# Updated Big Bang Nucleosynthesis Bounds on Long-lived Particles from Dark
Sectors

J. R. Alves1,2<EMAIL_ADDRESS>L. Angel1,2<EMAIL_ADDRESS>L.
Guedes1,2<EMAIL_ADDRESS>R. M. P. Neves2
<EMAIL_ADDRESS>F. S. Queiroz1,2,3<EMAIL_ADDRESS>D. R.
da Silva2,5<EMAIL_ADDRESS>R. Silva1<EMAIL_ADDRESS>Y. Villamizar6<EMAIL_ADDRESS>1Departamento de Física, Universidade
Federal do Rio Grande do Norte, 59078-970, Natal, RN, Brasil  2International
Institute of Physics, Universidade Federal do Rio Grande do Norte, Campus
Universitario, Lagoa Nova, Natal-RN 59078-970, Brazil  3Millennium Institute
for Subatomic Physics at the High-Energy Frontier (SAPHIR) of ANID, Fernández
Concha 700, Santiago, Chile  4Departamento de Fisica, Universidade Federal da
Paraiba, 58051-970, João Pessoa, PB, Brasil  5Centro Brasileiro de Pesquisas
Físicas, Rua Dr. Xavier Sicaud 150, Urca, CEP-22290-180, Rio de Janeiro, RJ,
Brasil  6 Centro de Ciências Naturais e Humanas, Universidade Federal do ABC,
09210-580, Santo André, São Paulo, Brasil

###### Abstract

As electromagnetic showers may alter the abundance of Helium, Lithium, and
Deuterium, we can place severe constraints on the lifetime and amount of
electromagnetic energy injected by long-lived particles. Considering up-to-
date measurements of the light element abundances that point to
$Y_{p}=0.245\pm 0.003$,$({\rm D/H})=(2.527\pm 0.03)\times 10^{-5}$,$(^{7}{\rm
Li/H})=1.58^{+0.35}_{-0.28}\times 10^{-10}$, $(^{6}{\rm Li/^{7}Li})=0.05$, and
the baryon-to-photon ratio obtained from the Cosmic Microwave Background data,
$\eta=6.104\times 10^{-10}$, we derive upper limits on the fraction of
electromagnetic energy produced by long-lived particles. Our findings apply to
decaying dark matter models, long-lived gravitinos, and other non-thermal
processes that occurred in the early universe between $10^{2}-10^{10}$
seconds.

##  I Introduction

There are four pillars that validate the hot Big Bang Theory, namely the
Planck spectrum of the Cosmic Microwave Background (CMB), the density
fluctuations imprinted in the CMB and the distribution of galaxies, the
confirmed expansion of the universe, and the Big Bang Nucleosynthesis (BBN).
Collectively, these observations support the notion that the universe
originated from a state of high temperature and density. BBN is rooted in
thoroughly studied physics processes, encompassing the spectra, reactions, and
weak decays of light elements. The BBN prediction for the light elements
abundances such as Deuterium, Helium-3, and Helium-4 and Lithium-7 rely solely
on one parameter, the baryon-photon ratio, $\eta=n_{b}/n_{\gamma}$. The ratio
of any two primordial abundances should give $\eta$, and the measurement of
the other three elements tests the BBN theory. The abundances of the light
elements have been measured in several terrestrial and astrophysical
environments. Although, it has often been hard to determine when these
abundances are close to the primordial ones. Nevertheless, there is a clear
agreement with the BBN predictions for all the light nuclei. The significant
improvement experienced in the measurements of the light element abundances
has deepened our knowledge of the primordial universe. For instance, for a
long time the error on the baryon density, $\Omega_{b}$ was large enough to
allow a dark baryonic component in the universe. In the late 90’s the error
was too large, $\Omega_{b}h^{2}=0.009-0.02$ Copi:1994ev . New measurements of
the Deuterium abundance, D/H, using quasars observations point to
$\Omega_{b}h^{2}=0.022$, with $h=0.674$ ParticleDataGroup:2022pth , solidly
confirming the existence of a dark matter component in the universe, which is
essentially non-baryonic. In other words, BBN stands as evidence for dark
matter in the universe. We highlight that Planck data on the CMB point to
$\Omega_{b}h^{2}=0.02230\pm 0.00021$ and $\eta=6.104\pm 0.058\times 10^{-10}$
Planck:2018vyg ; Fields:2019pfx , confirming BBN predictions.

