Limits from BBN on light electromagnetic decays

Injection of electromagnetic energy — photons, electrons, or positrons — into the plasma of the early universe can destroy light elements created by primordial Big Bang Nucleosynthesis (BBN). The success of BBN at predicting primordial abundances has thus been used to impose stringent constraints on decay or annihilation processes with primary energies near or above the electroweak scale. In this work we investigate the constraints from BBN on electromagnetic decays that inject lower energies, between 1–100 MeV. We compute the electromagnetic cascade from such injections and we show that it can deviate significantly from the universal spectrum commonly used in BBN calculations. For electron injection below 100 MeV, we find that the final state radiation of photons can have a significant impact on the resulting spectrum relevant for BBN. We also apply our results on electromagnetic cascades to investigate the limits from BBN on light electromagnetic decays prior to recombination, and we compare them to other bounds on such decays.


JHEP01(2019)074 1 Introduction
Big Bang Nucleosynthesis (BBN) is one of the most powerful probes of the very early universe [1][2][3][4]. Over the course of BBN, free protons and neutrons assemble into a handful of light elements [5][6][7]. Assuming a standard ΛCDM cosmological history, the primordial abundances of these elements can be predicted using known nuclear reaction rates in terms of a single input parameter, the overall baryon density. These predictions agree well with observational determinations of primordial abundances up to plausible uncertainties in astrophysical determinations and nuclear rates [8]. 1 The success of BBN gives very strong evidence for the ΛCDM cosmological model up to radiation temperatures near the MeV scale [13][14][15], which extends much earlier than other known tests [16]. BBN also places stringent constraints on new physics beyond the Standard Model that injects energy into the cosmological plasma or influences the expansion rate at early times. This includes the decays of massive particles with lifetimes greater than τ 0.1 s [17][18][19][20][21][22][23][24][25][26][27][28][29][30], dark matter (DM) annihilation with an effective cross section near the critical value for thermal freeze-out [31][32][33][34], and any new thermalized species with mass below a few MeV [35][36][37][38].
Limits from BBN on the decays of long-lived massive particles have been studied in great detail [26][27][28][29][30]. In the majority of this work, often motivated by new physics connected to the electroweak hierarchy puzzle or weakly-interacting massive particle (WIMP) dark matter, the energy injected by the decay has been assumed to be close to or greater than the weak scale. Thus, the decay products typically have initial energies that are much larger than the thresholds for nuclear reactions relevant to BBN which are typically on the order of several MeV. Weak-scale decay products typically also have both hadronic and electromagnetic components, if only through radiative effects.
Hadronic energy injection can modify the light element abundances at times as early as t ∼ 0.1 s [26,27]. Initially, these products scatter with protons and neutrons and alter the ratio of these baryons and thus the resulting helium abundance. At later times, injected hadrons destroy and modify the abundances of helium and other light elements through hadrodissociation. Since the initial hadronic energies are usually assumed to be much larger than the MeV scale, thresholds for these reactions are easily overcome.
Electromagnetic energy -photons, electrons, and positrons -injected into the cosmological plasma does not have a significant effect on the light element abundances until much later. The main effect of electromagnetic injection on the light elements is photodissociation (unless the amount of energy deposited is enormous). However, being much lighter than hadrons, photons and electrons lose their energy very efficiently by scattering off the highly-abundant photon background. The electromagnetic cascade initiated by this scattering is strongly suppressed for energies above E c , given by [39,40]

