Probing relic neutrino decays with 21 cm cosmology
Marco Chianese, Pasquale Di Bari, Kareem Farrag, Rome Samanta
PProbing relic neutrino radiative decays with21 cm cosmology
M. Chianese , P. Di Bari , K. Farrag , and R. Samanta Physics and Astronomy , University of Southampton,Southampton, SO17 1BJ, U.K. School of Physics and Astronomy , Queen Mary, University of LondonLondon, E1 4NS, U.K.
September 21, 2018
Abstract
We show how 21 cm cosmology can test relic neutrino radiative decays into sterileneutrinos. Using recent EDGES results, we derive constraints on the lifetime of thedecaying neutrinos. If the EDGES anomaly will be confirmed, then there are twosolutions, one for much longer and one for much shorter lifetimes than the age ofthe universe, showing how relic neutrino radiative decays can explain the anomalyin a simple way. We also show how to combine EDGES results with those fromradio background observations, showing that potentially the ARCADE 2 excesscan be also reproduced together with the EDGES anomaly within the proposednon-standard cosmological scenario. Our calculation of the specific intensity at theredshifts probed by EDGES can be also applied to the case of decaying dark matterand it also corrects a flawed expression used in previous literature.
With 21 cm cosmology we are entering a new exciting phase in the study of the history ofthe universe and how this can be used to probe fundamental physics [1, 2]. Observationsof the redshifted 21 cm line of neutral hydrogen from the emission or absorption of thecosmic microwave background radiation (CMB) by the intergalactic medium, can test We refer to hydrogen-1 (protium). Deuterium has an analogous line but at 92 cm. a r X i v : . [ h e p - ph ] S e p he cosmic history at redshifts z ∼ This range corresponds to those threeperiods, after recombination, on which we have fragmentary information: the dark ages ,from recombination at z rec (cid:39) z (cid:39)
30, when first astrophysical sources start toform; the cosmic dawn , from z (cid:39)
30 to the time when reionisation begins at z (cid:39) Epoch of Reionisation (EoR), from z (cid:39)
15 to z (cid:39) . Inthis way observations of the cosmological 21 cm line global signal can test the standardΛCDM cosmological model during a poorly period of the cosmic history, considering thatthe most distant galaxy is located at z = 11 . Experiment to Detect the Global Epoch of ReionisationSignature ) collaboration claims to have discovered an absorption signal in the CMBradiation spectrum corresponding to the redshifted 21 cm line at z (cid:39) . ∼ . σ deviation fromthe predictions of the ΛCDM model and for this reason the EDGES anomaly has drawngreat attention. It should be said that another group [6], re-analysing publicly availableEDGES data and using exactly their procedures, finds almost identical results but theyclaim that ‘the fits imply either non-physical properties for the ionosphere or unexpectedstructure in the spectrum of foreground emission (or both)’ concluding that their results‘call into question the interpretation of these data as an unambiguous detection of thecosmological 21-cm absorption signature.’ Therefore, more observations will be necessaryto confirm not only the anomaly but even the absorption signal.In the light of these recent experimental developments, it is anyway interesting tothink of possible non-standard cosmological scenarios that can be tested with 21 cmsignal observations at high redshifts and that might either explain the EDGES anomaly(if confirmed) or in any case be constrained. The EDGES anomaly can be expressed interms of a value of the photon-to-spin temperature ratio T γ ( z ) /T S ( z ) at redshifts z =15—20, where the absorption profile is observed, that is about twice what is expectedin a standard cosmological scenario. This can be of course due either to a larger valueof T γ ( z ) or a smaller value of T S ( z ) or some combination of the two. In this Letter,we show how radiative decays of the lightest relic neutrinos can explain the EDGESanomaly producing, after recombination, a non-thermal early photon background ableto rise T γ ( z ) above the CMB value. A similar scenario, recently revisited in [8], whereheavier relic neutrinos decay radiatively into lighter ordinary neutrinos [9, 10, 11] is ruled We will discuss this range in more detail in Section 2. The redshift boundaries of these stages are, in fact, very model dependent. Definite values stronglydepend on astrophysical parameters (e.g., see [3] for a recent parameter study). The approximate value z (cid:39) . Planck upper bound (cid:80) i m i (cid:46) .
17 eV (95% C . L . ) [12] and since, it requires a too large effective magneticmoment responsible for the decay. In our scenario the lightest relic neutrinos decayradiatively into sterile neutrinos and this allows to circumvent both bounds. The paper is organised as follows. In Section 2 we briefly review 21 cm cosmology andthe EDGES results. In Section 3 we discuss how lightest relic neutrinos radiative decayscan explain the EDGES anomaly. Finally, in Section 4, we draw the conclusions.
