models, we adopt similar end products. Their end-product mass and the BH Schwarzschild radii ${R}_{{\rm{s}}}$, which are used in our caustic transit calculations for stellar-mass BH accretion disks, are listed in Table 5.

Table 5.  Pop III Stellar Mass black hole Accretion Disk Parameters Adopted for Caustic Transit Calculations

Massa	${M}_{\mathrm{compact}}$ b	${R}_{s}$ c	Radiusd	${L}_{\mathrm{bol}}$ e	${M}_{\mathrm{bol}}$ f	Bolo+IGM+K-corrg			${m}_{\mathrm{AB}}$ Limits ath			${t}_{\mathrm{rise}}$ i	Transitj
ZAMS		BH	of the UV Accretion Disk			z = 7	z = 12	z = 17	z = 7	z = 12	z = 17	(z = 12)	Rate
(${M}_{\odot }$)	(${M}_{\odot }$)	(km)	(${R}_{\odot }$)	(${L}_{\odot }$)	AB-mag	(AB-mag)			(AB-mag)			(hr)	(/cl/yr)
BH accretion-disk bolometric luminosities and UV half-light radii scaling from microlensed quasars (Blackburne et al. 2011)
30	∼5.0 BH	15	1.4	≲ 4.2 × 104	≳−6.8	−0.6	−1.4	−1.7	≳41.8	≳42.4	≳42.9	0.27?	≳0.58?
50	∼24 BH	72	3.0	≲2.0 × 105	≳−8.5	−0.4	−1.2	−1.5	≳40.3	≳40.9	≳41.4	0.58*	≳0.15*
100	∼65 BH	195	4.9	≲5.4 × 105	≳−9.6	−0.2	−0.9	−1.3	≳39.4	≳40.0	≳40.5	0.95*	≳0.06*
300	∼230 BH	690	9.2	≲1.9 × 106	≳−11.0	−0.2	−1.0	−1.3	≳38.1	≳38.6	≳39.2	1.8*	≳0.02*
1000	∼720 BH	2160	16.3	≲6.0 × 106	≳−12.2	−0.2	−0.9	−1.3	≳36.8	≳37.5	≳37.9	3.2*	≳0.01*
BH accretion-disk bolometric luminosities and UV half-light radii estimated from multi-color thin-disk model
30	∼5.0 BH	15	1.9	≲3.1 × 104	≳−6.5	−0.6	−1.4	−1.7	≳42.1	≳42.8	≳43.2	0.37?	≳0.84?
50	∼24 BH	72	4.5	≲1.8 × 105	≳−8.4	−0.4	−1.2	−1.5	≳40.4	≳41.1	≳41.5	0.87*	≳0.18*
100	∼65 BH	195	7.8	≲5.9 × 105	≳−9.7	−0.2	−0.9	−1.3	≳39.3	≳40.0	≳40.4	1.51*	≳0.06*
300	∼230 BH	690	15.8	≲2.0 × 106	≳−11.0	−0.2	−1.0	−1.3	≳38.0	≳38.6	≳39.1	3.1*	≳0.02*
1000	∼720 BH	2160	29.8	≲6.6 × 106	≳−12.3	−0.2	−0.9	−1.3	≳36.7	≳37.4	≳37.8	5.8*	≳0.01*

Notes.

^aPop III ZAMS stellar mass in M_⊙ from Table 1. ^bResulting Pop III stellar BH mass in M_⊙, following Woosley et al. (2002). Note that for Pop III stellar masses of 100 ≲ M ≲ 200 M_⊙ there are likely no BH leftovers (see Section 5.1), which the weighting in Section 6.2 includes. ^cResulting Pop III BH Schwarzschild radius ${R}_{{\rm{s}}}$ in kilometers. ^dAdopted Pop III BH rest-frame UV-accretion disk half-light radius ${r}_{\mathrm{hl}}$ in R_⊙. The top tier of BH UV accretion radii (and bolometric luminosities) is inferred by scaling from observed microlensed quasars (Blackburne et al. 2011), the bottom tier was estimated from the multicolor thin-disk model discussed in Section 5.5.2. For a standard multicolor accretion disk around a BH of mass M, we get about ${R}_{\mathrm{UV}}$ ≲ 40,000 ${R}_{{\rm{s}}}$. ^eAdopted Pop III BH accretion disk bolometric luminosity in L_⊙. The quoted luminosities and resulting rest-frame UV magnitudes are upper limits, since they assume that the BH accretion disk is constantly feeding at the stated luminosities for maximum lifetimes discussed in Section 5.3–5.4. Therefore, the resulting caustic BH accretion disk transit rates in column 14 are lower limits. ^fResulting Pop III BH accretion disk absolute bolometric AB-magnitude ${M}_{\mathrm{bol}}$. ^gCombined bolometric+IGM+K-correction to the Pop III star ${M}_{\mathrm{bol}}$ at z = 7, z = 12, and z = 17, respectively, calculated as in Sections 3.3 and 5.5.2. ^hBH accretion disk apparent AB-magnitudes at z = 7, z = 12, and z = 17 in 2016 Planck cosmology (Planck Collaboration et al. 2016b) using the NIRCam filters that sample rest-frame UV 1500 Å, assuming K-corrections as in columns 7–9 and no dust (see Section 6.1). Distance moduli used are DM = 49.24, 50.58, and 51.42 mag at z = 7, z = 12, and z = 17, respectively. ⁱPop III BH accretion disk caustic transit rise time t_rise at z = 12 as estimated in Section 4.4. Asterisks (*) indicate BH masses M ≳ 24–65 M_⊙. For their accretion disks, caustic transit events are possibly observable to the detection limits of JWST medium-deep to deep survey epochs reaching AB ≲ 28.5–29 mag, assuming caustic transit magnifications of $\mu \simeq {10}^{4}\mbox{--}{10}^{5}$ can elevate Pop III stellar-mass BH accretion disks with AB ≲ 41.5 mag temporarily above these JWST detection limits. Details are in Section 6.2. ^jThe cluster caustic transit rate of stellar-mass BH UV accretion disks as estimated in Sections 4.4 and 6.2, but directly applying Equations (19) and (25), rather than the general expression in Equation (27), which is only valid for the M/L relation in Equation (3) and Table 2 for Pop III stars.

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In this context, we briefly consider possible constraints from the recent LIGO detections on stellar-mass BHs at z ≲ 0.1 (Abbott et al. 2016a, 2016c). These are very plausibly examples of merging BH pairs with M ≃ 29–36 M_⊙, 14–21 M_⊙, and 19–31 M_⊙, respectively, about 1–3 Gyr ago (Abbott et al. 2016b, 2016d, 2016e, 2017a). de Mink & Mandel (2016) suggest that these BHs are possibly left over from later (Pop II) starbursts about 5–12 Gyr before z ≃ 0.1, with a median age of ∼7 Gyr for these mass pairs, which in 2016 Planck cosmology corresponds to a range in their formation redshift of ${z}_{f}\simeq 0.7\mbox{--}10$ with a median of ${z}_{f}\simeq 1.1$. If true, such BHs may not have had significant accretion rates since their progenitor-star supernovae (SNe) went off 5–12 Gyr before their binary merger produced gravitational waves at their detection distance of z ≲ 0.1. In each LIGO case, a pair of massive stars of somewhat unequal masses formed, and so their evolutionary scenarios may have resulted in accretion onto the black holes left by the more massive parent stars with M ≳ 30–80 M_⊙ after it produced an SN.

Another issue that we need to consider in this section is the lowest ZAMS mass that can with some fidelity produce a BH, also for Pop III stars at z ≳ 7. This is a very active topic of research where different groups are getting different results (Sukhbold & Woosley 2014, 2016; I. Petermann & F. X. Timmes 2018, private communication). Depending on the models used (1D, 2D, or 3D, with or without rotation), the "compactness" of the end product is rather uncertain in the mass range of 10  ≲ M ≲ 30 M_⊙. Rotation and binary interaction can produce different initial–final mass landscapes (e.g., Yoon et al. 2008). For 10 ≲ M ≲ 30 M_⊙, not all models get a clean explosion. On the other hand, at M ≳ 30 M_⊙ nature can produce SNe with BH remnants, since LIGO has already seen 14–36 M_⊙ BHs at z ≲ 0.1–0.2. Hence, Pop III stars with 10 ≲ M ≲ 30 M_⊙ may yield BHs, while for M ≳ 30 M_⊙ they most likely do.

For our calculations of Pop III BH accretion disk caustic transits, we will assume that Pop III stars with M ≳ 30 M_⊙—with the exception of the mass range of 100 ≲ M ≲ 200 M_⊙—can and will produce BHs of roughly 15%–70% of the ZAMS Pop III stellar mass, or M ≃ 5–720 M_⊙ (deduced from Figure 12 of Woosley et al. 2002; see column 2 of Table 5 here). A full treatment of the evolution of Pop III binary or multiple stars, their end products, and their impact on Pop III BH accretion disks is beyond the scope of this study, and needs to be the focus of more detailed modeling in future work.

The actual resulting BH masses themselves are not as relevant for our caustic transit calculations. It only matters that such BHs exist—and for M ≳  14 M_⊙ LIGO has clearly shown that they do—and that they accrete while producing a sufficiently high UV luminosity to be detected by JWST during a caustic transit. Any accretion (Frank et al. 2002) would have to be maintained for ≳0.1 yr in the rest frame at z ≳ 7 (i.e., ∼ 1 yr in the observed frame) with ${L}_{\mathrm{bol}}$ values ≳ 10⁵ L_⊙ (Tables 2–4) to be possibly seen transiting across a cluster caustic by JWST and decay for about a year or less above the JWST detection threshold (Section 4.4).

