SUPERMASSIVE DARK STARS: DETECTABLE IN JWST

Spolyar et al. (2008) first considered the effect of dark matter (DM) particles powering the first stars. The first stars formed when the universe was about 200 million years old, at z = 10–50, in 10⁶ M_☉ halos consisting of 85% DM and 15% baryons in the form of H and He from big bang nucleosynthesis; for reviews of the standard picture of the formation of the first stars, see Barkana & Loeb (2001), Yoshida et al. (2003), Bromm & Larson (2004), and Ripamonti & Abel (2005). The canonical example of particle DM is weakly interacting massive particles (WIMPs). In many theories, WIMPs are their own antiparticles and annihilate among themselves wherever the DM density is high. Recently, there has been much excitement in the DM community about possible detections of WIMPs via annihilation to positrons seen by the PAMELA satellite (Adriani et al. 2009); annihilation to γ-rays seen by Fermi (Abdo et al. 2009; Dobler et al. 2009), and in the direct detection experiments DAMA and CDMS (Bernabei et al. 2010; Ahmed et al. 2009). The annihilation rate is n²_χ〈σv〉 where n_χ is WIMP density and we take the standard annihilation cross section⁵

$$\begin{equation} \langle \sigma v \rangle = 3 \times 10^{-26}\,{\rm cm}^3\,{\rm s}^{-1}, \end{equation} \tag{ 1 }$$

and WIMP masses in the range 1 GeV–10 TeV. The first stars are particularly good sites for annihilation: they form in the right place—in the high density centers of DM halos—and at the right time—at high redshifts (density scales as (1  +   z)³). Spolyar et al. (2008) found that above a certain density (≈10¹³ cm⁻³) the WIMP annihilation products remain trapped in the star, thermalize with the star, and thereby provide a heat source. These first dark stars (DSs) are stars made primarily of hydrogen and helium with only ∼0.1% of the mass in the form of DM; yet they shine due to DM heating. Note that the term "dark" refers to the power source, not the appearance or the primary matter constituent of the star.

DSs are born with masses ∼1 M_☉. They are giant puffy (∼10 AU), cool (surface temperatures <10, 000 K), yet bright >10⁶ L_☉ objects (Freese et al. 2008a). They reside in a large reservoir (∼10⁵ M_☉) of baryons, i.e., ∼15% of the total halo mass. These baryons can start to accrete onto the DSs. Previous work (Freese et al. 2008a; Spolyar et al. 2009) followed the evolution of DSs from their inception at 1 M_☉, as they accreted baryons from the surrounding halo, up to ∼1000 M_☉. DSs can continue to grow in mass as long as there is a supply of DM fuel.

The exciting new development of this paper is that we follow the growth of DSs to become supermassive dark stars (SMDSs) of mass M_* > 10⁵ M_☉. Specifically, we study the formation of 10⁵ M_☉ SMDS in 10⁶ M_☉ DM halos and 10⁷ M_☉ SMDS in 10⁸ M_☉ halos; perhaps the SMDSs become even larger. Hoyle & Fowler (1963) first postulated the existence of such large stars but were not aware of a mechanism for making them. Now the confluence of particle physics with astrophysics may be providing the answer. The key ingredient that allows DSs to grow so much larger than ordinary fusion-powered Population III stars is the fact that DSs are so much cooler. Ordinary Population III stars have much larger surface temperatures in excess of 50,000 K. They produce ionizing photons that provide a variety of feedback mechanisms that cut off further accretion. McKee & Tan (2008) have estimated that the resultant Population III stellar masses are ∼140 M_☉. DSs are very different from fusion-powered stars, and their cooler surface temperatures allow continued accretion of baryons all the way up to enormous stellar masses, M_* > 10⁵ M_☉.

WIMP annihilation produces energy at a rate per unit volume

$$\begin{equation} \hat{Q}_{\rm DM} = n_\chi ^2 \langle \sigma v \rangle m_\chi = \langle \sigma v \rangle \rho _\chi ^2/m_\chi, \end{equation} \tag{ 2 }$$

where n_χ is the WIMP number density, m_χ is the WIMP mass, and ρ_χ is the WIMP energy density. The annihilation products typically are electrons, photons, and neutrinos. The neutrinos escape the star, while the other annihilation products are trapped in the DS, thermalize with the star, and heat it up. The luminosity from the DM heating is

$$\begin{equation} L_{\rm DM} \sim f_Q \int \hat{Q}_{\rm DM} dV, \end{equation} \tag{ 3 }$$

where f_Q is the fraction of the annihilation energy deposited in the star (not lost to neutrinos) and dV is the volume element. We take f_Q = 2/3 as is typical for WIMPs.