The 21 cm line is associated with the hyperfine energy splitting between the two energylevels of the 1s ground state of the hydrogen atom characterised by a different relativeorientation of electron and proton spins: anti-parallel for the singlet level with lowerenergy, parallel for the triplet level with higher energy. The energy gap between thetwo levels and, therefore, of the absorbed or emitted photons at rest, is E = 5 . µ eVcorresponding to a 21 cm line rest frequency ν rest21 = 1420 MHz.A shell of neutral hydrogen at a given redshift z (cid:46) z rec , after recombination, can thenact, thanks to the 21 cm transitions, as a detector of the background photons producedat higher redshifts. In standard cosmology this background is just given by the CMBthermal radiation with temperature T CMB ( z ) = T CMB, (1+ z ), where T CMB, = 2 . K (cid:39) . × − eV.This possibility relies on the brightness contrast between the intensity of the 21 cmsignal from the shell of neutral hydrogen gas at redshift z and the background radiationat the observed (redshifted) frequency ν ( z ) = ν rest21 / (1 + z ). The brightness contrastcan be expressed in terms of the
21 cm brightness temperature (relative to the photonbackground) [14]: T ( z ) (cid:39)
23 mK (1 + δ B ) x H I ( z ) (cid:18) Ω B h . (cid:19) (cid:20)(cid:18) . m h (cid:19) (cid:18) z (cid:19)(cid:21) / (cid:20) − T γ ( z ) T S ( z ) (cid:21) , (1)where Ω B h = 0 . m h = 0 . δ B is the baryon overdensity, x H I is the fraction of neutral hydrogen, T γ ( z )is the effective temperature, at frequency ν ( z ), of the photon background radiation(coinciding with T CMB ( z ) in standard cosmology) and T S ( z ) is the spin temperature See end of Section 3 for more details on this point while for an updated review on neutrino massconstraints see for example [13]. n to that one of theground state n in such a way that n n ( z ) ≡ g g e − E TS ( z ) , (2)where g /g = 3 is the ratio of the statistical degeneracy factors of the two levels. Clearlyif x H I vanishes, then there is no signal, since in that case all hydrogen would be reionisedand there cannot be any 21 cm transition. The spin temperature is related to T gas , thekinetic temperature of the gas, by (cid:18) − T γ T S (cid:19) (cid:39) x c + x α x c + x α (cid:18) − T γ T gas (cid:19) , (3)where x α and x c are coefficients describing the coupling between the hyperfine levels andthe gas. In the limit of strong coupling, for x α + x c (cid:29)
1, one has T S = T gas , while inthe limit of no coupling, for x α = x c = 0, one has T S = T γ and in this case there is nosignal. The evolution of T with redshift can be schematically described by five mainstages [1, 2]:(i) In a first stage after recombination, during the dark ages, the gas is still coupledto radiation thanks to a small but non negligible amount of free electrons thatstill interacts via Thomson scatterings with the photon background. In this caseone has T γ = T gas = T S and consequently T = 0, i.e., there is no signal. Thisstage lasts until the gas starts decoupling from radiation above z gasdec (cid:39) T gas ( z ) = T ( z dec ) (1 + z dec ) .(ii) In a second stage, approximately for 250 (cid:38) z (cid:38)
30, still during the dark ages andwith the precise boundary values depending on different cosmological details, onehas approximately T S (cid:39) T gas , since gas collisions are efficient enough to couple T S to T gas . In this case one has T < z (cid:39)
30 the gas becomes so rarified that the collision rate becomes too low toenforce T S (cid:39) T gas and in this case one enters a regime where x a + x c (cid:28) This is an approximated relation valid for T gas (cid:39) T c , where T c is the colour temperatureparameterising the intensity of the UV radiation emitted by the astrophysical sources. Notice thatout of the four different temperatures we introduced, only T CMB and T gas are genuine thermodynamictemperatures associated to a thermal distribution. This conclusion is approximate and a very small signal is present even at high redshifts mainly dueto the fact that T c deviates slightly from T gas . This has been studied recently in detail in [16] and it wasfound − T (cid:39) . z (cid:39) S (cid:39) T γ . In this stage, during the cosmic dawn, one has T (cid:39) (iv) At z (cid:39)
30, gas also starts collapsing under the action of dark matter and firstastrophysical sources start to form with emission of Ly α radiation that is able,through Wouthuysen-Field effect [17], to gradually couple again T S to T gas . In theredshift range z h (cid:46) z (cid:46)
25, where z h (cid:39) z (cid:39) T <
0, implying anabsorption signal. This is within the range tested by EDGES whose results seem toconfirm the existence of the absorption signal.(v) In a fifth stage, for z (cid:46) z h (depending on the precise value of z h this stage can eitherstart during cosmic dawn and ending during the epoch of reionization or entirelyoccurring during the latter), the gas gets reheated by the astrophysical radiationand T S (cid:39) T gas > T γ , so that T turns positive and one has an emission signalfrom the regions that are not fully ionised. Eventually all gas gets ionised until thefraction of neutral hydrogen vanishes and the signal switches off again. EDGES High and Low band antennas probe the frequency ranges 90-200 MHz and50-100 MHz respectively overall measuring the 21-cm signal from between redshift 6 and27, which corresponds to an age of the universe between 100 Myr and 1 Gyr and includesthe epochs of reionization and cosmic dawn, when first astrophysical sources form and asecond stage of absorption signal is predicted (the fourth and fifth stage in the descriptionabove). The EDGES collaboration found an absorption profile approximately in the range z = 15—20 with the minimum at z E (cid:39) .
