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Christian Schultz, Jose Oñorbe, Kevork N. Abazajian, James S. Bullock, The high-z universe confronts warm dark matter: Galaxy counts, reionization and the nature of dark matter, Monthly Notices of the Royal Astronomical Society, Volume 442, Issue 2, 1 August 2014, Pages 1597–1609, https://doi.org/10.1093/mnras/stu976
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Abstract
We use N-body simulations to show that high-redshift galaxy counts provide an interesting constraint on the nature of dark matter, specifically warm dark matter (WDM), owing to the lack of early structure formation these models. Our simulations include three WDM models with thermal-production masses of 0.8, 1.3 and 2.6 keV, as well as CDM. Assuming a relationship between dark halo mass and galaxy luminosity that is set by the observed luminosity function at bright magnitudes, we find that 0.8 keV WDM is disfavoured by direct galaxy counts in the Hubble Ultra Deep Field at >10σ. Similarly, 1.3 keV WDM is statistically inconsistent at 2.2σ. Future observations with JWST (and possibly HST via the Frontier Fields) could rule out 1.3 keV WDM at high significance, and may be sensitive to WDM masses greater than 2.6 keV. We also examine the ability of galaxies in these WDM models to reionize the universe, and find that 0.8 and 1.3 keV WDM produce optical depths to the cosmic microwave background (CMB) that are inconsistent at 68 per cent C.L. with current Planck results, even with extremely high ionizing radiation escape fractions, and 2.6 keV WDM requires an optimistic escape fraction to yield an optical depth consistent with Planck data. Although CMB optical depth calculations are model dependent, we find a strong challenge for stellar processes alone to reionize the universe in a 0.8 and 1.3 keV WDM cosmology.
1 INTRODUCTION
Dark matter dominates the evolution of gravitational perturbations, leading to the formation of haloes and galaxies. In the prevalent paradigm of cold dark matter (CDM), the primordial perturbation spectrum extends to very small scales; galaxy formation proceeds from the bottom up, commencing in the smallest dark matter haloes where gas cooling can occur. If instead there exists a non-negligible minimal scale for primordial perturbations as in the case of warm dark matter (WDM), halo formation is delayed, and early galaxy formation is suppressed considerably.
Early galaxy formation has been understood to be a challenge for WDM models for some time (Barkana, Haiman & Ostriker 2001; Somerville, Bullock & Livio 2003). Today, the tension is only heightened by mounting evidence that structure formation is proceeding in earnest at very early cosmic times. There are now direct detections of galaxies at redshifts as high as ∼10 (Ellis et al. 2013; Oesch et al. 2013), clearly indicating that there are collapsed structures at this time. More indirectly, studies of quasar spectra show that the intergalactic medium (IGM) was almost fully ionized by redshift z ∼ 6 (Fan et al. 2006) and the measured electron scattering optical depth from the cosmic microwave background (CMB) may could imply reionization as early as z ∼ 10 (Ade et al. 2013). The maintenance of reionization back to these early times seems to require contributions from numerous, low-mass galaxies (Kistler et al. 2009; Kuhlen & Faucher-Giguere 2012; Robertson et al. 2013). In this paper, we examine how current and future observations of high-z galaxies, together with observational probes of reionization, can constrain the dark matter power spectrum on small scales, and by extension the particle nature of dark matter.
There has been considerable interest in the WDM paradigm for galaxy formation, owing to potential problems with the LCDM model on sub-galactic scales. Most recently, it has been recognized that the observed central densities of low-luminosity Milky Way dwarf satellite are significantly lower than expected in dissipationless CDM simulations (Boylan-Kolchin, Bullock & Kaplinghat 2012). This issue can be alleviated if the dark matter is warm (Lovell et al. 2012, 2014; Polisensky & Ricotti 2014). Here, we specifically study a WDM model (1.3 keV thermal particle mass) that corresponds to the cutoff scale that alleviates the central-density problem. It should be noted here that models including ultralight axions alongside a CDM component can also alleviate the sub-galactic problems in the pure CDM models (Marsh & Silk 2014).
The two most popular classes of WDM particle candidates are ‘thermal’ particles and sterile neutrinos. Thermal WDM is coupled to the primordial plasma in the early Universe, and is diluted to the proper (observed) dark matter density by an unspecified process. Sterile neutrinos, on the other hand, can be produced at the proper dark matter abundance through scattering processes due to their mixing with active neutrinos with the Dodelson–Widrow mechanism (Dodelson & Widrow 1994, sometimes referred to as non-resonant production), through resonant production in the case of a large cosmological lepton asymmetry (Shi & Fuller 1999), or through coupling with other fields (Kusenko 2006; Shaposhnikov & Tkachev 2006). An important characteristic of the different models is the free-streaming length they introduce, with a given particle mass having a different free-streaming length for the different WDM particles and for the different sterile neutrino production mechanisms. In this paper we primarily state particle masses in terms of thermal WDM particles, i.e. the ‘thermal mass’, but we also provide conversions to the Dodelson–Widrow sterile neutrino mass in summary statements and tables. We also quote the wave numbers where the associated power spectra fall to half the value of a standard CDM model, which allows our results to be interpreted generally for any model that results in truncated small-scale power, as can arise for standard CDM particles in the case of non-standard inflation (e.g. Kamionkowski & Liddle 2000; Zentner & Bullock 2002).