Typically (100–10⁴) M_☉ of DM (up to ∼1% of the halo mass) must be consumed by the star in order for large SMDS masses M_* ∼ 10⁵ M_☉ to be reached. We will consider two different scenarios for supplying this amount of DM:

• 1.  
  Extended adiabatic contraction (AC), labeled "without capture" below. In this case, DM is supplied by the gravitational attraction of the baryons in the star. The amount of DM available for DM annihilation due to AC may be larger than our previous estimates which were based on the assumption that DM halos are spherical. In a non-spherical DM halo, the supply of DM available to the star can be considerably enhanced, as we discuss in more detail in Section 2.2. In any case, this mechanism relies solely on the particle physics of WIMP annihilation and does not include capture of DM by baryons (discussed below).
• 2.  
  Extended capture, labeled "with capture" below. Here, the star is initially powered by the DM from AC, but the AC phase is taken to be short ∼300,000 yr; once this DM runs out, the star shrinks, its density increases, and subsequently the DM is replenished inside the star by capture of DM from the surroundings (Freese et al. 2008b; Iocco 2008) as it scatters elastically off of nuclei in the star. In this case, the additional particle physics ingredient of WIMP scattering is required. This elastic scattering is the same mechanism that direct detection experiments (e.g., CDMS, XENON, LUX, DAMA) are using in their hunt for WIMPs. In previous work (Freese et al. 2008a; Spolyar et al. 2009), we assumed minimal capture, where DM heating and fusion contributed equally to the luminosity once the star reached the main sequence. Here, we consider the more sensible case where DM heating dominates completely due to larger ambient DM density, and the star can grow to become supermassive.

SMDSs can result from either of these mechanisms for DM refueling inside the star. The SMDS can live for a very long time, tens to hundreds of million years, or possibly longer (even to today). We find that ∼10⁵ M_☉ SMDSs are very bright ∼3 × 10⁹ L_☉, which makes them potentially observable by the James Webb Space Telescope (JWST). We also note that SMDS may become even more massive (1) if they form in larger halos or (2) the DM halos in which they initially form merge with other halos so that there is even more matter to accrete (note that alternatively these mergers may remove the DS from their high DM homes and stop the DM heating). For example, if DSs form in 10⁸ M_☉ halos, then they could in principle grow to contain all the baryons in the halo, i.e., M_* > 10⁷ M_☉. Since the luminosity scales as L_* ∝ M_* these SMDSs would be even brighter, L_* > 10¹¹ L_☉ and are hence even better candidates for discovery in JWST.

Once the SMDSs run out of DM fuel, they contract and heat up. The core reaches 10⁸ K and fusion begins. As fusion-powered stars they do not last very long before collapsing to black holes (BHs). Again, this prediction is different from standard Population III stars, many of which explode as pair-instability supernovae (Heger & Woosley 2002) with predicted even/odd element abundance ratios that are not (yet) observed in nature. The massive BH remnants of the SMDSs are good candidates for explaining the existence of 10⁹ M_☉ BHs which are the central engines of the most distant (z > 5.6) quasars in the Sloan Digital Sky Survey (SDSS; Fan et al. 2001, 2004, 2006).⁶

The idea of supermassive DS and the resultant >10⁵ M_☉ BH was originally proposed by Spolyar et al. (2009). Subsequently, Umeda et al. (2009) took their existing stellar codes and added DM annihilation to allow the mass to grow. They started from Population III stars in which fusion was already present, assuming they then encounter a reservoir of DM. Here, we start from the very beginning with collapsing protostellar clouds that transition into DSs, which can be DM powered for millions to billions of years before fusion ever begins. The SMDSs in this paper are primordial supermassive stars. More generally, the capture mechanism in particular lends itself to growth of DSs, whether or not fusion has already set in, so that SMDS can result at various epochs in the universe.

Begelman (2009) presents another alternative for the formation of supermassive stars: rapid accretion onto stars which already have hydrogen burning in them. His "quasistars," another possible route to large BH, are quite different from the SMDS discussed in this paper.

Various other authors that have explored the repercussions of DM heating in the first stars (Taoso et al. 2008; Yoon et al. 2008; Ripamonti et al. 2009; Iocco et al. 2008). Recently, in a paper that appeared after we submitted this work, Ripamonti et al. (2010) study the effects of DM heating on the evolution of primordial gas clouds using a one-dimensional code. We are currently not in disagreement with the basic results of their work. We agree that the cloud continues to shrink to a smaller radius and higher density once dark matter heating dominates over cooling—but not for long, only until it reaches thermal and hydrostatic equilibrium. For example, once our canonical 100 GeV case reaches a central density of 10¹⁶ cm⁻³, then our equilibrium dark star begins (and not at a lower density). Such a high density is beyond the reach of the code run by Ripamonti et al. and requires further work.

