21-cm observations and warm dark matter models

TL;DR

This paper explores how 21-cm observations can inform us about warm dark matter models, showing that current bounds are weaker than other methods and that sterile neutrino models remain viable.

Contribution

It demonstrates that 21-cm data, when considering baryonic physics, provide weaker constraints on warm dark matter than traditional structure formation methods.

Findings

Current 21-cm bounds are weaker than Lyman-alpha constraints.

Sterile neutrino dark matter models are consistent with 21-cm data.

Holistic modeling of WDM could enhance future dark matter studies.

Abstract

Observations of the redshifted 21-cm signal (in absorption or emission) allow us to peek into the epoch of "dark ages" and the onset of reionization. These data can provide a novel way to learn about the nature of dark matter, in particular about the formation of small size dark matter halos. However, the connection between the formation of structures and 21-cm signal requires knowledge of stellar to total mass relation, escape fraction of UV photons, and other parameters that describe star formation and radiation at early times. This baryonic physics depends on the properties of dark matter and in particular in warm-dark-matter (WDM) models, star formation may follow a completely different scenario, as compared to the cold-dark-matter case. We use the recent measurements by the EDGES [J. D. Bowman, A. E. E. Rogers, R. A. Monsalve, T. J. Mozdzen, and N. Mahesh, An absorption profile centred at 78 megahertz in thesky-averaged spectrum,Nature (London) 555, 67 (2018).] to demonstrate that when taking the above considerations into account, the robust WDM bounds are in fact weaker than those given by the Lyman-$\alpha$ forest method and other structure formation bounds. In particular, we show that resonantly produced 7 keV sterile neutrino dark matter model is consistent with these data. However, a holistic approach to modelling of the WDM universe holds great potential and may in the future make 21-cm data our main tool to learn about dark matter clustering properties.

Equations

$$f_{\mathrm{coll}}(z)=\frac{1}{{\rho }_m}{\int }_{M_{\min }}^{\infty }dM\frac{dn}{d\ln M},$$

$$\begin{gathered} M_{min}=1.0\times 10^8{\left(\frac{1+z}{10}\right)}^{-3/2}{\left(\frac{\mu }{0.6}\right)}^{-3/2} \\ \times {\left(\frac{T_{\text{vir}}}{1.98\times 10^4\mathrm{K}}\right)}^{3/2}{\left(\frac{{\Omega }_m}{{\Omega }_m^z}\frac{{\Delta }_c}{18{\pi }^2}\right)}^{-1/2}{\mathrm{M}}_{\odot }/h, \end{gathered}$$

Full text

The hyperfine splitting of the lowest energy level of the neutral hydrogen atom leads to a cosmic 21-cm signal thanks to the abundance of primordial hydrogen. The 21-cm signal from the post-reionization Universe has been studied by a number of experiments (e.g., LOFAR Patil et al. (2017); Gehlot et al. (2019), GMRT Paciga et al. (2013), PAPER Ali et al. (2015) (see however Ali et al. (2018)), MWA Ewall-Wice et al. (2017)), but the only tentative detection of the 21-cm signal in absorption against the CMB background at $z\sim 16-19$ has recently been claimed by the EDGES experiment Bowman et al. (2018).111Note however that the result is still uncertain, and there are alternative, noncosmological explanations Hills et al. (2018); Bradley et al. (2019). It is clear that the forthcoming experiments, such as the staged HERA DeBoer et al. (2017) or future SKA Koopmans et al. (2015); Bull et al. (2018) will offer detailed information about the distribution of the 21-cm signal, thus allowing for the full 3D tomography of the signal, offering an unprecedented reach into the early Universe. This makes the study of the 21-cm signal a promising tool to learn not only about cosmological parameters (see, e.g. McQuinn et al. (2006); Mao et al. (2008); Oyama et al. (2016)) but also about different properties of dark matter, including its decays and annihilations D’Amico et al. (2018); Yang (2018); Clark et al. (2018); Liu and Slatyer (2018); Cheung et al. (2019); Mitridate and Podo (2018), dark matter-baryon interactions Barkana (2018); Fialkov et al. (2018); Berlin et al. (2018); Barkana et al. (2018); Fraser et al. (2018); Slatyer and Wu (2018), and the formation of gravitationally bound structures Madau et al. (1997); Furlanetto et al. (2006); Zaldarriaga et al. (2004); Ciardi and Ferrara (2005); Pritchard and Loeb (2012); Barkana (2016).

