The impact of EDGES 21-cm data on dark matter interactions

The recently announced results on the 21-cm absorption spectrum by the EDGES experiment can place very stringent limits on dark matter annihilation cross-sections. We properly take into account the heating energy released from dark matter annihilation from the radiation epoch to the 21-cm observation redshifts in the radiative transfer to compute the evolution of the gas temperature. Our results show that the global 21-cm absorption profile is a powerful cosmological probe of the dark matter interactions. For dark matter annihilating into electron-positron pairs, the EDGES results give a more stringent upper limit than the PLANCK result on the annihilation cross section at the lower dark matter mass region.


I. INTRODUCTION
The Big Bang theory is the most accepted theory for the beginning of the Universe. The theory has been established by astrophysical and cosmological observations, especailly the cosmic microwave background (CMB) in 90's [1]. The CMB, also called the "afterglow" of the Big Bang, was the earliest light coming out from the soup of free electrons, protons, neutrons, and nuclei. Only when the free electrons were caught by protons, neutrons, and nuclei to form neutral atoms, mostly hydrogen and helium, the photons can shine through the matter and form the CMB. It happened about 380,000 years after the Big Bang. Such footprints of the early Universe can tell us a lot of information about the ingredients of the Universe.
Subsequently, the WMAP in the last decade pinned down the valid parameters of the most popular cosmological model, namely, the ΛCDM model [2]. The Planck collaboration [3] continued the mission of measuring more precisely the model parameters in this decade, and particularly provided the measurements of the CMB polarization with improved precision.
After the recombination epoch, the Universe entered into the so-called dark ages, containing mainly neutral hydrogen gas. It is believed that during this period the dark matter (DM) first began to cluster to form halos that accrete normal matters. The Universe then entered into the "reionization" epoch when the first galaxy or star was being formed in the halo. The Gunn-Peterson test on quasar absorption lines has revealed that the Universe has been reionized at redshift z ∼ 7 [4]. More precise measurements of the reionization bump in the CMB E-mode polarization power spectrum have inferred an optical depth integrated to the recombination, τ ∼ 0.08, and a full reionization at z ∼ 9 [3]. Nevertheless, how the reionization happened and when exactly it happened are uncertain. It is generally believed that radiation from first stars and/or galaxies ionize the neutral gas beginning at z ∼ 15 − 20 [5].
A direct measurement of the reionization is made possible owing to the hyperfine splitting of neutral hydrogen atoms, which emit or absorb photons of wavelength at 21-cm in the rest frame dependent on the spin temperature of the hydrogen gas, T s . This spin temperature is determined by the background temperatures of the CMB (T γ ) as well as the surrounding thermal gas (T g ). Thus, measurements of redshifted 21-cm lines allow us to perform a tomographic study of the reionization process. There have been a lot of observational efforts to map the primeval hydrogen gas distributions; current experiments include LOFAR [6], MWA [7], SKA [8], and many more [5]. Recently, the EDGES experiment has detected the global signal of 21-cm absorption [9], if confirmed, opening a new chapter of the 21-cm cosmology.
EDGES has claimed that the measured amplitude of the 21-cm absorption profile is more than a factor of two greater than the largest predictions [9]. To explain this anomaly, one may have to reduce the spin temperature T s by introducing novel photon-DM interactions to cool down the gas [10][11][12][13][14][15]. An alternative way is to add a strong radio background such that the effective T γ is higher than the original CMB temperature [16][17][18][19].
In this work, instead of explaining the anomaly, we will use the EDGES result to constrain weakly-interacting massive DM (WIMP) annihilation in the early Universe. In our approach we assume that the known or unknown sources more or less reproduce the EDGES signal.
The DM annihilation that we are considering is additional, though small, contributions to the EDGES signal. We can then use the uncertainty of the EDGES to constrain the DM interaction. Namely, the allowed parameter space of WIMP annihilation totally depends on the size of the EDGES experimental uncertainties.
Generally speaking, as far as the WIMP annihilation is considered, one may naturally wonder if there is a sizable scattering cross section between DM and SM particles. Especially, they usually share the same couplings. If the size of the scattering cross section between DM and proton is considerable, the energy of gas can be taken away by the scattering process.