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where T is the cosmological photon temperature. As a result, even for initial energies orders of magnitude above the MeV-scale thresholds for photodissociation, the fraction of energy available for photodissociation is tiny until the background temperature falls below T 10 keV, corresponding to t ∼ 10 4 s. While much of the focus on new sources of energy injection during BBN has been on decays or annihilations at or above the weak scale, there exist many well-motivated theories that also predict new sources well below the weak scale. Specific examples include dark photons [41,42], dark Higgs bosons [42,43], dark gluons and glueballs [44][45][46], light or strongly-interacting dark matter [47,48], and MeV-scale neutrino decays [49,50]. As the injection energy falls below the GeV scale, hadronic decay channels start to become kinematically unavailable and disappear entirely below the pion threshold. This leaves electromagnetic and neutrino injection as the only remaining possibilities. Even more importantly, it was shown in refs. [51,52] that the development of the electromagnetic cascade at these lower energies can differ significantly relative to injection above the weak scale. Furthermore, as the injection energy falls below a few tens of MeV, photodissociation reactions begin to shut off.
In this work we expand upon the analysis of refs. [51,52] and investigate the effects of electromagnetic energy injection below 100 MeV on the primordial element abundances created during BBN. A major focus of this study is the development of the electromagnetic cascade from initial photon or electron (e + e − ) injection. For high energy injection, the resulting spectrum of photons is described very well by the so-called universal spectrum rescaled by a temperature-and energy-dependent relaxation rate. This spectrum is used widely in studies of photodissociation effects on BBN, it can be parametrized in a simple and convenient way, and has the attractive feature that it only depends on the total amount of electromagnetic energy injected. However, for lower-energy electromagnetic injection, the universal spectrum does not properly describe the resulting electromagnetic cascades.
The universal spectrum fails for lower-energy injection in two significant ways. First, the universal spectrum is based on a fast redistribution of the initial energy E X E c to a spectrum populated at E ≤ E c through Compton scattering and photon-photon pair production. As shown in refs. [51,52], this picture does not hold for initial injection energies E X < E c , which can easily occur for smaller E X and larger decay lifetimes. And second, as argued in ref. [42] the Compton scattering with background photons that dominates electron interactions is qualitatively different at high energies compared to low. At higher energies, s ∼ E T m 2 e , electrons scatter in the Klein-Nishina limit and typically lose an order unity fraction of their energy in each scattering event. In contrast, lower energy scattering with s ∼ E T m 2 e enters the Thomson regime where the fractional change in the electron energy per collision is very small and the up-scattered photon energy is much less than the initial electron energy.
To address the breakdown of the universal spectrum for lower-energy electromagnetic injection, we compute the full electromagnetic cascade for photon or electron (e + e − ) injection with initial energies E X ∈ [1, 100] MeV following the methods of ref. [40]. Our work expands upon refs. [51,52] that studied photon portion of the cascade for photon injection. We compare and contrast our results to the universal spectrum, and study their implica-JHEP01(2019)074 tions for BBN. In addition to finding important differences from the universal spectrum at these lower energies, we also demonstrate that final-state radiation (FSR) from electron injection can have a very significant impact on the resulting photon spectrum. For very low injection energies approaching the MeV scale, we also study the interplay of the spectrum with the thresholds for the most important nuclear photodissociation reactions.
The outline of this paper is as follows. After this introduction, we present our calculation of the electromagnetic cascade in section 2. Next, in section 3 we study the impact of such electromagnetic injection on the light element abundances. In section 4 we contrast the bounds from photodissociation of light elements with other limits on late electromagnetic injection. Finally, section 5 is reserved for our conclusions. Some technical details are listed in appendix A for completeness.

Development of the electromagnetic cascade
In this section we compute the electromagnetic cascade in the early universe following the injection of photons or electrons (e + e − ) with initial energy E X < 100 MeV.

Computing the electromagnetic cascade
Energetic photons or electrons injected into the cosmological plasma at temperatures below the MeV scale interact with background photons and charged particles leading to electromagnetic cascades that produce spectra of photons and electrons at lower energies. Since the development of the cascade is much faster than the typical interaction time with the much more dilute light elements created in BBN, these spectra can be used as inputs for the calculation of photodissociation effects.
The most important reactions for the development of the electromagnetic cascade in the temperature range of interest T ∈ [1 eV, 10 keV] are [40]: • photon photon pair production (4P): γ + γ BG → e + + e − Of these processes, IC and 4P are typically the fastest provided there is enough energy for them to occur. We define N a = dn a /dE to be the differential number densities per unit energy of photons (a = γ) and the sum of electrons and positrons (a = e). The Boltzmann equations for the evolution of these spectra take the form

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where Γ a (E) is a relaxation rate at energy E, and S a (E) describes all sources at this energy. Since the relaxation rates are typically much faster than the Hubble rate, the Hubble dilution term has been omitted. Furthermore, the relaxation rate is also much larger than the mean photodissociation rates with light nuclei, so a further quasistatic approximation can made with dN a /dt → 0 [40]. This gives the simple solution Note that N a (E) evolves in time in this approximation through the time and temperature dependences of the sources and relaxation rates. The source terms are discussed in more detail below while explicit expressions for the contributions to the relaxation rates are given in appendix A.

