Constraining the AGN Luminosity Function from JWST with the X-Ray Background

We predict the X-ray background (XRB) expected from the population of quasars detected by the James Webb Space Telescope spectroscopic surveys over the redshift range z ∼ 4–7. We find that the measured UV emissivities imply a ∼10 times higher unresolved XRB than constrained by current experiments. We illustrate the difficulty of simultaneously matching the faint end of the quasar luminosity function and the XRB constraints. We discuss possible origins and consequences of this discrepancy.


INTRODUCTION
The recent James Webb Space Telescope (JWST) photometric and spectroscopic surveys have discovered a large number of Active Galactic Nuclei (AGN) candidates over the redshifts z ∼ 4.5 − 7, especially helping to bridge the gap of missing faint AGN with M UV < −25 (Matsuoka et al. 2023).The AGN candidates are identified based on X-ray, broad Hβ, and high ionization lines.A large number of faint AGN have also been identified at z ∼ 4 − 7 (e.g., Onoue et al. 2023;Kocevski et al. 2023;Übler et al. 2023;Harikane et al. 2023;Maiolino et al. 2023a;Matthee et al. 2023), with steeper faint ends in their luminosity functions than expected from extrapolating previous results.Such candidates are also detected out to higher redshifts, z ∼ 9 − 12 (Fudamoto et al. 2022;Larson et al. 2023;Goulding et al. 2023;Maiolino et al. 2023b).Recently, the EPOCHS survey covering the JWST PEARLS and ERO fields identified nine new AGN in the 6.5 < z < 12 redshift range (Juodžbalis et al. 2023), and the UNCOVER survey used the JWST NIRSpec to provide the UV luminosity functions from 15 AGN candidates over z ∼ 4 − 8 (Greene et al. 2023).The UV luminosity functions (UV LFs) measured at wavelengths 912 Å and 1450 Å are hamsa.padmanabhan@unige.ch,aloeb@cfa.harvard.eduabout factors of ∼ 10 − 100 greater than the faintest UV-selected quasars observed previously.
The measured bolometric luminosities of the AGN are used to derive the properties of their central supermassive black holes (e.g., Pacucci et al. 2023;Maiolino et al. 2023a;Larson et al. 2023;Kocevski et al. 2023;Natarajan et al. 2023).AGN at z ∼ 4 − 7 identified in JWST and previous literature have bolometric luminosities covering a wide range and inferred black hole masses of 10 6 − 10 9 M ⊙ .Many of the derived black-hole to stellar mass ratios are about 10 to a 100 times greater than expected from the local population (e.g., Pacucci et al. 2023;Yue et al. 2023;Stone et al. 2023), with one candidate black hole (Bogdan et al. 2023) also showing evidence for being as massive as the stellar host.
A population of black holes at high redshifts would contribute to a copious production of X-ray photons with energies E > E max = 1.8[(1 + z)/15] 0.5 x 1/3 HI , where x HI is the neutral hydrogen fraction in the IGM (Dijkstra et al. 2004).Here, E max represents the energy above which the IGM becomes optically thin to photons over a Hubble length.Following the redshifting of these photons without absorption, they would be observed today as the present-day X-ray background (XRB) in the relevant energy regime (measured with, e.g., the Swift, INTEGRAL and Chandra surveys; e.g., Ajello et al. 2008;Moretti et al. 2012;Churazov et al. 2007;Cappelluti et al. 2017).Previous theoretical and observational studies (Shen et al. 2020;Haiman & Loeb 1999;Dijk-arXiv:2310.08633v1 [astro-ph.CO] 12 Oct 2023Oct stra et al. 2004;;Faucher-Giguère 2020) have found that the current constraints on the quasar population saturate the unresolved component of the measured XRB (Moretti et al. 2012;Cappelluti et al. 2017).
In this paper, we predict the level of the XRB implied by the measured AGN UV LFs from the recent JWST surveys over z ∼ 4−7.We find that the JWST measurements -assuming a quasar SED best-fitted to the current data -would oversaturate the Swift XRT/Chandra constraints on the unresolved XRB by about an order of magnitude.In so doing, we illustrate the difficulty of simultaneously matching both the UV LF and the XRB constraints.We discuss the possible reasons and consequences for the discrepancy.