In this work we focus on the global (sky-averaged) 21-cm absorption signal that appears when the spin temperature (logarithm of the ratio of population of two levels of the hydrogen’s $1S$ state) becomes smaller than the CMB temperature (for a review see, e.g., Furlanetto et al. (2006); Pritchard and Loeb (2008, 2012)). The standard explanation for this difference of temperatures is the presence of a bath of Ly-$\alpha$ photons which induce transitions between $1S_{1}$ and $1S_{3}$ levels: Ly-$\alpha$ pumping. Therefore, a detection of the global 21-cm absorption signal at some redshift $z_{0}$ implies that sources of radiation have already been active at that epoch.

With our current knowledge of baryonic physics, we can robustly state that such radiation sources can only form inside dark matter overdensities. Hence, to predict the 21-cm signal one has to follow several steps:

1. a)

Start from the description of bound gravitational structures at a given redshift $z$. 2. b)

Continue with the description of how baryons collapse into these structures (which depends both on the size of the structures, on redshift and on cosmology). 3. c)

Assuming a particular type of radiation sources (as they cannot be modeled from first principles), estimate the number of produced photons and model (usually through a combination of semianalytical and numerical methods) how radiation escapes from the bound structures and heats the ambient medium; 4. d)

Given the resulting function of radiation density $d\rho_{\rm rad}/dz$ one can then use available codes (such as ARES Mirocha (2014) or 21CMFAST Mesinger et al. (2011)) to predict the 21-cm signal.

Uncertainties as well as differences in predictions, between DM models are introduced at every step in this process.

(a) Bound DM structures.

Historically, the first warm dark matter models were those of sufficiently massive Standard Model neutrinos (see, e.g, Bond et al. (1980)). Such particles were in thermal equilibrium in the early Universe and froze out while still being relativistic. They remained relativistic for some period in the radiation dominated epoch and homogenized primordial density perturbations on scales below the free-streaming horizon, $\lambda_{\rm fs}$ (for a proper definition see, e.g., Boyarsky et al. (2009a, 2019)). The number density of such WDM thermal relics is uniquely determined by the temperature of freeze-out or, equivalently, by their mass, $m_{\rm TH}$. This mass of the thermal relic is the most typical parametrization of the WDM models.222An alternative parametrization is given by the mass of non-resonantly sterile neutrinos Dodelson and Widrow (1994). The two models lead to an almost identical shape of the matter power spectrum and therefore their masses are related to one another in a nonlinear way; see Viel et al. (2005); Boyarsky et al. (2009a) for details. In this work we always indicate what definition of mass we are using. All WDM models have suppressed (as compared to CDM) number of halos with masses below the free-streaming cutoff scale, $M_{\rm cut}=\frac{\pi}{6}\lambda_{\rm fs}^{3}$, where $\lambda_{\rm fs}$ is the free-streaming horizon (see, e.g., Boyarsky et al. (2019)). This leads to a large difference between a number of collapsed halos, especially at high redshifts, between CDM and WDM models (see Fig. 1 for our halo mass functions calculated by using the standard prescription proposed in Benson et al. (2013), also fully consistent with Fig. 1 of Schneider (2018)). Naively, one could also expect a big difference between two models in terms of produced starlight. However, only the halos with masses down to $10^{7}-10^{8}\,\mathrm{M_{\odot}}/h$ contribute to the formation of stars in CDM at redshifts of interest. Indeed, these masses correspond to virial halo temperatures $\sim 10^{3}-10^{4}\,\mathrm{K}$ – temperatures that are needed for the hydrogen to cool sufficiently fast, in order to collapse and form compact radiative sources Haiman et al. (2000); Barkana and Loeb (2001), see Eq. (2) below.