Monochromatic photon injection
For monochromatic photon injection at energy E X from a decay with rate per volume R, the source terms are where ξ γ is the number of photons injected per decay, and the K ab (E, E ) functions describe scattering processes that transfer energy from species b at energy E to species a at energy E ≤ E . Explicit expressions for these transfer functions are given in appendix A. Note that in the case of decays of species X with lifetime τ X , the rate is R = n X (t)/τ X . These equations can also be applied to annihilation reactions of the form X +X → n γ with cross section σv by setting R = σv n X nX and ξ γ = n.
It is convenient to describe the cascades resulting from the initial monochromatic (delta function) injection with smooth functions that are independent of the injection rate. To this end, we definef The functionsf γ andf e are expected to be smooth, and can be used to reconstruct the full spectra N γ (E) and N e (E) uniquely for any given injection rate R.

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Determining the electromagnetic cascade from monochromatic photon injection is therefore equivalent to solving eqs. (2.7), (2.8). We do so using the iterative method of ref. [40], with an important modification to account for the Thomson limit of IC scattering. In this method, the spectraf a (E) are determined on a grid of energy points E i given by where we use E 0 = 1 MeV, E N = E X , i = 0, 1, . . . , N , and N 1. For the top point i = N , eqs. (2.7), (2.8) givē (2.10) To compute the spectra at lower points, we use the fact that the transfer integrals at a given energy E only depend on the spectra at energies E > E. Thus, at any step i the integrals in eqs. (2.7), (2.8) can be approximated numerically (e.g. Simpson's rule) using the spectra already determined at points j = i + 1, . . . , N . Relative to ref. [40] we also apply a finer grid to compute the top two energy points.
This approach to computing the cascades works well for y e = E e T /m 2 e 1, but becomes numerically challenging for y e 0.1. The problem comes from the contribution of inverse Compton (IC) scattering to K ee . As y e becomes small, IC scattering enters the Thomson regime in which the cross section is large but the fractional change in the electron energy per scattering is much less than unity, and thus the function K ee (E, E ) develops a strong and narrow peak near E E. To handle this we follow refs. [39,53] and treat the electron energy loss due to IC in the Thomson limit as a continuous process by replacing Here,Ė is the rate of energy loss from IC of a single electron in the photon background, given by [53]Ė where σ T = (8π/3)α 2 /m 2 e is the Thomson cross section, n γ = [2ζ(3)/π 2 ]T 3 is the thermal photon density, and ζ(z) is the Riemann zeta function. The approximation of eq. (2.11) is valid provided the fractional energy loss rateĖ/E is much smaller than the total scattering rate σ T n γ , which coincides with y e 0.1. In this limit, the two terms on the left-hand side of eq. (2.11) are much larger than their difference leading to a numerical instability in the original iterative approach.
When computing the electromagnetic spectra, we use the iterative method described above with eqs. (2.7), (2.8) until y j = E j T /m 2 e < 0.05 is reached. For lower energy bins we keep eq. (2.7) forf γ but apply the replacement of eq. (2.11) forf e , yielding the solution and Again, this can be evaluated iteratively, from high to low. While we use the specific value y e < 0.05 to match from one method to the other, we find nearly identical results from matching within the range y e ∈ [0.001, 0.1].

Monochromatic electron injection
Monochromatic injection of electrons (and positrons) at energy E X can be treated nearly identically to monochromatic photon injection, with the only major change being in modifying the sources to where R is the decay (or annihilation) rate per unit volume, ξ e is the number of electrons plus positrons injected per decay, and S FSR γ (E) is a contribution to photons from finalstate radiation to be discussed in more detail below. For decays of the form X → e + + e − we have R = n X (t)/τ X and ξ e = 2, while for annihilation X +X → e + + e − the rate is R = σv n X nX and ξ e = 2.
Given these source terms, it natural to define the reduced spectraf a (E) bȳ Applying this to eq. (2.2) with the sources of eqs. (2.16), (2.17), we obtain the relations These equations can be solved using the same methods as described above for photon injection, including a matching in the Thomson limit using eq. (2.11).
A new feature that we include for electron injection is a contribution to the photon spectrum from final-state radiation (FSR) off the injected electron; S FSR γ (E) in eq. (2.16).