Interestingly, BBN is a phenomenon that unfolds within a well-defined sequence
of events during the first minutes of the Universe. The remarkable agreement
between BBN predictions and observational data establishes BBN as an important
probe for new physics events that might alter the abundance of those elements,
which is a fact that will be explored in this work. In particular, BBN places
stringent constraints on new physics beyond the Standard Model that injects
energy into the cosmological plasma, such as the decays of massive particles
with lifetimes greater than $0.1$ s Forestell:2018txr .

There have been many works that discussed in great detail the impact of energy
injection in the BBN, and used this information to derive limits on the
lifetime of long-lived massive particles whose decays induce electromagnetic
or hadronic interactions with the plasma, leading to processes that alter the
abundance of the light elements. Hadrons produced from the decay of a long-
lived particle scatter off protons and neutrons modifying the ratio of these
baryons, and consequently the helium abundance. They can also destroy and
modify the abundance of helium and other light elements through
hadrodissociation. The chain reactions are non-trivial and the final abundance
of the elements are highly sensitive to the energy injected in the form of
hadrons into the plasma Forestell:2018txr .

Supersymmetric models Feng:2003uy ; Feng:2004zu ; Allahverdi:2014bva , and
enlarged gauge groups Kelso:2013nwa , and others Hooper:2011aj  feature
particles with a long lifetime, $\tau>10^{2}$ s. When long-lived particles
inject electromagnetic energy into the plasma, i.e., photons, electrons, and
positrons, the abundance of light elements is only meaningfully affected when
the decay happens at later times, $\tau>10^{4}$ s. Photodissociation plays a
major role in the abundance of light elements compared to other processes. In
this work, we do not aim to go into detail about the physical processes and
review all technical computations, for this matter, we refer the reader to
Tytler:2000qf . We will rather present updated limits on the lifetime and the
amount of electromagnetic energy that long-lived particles could inject into
the early universe using the most updated values of the light elements
abundances. Previous works placed limits on long-lived particles using the
abundance of D, 4He, and 7Li only. Bearing in mind the large uncertainties
involved in the primordial abundance of 6Li, we will additionally set a limit
on its primordial abundance using the 6Li/7Li ratio Cyburt:2002uv , which will
allow us to constrain a larger region of parameter space, for $\tau>10^{6}$s.

In summary, our work is structured as follows: In section II we briefly review
the primordial abundances of the elements; in section III we discuss the key
ingredients for computing the electromagnetic cascade resulting from the decay
of the long-lived particles; in section IV we present our updated limits; in
section V we draw our conclusions.

##  II Abundance of the Light Elements

The abundance of the light elements is determined by a set of first-order
differential Boltzmann equations originally developed in Alpher:1948ve ;
Gamow:1948pob ; Hayashi:1950lqo ,

$\displaystyle\frac{dY_{i}}{dt}=-H(T)T\frac{dY_{i}}{dT}=\sum(\Gamma_{ij}Y_{j}+\Gamma_{ikl}Y_{k}Y_{l}+...),$
(1)

where $Y_{i}=n_{i}/n_{b}$ is the ratio between the number density of the
species $i$ and baryon number density. The relevant interaction and decay
rates are encoded in $\Gamma_{ij}$ and $\Gamma_{ikl}$, respectively. $H(T)$ is
the hubble rate given by,

$H(T)=T^{2}\left(\frac{8\pi^{3}g_{*}G_{N}}{90}\right)^{1/2}$ (2)

with $g_{*}=g_{bosons}+\frac{7}{8}g_{fermions}$ corresponding to the
relativistic degrees of freedom of the species at the time, $G_{N}$ being the
gravitation constant, and T the plasma temperature. Eq.1 must be solved
considering that after electron-positron annihilation, neutrinos, and photons
have different temperatures, $T_{\nu}\simeq(4/11)^{1/3}T_{\gamma}$.