Since we do not know the duplicity nor the separation distribution of Pop III stars, nor of the first polluted O-stars in mini halos, we need to consider a range of possibilities. Pop III BHs with 5 ≲ M ≲ 720 M_⊙ may accrete more steadily via Roche-lobe overflow from a (slightly polluted) Pop II.5 companion star of lower ZAMS mass, as discussed in Section 3.2.

The second scenario is much more common for O stars nearby, given their very high multiplicity, but may not be common for Pop III stars at z ≳ 7. Trenti & Stiavelli (2009) suggest that as soon as a massive Pop III star first forms in a mini halo, its powerful Lyman–Werner UV radiation field may prevent lower-mass Pop III stars from forming in its immediate surroundings. Self-shielding by very dense surrounding hydrogen gas against this UV radiation may allow some neighboring lower-mass Pop III stars to still form. Trenti & Stiavelli (2009) therefore also discuss mini halos that may have more than one Pop III star. In their models, Pop III stars generally start forming at z ≲ 30–40 (cosmic ages ≳99–65 Myr, respectively), followed by slightly polluted Pop II.5 stars that quickly ramps up at z ≲ 28–35 (cosmic ages ≳109–79 Myr, respectively), or about ∼10–15 Myr later in cosmic time. Sarmento et al. (2018) present hydrodynamical simulations that narrow the Pop III star redshift range from z ≃ 20 to z ≃ 7. In their models, pristine Pop III stars are still the dominant population at z ≃ 20, while at z ≃ 7, slightly polluted (Z≲ 10⁻⁴ Z_⊙) "Pop II.5" stars outnumber Pop III stars by ∼10:1. In other words, Pop III stars may have polluted their surroundings quickly enough that within 10–15 Myr, many lower-mass stars that formed in their neighborhood already have somewhat non-zero metallicities.

Comparing the estimated pre-MS lifetimes (${\tau }_{\mathrm{preMS}}$) to the MESA ZAMS–AGB lifetimes in Table 1, we found in Section 3.1 that Pop III stars with M ≃ 20–1000 M_⊙ live short enough (≲8 Myr) that they may have polluted the material from which stars with M ≃ 1–1.5 M_⊙ formed at z ≳ 7, since their pre-MS lifetimes are longer than 6–9 Myr. Hence, it is possible that most early low-mass stars (M ≃ 1–1.5 M_⊙) may have been polluted by massive Pop III stars as early as z ≲ 20, and certainly at lower redshifts down to z ≃ 7. This then also means that it is possible that very low-metallicity Pop II.5 stars may have formed at z ≲  20 in the vicinity of Pop III stars, perhaps some close enough to form binaries or multiple-star systems with those Pop III stars. In any case, the first polluted O stars likely also appeared at z ≲ 20. For the latter, the duplicity fraction may have quickly increased from the very low duplicity values expected for true zero-metallicity Pop III stars—with lower-mass companions not forming due to their significant LW radiation (Trenti & Stiavelli 2009)—to the much higher duplicity fraction seen in O stars today (see Section 3.2).

The metallicity evolution of stellar populations is not well-known at high redshifts (z ≳ 4; Maiolino et al. 2008; Kim et al. 2017). Trenti & Stiavelli (2007) and Sarmento et al. (2018) suggest that mini halos and the IGM can get quickly enriched (to $Z\simeq {10}^{-4}$ Z_⊙) by a progenitor Pop III SN. The hydrodynamical simulations of Sarmento et al. (2018) use Adaptive Mesh Refinement (AMR) to sample the mass range of M ≃ 10^5.5–10⁸ M_⊙ over the redshift range of z ≃ 8 to z ≃ 16, where their predicted metallicities range from Z ≃ 0.1 Z_⊙ at M ≃ 10⁸ M_⊙ to Z ≃ 0.003 Z_⊙ at M ≃ 10^5.5 M_⊙. Over this mass and redshift range, their mass–metallicity relation has a slope of Δlog(Z/Z_⊙)/log(M/M_⊙) ≃ 0.5–0.6. At masses below M ∼ 10^5.5 M_⊙, their AMR simulations have insufficient mass resolution, but if the mass–metallicity relation were to continue with this slope to single stellar masses as low as M ≲ 10³ M_⊙, then the non-pristine stars at z ≳ 7 could indeed have metallicities as high as Z ≃ 10^−3.5 Z_⊙. Madau & Fragos (2017) suggest that at z ≃ 7–10, the metallicity of massive (M$\sim {10}^{8}$ M_⊙) star-forming objects may be as high as 0.03–0.1 Z_⊙. For the low-mass environments in which slightly polluted Pop II.5 stars form, a metallicity of Z ≳ 10⁻⁴Z_⊙ is thus plausible.

Recent observation (e.g., Badenes et al. 2018) has shown that metal-poor (Z ≲ 0.3 Z_⊙) stars have a multiplicity fraction 2×–3× higher than metal-rich (Z ∼ Z_⊙) stars. Theoretical work on star formation (e.g., Machida et al. 2009) suggested a higher binary frequency in lower-metallicity gas, and that a majority of stars are born as members of binary/multiple systems for Z ≲ 10⁻⁴ Z_⊙. Hence, for non-zero metallicities, at least the binary fraction increases with decreasing metallicity. Physically, this occurs because metal-line cooling becomes significant above a threshold of Z ≳ 10⁻⁴ Z_⊙, which decreases the fragmentation of the gas clouds that form the stars. We do not know if this trend continues to hold for truly zero-metallicity Pop III stars at z ≳ 7, but it seems possible that any non-zero-metallicity massive star will form and evolve in an environment with a significant binary fraction (see, e.g., Adams et al. 2006; Adams 2010 for a discussion).

What matters for the current work is that, while some massive stars with zero or very low metallicity may still exist at z ≃ 7, at the same time, a sufficient fraction of polluted stars (Z ≳ 10⁻⁴ Z_⊙) already exists at z ≃ 7–17. The latter are critical, since they likely formed with a significant fraction of binaries and so play an essential role in BH accretion disk feeding via Roche-lobe overflow during its post-MS evolution.

Mass transfer is not currently considered in the MESA code. Future work needs to include detailed star formation scenarios with full metallicity evolution in the ISM at z ≳ 7, their subsequent evolutionary tracks at very low metallicities, and include scenarios of binary evolution that incorporate mass transfer and address how mass transfer affects the BH-feeding timescales.

We will consider here that any BHs left over after a massive Pop III star's death may accrete from a surrounding lower-mass, low-metallicity star filling its Roche lobe during its post-MS evolution, causing a UV-bright accretion disk. The accretion timescales onto these BHs in stellar binaries are not well-known, but may have plausibly lasted as long as the GB lifetimes of the less massive star in a binary when it fills its Roche lobe. Following the Pop III MS lifetimes from Section 3.1 and Table 1, this can happen within ≲12% of their MS ages, or within 0.3–60 Myr after the first SN of the more massive star in the pair has occurred. The question then arises: how often can this scenario have happened for Pop III stars at z ≳ 7, whose stellar-mass BH remnants would still be around today as leftovers from the First Light epoch?

If a fraction $(1-\epsilon )$ of the matter is accreted at the Eddington rate, where denotes the radiative efficiency, then the mass of the BH will increase exponentially with a characteristic timescale of ${t}_{E}=4\pi G\mu {{\rm{m}}}_{p}/({\sigma }_{e}c\epsilon )\simeq 45\,\mathrm{Myr}$. If all remnants of Pop III stars accreted at the Eddington rate for ≳10⁸ yr, then this would increase the BH mass by orders of magnitude, which would increase the mass density of BHs to values that are excluded by constraints on the present-day mass density of BHs (see, e.g., Tanaka et al. 2012). Steady BH feeding from accretion disks for ≳10⁸ yr would have likely given rise to BHs that will grow catastrophically to $\gg {10}^{2}$ M_⊙, and may quickly produce massive BHs with M ≳ 10³–10⁵ M_⊙ or more and become Ultra-Luminous X-ray sources (ULXs). Although UV-bright accretion disks around such massive BHs would be easier to detect with JWST during caustic transits (see Sections 4.4 and 6.2), they will likely also be much rarer.

For Pop III stellar-mass BH accretion disks, we will therefore consider lifetimes of at least ≳0.3 Myr from the massive binary argument in Section 3.2.2. In Section 5.5, we will assume that the BH accretion disks are constantly feeding at the luminosities predicted for maximum lifetimes of ≲60 Myr, during which the lowest-mass (M ≳ 2.0 M_⊙) companion AGB stars would fill their Roche lobes before reionization is complete at z ≃ 7 (Section 3.2.1). Given the uncertain accretion times, the BH accretion disk UV luminosities derived in Section 5.5 are upper limits, so their caustic transit rates in Section 6.2 are lower limits.