2, corresponding to ν ( z E ) (cid:39)
78 MHz, witha 21 cm brightness temperature T ( z E ) = − . +0 . − . K at 99% C . L . , including estimatesof systematic uncertainties. From Eq. (1), since (1 + δ b ) x H I ( z E ) (cid:39)
1, this translatesinto T γ ( z E ) /T S ( z E ) = 15 +15 − . . On the other hand, at the centre of the absorption profiledetected by EDGES, one expects, assuming the ΛCDM model, T γ ( z E ) = T CMB ( z E ) = T CMB, (1 + z E ) (cid:39)
50 K and T gas ( z E ) (cid:39) T CMB ( z gasdec ) (1 + z E ) / (1 + z gasdec ) (cid:39) z gasdec (cid:39)
150 and T gasdec (cid:39)
410 K respectively the redshift and the temperatureat the time when the gas decoupled from radiation. From Eq. (1) one then immediatelyfinds T ( z E ) (cid:38) − . T S ( z E ) = T gas ( z E ) and A detailed description and in particular how suppressed the signal is in this stage depends on variousastrophysical parameters [3]. In this stage the signal crucially depends on astrophysics and it should be said that not in all scenarios T gas becomes larger than T γ and in this case the emission signal is missing [18]. T γ ( z E ) /T gas ( z E ) (cid:39)
7. Therefore, the best fit value for T ( z E ) is about 2 . . L . it is still 50% lower.If this anomalous result will be confirmed and astrophysical solutions ruled out, then,very interestingly, it can be regarded as the effect of some non-standard cosmologicalmechanism. For example, it has been proposed that a (non-standard) interaction of thebaryonic gas with the much colder dark matter component would cool down T gas , andconsequently T S , below the predicted ΛCDM value [7]. Another possibility is that T gas islower because the gas decouples earlier so that z gasdec > z gasdec (cid:39) T gas ( z E ) (cid:39) . T γ above T at frequencies around ν ( z E ). For example,these could be produced by dark matter annihilations and/or decays [20, 21] and alsogive a signal at other frequencies for example addressing the ARCADE 2 excess at higher( ∼ GHz) frequencies [22] that, however, has not been confirmed by another group usingATCA data [23]. Conversion of dark photons into soft photons has also been proposed asa solution to the EDGES anomaly [24].In the next section we present a mechanism for the production of a non-thermal softphoton component relying on relic neutrinos radiative decays into sterile neutrinos. Evenif the EDGES anomaly will not be confirmed, we show that the EDGES results tightenthe existing constraints [10, 25] on the parameters of the scenario.
The 21 cm CMB photons absorbed at z E fall clearly in the Rayleigh-Jeans tail since E (cid:28) T ( z E ). In this regime the specific intensity depends linearly on temperature,explicitly I CMB ( E, z ) ≡ d F γ CMB E dA dt dE d Ω = 14 π dε
CMB dE = E π [ e E/T
CMB − − E (cid:28) T CMB −→ π T CMB ( z ) E . (4)Only photons with energy E at z (cid:39) z E could be absorbed by the neutral hydrogenproducing a 21 cm absorption global signal. The EDGES results can be explained byan additional non-thermal photon background with I nth ( E , z E ) (cid:39) I CMB ( E , z E ). Theeffective photon temperature T γ ( E, z E ) at an arbitrary energy E can be simply calculated6s T γ ( E, z E ) = E ln − (cid:18) E π I γ ( E, z E ) (cid:19) E (cid:28) T γ −→ π E I γ ( E, z E ) , (5)where we defined I γ ( E, z E ) = I nth ( E, z E ) + I CMB ( E, z E ). The EDGES anomaly can beexplained imposing T γ ( E , z E ) /T CMB ( z E ) = 2 . +2 . − . , or, equivalently, R ≡ I nth ( E , z E ) I CMB ( E , z E ) = T γ nth ( E , z E ) T CMB ( z E ) = R E ≡ . +2 . − . , (6)where T γ nth is defined in terms of I nth in the same way as T γ is defined in terms of I γ inEq. (5). We consider the radiative decay of active neutrinos ν i with mass m i and lifetime τ i into a sterile neutrino ν s with mass m s , i.e., ν i → ν s + γ . For definiteness we will refer tothe case of lightest neutrino decays corresponding to i = 1. We will comment at the endhow our results simply change if one considers heavier neutrinos. If decays occur aftermatter-radiation decoupling, photons produced from the decays will not distort CMBthermal spectrum but will give rise to a non-thermal γ background [10] contributing to R . For a given m one has two limits for m s : a quasi-degenerate limit for m (cid:39) m s anda limit m s (cid:28) m . For m s (cid:28) m the bulk of neutrinos, with E ∼ T CMB , necessarily decay when they arerelativistic. This is easy to understand. Let us introduce the scale factor a = (1 + z ) − and its corresponding value a E ≡ (1 + z E ) − at z E . In the matter-dominated regime wecan write a ( t ) (cid:39) a E ( t/t E ) , where t E (cid:39)
