Limits from BBN on Light Electromagnetic Decays
Lindsay Forestell, David E. Morrissey, and Graham White
(a) Department of Physics and Astronomy, University of British Columbia, Vancouver, BC V6T 1Z1, Canada
(b) TRIUMF, 4004 Wesbrook Mall, Vancouver, BC V6T 2A3, Canada
email: firstname.lastname@example.org, email@example.com, firstname.lastname@example.org
July 20, 2019
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.
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 CDM 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 .111 The extrapolated densities of Li and Li give a particularly acute puzzle in this regard [9, 10, 11, 12].
The success of BBN gives very strong evidence for the CDM cosmological model up to radiation temperatures near the MeV scale [13, 14, 15], which extends much earlier than other known tests . 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 [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 [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 , given by [39, 40]
where 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 , corresponding to .
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 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 () injection. For high energy injection, the resulting of 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 to a spectrum populated at through Compton scattering and photon-photon pair production. As shown in Refs. [51, 52], this picture does not hold for initial injection energies , which can easily occur for smaller and larger decay lifetimes. And second, as argued in Ref.  the Compton scattering with background photons that dominates electron interactions is qualitatively different at high energies compared to low. At higher energies, , 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 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 () injection with initial energies following the methods of Ref. . 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 implications 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 Sec. 2. Next, in Sec. 3 we study the impact of such electromagnetic injection on the light element abundances. In Sec. 4 we contrast the bounds from photodissociation of light elements with other limits on late electromagnetic injection. Finally, Sec. 5 is reserved for our conclusions. Some technical details are listed in Appendix A for completeness.
2 Development of the Electromagnetic Cascade
In this section we compute the electromagnetic cascade in the early universe following the injection of photons or electrons () with initial energy .
2.1 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 are :
photon photon pair production (4P):
photon photon scattering (PP):
pair creation on nuclei (PCN):
Compton scattering (CS):
inverse Compton (IC):
final state radiation (FSR):
Of these processes, IC and 4P are typically the fastest provided there is enough energy for them to occur.
We define to be the differential number densities per unit energy of photons () and the sum of electrons and positrons (). The Boltzmann equations for the evolution of these spectra take the form
where is a relaxation rate at energy , and 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 smaller than the mean photodissociation rates with light nuclei, so a further quasistatic approximation can made with . This gives the simple solution
Note that 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 App. A.
2.1.1 Monochromatic Photon Injection
For monochromatic photon injection at energy from a decay with rate per volume , the source terms are
where is the number of photons injected per decay, and the functions describe scattering processes that transfer energy from species at energy to species at energy . Explicit expressions for these transfer functions are given in App. A. Note that in the case of decays of species with lifetime , the rate is . These equations can also be applied to annihilation reactions of the form with cross section by setting and .
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 define
The functions and are expected to be smooth, and can be used to reconstruct the full spectra and uniquely for any given injection rate .
Determining the electromagnetic cascade from monochromatic photon injection is therefore equivalent to solving Eqs. (2.1.1,9). We do so using the iterative method of Ref. , with an important modification to account for the Thomson limit of IC scattering. In this method, the spectra are determined on a grid of energy points given by
To compute the spectra at lower points, we use the fact that the transfer integrals at a given energy only depend on the spectra at energies . Thus, at any step the integrals in Eqs. (2.1.1,9) can be approximated numerically (e.g. Simpson’s rule) using the spectra already determined at points . Relative to Ref.  we also apply a finer grid to compute the top two energy points.
This approach to computing the cascades works well for , but becomes numerically challenging for . The problem comes from the contribution of inverse Compton (IC) scattering to . As 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 develops a strong and narrow peak near . To handle this we follow Refs. [53, 39] and treat the electron energy loss due to IC in the Thomson limit as a continuous process by replacing
Here, is the rate of energy loss from IC of a single electron in the photon background, given by 
where is the Thomson cross section, is the thermal photon density, and is the Riemann zeta function. The approximation of Eq. (12) is valid provided the fractional energy loss rate is much smaller than the total scattering rate , which coincides with . In this limit, the two terms on the left-hand side of Eq. (12) 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. (2.1.1,9) until is reached. For lower energy bins we keep Eq. (2.1.1) for but apply the replacement of Eq. (12) for , yielding the solution
Again, this can be evaluated iteratively, from high to low. While we use the specific value to match from one method to the other, we find nearly identical results from matching within the range .