Following the arguments of Section 3.5, if N ≲ 10³ massive Pop III stars per arcsec² contribute to the near-IR sky-SB of AB ≳ 31 mag arcsec⁻² at 2.0 μm, then a large fraction (${f}_{\mathrm{BH}}$) of them will leave behind BHs. Accretion onto these BHs will give rise to additional flux in the IRB. The ratio of the Pop III to Pop III remnant contribution can be estimated from

$$\begin{eqnarray}&&\displaystyle \frac{{S}_{\mathrm{Pop}\mathrm{III}}}{{S}_{\mathrm{BH}}}=\displaystyle \frac{1}{{f}_{\mathrm{BH}}}\displaystyle \frac{{t}_{\mathrm{PopIII}}}{{t}_{\mathrm{acc}}}\displaystyle \frac{{L}_{\mathrm{Pop}\mathrm{III}}}{{L}_{\mathrm{BH}}}.\end{eqnarray} \tag{ 34 }$$

Figure 2 shows that a 300 M_⊙ Pop III star has a luminosity of ${L}_{\mathrm{bol}}$ $\simeq 2.5\times {10}^{40}$ erg s⁻¹, which agrees quite closely with the Eddington luminosity associated with an almost equal-mass BH, which is ∼10⁴⁰ erg s⁻¹ following Equation (34). If we assume that the fraction ${f}_{\mathrm{BH}}$ of Pop III stars that collapses into BHs produces a BH of ∼15%–70% of the original stellar ZAMS mass, then we expect ${L}_{\mathrm{Pop}\mathrm{III}}\simeq {L}_{\mathrm{BH}}$ at least at early times. We then obtain

$$\begin{eqnarray}&&\displaystyle \frac{{S}_{\mathrm{Pop}\mathrm{III}}}{{S}_{\mathrm{BH}}}=\displaystyle \frac{1}{{f}_{\mathrm{BH}}}\displaystyle \frac{{t}_{\mathrm{PopIII}}}{{t}_{\mathrm{acc}}}.\end{eqnarray} \tag{ 35 }$$

The efficiency of gas accretion onto stellar-mass BHs formed by Pop III stars is discussed by Milosavljević et al. (2009). It is possible that radiative feedback seriously limits the efficiency of gas accretion. Time-averaged Eddington ratios of $\sim 1 \%$ have been reported by, e.g., Park & Ricotti (2012), although this ratio could be smaller. If accretion occurs during the typical ∼0.3–60 Myr adopted for early massive binaries, then these accretion times are less than 1%–10% of the available Hubble time at z ≳ 7. Hence, BHs may have been feeding with a duration of ≲1%–10% of the total available time.

If we take into account that the mass of the BH grows with time, then it is plausible that $({S}_{\mathrm{Pop}\mathrm{III}}/{S}_{\mathrm{BH}})$ ≲ 1, i.e., the remnants of Pop III stars may contribute more to the near-IR sky-SB than the Pop III stars themselves. If Pop III remnants form the seeds for SMBHs, including the rare ${M}_{\mathrm{BH}}\simeq {10}^{9}$ M_⊙ black holes that are seen in quasars at z ≳ 6 (Willott et al. 2003; Jiang et al. 2007; Kurk et al. 2007), then at least a small fraction of them must accrete at practically the Eddington rate with a duty cycle of $\sim 100 \%$ (e.g., Willott et al. 2010). If a small fraction of the remnants accrete so efficiently, then it is not unexpected that a much larger fraction will accrete with duty cycles intermediate between 1% and 100%. Depending on how large the fraction of more slowly accreting BHs is, this population could contribute significantly more to the near-IR sky-SB and to caustic transits than the Pop III stars themselves.

Pop III stars with masses M ≃ 30–1000 M_⊙ can leave BHs behind with M ≃ 5–720 M_⊙ (Section 5.1), except for the mass range around 100–200 M_⊙, where they seem to produce no BHs (Woosley 2017). The Schwarzschild radii of these Pop III BHs will thus be in the range of ${R}_{{\rm{s}}}$ ≃ 15–2200 km, as listed in column 3 of Table 5. Using this range of BH masses and Schwarzschild radii, this section summarizes available constraints on the resulting sizes and luminosities of stellar-mass BH accretion disks. Since these parameters are more uncertain than those of Pop III stars, we will estimate them in two independent ways to permit a consistency check. The resulting UV accretion disk radii, bolometric luminosities, and corresponding ${M}_{\mathrm{AB}}$ magnitudes are listed in columns 3–6 of Table 5, which are described for both methods in the next two subsections.

5.5.1. Estimates by Scaling from Observed Microlensed Quasar Results

A first estimate of ${R}_{\mathrm{accr}}$ and ${L}_{\mathrm{accr}}$ can be made from observed microlensing results on strongly lensed quasars at z ≃ 1–2 by Blackburne et al. (2011). These authors present accretion disk sizes, temperatures, and luminosities from their quasar images that were monitored extensively with ground-based telescopes and through Chandra X-ray fluxes. Their Equation (2) gives a simple relationship among accretion disk half-light radius (${r}_{\mathrm{hl}}$ or ${R}_{\mathrm{accr}}$), the quasar SMBH mass, and the observed wavelength, which in their case is the observed optical that samples rest-frame ∼2500 Å:

$$\begin{eqnarray}&&{R}_{\mathrm{accr}}\simeq 1.68\times {10}^{14}\ {\left(\displaystyle \frac{{M}_{\mathrm{BH}}}{{10}^{9}{M}_{\odot }}\right)}^{2/3}\ {\left(\displaystyle \frac{\lambda }{\mu {\rm{m}}}\right)}^{4/3}\ m.\end{eqnarray} \tag{ 36 }$$

We rescale this for the JWST NIRCam near-IR filters F115W–F277W, which sample Pop III objects at z ≃ 7–17 approximately in the rest-frame UV at $\lambda \simeq 1500\,\mathring{\rm A}$. From their multicolor microlensing photometry, Blackburne et al. (2011) derive SMBH masses of order (0.04–2) × 10⁹ M_⊙ and bolometric luminosities in the range of ${L}_{\mathrm{bol}}$ ≃ (0.1–4) × 10⁴⁶ erg s⁻¹. Their Table 8 suggests that for all 12 quasars, the bolometric accretion disk luminosity scales with the SMBH mass approximately as

$$\begin{eqnarray}&&{L}_{\mathrm{bol}}\simeq 3.2\times {10}^{46}\ (\displaystyle \frac{{M}_{\mathrm{BH}}}{{10}^{9}\ {M}_{\odot }})\ \mathrm{erg}\,{{\rm{s}}}^{-1}.\end{eqnarray} \tag{ 37 }$$

For a solar luminosity of 3.828 × 10³³ erg s⁻¹, this corresponds to quasar accretion disk luminosities of ∼(0.3–10) × 10¹² L_⊙.

These are remarkable direct constraints to quasar rest-frame UV accretion disk sizes and their luminosities. We do not know if we can scale these values down from their observed mass range to our range of BH masses of ∼5–720 M_⊙ adopted in Table 5. Blackburne et al. (2011) suggest from the data over their mass range that their half-light radii scale as

$$\begin{eqnarray}&&{r}_{\mathrm{hl}}\propto {M}_{\mathrm{BH}}{}^{\rho },\end{eqnarray} \tag{ 38 }$$

with a best fit of $\rho \simeq 0.27\pm 0.17$. This is flatter than the ρ = 2/3 slope implied by the multicolor accretion disk theory in Equation (36). If we scale our UV accretion disk radii down with $\rho \simeq 0.27$ from their SMBH mass range, then we obtain very large radii (${R}_{\mathrm{UV}}$≳ 10³ R_⊙) and luminosities for Pop III stellar-mass BH accretion disks. This suggests that the flat ρ slope derived from their quasar sample may not hold down to Pop III BH masses, as may be caused by the strong dependence on BH mass of the tidal forces around each BH. We therefore adopt a slope between these values of $\rho \simeq 0.5$, which is consistent with the Blackburne et al. (2011) value within their errors and still provides a good fit to their data given the small dynamic range in ${M}_{\mathrm{BH}}$ in their sample. In Section 5.5.2, we suggest that $\rho \simeq 0.5$ produces more consistent overall results for the multicolor thin-disk accretion model. When we scale the Blackburne et al. (2011) UV accretion-disk radii down with $\rho \simeq 0.5$, we obtain the BH UV half-light radii listed in the top tier of Table 5. These are in the range of ${R}_{\mathrm{UV}}$ ≃ 1–16 R_⊙ for ${M}_{\mathrm{BH}}$ ≃ 5–720 M_⊙. The bolometric luminosities listed in the top tier of Table 5 were scaled down directly with Equation (37) and are in the range of 4 × 10⁴–6 × 10⁶ L_⊙ for ${M}_{\mathrm{BH}}$ ≃ 5–720 M_⊙, respectively.

5.5.2. Estimates from Multicolor Accretion Disk Theory

In this section, we compare the stellar-mass BH accretion disk sizes and luminosities as scaled down from the quasar observations in Section 5.5.1 to theoretical estimates.