Several well-known processes occur and dictate the production of light
elements. We briefly review them below.

### Helium-4

Weak interactions govern the n-p conversion, and they scale with
$\Gamma_{n-p}\sim G_{F}^{2}T^{5}$, where $G_{F}$ is the Fermi constant. The
universe cools down with $H(T)\sim\sqrt{g_{\ast}G_{N}}T^{2}$, and eventually,
the interaction rate fell faster than the Hubble rate, resulting in a
departure from chemical equilibrium (freeze-out) around $T\sim 1$ MeV. At this
temperature, the ratio between the number of neutrons to proton is
$n/p\sim\exp^{m_{n}-m_{p}/1MeV}\sim 1/6$, but continues to decrease down to
$1/7$ due to $\beta$ decay.

All neutrons form Helium-4 at this time, because many background photons delay
deuterium formation through photon-dissociation. Hence, the 4He mass fraction
$Y_{p}$ is found to be Pospelov:2010hj ,

$Y_{p}\simeq\frac{2n/p}{1+n/p}=0.248$ (3)

Therefore, 4He mass fraction is amenable to the neutron-to-proton freeze-out
and timing at which Deuterium production becomes efficient.

### Deuterium

When the temperature fall below $70$ keV, the exponential Boltzmann
suppression on the number of photons is sufficient to significantly induce the
production of Deuterium (D) and consequently initiate other nuclear reactions
(See the illustration in Fig.1). The Deuterium and 3He productions occur
concurrently. Most of the deuterium then collided with other protons and
neutrons to produce helium and a small amount of tritium. Deuterium also plays
a minor role in the production of Lithium 7.

### Lithium-7

The rate for producing elements with A=6,7 is smaller than the Hubble rate.
Consequently, the abundances are tiny. As far as 7Li is concerned, around 90%
of the primordial 7Li stems 7Be having in mind the $\eta$ value inferred from
the CMB.

### Lithium-6

The primordial abundance of 6Li is often not used to constrain new physics due
to the uncertainties involved. Similarly to 7Li, the abundance of 6Li is also
measured in halo stars. Observations lead to 6Li/7Li $\sim 0.05$
Asplund:2005yt , which will be the value adopted in our work. The observed 6Li
abundance is understood to be of galactic origin; thus one can use its
abundance to set constraints on new physics. The reader should keep in mind
that the assessment of the 6Li primordial abundance is non-trivial, because it
is related to the 7Li abundance, and there are ways to deplete 6Li while
keeping 7Li unaltered due to its relatively small binding energy. We will
present limits based on the 6Li abundance, which is strongly sensitive to
electromagnetic injection episodes after $10^{7}$ s.

Figure 1:  Representation of the relevant chain reactions for BBN.

After this brief review of the light element abundances, we will address the
electromagnetic injection.

##  III Electromagnetic Injection

New physics processes that inject electrons, positrons, or photons can induce
photodissociation reactions, which is the most important effect on the light
elements abundances. These particles lose energy very efficiently by
scattering off the highly abundant photon background. The scattering process
has an energy threshold. If the energy of the injected photon is sufficiently
large ($E_{\gamma}>m_{e}^{2}/E_{\gamma,BG}$), typically the pair production
$\gamma+\gamma\rightarrow e^{+}e^{-}$ dominates because both cross-section and
number density of the background photon are large Protheroe:1994dt ;
Kawasaki:1994sc . Below this energy threshold, photons lose energy mostly via
photon-photon scattering $(\gamma+\gamma_{BG}\rightarrow\gamma+\gamma)$,
Compton scattering ($\gamma+e^{\pm}\rightarrow\gamma e^{\pm}$) and pair
creation on nuclei ($\gamma+N_{BG}\rightarrow e^{+}e^{-}N_{BG}$). However, if
the new physics process injects high-energy electrons or positrons, then
inverse Compton scattering governs the energy loss Kawasaki:1994sc . Comparing
the interaction rate with the expansion rate, we conclude that these processes
are not very efficient at early times, $\tau<10^{4}$ s. Therefore, these
energy injection episodes are seen as a post-event that can alter the BBN
predictions.