The possibility that DM annihilation might have effects on today's stars was initially considered in the 1980s and early 1990s (Krauss et al. 1985; Bouquet & Salati 1989; Salati & Silk 1989; Graff & Freese 1996) and has recently been studied in interesting papers by Moskalenko & Wai (2007), Scott et al. (2007), Bertone & Fairbairn (2007), and Scott et al. (2009). Other constraints on DS will arise from cosmological considerations. A first study of their effects (and those of the resultant main-sequence stars) on reionization has been done by Schleicher et al. (2008, 2009) and further work in this direction is warranted.

In this paper, we examine the SMDSs that result from the two mechanisms discussed above for DM refueling inside the star. In Section 2, we discuss the procedure for calculation of models; in Section 3 we present results; and in Section 4 we end with a discussion.

DM heating is very different from fusion. In order to overcome the Coulomb barriers between nuclei, fusion requires very high temperatures and densities in the star. Fusion is not very efficient in that only <1% of the nuclear mass is converted to heat. WIMP annihilation, on the other hand, takes place at high DM densities regardless of the temperature. It is almost 100% efficient since O(1) of the WIMP mass is converted to useful energy. Thus, in the evolution of the first protostars, DM heating becomes important early. Here, we start the calculation when the DS is massive enough (3 M_☉) so that it is in hydrostatic equilibrium and most of the hydrogen and helium is ionized. The contribution to DM luminosity is roughly constant as a function of radius throughout the DS, unlike fusion which takes place only at the (high temperature) core of the star.

2.1. Basic Equations

We use the numerical code previously discussed in detail by Freese et al. (2008a) and Spolyar et al. (2009). We make the assumption that a DS can be described as a polytrope

$$\begin{equation} P=K\rho ^{1+1/n} \end{equation} \tag{ 4 }$$

in hydrostatic equilibrium. Here, P is the pressure, ρ is the density, and K is a constant. We solve the equations of stellar structure with polytropic index n initially 1.5, as appropriate for convective stars, and made a gradual transition to n = 3 as the star becomes radiative in the later phases. We require that at each time step during the accretion process the star is in hydrostatic equilibrium,

$$\begin{equation} \frac{{\it dP}}{dr}=-\rho (r)\frac{{\it GM}_r}{r^2}, \end{equation} \tag{ 5 }$$

where ${dM_r \over dr} = 4 \pi r^2 \rho (r)$, and M_r is the mass enclosed in a spherical shell of radius r. The equation of state includes radiation pressure,

$$\begin{equation} P(r)=\frac{k_B\rho (r)T(r)}{m_u\mu }+\frac{1}{3}aT(r)^4\equiv P_g+P_{\rm rad}, \end{equation} \tag{ 6 }$$

where k_B is the Boltzmann constant, m_u is the atomic mass unit, and μ = 0.588 is the mean atomic weight. The opacity is obtained from a zero metallicity table from OPAL (Iglesias & Rogers 1996) supplemented at low temperatures by opacities from Lenzuni et al. (1991) for T < 6000 K. The further assumption is made that the radiated luminosity of the star L_* is balanced by the rate of energy output by all internal sources, L_tot, as described below in Section 2.3.

$$\begin{equation} L_*=4\pi R_*^2\sigma _BT_{\rm eff}^4 = L_{\rm tot}, \end{equation} \tag{ 7 }$$

where T_eff is the surface temperature and R_* is the total radius.

Starting with a mass M and an estimate for the outer radius R_*, the code integrates the structure equations outward from the center. The total rate of energy production L_tot is compared to the stellar radiated luminosity, as in Equation (7) and the radius is adjusted until the condition of thermal equilibrium is met (a convergence of 1 in 10⁴ is reached).

2.2. Dark Matter Densities

2.3. Energy Sources

There are four possible contributions to the DS luminosity:

$$\begin{equation} L_{\rm tot}=L_{\rm DM}+L_{\rm grav}+L_{\rm nuc}+L_{\rm cap}, \end{equation} \tag{ 8 }$$

from DM annihilation, gravitational contraction, nuclear fusion, and captured DM, respectively. The heating due to DM annihilation in Equations (2) and (3) dominates from the time of DS formation until the adiabatically contracted DM runs out. As described previously, in our "without capture" models this stage never ends due to extended AC. In the models "with capture," on the other hand, we take this phase to end after ∼300,000 yr, so that the DS has to contract in order to maintain pressure support. The contribution L_grav due to gravitational energy release is calculated as in Spolyar et al. (2009). As the DS contracts, the density and temperature increase to the point where nuclear fusion begins. We include deuterium burning starting at T ∼ 10⁶ K, hydrogen burning via the equilibrium proton–proton cycle (Bahcall 1989), and helium burning via the triple-alpha reaction (Kippenhahn & Weigert 1990). During the later stages of the pre-main-sequence evolution in the cases "with capture," the DS becomes dense enough to capture DM from the ambient medium via elastic scattering. Already before fusion can begin, and possibly again after the onset of fusion, captured DM can provide an important energy source with accompanying luminosity

$$\begin{equation} L_{\rm cap}=2m_{\chi }\Gamma _{\rm cap}=2m_{\chi }f_Q\int dV \rho _{\rm cap}^2\langle \sigma v\rangle /m_{\chi }^2 \end{equation} \tag{ 9 }$$

and again f_Q = 2/3.