However, a null signal of the traditional WIMP (having mass between GeV to several TeV) has been obtained by current XENON1T [20] and PandaX [21] and hence a severe limit has been reported. With such a low scattering cross section, we therefore do not need to consider the effect of gas cooling resulting from WIMP-baryon scatterings in the reionization epoch.
If the DM annihilates into standard model (SM) particles such as quarks, leptons, and photons, they will heat up the gas and increase the gas temperature T g . Since T s somewhat traces T g , the DM annihilation will affect the evolution of T s and hence the T 21 signal probed by EDGES. One may suspect that any limits on the DM interaction are entirely degenerate with the unknown mechanism that produces the EDGES absorption feature. Firstly, the DM cooling of the gas has been severely constrained [13,15], it is unlikely that it can further compensate the heating by DM annihilation. Secondly, although increasing T γ also cancels the effect of DM heating of the gas, the amount of radiation excess present at z ∼ 15 − 20 would contribute to the radio radiation today and thus can be probed by measurements of the radio background. This has been studied most recently in Ref. [22], where they have found support for the existence of a strong diffuse radio background and suggested that additional radio data would help understanding the Galactic foregrounds so as to constrain the excess T γ . Thus, the DM annihilation rate would be excluded as T s is raised to a value beyond the experimental uncertainty of the EDGES absorption profile. Even though the EDGES signal turns out to be false or rectified to a predicted level, it is indeed its experimental sensitivity that can be used as a gauge for constraining DM annihilation.

II. METHODOLOGY
EDGES has recently measured an absorption feature for 21-cm emission [9]. At the redshift z = 17.2, the temperature T 21 at 99% confidence level (C.L.) is reported by where the errors +200 −500 mK present the systematic uncertainties. On the other hand, the theoretical prediction is given by where Ω b h 2 and Ω m h 2 are the relic densities of baryon and matter, respectively. The number fraction of neutral hydrogen x HI is approximately equal to 1 − x e , where x e is the ionization fraction. The photon temperature T γ (z) can be the same as the CMB temperature, T CM B = 2.7(1+z)K, and the spin temperature T s (z) controls the atomic excitation between the s = 0 and s = 1 states of the neutral hydrogen. The value of T s (z) lies between the gas temperature and the CMB temperature. Therefore, the precise value of T s (z) would be sensitive to the gas heated by DM annihilation.
In the early Universe, DM annihilates or decays into SM particles and ultimately produces some amount of energetic electrons/positrons (e − /e + ) and gamma rays (γ). Those of highenergy e ± and γ can ionize, heat, and excite the hydrogen atoms. We follow the formalism and methodology developed in Ref. [23][24][25], which does not rely on any assumption on the energy fractions f eff [26,27] as used in the literature related to CMB constraints on DM annihilation. The DM contribution to the ionization fraction of hydrogen atom (x e ) via DM annihilation is and the gas temperature T g is modified as where the subscript 'ch' represents the DM annihilation or decay channels with the branching ratio BR ch and i or h refers to ionization or heating. Note that in the above expressions the energy is injected by DM annihilation at redshift z while absorbed by the neutral hydrogen at redshift z. We define the parameters: the Rydberg energy E Ry = 13.6 eV, the DM mass m χ , the DM annihilation cross section σv , the DM number density n χ , and the number density of hydrogen atoms n H . The fraction of injected energy for a given m χ and an annihilation channel 'ch' can be obtained by integrating out the energy E, where the injected energy fractions for electron and photon are, respectively, given by Ref. [23][24][25]. The energy spectrum per DM annihilation at the source, dNe(ch,mχ) dE for electrons and dNγ (ch,mχ) dE for photons, are calculated by LikeDM [28] by using the tables from PPPC4 [29]. Our calculation of gas temperature evolution is similar to those in Refs. [34,35,47], except that we are using a set of tables for the transfer functions derived in Ref. [23][24][25], but Refs. [34,35,47] used the publicly available results of f eff in Ref. [27].