In summary, one can account for these possibilities by including the non-
thermal photon population and abundances of nuclei from,

$\displaystyle-
HT\frac{dY_{B}}{dT}=\sum_{A}Y_{A}\int_{0}^{\infty}dE_{\gamma}f_{\gamma}^{\rm
qse}(E_{\gamma})\sigma_{\gamma+A\rightarrow B}(E_{\gamma})$ $\displaystyle-
Y_{B}\sum_{C}\int_{0}^{\infty}dE_{\gamma}f_{\gamma}^{\rm
qse}(E_{\gamma})\sigma_{\gamma+B\rightarrow C}(E_{\gamma}).$ (4)

where the energy spectrum is found to be Cyburt:2002uv ,

$f_{\gamma}^{\rm
qse}=\frac{n_{X}p_{\gamma}(E_{\gamma})}{\tau_{X}\Gamma_{\gamma}(E_{\gamma})},$
(5)

with $n_{X}$ being the number density of the long-lived particle, and
$\tau_{X}$ its lifetime, and $p_{\gamma}$ the photon spectrum which we assumed
to be approximated by a power law Protheroe:1994dt ,

$\displaystyle
p_{\gamma}(E_{\gamma})=\begin{cases}K_{0}(E_{\gamma}/E_{\mathrm{low}})^{-1.5}&\mathrm{for}\
E_{\gamma}<E_{\mathrm{low}}\\\
K_{0}(E_{\gamma}/E_{\mathrm{low}})^{-2.0}&\mathrm{for}\
E_{\mathrm{low}}<E_{\gamma}<E_{C}\\\ \qquad\quad 0&\mathrm{for}\
E_{\gamma}>E_{C}\end{cases}$ (6)

where $E_{\rm low}\simeq m_{e}^{2}/(80T)$, and the energy threshold of pair
production $E_{C}\simeq m_{e}^{2}/22T$ Kawasaki:1994sc . The normalization
constant $K_{0}$ in Eq.6 is obtained by requiring that the electromagnetic
injected is given by $E_{X}=\int dE_{\gamma}E_{\gamma}p_{\gamma}$. We
emphasize that the power law approximation is a good approximation for
energies larger than $E_{C}$ Poulin:2015opa ; Forestell:2018txr  as we
consider here. That said, we can compute the change in the abundance of the
light elements as a function of the lifetime $\tau$ and the total
electromagnetic energy released $\zeta_{X}$, where $\tau$ is the lifetime of
the decaying particle. The total electromagnetic energy released in a process
of the type $X^{\prime}\rightarrow X+\gamma$ as we will consider here is given
by,

$\zeta_{X}=E_{\gamma}Y_{X}$ (7)

where $E_{\gamma}$ is the energy of the final state photon,

$E_{\gamma}=\frac{1}{2M_{X^{\prime}}}(M^{2}_{X^{\prime}}-M^{2}_{X}),$ (8)

with $Y_{X}=n_{X}/n_{\gamma}^{CMB}$ being the ratio between the number density
of species $X$ over the number density of CMB photons. If for each
$X^{\prime}$ particle we have the production of the pair $X+\gamma$, then
$Y_{X^{\prime}}=Y_{\gamma}=Y_{X,\tau}=Y_{X,0}$, where $Y_{X,\tau}$ sets the
number density of particles at a time $t=\tau$. $Y_{X,0}$ is the yield of the
species $X$ today. To simplify our analytical expressions, we will assume the
number density of the X is proportional to the one from dark matter. In other
words, we take $n_{X}=f\,n_{DM}$, where $f$ is a constant factor that defines
the relation between the abundance of dark matter and X particles. We include
this factor to be more general, and account for the possibility of subdominant
dark matter fields. One can easily rescale the total amount of energy released
for any long-lived particles. Knowing that
$n^{CMB}_{\gamma}=2\zeta(2)\pi^{-2}T^{3}$, we get,