Recent work has constrained the warm dark particle mass by a number of methods. Some of the currently tightest constraints come from observations of the Lyman α (Lyα) forest produced by neutral gas along the line of sight to distant quasars. The neutral gas follows the gravitationally dominant dark matter clustering in the mildly non-linear regime probed by the Lyα forest, and therefore it can be a powerful probe of the dark matter perturbation spectrum at small scales. However, the Lyα forest is a challenging tool, requiring disentangling the effects of pressure support and thermal broadening of the Lyα forest features from the effects of dark matter perturbation suppression from WDM. In addition, modelling the dependence on the physics of the neutral gas requires assumptions of the thermal history of the IGM and its ionizing background, which are done as parametrized fitting functions. Many of the limitations of the Lyα forest on constraints of the primordial power spectrum are discussed in Abazajian et al. (2011). Setting aside the limitations of the method, the Lyα forest provides stringent constraints, with recent quoted limits at mWDM > 3.3 keV (2σ; Viel et al. 2013).
The lack of early structure formation in WDM has motivated limits from the rate of high-z gamma-ray bursts (de Souza et al. 2013). Similarly, Pacucci, Mesinger & Haiman (2013) utilize strongly lensed ultrafaint, high-redshift galaxies to constrain the particle mass by halo mass function considerations. At low redshift, WDM models can be constrained by studying the abundance of small galaxies. Work by Polisensky & Ricotti (2011) and Lovell et al. (2014) uses N-body simulations of Milky Way-sized dark matter haloes and constrains the particle mass by assuming that the number of simulated dark matter satellites equals or exceeds the number of observed Milky Way satellites, and report limits on thermal WDM particle masses of mWDM > 2.3 keV and >1.5 keV, respectively. It should be recognized that constraints from satellite counts are sensitive to halo-to-halo variation in sub-structure counts as well as assumed completeness corrections to the observed Milky Way satellite luminosity function. More recently, Horiuchi et al. (2014) have tried to adequately account for the halo-to-halo scatter and focused on counts around M31 (which are higher than around the MW at fixed luminosity) and find mWDM > 1.8 keV.
Given the potential systematic problems with known WDM constraints, it is useful to explore alternative probes. In the rapidly evolving field of high-z galaxy surveys, the Lyman-break technique has proven useful for discovering galaxies and estimating the UV luminosity function out to redshifts z ∼ 9, although there are candidates in the literature at redshifts as high as z ∼ 12 (Bouwens et al. 2007, 2011; McLure et al. 2013; Oesch et al. 2013; Schenker et al. 2013b). New Fourier techniques seem promising in finding fainter candidates below the normally required detection threshold S/N ∼ 4.5 (Calvi et al. 2013). Furthermore, Lyman-break galaxies seem to be fair tracers of the overall halo population (Conroy, Wechsler & Kravtsov 2006). Thus, the UV luminosity function interconnects with the halo mass function of dark matter, a quantity which is readily constructed from simulations and from which different dark matter models can be distinguished.
Additional physical mechanisms may ease the tension between simulations and observations presented by, e.g. the missing dwarf galaxies or the too-big-to-fail problem (Boylan-Kolchin et al. 2012). One example is the work Bovill & Ricotti (2011). Here, it is noted that the Milky Way halo may be populated by fossils of early dwarf galaxies that formed before reionization, and that these fossils today have very low surface brightness rendering them outside of current observational bounds. Therefore, there may indeed be a population of low-luminosity dwarf galaxies near the Milky Way. Katz & Ricotti (2013) expand on this result, and argues that the bulk of the old globular clusters in the Milky Way formed in these first (now fossil) dwarf galaxies. Furthermore, the protoglobular clusters were an important mode of star formation in these galaxies, and could in fact be the main driver for reionization.
In this work, we study in detail the effects of WDM models on high-z dark matter halo counts using high-resolution cosmological simulations, and extend these results empirically to infer the observable effects on galaxies and reionization. We compare the high-z luminosity functions of galaxies recently measured from the Hubble Ultra Deep Field (HUDF) with the inferred luminosity function derived using N-body simulations in WDM and CDM cosmologies. Our predicted luminosity functions are normalized to match observed bright galaxy counts using abundance matching. All magnitudes quoted below are in the AB system. The same models allow us to study cosmological reionization in WDM models and compare them to CDM.