Similar to Ref. [30], we also introduce a cosmological boost factor B to account for the effect of DM inhomogeneities and structures [31], In Ref. In this paragraph, we would like to explicitly demonstrate how we compute the theoretical prediction of the DM modified T 21 signal. First of all, after all the DM ingredients are included, we insert Eqs. (3) and (4) into the RECFAST code [32] to obtain the DMmodified matter temperature T g (z). Secondly, we perform the RECFAST computation again by switching off the DM contribution in order to calculate matter temperature without DM annihilation. Then, comparing the matter temperature from both scenarios, one can obtain the change of T g by DM annihilation δT g . This quantity is useful in the next step. Before performing the third step, let us make a reasonable assumption that T s is fully coupled to The best-fit model for T 21 (lower panel) given in the extended data in Fig. 8 of Ref. [9].
The upper panel presents the T s which is converted from the best-fit model of T 21 by using Eq. (2).
step, we can simply obtain the DM-modified spin temperature by adding the difference δT g to the best-fit spin temperature T BF s which is converted from the EDGES best-fit T 21 signal shown in Fig. 2. Finally, plugging this new spin temperature T BF s + δT g into Eq.
(2), one can simply compute DM modified T 21 without involving any unknown astrophysical source other than the DM annihilation. In some sense, our approach can be treated as a backgroundfree approach because all the known or unknown astrophysical sources are absorbed in the EDGES best-fit T 21 signal.
Assuming that DM annihilation can only contribute sub-dominantly to the EDGES signal, one can invert Eq. (2) to obtain T s , to which DM does not contribute much. In Fig. 2, we present the best-fit residual model for T 21 [9] (lower panel) after the foreground is properly removed. In the upper panel of Fig. 2, we present the resulted T s in comparison with T CM B . As such, the data-derived T s can be treated as a base model to derive upper limits on DM annihilation. There lacks likelihood information in Ref. [9]. However, Ref. [10] claimed that the standard value predicted by the first-star model T 21 = −209 mK is 3.8σ away from the experimental value T EDGES 21 = −500 mK. By assuming a Gaussian distribution, one can approximate the standard deviation as σ EDGES = (500 − 209)/3.8 mK and hence the likelihood can be rewritten as The standard deviation captures the EDGES experimental uncertainty that limits the relative T 21 . As mentioned above, the absolute level of the 21-cm absorption is irrelevant in the present consideration.
Here are some comments on the choice of our likelihood. Firstly, one might concern that the usage of the Gaussian likelihood would be improper if the analysis is with a tiny signal.
However, one should bear in mind that the signal T TH. 21 in Eq. (8) includes both contributions from the EDGES best-fit model and the DM annihilation, despite the nonlinearity. For the EDGES best-fit model, the reported signal-to-noise ratio is 37 at frequency of 78.1 MHz with the amplitude of 0.53 K which is strong enough to treat its likelihood as Gaussian. When adding the DM annihilation contribution, T TH. 21 will only be larger because of more energy injection. Hence, it is reasonable to use the Gaussian likelihood. Secondly, under such a likelihood function, we directly use the experimental error bar of T 21 measured by EDGES to constrain DM annihilation. Namely, it is a background-only likelihood that the background refers to everything else beside the contribution from WIMP DM annihilation. Being not to explain the excess with DM annihilation, the upper limit on the annihilation cross section derived from the likelihood in Eq. (8) is expected to be weaker but more conservative than the true one.

III. RESULTS
As aforementioned, WIMP DM has a tiny scattering cross section with SM nucleon. Therefore, one can safely ignore the gas cooling effect and put an upper limit on the DM annihilation that contributes to the gas heating. To consequently constrain the annihilation cross section, the energy spectra (  channels. We take DM mass at 200 GeV and annihilation cross section of σv = 3 × 10 −25 cm 3 s −1 as an example. To explicitly show the differences between different channels, we do not include the boost factor in these results. The EDGES best-fit data is depicted with a black solid line for comparison. Clearly, DM annihilation to an electron-positron pair (red solid line) is the strongest one to increase the temperature while the τ + τ − final-state channel is the weakest one. Therefore, we choose the strongest lepton channel χχ → e + e − and one of the strongest quark channels χχ → bb as the representative cases, while other channels are weaker than these two.