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For processes of the form X → e + + e − or X +X → e + + e − with X uncharged and E X m e , this new source can be approximated by [54,55] S FSR To be fully consistent, a corresponding subtraction should be made from the electron source. However, we find that this modifies the spectra by less than a percent. In contrast, we show below that the direct contribution to the photon spectrum from FSR can be the dominant one at higher energies when E X T /m 2 e 1, when the initial electrons scatter via IC with the photon background mainly in the Thomson regime.

Review of the universal spectrum
Many studies of the effects of electromagnetic energy injection on BBN approximate the photon spectrum with the so-called universal spectrum. This is a simple parametrization of the full calculations of the photon spectrum in refs. [39,40]. It replaces the source terms (direct and cascade) in eq. (2.2) with a zeroeth generation spectrum S γ (E)/R → p γ (E) based on the assumption that 4P and IC processes instantaneously reprocess the initial injected electromagnetic energy.
The standard parametrization used for the zeroeth generation spectrum is [4,25,39] where E c m 2 e /22 T and E m m 2 e /80 T are derived from ref. [40], and K 0 is a normalization constant. For monochromatic injection of ξ photons, electrons, and positrons each with energy E X , it is fixed by the requirement An important feature of the spectrum is that it is proportional to the total injection energy (for either photons or electrons) provided E X E c , up to an overall normalization by the total amount of energy injected.
Within the universal spectrum approximation, the final spectra are given by where f γ (E) = N γ (E)/R, and the relaxation rate Γ γ (E) accounts for the further reprocessing of the spectrum by slower processes like Compton scattering, pair creation on nuclei, and photon-photon scattering.   the initial injection is assumed to be fully reprocessed into the zeroeth-order spectrum by 4P and IC scatterings.

Results for photon injection
To validate our electromagnetic spectra, we compare our results to previous calculations and the universal spectrum at high injection energies. In figure 1 we show our photon spectraf γ (E) for single photon injection with E X = 1000 GeV (left) and 100 GeV (right) at temperatures T = 1, 10, 100 eV. Also shown in the figure are the predictions from the universal spectrum and parametrizations of the results of Kawasaki and Moroi listed in ref. [40]. In all cases here, E X E c and the universal spectrum is expected to be a good approximation. Our spectra agree well with the results of ref. [40] but are somewhat larger than the universal spectrum. We have also checked that our spectra scale proportionally to the total energy injected provided E X E c . In all cases shown in the figure, the electron spectra are smaller than the photon spectra by orders of magnitude due to efficient IC scattering. Also visible is the strong suppression of the photon spectra for E > E c where the 4P process is active. In contrast to electromagnetic injection at high energies with E X E c , injection at lower energies with E X E c has received much less attention. In figure 2 we show our computed photon spectra for single photon injection with E X = 100 MeV (left), E X = 30 MeV (middle), and E X = 10 MeV (right) for T = 1, 10, 100 eV. Also shown are the predictions of the universal spectrum (normalized according to eq. (2.24)) and the prescription by Poulin and Serpico of ref. [52]. Since E X < E c , the assumptions that go into the universal spectrum are not met and it is not expected to be accurate in this regime, as first pointed out in ref. [52]. Our spectra agree fairly well with the results of ref. [52], which only kept the photon part of the spectrum. Some deviations are seen at lower energies where photon regeneration by IC becomes significant. Note as well that the full cascade also contains a moderately damped delta-function part that is not shown here (and was explicitly removed in our definition off γ in eq. (2.5)).