In the simplest form, accretion disks around BHs are assumed to be "multicolor" thin disks, which consist of a series of concentric shells, each of which emit blackbody radiation characterized by its radially dependent temperature (e.g., Shakura & Sunyaev 1973; Remillard & McClintock 2006; Blackburne et al. 2011). In the rest-frame UV–optical (at ${\nu }_{\mathrm{Ly}\alpha }$ ≲  2.466 × 10¹⁵ Hz), the spectrum of the accreting BH is dominated by the thermal disk component. In the very inner part of the accretion disk, other radiation mechanisms will likely produce significant X-ray emission, such as synchrotron radiation in the presence of strong central magnetic fields, inverse Compton radiation, or thermal bremsstrahlung (Shakura & Sunyaev 1973, 1976). Only the harder part of this redshifted X-ray emission will make it past the IGM and potentially be detected by Chandra, but what matters for any JWST detection is the amount of associated rest-frame UV emission that makes it past the IGM at λ ≳ 1216 Å. In the multicolor thin accretion disk model, the temperature increases with radius as

$$\begin{eqnarray}&&T\propto {r}^{-3/4}.\end{eqnarray} \tag{ 39 }$$

Using Equation (38), gas in the innermost stable orbit at R ≃ 3${R}_{{\rm{s}}}$ has a maximum temperature of about

$$\begin{eqnarray}&&{T}_{\max }\ \simeq 10\ {\left(\displaystyle \frac{{M}_{\mathrm{BH}}}{100{M}_{\odot }}\right)}^{-\tau }\ \mathrm{keV}.\end{eqnarray} \tag{ 40 }$$

Standard thin-disk accretion theory suggests a slope of τ = 1/4, but since we adopted $\rho \simeq 1/2$ in Equation (38), we need to use τ = 3/8 here to maintain consistency with Equation (39). The multicolor accretion disk models predict similar UV half-light radii ${R}_{\mathrm{UV}}$ for either slope τ, since the largest SED differences occur well below rest-frame 1216 Å, and this part of the SED does not make it past the IGM at z ≳ 7.

We will assume here that the maximum temperature of the inner accretion disk in Equation (40) for a 100 M_⊙ BH needs to be at least 10 keV, or ${T}_{\max }$ ≃ 3.87 × 10⁷ K. This is so that their hard X-ray photons can make it past the neutral hydrogen at z ≳ 7 (Haardt & Madau 2012), and when redshifted from z ≃ 7–17, still be in principle observable in the Chandra soft X-ray band, which covers 0.5–2.0 keV. This argument is based on the following: if part of the Spitzer–Chandra cross-correlation power-spectrum signal (Cappelluti et al. 2013; Mitchell-Wynne et al. 2016) came from redshift z ≳ 7, then the sources that cause it must be both Spitzer 3–4 μm and Chandra X-ray sources, as discussed in Section 2.3.2. Both of these papers discussed PBHs as possible candidates for the Spitzer–Chandra cross-correlation signal. As discussed in Section 3.1, Pop III stars alone cannot cause this Spitzer–Chandra cross-correlation signal, since they reach only a maximum temperature of ${T}_{\mathrm{eff}}$ ≲ 10⁵ K.

As we move out in radius, the temperature drops as in Equation (39). We assume that each concentric radius interval in the multicolor accretion disk emits as a blackbody with its own temperature. The largest radius that will contribute to the UV emission is the one where the blackbody curve peaks in the UV longwards of Lyα. For our 1500 Å rest-frame UV reference, this largest ring needs to have a temperature of T ≳ 3.2 × 10⁴ K. This suggests that we need to go out in radius where the temperature is a factor of ≳ 1200× lower than in the inner ring. Hence, we need to integrate out to r ≃ 13,000 R_min, where ${R}_{\min }$ ≃ 3 ${R}_{{\rm{s}}}$ is the radius of the innermost stable orbit around the BH. Plugging in the numbers above then yields ${R}_{\mathrm{accr}}$(UV) ≃ 1.6 × 10⁷ km for the maximum radius of the UV-emitting region, or ∼17 R_⊙ for ${M}_{\mathrm{BH}}$ = 100 M_⊙. Integration of the actual multicolor thin-disk light profiles for a 100 M_⊙ BH yields a half-light radius ${r}_{\mathrm{hl}}$that is about 1.7× smaller than this, as shown below.

We use the multicolor accretion disk model in Equations (39)–(40) for the Pop III BH mass range of 5–720 M_⊙ in Table 5 to predict their UV half-light radii ${r}_{\mathrm{hl}}$ and their bolometric and UV luminosities. Their UV half-light radii ${r}_{\mathrm{hl}}$ are then simply integrated from the part of the radially dependent UV accretion disk SED that makes it past the IGM at z ≳ 7. These results are listed in the bottom tier of Table 5 and shows UV half-light radii in the range of ${R}_{\mathrm{UV}}$ ≃ 2–30 R_⊙.

At these ${r}_{\mathrm{hl}}$ values, the multicolor accretion disks have an effective temperature of ${T}_{\mathrm{eff}}$ ≃ 47,500–48,000 K for M ≃ 5–720 M_⊙. Bolometric+IGM+K-corrections were applied to the bolometric luminosities in Table 5, as for Pop III stars in Section 3.3. For multicolor accretion disks with ${T}_{\mathrm{eff}}$ ≃ 47,700 K, these combined BIK-corrections are –0.3, −1.1, and −1.5 mag at z = 7, z = 12, and z = 17, respectively. These are comparable to the values in Tables 3–4 for Pop III RGB and AGB stars with monochromatic blackbody disks of similar temperatures. The bolometric luminosities predicted for the multicolor thin accretion disks are in the range of 3 × 10⁴–7 × 10⁶ L_⊙ for ${M}_{\mathrm{BH}}$ ≃ 5–720 M_⊙, and are listed in the bottom tier of Table 5.

To check our multicolor accretion disk models for consistency, we first verified that they reproduce the Blackburne et al. (2011) UV half-light radii obtained for accretion disks of z ≃ 1–2 quasars from Equation (36). Second, we apply our multicolor accretion disk model to M ≃ 10⁹ M_⊙ SMBHs known to be present in quasars at z ≳ 6 (e.g., Willott et al. 2003; Jiang et al. 2007; Kurk et al. 2007). The above equations imply a UV accretion-disk diameter of 2${R}_{\mathrm{UV}}$ ≃ 2 × 10⁵ R_⊙ for a 10⁹ M_⊙ SMBH, which is ∼1000 au or 0.005 pc across, corresponding to a light travel time of ≲5 days in the rest frame. This is comparable to the accretion-disk sizes of QSOs inferred from variability studies, where the somewhat larger broad-line region can be light-days–weeks across (e.g., Kozłowski et al. 2010; Butler & Bloom 2011). Our multicolor thin accretion disk model also predicts the unobscured rest-frame UV luminosity for the rare quasars with a 10⁹ M_⊙ SMBH at z ≳ 6, which is ${M}_{\mathrm{UV}}$ ≃ −27 AB-mag (Fan et al. 2001, 2003). Hence, their BIK-corrected near-IR fluxes are predicted to be ${m}_{\mathrm{AB}}$ ≳ 21 mag at z ≳ 7, which is comparable to what is observed for the highest-redshift quasars (e.g., Mortlock et al. 2011). This extrapolation to QSOs at z ≳ 6 then justifies the slightly modified choices of $\rho \simeq 1/2$ (instead of 2/3) and $\tau \simeq 3/8$ (instead of 1/4) above.

In summary, Table 5 shows that both estimates of the ${R}_{\mathrm{UV}}$ and ${L}_{\mathrm{bol}}$ of Pop III stellar-mass BH accretion disks in Section 5.5.1 and in this section are within a factor of two or less. We will therefore adopt the two scaling methods in the equations above and assume that the resulting range of properties in Table 5 captures the properties of Pop III BH UV accretion disks sufficiently well to make an order-of-magnitude estimate of the cluster caustic transit rates for Pop III BH accretion disks.

Given the unknown accretion efficiencies compared to Eddington, or the unknown accretion lifetimes compared to the maximum accretion lifetimes possible (Section 3.1), the values in Table 5 are upper limits to the Pop III BH UV accretion disk luminosities. That is, the luminosities and resulting ${M}_{\mathrm{bol}}$ and ${m}_{\mathrm{AB}}$ values in Table 5 assume that BH accretion disks radiate at steady-state levels inferred by the multicolor accretion-disk model for maximum lifetimes as discussed in Section 5.3.

In conclusion, the inner stellar-mass BH accretion disks may be significantly hotter than the typical T ≃ 10⁵ K temperatures of Pop III stars, plausibly reaching X-ray temperatures at the innermost radii, and reaching ∼30,000 K at the outermost radii. Their UV-bright accretion disks—if unobscured by surrounding dust—have SEDs that can make it in part through the neutral IGM at z ≳ 7 with UV radii ≲40,000 ${R}_{{\rm{s}}}$. Their rest-frame UV radii are ${R}_{\mathrm{UV}}$ ≃ 1–30 R_⊙, and their UV luminosities are at most 3 × 10⁴–7 × 10⁶ L_⊙ for ${M}_{\mathrm{BH}}\simeq 5\mbox{--}720$ M_⊙, respectively. Pop III stellar-mass BH accretion disk radii may thus be similar to, or somewhat larger than, the 1–13 R_⊙ radii of the ZAMS Pop III stars in Tables 1–2, but no larger than the Pop III RGB or AGB star radii in Tables 3–4. They would fit well within the ∼7–55 R_⊙ Roche lobe sizes seen in massive binaries discussed in Section 3.2.2, and so are eligible for feeding from a less massive RGB/AGB star in the binary that is filling its Roche lobe. This assumes that subsequent generations of (slightly) polluted massive stars at z ≳ 7 already have high-enough metallicity to form binaries. The predicted stellar-mass BH accretion disk UV radii and maximum luminosities are similar to those of Pop III RGB–AGB stars in the 10–300 M_⊙ range. We use this to estimate the BH accretion disk caustic transit time and rates in Section 6.2.

