Cosmological origin of anomalous radio background

# Cosmological origin of anomalous radio background

James M. Cline Department of Physics, McGill University, 3600 Rue University, Montréal, Québec, Canada H3A 2T8    Aaron C. Vincent Instituto de Física Corpuscular, Universitat de València - CSIC, 46071, Valencia, Spain
###### Abstract

The ARCADE 2 collaboration has reported a significant excess in the isotropic radio background, whose homogeneity cannot be reconciled with clustered sources. This suggests a cosmological origin prior to structure formation. We investigate several potential mechanisms and show that injection of relativistic electrons through late decays of a metastable particle can give rise to the observed excess radio spectrum through synchrotron emission. However, constraints from the cosmic microwave background (CMB) anisotropy, on injection of charged particles and on the primordial magnetic field, present a challenge. The simplest scenario is with a GeV particle decaying into at a redshift of , in a magnetic field of G, which exceeds the CMB -field constraints, unless the field was generated after decoupling. Decays into exotic millicharged particles can alleviate this tension, if they emit synchroton radiation in conjunction with a sufficiently large background magnetic field of a dark U(1) gauge field.

## I Introduction

The ARCADE 2 collaboration has measured the diffuse radio background at several frequencies between 3 and 90 GHz, with relatively small errors up to 10 GHz Fixsen:2009xn (). Their measurements extend earlier ones at lower frequencies GHz roger ()-reich () that are well fit by a power-law dependence of the brightness temperature on frequency,

 T∼TR(ν/ν0)β (1)

with . The measured value of is within the ballpark of expectations () for sources in which synchrotron emission dominates (e.g., star-forming galaxies), but the amplitude K (with GHz) is times too large to be explained by extrapolations of populations of known resolved sources seiffert (). These conclusions have received further support in ref. Condon:2012ug (). Attempts to identify the excess with previously overlooked standard astrophysical sources have so far not met with success galactic ()-Vernstrom:2011xt (). There have been several attempts to provide a new origin through the annihilation of dark matter into charged particles, which subsequently undergo synchrotron emission Fornengo:2011cn ()-Yang:2012qi (). It was recently pointed out that there is a serious challenge to such mechanisms: the spatial fluctuations in the observed radio signal are much too small to be consistent with sources that have undergone clustering Holder:2012nm (), as would be the case for dark matter annihilating near the present time.111Ref. Yang:2012qi () studied the possibility of dark matter annihilating into charged particles at earlier times, assuming that ultracompact minihalos and very large magnetic field exist, finding that Compton scattering leads to overproduction of the diffuse x-ray background relative to observed values. Ref. Holder:2012nm () conservatively uses the linear theory power spectrum for its calculations, whereas nonlinear structure formation would be expected to enhance the clustering by a (possibly large) factor that is currently highly uncertain Ando:2005xg (). A possible caveat to the analysis of Holder:2012nm () is its assumption of Gaussian fields in the computation of error bars, whereas the intrinsic fluctuations at the relevant scales are likely dominated by non-linear (hence non-Gaussian) structure. However, the measurements are most likely noise-dominated; thus the statement that clustered dark matter sources are disfavored by the observed smoothness of the signal seems to be robust.

These considerations suggest a cosmological origin of the radio excess, from an epoch prior to the formation of structure and consequent large inhomogeneities in the density of decaying dark matter or of the cosmological magnetic fields. On the other hand, the excess photons should have been produced relatively late, after redshift when recombination occurred, to avoid thermalizing away such a spectral feature. When one tries to imagine a mechanism for producing excess photons with the right spectrum at this epoch or later, it proves to be highly constrained, due to cosmic microwave background (CMB) constraints G on primordial magnetic fields and on injection of ionizing charged particles. The purpose of the present work is to point out some mechanisms that could be promising and the challenges that they must be overcome. We show that a spectrum of excess diffuse radio background consistent with observations can be generated either through synchrotron emission from electrons injected into the plasma at some time after decoupling, depending upon the mass of the decaying particle that produces them. However, CMB constraints on the magnetic field (if it was generated before decoupling) and on the amount of injected energy turn out to be in conflict with this hypothesis.

For example, we find that a metastable particle that decays via can produce the desired radio background through synchrotron emission if MeV, consistent with the CMB constraint on ionizing radiation, but it requires a magnetic field some 100 times greater than the CMB constraint on primordial fields, necessitating a late-time magnetogenesis mechanism. A more complicated model may be viable: if decays into millicharged particles that also carry a dark U(1) gauge charge, and if there is a corresponding background of dark photons with a sufficiently large dark magnetic field, the CMB constraint on charged particle injection can be robustly overcome.