Results for electron injection
For electron and positron (e + e − ) injection with energies E X E c , we find the same photon (and electron) spectra as from photon injection with an equal total input energy, and thus our results agree reasonably well with ref. [40] and the universal spectrum in this limit. However, for E X E c we find very significant variations from the universal spectrum as well as from pure photon injection. Photon spectraf γ resulting from e + e − injection are shown in figure 3 for input energies E X = 100 MeV (left), E X = 30 MeV (middle), and E X = 10 MeV (right) and temperatures T = 1, 10, 100 eV. The solid lines show the full spectra, while the dashed lines show the corresponding result when FSR off the initial decay electrons is not taken into account. Also shown is the universal spectrum for the same total energy injection (normalized according to eq. (2.24)). Let us also mention that the photon spectra do not have a delta function component for electron or positron injection.
The strong suppression of the photon spectrum from electron injection at lower energies in the absence of FSR was pointed out in ref. [42]. As argued there, this suppression can be understood in terms of the behavior of IC scattering at low energy, which is the main
mechanism for electrons to transfer energy to photons in this context. For smaller E X and T , the dimensionless combination y e = E e T /m 2 e 1 is small, and IC scattering lies in the Thomson regime where each collision only slightly reduces the initial electron energy. Correspondingly, the maximal scattered photon energy E γ in the Thomson limit is E γ ≤ 4(E e /m e )E γ , where E γ is the energy of the initial photon. Since the initial photon comes from the CMB, E γ ∼ T is expected so that Higher scattered photon energies are possible, but they come at the cost of an exponential Boltzmann suppression. In this regime, FSR from the injected electrons and positrons can be the dominant contribution to the photon spectrum, as illustrated in figure 3. Relative to the rest of the cascade, the distribution of photons from FSR is hard, falling off roughly as 1/E instead of as 1/E 2 . Despite the suppression of FSR by (α/π) × log (with log ∼ few), it can easily overcome the exponential suppression of IC for photon energies above the bound of eq. (2.26). We show below that this has a very important implication for the effects of lower-energy electron injection on the primordial light element abundances. Note, however, that FSR has only a very minor effect on the spectra for photon injection or when E X E c .

Effects of electromagnetic injection on BBN
Having computed the electromagnetic cascades from lower-energy injection, we turn next to investigate the effects of such injection on the primordial element abundances from BBN.

Photodissociation of light elements
Photodissociation of light element begins when the temperature of the cosmological plasma falls low enough for MeV photons to populate the electromagnetic cascade. From eq. (1.1),

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this does not begin until temperatures fall below about 10 keV (corresponding to t ∼ 10 4 s). By this time element creation by BBN has effectively turned off, and thus we can compute the effects of photodissociation as a post-processing of the outputs of standard BBN [3,25]. The effects of photodissociation on the light element abundances can be described by a set of coupled Boltzmann equations of the form where N γ (E γ ) are the photon spectra calculated above, A and the sums run over the relevant isotopes, and Y A are number densities normalized to the entropy density, Note that we do not include reactions initiated by electrons because the electron spectra are always strongly suppressed by IC scattering. In our analysis we include the nuclear species hydrogen (H), deuterium (D = 2 H), tritium (T = 3 H), helium-3 ( 3 He), and helium (He = 4 He). Heavier species including lithium isotopes could also be included, but these have much smaller abundances and they would not alter the results for the lighter elements we consider. The nuclear cross sections included in our study are listed in table 3.1, for which we use the simple parametrizations of ref. [25]. All these cross sections have the same general shape as a function of energy, with a sharp rise at the threshold up to a peak followed by a smooth fall off. We list the threshold energies and peak values of the cross sections in the table to give an intuitive picture of their relevant strengths and ranges of importance. Of the nine cross sections listed, it is helpful to group them into processes that destroy helium and create deuterium and helium-3 with thresholds above 20 MeV, and processes that destroy the lighter isotopes with significantly lower thresholds.
It is straightforward to solve the evolution equations of eq. In the analysis to follow, we compare the computed output densities to the following observed values, quoted with effective 1σ uncertainties into which we have combined theoretical and experimental uncertainties in quadrature: For the helium mass fraction Y p , the value we use is consistent with ref. [68] and previous determinations but significantly lower than the determination of ref. [69]. The quoted JHEP01(2019)074 uncertainty on the ratio n D /n H is dominated by a theory uncertainty on the rate of photon capture on deuterium from ref. [70]. For n3 He /n H , we use the determination of (n D + n3 He )/n H of ref. [67] together with the value of n D /n H from ref. [66]; the resulting upper bound (with uncertainties) is similar to but slightly stronger than what is used in ref. [30].
The uncertainties quoted here are generous, and in the analysis to follow we implement exclusions at the 2σ level.