For completeness, we will briefly consider here the potential impacts of White Dwarfs (WDs) that likely result from low-mass stars (M ≲ 5 M_⊙) at z ≳ 7 (see, e.g., the Z = 5 × 10⁻³ Z_⊙ sample of Romero et al. 2015), and of Neutron Stars (NSs) that likely result from Pop III stars at ZAMS masses M ≲ 20 M_⊙, since both will be far more common than Pop III stellar-mass BHs (see Table 5 and the IMF slopes in Figure 3).

Table 1 implies that NSs would not appear until 8–70 Myr after their progenitor stars with 5 ≲ M ≲ 20 M_⊙ form, while WDs would appear at least ≳230 Myr after their progenitor stars with M ≲ 5 M_⊙ form at z ≃ 7–17. If the first stars form at z ≃ 35–40, then the first NSs would thus appear soon thereafter, but the first WDs would not appear until z ≲ 14. In either case, the first NS or WD mergers at z ≳ 7 would have only 500–700 Myr to occur. Hence, we will not consider NS–NS mergers such as those recently found by LIGO (Abbott et al. 2017b, 2017c) and identified by ground-based follow-up campaigns (e.g., Chornock et al. 2017; Cowperthwaite et al. 2017) in a nearby galaxy, nor potential NS–WD or WD–WD mergers, as these are far more rare than regular accretion onto either a WD or an NS.

The duration of regular accretion onto WDs or NS before they undergo a nuclear explosion on their surfaces depends mainly on their accretion rates. These in turn depend on the binary separation, masses of the two components, evolutionary state of the companion, and the nature of the explosion. For WDs (novae and super-soft X-ray sources), the recurrence timescales are ∼20 to ∼10,000 years (Shara et al. 1986; Cannizzo et al. 1988; Wolf et al. 2013; Henze et al. 2015; Shafter et al. 2015; Shafter 2017), and are likely too rare to average out to a flux that could be detected during a cluster caustic transit. For NSs (X-ray bursters), the recurrence timescales can be hours to weeks (Tanaka & Shibazaki 1996; Watts 2012). Their luminosities when averaged over ≳0.1 years at z ≳ 7 would determine if such objects could be seen via caustic transits by JWST. In all cases, their surface layers explode, after which they may resume accretion and may approach their previous steady-state luminosity. A proper description of WD and NS accretion will thus not only require the multicolor thin-disk models that we use for BH accretion disks in Section 5.5.2, but also a quantitative modeling of these episodic nuclear detonation events, which is beyond the scope of the current paper. Future work will need to consider if accretion onto Pop III NS or WDs can be steady enough and luminous enough to be a source of caustic transits that are potentially observable by JWST.

True zero-metallicity massive stars, with all modeling investigations to date, have significantly reduced mass loss. The normal driver of massive-star winds—radiation pressure from scattering off metals, is not present. Alternatives such as rotational mixing, some dredge-up scenario to bring core material to the photosphere, or (epsilon- and kappa-) pulsation mechanisms, are too weak to cause much mass loss (Castor et al. 1975; Götberg et al. 2017; Renzo et al. 2017). Thus, zero-metallicity massive stars may not be shrouded by dusty circumstellar material. For metallicities of Z ≲ 10⁻⁴ Z_⊙ (or even ≲10⁻⁵ Z_⊙), the winds (hence dust) will be at levels more common for massive stars seen nearby, although still significantly reduced.

If Pop III stars—or the slightly polluted Pop II.5 stars—did manage to produce stellar winds during their main sequence and Blue–Red Supergiant (BSG-RSG/AGB) phases, this could have deposited dust into the surrounding medium. When a fraction of Pop III stars goes off as Pair Instability SuperNovae (PISNe), they would deposit additional metals into their immediate surroundings. Dust formation in the circumstellar material of initially zero-metallicity Pop III stars could thus have added a non-trivial extinction/reddening factor, especially in their late stellar evolution and subsequent BH accretion disk stages. Hence, we should consider possible cases where either Pop III stars or their stellar-mass BH accretion disks are significantly reddened by dust, or both.

For non-rotating Pop III stars, this dust could be distributed rather uniformly and obscure most of the Pop III stars and their BH accretion disks, but for rotating stars, the situation may be quite different. We do know that Gamma-Ray Bursters (GRBs) are quite visible from γ-ray to radio waves when viewed from the right direction. The same is true for unobscured versus obscured AGNs—much of their visibility is viewing-angle dependent with respect to the dust torus. We therefore must consider that at least a fraction of Pop III stars with significant stellar rotation produced BH accretion disks that are visible under certain viewing angles and produced an equal amount of UV-continuum radiation as the Pop III stars themselves, or perhaps more. Evolving rotating Pop III star models, dust production, and their likely non-uniform dust-expulsion mechanism are currently too uncertain to take into account in the model calculations and will require more detailed numerical modeling in future work.

If both Pop III stars and their stellar-mass BH accretion disks were fully unobscured, then the average ≳2 × 10⁶ yr MS lifetime of Pop III stars (Section 3.1) and maximum BH accretion disk lifetimes—as visible in the rest-frame UV—of ≲60 Myr would determine their visible ratio. Some fraction of Pop III BH accretion disks may not be fully obscured, as would be implied by the Spitzer–Chandra power spectrum results in Section 2.3.2, if some of this signal came from z ≳ 7 (Cappelluti et al. 2013, 2017). In reality, nature may have well-produced some observable combination of obscured and unobscured Pop III stars and their BH accretion disks, as it has for the iEBL from spheroids, disks, and unobscured AGNs at lower redshifts in Figure 1. For that reason, we allowed the maximum sky-SB of ≳31.0 mag arcsec⁻² of Section 2.3 to be either fully caused by Pop III stars or by their BH accretion disks, or by a combination of the two not exceeding this SB level. JWST may be able to distinguish between the two through chromatic effects of caustic transits, as discussed in Section 7.2.

In this section, we present estimates of the cluster caustic transit rates resulting from stellar-mass BH accretion disks as described in Section 5. To first order, for Pop III stellar-mass BHs, the same principles apply as above, so unless stated otherwise, we use the equations in Section 4.4.

As discussed in Section 5.5.2, the expected BH accretion disk radii are similar to, or somewhat larger than, the 1–13 R_⊙ radii of the ZAMS Pop III stars in Tables 1–2, but no larger than Pop III RGB or AGB star radii in Tables 3–4. The maximum BH accretion disk luminosities are in general similar to those of Pop III RGB–AGB stars in the 10–100 M_⊙ range, or ∼10⁴–10⁷ L_⊙.

Pop III stellar-mass BH accretion disks—when lensed through cluster caustic transits—thus also have rise times of order one to several hours. For their similar luminosities, the decline times will then be also of the order of a year, as discussed in Section 4.4. These, together with their transit rates predicted for the Pop III BH radii and luminosities in Section 5.5, are listed in Table 5.

For a Pop III ZAMS mass function slope of $\alpha \simeq 2$, the weights for each of the mass bins for BH accretion disks in Table 5 are very similar, following Equation (32). The resulting total transit rates for Pop III stellar-mass BH accretion disks with ${M}_{\mathrm{BH}}$≳ 24–720 M_⊙ that are in principle observable with JWST to AB ≲ 28.5–29 mag across the caustics are predicted to be ≳0.18 per cluster per year for the top tier in Table 5, and ≳0.24 per cluster per year for the bottom tier, respectively.

Because the luminosities and the resulting ${M}_{\mathrm{bol}}$ and ${m}_{\mathrm{AB}}$ values in Table 5 are upper limits (Section 5.5), the inferred BH accretion-disk transit rates are lower limits, as indicated in Table 5. That is, if the actual accretion efficiencies were 10× lower, then the BH accretion luminosities would be ∼2.5 mag fainter, and the caustic transit rates could be several times higher. This is because there would be 10× as many faint objects per mass bin that make up the near-IR SB adopted in Section 2.3 that contribute to caustic transits above the detection limit, but there would also be fewer mass bins contributing above the JWST detection limit.

The limits to the caustic transit rates of stellar-mass BH accretion disks of ∼0.2 per cluster per year are similar to caustic transit rate of ∼0.32 per cluster per year obtained for Pop III ZAMS+RGB+AGB stars (Section 4.4). For BHs, they could be several times higher, depending on their actual accretion efficiency.

We briefly discuss this in the context of the lifetime differences between Pop III stars and their stellar-mass BH accretion disks that could affect the mix of caustic transits JWST may observe. In Table 5, BHs with UV accretion disks bright enough to be detected by JWST during caustic transits have ${M}_{\mathrm{BH}}\simeq 24\mbox{--}720$ M_⊙ and AB ≃ 37–42 mag. Pop III stars with 30 ≲ M ≲ 1000 M_⊙ that produce BHs have ZAMS ages of 5.6–2.1 Myr (Table 1) with an average of ∼3 Myr. Pop III stars of masses M ≃ 2–20 M_⊙ live considerably longer than these during their AGB stage, where they could fill their Roche lobes for up to 0.6–60 Myr, with an IMF-weighted average GB age of ∼6 Myr. Hence, during their AGB stage 2–20 M_⊙ stars could feed the BH that is leftover from a 30–1000 M_⊙ star for a maximum duration that is significantly longer than the ZAMS lifetime of the massive Pop III star that produced this BH.