Using our polytropic model for DSs, we have started with 3 M_☉ stars and allowed baryonic matter to accrete onto them until they become supermassive with M_* > 10⁵ M_☉. We display results for the case without capture (but with extended AC) as well as the case with capture for a variety of WIMP masses m_χ = 10 GeV, 100 GeV, and 1 TeV. We have run models for a variety of accretion rates of baryons onto the star including constant accretion rates of $\dot{M} = 10^{-1}, 10^{-2}, 10^{-3} \,M_{\odot }$ yr⁻¹. We will present results for $\dot{M} = 10^{-3} \,M_{\odot }$ yr⁻¹, which is approximately the average rate calculated by Tan & McKee (2004) and by O'Shea & Norman (2007).

Our stellar evolution results can be seen in the Hertzsprung–Russell (H-R) diagram of Figure 1 for the case of a 10⁶ M_☉ halo. The DS travels up to increasingly higher luminosities as it becomes more massive due to accretion. We have labeled a sequence of ever larger masses until all the baryons (150,000 M_☉) in the original halo are consumed by the SMDS. As the mass increases, so do the luminosity and the surface temperature. In the cases "without capture," the radius increases continuously until all the baryons have been eaten. In the cases "with capture," we have taken the (overly conservative) assumption that the DM from AC is depleted after ∼300,000 yr as in our earlier papers; then the luminosity plateaus for a while the DS contracts until eventually it is dense enough to capture further DM.

Zoom In Zoom Out Reset image size

We note that, for the case "without capture," the H-R diagram is unchanged by varying the accretion rate: only the time it takes to get from one mass stage to the next changes, but the curves we have plotted apply equally to all accretion rates. Similarly, given m_χ, the following quantities are the same regardless of accretion rate: R_*, T_eff, ρ_c, and T_c.

In a beautiful paper, Hoyle & Fowler (1963) studied supermassive stars in excess of 10³ M_☉ and found results germane to our work. They treated these as n = 3 polytropes (just as we do) dominated by radiation pressure, and found the following results: R_* ∼ 10¹¹(M_*/M_☉)^1/2(T_c/10⁸ K)⁻¹ cm, L_*/L_☉ ∼ 10⁴M_*/M_☉, and T_eff ∼ 10⁵(T_c/10⁸ K)^1/2 K. While some of the details of their calculations differ from ours, taking the central temperature appropriate to DS in the above relations roughly reproduces our results (to O(1)). For example, by using the temperature appropriate to DSs with extended AC (T_c ∼ 10⁶ K) rather than the much higher central temperature (T_c > 10⁸ K) appropriate to nuclear power generation, the above relations show that DSs have much larger radii and smaller surface temperatures than fusion-powered stars. We wish to draw particular attention to the fact that luminosity scales linearly with stellar mass, and is independent of power source.

Figure 2 plots the H-R diagram "with capture" for a single WIMP mass of 100 GeV, for $\dot{M} = 10^{-3} \,M_{\odot }$ yr⁻¹, and for σ_c = 10⁻³⁹ cm², but for a variety of ambient densities ranging from $\bar{\rho }_\chi = (10^{10} \hbox{--} 10^{14})$ GeV cm⁻³. The latter is the density one finds due to AC at the photosphere of the DS, and seems the largest sensible starting point for the value of the ambient density. Our previous paper (Spolyar et al. 2009) considered the case of minimal capture with 10¹⁰ GeV cm⁻³, which was artificially chosen so that the growth of the DS ceases at 680 M_☉, the radius shrinks, and then fusion and DM heating play equal roles. For ambient densities below 5 × 10¹⁰ GeV cm⁻³, the DS mass growth shuts off well before the star becomes supermassive for the following reason. The cases with capture all take place at higher stellar densities than the cases without; the density must be high enough to be able to capture WIMPs. Consequently, the surface temperature is larger and accretion is shut off more easily by radiation coming from the star. The case of ambient density 10¹⁰ GeV cm⁻³ (from our previous paper) is a very carefully chosen (delicate) situation. On the low side of this density, DM heating is completely irrelevant and fusion tells the whole story; on the other hand, for any density $\bar{\rho }_\chi \ge {\rm few} \times 10^{10}$ GeV cm⁻³, DM heating is so dominant over fusion that the DS can just continue growing in mass. At these higher densities, the surface temperature never becomes hot enough (≈100,000 K) for feedback effects from the star to cut off accretion. Between 50,000 K and 100,000 K, feedback effects are included, and they act to reduce the accretion rate, but they never shut it off entirely for densities above 5 × 10¹⁰ GeV cm⁻³. In reality, a star that is moving around can sometimes hit pockets of high $\bar{\rho }_\chi$ (where it is DM powered and grows in mass) and sometimes hit pockets of low $\bar{\rho }_\chi$ (where fusion takes over). As long as the ambient density remains at least this large, the star can reach arbitrarily large masses and eat the entire baryonic content of the halo.