BBN constraints on photon injection
Following the methods described above and the electromagnetic cascades computed previously, we derive BBN bounds on monochromatic photon injection from late decays with lifetime τ X and initial injection energy E X . In figure 4 we show the resulting limits on the combination E X Y X , where Y X is the predecay yield of the decaying species X (assumed to produce one photon per decay) for injection energies E X = 10, 30, 100 MeV. The bounds coming from D, 3 He, and 4 He are shown individually, and correspond to 2σ exclusions. Early on, when E c is small, the dominant effect is destruction of D since it has the lowest photodissociation threshold. Later on, as E c increases, it becomes possible to create excess D and 3 He through the destruction of 4 He provided the injection energy is larger than the 4 He threshold of about 20 MeV. Destruction of D is the dominant effect at all times for E X below the helium threshold, as can be seen in the leftmost panel of figure 4. In figure 5 we show maximal values of E X Y X from monochromatic photon injection at energy E X from the decay of species X as a function of τ X and E X . The combined exclusion is based on the union of 2σ exclusions of the individual species. Clear features are visible in this figure at τ X 10 6 s and E X 20 MeV. These coincide with the structure of the exclusions shown in figure 4, with both corresponding to where the photodissociation of 4 He turns off, either because E c or E X is too small.

BBN constraints on electron injection
In figure 6 we show the limits for e+e − injection from the decay of a species X with lifetime τ X on E X Y X , where Y X is the predecay yield of the decaying species X (assumed to produce log 10 E X Y X (GeV) Figure 5. Combined limits on E X Y X as a function of τ X and E X for the decay of a species X with lifetime τ X injecting a single photon with energy E X . one e + e − pair per decay) for injection energies for each electron of E X = 10, 30, 100 MeV (from left to right). The bounds coming from D, 3 He, and 4 He are shown individually, and correspond to 2σ exclusions. The electromagnetic spectra used in this calculation include FSR from the injected e + e − pair. The resulting bounds are somewhat weaker than for photon injection and follow a similar pattern, and remain quite strong even down to E X = 10 MeV. For comparison, we show the corresponding results when FSR effects are not included in figure 7. As expected, the exclusions are significantly weaker, particularly for larger τ X and lower E X where the relevant IC scattering is deep in the Thomson regime.
In figure 8 we show maximal values of E X Y X from monochromatic e + e − injection at energy E X from the decay of species X as a function of τ X and E X , with FSR effects JHEP01(2019)074  log 10 E X Y X (GeV) Figure 8. Combined limits on E X Y X as a function of τ X and E X for the decay of a species X with lifetime τ X injecting an electron-positron pair each with energy E X , with FSR effects included.
included in the electromagnetic cascade. The combined exclusion is based on the union of 2σ exclusions of the individual species. Again, the exclusions become weaker for τ X 10 6 s or E X 20 MeV where the photodissociation of 4 He turns off. The bounds on e + e − injection are also typically weaker than for photon injection, but not drastically so when FSR is taken into account.