Our paper starts with a brief recapitulation of the observed radio spectrum, reviewing in section II the data and the possibility that a spectral index for the excess temperature provides a consistent description. In section III we note that direct production of photons through decays or annihilations does not give this kind of spectrum, nor do processes involving low-energy electrons, motivating the alternative of relativistic charged particle injection. Section IV discusses the rate of energy loss of charged particles through Compton scattering (CS) and synchrotron emission, and in it we derive an upper bound on the primordial magnetic field around the decoupling epoch, from CMB constraints. This bound implies that CS is by far the dominant means of energy loss. Section V computes the spectrum of Compton-scattered photons and consequent bounds from x-ray observations. Section VI shows that synchrotron emission can produce a radio background with the desired properties, if the initial electron energy is sufficiently large. In sections V-VII we make a simplifying assumption that the charged particles are injected at specific time (sudden decays). In section VII we show that this approximation works very well for decays, and we generalize it to the case of annihilations. In section VIII we show that CMB constraints are in conflict with the above mechanisms if the injected particles are electrons, and if the primordial field was generated before decoupling. In section IX we suggest a more exotic scenario in which millicharged particles are instead injected, as a second example of how the challenges might be overcome. Conclusions are given in section X. A possibly novel way of generating primordial magnetic fields (though too weak for our purposes) is elaborated in appendix A. In appendix B we show that brehmsstrahlung cannot yield the observed spectral shape.

## Ii The observed spectrum

Taking data from table 4 of ref. Fixsen:2009xn (), which also compiles results from the earlier experiments roger ()-reich (), we have replotted the antenna temperature222The antenna temperature takes the place of in the familiar formulas for fluxes, intensities, etc., of blackbody radiation. We use units . data versus frequency and display them in fig. 1. The ARCADE-2 data are the 6 highest frequency points (we do not show those above GHz, where the error bars are much larger). Also shown are power law dependences, with the ARCADE-2 best fit value , and the theoretical prediction that we will derive in section VI. From fig. 1, it is clear that the preference for is driven by the measurements near MHz and 3.3 GHz that have the smallest errors. This assumes there is no systematic miscalibration between the two different experiments maeda () and Fixsen:2009xn () that determined the respective values. Even if not, is within of ARCADE-2’s best-fit value. In the present work, we will consider to be adequate for describing the observations. It will be seen that there are more challenging problems for finding a working mechanism than the discrepancy between and .

To put the radio observations into a broader perspective we have taken the compendium of diffuse radiation backgrounds presented in fig. 1 of Hauser:2001xs () and translated it into antenna temperature using to indicate how a new source in the radio region (energies less than eV) would compare to higher-energy backgrounds if it continued with the same power law indefinitely. At high energies eV, the index or would not be sufficiently steep to remain within observational bounds, but we will see that this need not be a serious limitation. First, the mechanism we propose has a natural cutoff energy above which the spectrum of the new contribution drops sharply, and this can be dialed by adjusting the initial energy of the injected charged particles. Even in the absence of such a cutoff, one must consider whether the universe is transparent to radiation in a given energy range and at a given redshift. The transparency window as a function of and is given in refs. Chen:2003gz (); Slatyer:2009yq (). At low redshifts, this window extends from roughly 1 keV to GeV, and it is also open at eV below the energy of Lyman- absorption. Thus any x-rays produced in conjunction with the radio excess will be absorbed by the baryonic plasma for energies below keV. At higher redshifts this cutoff increases to keV. Depending upon the details, this effect could weaken the constraining power of experiments like Chandra Hickox:2007gj () and XMM-Newton xmm-newton () that we will discuss in section V.

## Iii Motivation for relativistic charged particle injection

One might hope to bypass the complications of synchrotron and Compton emission by a mechanism that directly produces photons after decoupling. An axion-like particle with a very long lifetime and which decays into two photons is the simplest possibility. Although it produces monochromatic photons at any given time, their energies are redshifted and so a continuous spectrum results. Ref. Bertone:2007aw () finds, in the case where the decaying particle lifetime is greater than the age of the universe, that the flux of such photons is given by

 dJdEγ=2Amχ(1+κ(2Eγmχ)3)−12(2Eγmχ)12Θ(1−2Eγmχ) (2)

where cm s str)s (10 GeV/) assuming is the dark matter, and . The energy-dependence can be derived from the integral over redshifts Overduin:2004sz () up to that at decoupling ()

 dJdEγ∼∫zdec0dz1H(z)δ[mχ2Eγ(1+z)−1] (3)

where the Hubble parameter is proportional to and the delta function yields the spectrum of photons from a given decay.333Ref. Overduin:2004sz () derives the spectral intensity rather than the flux, giving an extra factor of in the denominator of the integrand.

