Reionization and Galaxy Formation in Warm Dark Matter Cosmologies
We compare model results from a semi-analytic (merger-tree based) framework for high-redshift () galaxy formation against reionization indicators, including the Planck electron scattering optical depth () and the ionizing photon emissivity (), to shed light on the reionization history and sources in Cold (CDM) and Warm Dark Matter (WDM; particle masses of and 5 keV) cosmologies. This model includes all the key processes of star formation, supernova feedback, the merger/accretion/ejection driven evolution of gas and stellar mass and the effect of the ultra-violet background (UVB), created during reionization, in photo-evaporating the gas content of galaxies in halos with . We find that the delay in the start of reionization in light (1.5 keV) WDM models can be compensated by a steeper redshift evolution of the ionizing photon escape fraction and a faster mass assembly, resulting in reionization ending at comparable redshifts () in all the Dark Matter models considered. We find the bulk of the reionization photons come from galaxies with a halo mass and a UV magnitude in CDM. The progressive suppression of low-mass halos with decreasing leads to a shift in the “reionization” population to larger halo masses of and for 1.5 keV WDM. We find that current observations of and the Ultra-violet luminosity function are equally compatible with all the (cold and warm) Dark Matter models considered in this work. Quantifying the impact of the UVB on galaxy observables (luminosity functions, stellar mass densities, stellar to halo mass ratios) for different DM models we propose that global indicators including the redshift evolution of the stellar mass density and the stellar mass-halo mass relation, observable with the James Webb Space Telescope, can be used to distinguish between CDM and WDM (1.5 keV) cosmologies.
According to the standard Lambda Cold Dark Matter (CDM) cosmological model the universe consists of three main components: Dark Energy, Cold Dark Matter (CDM) and baryons with energy densities corresponding to , and respectively (Planck Collaboration et al., 2015). Dark Matter forms the basis of the hierarchical model of galaxy formation, providing the halos into which baryonic matter collapses, merging and accreting matter to form successively larger structures through cosmic time. With its property of clustering on all scales, CDM has been remarkably successful in predicting the large scale structure of the universe, the temperature anisotropies measured by the Cosmic Microwave Background (CMB) and Lyman- forest statistics (e.g. Peebles, 1971; Blumenthal et al., 1984; Bond & Szalay, 1983; Cole et al., 2005; Lange et al., 2001; Hinshaw et al., 2013; Planck Collaboration et al., 2014a; Slosar et al., 2013). However, as recently reviewed by Weinberg et al. (2013), CDM exhibits a number of small scales problems including: (i) producing halo profiles that are cuspy as opposed to the observationally preferred constant density cores (Navarro et al., 1997; Subramanian et al., 2000), (ii) over-predicting the number of satellite and field galaxies as compared to observations - the “missing satellite problem”(Klypin et al., 1999; Moore et al., 1999), (iii) predicting massive (Large Magellanic Cloud mass) concentrated Galactic sub-halos inconsistent with observations (e.g. Boylan-Kolchin et al., 2012) and (iv) facing difficulties in producing typical disks due to ongoing mergers down to (Wyse, 2001). The limited success of baryonic feedback in solving these small scale problems (e.g. Boylan-Kolchin et al., 2012; Teyssier et al., 2013) has prompted questions regarding the nature of dark matter itself. Indeed, a popular solution to the small scale problems in CDM cosmology involves invoking ( keV) Warm Dark Matter (WDM) particles that erase small-scale power (Blumenthal et al., 1984; Bode et al., 2001, e.g)
The most stringent astrophysical constraints on the WDM particle mass of have been obtained using the low- () Lyman- flux power spectrum (Viel et al., 2013). However, allowing for the intergalactic medium (IGM) temperature-density relation to vary independently as a function of redshift, Garzilli et al. (2015) have obtained a lower bound of using the same data. A number of studies have used alternative high- probes to obtain constraints on the WDM particle mass ranging from using number counts of high- gamma ray bursts (de Souza et al., 2013) to using dwarf spheroidal galaxy observations (de Vega & Sanchez, 2010), simultaneously reproducing stellar mass functions and the Tully-Fisher relation for galaxies (Kang et al., 2013) and comparing the observed number density of galaxies to that expected from the halo mass function (Pacucci et al., 2013).