2 THEORY AND SIMULATIONS
2.1 Power spectrum
mT . | |$m_{\nu _{\rm s}}$| . | k1/2 [h−1 Mpc] . | kmax [h−1 Mpc] . |
---|---|---|---|
2.6 keV | 15.5 keV | 21.4 | 2.62 |
1.3 keV | 6.0 keV | 9.47 | 1.28 |
0.8 keV | 3.0 keV | 5.24 | 0.764 |
mT . | |$m_{\nu _{\rm s}}$| . | k1/2 [h−1 Mpc] . | kmax [h−1 Mpc] . |
---|---|---|---|
2.6 keV | 15.5 keV | 21.4 | 2.62 |
1.3 keV | 6.0 keV | 9.47 | 1.28 |
0.8 keV | 3.0 keV | 5.24 | 0.764 |
mT . | |$m_{\nu _{\rm s}}$| . | k1/2 [h−1 Mpc] . | kmax [h−1 Mpc] . |
---|---|---|---|
2.6 keV | 15.5 keV | 21.4 | 2.62 |
1.3 keV | 6.0 keV | 9.47 | 1.28 |
0.8 keV | 3.0 keV | 5.24 | 0.764 |
mT . | |$m_{\nu _{\rm s}}$| . | k1/2 [h−1 Mpc] . | kmax [h−1 Mpc] . |
---|---|---|---|
2.6 keV | 15.5 keV | 21.4 | 2.62 |
1.3 keV | 6.0 keV | 9.47 | 1.28 |
0.8 keV | 3.0 keV | 5.24 | 0.764 |
2.2 Numerical simulations
Our simulations were performed with the gadget-2 code, in TreePM mode (Springel 2005). In order to generate the ICs, we have used the music code (Hahn & Abel 2011). The method uses an adaptive convolution of Gaussian white noise with a real-space transfer function kernel together with an adaptive multigrid Poisson solver to generate displacements and velocities following second-order Lagrangian perturbation theory (2LPT). For more specific details on the music code, we refer the reader to Hahn & Abel (2011). The camb (Lewis, Challinor & Lasenby 2000; Howlett et al. 2012) package was used to generate the CDM transfer functions used to generate the ICs for this cosmology. The WDM transfer functions were obtained from CDM using equations (2) and (3). Only the ICs were modified, and the thermal velocities of the WDM particles are not included in the simulations, since they have not been found to be significant in affecting WDM structure formation (Bode, Ostriker & Turok 2001; Villaescusa-Navarro & Dalal 2011).
The cosmological parameters used were h = 0.71, Ωm = 0.266, |$\Omega _\Lambda =0.734$|, ns = 0.963 and σ8 = 0.801. All simulations are 10243 particles in (50 Mpc h−)3 boxes started at zini = 125. The implied particle mass is mp = 8.6 × 106 h−1 M⊙. We employ three WDM models with thermal particle masses of 0.8, 1.3 and 2.6 keV, which are equivalent to oscillation-produced Dodelson–Widrow sterile neutrino particle masses of 3, 6 and 15.5 keV. Note that the 1.3 keV (thermal; 6 keV sterile) case is equivalent in the structure formation cutoff scale of the M2L25 model of Boyarsky et al. (2009) and Lovell et al. (2012).
Three separate issues require special attention when running these simulations: (1) the dependence of the halo mass function at high redshift on the chosen starting redshift, zini; (2) systematic errors induced by the finite volume of the simulation; and (3) artificial haloes that emerge in WDM as a result of shot noise in regimes where the underlying power spectrum is suppressed. We discuss each of these issues in turn.
Concerning the initial redshift zini, recent advances in the techniques used for numerical calculation of perturbations, such as the 2LPT (see, for example Jenkins 2010), have improved the convergence of simulations using different zini. Several groups (e.g. Lukić et al. 2007; Prunet et al. 2008; Knebe et al. 2009; Jenkins 2010; Reed et al. 2013) have worked to quantify the effect of zini on the final results of cosmological simulations. These works stress the point that not using 2LPT algorithm leads to simulations that converge very slowly as the start redshift is increased. In order to reduce as much as possible any zini effect we have used the 2LPT algorithm incorporated in music (Hahn & Abel 2011) to generate all the ICs of our simulations. Additionally, all the simulations presented in this work use the same initial redshift (zini = 125). Therefore, any systematic effects associated with starting redshift will be present in all cases and cancel when considering the ratios between the WDM and CDM halo mass functions.
Systematic errors from the finite volume of the simulation box can be divided into three categories: shot noise, sampling variance and lack of power from modes larger than the simulation box. Shot noise is especially important for the most massive haloes since only a few exists in the simulation volume, and it generally decreases as |$1/\bar{n}V$| where V is the simulation volume and |$\bar{n}$| the number count. However, for smaller halo masses, shot noise is dwarfed by sample variance (Hu & Kravtsov 2003). The average density in the simulation volume may happen to be an over- or underdense part of the universe, and since haloes are biased tracers of the density field, this will lead to differences in the halo mass function. The best way to correct this is to run independent samples of the underlying density field (different seeds for the ICs), but this comes at a considerable cost in terms of CPU hours. Alternatively, the sampling variance can be estimated by analytic methods as given by equation 4 in Hu & Kravtsov (2003), with a Sheth–Tormen bias for example. However, such a bias is based on fits to ΛCDM simulations. It seems plausible that such a bias would not change significantly if used in a WDM cosmology as it is primarily determined by nature of halo collapse, but to avoid any complications with the error estimate, we directly calculate the sample variance in the halo mass function by the jackknife technique. We do not consider any contributions to the halo mass function from scales larger than the simulation box, since we are mostly interested in the low-mass end, and the simulated volume is significantly larger than the scale of clusters at the redshifts of interest.
Finally, below a specific mass scale dependent on numerical resolution, it has been well established that WDM produces artificial haloes in simulations (Wang & White 2007; Angulo, Hahn & Abel 2013), an effect of the shot noise due to the finite particle count. These haloes are usually visible as regularly spaced clumps in the filaments of the cosmic web, and they form below a mass scale proportional to |$m_{\rm p}^{-1/3}$|, where mp is the simulation particle mass. However, force resolution also plays a role, and an excessive force resolution, as compared to the mass resolution, can increase the number of artificial haloes (Angulo et al. 2013). Schneider et al. (2013a) showed that the artificial haloes can be modelled by a power law increase in the WDM halo mass function below the mass scale. Most attention has been given to correcting the halo mass function for low redshift, since contamination of the halo population is the largest here, due to the fact that the artificial haloes have had more time to form and accrete.