222 Myr is the age of the universe at z = z E [26]. For neutrinos that decay at rest at time t one has to impose m = 2 E a E /a ( t )in order to have photons with the correct energy at t E . Imposing that decays occurafter recombination, otherwise the non-thermal component would thermalise or produceunacceptable distortions to the CMB spectrum, and of course before the time whenphotons are absorbed by neutral hydrogen, corresponding to a condition z E < z ( t ) 012 meV (cid:46) m (cid:46) . 71 meV, showing that the ν ’s are too light tobe treated non-relativistically for m s (cid:28) m . The origin and properties of neutrino masses and mixing would be related to extensions of the SM(e.g., grand-unified theories). Simplest models usually require the existence of very heavy sterile neutrinos( m s (cid:29) 100 GeV) in the form of right-handed neutrinos. However, the existence of light sterile neutrinoscannot be excluded and many models have been proposed especially in connection with different neutrinomixing anomalies (for a review see for example [27]). This also shows that the two heavier neutrinos radiative decays would produce photons at too highfrequencies, considering that m ≥ m sol (cid:39) m ≥ m atm (cid:39) 50 meV, where the lower boundsare saturated in the normal hierarchical limit. One could consider radiative decays ν , → ν + γ and inthe quasi-degenerate limit m (cid:38) . 12 eV photons with the correct energy would be produced. However,the upper bound m (cid:46) . 07 eV placed by the Planck collaboration now rules out this possibility [12]. 7n the other hand the non-relativistic case can be realised in the quasi-degeneratelimit for m (cid:39) m s since one can then have m (cid:29) T ν ( z ) (cid:39) (4 / / T ( z ) at the timewhen they decay. Indeed at z = z E one has T ν ( z E ) (cid:39) m (cid:46) 50 meV, one can well have m (cid:39) m s (cid:29) (cid:38) m / meV (cid:38) a window that will be fully testedby close future cosmological observations [28]. Moreover in this (testable) non-relativisticand quasi-degenerate case not only it is easy to calculate R , as we will see, but also oneobtains the most conservative constraints on τ and ∆ m ≡ m − m s since, as we willpoint out, if neutrinos decay relativistically, constraints get more stringent.Let us then calculate R for m (cid:39) m s (cid:29) T ν ( z ). An emitted photon has an energy atdecay E = ∆ m . Moreover let us suppose first that all neutrinos decay instantaneously at t = τ corresponding to a redshift z decay such that a decay = (1 + z decay ) − (cid:39) a E ( τ /t E ) / .A sketchy representation of this toy model is given in Fig. 1. Requiring that photonsproduced from neutrino decays are then (21 cm) absorbed at z = z E , one has to impose∆ m = E a E /a decay , implying an unrealistic fine-tuned relation between τ , E and∆ m . In addition it is easy to see that, since all photons produced in the decay contributeto the signal, one obtains a far too high value of R . This is because in this instantaneousdecay description one has simply I nth ( E , z E ) = n ∞ ν ( z E )4 π = 311 ζ (3)2 π T CMB ( z E ) , (7)where n ∞ ν ( z E ) = (6 / ζ (3) /π ) T CMB ( z E ) is the relic neutrino number density at z E inthe standard stable neutrino case. This gives straightforwardly: R (cid:39) R (cid:63) ≡ ζ (3)11 (cid:20) T CMB, (1 + z E ) E (cid:21) (cid:39) . × , (8)many orders of magnitude larger than R E . However, this simplistic instantaneous descriptionhas the merit to show the natural normalisation for the specific intensity of the non-thermalphotons in terms of n ∞ ν ( z E ) and for R itself in terms of R (cid:63) .Let us now calculate R removing the instantaneous assumption. Writing the fluidequation for the energy density of non-thermal photons produced by ν decays [10, 25, 29], dε γ nth dt = ∆ m τ n ∞ ν ( t ) e − tτ − ε γ nth H , (9) Moreover these processes need values of the effective magnetic moment ruled out by current experimentalupper bound [11, 8]. The lower bound m (cid:38) 10 meV corresponds to m (cid:38) T ν ( z E ) that is quite a conservative conditionto enforce that the bulk of neutrinos are non-relativistic when they decay since remember that for aMaxwell-Boltzmann distribution (cid:112) (cid:104) v (cid:105) = (cid:112) T /m . In this way the bulk of neutrinos have a kineticenergy that is negligible compared to the rest energy. z γ CMB (cid:24)(cid:24)(cid:24)(cid:24) γ γ nth 10 100 1000 + z E + z d ec + z d ec a y ???? ? ? ????? ? ? Figure 1: Schematic picture describing the generation of the 21 cm absorption signalwithin a non-standard history of the universe including ν radiative decays producing anon-thermal photon background. At redshifts z ∼ z E the first astrophysical sources couplethe spin temperature to the gas temperature (via Wouthuysen–Field effect) inducing theabsorption of (redshifted) γ photons.where H ≡ ˙ a/a is the expansion rate, one easily finds a solution in terms of a Eulerintegral ε γ nth ( t E ) = ∆ m τ n ∞ ν ( t E ) (cid:90) t E dt e − tτ a ( t ) a ( t E ) . (10)The integral is done over all times t when photons are produced by neutrino decays withenergy ∆ m that is redshifted to an energy E ( t, t E ) = ∆ m a ( t ) /a E at t E . Photonswith the correct energy E at t E are produced at a specific time t such that a /a E = E / ∆ m , where a ≡ a ( t ). Of course notice that imposing z rec > z ≡ a − − > z E ,one would find E (cid:46) ∆ m (cid:46) . 