With its property of smearing out small-scale density perturbations, WDM leads to a delay in the assembly of the lowest mass galaxies at early epochs, resulting in a corresponding delay in the appearance of the first metal-free Population III (PopIII) stars. In addition, since low-mass high- galaxies are believed to be the main sources of reionizing photons in the early universe (e.g. Barkana & Loeb, 2001; Ciardi & Ferrara, 2005; Choudhury & Ferrara, 2007; Choudhury et al., 2008; Bouwens et al., 2015; Robertson et al., 2015), their delayed and accelerated assembly would naturally lead to a corresponding delay and acceleration in the reionization history. A number of works have used such arguments to constrain the WDM particle mass: Yoshida et al. (2003) have used simulations of PopIII star formation to show that they can yield a maximum electron scattering optical depth of in WDM models with masses as large as . While such models could be ruled out using WMAP data, this number is in good agreement with the latest Planck estimates (Planck Collaboration et al., 2015, 2016). Magg et al. (2016) have recently proposed observing the PopIII supernovae (SN) rate with the forthcoming James Webb Space telescope (JWST) and the European-Extremely Large Telescope (E-ELT) to differentiate between CDM and WDM, and constrain the mass of the latter. Somerville et al. (2003) have used the global star formation rate density (SFRD) to show that reionization is delayed until in models, a result that is broadly consistent with the latest Planck results. Yue & Chen (2012) have used a minimum halo mass for star formation embedded in a bubble-model for reionization to show that although WDM can delay the start of reionization, the end might not necessarily be delayed; however, these authors assume a fixed ionizing photon production rate per baryon and do not explicitly model any baryonic physics. Recently, using a constant relation linking the ultra-violet (UV) and ionizing photon production rates to the DM halo mass, Schultz et al. (2014) have ruled out ; however, even WDM yields an optical depth consistent with the latest Planck results (Planck Collaboration et al., 2014b, 2016) using their model. Finally, using a semi-analytic model of galaxy formation where 10% of the baryons that collapsed into halos, before virialization of the outermost DM shell, are converted into stars Barkana et al. (2001) obtain constraints of if the star formation efficiency is the same (10 times larger) at high- as at .
This last work is most similar in spirit to our approach, although it does not include gas loss from SN winds which progressively decreases the gas fraction in galaxies as shown in Dayal et al. (2014a). A major caveat in many of these existing works is their assumption of a cosmology-independent mass-to-light (M/L) or ionizing photon production rate-to-mass ratio. However, as shown in Dayal et al. (2014b), the M/L ratio sensitively depends on the cosmology considered: a delay in structure formation in WDM models results in a younger stellar population in low-mass halos that, as a result, produces more light per unit stellar mass compared to CDM. The same behaviour is expected to hold for the amount of reionization photons produced per unit stellar mass at any cosmic epoch.
Over the past few years, ground and space-based observations have allowed a statistically significant data-set to be collected for Lyman Break galaxies (LBGs; Oesch et al., 2010; Bouwens et al., 2010, 2011; Castellano et al., 2010; McLure et al., 2010; Bradley et al., 2012; Oesch et al., 2013; McLure et al., 2013; Bowler et al., 2014b; Bouwens et al., 2014; Bowler et al., 2014a; Atek et al., 2015; Livermore et al., 2016; Oesch et al., 2016). In this work, our aim is to revisit the constraints allowed on the WDM particle mass by combining these “galaxy” data sets with the latest limits on the cosmological electron scattering optical depth inferred by Planck. We build upon a semi-analytic model that traces both the DM and baryonic assembly of high- () galaxies, reproducing a number of key observables, presented in Dayal et al. (2014a, b) to show how the reionization history and sources depend on the DM model considered. The strength of the model lies in the fact that in addition to internal feedback from SN (see Sec. 2.3 Dayal et al., 2014b) our model also includes the effects of “external” feedback (i.e. reionization photo-evaporating gas from DM halos).
The cosmological parameters used in this work correspond to (, consistent with the latest results from the Planck collaboration (Planck Collaboration et al., 2015) and we quote all quantities in comoving units unless stated otherwise.
2. Modelling galaxy formation and reionization
In this work, we explore 4 DM models: CDM and WDM with and 5 keV. Although we cite values assuming thermally decoupled relativistic particles, these numbers can be converted into sterile neutrino masses () using the expression (Viel et al., 2005):
This yields corresponding to respectively.
2.1. Merger trees and the baryonic implementation
We now briefly summarise the theoretical model and interested readers are referred to Dayal et al. (2014a, b) for complete details. We construct 400 merger trees starting at , linearly distributed across the halo mass range for the 4 DM models considered. The merger-trees use 320 equal redshift steps () between and with a mass resolution of using the modified binary merger tree algorithm with smooth accretion detailed in Parkinson et al. (2008) and Benson et al. (2013). We scale the relative abundances of the merger tree roots to match the Sheth-Tormen halo mass function (HMFs; Sheth & Tormen, 1999) and have verified that these yield HMFs in good agreement with the Sheth-Tormen HMF at all .