In addition to halos another bound DM structures – filaments – can exist in the early Universe. Near the cutoff mass formation of filaments and their subsequent fragmentation may be the dominant structure formation process in WDM Gao and Theuns (2007); Paduroiu et al. (2015), as opposed to the CDM model. The impact of filaments on the 21-cm signal is studied by Leo et al. (2019) (see also Chatterjee et al. (2019)), with the outcome that the lower bound on the WDM mass should be weakened compared with $\gtrsim 6$ keV in earlier works Schneider (2018); Lopez-Honorez et al. (2019) that did not take into account this effect. In addition to this difference, the presence of filaments also interferes with the structure formation processes, as discussed below.

(b) Baryonic collapse and star formation in different DM universes

In general, the naive expectation that what is known from CDM simulations would also apply to WDM universes does not hold up. Let us point out two remarkable differences between star formation in CDM and WDM.

First, in WDM universes star formation in filaments may dominate over star formation in halos at redshifts $z\gtrsim 6$ Gao and Theuns (2007); Gao et al. (2015), producing different populations of stars and different amounts of Lyman-$\alpha$ photons. The star-formation efficiency of these processes is still highly uncertain, but it is clear that they can play a role. Such a mechanism is absent in CDM.

Second, both hydrodynamical simulations of galaxy formations (cf. Herpich et al. (2014); Maio and Viel (2015); Colin et al. (2015); Power and Robotham (2016); Lovell et al. (2017a)) and semianalytical models (cf. Menci et al. (2012); Kang et al. (2013); Menci et al. (2013); Nierenberg et al. (2013); Lovell et al. (2016, 2017b)) are tuned to reproduce galaxy observables (e.g., luminosity or stellar mass functions, etc.) at $z=0$. Not surprisingly this leads to galaxy populations in CDM and WDM having similar properties in recent epochs Wang et al. (2017). However, in order to achieve this agreement one has to choose quite different star-formation prescriptions in CDM and WDM at high redshifts Wang et al. (2017) , especially for halos close to $M_{\rm fs}$ Bose et al. (2016). As the halo formation in the WDM Universe often starts later, one generically requires higher star-formation efficiencies for WDM (consistent with what we infer in our work).

(c) Modeling radiation.

According to the well-developed theory of the 21-cm signal in the early Universe (see, e.g., Pritchard and Loeb (2012)), the key driver of the timing of 21-cm absorption is the emission rate of Ly-$\alpha$ photons that excite the electrons in hydrogen and result in a spin flip of such electrons after deexcitations (Ly-$\alpha$ pumping). The most common mechanism for emitting Ly-$\alpha$ photons at high redshifts is early star formation Pritchard and Loeb (2012) (note however that the QSO contribution can also be significant; see, e.g., Ross et al. (2019)).

In a CDM universe, the bulk of stars is formed in halos. Therefore, the star-formation rate density is usually parametrized by the ansatz (see, e.g., Pritchard and Loeb (2012); Mirocha (2014); Safarzadeh et al. (2018); Schneider (2018); Lopez-Honorez et al. (2019)) $\dot{\rho}_{*}(z)=f_{*}\bar{\rho}_{{\rm b},0}\dot{f}_{\rm coll}(z),$ for redshift $z$, star density (calculated in comoving volume) $\rho_{*}$, $\dot{\,}\equiv\frac{d}{dt}$ with time $t$, $\bar{\rho}_{{\rm b},0}$ the homogeneous baryon density today, $f_{\rm coll}(z)$ the fraction of baryons in collapsed structures, and $f_{*}$ the fraction of collapsed baryons that form stars.

The fraction $f_{\rm coll}(z)$ is derived from the halo mass function of a model as

[TABLE]

with a cutoff for halos below mass $M_{\rm min}$ which are expected not to be able to form stars. This cutoff is set by the halo’s virial temperature $T_{vir}$, the temperature which the gas reaches during the virialization of the halo (Barkana and Loeb, 2001):

[TABLE]

where $z$ is the halo redshift, $\mu\simeq 0.60$ is the mean molecular weight, $\Omega_{m}^{z}=1-\Omega_{\Lambda}/[\Omega_{m}(1+z)^{3}+\Omega_{\Lambda}]$ and $\Delta_{c}=18\pi^{2}+82(\Omega_{m}^{z}-1)-39(\Omega_{m}^{z}-1)^{2}$ (Bryan and Norman, 1998). Depending on which mechanism is responsible for cooling, this cutoff may vary: atomic cooling is associated with a cutoff $T_{vir}\simeq 10^{4}$ K, while molecular cooling leads to a cutoff $T_{vir}\simeq 10^{3}$ K, see, e.g., Fig. 12 of Barkana and Loeb (2001). The consequences of this parameter are discussed later, and visualized in Fig. 1.