Other constraints on electromagnetic decays
In addition to modifying the primordial light element abundances, energy injection in the early universe can produce other deviations from the standard cosmology. Electromagnetic decays near or after recombination at t rec 1.2 × 10 13 s can modify the temperature and polarization power spectra of the CMB [71][72][73][74]. Since current CMB observations are found to constrain such decays much more strongly than BBN [75,76], we focus here on decays JHEP01(2019)074 prior to recombination. The best limits in this case, aside from BBN, come from entropy injection and modifications to the CMB frequency spectrum. In this section we estimate these other limits on late energy injection and compare them to our results for BBN.
Entropy injection after the start of BBN leads to a lower measured baryon density today relative to the value deduced from BBN. This was studied in ref. [77] with the result ∆s s 7.8 × 10 −5 ∆E Y X 10 −10 GeV where ∆E is the total electromagnetic energy injected per decay and τ X 1 s. A related constraint can be derived for variations in the effective number of neutrinos N eff from photon heating after neutrino decoupling [37,38,50,78]. Late decays releasing electromagnetic energy can also distort the frequency spectrum of the CMB [79,80], which is observed to be a nearly-perfect blackbody [81]. The effect depends on the decay time τ X relative to the times τ dC 6.1 × 10 6 s when double-Compton scattering freezes out and τ C 8.8 × 10 9 s when Compton scattering turns off [79,80]. Decays with τ dC < τ X < τ C yield products that thermalize through Compton scattering and generate an effective photon chemical potential µ given by [79,80,82] µ 5.6 × 10 −4 ∆E Y X 10 −10 GeV For τ X > τ C , electromagnetic injection produces a distortion that can be described by the Compton parameter y = ∆ρ γ /4ρ γ , with the approximate result [79,80,82] y 5.7 × 10 −5 ∆E Y X 10 −10 GeV where C(τ ) = 1 for τ < t eq and C(τ ) (τ /t eq ) 1/6 for τ > t eq . The current limits on µ and y are [81] µ < 9 × 10 −5 , |y| < 1.5 × 10 −5 , (4.4) while the proposed PIXIE satellite is to have sensitivity to constrain [83] µ < 1 × 10 −8 , |y| < 2 × 10 −9 . (4.5) In the left and right panels of figure 9 we show the limits from entropy injection and CMB spectral distortions. The solid red line shows ∆s/s = 0.1, and demanding variations below this is a conservative requirement relative to those imposed in refs. [50,52]. For CMB spectral distortions we show bounds on the µ and y parameters in blue based on the approximate estimates above based on measurements by COBE/FIRAS (solid) and the projected sensitivity of PIXIE (dotted). For comparison, we show in green the limits derived above for monochromatic photon injection (left) and monochromatic e + e − injection (right). In both panels, the dotted, dashed, and solid lines correspond to injection with E X = 10, 30, 100 MeV. Even for low injection energies, BBN constraints currently dominate for τ 10 4 s until being replaced by bounds from either CMB frequency or power spectrum variations. Even with the vast improvement expected from PIXIE, BBN will continue to provide the strongest limit on electromagnetic decays in the early universe with lifetimes 10 4 s τ X 10 6 s and energy injections above a few MeV.

Conclusions
In this paper we have investigated the electromagnetic cascades induced by electromagnetic energy injection in the range E X = 1−100 MeV and we have studied its effects on the light elements abundances created during BBN. As in ref. [52], we find significant deviations from the universal photon spectrum for monochromatic initial photon injection with energy E X E c = m 2 e /22T . We also study how this impacts BBN. Our study also expands on previous work by computing the full electromagnetic cascade including electrons. For either photon or electron injection, we find that BBN provides the strongest constraint on late-decaying particles with lifetimes between 10 4 s τ 10 13 s for electromagnetic energies nearly all the way down to the photodissociation threshold of deuterium near E th 2.22, MeV.
Photon and electron injection produce very similar electromagnetic cascades for E X E c but differ in important ways for E X E c . Initial hard photons induce a smooth population of lower-energy photons through Compton and photon-photon scattering. In contrast, electrons injected with E X m 2 e /10T interact mainly through inverse Compton (IC) scattering off the CMB, which lies in the Thomson regime at such energies. The upscattered photons from Thomson scattering have much lower energy than the initial electron, and can easily fall below the MeV scales needed to induce photodissociation. However, in this regime we find that photons radiated off the initial hard electrons can populate and dominate the induced photon spectrum up to near the initial electron energy. To our knowledge, the contribution of FSR to the photon spectrum has not been considered before in this  Table 2. Summary of the contributions to relaxation rates and energy transfer functions.
context since its effects are very small at the higher initial injection energies that have been investigated in the greatest detail.
While this work has concentrated on decays, our results for electromagnetic cascades are also applicable to annihilation in the early universe. Our results could also be used to investigate potential solutions to the apparent anomalies in the lithium abundances, which was studied in refs. [50,84] using the universal spectrum.
Note added: as this work was nearing completion, ref. [85] appeared investigating many of the same topics including the development of the electromagnetic cascades from sub-100 MeV decays. In the cases that are directly comparable, our results appear to be in substantial agreement.

A Relaxation and transfer rates
In this appendix we collect explicit expressions for the relevant relaxation rates and transfer functions. The processes included for each are listed in table 2 together with links to the corresponding equations.