On the other hand, the radio signal of interest has the observed flux

 dJdEγ=EγT(Eγ)2π2∼E−1.6γ (4)

where with eV (corresponding to GHz) and eV from (1). The shapes of (2) and (4) are compared in fig. 3. The slope of the decay spectrum is never as negative as the observed one. We find that the agreement is not improved by considering decays into three photons, approximating the spectral distribution as a box in place of the delta function in (3), nor by decreasing the lifetime such that the ’s disappear before the present time (which also changes the shape of the present-day spectrum via an extra factor in (3)).444To avoid the problem of large angular fluctuations in the radio excess, we should insist on lifetimes shorter than the time scale for structure formation  Neither do annihilations produce better results, since the input photon spectrum is monochromatic just like for decays. We conclude that particle physics mechanisms for direct production of photons do not work. On the other hand, the radio excess has a spectrum that appears to be consistent with synchrotron emission. Production of charged particles thus seems like a more promising approach.

In order to generate the correct spectrum, we will furthermore require the injected charged particles to be relativistic: . The cyclotron output from nonrelativistic electrons in the cosmic magnetic field will be in the form of peaks around the gyrofrequency. This is at most Hz for values of nG allowed by CMB constraints—far below  the MHz-GHz region of the ARCADE-2 excess.

Compton scattering with the CMB is a second mechanism which may produce low-energy photons. By computing the lower-limit to the scattered photon energy, we can show that this mechanism is not an efficient means of producing radio photons, if the injected electron is nonrelativistic. A CMB photon with energy scattering head-on with a low energy electron of velocity follows the Compton formula, producing a photon of energy :

 Eγ=Eγ′1+√1+ve1−veEγ′me(1−cosθ), (5)

We are computing an upper limit to energy loss which we therefore maximise by choosing . This yields:

 Eγ=Eγ′⎛⎝1−√1+ve1−veEγ′me+O⎛⎝[Eγ′me]2⎞⎠⎞⎠. (6)

To explain the ARCADE excess, we are seeking a mechanism to produce photons (the average CMB photon energy). This is clearly not possible in the regime of (6).

## Iv Injection of electrons and magnetic field constraints

We have argued that a more likely mechanism for producing the radio excess would involve injecting charged particles with excess energy into the primordial plasma at some time after recombination but before structure formation, and relying upon the subsequent synchrotron emission. For example, a metastable particle of mass could decay into pairs at the appropriate time. Alternatively a first order phase transition of some scalar field coupled to the Higgs boson could cause a small change in the Higgs vacuum expectation value, leading to a change in the mass of the electron and hence its kinetic energy. However it seems difficult to obtain relativistic electrons as required through the latter mechanism. The decaying scenario on the other hand naturally provides relativistic electrons.

It is enlightening to estimate how much energy is needed to produce the observed excess. Supposing that the spectral anomaly continues up to some maximum energy , its energy density is given by

 ρr = 1π2∫Emax10−2E0dEγE2γT(Eγ) (7) ≅ 5×10−21eV4(Emax10E0)0.4

(Recall that corresponds to the frequency GHz.) As a fraction of the critical density, this is .

Eq. (7) gives a lower bound on the excess kinetic energy in electrons that needs to be injected. But in fact much more is required if we rely upon synchrotron emission, since it is not the most efficient means of dispersing the excess energy in the early universe; Compton scattering on CMB photons is much more important. The total energy loss rate is due to the sum of the two processes, and their relative importance depends upon the energy density in CMB photons versus that in magnetic fields:

 dEedt=43γ2σT(ργ′+ρB) (8)

where and is the Thomson cross section. The relative efficiency for producing synchrotron radiation is therefore

 ρBργ′ = 12B2π215T4γ′ ≅ 10−7(B10−9G)2 (9)

Here we have alluded to the CMB constraint on the comoving magnetic field G Durrer:1999bk ()-Paoletti:2010rx (). Taken at face value, a limit of nG combined with (7) implies that times more energy in Compton photons is produced compared to synchrotron. If we wish to use the latter to produce the radio excess, it would then lead to an extra contribution to in photons via CS of order , which is 25 times greater than that in the CMB.