$Y_{X}=\frac{f\,n_{DM}}{n^{CMB}_{\gamma}}=\frac{f\,\Omega_{DM}\rho_{c}}{M_{X}n^{CMB}_{\gamma,0}},$
(9)

which simplies to,

$Y_{X}\simeq 3\cdot 10^{-14}\left(\frac{\rm
TeV}{M_{DM}}\right)\left(\frac{\Omega_{DM}}{0.227}\right)\left(\frac{f}{0.01}\right).$
(10)

Using Eq.(7) we obtain the total electromagnetic energy released,

$\displaystyle\zeta_{X}$ $\displaystyle=$ $\displaystyle 1.5\cdot 10^{-11}\
\mbox{GeV}\times$ (11)
$\displaystyle\left(\frac{\Omega_{DM}}{0.227}\right)\left(\frac{f}{0.01}\right)\left(\frac{M_{X^{\prime}}}{M_{X}}-\frac{M_{X}}{M_{X^{\prime}}}\right).$

Therefore, one can use Eq.(11) to determine the amount of electromagnetic
energy released by a long-lived particle. Moreover, one can then pick their
favorite dark sector, use Eq.(11), and map our bounds on the pair
$\zeta_{X}-\tau$ to constrain their models. We will see further that BBN
places severe constraints on dark sectors that induce electromagnetic cascades
in the early universe. This reasoning will clearly show that one can explore
the interplay between cosmology and particle physics and use BBN as a probe
for new physics.

Hence, in this work, we compare the results of Big Bang Nucleosynthesis (BBN)
old data against the most recent Planck data, considering Planck’s measurement
of $\eta_{10,\textrm{CMB}}$111The subscript 10 is a reference to the fact that
we are considering the values of $\eta\times 10^{10}$., which is in agreement
with the current data ParticleDataGroup:2022pth  provided by

$\eta_{10}=\frac{n_{b}}{n_{\gamma}}=6.104\pm 0.058.$ (12)

This measurement of the baryon-to-photon ratio can be used to derive a bound
on the baryon relic abundance $\Omega_{b}h^{2}$ Planck:2018vyg ,

$\Omega_{b}h^{2}=0.022298\pm 0.000212,$ (13)

where $h$ is the reduced Hubble constant.

The BBN theory predicts the abundance of light elements like D/H, 3He,4He and
7Li/H that can be inferred from astronomical observations. However, it is
important to emphasize that these results depend on the assumed baryon-photon
relationship, which can be determined by the anisotropy spectrum of the CMB or
by the concordance of the abundances generated by the BBN. In Tab. 1 we
compare the theoretical values obtained from the BBN predictions with the old
observational data Cyburt:2002uv  and with the new results provided by the PDG
ParticleDataGroup:2022pth . As we can see for YP, D/H and 3He/H the
theoretical and observational results converge, with only a small fluctuation
between the values. However, when comparing the theoretical prediction with
the observational data of 7Li we verify a marked divergence between the
results, where 7Li/H${}_{(teo)}\sim 3\times^{7}$Li/H(obs). In our work, we
took the 7Li abundance extracted from recent observations
ParticleDataGroup:2022pth . The values adopted for the abundance of the light
elements are summarized in Table.1. In Table.1 we show the values resulting
from BBN theory, old and new observations. The latter was used to derive our
bounds, as we describe below.

