New Physics in the Rayleigh-Jeans Tail of the CMB

New Physics in the Rayleigh-Jeans Tail of the CMB


We show that despite stringent constraints on the shape of the main part of the CMB spectrum, there is considerable room for its modification within its Rayleigh-Jeans (RJ) end, . We construct explicit New Physics models that give an order one (or larger) increase of power in the RJ tail, which can be tested by existing and upcoming experiments aiming to detect the cosmological 21 cm emission/absorption signal. This class of models stipulates the decay of unstable particles to dark photons, , that have a small mass,  eV, non-vanishing mixing angle with electromagnetism, and energies much smaller than . The non-thermal number density of dark photons can be many orders of magnitude above the number density of CMB photons, and even a small probability of oscillations, going down to values of the mixing as small as , can significantly increase the number of RJ photons. In particular, we show that resonant oscillations of dark photons into regular photons in the interval of redshifts can be invoked as an explanation of the recent tentative observation of a stronger-than-expected absorption signal of 21 cm photons. We present a realistic model that realizes this possibility, where micro-eV mass dark matter decays to dark photons, with a lifetime longer than the age of the Universe.




Modern cosmology owes much of its advance to precision observations of the Cosmic Microwave Background (CMB). By now, both the spectrum of the CMB and its angular anisotropies are precisely measured by a number of landmark experiments Mather et al. (1994); Hinshaw et al. (2013); Ade et al. (2016). CMB physics continues its advance Abazajian et al. (2016) into probing both the standard CDM model to higher precision and possible New Physics that can manifest itself in small deviations from theoretical expectations. In addition, a qualitatively new cosmological probe, the physics of 21 cm emission/absorption at the end of the “dark ages,” may come into play in the very near future Furlanetto et al. (2006).

Figure 1: Illustration of our mechanism. The decays of an unstable relic, , which may constitute dark matter, produce a non-thermal population of soft massive dark photons, (blue). At some —after recombination but before the redshift relevant for the 21 cm absorption signal—a fraction of these dark photons is resonantly converted into ordinary photons (red). The latter add to the CMB photon count (green) in the RJ tail, resulting in a more negative temperature contrast.

Cosmology has been a vital tool for learning about physics beyond the Standard Model (SM). In particular, we know that about a quarter of our Universe’s energy budget is comprised of cold Dark Matter (DM), which probably cannot be identified with any known particles or fields. The precision tools of cosmology, on the other hand, provide serious constraints on the properties of DM, which instead of coming “alone,” may be a part of an extended dark sector, comprising new matter and radiation fields, and potentially new forces. Recent years have seen a significant increase in studies of dark sectors, both in connection with terrestrial experiments, and in cosmological settings Jaeckel and Ringwald (2010); Alexander et al. (2016); Battaglieri et al. (2017). Both the spectral shape and angular anisotropies of the CMB radiation significantly restrict the amount of additional energy that dark sectors can deposit into the SM bath, as a function of injection time.

If such light fields are thermally excited, they can be detected through their gravitational interaction alone, as they would modify the Hubble expansion rate, affect the outcome of Big Bang Nucleosynthesis (BBN), and modify the statistics of the CMB angular anisotropy patterns. The resulting constraint, purely for historical reasons, is phrased in terms of the number of effective neutrino degrees of freedom which, according to the latest observational bounds,  Ade et al. (2016), is consistent with the expectations of the standard cosmology. Non-thermal Dark Radiation (DR) is considered in the literature less often, although many processes occurring solely in the dark sector may lead to its appearance.

In recent papers Cherry et al. (2015); Cui et al. (2017), interacting DR was examined in the regime where the individual quanta are much fewer in number but much harder in frequency than the typical CMB photons, , but such that the constraint is satisfied. This type of DR may arise as a consequence of the late decays or annihilations of massive DM particles. In this paper we study the alternative, a much softer than CMB, but more numerous DR quanta,


In this formula, stands for the total energy density of radiation and DM, is the number density of DR quanta, while represents the low-energy Rayleigh-Jeans (RJ) tail of the standard CMB Planck distribution,


where we find it convenient to define the normalized photon frequency, , which is redshift-independent. In this formula, is the full Planckian number density, while is a (somewhat arbitrary) maximum frequency of the low-energy RJ interval, . If for example we take , then we find . It is easy to see that the number density of DR quanta may indeed significantly exceed . Saturating the constraint on for the DR that matches the CMB frequencies with , or alternatively letting of DM energy density Berezhiani et al. (2015); Poulin et al. (2016) be converted to DR in the same frequency range after the CMB decoupling, we arrive at the maximum number densities given by


Thus, soft DR quanta have a potential to outnumber the RJ CMB photons by up to orders of magnitude.