35 meV, analogously to the range found for m in thecase m s (cid:28) m . However, since we are assuming that neutrinos are non-relativistic atdecays, and this implies T ν ( z ) (cid:39) . 18 eV (1 + z ) / (1 + z dec ) (cid:46) m (cid:46) 50 meV, one finds z (cid:46) E (cid:46) ∆ m (cid:46) . × − eV . (11)We can now easily switch from time to energy derivative finding I nth ( E , z E ) = 14 π dε γ nth dE = n ∞ ν ( z E )4 π (cid:18) E ∆ m (cid:19) / e − tEτ (cid:16) E m (cid:17) / H E τ , (12)9here H E ≡ H ( t E ) (cid:39) / (3 t E ). From the definition of R (see Eq. (6)) and R (cid:63) (see Eq. (8)),one immediately obtains R = R (cid:63) (cid:18) E ∆ m (cid:19) / e − tEτ (cid:16) E m (cid:17) / H E τ . (13)The condition R ≤ R E , where the equality corresponds to the condition to explain theEDGES anomaly and the inequality implies constraints on τ and ∆ m , can be put in thesimple form x e − x = 23 R E R (cid:63) = 2 . +4 − × − , (14)where we defined x ≡ t E τ (cid:18) E ∆ m (cid:19) / . (15)There are clearly two solutions. A first one (referred to as ‘EDGES A’ in Fig. 2) for τ (cid:29) t E is simply given by x = 2 . +4 − × − , from which one finds τ ≥ +300 − t (cid:18) − eV∆ m (cid:19) / , (16)where t = 13 . . × s is the age of the universe. A second solution (referredto as ‘EDGES B’ in Fig. 2) is found for τ (cid:28) t E and is given by x = 15 . +1 . − . , from whichone finds τ ≤ . +0 . − . × yr (cid:18) − eV∆ m (cid:19) / . (17)For this second solution one has to consider that decays should occur mainly aftermatter-radiation decoupling time in order to have a photon non-thermal background andone has to impose τ (cid:38) t dec (cid:39) . × yr. Moroever, though photon energies are muchbelow the thermal bath temperature, they might produce too large deviations of CMBfrom a thermal spectrum. Even though this second solution is less appealing and likelynot viable, it is also interesting that one could in principle expect a number of neutrinospecies at recombination lower than three if only a fraction of the decays are allowed tooccur before recombination.We have now also to consider whether photons produced from neutrino decays mightgive visible (wanted or unwanted) effects at other frequencies. First of all one shouldworry of the CMB spectrum tested by the the COBE-FIRAS instrument in the range offrequencies (2—21) cm − corresponding to (60—600) GHz or to energies (0 . . 5) meV Of course one should also not forget that ∆ m is constrained within the range in Eq. (11). m Δ m = E EDGES B EDGES AARCADE AARCADE B EXCLUDED by EDGESEXCLUDED by ARCADEEXCLUDED by ATCA t E t dec t - - - - - τ ( s ) Δ m ( e V ) τ ( Myr ) Figure 2: The blue lines delimit the excluded region in the plane τ − ∆ m from EDGESdata and the two solutions (A and B) addressing the EDGES anomaly. The orange areaindicates the approximate solution to the ARCADE 2 excess. The cyan area indicatesthe ATCA constraint.[30]. However, since ∆ m < . 09 meV (see Eq. (11)), in this non-relativistic scenario onecompletely circumvents the constraints from CMB thermal spectrum measurements.Radio background observations in the GHz frequencies can also test the scenario eitherconstraining it, as ATCA data do [23], or even providing, with the ARCADE 2 excess[22], another signal to be explained together with the EDGES anomaly. Let us see how they can be combined with 21 cm observations to test relic lightestneutrino decays. In this case the results are given in terms of an effective temperature T rb ( E rb ) of the radio background compared to the Rayleigh-Jeans tail of the CMB spectrum.This time the detection of the produced photons is made directly at the present time, whilein the case of EDGES, as we discussed, photons produced by the decays are absorbed bythe intergalactic medium at the time t E . Therefore, now we have to impose I nth ( E rb , z = 0) = n ∞ ν ( z = 0)4 π e − t ( a rb) τ H ( a rb ) τ = 14 π T rb E , (18)where this time a rb = E rb / ∆ m . If we focus on the solution at τ (cid:29) t , then theexponential can be neglected and using H ( a rb ) = (cid:113) Ω M a − rb + Ω Λ0 and defining a M Λeq ≡ Notice that in [31] the existence of the ARCADE 2 excess is questioned and it is proposed that a amore realistic galactic model can reconcile measurements of uniform extragalactic brightness by ARCADE2 with the expectations from known extragalactic radio source populations. M / Ω Λ0 ) / (cid:39) . 