There have been no focused studies on artificial halo contamination is at redshifts z ≳ 5. In the results presented below, down to the halo mass scale adopted for our completeness limit, we see little if any indication of a low-mass upturn in our WDM halo mass functions; such an upturn would be indicative of significant artificial halo contamination. Moreover, since any artificial haloes present would provide an increase in the halo mass function (thus making WDM more like CDM) ignoring them only makes our WDM constraints more conservative. In what follows, we have conservatively chosen to ignore any corrections for artificial haloes in our catalogues.
Fig. 2 provides a qualitative depiction of the differences inherent in WDM compared to CDM simulations. Shown are 10 × 10 × 6 (comoving h−1 Mpc) slices of each of our simulation volumes, centred on the most massive halo at a redshift of z = 6. On large scales the slices look similar, but on smaller scales there is a clear lack of structure in the WDM models.
2.3 Halo catalogues
We used the Amiga Halo Finder (AHF; Knollmann & Knebe 2009) to identify haloes in our simulations. The halo mass Mh used in this work is calculated using the overdensity (Δvir) formula from Bryan & Norman (1998) for our cosmology at each specific redshift. Note that our conclusions do not change when using different overdensity definitions, e.g. Δ200 = 200ρcrit. As explained above, to build our mass–luminosity relation using the abundance-matching technique we took into account the merger history of each halo and used its maximal mass obtained over its lifetime Mpeak instead of Mh. In any case, this correction turned to be small due to the lack of sub-structure at high redshifts. We used a requirement of at least 40 simulation particles to constitute a halo, setting a halo mass completeness limit of Mh = 3.4 × 108 h−1 M⊙.
Compared to the density maps shown in Fig. 2, the differences between WDM and CDM become even more apparent when we compare halo counts. Fig. 3 shows two of the same density slices overlaid with white circles to indicate identified dark matter haloes more massive than our Mh = 3.4 × 108 h−1 M⊙ completeness limit. Circle sizes are proportional to the virial radius of each identified halo. The difference in collapsed structures is striking between these two simulations. For example, the void in the upper-left corner is completely empty of any haloes in the 0.8 keV WDM run.
As can be seen in the z = 13 panel of Fig. 4, no haloes at all exist in 0.8 keV WDM model. Indeed, we find that no haloes have formed before z = 12 for 0.8 keV WDM and none before z = 15 in the 1.3 keV model. Detections at these epochs should be robust in the future with JWST. However, even current detections offer an interesting test: the point with error bar (2σ) corresponds to the lower limit on the cumulative abundance of galaxies at those redshifts, as set by the faintest galaxies observed in the HUDF (Bouwens et al. 2007; McLure et al. 2013; Oesch et al. 2013). Its horizontal position (corresponding halo mass) is based on the luminosity limit and our adopted Mh–L relation presented in the next section. Importantly, the total abundance of galaxies at each redshift must be above the data point shown (regardless of its horizontal positioning on the plot). One can see without any further analysis that the 0.8 keV WDM model will have trouble producing enough galaxies to match current observations at z > 8; there are simply not enough collapsed objects of any mass to account for the known galaxies at this epoch. The viability of the other WDM models is not immediately apparent from Fig. 4 in itself, since the halo mass function is not directly observed.
In order to provide a more precise connection with observations we will need a mapping between halo mass and galaxy luminosity. This is a primary subject of the next section.
3 PREDICTING OBSERVABLES
3.1 Observed luminosity functions
We parametrize the evolution of the luminosity function with redshift by fitting quoted observational results for log ϕ*, L* and α and fitting them linearly as a function of z from z = 4-8. Fig. 5 shows the fit used in this work in comparison with fits from other authors. The data points used for this fit (plotted) are taken from Bouwens et al. (2007) for z = 4-6 and from McLure et al. (2013) for z = 7-8. Points at higher z (which assumed fixed values for α and ϕ*) are shown for reference from Oesch et al. (2013). Note that formally the luminosity density becomes divergent if α < −2; however, due to the introduction of a minimum cutoff scale in halo masses in equation (12), this is not a cause for concern. It is important to stress here that even small changes in the fit parameters provides drastic changes in the reionization history. This is especially true for changes to the faint end slope α of the luminosity function (Bouwens et al. 2011). Future observations from the JWST can hopefully much better constrain α at high redshift.
3.2 Connecting haloes to galaxies
In assigning luminosities to dark matter haloes we assume that brighter galaxies reside in more massive haloes and that the relationship between halo mass and galaxy luminosity is monotonic, following the same relation for all dark matter models.
Fundamentally, we rely on the abundance-matching technique (Kravtsov et al. 2004; Vale & Ostriker 2004), which defines the relationship between halo mass and galaxy luminosity (or alternatively stellar mass) by equating the cumulative number density of haloes to the cumulative number density of galaxies observed. The power of this approach is that the observed luminosity function is fully reproduced (at least down to luminosities where the observations are complete or to where the matching is performed) while sweeping all uncertainties galaxy formation physics under the rug. In principle, other halo parameters could be used as the rank order of choice (e.g. maximal circular velocity; Trujillo-Gomez et al. 2011). Recently, Behroozi and collaborators have argued that halo mass is the most robust variable to use for these purposes (Behroozi et al. 2014).