Zoom In Zoom Out Reset image size

As described previously, the capture mechanism depends on the product of scattering cross section times ambient density, $\sigma _c \bar{\rho }_\chi$, rather than on either of these quantities separately. Hence, our current discussion could trade off ambient density versus cross section. For example, the product is the same for $\bar{\rho }_\chi = 5 \times 10^{10}$ GeV cm⁻³ and σ_c = 10⁻³⁹ cm² as it is for $\bar{\rho }_{\chi } = 10^{14}$ GeV cm⁻³ and σ_c = 5 × 10⁻⁴³ cm². Thus, for the highest reasonable ambient density, the scattering cross section can be several orders of magnitude lower than the experimental upper bound for SD scattering and still provide substantial capture in DS. While the required σ_c is ruled out (Ahmed et al. 2009) for SI scattering at m_χ = 100 GeV, it is below the bounds at low masses m_χ ∼ 10 GeV and in this case can lead to significant DM capture in the stars.

Above ∼100 M_☉, one can see that the stellar luminosity scales as L_* ∝ M_* and is the same for all models for a given stellar mass; this statement is essentially true for all stars no matter the power source. The reason is that at these masses, the star is essentially radiation pressure supported throughout.⁷ This same scaling in supermassive stars was already noticed by Hoyle & Fowler (1963). The luminosity essentially tracks (just below) the Eddington luminosity which scales as L ∝ M_*.

The curves with higher values of WIMP mass m_χ lie to the left of the curves with lower m_χ. This can be understood as follows. The DM heating rate in Equation (1) scales as Q ∝ ρ²_χ/m_χ. Hence to reach the same amount of heating and achieve the same luminosity, at higher m_χ the DS must be at higher WIMP density, i.e., the stellar radius must be smaller, the DS is hotter, and the corresponding surface temperature T_eff is higher. Also, for higher m_χ the amount of DM in the star is smaller since the star is more compact for the same number of baryons but ρ_χ ∝ n^0.8 where n is the hydrogen density.

Tables 1 and 2 show various stellar properties for a DS that forms in a 10⁶ M_☉ DM halo, as the star evolves to higher mass for the case of m_χ = 100 GeV, for the two cases "without" and "with" capture, respectively. While the DM density is a gently decreasing function of radius for the case without capture, it is extremely sharply peaked at the center of the DS for the case "with capture."

One can see that, in the case "without capture," ∼10⁴ M_☉ of DM must be annihilated away in order for the DS to reach 10⁵ M_☉ for accretion rate $\dot{M} = 10^{-3} \,M_{\odot }$. The time to reach this mass is ∼100 Myr. For an alternative faster accretion rate $\dot{M} = 10^{-2}\,M_{\odot }$, a smaller amount of DM must be annihilated away, ∼100 M_☉; then it takes the DS ∼ 10 Myr to reach ∼10⁵ M_☉. The caveat is that the DM orbits must continue to penetrate into the middle of the DS for this length of time in order for the DM abundance and consequent heating to continue; it is the DM heat source that keeps the DS cool enough to allow it to continue to accrete baryons. Additionally, the assumption that baryons continue to accrete onto the DS must continue to hold. Yet, in the time frame required, the original 10⁶ M_☉ halo will merge with its neighbors, so that both the baryon and DM densities are disturbed. These mergers could affect the DS in one of two ways: either they provide more baryons and DM to feed the SMDS so that it ends up being even larger, or they disrupt the pleasant high DM environment of the SMDS so that it loses its fuel and converts to an entirely fusion-powered star. Continued growth of the DS is quite plausible since simulations with massive BHs in mergers show that they prefer to sit close to the center of the density distribution or find the center in a short time after the merger.

Someday detailed cosmological simulations will be required to answer this question. Individual DS in different halos may end up with a variety of different masses depending on the details of the evolution of the halos they live in. The case studied in this paper is clearly a simplistic version of the more complicated reality, but illustrates the basic idea that supermassive stars may be created by accretion onto DS, either with or without capture.