75, one arrives to the condition τ ≥ ζ (3)11 √ Ω M T CMB, (cid:16) E rb ∆ m (cid:17) / T rb E (cid:18) a a M Λeq (cid:19) − / t , (19)where again the equality holds in the case one wants to explain the ARCADE 2 excessor the inequality in the case one impose the constrain from the ATCA data. TheARCADE 2 collaboration claims an excess with T rb = (62 ± 10) mK at a frequency3 . E rb = 13 . × − eV and from the condition (19) one finds τ (cid:39) (800 ± (cid:18) − eV∆ m (cid:19) / (cid:18) a a M Λeq (cid:19) − / t , (20)a solution shown in Fig. 2 (‘ARCADE A’) in orange (at 99% C.L.) together with thecorresponding allowed range for ∆ m found similarly to Eq. (11) with the difference thatnow the energy at the production has to be redshifted at z = 0 instead of z = z E . If this iscompared with the condition we found to explain the EDGES anomaly Eq. (16), one cansee that within the errors the two anomalies can be reconciled, in particular for the lowestvalues of R E corresponding to the highest τ values, and of course with the help of a rb asclose as possible to unity (its maximum value), corresponding to ∆ m = E rb . As in thecase of the EDGES one can also consider a solution for τ (cid:28) t and in this case one finds,neglecting this time the small correction from Ω Λ , τ (cid:39) (70 ± . 7) Myr (10 − eV / ∆ m ) / .This is also show in Fig. 2 (‘ARCADE B’) in orange (at 99% C.L.). However, this time it isclear that there is no overlap with the ‘EDGES B’ solution and this somehow makes evenmore remarkable that in the case of τ (cid:29) t we could find two overlapping solutions. Onecan also consider the ATCA constraints [23] that place a (3 σ ) upper bound T rb (cid:46) 100 mKat a frequency of 1 . 75 GHz. In this case one finds the following (99%) excluded region3 . × yr (cid:18) − eV∆ m (cid:19) / (cid:46) τ (cid:46) (cid:18) − eV∆ m (cid:19) / (cid:18) a a M Λeq (cid:19) − / t . (21)shown in Fig. 2 in light green. Notice that this constraint does not apply to the narrowrange E < ∆ m < E rb (cid:39) × − eV (so that EDGES allows to extend the constraintsat slightly lower values of ∆ m ). Another interesting observation is that in the secondstage of the evolution of T that we outlined in Section 2, for redshifts 250 (cid:38) z (cid:38) Recently a study of the radio background data from the LWA1 Low Frequency Sky Survey (LLFSS)at frequencies between 40 MHz and 80 MHz [33] has found an excess well described by a power law T rb (cid:39) T rb , ( ν/ν ) β with ν = 310 MHz and β (cid:39) − . 58, also fitting the ARCADE 2 results at muchhigher frequencies. For example at ν = 80 MHz the survey finds T rb = (1188 ± z (cid:39) R at a generic redshift z absorption , one can easily see from our expressions that this scalesas ∝ (cid:112) z absorption . Therefore, the scenario predicts a doubled value of R ( z (cid:39) z E . This would be a powerful test of the scenario,though consider that in order to have a signal also in the early absorption stage, theupper bound on ∆ m in Eq. (11) gets more stringent: from the requirement z (cid:46) m (cid:46) E .The derivation of the constraints could be further extended going beyond the quasi-degenerate limit m (cid:39) m s implying necessarily going beyond the non-relativistic regime.In this case one has to take into account the thermal distribution function of neutrinosand from this derive the non-thermal distribution of photons solving a simple Boltzmannequation [32]. The factor R gets reduced for fixed τ since the photon energy spectrumspreads at higher energies and at the energy E at z E there are less photons and so thevalues of the lifetime that are necessary to explain the EDGES anomaly become shorterand this tends to generate a conflict with the constraints from radio observations andlikely with the FIRAS-COBE data as well since photon energies can be much higher.Finally, let us comment that though we have considered for definiteness decays ofthe lightest neutrinos, the results are also valid for heavier neutrinos of course with thereplacement ( τ , ∆ m ) → ( τ , , ∆ m , ). The only difference is that now they automaticallyrespect the condition m , (cid:29) m (cid:38) 10 meV does not hold so that the lightest neutrino mass can bearbitrarily small since lightest neutrinos do not play any role.We should also say that of course, even though for definiteness we considered radiativedecays into sterile neutrinos, our results are valid for any other decay mode involving alight exotic particle. Our results can also be easily exported to the case of quasi-degeneratedark matter recently proposed in [21] though notice that the correct way to calculatethe specific intensity is Eq. (12) (replacing of course neutrino with dark matter number cannot be explained by our model since from Eq. (19) one can see that it predicts T rb ∝ E − . . If we fitthe ARCADE 2 results, then we have a signal at ν = 80 MHz, approximately the same frequency testedby EDGES, that is about 100 times smaller than what LLFSS finds. Of course the