To understand whether synchrotron emission can be more efficient than indicated by the above discussion requires us to examine the CMB contraint on in greater detail. The primordial field is believed to be stochastic, with a spectrum that in general depends on the length scale. The limit is thus a bound on the average value at some comoving scale , which is usually taken to be Mpc for CMB constraints. If the spectrum of is scale invariant (meaning that in ), then there is no possibility that could be larger than the nominal bound at some smaller scale. But if it has a blue-tilted spectral index , then could conceivably exceed by some factor at sufficiently small scales (yet large enough to be coherent over the Larmor radius of an electron and thus produce synchrotron radiation).

A limitation is that magnetic fields are damped at small scales by radiative viscosity, and the damping scale is a function of the field itself. The result of Subramanian:1997gi () for the damping scale can be expressed as Mack:2001gc ()

 λD2π≅(Bλ170×10−9G)2n+5(λ2πMpc)n+3n+5h−1n+5Mpc (10)

(where ). This applies to vector perturbations, which were shown in ref. Paoletti:2010rx () to dominate over scalar perturbations. At G, ranges from to Mpc as goes from to (the maximum value usually considered for a causal mechanism of primordial magnetogenesis). For a given , the maximum field value will be attained at the minimum undamped scale, just above .

CMB constraints on and are correlated, as shown in ref. Yamazaki:2010nf (); Paoletti:2010rx (); Shaw:2010ea (). We reproduce the 95% C.L. limit on versus from ref. Paoletti:2010rx () in fig. 4(a). The upper limit on is strongest for the largest values of . We have evaluated at the damping scale for points along this contour to find the maximum allowed as a function of . The result, plotted in fig. 4(b), is that as large as 50 nG is allowed for a spectral index in the range , corresponding to damping scales Mpc. Using this field strength in (9), the relative efficiency to produce synchrotron radiation is enhanced by a factor of 2500, and the energy density produced by associated Compton scattering becomes around 0.01 of that in the CMB, if the excess radio signal is due to synchrotron emission. This relaxation of the nominal bound of a few nG is only useful if synchrotron emission occurs close to the decoupling era since thereafter we expect the damping scale to increase again as the free charge density decreases. This caveat will not affect our conclusions in the end since we will have to invoke either late-time magnetogenesis or other exotic particle physics to construct a working scenario.

Stronger limits on for scale-noninvariant spectra can be obtained by requiring that the energy density in not exceed the critical density at earlier times Caprini:2001nb (); the fraction of energy in increases at earlier times because depends upon time (eq. (10) is valid around the epoch of matter-radiation equality). However these limits would be evaded if either the time of magnetogenesis was not too early, or if there is an intrinsic cutoff in the initial spectrum (as one would expect there must be) that falls below the damping scale before becomes too large. Thus the upper limit nG is fairly conservative, from the perspective of making the fewest assumptions about the unknown origin of the primordial magnetic field. Of course, if magnetogenesis occurs even later than recombination, we can evade this constraint as well.

A single mechanism that could accomplish both of these at once might interesting for explaining the radio excess. For example, the decay process produces a small electric current at each decay, which gives rise to stochastic fields. We can estimate their strength as

 ⟨B(x)B(y)⟩∼4πe2ne|x−y| (11)

for relativistic electrons where is the electron density. In appendix A we derive this, and show that it is too small for our purposes. In the following, we will assume that magnetogenesis occurred before decoupling, so the maximum field was no more than 50 nG, although the possibility of relaxing this assumption should be kept in mind if one is looking for loopholes.

## V Compton Scattering

Since we have argued that much more energy will be produced by Compton scattering than by synchrotron emission, it is necessary to compute the spectrum of CS photons resulting from injected electrons. Although this spectrum itself does not have the right properties to explain the excess radio background, it has the potential to make a larger contribution to the diffuse x-ray background than is observed, and thus provides a constraint on mechanisms that rely upon synchrotron emission to account for the radio excess. We derive such a constraint in this section.

The rate of Compton emission of photons of energy per scattered electron of initial energy , involves an integral over the energy spectrum of CMB photons:

 dNγdtdEγ=2πe4E2e∫dEγ′1Eγ′dndEγ′fC(q,Γ)gC(Eγ/Eγ′) (12)

where (following ref. Jones:1968zza ())

 fC(q,Γ)=2qlnq+(1+2q)(1−q)+(1−q)(Γq)22(1+Γq) (13)

with and and

 gC(Eγ/Eγ′)={1,Eγ>Eγ′Eγ/Eγ′,Eγ≤Eγ′ (14)