A.1 Rates for Photon Photon Pair Production (4P)
The 4P process plays a key role in determining the electromagnetic cascade for E > E c and contributes to Γ γ and K eγ .

A.1.1 4P photon relaxation
The relevant expression is given in ref. [40] ∆Γ γ (E γ ) = 1 8 This expression can be written in the form where y γ = E γ T /m 2 e and a function G(y γ ). A good numerical fit to this function is which is valid to better than about 10% for y γ ∈ [10 −5 , 10 2 ]. A similar approximate expression is given in ref. [39].

A.1.2 4P transfer to electrons
The 4P process also produces high energy electrons and positrons, contributing to the transfer function K eγ by [40,86]: 3 and Θ({E i ; T }) is zero unless B ≥ 1 and These inequalities reflect kinematic constraints on the electron and positron energies. 4

A.2 Rates for Inverse Compton scattering (IC)
Inverse Compton (IC) scattering in this context corresponds to the scattering of high energy electrons (or positrons) off CMB photons, e ± + γ BG → e ± + γ. This reduces the electron energy and produces an energetic photon. We evaluate the corresponding contributions to Γ e , K ee , and K γe .

A.2.1 IC electron relaxation
The relaxation rate for electrons due to IC was computed in ref. [87] and reproduced in ref. [40]. For e(E e ) + γ(Ē γ ) → e(E e ) + γ(E γ ), the expression is Note that this expression is based on an approximation that is only valid forĒ γ < E γ and q ≥ (m e /2E e ) 2 [87]. However, for the energies and temperatures relevant for photodissociation, we find that applying q ≥ 0 is a good approximation. The IC electron relaxation rate can be written in the simpler form where the second line is a numerical fit valid to within about 5% over y e ∈ [10 −5 , 10 2 ].
For y e 1 this coincides with IC in the Thomson limit: ∆Γ e = σ T n γ (T ), where σ T = (8π/3)α 2 /m 2 e is the Thomson cross section and n γ is the photon density.

A.2.2 IC transfer to electrons
The IC process also produces an electron with a lower energy than the initial value. The corresponding transfer kernel for e(E e ) + γ(Ē γ ) → e(E e ) + γ(E γ ) is

A.3 Photon Photon Scattering (PP)
Photon photon (PP) scattering contributes to photon relaxation and transfer.

A.3.1 PP photon relaxation
The contribution to the relaxation rate is approximately [52,88] ∆Γ γ (E γ ) (0.1513)α 4 m e T m e 3 y 3 γ e −yγ , (A. 18) with y γ = E γ T /m 2 e . Note that this form only applies for y γ < 1, and thus we add an exponential by hand to provide a smooth cutoff at larger values.

A.4 Pair Creation on Nuclei (PCN)
Photon scattering on background nuclei can create e + e − pairs, γ(E γ ) + N → N + e + + e − . This reduces the photon energy and injects energy into electrons and positrons.

A.4.1 PCN photon relaxation
The contribution to photon relaxation is [40] ∆Γ γ (E γ ) = Note that x = 4 is the threshold for the process to occur.

A.4.2 PCN electron transfer
The contribution to the electron transfer kernel is [40] ∆K eγ (E e , E γ ) = Z n Z dσ where E e = (E γ − E e ), p ( ) = E As before, the relevant nuclei are H and 4 He.

A.5 Compton Scattering (CS)
Compton scattering in this context refers to high-energy photons colliding with background electrons, γ + e − BG → γ + e − . This reduces the photon energy and transfers it to the scattered electron.

A.5.1 CS photon relaxation
The contribution to the relaxation rate is ∆Γ γ (E γ ) = n e σ CS , (A. 28) where n e is the background electron density given and

A.5.2 CS photon transfer
The energy of the initial CS photon E γ is partially transferred to the energy E γ of the outgoing photon. This contributes to the transfer kernel by [40,52,75] This expression is valid only for E γ /(1 + 2E γ /m e ) ≤ E γ ≤ E γ and zero otherwise. 7

A.5.3 CS electron transfer
Energy transfer to electrons by CS is given by where E γ = (E γ +m e −E e ), E γ ≥ E e , and the differential cross section is given in eq. (A.31).
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