Galaxies or galaxy candidates have been observed for $z\lesssim 10$ Oesch et al. (2018), and we can only extrapolate the aforementioned ansatz for the redshifts of interest. The star-formation efficiency in halos can be estimated from the observed ultraviolet luminosity function (UV LF) (see, e.g., (Dayal et al., 2014; Sun and Furlanetto, 2016; Mirocha et al., 2017; Mirocha and Furlanetto, 2019; Park et al., 2019)). The dependency $f_{*}(M,z)$ on halo mass and redshift relies on the model of star formation, and possible values of $f_{*}$ vary in a wide range. For example, in CDM halos $f_{*}$ may reach $0.3$ at $z=5-8$ for $10^{11}-10^{12}\,\mathrm{M_{\odot}/h}$ halos, increase with redshifts, and be close to unity during the Dark Ages (Sun and Furlanetto, 2016). In addition the observational estimates of star-formation efficiency depend on assumed cosmology and $f_{*}$ in low-mass galaxies may be higher in WDM compared to CDM (see, e.g., Sawala et al. (2015); Corasaniti et al. (2017); Menci et al. (2018)).

Apart from observations, $f_{*}$ can be predicted in CDM by use of detailed numerical simulations of the Universe during redshifts $z\sim 6-15$ Sawala et al. (2015); Wise et al. (2014); Xu et al. (2016); Ma et al. (2018); Rosdahl et al. (2018); Sharma and Theuns (2019). However, there is a three-orders-of-magnitude scatter among the values of $f_{*}$ in individual simulated galaxies. As Figs 15 and 16 of Xu et al. (2016) demonstrate, a few galaxies with $f_{*}\simeq 0.3$ produce an amount of starlight which is several times larger than that of the bulk of galaxies with $f_{*}\simeq 0.01$. As a result, it is currently impossible to derive a robust constraint on $\dot{\rho}_{*}(z\sim 17)$.

An escape fraction of ionizing photons in galaxies during the reionization and Dark Ages has not been determined directly and is still uncertain (see, e.g., Sec. 7.1 in Dayal and Ferrara (2018)). However, varying the ionizing photon escape fraction in a wide range does not change the redshift of the 21-cm absorption signal significantly. The escape fraction of photons in the band $10.2-13.6$ eV is usually assumed to be close to unity (see Sec. 3.5 of Mirocha et al. (2017) and references therein).

(d) Predicting the 21-cm signal

The above-mentioned uncertainty on $f_{*}$ translates into a strong systematic uncertainty on WDM parameters that can be probed with a 21-cm absorption signal. In order to demonstrate this, we computed the 21-cm absorption signal using the ARES code for three models: CDM, thermal relics with a mass $m_{\text{TH}}=6$ keV (claimed to be excluded in Schneider (2018); Lopez-Honorez et al. (2019)) and the resonantly produced sterile neutrino, with particle mass of 7 keV and lepton asymmetry $L_{6}=10$.333Lepton asymmetry $L_{6}\equiv 10^{6}(n_{\nu_{e}}-n_{\bar{\nu}_{e}})/s$, where $n_{\nu_{e}}$ and $n_{\bar{\nu}_{e}}$ are the number densities of electron neutrinos and antineutrinos, and $s$ is the total entropy density in early Universe Laine and Shaposhnikov (2008); Boyarsky et al. (2009b) This sterile neutrino model is consistent with all astrophysical and cosmological bounds: x-ray bounds on decaying DM Bulbul et al. (2014); Boyarsky et al. (2014, 2015); Iakubovskyi et al. (2015); Ruchayskiy et al. (2016); Franse et al. (2016); Drewes et al. (2017); Abazajian (2017); Boyarsky et al. (2019), suppression of the power spectrum as inferred from the Lyman-$\alpha$ forest Garzilli et al. (2017); Baur et al. (2017); Garzilli et al. (2019), cosmic reionization Rudakovskiy and Iakubovskyi (2016); Bose et al. (2016); Rudakovskyi and Iakubovskyi (2019), and Milky Way satellite and galaxy counts Lovell et al. (2016, 2017a).