Tables 1 and 2 present models for DSs which form in 10⁶ M_☉ DM halos, but SMDS could also form in a variety of halo masses with different final stellar masses. For example, a hydrogen/helium molecular cloud may start to contract in a 10⁸ M_☉ halo and produce a DS. Here, the situation is more complicated. The virial temperature of the halo exceeds 10⁴ K, the surface temperature we have found for a DS in equilibrium. Hence, it is not clear how accretion onto the DS will proceed. This is the subject of future work. The accretion is expected to be faster due to the increased ambient temperature. We extended our models to 10⁸ M_☉ halos in which a more extended period of accretion can lead to SMDS with even larger masses. Tables 3 and 4 show examples of "without" and "with" capture cases, respectively, for 10⁸ M_☉ halos. The baryonic mass in the halo is 1.5 × 10⁷ M_☉. Potentially all of this mass could go into the SMDS. Then once the DM runs out it becomes an enormous Population III fusion-powered star, which soon burns out and becomes a BH with mass >10⁷ M_☉. Such a large BH at early times would clearly help to explain the many large BH found in the universe at early times and today. In addition, since L_* ∝ M_*, we can predict that the luminosity of the M_* ∼ 10⁷ M_☉ SMDSs would be L_* ∼ 10¹¹ L_☉, even easier to detect than a 10⁵ SMDS.

It can be seen in Tables 2 and 4 that in the case "with capture" the amount of DM inside the star M_χ is very low at any given moment after the star transitions from adiabatically contracted DM to captured DM. The reason is simple: after a very brief time an equilibrium regime is reached where the annihilation rate is equal to the capture rate. In other words, as soon as a WIMP gets into the DS, it quickly thermalizes and annihilates. This leads to very small amounts of DM inside the star at any one time. Notice, however, that the total amount of DM annihilated (integrated over time) is still very large, as can be seen from the last three entries in the column labeled M_Ann of Tables 2–4.

General relativistic instability. The pulsational stability of supermassive stars is an interesting issue. They are radiation pressure dominated with adiabatic index close to γ = 1 + 1/n = 4/3, the value that yields neutral stability to radial pulsations for Newtonian bodies with no rotation. Indeed general relativistic corrections (which scale as GM_*/R_*) act in the direction of destabilizing stars and are particularly important for high mass stars. Fowler (1966) examined the stability of supermassive stars using polytropes with n = 3 (see Wagoner 1969 for a review). Fowler found that, for the case of no rotation, radial oscillations become dynamically unstable and prevent standard stars more massive than 10⁵ M_☉ from reaching a phase of hydrogen burning before collapse. Yet he also found that a small amount of rotation can stabilize the stars, so that rotating stars as heavy as 10⁸ M_☉ could be stable en route to reaching hydrogen burning.

In the case of DS, stability to radial pulsations is much easier to achieve. DSs have much larger radii and lower temperatures than fusion-powered stars, so that the general relativistic corrections ∼GM_*/R_* are much smaller. The upper limit on the allowed stellar mass will be larger. In any case, SMDSs are undoubtedly rotating, so that very large stable masses can be achieved (even in the case of rotating ordinary stars, the mass limit is 10⁸ M_☉).

In the future, we suggest a stability analysis of our models. The very interesting possibility of short term pulsational instabilities exists, which leads to SMDS as possible standard candles for learning about dark energy or other evolutionary properties of the universe.

We discuss the capabilities of JWST to discover DSs, following the properties of the telescope described by Gardner et al. (2006, 2009). The telescope is designed to be diffraction limited at a wavelength λ_obs = 2 μm. The Near Infrared Camera (NIRCam) will operate in the wavelength range λ = (0.6–5) μm and the Mid-Infrared Camera (MIRI) will operate in the wavelength range λ = 5–27 μm. In an exposure of duration 10⁴ s, NIRCam will have a limiting sensitivity of 11.4 nJy (1 nJy = 1 × 10⁻³² erg s⁻¹ cm⁻² Hz⁻¹) in the 2 μm band, and 13.8 nJy in the 3.5 μm band and MIRI will have a sensitivity of 700 μJy in the λ = 10 μm band (in all cases limiting sensitivities are for a signal-to-noise ratio (S/N) = 10). With longer exposure times, the limiting flux detectable will scale as $\sqrt{t}$ for the same S/N. DS will be characterized by blackbody spectra with surface temperatures T_eff ≲ 5 × 10⁴ K. In addition, DSs are also predicted to have hydrogen lines.

We determine the detectability of DSs located at various redshifts z = 5, 10, and 15 using the standard Planck spectrum of blackbody with surface temperature T_eff and radius R_* (for DSs similar to those from Tables) in a cosmology with H₀ = 74, Ω_Λ = 0.71, andΩ_M = 0.29.

