New Physics in the Rayleigh-Jeans Tail of the CMB

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 $\Lambda$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).

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, $N_{eff}=3.04\pm 0.33$ 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.

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), $A^{\prime }$, that develop mixing angles with ordinary photons, $ϵF_{\mu \nu }^{\prime }{F}_{\mu \nu }$  Holdom (1986). The apparent number counts of the CMB radiation can be modified by photon/dark photon oscillations:

where $P_{A\to A}=1-P_{A\to A^{\prime }}$ is the photon survival probability, while $P_{A^{\prime }\to A}$ is the probability of $A^{\prime }\to A$ oscillation. Previously the constraints on the $\{m_{A^{\prime }},ϵ\}$ 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 $z=15-20$ Bowman et al. (2018). The strength of the absorption signal is expected to be proportional to $1-T_{CMB}/T_s$ Zaldarriaga et al. (2004), where $T_{CMB}$ counts the number of CMB photons interacting with the two-level hydrogen hyperfine system, and $T_s$ is the spin temperature. The relevant photon energy is ${\omega }_0=5.9\mu eV$, and photons with this energy at the redshift of $z=17$ reside deep within the RJ tail, $x_0\equiv {\omega }_0/T_{CMB}=1.4\times {10}^{-3}$. This corresponds to much lower energy than direct measurements such as COBE-FIRAS, that measures above $x=0.23$ Fixsen et al. (1996), and ARCADE 2, which probes as low as $x=0.053$ (and finds an excess above the CMB prediction) Fixsen et al. (2009). There are also earlier measurements that constrain $x\sim 0.02-0.04$, although with larger uncertainties Staggs et al. (1995); Bersanelli et al. (1995).

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 $1-T_{CMB}/T_s$ temperature contrast than expected. Given that the spin temperature $T_s$ cannot drop below the baryon temperature $T_b$, a naive interpretation of this result could consist in lower-than-expected $T_b$, or higher $T_{CMB}$. 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.

The decay rate of $a\to 2A^{\prime }$ is

The lifetime, ${\tau }_a=1/{\Gamma }_a$, can be either much longer or much shorter than the present age of the Universe, ${\tau }_U\approx 13.8\times {10}^{9}y$, depending on the choice of parameters in (5).

For the case of short lifetimes, ${\tau }_a\ll {\tau }_U$, we require that the mass of $a$ is such that at the time of decay, ${\Gamma }_a\sim H\left(T\right)$, the energy of the resulting $A^{\prime }$ matches the CMB energy in the RJ tail, $x\sim {10}^{-3}$. Here, $H\left(T\right)$ is the Hubble expansion rate as a function of photon temperature. Assuming decays during radiation domination, this condition amounts to $g_{\ast }^{1/4}{f}_a\sim {10}^5-{10}^{6}GeV(m_a/GeV{)}^{1/2}$, in terms of parameters in (5), where $g_{\ast }$ is the effective number of degrees of freedom. If this condition is satisfied, the decays of $a$ to $A^{\prime }$ can happen arbitrary early, but the energy of the $A^{\prime }$ still approximately match RJ photons.

The case of a cosmologically long-lived particle, ${\tau }_a\gg {\tau }_U$, is especially attractive as $a$ can also naturally serve as DM. For the remainder of this paper, we will concentrate on this possibility. If the mass of $a$ falls in the range ${10}^{-5}eV<m_a<{10}^{-1}eV$, its decay can create significant modifications to the RJ tail of the CMB spectrum via $A^{\prime }\to A$ oscillations. It is worth noting that this overlaps the mass range often invoked for axion DM.

All constraints on parameters of (5) and (6) can be divided into two groups: those that decouple as $m_{A^{\prime }}\to 0$, and those that persist even in the limit of a massless dark photon. The stellar energy loss constraint due to $A^{\prime }a$ pair production is in this second category, as the in-medium transverse modes of photons can decay via $A_T^{\ast }\to A^{\prime }a$ even in the $m_{A^{\prime }}\to 0$ limit. We calculate the approximate emission rate to be

where $n_T$ is the number density of transverse plasmons (photons) and $m_A$ is the standard plasma frequency, $m_A^2=4\pi \alpha n_e/m_e$. Observing that it has the same scaling as the emission rate for a pair of Dirac neutrinos due to their magnetic moment $\mu$, $Q_{A^{\ast }\to \nu \overline{\nu }}={\mu }^2{m}_A^4{n}_T(24\pi {)}^{-1}$ Bernstein et al. (1963); Raffelt and Weiss (1992), we recast the corresponding bound $\mu \le 3\times {10}^{-12}\left(e/2m_e\right)$ Haft et al. (1994) to obtain