LLFSS results do notexclude our model, they simply require an alternative explanation. More generally, they can be hardlyreconciled with the EDGES anomaly within a realistic model since one would need a mechanism wherethe intensity of the produced radiation increases by about 20 times between z (cid:39) z E and today and thisdespite the fact that the expansion dilutes a matter fluid number density, such as primordial black holes,by a factor (1 + z E ) . Even if one finds a model where this huge enhancement of the intensity is realised,this has to be strongly fine-tuned to match both results and this without considering the ARCADE 2excess. t E can beresponsible for the signal while the authors of [21] incorrectly use an expression validfor photons detected at the present time. However notice that in the case of decayingdark matter the fact that the intensity of non-thermal photons has to be comparable tothat of CMB photons, as required by EDGES, is a coincidence. On the other hand, in thecase of decays of active to light sterile neutrinos, the abundance of relic active neutrinos isfixed by thermal equilibrium and this naturally produces a non-thermal photon intensitycomparable to that of CMB photons. One can think of a simple model for example interms of singular seesaw [34] extended with a type II contribution [35]. In this case anactive-sterile neutrino mixing is expected and one can have interesting phenomenologicalconsequences that can help testing the scenario. For example, in addition to obviouspossible effects in neutrino oscillation experiments and in particular in solar neutrinos, thefact that m s < m makes possible a mechanism of generation of a large lepton asymmetryin the early universe [38] with possible testable effects in big bang nucleosynthesis andCMB acoustic peaks [37]. We discussed a scenario where relic neutrinos can radiatively decay into sterile neutrinos.This can be probed with 21 cm cosmology and, from EDGES results, we derived constraintson the mass and lifetime of the decaying active neutrino and on the difference of massesbetween active and sterile neutrino in the quasi-degenerate case. Interestingly, the scenariocan explain the EDGES anomaly if this will be confirmed. The scenario could alsopotentially have other testable phenomenological effects such as the excess at higherradio frequencies claimed by the ARCADE 2 collaboration. Our results can be alsostraightforwardly extended to the case of decaying quasi-degenerate dark matter. Additionalindependent results on the global 21 cm signal from experiments such as SARAS [40] andLEDA [41] might provide independent tests of the EDGES anomaly. If this will beconfirmed, a precise determination of the dependence of the absorption signal on redshiftcould potentially be used to test even more strongly our proposed scenario. Certainly21 cm cosmology opens new fascinating opportunities to test models of new physics and Radiative decays would still generate an effective magnetic moment for active neutrinos but if themixing the sterile neutrino is sufficiently small, a condition easily realised especially for the A solutionwith very long lifetime, this can be well below the upper bound from stellar cooling. One could investigate whether such dynamical generation of the asymmetry might suppress thethermalisation of a ∼ eV sterile neutrino [38] required by the solution to the various anomalies [39]. Acknowledgments PDB and MC acknowledge financial support from the STFC Consolidated Grant L000296/1.RS is supported by a Newton International Fellowship from Royal Society (UK) and SERB(India). KF acknowledges financial support from the NExT/SEPnet Institute. Thisproject has received funding/support from the European Union Horizon 2020 researchand innovation programme under the Marie Sk(cid:32)lodowska-Curie grant agreement number690575. We wish to thank Teppei Katori for useful comments. References [1] J. R. Pritchard and A. Loeb, Rept. Prog. Phys. (2012) 086901 [arXiv:1109.6012[astro-ph.CO]].[2] S. Furlanetto, S. P. Oh and F. Briggs, Phys. Rept. (2006) 181 [astro-ph/0608032].[3] A. Cohen, A. Fialkov, R. Barkana and M. Lotem, Mon. Not. Roy. Astron. Soc. (2017) no.2, 1915 doi:10.1093/mnras/stx2065 [arXiv:1609.02312 [astro-ph.CO]].[4] P. A. Oesch et al. , Astrophys. J. (2016) 129 [arXiv:1603.00461 [astro-ph.CO]].[5] J. D. Bowman, A. E. E. Rogers, R. A. Monsalve, T. J. Mozdzen and N. Mahesh,Nature (2018) no.7694, 67.[6] R. Hills, G. Kulkarni, P. D. Meerburg and E. Puchwein, Concerns about Modelling ofForegrounds and the 21-cm Signal in EDGES data , arXiv:1805.01421 [astro-ph.CO].[7] R. Barkana, Possible interaction between baryons and dark-matter particles revealedby the first stars , Nature (2018) no.7694, 71 [arXiv:1803.06698 [astro-ph.CO]].[8] D. Aristizabal Sierra and C. S. Fong, The EDGES signal: An imprint from the mirrorworld? , arXiv:1805.02685 [hep-ph].