In summary, depending on how steady and efficient BH feeding by a lower-mass AGB star in its Roche lobe is, stellar-mass BH accretion disks may be about as likely as Pop III stars at z ≳ 7 to cause cluster caustic transits that could be observed by JWST, and possibly more likely. Stellar-mass BH accretion disks with an SB ≃ 31.0 mag arcsec⁻² (or ∼1 nW m⁻² sr⁻¹) could produce about one caustic transit per five clusters per year, and perhaps as many as one event per two clusters per year. As for the Pop III stars in Section 4.4, a dedicated JWST program that monitors three clusters per year for a number of years could possibly detect several caustic transits for stellar-mass BH accretion disks. If their SB were to be as dim as ∼36.0 mag arcsec⁻², which corresponds to ∼10 Pop III BH accretion disks per arcsec² (see Figure 1), then 30 clusters would have to be monitored for up to 10 years to detect any caustic transits from BH accretion disks at z ≳ 7.

Appendix D discusses the uncertainty estimates in the main parameters that determine the caustic transit rates and rise times of Pop III stellar-mass BH accretion disks. As in Section 4.5 and Appendix C, the combined uncertainty in their caustic transit rates follows from the adopted effective caustic length L_caust (with ∼0.3 dex uncertainty), the cluster transverse velocity v_T (∼0.3 dex), and the uncertainty from the presence of microlensing in the ICL (≳ 0.5 dex). The uncertainty in the Pop III stellar-mass BH accretion disk luminosity L is larger than for Pop III stars. This is due to their uncertain accretion efficiency or accretion duration, which we assume is uncertain by at least 0.5 dex, as discussed in Section 5.4 and Appendix D. On the other hand, the uncertainty in the 1–4 μm sky-SB from Pop III stellar-mass BH accretion disks may be smaller than that of Pop III stars, since the Spitzer–Chandra power spectrum results (Section 2.3.2) hint at a possible contribution from (stellar-mass) BHs at z ≳ 7. As discussed in Appendix D, the error in their power spectrum signal is estimated at ≳0.15 dex.

Since these parameters are independent, the combined uncertainty for the caustic transit rates of stellar-mass BHs is thus at least ∼0.7 dex, but for somewhat different reasons than for Pop III stars. This is indicated by the vertical black error range in Figure 1. Given the Spitzer–Chandra power-spectrum signal discussed in Section 2.3.2 and Appendix D, and the possibility that their non-steady-state luminosities may increase their caustic transit rates for a given near-IR sky-SB, as discussed in Section 5.4, the caustic transit rates for stellar-mass BH accretion disks may be closer to the upper value indicated in black in Figure 1. Within the uncertainties detailed in Appendices C–D, it is thus possible that stellar-mass BH accretion disks at z ≳ 7 may outshine the sky-SB from Pop III stars in the observed near-IR, and that they may produce correspondingly more caustic transits. Only a long-term, dedicated observing program may be able to tell the difference between these two possible sources of caustic transits at z ≳ 7, as discussed in Section 7.

In conclusion, Pop III star rotation, the way dust is produced and expelled during and after the Pop III star evolutionary sequence, the massive star binary fraction, and the subsequent stellar-mass BH accretion timescales and accretion efficiency may well in the end determine which of the two has the best chance to be detected by JWST via cluster caustic transits.

To observe caustic transits from First Light objects, a dedicated JWST observing program will be required of at least several and up to 30 clusters for a duration of 1–10 years (see Figure 1). Depending on their exact contribution to the diffuse 1–4 μm sky-SB (≲0.01–0.1 nW m⁻² sr⁻¹), such a JWST observing program to detect individual Pop III stars and/or their stellar-mass BH accretion disks at z ≳ 7 may well require monitoring—in the optimistic case that most of the NIR power-spectrum signal comes from z ≳ 7—a few suitable galaxy clusters during a year. In the most pessimistic case where there exist really only a few Pop III objects per square arcsecond and/or that most of them are shrouded by dust, JWST may need to monitor 30 clusters twice a year for a good fraction of JWST's 5–10 year lifetime to detect a few Pop III caustic transits. All of these cluster observations would require coeval images in four NIRCam filter pairs and/or four NIRISS filters to constrain the spectral signature and redshift of a Pop III caustic transit candidate. These would appear as z ≳ 7 dropout candidates that vary with time, either increasing rapidly and then slowly fading, or vice versa. Their rise time are of the order of hours, while their fading times are a good fraction of a year.

Both cases pose interesting challenges to JWST IR-array data reduction techniques: great care must be taken that a sudden increase in magnified object flux during a caustic transit does not get rejected as an artifact or as a cosmic ray in the series of images taken that day. Also, care must be taken that a slow increase in magnified flux of an object that approaches the caustic from the other side does not get misinterpreted as a slowly variable faint Galactic brown dwarf star or a weak variable AGN (e.g., Cohen et al. 2006). The nature of such "reverse transits" may therefore not be obvious when first identified observationally by JWST.

Could Pop III caustic transits cause a real difference in the luminosity function at z ≳ 7 in the field (e.g., Bouwens et al. 2017) compared to clusters (Livermore et al. 2017)? If in the most optimistic case several Pop III objects at z ≳ 7 were always seen transiting a cluster caustic in any given year, then this could artificially boost the number of z ≳ 7 objects seen behind clusters. This may not be obvious if such Pop III objects resided in small star-forming objects that are well below the HST or JWST detection limits, so one would not know in advance to expect caustic transits at these locations. This is unlike the caustic transiting objects detected by Kelly et al. (2018) and Rodney et al. (2017), where there was a known faint galaxy at a given location on the cluster caustic. Although such caustic transit detections of Pop III objects at z ≳ 7 behind clusters could be real, they may need additional lensing magnification corrections in order to represent the unlensed background universe at z ≳ 7, and so could affect the derived steady-state LF. Detailed JWST studies of high-quality LFs at z ≳ 7 that are well-sampled behind different clusters may reveal cluster-to-cluster differences in caustic properties. Cosmic variance of the z ≳ 7 population will also require averaging over a significant number of lines of sights, by observing a number of clusters with JWST.

Microlensing by faint stars in the lensing cluster ICL may decrease the magnifications from ∼10⁴–10⁵ to ≳10³, but greatly lengthen the visibility time of the caustic transit, where a transiting microlensed object may be visible for many decades or longer. We outlined an observing strategy that JWST may use to observe these objects. To minimize the effects from microlensing in the modeling of caustic transits, one could also target some compact galaxy clusters at 0.3 ≲ z ≲ 0.5 that have a smaller fraction of ICL compared to their total galaxy light at the z ≳ 7 lensing contours, but that—due to their compactness—have excellent lensing properties (e.g., Griffiths et al. 2018).

A dedicated multiband JWST monitoring program of well-studied lensing clusters may be able to detect the chromatic differences expected between caustic transits of stellar-mass BH accretion disks and those of Pop III stars, perhaps including spectroscopic confirmation.

The one significant difference between Pop III stellar-mass BH accretion disks and Pop III stars is likely the presence of a hard X-ray component that contributes very significantly at the inner accretion disk radii and that will also have a significant energy tail longwards of Lyα 1216 Å. No such X-ray component would exist for the Pop III stars themselves, since, ignoring their limb darkening and any starspots, their stellar photospheres have nearly uniform temperatures of T ≃ 10⁵ K (Section 3.1). Hence, Pop III stars will not show significant chromatic behavior that may be traced during a caustic transit, but BH accretion disks could show such chromaticity if they were detected close to the actual caustic transit.

Any differences in the effective UV radii between BH accretion disks and Pop III stars are important, since the maximum magnification obtained during a caustic transit increases strongly for objects with smaller effective UV radii (Equation (21) in Section 4.4). A very hot BH accretion disk crossing a caustic could have much larger magnifications at its smallest intrinsic X-ray–UV-bright radii, since these radii contribute a larger fraction of the total energy longwards of Lyα than they do for Pop III stars. Since the maximum magnification scales as 1/$\sqrt{{R}_{\mathrm{UV}}}$ (Equations (16) and (21)), their smallest UV-bright radii (Section 5.5) could undergo a maximum magnification, ${\mu }_{\max }$, that is considerably larger during a caustic crossing, which could boost their observed rates accordingly compared to Pop III stars.

Specifically, the inner (bluer) part of the BH accretion disk would be magnified much more than its outer (redder) part. The ratio in magnifications should follow $\sqrt{({r}_{\mathrm{out}}/{r}_{\mathrm{in}})}$, where ${r}_{\mathrm{out}}\gt {r}_{\mathrm{in}}$ are the largest and smallest BH UV-accretion disk radii discussed in Section 5.5.2, respectively. This will result in chromaticity due to lensing, where the shape of the BH light curve peaks during a caustic crossing would depend more strongly on rest-frame UV wavelength, unlike that of the Pop III stars. For JWST, there could be a ∼1 dex difference in magnification between the bluer and redder filters for an object undergoing a caustic transit at z ≳ 7. If a caustic transit maximum is observed almost simultaneously in different JWST filters, we could then constrain the BH mass using Equation (38), assuming its scaling holds with slope $\rho \simeq 1/2$ to stellar BH masses (see Section 5.5). This would be an indirect way of confirming that part of the light observed from a z ≳ 7 object undergoing a caustic transit originates in accretion disks around stellar-mass BHs.

Pop III stars may also be detected or confirmed by JWST in other ways. For instance, Macpherson et al. (2013) consider the prospect of finding a Pop III hypernova "in flagrante," and suggest a detection rate of 2.78 × 10⁻⁶ per JWST field of view (FOV) and a probability of 37% that JWST will serendipitously image an afterglow during its lifetime. What JWST truly will find from the Pop III epoch may include these and other unexpected surprises. It is therefore critical that JWST First Light surveys are well-designed to optimize the possible detection of Pop III objects directly.