2.1.2 Monochromatic Electron Injection
Monochromatic injection of electrons (and positrons) at energy can be treated nearly identically to monochromatic photon injection, with the only major change being in modifying the sources to
where is the decay (or annihilation) rate per unit volume, is the number of electrons plus positrons injected per decay, and is a contribution to photons from final-state radiation to be discussed in more detail below. For decays of the form we have and , while for annihilation the rate is and .
Given these source terms, it natural to define the reduced spectra by
These equations can be solved using the same methods as described above for photon injection, including a matching in the Thomson limit using Eq. (12).
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; in Eq. (17). For processes of the form or with uncharged and , this new source can be approximated by [54, 55]
where . 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 , when the initial electrons scatter via IC with the photon background mainly in the Thomson regime.
2.2 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. (3) with a zeroeth generation spectrum based on the assumption that 4P and IC processes instantaneously reprocess the initial injected electromagnetic energy.
where and are derived from Ref. , and is a normalization constant. For monochromatic injection of photons, electrons, and positrons each with energy , it is fixed by the requirement
implying for . An important feature of the spectrum is that it is proportional to the total injection energy (for either photons or electrons) provided , up to an overall normalization by the total amount of energy injected.
Within the universal spectrum approximation, the final spectra are given by
where , and the relaxation rate accounts for the further reprocessing of the spectrum by slower processes like Compton scattering, pair creation on nuclei, and photon-photon scattering.222In practice, this is effectively equal to the full relaxation rate that also includes 4P scattering since this process is very strongly Boltzmann-suppressed for . These spectra have no residual delta-function parts since the initial injection is assumed to be fully reprocessed into the zeroeth-order spectrum by 4P and IC scatterings.
2.3 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 Fig. 1 we show our photon spectra for single photon injection with (left) and (right) at temperatures . Also shown in the figure are the predictions from the universal spectrum and parametrizations of the results of Kawasaki and Moroi listed in Ref. . In all cases here, and the universal spectrum is expected to be a good approximation. Our spectra agree well with the results of Ref.  but are somewhat larger than the universal spectrum. We have also checked that our spectra scale proportionally to the total energy injected provided . 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 where the 4P process is active.
In contrast to electromagnetic injection at high energies with , injection at lower energies with has received much less attention. In Fig. 2 we show our computed photon spectra for single photon injection with (left), (middle), and (right) for . Also shown are the predictions of the universal spectrum (normalized according to Eq. (25)) and the prescription by Poulin and Serpico of Ref. . Since , 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. . Our spectra agree fairly well with the results of Ref. , 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 of in Eq. (6)).
2.4 Results for Electron Injection
For electron and positron () injection with energies , 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.  and the universal spectrum in this limit. However, for we find very significant variations from the universal spectrum as well as from pure photon injection. Photon spectra resulting from injection are shown in Fig. 3 for input energies (left), (middle), and (right) and temperatures . 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. (25)). 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. . 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 and , the dimensionless combination 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 in the Thomson limit is , where is the energy of the initial photon. Since the initial photon comes from the CMB, 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 Fig. 3. Relative to the rest of the cascade, the distribution of photons from FSR is hard, falling off roughly as instead of as . Despite the suppression of FSR by (with ), it can easily overcome the exponential suppression of IC for photon energies above the bound of Eq. (2.4). 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 .
3 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.
3.1 Photodissociation of Light Elements
|Process||Threshold (MeV)||Peak value (mb)|
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), this does not begin until temperatures fall below about (corresponding to ). 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 are the photon spectra calculated above, and the sums run over the relevant isotopes, and 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 (), tritium (), helium-3 (), and helium (). 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 1, for which we use the simple parametrizations of Ref. . 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 , and processes that destroy the lighter isotopes with significantly lower thresholds.
It is straightforward to solve the evolution equations of Eq. (28) numerically following the standard convention of converting the dependent variable from time to redshift. For standard BBN values of the primordial abundances, we use the predictions of PArthENoPE [63, 64]:
In the analysis to follow, we compare the computed output densities to the following observed values, quoted with effective uncertainties into which we have combined theoretical and experimental uncertainties in quadrature:
For the helium mass fraction , the value we use is consistent with Ref.  and previous determinations but significantly lower than the determination of Ref. . The quoted uncertainty on the ratio is dominated by a theory uncertainty on the rate of photon capture on deuterium from Ref. . For , we use the determination of of Ref.  together with the value of from Ref. ; the resulting upper bound (with uncertainties) is similar to but slightly stronger than what is used in Ref. . The uncertainties quoted here are generous, and in the analysis to follow we implement exclusions at the 2 level.