We implement the merger trees with baryonic physics including star formation, SN feedback, and the merger/accretion/ejection driven evolution of gas and stellar masses. Our model is based on the simple premise that any galaxy can form stars with a maximum efficiency () that provides enough energy to expel all the remaining gas, quenching further star formation, up to a threshold value of (see Dayal et al., 2014a). This implies the effective star formation efficiency () is the minimum between and . This model has two - and mass-independent free parameters whose values are selected to match the evolving ultra-violet luminosity function (UV LF): the maximum threshold star formation efficiency () and the fraction of SN energy that goes into unbinding gas (). We implement this simple idea proceeding forward in time from the highest merger tree output redshift, . At any step, the gas mass in a galaxy is determined both by the gas mass brought in by merging progenitors as well as that smoothly-accreted from the IGM. A part () of this gas forms new stellar mass, , with the final gas mass depending on the ratio of the (instantaneous) energy provided by exploding SN and the potential energy of the halo. Further, at any step the total stellar mass in a galaxy is the sum of newly-formed stellar mass, and that brought in by its progenitors. In this work we also explore the effects of the ultra-violet background (UVB) in photo-evaporating gas from low-mass halos, impeding their star-formation capabilities, and its impact on both galaxy assembly and reionization as explained in Sec. 2.2 that follows.
For simplicity, we assume every new stellar population to have a fixed metallicity of and an age . Using the population synthesis code STARBURST99 (Leitherer et al., 1999), the initial UV luminosity (at Å) can be calculated as and the initial output of ionizing photons can be calculated as . Further, the time evolution of these quantities can be expressed as
For any galaxy along the merger tree the UV luminosity and ionizing photon output rate are the sum of the values from the new starburst and the contribution from older populations accounting for the drop with time.
As shown in Dayal et al. (2014a, b), our model reproduces the observed UV LF for all DM models (CDM and WDM with and 5 keV) at over 7 magnitudes in luminosity and predicts the -evolution of the faint end UV LF slope, in addition to reproducing key observables including the stellar mass density (SMD) and mass-to-light ratios using fiducial parameter values of and . We maintain these fiducial parameter values in all the calculations carried out in this work.
2.2. Modelling reionization
where the first term on the right hand side represents the growth of H regions while the second term accounts for the decrease in due to recombinations. Here, represents the hydrogen ionizing photon rate density (per comoving volume) contributing to reionization, with accounting for the fraction of ionizing photons that escape out of the galactic environment. Further, is the comoving hydrogen number density, and is the recombination time that can be expressed as (e.g. Madau et al., 1999)
Here is the hydrogen case-B recombination coefficient, accounts for the excess free electrons arising from singly ionized helium and is the IGM clumping factor. We use the results of Pawlik et al. (2009) and Haardt & Madau (2012) who show that the UVB generated by reionization can act as an effective pressure term, reducing the clumping factor with such that
|DM Model||SN feedback||UVB feedback||UV cut||relation|
|CDM||yes||yes||all galaxies (Eqn. 10)|
|3 keV WDM||yes||yes||all galaxies (Eqn. 10)|
|1.5 keV WDM||yes||yes||all galaxies (Eqn. 10)|
|3 keV WDM||yes||yes|
|1.5 keV WDM||yes||yes|
While reionization is driven by the hydrogen ionizing photons produced by early galaxies, the UVB built up during reionization suppresses the baryonic content of galaxies by photo-heating/evaporating gas at their outskirts (Klypin et al., 1999; Moore et al., 1999; Somerville, 2002), suppressing further star formation and slowing down the reionization process. In order to account for the effect of UVB feedback on , in the fiducial model, we assume total photo-evaporation of gas from halos below embedded in ionized regions at any ; we vary this limit between to check the robustness of our results. In this “maximal external feedback” scenario, halos below in ionized regions neither form stars nor contribute any gas in mergers. The globally averaged can then be expressed as
where and account for the intrinsic hydrogen ionizing photon production rate density within ionized and neutral regions respectively. While contains contribution from all sources, represents the case where sources below do not contribute to the ionizing photon budget. At the beginning of the reionization process the volume filled by ionized hydrogen is very small () and most galaxies are not affected by the UVB, so that . As increases and reaches a value , all galaxies less massive than are feedback-suppressed so that .
Given that is an output of the model, with calculated as a function of , is the only free parameter in our reionization calculations. In this work is assumed to be independent of the halo mass. As shown in Sec. 3 that follows, simultaneously fitting to observations of the CMB electron scattering optical depth () and the ionizing emissivity require a -dependent .
3. Reionization in different DM cosmologies
We now discuss model constraints on using observed values of the CMB electron scattering optical depth () and ionizing emissivity (). We then present the corresponding reionization history and sources for both CDM and WDM, as described in what follows.