[9] Z. G. Berezhiani, M. I. Vysotsky, V. P. Yurov, A. G. Doroshkevich andM. Y. Khlopov, Anomaly in Wien region of background radiation spectrum andradiative decay of relic neutral particles. (In Russian), Sov. J. Nucl. Phys. (1990)1020 [Yad. Fiz. (1990) 1614]. 1510] M. T. Ressell and M. S. Turner, The Grand Unified Photon Spectrum: A CoherentView of the Diffuse Extragalactic Background Radiation , Comments Astrophys. (1990) 323 [Bull. Am. Astron. Soc. (1990) 753].[11] A. Mirizzi, D. Montanino and P. D. Serpico, Revisiting cosmological boundson radiative neutrino lifetime , Phys. Rev. D (2007) 053007 [arXiv:0705.4667[hep-ph]]; J. L. Aalberts et al. , Precision constraints on radiative neutrino decaywith CMB spectral distortion , arXiv:1803.00588 [astro-ph.CO].[12] N. Aghanim et al. [Planck Collaboration], Astron. Astrophys. (2016) A107[arXiv:1605.02985 [astro-ph.CO]].[13] K. Nakamura and S. Petcov, Neutrino masses, mixing and oscillations , inC. Patrignani et al. [Particle Data Group], Chin. Phys. C (2016) no.10, 100001.[14] M. Zaldarriaga, S. R. Furlanetto and L. Hernquist, 21 Centimeter fluctuations fromcosmic gas at high redshifts , Astrophys. J. (2004) 622 [astro-ph/0311514].[15] P. A. R. Ade et al. [Planck Collaboration], Planck 2015 results. XIII. Cosmologicalparameters , Astron. Astrophys. (2016) A13 [arXiv:1502.01589 [astro-ph.CO]].[16] P. C. Breysse, Y. Ali-Hamoud and C. M. Hirata, arXiv:1804.10626 [astro-ph.CO].[17] S. A. Wouthuysen, Astron. J. (1952) 32; G. B. Field, Proc. IRE (1958) 240.[18] A. Cohen, A. Fialkov, R. Barkana and M. Lotem, Mon. Not. Roy. Astron. Soc. (2017) no.2, 1915 [arXiv:1609.02312 [astro-ph.CO]].[19] J. C. Hill and E. J. Baxter, Can Early Dark Energy Explain EDGES? ,arXiv:1803.07555 [astro-ph.CO].[20] N. Fornengo, R. Lineros, M. Regis and M. Taoso, Possibility of a Dark MatterInterpretation for the Excess in Isotropic Radio Emission Reported by ARCADE ,Phys. Rev. Lett. (2011) 271302 [arXiv:1108.0569 [hep-ph]]; L. Lopez-Honorez,O. Mena, A. Moline, S. Palomares-Ruiz and A. C. Vincent, The 21 cm signal andthe interplay between dark matter annihilations and astrophysical processes , JCAP (2016) no.08, 004 [arXiv:1603.06795 [astro-ph.CO]].[21] S. Fraser et al. , Phys. Lett. B (2018) 159 [arXiv:1803.03245 [hep-ph]].[22] D. J. Fixsen et al. , ARCADE 2 Measurement of the Extra-Galactic Sky Temperatureat 3-90 GHz , Astrophys. J. (2011) 5 [arXiv:0901.0555 [astro-ph.CO]].1623] T. Vernstrom, R. P. Norris, D. Scott and J. V. Wall, The Deep DiffuseExtragalactic Radio Sky at 1.75 GHz , Mon. Not. Roy. Astron. Soc. (2015) 2243[arXiv:1408.4160 [astro-ph.GA]].[24] M. Pospelov, J. Pradler, J. T. Ruderman and A. Urbano, New Physics in theRayleigh-Jeans Tail of the CMB , arXiv:1803.07048 [hep-ph]. T. Moroi, K. Nakayamaand Y. Tang, Axion-Photon Conversion and Effects on 21cm Observation ,arXiv:1804.10378 [hep-ph].[25] E. W. Kolb and M. S. Turner, The Early Universe , Front. Phys. (1990) 1.[26] For an updated calculation of cosmological quantities such age of the universe andexpansion rate, see for example P. Di Bari, Cosmology and the early universe , CRCPress, May 2018 (ISBN 9781498761703).[27] C. Giunti, Nucl. Phys. B (2016) 336 [arXiv:1512.04758 [hep-ph]].[28] T. Sprenger, M. Archidiacono, T. Brinckmann, S. Clesse and J. Lesgourgues,arXiv:1801.08331 [astro-ph.CO].[29] P. Di Bari, S. F. King and A. Merle, Dark Radiation or Warm Dark Matterfrom long lived particle decays in the light of Planck , Phys. Lett. B (2013) 77[arXiv:1303.6267 [hep-ph]].[30] D. J. Fixsen, E. S. Cheng, J. M. Gales, J. C. Mather, R. A. Shafer and E. L. Wright, The Cosmic Microwave Background spectrum from the full COBE FIRAS data set ,Astrophys. J. (1996) 576 [astro-ph/9605054].[31] R. Subrahmanyan and R. Cowsik, Astrophys. J. (2013) 42 [arXiv:1305.7060[astro-ph.CO]].[32] E. Masso and R. Toldra, Phys. Rev. D (1999) 083503.[33] J. Dowell and G. B. Taylor, Astrophys. J. (2018) no.1, L9 [arXiv:1804.08581[astro-ph.CO]].[34] E. J. Chun, C. W. Kim and U. W. Lee, Phys. Rev. D (1998) 093003.[35] K. L. McDonald and B. H. J. McKellar, Int. J. Mod. Phys. A (2007) 2211[hep-ph/0401073]. 1736] R. Foot, M. J. Thomson and R. R. Volkas, Large neutrino asymmetries from neutrinooscillations , Phys. Rev. D (1996) R5349 [hep-ph/9509327]; P. Di Bari and R. Foot, Active sterile neutrino oscillations in the early universe: Asymmetry generation atlow | δm | and the Landau-Zener approximation , Phys. Rev. D (2002) 045003[hep-ph/0103192].[37] P. Di Bari and R. Foot, Active sterile neutrino oscillations and BBN + CMBRconstraints , Phys. Rev. D (2001) 043008 [hep-ph/0008258].[38] R. Foot and R. R. Volkas, Phys. Rev. D (1997) 5147 [hep-ph/9610229]; P. Di Bari,P. Lipari and M. Lusignoli, Int. J. Mod. Phys. A et al. , Astrophys. J. (2017) no.2, L12 [arXiv:1703.06647 [astro-ph.CO]].[41] G. Bernardi et al. , Mon. Not. Roy. Astron. Soc.461