3.2 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 and initial injection energy . In Fig. 4 we show the resulting limits on the combination , where is the predecay yield of the decaying species (assumed to produce one photon per decay) for injection energies . The bounds coming from , , and are shown individually, and correspond to exclusions. Early on, when is small, the dominant effect is destruction of since it has the lowest photodissociation threshold. Later on, as increases, it becomes possible to create excess and through the destruction of provided the injection energy is larger than the threshold of about . Destruction of is the dominant effect at all times for below the helium threshold, as can be seen in the leftmost panel of Fig. 4.
In Fig. 5 we show maximal values of from monochromatic photon injection at energy from the decay of species as a function of and . The combined exclusion is based on the union of exclusions of the individual species. Clear features are visible in this figure at and . These coincide with the structure of the exclusions shown in Fig. 4, with both corresponding to where the photodissociation of turns off, either because or is too small.
3.3 BBN Constraints on Electron Injection
In Fig. 6 we show the limits for injection from the decay of a species with lifetime on , where is the predecay yield of the decaying species (assumed to produce one pair per decay) for injection energies for each electron of (from left to right). The bounds coming from , , and are shown individually, and correspond to exclusions. The electromagnetic spectra used in this calculation include FSR from the injected pair. The resulting bounds are somewhat weaker than for photon injection and follow a similar pattern, and remain quite strong even down to . For comparison, we show the corresponding results when FSR effects are not included in Fig. 7. As expected, the exclusions are significantly weaker, particularly for larger and lower where the relevant IC scattering is deep in the Thomson regime.
In Fig. 8 we show maximal values of from monochromatic injection at energy from the decay of species as a function of and , with FSR effects included in the electromagnetic cascade. The combined exclusion is based on the union of exclusions of the individual species. Again, the exclusions become weaker for or where the photodissociation of turns off. The bounds on injection are also typically weaker than for photon injection, but not drastically so when FSR is taken into account.
4 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 can modify the 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 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.  with the result
where is the total electromagnetic energy injected per decay and . A related constraint can be derived for variations in the effective number of neutrinos 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 . The effect depends on the decay time relative to the times when double-Compton scattering freezes out and when Compton scattering turns off [79, 80]. Decays with yield products that thermalize through Compton scattering and generate an effective photon chemical potential given by [79, 80, 82]
where for and for . The current limits on and are 
while the proposed PIXIE satellite is to have sensitivity to constrain 
In the left and right panels of Fig. 9 we show the limits from entropy injection and CMB spectral distortions. The solid red line shows , 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 and 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 injection (right). In both panels, the dotted, dashed, and solid lines correspond to injection with . Even for low injection energies, BBN constraints currently dominate for 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 and energy injections above a few MeV.
In this paper we have investigated the electromagnetic cascades induced by electromagnetic energy injection in the range and we have studied its effects on the light elements abundances created during BBN. As in Ref. , we find significant deviations from the universal photon spectrum for monochromatic initial photon injection with energy . 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 for electromagnetic energies nearly all the way down to the photodissociation threshold of deuterium near .
Photon and electron injection produce very similar electromagnetic cascades for but differ in important ways for . Initial hard photons induce a smooth population of lower-energy photons through Compton and photon-photon scattering. In contrast, electrons injected with 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 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.  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.
We thank Sonia Bacca, Nikita Blinov, David Curtin, Barry Davids, David McKeen, Maxim Pospelov, and Adam Ritz for helpful discussions. DEM and GW thank the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-1607611, for their hospitality while this work was being completed. This work is supported by the Natural Sciences and Engineering Research Council of Canada (NSERC), with DEM supported in part by Discovery Grants and LF by a CGS D scholarship. TRIUMF receives federal funding via a contribution agreement with the National Research Council of Canada.
Appendix 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 and contributes to and .
a.1.1 4P Photon Relaxation
a.1.2 4P Transfer to Electrons
The 4P process also produces high energy electrons and positrons, contributing to the transfer function by [86, 40]:333This than Ref.  by a factor of two to account each collision producing an electron and a positron.
and is zero unless and
These inequalities reflect kinematic constraints on the electron and positron energies.444These limits were not specified explicitly in Ref. .
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, . This reduces the electron energy and produces an energetic photon. We evaluate the corresponding contributions to , , and