3.1. Ionizing photon production efficiency in different DM cosmologies
The delay in structure formation in WDM models results in a much younger stellar population hosted by low-mass halos at any , when compared to CDM. This naturally results in low-mass WDM halos producing more ionizing photons, both considering and ignoring the effects of the UVB. This is the behaviour shown by as a function of stellar mass, , in Fig. 1. We start by focusing on the left panel corresponding to where the relation is essentially independent of the underlying DM model for galaxies with , corresponding to currently observed LBGs. However, given their younger ages, galaxies with produce about 4 times more ionizing photons in the 1.5 keV WDM model as compared to CDM. Quantitatively we find that for the 1.5 keV scenario, steepening to for CDM, as also quantified in Table 1; while the exact coefficients change, the same trends are seen at (right panel of the same figure).
According to the fiducial model, including the effects of the UVB results in a loss of all gas content, switching-off star formation in all halos with . This naturally results in a lower stellar mass for the successors of such systems. On the other hand, effectively only depends on the ongoing star formation rate (i.e. the instantaneous gas mass) given its steep decay with time as shown in Eqn. 3. As noted in Dayal et al. (2014a), galaxies with are feedback limited, contributing only stellar mass but no gas mass to their successors. So although the final stellar mass decreases in the presence of the UVB, remains unchanged, resulting in a higher relation as seen from the same figure. With the largest number of low-mass halos, the effect of UVB suppression is most strongly seen in CDM where halos produce about 2-2.5 times more ionizing photons as compared to the case with SN-feedback only; the effect of the UVB is the weakest in the 1.5 keV case where only increases by a factor of 1.5 for low-mass galaxies. Quantitatively, the relation including the UVB is well-fit by for CDM, changing to for the 1.5 keV scenario. Again, galaxies at follow the same qualitative trends.
3.2. Joint reionization constraints from the CMB optical depth and UVB emissivity
Combining Planck polarisation measurements with temperature and lensing data, the electron scattering optical depth corresponds to (Planck Collaboration et al., 2015), compared to the value of obtained by combining Planck and WMAP low-l polarization data (Planck Collaboration et al., 2014b); low multipole EE data has been used to infer an even lower value of (Planck Collaboration et al., 2016). We calculate as a function of by solving the equation:
A second observable that needs to be fit simultaneously is the emissivity of ionizing photons, which has been measured for . In this work, we follow the approach of Kuhlen & Faucher-Giguere (2012) and combine the observational constraints on the H photoionization rate () from Wyithe & Bolton (2011) with the mean free path of ionizing radiation () from Songaila & Cowie (2010) to obtain the observational estimate of . We have used the fiducial values from Kuhlen & Faucher-Giguere (2012): for the source spectral index and for the H column density distribution. Varying these within allowed would only result in larger error bars, leaving our results unchanged (see Kuhlen & Faucher-Giguere, 2012).
Once the parameters related to the galaxy formation model are fixed our model contains only one additional free parameter, , that can be varied to fit the two observational constraints mentioned above. Unfortunately there do not exist observations of at high redshifts which can be used as possible prior bounds. The indirect constraints on (e.g. Mitra et al., 2013; Robertson et al., 2015) have been obtained only for the standard CDM model and may not be valid for WDM cosmologies. We thus conservatively allow to take any value between 0 and 1 and constrain it by simultaneously fitting and for each DM model.
We start by assuming to have a redshift-independent constant value for all the DM models considered (e.g. Schultz et al., 2014). As shown in Fig. 2, the Planck 2014 value of (Planck Collaboration et al., 2014b) requires for the CDM and 5 keV models and for the 3 keV scenario; the 1.5 keV model can be ruled out using this data since it results in an optical depth lower than these Planck limits even in the maximal case of . The lower Planck-2015 value of is well-reproduced for CDM and WDM models for ; the 1.5 kev case requires a higher value of to be consistent with the observed bounds. We note that the most recent, and lowest, value of (Planck Collaboration et al., 2016) is matched by all DM models for values as low as .
Although matching the observed , even the minimum -independent constant value of over-predicts the observed value of by about 0.8 dex at as shown in Fig. 3. A scenario wherein is -independent is therefore ruled out by the emissivity constraints, irrespective of the DM model considered. In what follows, we show results using calibrated to the Planck 2015 results (Planck Collaboration et al., 2015) that has been obtained combining polarisation, lensing and temperature data; however these results are, within the error bars and hence, also valid for the latest Planck 2016 estimates as shown in Fig. 2.