Our fundamental assumption is that the halo mass–luminosity mapping is a power law at faint magnitudes. Furthermore we are conservatively assuming that the relationship between halo mass and galaxy luminosity is the same in CDM and WDM (obeying near power-law behaviour at faint luminosities). This approach demands that all models match the observations at the bright end (where halo counts overlap), and makes the assumption that there is no special break (towards more efficient galaxy formation) in the luminosity–halo mass relation in WDM for small haloes. This is conservative because there is no reason to expect that WDM haloes will be more efficient at making galaxies than CDM haloes. Indeed, WDM haloes collapse later and have had less time to form stars, so we might expect them to be less efficient at forming stars than their CDM counterparts.
Herpich et al. (2014) performed hydrodynamical simulations of several different WDM cosmologies. They found only slight differences between the stellar masses of the different dark matter models they considered, and the difference they did see was towards less efficient formation in WDM as discussed above. For the most extreme case, comparing a 1 keV WDM model to CDM, they found a ratio of |$M_{\star ,\rm CDM}/M_{\star ,\rm WDM}\sim 2$| in stellar masses at z = 0. Therefore, the main differences in the star formation histories are produced at late times, and therefore, this relatively small effect is more reduced at the high redshifts of interest for this work. Small variations in the stellar feedback implementation have a much greater impact on the final stellar mass of a galaxy than WDM particle mass. While the results of Herpich et al. (2014) are based on low-z simulations it seems reasonable to expect the same to be true at higher redshift. Appendix A further elaborates on using the CDM halo catalogue as the fiducial model.
The assumption that the power-law behaviour for the faint end of the halo mass–luminosity mapping translates into a suppression of the luminosity function in the corresponding WDM model. This suppression is then exactly equal to the ratio of the halo catalogues at the given mass scale, which is an aggressively conservative assumption of a cancelation of two disparate physical mechanisms. Furthermore, this requires that the WDM luminosity functions diverge from a pure Schechter fit. We stress that several other processes could also cause the luminosity function to diverge from a power law; however, since no such divergence is yet observed, this translates into constraints on the nature of dark matter.
Before moving on, we note that while the general abundance-matching approach has proven successful and robust at reproducing galaxy properties in the low-redshift universe, it is less well tested at higher redshifts. For example, the scatter at fixed halo mass appears to be fairly minimal at low-z (Behroozi et al. 2014) and the relationship between halo mass and luminosity is well described by a power law for faint systems (Moster, Naab & White 2013; Garrison-Kimmel et al. 2014). At high redshift however, the relationship between halo mass and UV luminosity could in principle exhibit significant scatter, though a power-law relationship for the smallest galaxies appears consistent with the data (Behroozi et al. 2014). We adopt a strict one-to-one relationship between halo mass and UV luminosity as a starting point in investigating the expected differences between CDM and WDM on galaxy counts in the high-z universe. Because we are looking at differential effects between the two models, driven by the declining number of low-mass haloes in WDM, we anticipate that this approach provides a fair starting point, though it would be useful to extend this approach to more complicated mappings in the future.2
3.3 Reionization
With predictions for luminosity functions in hand, we can directly connect those to expectations on reionization. Star forming galaxies at z ≳ 6 are the primary candidate for the main process driving the reionization of the IGM. Changes in the abundance of early galaxies therefore translates into different reionization histories.
In this work, we do not include an evolution in the reionization parameters. Kuhlen & Faucher-Giguere (2012) found that for their best-fitting scenario, evolution in the limiting luminosity alone is not enough to match both Lyα constraints and reionization constraints, and the data provide no conclusive evidence for an evolution in any case. Evolution in fescζion (resulting perhaps from evolution in the stellar initial mass function) may be more plausible. We will present our results with different values of fescζion and limiting luminosity, but they will remain fixed with redshift. Evolution in the clumping factor |$C_{\rm{H\,\small {II}}}$| may be expected, but no definitive determination of its evolution exists. For example, Finlator et al. (2012) presents a detailed analysis of the evolution, suggesting that the clumping factor rises from |$C_{\rm{H\,\small {II}}}<1$| for z > 10 to |$C_{\rm{H\,\small {II}}}\sim 3.3$| at z ∼ 6. It should be noted that our model equation (8) does not include the detailed distribution of hydrogen where some dense clumps reionize later than less dense clumps. This simplification is likely inaccurate in the final phase of reionization around z ∼ 6; however, we expect it to be an appropriate approximation on average for higher-z and in the large cosmological volumes of interest here.
In what follows, we adopt the reionization parameters |$C_{\rm{H\,\small {II}}}=3$|, ζion = 1025.3erg s− 1 Hz, Xp = 0.75, T0 = 2 × 104 K and αB = 1.6 × 10− 13 cm3 s−. The escape fraction and limiting luminosities vary, and will be indicated in the relevant figures.
4 RESULTS
Here, we explore the constraints on WDM models by direct number counting and the inferred reionization history, and illustrate how future galaxy count surveys can improve these constraints on WDM models.