JWST will be able to detect and monitor caustic transits during its 5–10 year lifetime. It is therefore useful to consider which other facilities can observe caustic transits on longer timescales. JWST's unique advantage is its very dark zodiacal sky in L2 (AB ≳ 23–24 mag arcsec⁻² at $\lambda \simeq 2.0\mbox{--}3.5$ μm; Figure 1), and its stable PSF over a relative wide FOV (Rieke et al. 2005, 22 × 44). Together with its 25 m² collecting area, JWST should be able to reach AB ≳ 28.5 mag routinely (Windhorst et al. 2008). The next generation 25–40 m ground-based telescopes—the European Extremely Large Telescope (E-ELT), the Giant Magellan Telescope (GMT), and the Thirty Meter Telescope (TMT)¹¹ —will have a much larger collecting area and narrower PSFs when using Multi-Conjugate (laser-assisted) Adaptive Optics, although perhaps not as stable as JWST's PSFs, and they will have lower Strehl ratios. They will also have a 1–2 μm sky foreground that is ≳7 mag brighter than JWST's in L2. As a consequence, the next-generation ground-based telescopes may be able to reach AB ≲ 29 mag in integrations of hours at 1–2 μm, but—given their adaptive optics—only over a smaller FOV (≲20'' × 20''–1' × 1'). Ground-based telescopes will have reduced sensitivity at wavelengths λ ≳ 2–2.2 μm because of the strongly increasing thermal foreground. For that reason, JWST will be able to better address any chromatic differences between caustic transits of Pop III stars and their stellar-mass BH accretion disks (Section 7.2), especially those at z ≳ 12 that require several very sensitive filters at λ ≳ 2 μm, where ground-based telescopes cannot reach AB ∼ 29 mag due to the much brighter thermal foreground.

Confirming spectra of caustic transits by Pop III stars or their stellar-mass BH accretion disks could be taken with the JWST NIRISS and NIRSpec spectrographs, and also with the next-generation near-IR spectrographs on the ELT, GMT, and TMT telescopes. Of particular interest would be detecting the 1640 Å He line, which is expected to be present in the ionized regions around Pop III stars, or their BH accretion disks, with T ≳ 10⁵ K (Schaerer 2002; Sobral et al. 2015).

We do not need to catch a caustic transit event at the precise moment of crossing the caustic. It may be sufficient if a Pop III star is seen ≲10 yr before or after a caustic crossing, when the typical magnification may well be of order 10⁴, which can make a Pop III star with AB ≃ 38 mag visible to JWST. In five years' time, the observed flux would increase (or decrease) steadily by a factor of $\sim \sqrt{2}$, which could be identified as a star heading toward (or away from) a caustic. Perhaps the caustic transit of such stars will not be observed during JWST's lifetime, but the next-generation ground-based telescopes will be able to continue to monitor such stars for a much longer period, when a given star appears to be heading toward a caustic in several years' time.

In summary, the next generation ground-based telescopes can monitor at 1–2 μm over a much longer period than JWST—individual Pop III caustic transits that JWST will have detected at 1–4 μm during its lifetime, and also discover new ones on timescales longer than JWST's lifetime. This capability would be particularly useful to follow up on caustic transits that may be affected by microlensing, and so may stretch out over many decades. Because of its much wider 1–4.5 μm wavelength range over which it can reach AB ≃ 29 mag, JWST will be essential to distinguish between possible chromatic differences between Pop III stars and BH caustic transits.

When microlenses are present, the rate of caustic transit events is sensitive to the mass function of low-mass stars in the cluster ICL (e.g., Miralda-Escude 1991). If brighter Pop III stars are more common, relatively modest microlensing peaks with ${\mu }_{\max }\simeq 5\times {10}^{3}$ can momentarily amplify a bright star (AB ≲ 37 mag) to above the detection limit of JWST. The rate of events will then be dominated by the microlens peaks of the brightest stars, and the rate of events will be proportional to the optical depth of microlenses. Significantly fainter stars may be sufficiently magnified only if the disruption of the cluster caustic by microlenses is moderate. In this case, most events will be produced by relatively faint Pop III stars crossing the lightly disrupted caustic, which can have a maximum magnification of ${\mu }_{\max }\simeq {10}^{5}$ (see Section 4). The rate of events will be dominated in this case by the anticipated, much more numerous fainter stars (AB ≃ 40–42 mag) when crossing the caustic, and will be proportional to the surface mass density of stars in the ICL. A rough estimate of the expected rate of events, R, in the presence of microlenses can be obtained based on the predictions from Kelly et al. (2018) and Diego et al. (2017), who modeled the HFF clusters with v_T ≃ 1000 km s⁻¹:

$$\begin{eqnarray}&&R({\mathrm{yr}}^{-1})=A(\gt \mu ){\rho }_{* }\,r({\rm{\Sigma }},{v}_{T})=\displaystyle \frac{{B}_{o}}{{\mu }^{2}}\,{\rho }_{* }\,r({\rm{\Sigma }},{v}_{T}).\end{eqnarray} \tag{ 41 }$$

Here, $A(\gt \mu )$ is the area in the source plane above a given magnification μ, which scales as ${B}_{o}/{\mu }^{2}$ in the presence of microlenses and at high magnification. Also, ${\rho }_{* }$ is the surface mass density of Pop III stars above z = 7, so that $A(\gt \mu )\ .\ {\rho }_{* }$ is the number of Pop III stars undergoing microlensing at a particular moment. Last, r(Σ,v_T) is the rate of microlens caustic (hereafter "microcaustic") events a moving object in the background ($z\gt 7$) would encounter if the surface mass density of microlenses is Σ, and the background object is moving with a transverse velocity v_T with respect to the network of microcaustics. Kelly et al. (2018) estimated that r(Σ,v_T) is of order 0.1 yr⁻¹ for an event like Icarus. Using the equations of Diego et al. (2017) and assuming the total length of the caustics to be L ≃ 100'', we estimate that ${B}_{o}\simeq 1.8\times {10}^{-4}$ arcsec², or A(μ ≳ $3\times \,{10}^{3}$)  ≃ 0.002 arcsec² for an HFF-like cluster in the presence of microlenses. The value $\mu \simeq 3\times {10}^{3}$ is adopted to select regions in the source plane associated with microlensing peaks that will reach ${\mu }_{\max }$ ≳ 10⁴. Then, if there are ${\rho }_{* }\simeq 100$ Pop III objects/arcsec² brighter than AB ≃ 38 mag, we would expect for each HFF-like cluster $R({\mathrm{yr}}^{-1})\simeq 0.2\times r({\rm{\Sigma }},{v}_{T})\simeq 0.02$ caustic transits yr⁻¹ if we extrapolate the results of Kelly et al. (2018) to the entire caustic region, i.e., one event when monitoring five HFF-like clusters for 10 years.

We note that both the estimates with microlensing in this appendix and those in Section 4.3 based on the adopted transverse velocity are in good agreement. These numbers should be compared with the expected caustic transit rate if we assume there are no microlenses. In this case, the magnification would be described by $\mu \simeq 20/\sqrt{d}$, which implies d = 1.6 × 10⁻⁵ arcsec for $\mu \simeq {10}^{4}$. Note that the above expression would give $\mu =(20/\sqrt{1.6\times {10}^{-5}}$) ≃ 5000, but the total magnification would be μ = 10⁴ when we account for the double image produced in the image plane. If the perimeter of the caustic region is L ≃ 100'', then the area over which a magnification larger than $\mu \simeq {10}^{4}$ can be attained is  ≃0.0016 arcsec². This number is comparable to the value estimated above when microlenses are included (∼0.002 arcsec²).

The similarity of results obtained with and without microlenses could have been anticipated from basic principles. Owing to flux conservation, the number of photons collected after integrating for a long period (tens to hundreds of years) should be the same independent of the presence (and number) of microlenses. The distribution of microlenses determines how this magnification is redistributed. A lens plane without microlenses results in large magnifications concentrated in a unique narrow region around the caustic (i.e., a single very bright peak; see Section 4.4), while a lens plane populated with microlenses will break apart the single caustic into multiple (smaller) microcaustics. Thus, the rate of high magnification events (${\mu }_{\max }$ ≳ 10⁴) would be similar whether or not there are microlenses, but in the case without microlenses, we would see a single very bright peak when the Pop III star crosses the caustic, while in the case with microlenses, we would see many (smaller) peaks hundreds of years before (or after) the star crosses the position of the cluster caustic (e.g., Diego et al. 2017). Extreme magnification (${\mu }_{\max }$ ≃ 10⁶) can be attained only when microlenses are not included, and may only occur for the lower-mass BH accretion disks whose inner X-ray–UV bright core radii may be much smaller than those of Pop III stars (Section 5.5.2), so that all large magnifications are concentrated in a single peak around the cluster caustic. When microlenses are included, the caustic region is expanded in size, as shown in Diego et al. (2017), so there is a higher probability for a star in the source plane to align with a microcaustic. However, the magnified peaks will be correspondingly fainter, so only the rarer, brighter Pop III stars or the brighter stellar-mass BH accretion disks may produce caustic transits that can be observed by JWST.