In addition, the $ϵ$-parameter is limited via $A\to A^{\prime }$ oscillations Essig et al. (2013), and depends rather sensitively on $m_{A^{\prime }}$. Stellar energy losses via these oscillations are important only for the higher mass range of $A^{\prime }$, $m_{A^{\prime }}>{10}^{-5}$ eV, as the emission is suppressed by $m_{A^{\prime }}^2/m_A^2$ inside stars, which is a small parameter An et al. (2013); Redondo and Raffelt (2013). Cosmological $A\leftrightarrow A^{\prime }$ oscillations may be significant if the resonant condition is met, $m_{A^{\prime }}=m_A\left(z\right)$, where $m_A\left(z\right)$ is the plasma mass of photons at redshift $z$ Mirizzi et al. (2009); Kunze and Vázquez-Mozo (2015). In the course of cosmological evolution $m_A\left(z\right)\simeq 1.6\times {10}^{-14}eV\times (1+z{)}^{3/2}{X}_e^{1/2}(z)$ scans many orders of magnitude; $X_e$ is the free electron fraction that we take from Kunze and Vázquez-Mozo (2015). For any $m_{A^{\prime }}$ in the range ${10}^{-14}-{10}^{-9}$ eV, the resonance happens at some redshift, $z_{res}$, 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 ${ϵ}^2$. 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 $P_{A^{\prime }\to A}\ll 1$. For large $ϵ$ the probability saturates, and in such cases we use its full expression. We notice that the probability of oscillation for RJ photons, $x\sim {10}^{-3}$, can be three orders of magnitude larger than for photons with $x\sim 1$, due to the ${\omega }^{-1}$ 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, $x_0=1.4\times {10}^{-3}$.

The left panel of the same plot shows $\{m_{A^{\prime }},m_a\}$ parameter space relevant for the 21 cm signal. Low values of $m_a$ 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 $m_{A^{\prime }}$ originate from the requirement of a resonance in the relevant redshift window, with some interval ${10}^{-13}-{10}^{-12}$ 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 $ϵ$ and ${\tau }_a$ required to make a significant modification to the 21 cm signal are consistent with other constraints. To that end, we select one point on $\{m_{A^{\prime }},m_a\}$ 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 $z_{res}=500$. By combining Eqs. (7), and (12), we first determine by how much the RJ photon count can be increased due to $A^{\prime }\to A$ conversion in the $1.2\times {10}^{-3}<x<1.6\times {10}^{-3}$ interval. The left panel of Fig. 4 shows the allowed values for the RJ photon increase, when the lifetime ${\tau }_a$ of the DM particle is varied, which is equivalent to scanning $f_a$. 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 $ϵf_a^{-1}$ in (9). If $n_{A^{\prime }\to A}/n_{RJ}$ is kept constant, while ${\tau }_a$ is increased, the required value of $ϵ$ eventually runs into the CMB spectral distortion constraint. Still, we find that the unexpected strength of the EDGES signal, which would require roughly $n_{A^{\prime }\to A}/n_{RJ}\simeq 1$ can be easily met over four orders of magnitude in the lifetime, ${\tau }_a\sim (10-{10}^5){\tau }_U$. Finally, the right panel of the same figure presents the $\{m_{A^{\prime }},ϵ\}$ slice of the parameter space, where ${\tau }_a$ is chosen such that $n_{A^{\prime }\to A}/n_{RJ}$ is set to 1. Since $m_{A^{\prime }}$ is allowed to vary, $z_{res}$ 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, $a$, to a scalar $\Phi$ that is charged under a $U(1{)}^{\prime }$ gauge group,

where $D_{\mu }={\partial }_{\mu }+i{g}^{\prime }{A}_{\mu }^{\prime }$, $L_a=({\partial }_{\mu }a{)}^2/2-m_a^2{a}^2/2$ as before, and $L_{A{A}^{\prime }}$ includes all terms as in (6) but the mass term for $A^{\prime }$ that will arise upon the condensation of the $\Phi$ field. The last term represents the Higgs-portal type coupling, leading to the decay of DM particle $a$ to two dark Higgs particles $h^{\prime }$, which in turn can decay to a pair of dark photons, $a\to 2h^{\prime }\to 4A^{\prime }$. This model fulfills our basic requirements for creating a population of non-thermal $A^{\prime }$, with the same implications for RJ photons. The stellar constraints will limit the product of $g^{\prime }ϵ$, but are not directly related to ${\tau }_a$. The DM abundance of $a$ may arise from the initial displacement of $a$ 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 $a$ inside galaxies and clusters of galaxies will lead to a flux of $A^{\prime }$, 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), $n_e\sim {10}^{-3}{cm}^{-3}$, this may create a sizable flux of photons, if the dark photon mass is on the order of ${10}^{-12}$ eV or lighter. Models with higher $m_{A^{\prime }}$ 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 $m_a$ and $m_{A^{\prime }}$, 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.