4.1 WDM constraints from galaxy counts
As explained in Section 3.2, given our (conservative) assumption of the star formation efficiency being the same in the different dark matter models, deviations in the implied WDM luminosity function from the observed Schechter fit translates into constraints on the WDM model.
Fig. 7 shows the implied luminosity functions for CDM (dotted black) and each of our WDM models (solid, colours indicated). The symbol with error bar is the known (observed) cumulative count of galaxies at the faint HUDF limit, with errors indicative of the 2σ uncertainty calculated as in Fig. 4. The shaded band corresponds to a 1σ uncertainty of the observed best-fitting Schechter function (seen in Fig. 5), extrapolated down to an approximate JWST deep field limit (indicated by the vertical line). We see clearly here that the 0.8 keV WDM model (solid cyan) is strongly disfavoured by current observations of the galaxy luminosity functions. The 1.3 keV model, while currently consistent with observations, demonstrates significant deviations from CDM at z = 13 at magnitudes observable with JWST. Deep galaxy counts at this and earlier epochs may be even sensitive to 2.6 keV WDM.
Had we considered WDM artificial haloes and the fact that the star formation efficiency is slightly lower in WDM than in CDM, the constraints would improve slightly, since there would then be fewer and less luminous galaxies. However, these effects are not likely to be very important at redshifts z ≥ 4, and are a priori unknown.
Faint galaxy counts at even higher redshift will be particularly sensitive to WDM models. We demonstrate this in Fig. 8. Here, we show the cumulative number density of galaxies brighter than MAB = −16 as a function of redshift for each of our models. The differences between CDM and WDM are significant, especially for the lower mass WDM cases. Deep JWST observations should be sensitive to galaxy detections at least this faint out to z = 15, and therefore will provide a direct probe of the small-scale power spectrum by counting galaxies.
4.2 WDM constraints from reionization
Fundamental to the ability of galaxies to reionize the universe is the production rate of ionizing photons, |$\dot{n}_{\rm ion}$|, which is proportional to the total UV luminosity density, ρUV, coming from these sources (see equation 11). For an underlying galaxy luminosity function with a steep faint-end slope α, the total luminosity density implied will be sensitive to the assumed faint-end cutoff used to calculate ρUV. Fig. 9 shows the luminosity density in our models as a function of faint-end cutoff at selected redshifts. Because WDM models have flatter faint-end luminosity function slopes, the total ρUV is less sensitive to the faint-end cutoff, i.e. the implied cumulative ρUV values flatten relative to CDM at fainter magnitudes. Importantly for our considerations, WDM predictions for reionization will be less sensitive to the adopted faint-end cutoff than CDM, owing to the lack of small galaxies in these models.
The points where the lines change to dashed in Fig. 9 mark the resolution limit in the simulations. The two WDM dashed lines bracket the following extreme cases: one, a power-law fit to the faint end, and, two, the constant value at the faintest point resolved in the simulations. The actual luminosity density would be somewhere between these two extremes, though for the 0.8 and 1.3 keV models the difference is negligible. All our analysis utilizes the power law extrapolations to get conservative estimates.
With the luminosity density in hand, the reionization history can now be determined by virtue of equation (8). Fig. 10 presents the volume filling fraction |$Q_{\rm{H\,\small {II}}}$| as a function of redshift for two choices of limiting magnitude in calculating the luminosity density. The fiducial line types (shown in the legend) correspond to a limiting magnitude of MAB = −10 while dashed lines cutoff at a brighter limit of MAB = −13. We used the initial condition |$Q_{\rm{H\,\small {II}}}=0$| at z = 20 and integrated forward in time. We choose an optimistic escape fraction of fesc = 0.5, higher than assumed in both Robertson et al. (2013) and Kuhlen & Faucher-Giguere (2012), and therefore more conservative with respect to WDM model constraints since we use the same ζion as in Kuhlen & Faucher-Giguere (2012). Robertson et al. (2013) uses a lower ζion, but instead assumes an almost constant luminosity density at the cutoff scale at high redshift.
All of the models in Fig. 10 except 0.8 keV WDM have completed reionization by z ∼ 5.8 as required by results inferred from the kinematic Sunyaev-Zel'dovich effect and CMB polarization observations (Zahn et al. 2012). For the 0.8 keV model shown, reionization is complete at z = 5.5. In general, the WDM cases produce a more rapid late-time build-up of ionized hydrogen due to the high-redshift suppression of haloes. It can also be seen that the difference between CDM and 2.6 keV WDM is larger when the fainter limiting magnitude is used, simply because the difference between the models is much larger here. Of course, these results are sensitive to the escape fraction. For example, if an escape fraction of fesc = 0.2 is used for the 1.3 keV model full reionization is not reached until z = 5.4 (not shown on figure), so fairly high escape fractions seem to be required for 1.3 keV to reach full reionization by z ∼ 6.
Another important probe of reionization is the integrated optical depth of electron scattering from the CMB. The shaded bands in Fig. 11 show the CMB optical depth range of 0.092 ± 0.013 from the most recent Planck results (Ade et al. 2013). The lines show predictions for the optical depth as contributed as a function of redshift for our WDM and CDM models assuming an escape fraction fesc = 0.5, with the two panels corresponding to different limiting magnitudes. Interestingly, with this choice of (fairly high) escape fraction, none of our WDM models can reproduce the measured optical depth, and even CDM requires a luminosity function extrapolation to a very faint limiting magnitude. This is consistent with the findings of Robertson et al. (2013).