The above parametrization is approximate, and can be inferred from fig. 3 of Jones:1968zza (); its main purpose is to show that the spectral index of Compton radiation changes by 1 between the upscattering and downscattering regimes. , which is meant to describe the regime where , becomes roughly constant at its maximum value for . This is true provided that the electrons are relativistic.  The factor is designed to give the correct behavior for . We will further approximate (12) by replacing with its average value , and where is the temperature at time when the electron was injected (the decay time of in the model where ) and is the density of CMB photons at this temperature. Later on we will relax this “sudden decay approximation” by averaging over with the appropriate dependence of on . The spectrum of emitted photons at the decay time is then given by

 dnγ,ddEγ≅4πe4nχ,dnγ,d¯¯¯¯Eγ′gC(Eγ/¯¯¯¯Eγ′)∫dtE−2efC(q,Γ) (15)

We ignore diffusion since the universe is homogeneous and isotropic during the epoch of interest, and we assume that the electron spectrum remains monoenergetic. The electron energy diminishes very quickly with time; by solving eq. (8) with , we find

 Ee(t)=Ed1+Ce(t−td) (16)

for , where is the initial electron energy when it was injected at time , and

 Ce=43σTργ,dEdm2e (17)

The time scale for energy loss is much smaller than the Hubble time, so we can ignore redshifting of the electron energy. It is convenient to change variable in the integral (15), using , where the minimum electron energy is from the kinematics of Compton scattering Blumenthal:1970gc (). To evaluate at the present time, we must correct for redshift by replacing and (where the redshift corresponding to is ) in and in , and dividing by an overall factor of (by dimensionality of ). Thus

 dnγ,0dEγ = 9nχ,0m4egC8Zd(2.7T0)2E30(Zd)∫EdE01d^Ee^E4efC(q(Zd),Γ) (18) ≅ 9nχ,0megC(Eγ/¯¯¯¯Eγ′)Zd(2.7T0)1/2E3/2γ×0.088

The approximation for the integral holds as long as is sufficiently larger than 1, i.e., for

 Eγ≲2.6T0(mχ/me)2 (19)

where eV is the current CMB temperature. For higher , the integral drops sharply as shown in fig. 5. To get a contribution at the lowest Chandra energies keV, for example, would thus require GeV.

At energies below that of the CMB photons, , (18) scales like which translates into a subdominant contribution of the form to the temperature of the excess radio background, where

 TA=π2EγdnγdEγ (20)

We can put an upper bound on the initial abundance such that this extra contribution to the radio spectrum does not exceed the observed contribution at the highest energies where it is reliably observed. Taking this to be eV, corresponding to frequency GHz, and the corresponding temperature K eV (see fig. 1), we obtain the bound

 Yχ<2×10−7(Zd1100) (21)

on the relative initial abundance of to photons, , before the decays. From (19) one finds that for greater than a few times , the Compton contribution to the spectrum extends well above the highest frequency GHz measured by ARCADE-2 before cutting off. For larger values of such that the Compton contribution extends into the x-rays, we can derive a much stronger constraint.

To compare (18) with observations such as those of Chandra Hickox:2007gj (), we compute

 ∫Iνdν=14π∫E2E1dEγEγdnγ,0dEγ≅85Yχ(Zd/1100)−1 ergcm2sdeg2 (22)

in an energy window over which surface brightness has been measured (multiplying by to convert from str to deg). The value is for the keV energy interval, in which the measured value is erg cm s deg. We thereby derive a limit on the abundance of decaying particles relative to photons of

 Yχ≲10−14(Zd1100) (23)

which is much more stringent than (21). However we emphasize that (23) applies only for GeV; otherwise Compton emission in the keV region is kinematically blocked, since then and the Compton spectrum cuts off as shown in fig. 5. In this case, a stronger constraint might ostensibly come from lower-energy x-ray observations, such as from XMM-Newton xmm-newton (). But in section 1 we noted that photons of energy lower than 1 keV are effectively attenuated through absorption by hydrogen. Hence the 600 MeV restriction is sufficient to generally circumvent the constraints from x-ray observations.

## Vi Synchrotron emission

To derive the spectrum of synchrotron emission from the injected electrons, we follow a similar procedure as for Compton in the previous section. The rate of production of synchrotron photons from a single electron of energy is given by Blumenthal:1970gc ()

 dNγdtdEγ = 1(4π)2νdEγdtdν (24) = e2m2e√3πE2eF(x)

where and is the integral over harmonics of the gyrofrequency of the electron:

 F(x) = ∫∞xK3/2(ξ)dξ (25) ≅ 4π312213Γ(13)x−23e−x

Although can be expressed exactly in terms of hypergeometric functions, we find the more tractable approximation in the second line, which is just the known small- asymptotic value times . This is accurate for while overestimating the true value by % near where is maximized, which is sufficient for our purposes.