Figure 2: Contours of the observations of the abundance of light elements in the plane $\zeta_{X}-\tau_{X}$, for $\eta_{10}=6$ considering different values for (a) mass fraction of 4He, (b) Deuterium abundance, (c) contours for 7Li/H and (d) observations for 6Li/7Li. The new contours correspond to the most recent data from the PDG Cyburt:2002uv  Table 1: Theoretical and observational analyses of the abundance of light elements in BBN. Yields | Yp | D/H $\left(\times 10^{-5}\right)$ |  3 He/H $\left(\times 10^{-6}\right)$ |  7 Li/H $\left(\times 10^{-10}\right)$ |  6Li/7Li | Ref.
---|---|---|---|---|---|---
Calculation | 0.248 ${}^{+0.001}_{-0.002}$ | 2.692 ${}^{+0.177}_{-0.070}$ | 9.441 ${}^{+0.511}_{-0.466}$ | 4.283 ${}^{+0.335}_{-0.292}$ |  |
Old Observation | 0.248 $\pm 0.001$ | 2.9 ${}^{\pm}0.3$ | 9\. $\pm 1.$ | 1.23 $\pm 0.06$ |  | Cyburt:2002uv
New Observation | 0.245 $\pm 0.003$ | 2.527 $\pm 0.03$ | $\geq 11.\pm 2.$ | 1.58 ${}^{+0.35}_{-0.28}$ |  | ParticleDataGroup:2022pth

##  IV New limits for the abundance of observed light elements

In section III we saw that it is important to consider the baryon-photon
number density of the early universe to study the BBN predictions. Thus, to
impose stronger restrictions on the abundance of light elements, we use the
results presented in section IV and set $\eta_{10}=6$, varying only the
lifetime between $T=10^{4}$ sec and $T=10^{11}$ sec C. We present the new
energy injection limits obtained for the new constraints of 4He, D/H, 7Li/H
and 6Li/7Li, as can be seen in Fig. 2.

The most probable value of the 4He abundance ($Y_{p}=0.2453$) was found
through analysis of extragalactic HII regions (ionized)
ParticleDataGroup:2022pth ; Aver:2020fon . However, the recommended confidence
interval for $Y_{p}$ is

$\displaystyle 0.2419<Y_{p}<0.2487.$ (14)

Thus, by imposing the lower bound of $Y_{p}>0.2419$, the constraint observed
for $Y_{p}$ is

$\displaystyle\zeta_{X}(Y_{p})<5.2\times 10^{-11}{\rm GeV}.$ (15)

As 4He is orders of magnitude more abundant than the other elements, no
significant production of 4He can take place. In other words, electromagnetic
energy injection can only destroy 4He. To understand the behavior of the curve
in Fig.2, we remind the reader of some fundamental aspects of electromagnetic
injection. In the sudden decay approximation, where all particles decay at
$t=\tau_{X}$, a reaction turns on when $\tau_{X}>10^{6}{\rm
s}\,(E_{th}/10MeV)^{2}$. Keep in mind that the photodestruction processes have
an energy threshold of 20MeV. Hence, the 4He destruction will be efficient
only for $\tau\gtrsim 10^{6}$ s. For this reason, any small value of
$\zeta_{X}$ will severely destroy 4He in contradiction to observations. One
needs a very large energy injection for a shorter lifetime to alter the 4He
abundance. This feature explains the weakening in sensitivity $\tau<10^{6}$ s
in Fig.2. In Fig.2, we also exhibit how BBN sensitivity to new physics
improves with the value of $Y_{p}$.

Unlike $Y_{p}$, the change of the Deuterium abundance is more subtle. The D/H
ratio will depend highly on the moment the energy injection took place. The
most important production process is 4He $+\ \gamma$, because the other
elements have relatively negligible abundances to produce any significant
impact on the Deuterium abundance. The reactions that produce Deuterium have
an energy threshold of 25 MeV, so again they are efficient only for
$\tau_{X}\gtrsim 10^{6}$ s. In other words, Deuterium production occurs mostly
when there is sizeable 4He destruction. If we increase $\zeta_{X}$ too much,
the photodestruction processes are so effective that Deuterium abundance
quickly decreases, leaving the universe filled with protons only. For
$\tau<10^{6}$ s, the Deuterium production does not happen, and only
destruction occurs. Eventually, for $\tau<10^{3}$ s, the Deuterium destruction
freezes. The competing processes, around $10^{6}$ s, explain the shape of the
curve in Fig.2. In Fig.2, we plot an old limit that delimited the Deuterium
abundance to lie between $10^{-4}<$ D/H $<10^{-6}$ and show ours which is
based on a much more restrictive statistical range for D/H, namely
ParticleDataGroup:2022pth ; Cooke:2017cwo ,