RELEVANT EQUATIONS
The data compilation of Shen et al. (2020) allow for a characterization of the bolometric quasar luminosity function from observations in the rest-frame IR, B, UV, soft and hard X-ray wavebands, covering redshifts z ∼ 0−7.A best-fitting quasar Spectral Energy Density (SED) model is constructed to convert the luminosities across various bands and fitted to the data.The bolometric quasar luminosity function, describing the comoving number density of quasars as a function of luminosity over all wavebands, is found to be well-modelled by a double power law (Shen et al. 2020) form.The same functional form is found to describe the luminosity function in specific wavebands (of rest frequencies ν with corresponding luminosities L ν and number densities n): (1) with the best-fitting parameters L * , γ 1 , γ 2 and ϕ * being matched to observations, both separately at each redshift and globally across the range of redshifts considered.The best-fitting form of the luminosity function for the UV (ultraviolet) regime (with rest wavelength 912 Å) over z ∼ 5 − 9 is found to be in good agreement with multiband observational data.The total emissivity of the AGN in this regime at a given redshift is found by integrating over the luminosity function above a minimum value L min : and has units of ergs s −1 Hz −1 Mpc −3 .Using the best-fitting Spectral Energy Density (SED) template of the AGN emission, the above emissivity can be converted from the UV regime into the X-ray waveband(s).Following Shen et al. (2020), this is done by normalising a power-law X-ray SED template over the soft (0.5-2 keV) and hard (2-10 keV) X-ray regimes using the nonlinear α ox conversion factor.From this, the comoving X-ray emissivity can be computed as a function of redshift z and used to define the cosmic X-ray background (XRB; Shen et al. 2020;Faucher-Giguère 2020).
with units erg (or keV) cm −2 s −1 sr −1 Hz −1 , and in which z min and z max are the minimum and maximum redshifts under consideration.In the above, the differential comoving volume element is denoted by The above formalism can be compared to recent measurements of the X-ray background (Cappelluti et al. 2017, e.g.,), specifically its unresolved component which is constrained from the Swift XRT and Chandra data (Moretti et al. 2012).This is plotted as the red dashed line in Fig. 1.