Reconciling the optical depth and emissivity data sets requires a -dependent value of such that
where and are -independent parameters, whose values are determined individually for each DM model considered, the results of which are shown in Table 2. This formalisation requires , which provides enough photons at high- to obtain the right value of , whilst yielding reasonably low values at later times; this point has already been noted in previous works (see e.g., Mitra et al., 2011, 2012). With the largest number of low mass halos available to provide H ionizing photons, CDM requires the most shallow -dependence on . A decrease in the number of low mass halos with decreasing requires successively larger values at early times, resulting in a steeper -dependence. Finally, varying the minimum halo mass below which the UVB photo-evaporates the entire gas content by an order of magnitude between to has a minimal effect on both and : as seen from Fig. 2, while changes by 0.005, changes by less than 0.1 dex. For this reason, we only show results for the fiducial value of in what follows.
We end by noting that while the older Planck (2014) results could have ruled out the 1.5 keV scenario all models are equally probable given the latest CMB results (Planck Collaboration et al., 2015, 2016), although the 1.5 keV model requires an exceptionally steep -evolution of .
|keV (33.3 keV)||4.1||3.7|
|keV (16.8 keV)||3.8||4.3|
|keV (6.7 keV)||4.8||6.2|
3.3. Reionization histories and sources in different DM models
As discussed in Sec. 2.2, given the source galaxy population derived from the semi-analytic model, the reionization history, expressed through the global -evolution of , is fixed once is determined. Given that CDM allows halo collapse on the smallest scales, it is the most affected by UVB feedback; the effect of UVB feedback naturally decreases with a decrease in the WDM particle mass. As shown in Fig. 4, reionization starts with the formation of the first galaxies as early as in CDM, corresponding to about 200 Myr after the Big Bang and is 50% complete by , initially driven by low mass halos (see also Bromm & Yoshida, 2011). Progressive UVB suppression of such halos with decreasing then results in halos providing most of the H ionizing photons and reionization is 90% complete by volume at .
Reionization is delayed by a small period about 60 Myr in models compared to CDM as shown in the same figure. However, this minor delay is easily compensated by the steeper behaviour (see Table 2) such that reionization is 50% and 90% complete at, almost the same epochs as in CDM, at and respectively. With its largest suppression of low-mass halos, reionization only starts at in the 1.5 keV scenario, being delayed by about 150 Myr compared to CDM. This delay in the start of reionization is compensated by (i) the larger, and hence less feedback-limited, galaxies that produce stars and H ionizing photons at a much faster rate as compared to CDM and models, and (ii) the steeper evolution required to match to the observed optical depth and emissivity constraints. The two effects results in a faster reionization in the 1.5 keV model which is 90% complete by . Interestingly, reionization finishes by about in all these different scenarios, in accord with the results of Yue & Chen (2012) who have shown that although WDM can delay the start of reionization, the end is not necessarily delayed.
With regards to the main reionization sources, we now calculate the fractional contribution of galaxies to the total ionizing emissivity at as a function of their halo masses and magnitudes. Given the similarity between CDM and 5keV demonstrated above, for clarity, we only show results for CDM, 3 keV and 1.5 keV WDM in what follows. As a function of halo mass, we find that CDM galaxies with provide only about of the total ionizing photons, with the dominant contribution coming from lower halo masses as shown in Fig. 5. Further, halos only provide of the total ionizing photons, dropping to less than 0.1 percent for halos. Given 3 keV WDM suppresses the formation of low-mass halos, galaxies provide about 63% of the total ionizing photons, with halos contributing only 15% (0.3%) to the total ionizing budget. With the suppression of all halos below , galaxies above this limit make up a 100% of the ionizing photon budget in the 1.5 keV WDM model, with halos providing an appreciable 32% of the total budget.
In terms of the observed UV magnitude, we find that an integration limit as faint as is required in order to catch the total ionizing photon budget for CDM and the 3 keV WDM models as shown in Fig. 6; integrating down to the generally used limit of provides 63% of the budget for CDM, rising to about 80% for the 3 keV scenario, with the rest coming from fainter sources. Given the lack of low-mass halos, on the other hand, galaxies brighter than provide all of the ionizing budget for the 1.5 keV model. As expected from the discussion above, massive galaxies brighter than that reside in halos (see Fig. 3; Dayal et al., 2014a) provide only about 10% of the total ionizing budget in CDM. This rises to about 20% (40%) for the 3 (1.5) keV models.
To summarize, we find the bulk of the reionization photons come from galaxies with and in CDM. The progressive suppression of low-mass halos with decreasing leads to a shift in the “reionization” population to larger halo masses of and for WDM.
4. Reionization impact on high- galaxy observables
We now quantify the impact of the UVB feedback on global observables such as the UV LF, the stellar mass density and the ionizing photon density, as well as observables measured for individual galaxies such as the stellar mass-halo mass relation. Given the small impact of changing between and on reionization noted in Sec. 3.2, we only show results for the fiducial case of in what follows.