$\displaystyle 2.227\times 10^{-5}<\left(\frac{{\rm D}}{{\rm
H}}\right)_{p}<2.827\times 10^{-5}.$ (16)

As we do not allow a large abundance for Deuterium, i.e. a high D/H, our limit
weakens for $\tau>10^{7}$ s. From Fig.2 we concluded that,

$\displaystyle\zeta_{X}({\rm D/H})\leq 6.0\times 10^{-10}{\rm GeV},{\rm
for}\,\tau\sim 10^{6}\leavevmode\nobreak\ s,$ (17)
$\displaystyle\zeta_{X}({\rm D/H})\leq 3\times 10^{-8}{\rm GeV},{\rm
for}\,\tau\sim 10^{8}-10^{10}\leavevmode\nobreak\ s.$ (18)

The 7Li primordial abundance stems from observations of low metallicity stars
in the halo of our Galaxy indicating ParticleDataGroup:2022pth ;
Bertulani:2022qly ; Coc:2011az ,

$\displaystyle 1.23\times 10^{-10}<\left(\frac{{}^{7}{\rm Li}}{{\rm
H}}\right)_{p}<1.93\times 10^{-10}.$ (19)

We highlight that the statistical errors in the 7Li abundance are overwhelmed
by systematic errors surrounding the primordial abundance of 7Li. Therefore,
one should use 7Li abundance as a probe to new physics with care
Alcaniz:2019kah . Anyway, from Fig.2, we get,

$\displaystyle\zeta_{X}(^{7}{\rm Li/H})<2\times 10^{-9}{\rm GeV},{\rm
for}\,\tau\sim 10^{6}-10^{10}s$ (20)

Looking at Fig.2, we notice that the upper bound on the energy injection
stemming from 7Li is similar to the one from 4He. As the production of 7Li is
insignificant, we can only destroy it. As 7Li has a smaller binding energy, it
is rather sensitive to electromagnetic energy injection for $\tau<10^{6}$ s
compared to 4He. These facts explain the curve shape in Fig.2. Nevertheless,
the 7Li abundance is subject to large uncertainties.

Regarding 6Li, we included the observed constraints on 6Li/7Li as seen in Fig.
2. Unlike 7Li, standard BBN does not produce 6Li in any observable quantity,
so any other production mechanism is important. The behavior of Fig.2 can be
understood in terms of the 4He and 7Li dynamics because these are the two 6Li
sources. For $\tau_{X}<10^{7}$ s, the 6Li production rises from 7Li and 7Be
due to their low energy threshold (see Fig.1). The dominant production channel
is from secondary reactions tied to the 4He destruction, otherwise. One should
bear in mind that there are even greater systematic uncertainties regarding
the 6Li abundance. Anyhow, it is clear from Fig.2 that 6Li has the potential
to be much more restrictive than any other element to an electromagnetic
energy injection in the early universe for $\tau>10^{6}$ s. Considering
${}^{6}{\rm Li}/^{7}{\rm Li}=(1.5\pm 0.3)\times 10^{-5}$ as indicated in
Table1, we find,

$\displaystyle\zeta_{X}(^{6}{\rm Li}/^{7}{\rm Li})<4\times 10^{-13}{\rm
GeV}\,{\rm for}\,\tau\sim 10^{9}-10^{11}$ (21)