X-RAY BACKGROUND FROM JWST MEASUREMENTS OF AGN LF
The quasar UV LF at z ∼ 4 − 11 compiled from the JWST ERO, PEARLS and CEERS surveys (Matthee et al. 2023;Maiolino et al. 2023a;Harikane et al. 2023;Greene et al. 2023) all indicate faint-end values of about a factor of 10 above those expected from extrapolating the fits in the literature (as shown in Fig. 2 for the case of the UV LF measured at rest wavelength 1450 Å at z ∼ 5).
We can use the measurements of Harikane et al. ( 2023) along with the best-fitting quasar SED to scale the luminosity in the X-ray regime with that observed in the 912 Å waveband (Shen et al. 2020).To do this, we compare the inferred emissivities ϵ 912 at four representative  2020) emissivity by factors of 2×, 5× and 10× respectively.The red dashed horizontal line denotes the constant value of comoving emissivity required (Haardt & Salvaterra 2015) to saturate the unresolved X-ray background measured by Moretti et al. (2012).
redshifts from the results of the JWST (Harikane et al. 2023) to the best-fitting form of Shen et al. (2020):  4), is proportional to the unresolved X-ray background implied by the corresponding measurement.Haardt & Salvaterra (2015) inverted the above analysis to set an upper limit on the (assumed constant for z > 5) comoving emissivity at 912 Å which, along with a quasar SED model consistent with Shen et al. (2020), saturates the unresolved X-ray background measured by Moretti et al. (2012).2This is shown by the dashed red horizontal line in Fig. 3.The X-ray background expected from the JWST is shown by the black solid line in Fig. 1.This is about an order of magnitude higher than the constraints from using the Swift XRT and Chandra observations (Moretti et al. 2012).
We can restate these findings in an equivalent manner by illustrating how the emissivity at 912 Å measured by JWST is discrepant with that required to saturate the current X-ray background limits.To quantify this discrepancy, we use the Cressie-Read χ 2 statistic (and its associated p-value) between the ϵ 912,JWST,i and ϵ 912,S,i quantities, denoting the 'observed' (Harikane et al. 2023) and 'expected' (Shen et al. 2020) emissivities, where i tracks redshift in the interval z ∈ {4, 7}.We also measure the χ 2 between the Harikane et al. ( 2023) emissivities and those obtained by multiplying the Shen et al. (2020) emissivities by factors of 2×, 5× and 10× respectively (shown by the dashed, dotted and dash-dotted blue lines in Fig. 3).The value of the χ 2 statistic for each case is plotted in the top panel of Fig. 4. Assuming a p-value of greater than 1% for the curves to be deemed comparable, we find that the 10× higher value comes closest to consistency with the JWST results (with pvalue 0.0101).
The comparison to the unresolved X-ray background data, however, runs in sharp contrast to the above behaviour for the emissivity.This is illustrated by predicting the X-ray background expected from the emissivity enhanced by the factors above and comparing it to the Moretti et al. (2012) best-fitting curve in Fig. 1.The expected backgrounds for the 2×, 5× and 10× multiplicative factors are illustrated by the dashed, dotted and dash-dotted blue lines in Fig. 1.The corresponding values of the two-dimensional Kolmogorov-Smirnov (KS) test statistic [d-value, which measures the deviation between the predicted XRB and that measured by Moretti et al. (2012)] is plotted in the lower panel of Fig. 4 as a function of the multiplicative factor (a higher d-value indicates larger deviation, with a value of unity ruling out the model).The associated p-value of the 10× enhancement is 8.8 × 10 −8 , reiterating the difficulty of simultaneously satisfying the JWST constraints on the 912 Å emissivity over z ∼ 4 − 7, and the XRB measurements.
this population would oversaturate the limits by about an order of magnitude.
The scatter in the curves arising from the bolometric data considered in Shen et al. (2020) is of the order of 0.2 dex, which is too low to allow consistency with the new results.It is difficult to invoke a stronger obscuring column that limits X-ray escape since any such column would have a greater attenuation effect (e.g., Masters et al. 2012) on the escape of UV photons (and thus, serve to decrease the overall UV emissivity).The hard X-rays, originating in the compact corona of the AGN (e.g., Reis & Miller 2013), are also expected to be less collimated than the optical/UV which originates from the accretion disk.In view of this, the results obtained here may be regarded as conservative.They may also point to a fairly different X-ray to UV SED for quasars in the JWST sample as compared to those from previous measurements (Shen et al. 2020;Faucher-Giguère 2020).
The JWST UV LF is systematically higher than that measured at z ∼ 7 with the SHELLQs survey (Matsuoka et al. 2023).At least 23 of the candidate AGN thus far have reported spectroscopic confirmations with NIR-Spec (Pacucci et al. 2023).However, classical diagnostics alone have been deemed insufficient to confirm secure AGN detections since the NIRSpec resolutions are borderline (R ∼ 300 compared to the required R ∼ 500), which could also contribute to the source of the discrepancy ( Übler et al. 2023).Other possibilities include biases from faint Hα lines which, when corrected, may bring down the number densities of the AGN by factors of ∼ 5 (Matthee et al. 2023).Newer studies with JWST NIRSpec find comparable or higher UV LFs for the AGN at z ∼ 9 than at z ∼ 6 (Fujimoto et al. 2023), with a 10-15% AGN fraction in galaxies.Forthcoming spectroscopic census of AGN will help reveal the presence of other factors such as star-formation driven outflows that may make a dominant contribution to the luminosity (Matthee et al. 2023;Zhang et al. 2023;Ferrara et al. 2023) or objects such as brown dwarfs which may account for up to a third of the AGN candidate sample (Greene et al. 2023;Langeroodi et al. 2023).
4)and d A (z), d L (z) are the angular diameter and luminosity distances respectively to redshift z.The overall redshift-dependent factor in Eq. (3) thus reduces to c/[H(z)(1 + z) 2 ].The I XRB can equivalently be expressed as a function of the energy E of emission corresponding to the rest frequency ν em .The unresolved component of the above background, amounting to about 8-9%(Cappelluti et al. 2017) is constrained to originate from sources above z ∼ 4(Haardt & Salvaterra 2015).

Figure 2 .Figure 3 .
Figure 1.Unresolved X-ray background constraints (EIXRB) as a function of rest energy E from fitting to the Swift XRT and Chandra observations by Moretti et al. (2012, red dashed line).Overplotted are the expected X-ray background from the measured JWST UV LFs (black solid line), and those from enhancing the UV emissivities in Shen et al. (2020) by factors of 2×, 5× and 10× (blue dashed, blue dotdashed and blue dotted lines) over the z ∼ 4 − 7 interval.