The desired quantity is obtained by multiplying (24) by the electron density and integrating with respect to time. As in the previous section, the relevant time dependence is that of the rapidly decreasing electron energy (16). Again changing variables and taking (note that also in order to reexpress at in terms of its comoving value), we find

 dnγ,0dEγ = 0.048YχB3/2m3/2e√eZ3/2dT0E3/2γ∫√CBEdme√CBduu2/3e−u2 (26)

where . We are interested in parameter values such that the upper limit of the integral in (26) is ; in that case the integral can be approximated by the incomplete gamma function . The latter approaches 1.13 as but falls exponentially with large . To insure that , it is necessary to keep over the relevant range of . This puts a constraint on the injected energy, here from ,

 mχ>6.7 GeV×(Zd1100)−1/2(B50 nG)−1/2 (27)

The distortion away from the pure power law for the case where , hence (27) is saturated, is shown in fig. 6.

Using and normalizing to the measured value at 45 MHz, we find that the required relative abundance is

 Yχ≅6×10−12(Zd1100)3/2(B50 nG)−3/2 (28)

which is inconsistent with the Chandra constraint (23) unless exceeds the CMB bound. To make them consistent, we would need . Here we have expressed the result with lower redshifts in mind, since this reduces the size of the required field somewhat. This assumes that MeV so that (23) applies. To evade the x-ray constraint by having MeV, (27) would require to be even larger, . However we shall see that CMB constraints on charged particle injection require yet larger values of than either of these bounds.

## Vii Dark matter annihilations and nonsudden decays

In the previous sections, for simplicity we considered the case in which electrons are injected suddenly at a specific time (the sudden decay approximation). It is straightforward to generalize this to a continuous injection of energy; one replaces by

 Yχ→∫dtddYχdt (29)

(taking into account the time-dependence of in the expressions for ) where is the relative abundance per unit time of particles that decay into electrons. The same logic can be applied to annihilations . The expressions for for decays or annihilations are, respectively,

 (30)

Let us consider the synchrotron photons coming from annihilation . We can convert the integral over into one over using , which is not accurate at late times when dark energy dominates, but is adequate during matter domination. Carrying out the integral over , where for a first estimate we approximate as a step function that goes to zero at , we find that in (26) is modified by replacing

 YχZ−3/2d→32Y2χ,0t0nγ,0⟨σv⟩ln(ZdecZend) (31)

where is the value of such that and . Here is the dark matter abundance after it has frozen out, which we can rewrite as , and is the cross section to produce . It is therefore no more than , where cm/s is the standard thermal cross section for obtaining the observed .555If there is some other annihilation channel determining the relic density, then for can be smaller than this bound. Here we only assume that since need not be the only species of dark matter. Now equating the modified (26) to the observed radio spectrum, we find the necessary condition (but not sufficient, as will become immediately apparent)

 mχ=0.6 MeV×√r(ΩχΩDM)1/2(B50nG)3/4 (32)

where we define to be the ratio of for to the total annihilation cross section. For simplicity we assume that any additional annihilation channels do not result in charged particles, since otherwise these will quickly decay into electrons and behave similarly to the electrons produced as primary annihilation products. The requirement (32) insures that the spectrum matches the observed one at arbitrarily low frequencies, but it does not take into account the need for the high-frequency cutoff to extend to the observed GHz region. This requires , which is only consistent if , which puts a second constraint on the mass (see (27)),

 mχ>6.7GeV×(B50nG)−1/2 (33)

These constraints begin to overlap (leaving no allowed parameter space) if the field is too small, leading to the lower bound

 B>90μG×(ΩDMrΩχ)2/5 (34)

In the case where this bound is saturated, the mass is MeV. We will see that this is strongly ruled out by CMB constraints on charged particle injection. Thus dark matter annihilations in the early universe cannot explain the radio excess even with late-time magnetogenesis.

For decays, we can carry out the time-averaging to find the more accurate dependence of the normalization of the spectra upon . This results in a correction factor of order unity:

 1Zpd→1Zpd∫t0/τtdec/τdxx2p/3e−x (35)

where y is the age of the universe at decoupling, and Gyr is the present age. For , the correction factor is approximately for , corresponding to the cases of Compton and synchrotron emission, respectively. Thus the sudden approximation works quite well for the case of decays when is in the range of interest.