Figure 3 shows the observed blackbody flux distribution of two SMDSs formed at z = 15 for a WIMP mass m_χ = 100 GeV for the case of extended AC (without capture). The star in the left panel is formed in a 10⁶ M_☉ halo and the star in the right panel is formed in a 10⁸ M_☉ halo and their stellar (baryonic) masses are 1.7 × 10⁵ M_☉ and 1.5 × 10⁷ M_☉, respectively. Curves are shown assuming the SMDS formed at z = 15 and survived to various redshifts, at which it is still producing blackbody radiation. The 1.7 × 10⁵ M_☉ star (left panel) will be detectable by JWST (NIRCam) in an exposure of a million seconds, but only if it survives intact till z = 10. The 1.5 × 10⁷ M_☉ star (right panel) will be detectable even in a shorter 10⁴ s exposure even at z = 15 in both the 2 μm and 3.5 μm bands. The star on the right may be marginally detectable in a million second exposure in the 10 μm band of MIRI. The relative flux levels in the three different bands will be important for distinguishing these objects from galaxies.

Zoom In Zoom Out Reset image size

The curves are not corrected for Lyα absorption by the intergalactic medium (IGM) but the red vertical lines show the location of the 1216 Å line redshifted from the rest-frame wavelength of the star at each of the three redshifts. Flux at wavelengths to the left of the redline at each redshift is expected to be absorbed to some extent by the IGM. Since the surface temperatures of our stars are ≃10⁴ K, the majority of the Lyα absorption (λ_rest = 1216 Å) is expected to occur at wavelengths shorter than that at which the peak flux is emitted. We note that for a DS at z = 15 the Lyα absorption line lies at 1.94 μm—roughly in the middle of the NIRCam 2 μm band. In this case, the flux in this band will be reduced by about a factor of 2 but will still be well above the detection limit (in 10⁶ s). A detailed calculation of the absorption by the IGM is outside the scope of the current paper.

We studied numerous other cases without capture as well: a variety of WIMP masses (10 GeV–2 TeV) as well as various formation redshifts (z_form = 10–20). We found that DSs with masses up to M_* ∼ 10⁵ M_☉ forming in 10⁶ M_☉ halos at z = 20 and shining at that redshift will in general not be detectable for any value of m_χ (they will become detectable only if they survive to and shine at much lower redshifts).

The smaller DS with M_* = 800 M_☉ discussed in our previous work (Freese et al. 2008a; Spolyar et al. 2009) will not be visible in JWST (see also P. Scott et al. 2010, in preparation); they are several orders of magnitude below the detection limit of JWST (in 10⁶ s).

Figure 4 shows the observed blackbody flux distribution of two DSs formed at z = 15 in halos of two different masses for the case "with capture." Since DSs formed via capture are smaller (in radius) and hotter (than DS formed via extended AC without capture), their peak wavelength tends to shift to lower wavelengths, in some cases out of the range detectable by JWST. 10⁵ M_☉ stars in 10⁶ M_☉ halos are only detectable if they survive until z = 5 at which time they could be detectable in a long (10⁶ s exposure). On the other hand, 10⁷ M_☉ stars formed in 10⁸ M_☉ halos will be easily detectable even in an exposure of 10⁴ s all the way out to z = 15. For most other WIMP masses and formation redshifts, DSs formed in 10⁶ M_☉ halos via capture are below the detection limit of JWST.

Zoom In Zoom Out Reset image size

The prospect of detecting SMDS in JWST and confirming the existence of a new phase of stellar evolution is exciting. In the most optimistic cases, detection in a number of different wavelength bands could be used to obtain a spectrum and differentiate these DSs from galaxies or other sources.

Using our polytropic model for DSs, we have considered accretion of baryonic matter onto the DS as they become supermassive, M_* > 10⁵ M_☉. Such large masses are possible because the DS is cool enough (as long as it is powered by DM) so that radiative feedback effects from the star do not shut off the accretion of baryons, as long as it is powered by DM. We considered two different scenarios for supplying the required amount of DM.

• 1.  
  The case of extended AC, labeled "without capture" in the figures. In this case, DM is supplied by the gravitational attraction of the baryons in the star. In triaxial halos, DM orbits are quite complex and the DM in the core is harder to deplete than previously estimated. This case does not include any captured DM, and relies solely on the particle physics of WIMP annihilation. To grow to a 10⁵ M_☉ SMDS in a 10⁶ M_☉ halo, or to grow to a 10⁷ M_☉ SMDS in a 10⁸ M_☉ halo, the amount of DM consumed can be as much as ∼1% of the total DM in the halo (depending on the accretion rate). This amount is not unreasonable, since Valluri et al. (2010) found that the fraction of box and chaotic DM orbits is as high as 85% in triaxial halos and remains over 10% when a significant compact baryonic component causes the halo to become axisymmetric at small radii. Future work will be required to accurately obtain WIMP orbits, densities, and timescales (work in progress). For now, we took the simplistic approach of using our previous prescription for AC in a spherical potential but not removing the annihilated DM.
• 2.  
  The case of extended capture, labeled "with capture" in the figures. Here, the original DM inside the star from AC is assumed to be depleted after ∼ 300,000 yr, then the star begins to shrink somewhat, and capture of DM from the surroundings takes place as it scatters elastically off of nuclei in the star. In this case, the additional particle physics ingredient of WIMP scattering is required.