4.1. Faint end of the UV luminosity functions
The evolving UV LF has now been measured over an unprecedented 10 (7) magnitudes at as shown in Fig. 7. At and 8, the faint end () data has been collected using the lensed Hubble Frontier fields (HFF; Atek et al., 2015; Livermore et al., 2016) while the brightest galaxies have been detected using ground-based Ultra-Vista surveys (e.g. Bowler et al., 2014b; Bradley et al., 2013); (unlensed) HST data has been used to obtain the UV LF as shown in the same figure.
We start by discussing the UV LF considering only internal (SN) feedback; this effectively implies setting when modelling the feedback effects of the UVB. As shown, the UV LFs for CDM and 3 keV WDM are in agreement at least down to at ; results from the 1.5 keV scenario are in agreement with these down to at and start peeling-away at a slightly brighter magnitude corresponding to at (see also Dayal et al., 2014b). However within error bars, all three UV LFs are in agreement with observations over the entire range probed so far. As shown in Dayal et al. (2014b), this overlap is the result of the absence of low-mass halos in WDM models being partly compensated by their younger, and hence brighter, stellar populations.
Including the maximal effects of UV feedback, i.e. and no star formation in all galaxies below the fiducial limit of at all , naturally leads the cut-off shifting to brighter luminosities by about 2-3 AB magnitudes in all DM models considered: the LFs for CDM and 3 keV WDM cut-off at while the 1.5 keV LF cuts-off at . It is interesting to see that probing down to , observations are beginning to enter the regime at which the 1.5 keV UV LF cuts-off, although all three DM models including maximal UV feedback are still in agreement with available data-sets. However, as shown in Fig. 4, roughly 85% of the IGM is ionized by volume at in the 1.5 keV scenario- this implies 15% of the volume would still be capable of hosting star formation in halos below , pushing up the LF. Interestingly, the faint end slope remains unchanged for all DM models with at , increasing to at (see also Dayal et al., 2014a), which are in excellent agreement with the values of observed at by Atek et al. (2015) and of observed at by Livermore et al. (2016).
To summarise, CDM and WDM UV LFs overlap over the entire observed range and therefore can not be used to rule out WDM models even using the faintest galaxies with detected so far, corresponding to halo masses . Further, the exact magnitude at which the LF cuts-off depends both on the DM model as well as the feedback prescription implemented - indeed, the 1.5 keV LF excluding UV feedback is essentially degenerate with that obtained in CDM including the effects of both SN and UV feedback.
4.2. Ionizing photon density
The intrinsic ionizing photon density and its -evolution is a key input for modelling reionization. We start by comparing obtained from the different DM models considering three scenarios each of which includes SN feedback: (a) ignoring UV feedback i.e. , (b) including maximal UV feedback, i.e. in halos with and (c) the more realistic case where we only suppress the gas mass of galaxies with in ionized regions. We model the third case as
Here, and represent the ionizing photon production rates including and ignoring UV feedback, respectively. This formalism leads to a situation where a progressive fraction of galaxies are photo-heating suppressed. UV feedback suppresses the largest amount of star formation and hence in CDM - as shown in Fig. 8, including UV feedback decreases by about order of magnitude in the initial reionization stages at . With a fewer number of low-mass halos, is suppressed by a factor of 3 in 3 keV WDM at the same redshifts, while 1.5 keV WDM galaxies are essentially impervious to this effect. The evolution for the fiducial model described in Eqn. 10 is tabulated in Table 3. As noted in Sec. 3.2, the delay in structure formation, and hence the intrinsic production of ionizing photons, in the 1.5 keV WDM model can be compensated by a larger at earlier epochs. Finally, we find that including/excluding UV feedback has no effect on for the massive galaxies that have been observed with . Their large masses also result in a minimal dependence on the DM model, as shown in Fig. 8 and tabulated in Table 3.
4.3. Stellar mass density (SMD)
Encoding information on the entire stellar mass assembled at any cosmic epoch, the SMD and its -evolution is a key test of theoretical models. We discuss the SMD relation considering the three key scenarios described in Sec. 4.3 above - no UV feedback, full baryonic suppression in galaxies with and the realistic scenario wherein
Here and represent the total SMD calculated using the maximal UV feedback scenario (case B above) and ignoring UV feedback (case A above), respectively. This formalism leads to a realistic situation where no galaxies are UV feedback suppressed in the initial reionization stages where . As reionization progresses, a progressively larger fraction of galaxies below are UVB suppressed, culminating in all such galaxies being fully suppressed when . The results of these calculations for all galaxies at any are shown in the left panel in Fig. 9 - as expected, including UV feedback has the largest effect on the SMD in the initial reionization stages for CDM, given it contains the largest number of low-mass halos below . The effect of UV feedback reduces with decreasing redshift since such low-mass halos lose most/all of their gas mass after a single epoch of star formation, according to our formalism. The importance of UV feedback on the SMD naturally reduces with decreasing in WDM models given the dearth of low-mass halos. As seen from the same plot, while the SMD decreases by about an order of magnitude by including UV feedback for CDM, it only changes by about 0.5 (0.2) magnitudes for 3 (1.5) keV WDM. The relation for all the above mentioned cases is detailed in Table 3. As clearly seen, the slope of the relation is extremely steep for the 1.5 keV model compared to CDM or 3 keV. As noted in (Dayal et al., 2014b) these relations are independent of the free-parameter (star formation efficiency threshold and fraction of SN energy coupling to winds) values used. Indeed, in the same work, the authors have shown that with its capability of pinning down the slope evolution of the SMD down to , the James Webb Space Telescope will be invaluable in differentiating between CDM and 1.5 keV WDM cosmologies.