## Viii CMB constraints on charged particle injection

There are stringent bounds on the injection of ionizing energy after decoupling due to its distortion of the CMB anisotropies, either through particle decays Chen:2003gz (); Zhang:2007zzh () or annihilations Galli:2009zc (); Slatyer:2009yq (). These depend mainly upon the total amount of kinetic energy of charged particles that is injected, and so are closely correlated with the requirements for producing the observed radio background. We first consider the case of decaying particles. Ref. Zhang:2007zzh () obtains

 Yχτ≲2×10−25s−1YDM (36)

for s, and a somewhat less stringent limit at shorter lifetimes; however the above approximation will be adequate for our purposes since the tension of this constraint with what is needed for the radio excess is minimized at the higher values of , despite the modest weakening of the constraint at low . Here is the standard abundance that would have if it was a stable particle constituting the full WMAP DM abundance. Substituting for we then find

 Yχ<2×10−17(6Zd)3/2(1 GeVmχ) (37)

We have taken to be the latest decay epoch consistent with generating the signal before significant clustering of the DM would occur and induce unwanted fluctuations. If we take the synchrotron mass bound (27) to be saturated, then (37) becomes G. Comparing this to the required value of (28), we get the lower bound on

 B>5μG×(Zd6)5/4 (38)

For this value of , the minimum mass from (27) is GeV, and the abundance is . This is compatible with the x-ray constraint (23), which can be written as . A summary of the bounds on electron injection from decay is presented in Figure 7.

For annihilations, we adapt the result of ref. Slatyer:2009yq () to obtain

 ⟨σv⟩≡r⟨σv⟩tot≲f−1⟨σv⟩0(mχ10GeV)(ΩDMΩχ)2 (39)

where is an efficiency factor that is close to 1 for annihilation into electrons, and was defined below eq. (32). Since the rate of injection of charged particles depends upon , we have appended the factor for the case to account for the weakening of the bound derived by Slatyer:2009yq (), where it was assumed that . Consider the case where the field is just barely large enough to give the radio excess, so that (34) is saturated. Then using the fact that and the value of determined below eq. (34), we find that the CMB constraint can be written as

 (rΩχΩDM)<6×10−3 (40)

which with eq. (34) then implies mG (milligauss) in order for annihilations to give the radio excess. This is such a strong field that it would overclose the universe.

## Ix Millicharged model

If we insist that was generated prior to decoupling and thus must obey the CMB constraints, additional new physics is required. In this section we present a somewhat more complicated model, in which the injected particles, instead of being electrons, are exotic particles that we denote by , carrying a fractional electric charge . In adddition, we assume that has a larger charge under a hidden, unbroken gauge group that comes with its own “dark photons,” (denoted by ) having a cosmological background analogous to the CMB. This scenario has been considered previously in the millicharged atomic dark matter model of ref. Cline:2012is (); Cline:2012ei ().

It is straightforward to generalize the Compton result (18) to the present model. For the rate of production of CS photons, the combination coming from is replaced by

 A≡g2(ϵe)2n~γ¯¯¯¯E1/2~γ+(ϵe)4nγ¯¯¯¯E1/2γ′ (41)

On the other hand in the energy loss rate that eventually appears in the denominator, the combination is replaced by

 A′=g4ρ~γ+g2(ϵe)2(ρ~γ+ργ)+(ϵe)4ργ (42)

Thus the CS spectrum will be proportional to . In addition the electron mass is replaced by . Assuming that the first term in each of (41) and (42) dominates, we find that the x-ray constraint (23) generalizes to

 Yχ<5×10−17(Zd6)(~T0T0)1/2(gϵe)2(mem~e) (43)

In contrast, the CMB constraint depends mainly upon the fraction of ionizing electromagnetic energy per decay. In the present model, this fraction is given by . The bound (37) is accordingly weakened to become

 (44)

where we have assumed that the leading terms in numerator and denominator of the reduction factor are dominant, and also that . These two assumptions would be consistent if for example , the maximum allowed by constraints on the Hubble rate during big bang nucleosynthesis. We need to choose the more restrictive of the two bounds (43) and (44) in what follows.

The millicharged model can also admit a primordial dark magnetic field whose magnitude is less constrained than that of the visible field, allowing for the possibility of early magnetogenesis that is unconstrained by the CMB. The modified synchrotron spectrum can be deduced from the derivation of section VI by making the following changes. (1) for the rate of emission in (24); (2) in the definition of , since the cyclotron frequency is now determined by this combination; (3) in the Thomson cross section and coming from the energy loss rate, which is dominated by Compton emission; (4) obviously everywhere as well. In this way, we find that the required abundance (28) generalizes to

 Yχ,sync = 2.7×10−17× ϵ−2(~T0T0)4(Zd6)32(~B1μG)−32(mem~e)32(ge)52

which is valid as long as . Similarly, the generalization of the mass constraint (27) is

 mχ≳20GeV(Zd6)−12(~B1μG)−12(eg)12(m~eme)32 (46)