The results are shown in Fig. 2. The results strongly depend on the range of assumed values of $f_{*}$. From the discussion above we see that it should be at least from $f_{*}\simeq 0.01$ to $f_{*}\simeq 0.3$ (see, e.g., Xu et al. (2016)). We see that for $f_{*}=0.09$ in both 7-keV sterile neutrinos and thermal relics with $m_{\text{TR}}=6$ keV, the minimum of $\delta T_{b}(z)$ happens around $z=17$, in agreement with the EDGES results. On the contrary, taking $f_{*}=0.03$ (as done in Schneider (2018)) would make CDM consistent with the EDGES data, while the two WDM models would have an insufficient number of Lyman-$\alpha$ photons at the redshifts of interest.

In Fig. 3 we plot the range of $f_{*}$’s that have the minimum of the absorption trough for $15.8\leq z\leq 18.7$. We see that starting from $m_{\text{TR}}\leq 4$ keV $f_{*}$ can be as large as 100% and that for masses of this order or above.

Given several orders of magnitude uncertainties in $f_{*}$ (as discussed above), the only robust bound can be obtained if one chooses $f_{*}=1$; at most all baryons enter star formation.

In this case, for example, thermal WDM masses as light as $m_{\text{TR}}\geq 2$ keV cannot be excluded (see Fig. 4). This puts the sensitivity of the EDGES signal in line with a number of previous bounds on WDM parameters (see, e.g., the Lyman-$\alpha$ constraints Garzilli et al. (2017), taking into account proper marginalizations over possible thermal histories; bounds Menci et al. (2016) from counting of high-$z$ galaxies; bounds Birrer et al. (2017); Vegetti et al. (2018) from strong gravitational lensing; bounds Lovell et al. (2014); Kennedy et al. (2014) from the Milky Way satellite counts, etc.). As Mesinger et al. (2013) demonstrates, future measurements of star-formation efficiency at high redshifts, as well as the 21-cm power spectrum, are required to improve the sensitivity for WDM particles.

In this paper we have concentrated on the redshift position of minimum of $\delta T_{b}(z)$ as an indicator of star-forming processes at high-redshifts. However, both the depth of the 21-cm absorption trough and its width carry important information about the underlying physics.

Much like the position, the width of the obtained profile also depends on the cosmology. When using $T_{vir}=10^{3}$ K (molecular cooling) and ignoring possible suppression due to the Lyman-Werner radiation background (see, e.g., Yue and Chen (2012)), we see that CDM predicts an absorption-trough width which is larger than the one observed by the EDGES experiment, Fig. 5. For the WDM and $\nu$MSM profiles the molecular cooling brings little to no effect due to the lack of substructures of the mass $\sim M_{\rm min}$.

The depth of the observed trough is much greater than what any of the models discussed in this paper predict. To date, only additional nongravitational baryon-DM interactions can accommodate such a strong spin-temperature cooling, which is beyond the scope of this paper Barkana (2018); Berlin et al. (2018); Muñoz and Loeb (2018); Fialkov et al. (2018).

To summarize, we discussed the large uncertainty in star formation at very high redshifts ($z\sim 17$), which are probed by recent EDGES observations of the global 21-cm signal. As a consequence, using only this signal it is impossible to robustly constrain the parameters of dark matter models, such as the mass of the warm dark matter particle. Conversely, various DM models need distinct star-formation scenarios to fit the signal. Detailed future studies of star formation at very high redshifts ($z\gtrsim 10$), together with detailed modeling of structure assembly and early star formation, will reduce the existing uncertainties. Ongoing and future studies of the 21-cm signal remain promising tools for inferring the key dark matter parameters.

Acknowledgements. We thank Tom Theuns for valuable comments on an earlier version of this paper and the authors of Leo et al. (2019) for sharing with us results of their work before publication. The work of D.I. and O.R. œwas supported by the Carlsberg Foundation. The work of A.R. was partially supported by grant for Young Scientists Research Laboratories of the National Academy of Sciences of Ukraine. A.R. also acknowledges the grant from the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 Research and Innovation Programme (GA 694896).