Finally, we check that the SMD from all 3 DM models are in accord with observed data, as shown in the right panel of the same figure. The observed SMDs have been inferred by integrating down to a limit of . As expected these galaxies, corresponding to , are large enough to be impervious to the effects of UV feedback, as a result of which . The theoretical relations for observed galaxies are also tabulated in Table 3.
4.4. Stellar mass-halo mass relations
We now discuss the relation between the stellar and halo mass for high- galaxies for the three DM models considered. As expected, the value of increases with increasing halo mass for all models for , signifying their larger star formation efficiencies. Starting by ignoring UVB feedback, in CDM, galaxies have - this rises by a factor of about 7 to for halos, with the slope flattening off above this value. While the value is independent of for galaxies, corresponding to , decreases with decreasing for lower mass halos by about 0.3 dex from to 7. This is a result of SN-driven winds progressively decreasing the gas mass, and hence star-formation capabilities, in low-mass halos through time. While a similar behaviour is observed for galaxies in 1.5 keV WDM, lower mass galaxies exhibit higher values at earlier cosmic epochs as a result of the larger (initial) potential wells - rises from about 0.2% at to about 0.6% at for galaxies in this model, compared to the marginal rise from 0.2% to 0.3% seen in CDM at the same redshifts.
Including the maximal UV feedback scenario (i.e. for halos) naturally results in a decrease in the ratio for low mass halos since their (gas devoid) progenitors are unable to form stars and build up their stellar content. This results in the values of low-mass () galaxies decreasing by about 0.3 dex in CDM and about 0.15 dex for 1.5 keV WDM at . We find that the effect of UV feedback loses its impact with decreasing as a result of low-mass progenitors losing gas due to SN feedback. While WDM galaxies show the same trends of with halo mass and , there are quantitative differences: firstly, the trend does not show any flattening at large masses which is a result of galaxies starting from larger, and hence less feedback-limited, progenitors in such models compared to CDM. Secondly, at a given halo mass, while the CDM and 3 keV relations are quite similar, the amplitude of this relation is higher by about 0.2-0.4 dex in the 1.5 keV scenario which is again driven by the fact that starting from larger progenitors, galaxies of a given halo mass are less feedback limited and hence able to assemble larger stellar masses. We end by noting that the “true” stellar mass value should lie between these limits of zero and maximal UV feedback; while most low-mass galaxies would be unaffected by UV feedback in the initial reionization stages, the results would move closer to the maximal UV feedback scenario in the end reionization stages.
We then study the evolution of the relation in different DM cosmologies, shown in Fig. 11. As seen from this plot, CDM shows the lowest value at any - this is the result of stronger feedback decreasing the total stellar mass of low-mass halos. As expected, the value increases with decreasing at any redshift. Indeed, integrating down to a limit of , CDM galaxies show a slope proportional to including (excluding) UV feedback. The corresponding, steeper, 1.5 keV WDM slopes have a value of about , given the minimal effect of UV feedback on galaxies in this model. As expected, the slopes steepen in all cases integrating down to a fainter magnitude limit of . In this case, the CDM slopes correspond to with (without) UV feedback which steepen to for the 1.5 keV model as tabulated in Table 3. Current estimates on the stellar mass have an error of about 0.3 dex, with abundance matching techniques, to infer halo masses, resulting in errors of the order of 0.5 dex. While current stellar mass-halo mass observations are probably unable to distinguish between these different slopes, more precise estimates, of both stellar and halo masses, in the future might offer a useful tool to distinguish between CDM and 1.5 keV WDM.