In this paper, we studied the formation of 10⁵ M_☉ SMDS in 10⁶ M_☉ DM halos and 10⁷ M_☉ SMDS in 10⁸ M_☉ halos. These stars become very bright, L_* ∼ (10⁹–10¹¹) L_☉. Figure 1 shows the H–R diagram for a variety of WIMP masses, and follows the DS as it climbs up to ever higher masses. They live millions to billions of years, depending on the merger history with other halos. Once the DM runs out, the SMDSs have brief lives as fusion-powered Population III stars before collapsing into >10⁵ M_☉ BHs, possible seeds for many of the big BH seen in the universe today and at early times. A proper study of the final mass of the DS and resultant BH will depend on cosmological simulations. The original halo containing the DS will merge with other halos. No one knows what exactly will happen to the DM density in the vicinity of the DS when this happens. The DS could end up even more massive. DS may also form in larger halos that form at later times, as long as the baryonic content is still only H and He. Localized regions with this property could exist even at redshifts z < 7 (Furlanetto & Loeb 2005; Choudhury & Ferrara 2007).

SMDS would make plausible precursors of the 10⁹ M_☉ BHs observed at z > 6 (Fan et al. 2004; Li et al. 2007) of intermediate mass BHs; of BH at the centers of galaxies; and of the BH inferred by extragalactic radio excess seen by the ARCADE experiment (Seiffert et al. 2009). In addition, the formation of BH from DS could be accompanied by high-redshift gamma ray bursts thought to take place due to the gravitational collapse of supermassive stars.

In the future, we suggest a stability analysis of our models. The very interesting possibility of short term pulsational instabilities exists, which leads to SMDS as possible standard candles for learning about dark energy or other evolutionary properties of the universe.

SMDSs could be detected by JWST for a variety of parameter ranges. These are extremely bright objects L_* ∼ (10⁹–10¹¹) L_☉ and yet are very cool T ∼ 10, 000 K, so that their emitted light is in the wavebands detectable by JWST. The longer they live, the more easy they are to detect. Figures 3 and 4 give examples of what one could look for in JWST. For the most optimistic cases, one could even test for the blackbody spectrum in a number of different wavebands. In principle, hydrogen or helium lines could be found to complement the blackbody emission. If, in addition, someday high energy neutrinos are found to emanate from these stars, then it will be a clincher that DM annihilation took place inside the DS.

It is interesting to speculate that the initial mass function of Population III fusion-powered stars may be determined by the cutoff of the DM supply, which may vary from one DS to another. DSs continue to accrete mass as long as the DM annihilation powers the star and keeps it cool enough. Once the DM fuel supply is exhausted, the star shrinks and heats up, fusion begins, and the mass growth of the star is quickly halted due to feedback from hot emitted photons. Hence, the details of the cutoff of the DM supply may determine the sizes of Population III stars entering the fusion era. The cutoff will take place at different DS masses in different halos, depending on the details of the cosmological merger history. Different final DS masses may result for different individual DS depending on the evolution of their parent halos.

After the submission of this paper, we became aware of another paper which discusses the prospects of detecting DSs with masses up to ≃800 M_☉ with JWST, Zackrisson et al. (2010). This paper considers the possibility that DSs might be detected individually if their flux is magnified by a foreground gravitational lensing cluster. These authors also show that the stellar atmospheres can quite significantly modify the observed spectral energy distributions of DSs due to the Balmer break for DSs with surface temperatures less than ∼6000 K but that these effects are relatively weak for T_eff ≳ 10⁴ K. Since most of our SMDSs are in fact hotter than 10⁴ K, we expect that our blackbody spectra are quite a good approximation to the observable spectral energy distributions. Zackrisson et al. (2010) also demonstrate that if DSs live for ≳10⁷yr, even a small fraction (0.7%)of such stars in a protogalactic star cluster at z ∼ 10 would result in a significantly redder color of this object and that it would be clearly distinguishable from a normal star forming galaxy, active galactic nucleus at high redshift, as well as from foreground brown dwarfs and cool stars in the Milky Way.

We acknowledge support from the DOE and MCTP via the University of Michigan (K.F. and C.I.), DOE at Fermilab (D.S.), and the NSF grant AST-0908346 (M.V). K.F. is grateful to M. Begelman, W. Freedman, J. Friedman, O. Gnedin, B. O'Shea, B. Schmidt, P. Shapiro, T. Tyson, and S. White for helpful discussions. We thank Jon Gardner for very helpful comments regarding JWST.