If a background source at z ≳ 7 were to be relatively "bright" (AB ≲ 37 mag for the unlensed source), virtually all microlensing peaks—with magnifications of about one to several thousand (Kelly et al. 2018)—can be observed if the star is sufficiently close to the critical curve (i.e., μ ≳ 10³). The number of events in that case will be approximately equal to the number of microcaustics that the background source encounters as it moves across the web of microcaustics. In this section, we therefore present an estimate of the case when JWST may observe when the background object is relatively bright. Instead of computing the probability of an event based on the area above a given magnification as in Appendix B.1, we can simply estimate the number of times a microcaustic is crossed, since a rare but very bright Pop III star (AB ≲ 35–37.5 mag; see Table 2–4) may be directly observed, if the microlensing magnification from the cluster ICL at its location is at least $\mu \simeq {10}^{3}$. In this case, even modest microlenses with subsolar masses can produce changes in flux of 0.5 magnitudes or more. Having a bright star undergoing such frequent encounters with microcaustics is possible, as discussed by, e.g., Kelly et al. (2018). These variations in flux may be observed with JWST when the star crosses the network of microlensing caustics.

The probability of having a microlensing event at a given distance, θ, from the critical curve is given by the effective optical depth to microlensing, which is defined as the fractional area at that location that is being affected by microlensing. Using the model in Diego et al. (2017), the effective optical depth from microlenses is given by

$$\begin{eqnarray}&&\tau (\theta )\simeq 21\times {10}^{-2}\ {\rm{\Sigma }}({M}_{\odot }\,{\mathrm{pc}}^{-2})/\theta (^{\prime\prime} ).\end{eqnarray} \tag{ 42 }$$

For sources at high redshift, the critical curve moves to distances of order 1' from the center of the Brightest Cluster Galaxy (BCG), where the impact of microlenses is expected to be small but still not necessarily negligible. At these angular distances, Diego et al. (2017) estimated for MACS1149 that the surface mass density of microlenses decreases by approximately two orders of magnitude with respect to the one estimated at the position of Icarus. This corresponds to a mass surface density of ${\rm{\Sigma }}\simeq 0.1\ {M}_{\odot }\,{\mathrm{pc}}^{-2}$. Hence, if we restrict our analysis to the region where the effective optical depth of microlenses reaches the saturation level (i.e., $\tau \simeq 1$), then this implies an angular distance of $\theta \simeq 20$ milliarcsec (mas). That is, the two counter-images of the lensed background object would appear on either side of the critical curve and be separated by ∼40 mas. At these separations, both counter-images would form a single unresolved—or at best a slightly resolved—object in the JWST mosaics. At 20 mas distance from the critical curve, the model in Diego et al. (2017) predicts that the magnification from the cluster is approximately $\mu \simeq 5000$, so that any star brighter than AB ≃ 38 mag transiting the caustic at this location could be detected by JWST to AB ≲ 29 mag.

In principle, we could see twice the caustic transit rate in this case, since this unresolved image would contain fluctuations form both sides of the critical curve (i.e., from its positive and negative parity). In reality, the rate on the side with negative parity is expected to be a factor $\sqrt{2}$ times smaller than the rate on the side with positive parity. This difference in rate can be obtained from Oguri et al. (2018), who estimated that the maximum dimension (or cross-section ${C}_{{\rm{S}}}$) of the microcaustic on the side with positive parity is ${C}_{{\rm{S}}}(M)={\theta }_{e}(M)\sqrt{{\mu }_{t}}/{\mu }_{r}$, while its shape is that of a stretched diamond. The Einstein radius of the microlens, ${\theta }_{e}$, depends on the mass of the microlens and the angular-diameter distances from the observer to the lens, from the lens to the background object, and from the observer to the background object. On the side with negative parity, this extension is smaller by a factor of $\sqrt{8}$, but the caustic is divided into two semi-diamond shapes, so that a star crossing the microcaustic would cross caustic lines four times instead of twice. Hence, the effective length (or cross-section) of the caustic on the side with negative parity is smaller by a factor of $\sqrt{8}/2=\sqrt{2}$.

With the above ingredients, it is possible to estimate the expected number of caustic crossings for relatively bright sources (AB ≲ 37 mag), which are now the microcaustics formed by the local microlenses. Each microcaustic is shaped as a diamond, or a double semi-diamond, on the sides with positive and negative parity, respectively (for details, see, e.g., Diego et al. 2017; Oguri et al. 2018). For simplicity, we assume that the microcaustic crossing events take place within the region where the saturation level to lensing is reached (i.e., the 40 mas region surrounding the critical curve mentioned above). In the regime where the saturation level to lensing has not been reached, one can still use the relation between the position in the source plane, β, and the position in the image plane, θ, given by standard lensing theory, $\beta ={\theta }^{2}/C$, where the constant C depends on the lens strength (i.e., the gradient of the lensing potential), and both β and θ are given with respect to the caustic and critical curves, respectively. For a cluster like MACS 1149, Diego et al. (2017) estimated $C\simeq 68$''. For this particular value of C, one obtains $\beta \simeq 6$ microarcsec, or $2.5\times {10}^{-2}$ pc at $z\simeq 10$. The distance traveled by a moving background star with respect to the caustic network is $d(v)=1\,\times \,{10}^{-4}({v}_{T}/1000\,\mathrm{km}\,{{\rm{s}}}^{-1})$ pc yr⁻¹. During this time, the star may encounter multiple microcaustics, depending on the surface density of microlenses. For simplicity, we assume that the background source is moving perpendicular to the maximum extension of the diamond-shaped macrocaustics. This is a reasonable assumption, since the microlens caustics are typically stretched by very large factors, so to first order they can be approximated by straight parallel lines. Since each microcaustic has a cross-section ${C}_{{\rm{S}}}(M)$ that scales with ${\theta }_{e}(M)$ (see Oguri et al. 2018 or the expressions above), the yearly rate of intersections with a microcaustic of mass M is then given by

$$\begin{eqnarray}&&r(M)=2(1+\sqrt{2})\,.\,n(M)\mu \,.\,d(v)\,.\,2{C}_{{\rm{S}}}(M).\end{eqnarray} \tag{ 43 }$$

Here, the first term, $2(1+\sqrt{2})$, accounts for the events produced on either side of the critical curve and the fact that a microcaustic is crossed twice (four times for the side with negative parity). The second term, $n(M)\mu$, is the number density of microlenses with mass M in the lens plane, n(M), which is increased by a factor μ in the source plane. The third term, d(v), is the distance traveled by the background object in one year. The last term, ${C}_{{\rm{S}}}(M)$, is the cross-section of a microcaustic of mass M. For realistic distributions of n(M), one should integrate r(M) to compute the caustic transit rate, but for our purposes we adopt the simple scenario where all microlenses have similar masses of $M\,\simeq 1$ M_⊙. In that case, we get $n(M)={\rm{\Sigma }}(M)=0.1\,{\mathrm{pc}}^{-2}$ and $r(m)=4.7\,\times \,{10}^{-7}{\mu }_{t}^{3/2}$ per year. Here we assumed that v_T ≲ 1000 km s⁻¹, z_lens = 0.5, and z_source = 10, for which one obtains ${\theta }_{e}=2.3\times {10}^{-6}$ arcsec, or 0.0097 pc at z = 10. Also, $\mu ={\mu }_{t}{\mu }_{r}$, so that the dependency with ${\mu }_{r}$ cancels out and the rate depends only on ${\mu }_{t}^{3/2}$. To estimate the value of ${\mu }_{t}$, we adopt the model in Diego et al. (2017), according to which ${\mu }_{t}=\mu /5$ and $\mu \simeq 100/\theta$, where θ is in arcseconds. At the point where the effective optical depth of microlenses reaches the saturation level, $\theta =0\buildrel{\prime\prime}\over{.} 02$, we get ${\mu }_{t}\simeq 1000$, and the caustic transit rate becomes $r(M)\simeq 0.015$ per year.

The above rate is the expected rate per year for one background star intersecting $n(M)={\rm{\Sigma }}\simeq 0.1\,{\mathrm{pc}}^{-2}$ stars per pc² with mass M ≃ 1 M_⊙. During the time the star is moving across the saturation region and toward the main caustic of the cluster, it will intersect many microcaustics until it reaches the main caustic, after which it fades away forever from our vantage point. We ignore the equally likely case where the star approaches from the main caustic from the other direction, which is discussed in Sections 4.4 and 7.1, but is observationally much harder to recognize. We can then estimate the time it takes to cross the saturation region as 0.0097 (pc)/10⁻⁴ (pc/year) = 97 years, so in this time the background star would cross 1.5 microcaustics before reaching the main cluster caustic. The final boosting factor is expected to be modest in the regions of the critical curve with a very small density of microlenses, which would amount to ∼2.5 caustic crossings instead of just one.

Finally, we note that the approximations made above assumed a very conservative low density of microlenses, about two orders of magnitude smaller than in the outskirts of the BCG region. There may be certain regions along the critical curve where the density of microlenses increases very significantly, for instance near an area with a larger fraction of ICL or near a cluster member galaxy. If in these areas the number density of microlenses, n(M), increases substantially, the rate would increase by a similar amount. Assuming one could estimate the luminosity function of Pop III stars at z ≳ 7 in the future from a long-term monitoring program of cluster caustic transits, one would expect their LF to show a significant excess at the highest luminosities, as a consequence of the caustic/microcaustic crossings boosting their observed luminosities the most. This would be akin to the lensing tail observed in the bright end of the high-redshift submillimeter galaxy luminosity function, such as, for instance, has been seen for the lensed Herschel sample (e.g., Negrello et al. 2017).