What are the observational consequence of such soft and numerous DR? Very light fields often have their interactions enhanced (suppressed) at high (low) energies. This is the case for neutrinos, that have Fermi-type interactions with atomic constituents, as well as of axions that have effective dimension 5 interactions with fermions and gauge bosons. This type of DR would be impossible or very difficult to see directly. There is, however, one class of new fields comprising DR that can manifest their interactions at low energies and low densities. These are light vector particles (often called dark photons), , that develop mixing angles with ordinary photons,  Holdom (1986). The apparent number counts of the CMB radiation can be modified by photon/dark photon oscillations:


where is the photon survival probability, while is the probability of oscillation. Previously the constraints on the parameter space were derived Mirizzi et al. (2009); Kunze and Vázquez-Mozo (2015) using COBE-FIRAS data Fixsen et al. (1996) (that is, considering the depletion of CMB photons due to the first term in eq. (4)). The point of the present paper is that the RJ tail of the CMB can get a significant boost due to the second term in (4) without contradicting the COBE measurement. While the reliable extraction of the primordial contribution to the RJ tail is challenging due to significant foregrounds, the physics of the 21 cm line can provide a useful tool to probe DR through the apparent modification of the low-energy tail of the CMB.

The EDGES experiment has recently presented a tentative detection of the 21 cm absorption signal coming from the interval of redshifts  Bowman et al. (2018). The strength of the absorption signal is expected to be proportional to  Zaldarriaga et al. (2004), where counts the number of CMB photons interacting with the two-level hydrogen hyperfine system, and is the spin temperature. The relevant photon energy is , and photons with this energy at the redshift of reside deep within the RJ tail, . This corresponds to much lower energy than direct measurements such as COBE-FIRAS, that measures above  Fixsen et al. (1996), and ARCADE 2, which probes as low as (and finds an excess above the CMB prediction) Fixsen et al. (2009). There are also earlier measurements that constrain , although with larger uncertainties Staggs et al. (1995); Bersanelli et al. (1995).

Figure 2: Left panel. Effective photon mass as function of redshift. In the region marked with diagonal green lines the resonant oscillation of dark photons into regular photons can modify the 21 cm absorption signal relative to the CMB at . Right panel. Conversion probability as a function of redshift, in the limit (for large enough the probability saturates, ). We consider photons with energy , which implies a wavelength of 21 cm at . The vertical blue band in the two graphs covers the region , corresponding to the width of the 21 cm absorption signal measured by the EDGES experiment. Photons with that are injected in the gray region, , are rapidly absorbed by the plasma.

The locations of the left and right boundaries of the claimed EDGES signal agree rather well with standard cosmological expectations, but the amount of absorption seems to indicate a more negative temperature contrast than expected. Given that the spin temperature cannot drop below the baryon temperature , a naive interpretation of this result could consist in lower-than-expected , or higher . Together with related prior work Tashiro et al. (2014); Muñoz et al. (2015), a number of possible models were suggested Barkana (2018); Ewall-Wice et al. (2018); Barkana et al. (2018); Fialkov et al. (2018); Fraser et al. (2018), most of which have difficulty to pass other constraints, Dvorkin et al. (2014); Gluscevic and Boddy (2017); Xu et al. (2018); Berlin et al. (2018); D’Amico et al. (2018). The mechanism that we point out, oscillation of non-thermal DR into visible photons, can accommodate the EDGES result without being challenged by other constraints. In the rest of this paper, we provide more details on the suggested mechanism, and identify the region of parameter space where 21 cm physics can provide the most sensitive probe of DR.

Decay of unstable relics into dark radiation:

The framework described in the introduction allows for significant flexibility with respect to the actual source of non-thermal soft DR. To give a concrete realization of the proposed mechanism to increase , we specify a model of unstable scalar particles, , that couple to dark dark photons via an effective dimension five operator,


where , and the last term describes the photon-dark photon Lagrangian with corresponding mass and mixing terms for :


We assume that an initial relic abundance of is present. The cosmology of is model-dependent, but to keep our discussion general we leave the study of production for future work.

The decay rate of is


The lifetime, , can be either much longer or much shorter than the present age of the Universe, , depending on the choice of parameters in (5).

For the case of short lifetimes, , we require that the mass of is such that at the time of decay, , the energy of the resulting matches the CMB energy in the RJ tail, . Here, is the Hubble expansion rate as a function of photon temperature. Assuming decays during radiation domination, this condition amounts to , in terms of parameters in (5), where is the effective number of degrees of freedom. If this condition is satisfied, the decays of to can happen arbitrary early, but the energy of the still approximately match RJ photons.