Proceeding as in section VIII, by inserting (46) into (44) and comparing to (IX), we find that all the dependence upon new parameters cancels out and yields (38) again, but now as a constraint on instead of . The advantage is that there are relatively few direct constraints on the dark field, and it could therefore have been generated prior to decoupling. For example taking , GeV, G, , , we find from (IX) that , which marginally satisfies the CMB constraint (44) for any value of . We can then take MeV (hence TeV) and be consistent with constraints on millicharged particles Davidson:2000hf (); Jaeckel:2010ni (); Prinz:1998ua (). The x-ray constraint (43) is weaker than (44) and provides no additional information. For such values of , the mass density of is sufficiently small before it decays, , that there is no danger of it significantly affecting the expansion history of the universe.

## X Conclusions

The diffuse excess radio background observed by the ARCADE-2 collaboration has a spectrum that appears to be consistent with observations of four other groups. This adds to the plausibility of it being a real anomaly, and the observation of ref. Holder:2012nm () concerning its homogeneity seems to constitutes a strong motivation for its origin to be cosmological in nature, in the absence of ubiquitous astrophysical sources of size Mpc.

Using approximate analytic methods,666notably the approximation of (25) and extending the upper limit of integration in (26). We expect this to introduce errors of order 20% in the normalization of the spectrum, and O(1) errors in the details of the spectrum in the high-frequency region where it is rapidly falling away from the power law behavior. we have shown that injection of relativistic electrons after decoupling, through late decays of a metastable particle , can give rise to a radio excess with the correct power law fall-off through synchrotron emission. However we find that injecting enough charge to get the right normalization of the excess background should also have distorted the CMB fluctuations relative to their observed properties, due to the deposition of electromagnetic energy into the intergalactic medium. This contradiction can be circumvented if the primordial field is larger than expected on the basis of other CMB constraints, G. The size of the required field is smallest when the decays are as late as possible (but still before the onset of structure formation, to avoid large fluctuations in the radio background). Such large fields would have to be generated by some unknown cosmological mechanism after decoupling in order to avoid the CMB -field constraint, several nG.

Based on our results, the simplest and most conservative cosmological scenario for producing the excess radio background is through the synchroton radiation emitted by electrons from decays at redshift , in a magnetic field of strength G, with GeV and an initial abundance relative to photons. Earlier decay epochs would require larger fields, given by (38). This requires magnetogenesis at low redshifts to circumvent CMB constraints on the primordial magnetic field. We find that the annihilation scenario, , is too inefficient and would require unacceptably large fields to produce the observed radio excess.

One might wonder whether decays into other known charged particles than the electron could give more promising results, since the CMB bound is weakened in that case. The weakening comes about because some of the initial energy eventually goes into neutrinos, which do not ionize the intergalactic medium. We do not expect any improvement in this way however, since the same loss of efficiency in the production of radio and other electromagnetic backgrounds should occur, and the amount of radiation produced by the unstable charged particles before they turn into electrons must be very small due to their short lifetimes, compared to the time scale for energy loss through CS.

To demonstrate an alternative example that does not require late-time magnetogenesis, we presented a model of small fractionally charged () decay products with mass MeV, produced from the decays of heavy TeV parent particles. If the fractionally charged particles undergo cyclotron motion due to their interaction with a dark background magnetic field G, the resulting synchrotron emission (in normal photons) can explain the radio anomaly without conflicting with CMB observations. The fractionally charged particles are stable relics, but with an abundance less than that of baryons, making them innocuous at present times. This example might be regarded as proof that it is possible to construct a viable, albeit exotic model; we hope it will motivate others to search for alternative possibilities.777Recently ref. Lawson:2012zu () appeared, which also proposes a cosmological dark matter mechanism.

Acknowledgments. We are grateful to F. Tavecchio for pointing out an important error in the first version of this manuscript. We thank M. Cirelli, R. Durrer, F. Finelli, M. Giovannini, G. Holder, Z. Liu, T. Kahniashvili, M. Kamionkowski, G. Moore, V. Poitras, P. Scott and T. Slatyer for helpful dicussions or correspondence. Our work was supported by the Natural Sciences and Engineering Research Council (NSERC, Canada). AV is supported by European contract PITN-GA-2011-289442-INVISIBLES and Fonds de recherche du Québec - Nature et technologies (FQRNT).

## Appendix A Stochastic B field from χ→e+e− decay

In this appendix we estimate the stochastic magnetic field spectrum that is generated from the decays of a heavy particle into . Consider a decay at time and position , producing an electron of velocity . The vector potential due to the pair is

 →A=e→v(|→r−→vt|