In Fig. 12, we show results for the optical depth, now assuming fesc = 1. In this case, the 2.6 keV model can reproduce the Planck value, though a fairly faint limiting magnitude seems to be required, even in this extreme case. Unsurprisingly, CDM severely overshoots the optical depth with these (rather high) reionization parameters. It is noteworthy that neither of the low-mass WDM models can reproduce the Planck optical depth within its 68 per cent confidence interval, even with very optimistic choices. If these WDM models are to be viable in the face of reionization constraints, they would require either significant contribution to the ionizing flux from non-stellar sources, or a significantly larger combination of fescζion than what is currently believed to be realistic. A smaller |$C_{\rm{H\,\small {II}}}$| could also help in this respect. Future observations will likely better constrain these parameters.
5 DISCUSSION
Future measurements of the luminosity function of faint galaxies at high-z, particularly those from JWST, and possibly with HST via the Frontier Fields initiative, will significantly improve the sensitivity to WDM models and the halo mass cutoff effects presented here. For a direct number count comparison, the mass resolution of our simulations is sufficient to connect with observations down to plausible detection limits with JWST.
In contrast, our results on reionization specifically for the 2.6 keV model would improve with greater mass resolution. This is because of our conservatively approximated increasing faint end luminosity density function, which likely flattens at low luminosities in this model, but remains unresolved in our simulations. This improvement can also be made, though to a lesser degree, for the lower WDM particle mass models. Our results for these particle mass models indicate that the luminosity density reaches an approximately constant level at the faint end, and thus the ionizing flux will remain constant at fainter luminosities. Deep observations with JWST certainly will much better determine the faint end slope α of the luminosity function. The reionization history is highly sensitive to this parameter. A future analysis could possibly circumvent the uncertainty stemming from fitting the evolution of the Schechter parameters, instead relying on direct observations of α for most redshifts.
We have neglected any evolution in the escape fraction with redshift. Naïvely, one might expect the escape fraction to decrease with redshift since the overall density scales as (1 + z)3. However, observations seem to indicate the opposite: the escape fraction increases with redshift. Mitra, Ferrara & Choudhury (2013) found that fesc ∼ 0.06 at z ∼ 6, and increases to at least fesc ≳ 0.146 by z ∼ 10. This could be caused by an initial mass function for star formation favouring high mass stars in the early universe. Alternatively, Ferrara & Loeb (2012) proposes a mechanism where mini haloes close to the cooling limit contributes appreciably to the ionizing flux, but their contribution diminishes over time due to feedback mechanisms. Since these small haloes have a relatively larger escape fraction the overall escape fraction decreases with time. This mechanism is especially interesting from a WDM perspective, since the lower abundance of small haloes directly counteracts this. The evolution of the escape fraction remains uncertain, and a typical constant value in previous work was fesc ∼ 20 per cent, and our conservative choice of fesc > 0.5 will very unlikely overestimate the ionizing flux.
Non-stellar processes can potentially contribute to the reionization history. Quasars might play an important role at high redshift, although current results seem to indicate the contribution from quasars is sub-dominant (Volonteri & Gnedin 2009). For example, Willott et al. (2010) found that at z ∼ 6 the ionizing flux from quasars is 20–100 times lower than what is needed for continued reionization. X-ray emission by black holes may also contribute appreciably. Ricotti & Ostriker (2004) and Ricotti, Ostriker & Gnedin (2005) analysed a pre-ionization contribution from a top-heavy initial mass function for the Population III stars. These Population III stars then collapse into black holes and subsequently accrete at nearly the Eddington limit. Accretion on to the black holes could partially reionize the IGM, although the primary effect would be heating the IGM. The Population III phase is rapidly self-limiting due to pollution by heavy elements and pair instability supernovae causing strong outflows, and thus the metal-poor Population III stars needs to collapse into black holes. This pre-ionization phase is then followed by a period of stellar reionization.
Bovill & Ricotti (2011) and Katz & Ricotti (2013) present a mechanism where the bulk of the star formation in the first dwarf galaxies happened in protoglobular clusters that were subsequently tidally stripped from the dwarf galaxy. Since the tidal stripping of globular clusters in the halo outskirts precede the stripping of the dark matter halo this can break the assumption of the abundance-matching technique since the dwarf galaxies would be stripped off much of their luminous matter. Consequently, the mass–luminosity mapping would no longer be strictly monotonic, and the scatter on any mass–luminosity relation would likely increase. Importantly, for mass scales larger than the masses of these dark dwarf galaxies, however, the abundance matching would still give a meaningful average mass–luminosity mapping. In any case, the abundance matching we employ here would be an upper bound on the luminosity of a dwarf galaxy and, therefore, our approach is highly conservative.