To compute the background evolution, we adopt the cosmological parameters as listed in Table I. Here we ignore the inherent uncertainties in these parameters because the EDGES for z > 300. This is a general feature of CMB constraints on DM annihilation that affects x e in z ∼ 600 − 1100 [34,35,47]. However, during the reionization epoch (z ∼ 20) the linearity disappears and the behavior becomes strongly dependent on the DM property and the injected energy propagation [36][37][38]. In the redshift region below z = 10, where T 21 is completely governed by the first stars, we rely on the reionization model for x e in Ref. [3].
Energetic e + e − pairs and photons can propagate a farther distance before they are absorbed, namely, energy absorption into the gas at lower redshifts more likely comes from lower-energy e + e − pairs and photons being created by energetic particles in the gas neighborhood. Hence, there are two factors that cause the growth of the bb contribution over the e + e − channel in heating the gas at low redshifts (z < 20). First, the DM annihilation to bb can yield more lower-energy e + e − pairs and photons than the e + e − channel. Second, the low-energy e + e − pairs and photons created at low redshifts will be shortly absorbed and successively affecting the recent epoch. Moreover, the boost factor can largely enhance the annihilation contribution, leading to higher T g (χχ → bb) temperatures at low redshifts.
In Fig. 4, we compare the experimental best-fit model of T 21 with those T 21 of four different DM annihilation scenarios. Here, we fix the DM mass to be 100 GeV and the annihilation cross sections to be 3 × 10 −26 cm 3 s −1 (solid line) and 3 × 10 −25 cm 3 s −1 (dash line). As explained previously, T 21 at low z should be more altered by χχ → bb (blue line) than χχ → e + e − (red line). When comparing the results from χχ → bb and χχ → e + e − , we found that the boost factor is not so effective for the e + e − channel. The 95% exclusion limits from PLANCK CMB [3], Fermi dSphs [39], and AMS02 data for electronpositron [40,41] and antiproton [42,43] are also shown. For comparison, we plot two recent results from Liu and Slatyer [44] (green long dashed line) and D'Amico et al. [30] (green short dashed line). The reference limits taken from [44] are based on the scenario: non-standard recombination allowing the gas to decouple thermally from the CMB earlier with T g (z = 17.2) = 5.2 K and (1 + z) td = 500 where "td" in the subscript stands for thermal decoupling.
Finally, we present the 95% exclusion limits in the (m χ , σv ) plane based on EDGES data in Fig. 5. For each DM mass bin, we normalize the background-only likelihood to one and depict a solid line in 95% C.L. corresponding to −2∆ ln L = 2.71 for a one-sided Gaussian likelihood distribution.
With the boost factor (red solid lines), one can see that the e + e − channel (left panel) is only slightly more stringent than the bb (right panel) channel in the lower-mass region but it becomes stronger at higher masses. However, for the case without the boost factor (red dashed lines), the e + e − channel is overall a factor of 2 − 3 more stringent than the bb channel. Interestingly, the boost factor does not affect the e + e − channel as significantly as the bb channel. This is totally due to the fact that more energetic e + e − pairs or photons are absorbed in a longer distance as aforementioned. For m χ 500 GeV, one can see no difference in the e + e − channel between the cases with and without the boost factor.
Both our EDGES limits for the e + e − and bb channels are stronger than the PLANCK We also depict the relevant results from D'Amico et al. [30] (green short-dashed lines) and Liu and Slatyer [44] (green long-dashed lines). The green short-dashed and the red solid lines are based on the same boost factor, whereas there is no boost factor applied in the green long-dashed and the red dashed lines. Note that the statistics between our constraints and those in Refs. [30] and [44] is clearly different. The green long-dashed line is taken out from is not washed out by the DM annihilation by half the amount (∆T 21 ≤ 100 mK). We note that our result is similar to that in Ref. [30] at the lower DM mass because the one-sigma uncertainty in T 21 is T 95%

21
∼ 126 mK in our statistical approach. In Ref. [44], their upper limits are obtained under the criteria that DM-modified gas temperature T g < 5.2 K at z = 17.2. On the other hand, our analyses are based on the best-fit model of the EDGES T 21 signal to derive the constraints on the DM annihilation. Our limits are slightly weaker than those in Ref. [30] but stronger than Ref. [44].