The case of a cosmologically long-lived particle, , is especially attractive as can also naturally serve as DM. For the remainder of this paper, we will concentrate on this possibility. If the mass of falls in the range , its decay can create significant modifications to the RJ tail of the CMB spectrum via oscillations. It is worth noting that this overlaps the mass range often invoked for axion DM.

oscillations and constraints:

Figure 3: Left panel. We show the values of the DM mass that are relevant for 21 cm, as a function of the dark photon mass (bottom axis) or, equivalently, as a function of resonant redshift (top axis). In the red region, , implying that the produced photons are too soft to impact 21 cm. In the region above the purple line, DM is heavy enough that it produces photons at the energies measured precisely by COBE-FIRAS, implying a tight constraint (depending on and the -lifetime). Note that the region delineated by the dotted purple line corresponds to resonant conversion after the redshift relevant for the 21 cm absorption signal. The vertical orange band shows the region of dark photon mass constrained by black hole superradiance Cardoso et al. (2018). The blue star corresponds to the benchmark case (, eV) further explored in the rest of the figures. Right panel. Comparison between the RJ tail of the CMB (orange) and the energy spectrum of the photon population generated from the resonant oscillation (solid black). In our benchmark example the resonance took place at , and we assume the decaying particle with mass eV and lifetime constitutes the whole DM in the Universe. For comparison, we also show–rescaled by a factor –the original number density distribution of dark photon before resonant oscillation (dotted black). The spectra are plotted as a function of the redshift-independent variable , and therefore apply to all redshifts . We choose , such that the number of photons is doubled in the window .

All constraints on parameters of (5) and (6) can be divided into two groups: those that decouple as , and those that persist even in the limit of a massless dark photon. The stellar energy loss constraint due to pair production is in this second category, as the in-medium transverse modes of photons can decay via even in the limit. We calculate the approximate emission rate to be


where is the number density of transverse plasmons (photons) and is the standard plasma frequency, . Observing that it has the same scaling as the emission rate for a pair of Dirac neutrinos due to their magnetic moment ,  Bernstein et al. (1963); Raffelt and Weiss (1992), we recast the corresponding bound  Haft et al. (1994) to obtain


In addition, the -parameter is limited via oscillations Essig et al. (2013), and depends rather sensitively on . Stellar energy losses via these oscillations are important only for the higher mass range of ,  eV, as the emission is suppressed by inside stars, which is a small parameter An et al. (2013); Redondo and Raffelt (2013). Cosmological oscillations may be significant if the resonant condition is met, , where is the plasma mass of photons at redshift  Mirizzi et al. (2009); Kunze and Vázquez-Mozo (2015). In the course of cosmological evolution scans many orders of magnitude; is the free electron fraction that we take from Kunze and Vázquez-Mozo (2015). For any in the range  eV, the resonance happens at some redshift, , within the cosmic dark ages, see the left panel of Fig. 2. The resonance ensures that the probability of oscillation is much larger than the vacuum value of . Following Kuo and Pantaleone (1989); Mirizzi et al. (2009), we take it to be


We remark that this expression is valid only in the limit . For large the probability saturates, and in such cases we use its full expression. We notice that the probability of oscillation for RJ photons, , can be three orders of magnitude larger than for photons with , due to the dependence. The redshift dependence of (10) is shown in the right panel of Fig. 2, assuming a dark photon energy that is relevant for 21 cm, .

Figure 4: Left panel. Allowed region in the ratio between ordinary CMB photons and photons generated from resonant conversion of dark photons in the energy window relevant for the 21 cm absorption signal as function of the DM lifetime. The region is bounded by stellar energy losses (blue, see eq. (9)), CMB distortion of the blackbody spectrum due to oscillations (red) Kunze and Vázquez-Mozo (2015), and by a limit on the DM lifetime,  Poulin et al. (2016). The red dashed line shows the projected sensitivity of PIXIE/PRISM to oscillations Kunze and Vázquez-Mozo (2015). The purple region is forbidden since it would require a conversion probability larger than . Right panel. The allowed parameter space as a function of the dark photon mass and . At each point, the DM lifetime is chosen such that , in the 21 cm signal region . We consider our benchmark case  eV. The shaded orange region is disfavored by black hole superradiance Cardoso et al. (2018), and the other limits are as on the left panel.

Dark age resonance and EDGES signal:

For , the Universe becomes transparent to photons that are converted into the RJ tail of the CMB, , whereas for these soft photons are efficiently absorbed Chluba (2015). Therefore, only dark photons with —possibly injected at a much earlier epoch—will yield excess radiation at 21 cm. Focusing on a mono-chromatic injection of with cosmologically long lifetime , the energy spectrum at redshift reads,


Here, is the critical density today and the Hubble rate, , is evaluated at redshift , where . In the following we saturate the -abundance (prior to decay) to the CMB-inferred DM density, . The total number of injected grows with cosmic time , and can eventually outgrow by a large margin, .