In the case of sterile neutrino WDM, it has been shown that the effects of the radiative decay of sterile neutrino WDM to X-ray photons may catalyse the formation of H2 and star formation (Biermann & Kusenko 2006). This does not affect the results presented here, because in this case the sterile neutrino WDM cosmology is constrained to produce the same observed high-z luminosity functions from which the reionization history is inferred. The only method by which such radiative decays would enhance the reionization rate is if they preferentially enhanced star formation for small mass haloes below the luminosity function cutoff magnitude. We are aware of no mechanism in the literature that would produce such an enhancement for low-mass haloes. Moreover, this X-ray photon catalyzation process would enhance star formation preferentially in metal free low-mass haloes that rely on H2 to cool. More massive haloes which are involved in reionization are insensitive to this mechanism as they can cool by Lyα line emission and are more likely to be metal rich. Therefore, since halo formation is suppressed at these low masses, we believe our results apply for the case of sterile neutrino WDM, with the 3 and 6 keV mass scales disfavoured at >10σ and 98.6 per cent C.L., respectively.
Recall that the 1.3 keV (thermal; 6 keV sterile) model we have considered corresponds to model discussed by Lovell et al. (2012) as a solution to the too-big-to-fail problem (Boylan-Kolchin et al. 2012) and the M2L25 model studied in Boyarsky et al. (2009). We have demonstrated that this model is disfavoured at 98.6 per cent C.L. by direct galaxy counts at high redshift and is unable to reproduce the CMB optical depth even with extreme assumptions about the escape fraction.
6 CONCLUSION
We have shown that the Lyman-break technique for galaxy surveys at high redshift can provide a direct method for constraining the nature of dark matter and its clustering at small scales, with sensitivity to the structure formation suppression present in WDM models. We have analysed CDM and WDM cosmological simulations in order to test WDM models using the luminosity function observations at high-z as well as a new analysis of cosmological reionization limits. Given the assumptions that the luminosity function of a ΛCDM universe is modelled by a Schechter function down to faint magnitudes and that the mass–luminosity relation of galaxies is independent of the dark matter model employed, we have modelled the luminosity function for several dark matter models to analyse the sensitivities to WDM dark matter models.
Using an approximate χ2 test of the faint end of the luminosity function, direct number counts of galaxies significantly disfavours a 0.8 keV WDM model at greater than 10σ, and a 1.3 keV model is disfavoured at approximately 98.6 per cent C.L. (2.2σ). Further, with highly optimistic values for the parameters that translate high-redshift galaxy luminosity to ionizing flux, the 0.8 and 1.3 keV models are inconsistent with the CMB optical depth at greater than 68 per cent C.L. Furthermore, for the conservative case of a limiting luminosity of MAB = −13, a 2.6 keV WDM model is only marginally consistent with the 68 per cent confidence region of the optical depth from Planck. Wherever possible, we have used conservative values on parameters, making WDM behave more like CDM. For this reason, we feel confident concluding that neither the 0.8 or 1.3 keV models are consistent at more than 68 per cent C.L. with reionization, even with the large uncertainty on the reionization process.
We expect upcoming deep surveys with JWST (and possibly HST via the Frontier Fields) to be able to reach luminosities and redshifts that can fully discern between a CDM model and a 1.3 keV model by direct number counts. Even 2.6 keV WDM might prove discernible if the observations are deep enough. Additionally, if the constraints on reionization parameters are improved, a 2.6 keV WDM model can be distinguished from CDM by its different reionization history. The study of galaxy formation and reionization in the high-z universe adds a complementary and competitive probe to the nature of dark matter.
We thank useful discussions with John Beacom and Richard Ellis. We would also like to thank the referee Massimo Ricotti for helpful feedback on the paper. KNA is partially supported by NSF CAREER Grant no. PHY-11-59224. JO and JSB were supported by grants from the NSF and NASA.
Strictly speaking Angulo et al. (2013) has α = 1 fixed; however, they also correct for artificial haloes. We find that keeping α as a free fitting parameter is necessary to provide reasonable fits, probably owing to a strong evolution with redshift.
Notably, we take the inherent scatter in the halo mass–luminosity mapping into consideration.
Commonly used cases A and B definitions differentiate mediums that allow the Lyman photons to escape or that are opaque to these lines (except Lyα) respectively. Case B is most appropriate for this reionization calculation.
We use the jackknife errors instead of the errors on the observed luminosity function, since they are larger (shown in Fig. 7). The χ2 test here is approximate, but our conclusions are robust regarding the models’ consistency.
REFERENCES
APPENDIX A: ABUNDANCE MATCHING WITH wDM
Herpich et al. (2014) found that the star formation in low-z Milky Way-like galaxies is slightly suppressed in WDM cosmologies. This seems to fit well with what one would expect: the small-scale cutoff in the WDM transfer function postpones the formation of dwarf galaxy haloes, and therefore the potential wells that act as seeds for the first galaxies are shallower. The effect is relatively small, only a factor of 2 for their most extreme WDM model at z = 0.
The star formation efficiency for different dark matter models can readily be inferred from abundance matching. Fig. A1 shows the halo mass–luminosity relation by using the different dark matter halo catalogues. For a fixed mass, a WDM halo is seen to be more luminous than a CDM halo, and thus WDM would, unsurprisingly, need to have an enhanced star formation efficiency relative to CDM in order to match observations. This is counterintuitive, and more importantly contradicts the low-z results of Herpich et al. (2014). A realistic WDM halo mass–luminosity mapping would give a slightly lower star formation efficiency: that is, a flatter slope than CDM in Fig. A1 instead of a steeper slope. In our analysis, we therefore conservatively assume that the halo mass–luminosity is a power law. Therefore, a halo mass can uniquely be mapped to the same luminosity independent of WDM model.