5. Conclusions and discussion
Although the Cold dark matter (CDM) paradigm has been enormously successful in explaining the large scale structure of the universe, it shows a number of small scale problems that can be alleviated by invoking ( keV) Warm Dark Matter (WDM) particles that erase small scale power, suppressing the formation of low-mass structures. We use a semi-analytic model traces the DM and baryonic assembly of high- () galaxies in 4 DM cosmologies - CDM and WDM with and - including the key baryonic processes of star formation, SN feedback and the merger/accretion/ejection driven evolution of gas and stellar mass. This model uses only two and mass-independent free parameters with fiducial values of 3.8% for the maximum star formation efficiency and 10% of the SN energy going into unbinding gas (Dayal et al., 2014a, b). Modelling reionization requires an additional free parameter which is the fraction of ionzing photons that can escape out of the galactic environment, . In this work, we also include the effects of external feedback from a UVB suppressing the baryonic content of galaxies below a threshold mass, for which we use a fiducial value of . The model is constrained using key observables for high- galaxies (UV LF, SMD, M/L ratios), the latest constraints on the cosmological electron-scattering optical depth () from Planck and the measured ionizing photon emissivity () to understand: (i) the reionization history and sources , and (ii) the effects of UV feedback on galaxy assembly in different DM cosmologies. Our main findings can be summarized as follows:
Although the start of the reionization process is delayed in light WDM models, a faster galaxy assembly and a steeper redshift dependence of result in reionization ending at almost the same epoch in all DM models (see also Yue & Chen, 2012). In terms of reionization sources, the bulk of the reionization photons come from galaxies with and in CDM. The progressive suppression of low-mass halos in light WDM models results in a shift in the reionization population to larger halo masses of and for WDM.
We quantify the effects of UV feedback on observables including the evolving UV LF, SMD, and the ratio and show that its effects are the most pronounced for CDM (with the largest number of low-mass halos) and decrease with a decreasing WDM particle mass. We find that even in the maximal UV feedback scenario (where galaxies have no gas content), all DM models (CDM, 3 and 1.5 keV WDM) are in agreement with each other down to the current observational limits of .
Interestingly, the UV LFs obtained from the different DM models considered are degenerate with the physics implemented - for example, the 1.5 keVLF excluding UV feedback is indistinguishable from that obtained in CDM including the effects of both SN and UV feedback. We find that the SMD and values are indistinguishable amongst the different DM models for observed galaxies, with the differences being the most pronounced for the smallest halos.
We find that, in addition to the slope of the relation which will be measured with the forthcoming JWST (Dayal et al., 2014b), another global estimate is the redshift evolution of the relation that can potentially be used to put constraints on the DM particle mass.
We end with a few caveats. Firstly, we have assumed reionization to be solely driven by the stars within galaxies. In recent times, it has been proposed that a population of relatively faint quasars at high- can complete reionization without any significant contribution from galaxies (Madau & Haardt, 2015; Khaire et al., 2016; Mitra et al., 2015; D’Aloisio et al., 2016). It would be interesting to explore the contribution of these quasars in the framework of the WDM models and see if these objects can relax the rapidly evolving required from 1.5 keV WDM models. Secondly, we have ignored SN radiative losses that could significantly reduce the total energy available to drive winds. However, a decrease in this total energy could be countered by scaling up the fraction of the total energy we put into driving winds from the fiducial value of 10% used in this work. Thirdly, Pawlik et al. (2009) have shown that UVB photo-heating reduces the clumping factor of the IGM since the additional pressure support from reionization smoothes out small-scales density fluctuations. While we have used their results for an over-density of 100, we have confirmed that our results do not change using threshold values of 50, or 200. Fourthly, while we assume that all halos with mass are feedback suppressed, we find our results are equally consistent with observations ( and ) using values ranging between . Finally, we have made the simplifying assumption of using a mass-independent at all . This is partly motivated by the uncertainty regarding the mass-dependence of : while some authors find to decrease with an increase in the halo mass (Razoumov & Sommer-Larsen, 2010; Yajima et al., 2011; Ferrara & Loeb, 2013), other works have found the opposite trend (Gnedin et al., 2008; Wise & Cen, 2009). Significant progress is expected to be made by comparing our model predictions with further LBG data expected from the Frontier Fields, and from forthcoming observatories such as the JWST. Furthermore, if the claim of keV X-ray line holds to be true, it will usher in an exciting era for WDM dominated cosmology.
|DM Model||SN feedback||UVB feedback||UV cut||relation at|
|3 keV WDM||yes||no||all galaxies|
|1.5 keV WDM||yes||no||all galaxies|
|3 keV WDM||yes||yes||all galaxies|
|1.5 keV WDM||yes||yes||all galaxies|
|SN feedback||UVB feedback||UV cut||relation|
|CDM||yes||yes||all galaxies (Eqn. 11)|
|3 keV||yes||yes||all galaxies|
|3 keV||yes||yes||all galaxies (Eqn. 11)|
|1.5 keV||yes||yes||all galaxies|
|1.5 keV||yes||yes||all galaxies (Eqn. 11)|
|1.5 keV keV||yes||yes|
|SN feedback||UVB feedback||UV cut||relation|