Once the resonance condition is met at , a fraction of will be converted as per Eq. (4). Those extra photons then redshift, and, overall, the spectrum of converted photons at becomes,


Here, is the blue-shifted energy at resonance. The right panel of Fig. 3 shows (12) (solid black line). It is compared to the RJ tail of the CMB (dashed orange), and to the spectrum of , Eq. (11), rescaled by a factor (dotted black). Furthermore,  eV, , and are chosen.

In order to identify models that can be tested with 21 cm observations, we compare the number density of converted photons, , to the RJ density of the CMB, , within a relevant energy window. We define this window to include all photons with a wavelength of 21 cm within the redshift interval . This is equivalent to requiring


The left panel of the same plot shows parameter space relevant for the 21 cm signal. Low values of give photons that are too soft to affect the 21 cm absorption line, while high values produce photons at energies that (depending on other parameters) could be probed by COBE-FIRAS. Low and high limiting values for originate from the requirement of a resonance in the relevant redshift window, with some interval  eV possibly disfavored by black hole superradiance Cardoso et al. (2018) (see also Pani et al. (2012); Baryakhtar et al. (2017)). The crucial question that remains to be investigated is whether values of and required to make a significant modification to the 21 cm signal are consistent with other constraints. To that end, we select one point on parameter space, shown by an asterisk on the left panel of Fig. 3, and analyze it in full detail.

The selected point corresponds to the resonance at . By combining Eqs. (7), and (12), we first determine by how much the RJ photon count can be increased due to conversion in the interval. The left panel of Fig. 4 shows the allowed values for the RJ photon increase, when the lifetime of the DM particle is varied, which is equivalent to scanning . We observe that the photon count can be increased by more than an order of magnitude with higher values limited by the stellar bound on in (9). If is kept constant, while is increased, the required value of eventually runs into the CMB spectral distortion constraint. Still, we find that the unexpected strength of the EDGES signal, which would require roughly can be easily met over four orders of magnitude in the lifetime, . Finally, the right panel of the same figure presents the slice of the parameter space, where is chosen such that is set to 1. Since is allowed to vary, is scanned along the horizontal axis. The white area shows the significant portion of the parameter space that is allowed, including the asterisk-marked point that corresponds to the spectrum plotted earlier in Fig. 12, right panel. It is worth noting that much of the allowed parameter space can be probed by future generation of experiments aimed to refine COBE-FIRAS measurements.


The concrete model of decaying light DM presented here is simple, but definitely not unique, opening an avenue of studying such models in greater detail. For example, a fully renormalizable model with a naturally achieved abundance of DM can be built using the coupling of dark matter, , to a scalar that is charged under a gauge group,


where , as before, and includes all terms as in (6) but the mass term for that will arise upon the condensation of the field. The last term represents the Higgs-portal type coupling, leading to the decay of DM particle to two dark Higgs particles , which in turn can decay to a pair of dark photons, . This model fulfills our basic requirements for creating a population of non-thermal , with the same implications for RJ photons. The stellar constraints will limit the product of , but are not directly related to . The DM abundance of may arise from the initial displacement of from its minimum, as for most models of light bosonic DM. Detailed analysis and classification of further models of light DM that can be probed using RJ photons is beyond the scope of the current paper.

Finally, we comment on other possible signatures of this type of models. In particular, the decay of inside galaxies and clusters of galaxies will lead to a flux of , that could be converted to regular photons, as dark photons travel outside such a cluster, from high to low density. If a typical density of electrons inside a cluster is employed Bahcall (1995), , this may create a sizable flux of photons, if the dark photon mass is on the order of  eV or lighter. Models with higher may also experience resonant conversion in cluster regions of higher electron density. Thus, radio and microwave emission from clusters may contain components that are not expected from models with stable DM. The resulting signals/constraints will have strong dependence on both and , may contain astrophysical uncertainties, and will be addressed separately.


The RJ tail of the CMB spectrum can be modified by light and weakly coupled New Physics particles/fields without contradicting any other cosmological or astrophysical constraints. We have presented one such example where the resonant conversion of non-thermal and numerous dark photons to ordinary photons leads to an enhancement in the RJ tail of the CMB. The upcoming era of 21 cm precision cosmology, as perhaps signaled by the first reported tentative detection Bowman et al. (2018), will provide an invaluable tool in testing such new physics.


We thank Yacine Ali-Haïmoud and Jens Chluba for helpful conversations. MP and JTR acknowledge the financial support provided by CERN. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Economic Development & Innovation. JP is supported by the New Frontiers program of the Austrian Academy of Sciences. JTR is supported by NSF CAREER grant PHY-1554858.


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