Beside the statistics, we do not assume immediate absorption of electrons, positrons, and photons during the propagation from z to z. This assumption is applied in Refs. [30], where it relies on fitting energy fractions f eff . On the other hand, Ref. [44] models the heating and ionization due to gradual cooling of the injected electron/positron/photon over time.
Note that we only adopt the non-standard recombination scenario 1 of Ref. [44] which the gas decouples thermally from the CMB earlier, see the right panel of Fig. 6 of Ref. [44].
There are three factors that can make the approximation of a simple f eff fraction invalid: (1) when one includes the boost factor B(z), taking delayed deposition into account is particularly important; (2) the efficiency of deposition at z ∼ 17 is quite different than at the high redshifts (z ∼ 600) that dominate the CMB constraints; (3) the f eff values in Refs. [26,27] are chosen to give equivalent energy transfers to the ionization of the gas, but not to the heating. 2 We have tried to extract redshift-dependent f eff fractions from our full numerical calculations using Eqs. (3) and (4). Our results show that the f eff 's for ionization vary slowly with redshift in the range of f eff ∼ 0.1−0.4, quantitatively consistent with those given in Refs. [26,27], and that the heating f eff 's rise monotonically with redshift from 0.1 to 0.5. Note that the f eff here has different definition with those given in Refs. [26,27]. We will report the details in the Appendix A.

IV. CONCLUSIONS
Regarding the absorption signal of 21-cm line probed by the EDGES experiment, we have assumed null DM contribution in signal to derive the modified evolution of T 21 by considering DM annihilation into e + e − and bb final states. We have further assumed that the process of DM annihilation is velocity independent (s-wave) and considered the boost factor as for approximating the enhancement of DM annihilation in the cosmic structures.
Different from other studies that simply make use of effective energy fractions f eff , we directly compute the propagation of the injected energy rather than assuming instant energy 1 There are three general scenarios discussed in Ref. [44]: (i) additional radiation backgrounds in the frequency range surrounding 21-cm, (ii) non-standard recombination allowing the gas to decouple thermally from the CMB earlier, and (iii) cooling of the gas through DM-baryon scattering. Our WIMP scenario can be either the scenario (i) or (ii). However, there is no public available bb annihilation limit for scenario (i) in Ref. [44]. We only present the scenario (ii) but the limits are similar, see Fig. (12c) in Ref. [44]. 2 We gratefully acknowledge the private communication with T. Slatyer, the author of Refs. [26,27,44]. deposition. It is precise enough to obtain the CMB constraints on DM annihilation by using f eff , since the effects take place linearly at high redshifts. However, for the 21-cm absorption at low redshifts, the linearity breaks down due to the formation of structures, and thus the assumption of using a simple f eff may not be adequate. We have shown the effects of the boost factor on the e + e − and bb channels and found that the enhancement is larger for the latter due to the fact that the bb final states produce more low-energy e + e − pairs and photons after a sequence of cascading decays. As expected, our EDGES limit for the e + e − channel is more stringent than that for the bb channel. Nevertheless, with the enhancement of the boost factor, the two channels are almost tantamount. Overall, we have derived the 21-cm constraints on the DM annihilation to e + e − and bb channels, which exceed the CMB constraints and in some low-mass regions better than the limits from the Fermi dSphs data and the AMS-02 antiproton data. χχ→e + e −
method. However, if including the boost factor, such a difference will be reduced as shown in Fig. 5.
For further discussion of the difference, we also compute the f eff based on our methodology. The result is shown in Fig. 7 and basically reflects that the bb channel is more effective and f eff,h (z, m χ ) = − dT g dz DM /{ 1 H(z)(1 + z) Note that f eff is independent of the the velocity-averaged cross section σv , since σv is a constant in the original integration with respect to the redshift z .