We combine all upper limits into Fig. 3, so one can clearly see the importance
of each element to constraining electromagnetic injections in the early
universe. The shaded regions delimit the exclusion limits from the 7Li/H ratio
(blue) and the D/H abundance (orange). The hashed red region comes from the
4He abundance and the pink line delimit the 6Li/7Li ratio. It is clear that
Deuterium and 4He which have reliable primordial abundances suffice to probe
electromagnetic energy injection episodes. Nearly the entire region which 7Li
is sensitive to, has already been excluded by D and 4He. This is not true for
6Li, though. Using the 6Li/7Li ratio, we can probe electromagnetic energy
injections down to $\zeta_{X}\sim 4\times 10^{-13}$ for $\tau>10^{9}$ s. It
represents two orders of magnitude improvement over the 4He sensitivity.
Therefore, a more solid development on the 6Li/7Li primordial abundance would
mean a significant improvement in the BBN sensitivity to electromagnetic
energy injection.

Figure 3: Summary of the upper limits on the electromagnetic energy injection,
$\zeta_{X}$, as a function of the lifetime of the decaying particle. It is
clear that Deuterium and 4He which have reliable primordial abundances suffice
to probe electromagnetic energy injection episodes that took place from
$10^{2}$ s to $10^{1}1$ s. 7Li offers important limits only for $\tau\simeq
4\times 10^{6}$ s. Using the 6Li/7Li ratio, we can probe electromagnetic
energy injections down to $\zeta_{X}\sim 4\times 10^{-13}$ for $\tau>10^{9}$
s. It represents two orders of magnitude improvement over the 4He sensitivity.
Nevertheless, Therefore, a more solid development on the 6Li/7Li primordial
abundance would mean a significant improvement in the BBN sensitivity to
electromagnetic energy injection

##  V Discussion and Conclusions

In this article, we have revisited the impact of electromagnetic injection
episodes on the abundance of light elements, such as 4He, D, 7Li, and 6Li. We
were particularly interested in the decaying scenario, $X^{\prime}\rightarrow
X+\gamma$, where $X$ had an abundance proportional to the dark matter one. We
have computed the change in the abundance of those elements as a function of
the total electromagnetic energy released and the lifetime of the decaying
particle, $X^{\prime}$. In our analysis, we adopted the primordial abundance
of the light elements reported from recent observations and used the baryon-
photon ratio indicated by Planck data, $\eta^{cmb}_{\gamma}=6.104\times
10^{-10}$. Firstly, we derived separately the bounds from 4He, $D$, 7Li, and
6Li abundances and later combined them all into one figure. We concluded that
Deuterium and 4He which have reliable primordial abundances can already rule
out energy injections down to $\zeta_{X}\sim 10^{-9}$ GeV for $\tau\sim
10^{5}-10^{6}$ s, and $\zeta_{X}\sim 10^{-10}$ GeV for $\tau\sim
10^{8}-10^{11}$ s, surpassing the sensitivity of 7Li to new physics nearly in
the entire parameter space. The ratio 6Li/7Li proved to be rather restrictive
to energy injection episodes for $\tau>10^{7}$ s, offering an upper limit on
$\zeta_{X}$ which is two orders of magnitude better than the one from 4He
abundance. Nevertheless, as BBN does not produce any meaningful amount of 6Li,
this bound from the 6Li/7Li should not be taken at face value.

Anyway, we point out that the scenario covered in our study appears in
supersymmetric models, extended gauge sectors, and decaying dark matter
models. Hence, it is clear that BBN represents a great laboratory for well-
motivated dark sectors that induce energy injection in the early universe.

###### Acknowledgements.

The authors thank Jacinto Paulo and Alvaro de Jesus for discussions. DRS
thanks for the support from CNPQ under grant 303699/2023-0. FSQ is supported
by Simons Foundation (Award Number:1023171-RC), FAPESP Grant 2018/25225-9,
2021/01089-1, 2023/01197-4, ICTP-SAIFR FAPESP Grants 2021/14335-0, CNPq Grants
307130/2021-5, and ANID-Millennium Science Initiative Program ICN2019_044.
Y.S.V. expresses gratitude to São Paulo Research Foundation (FAPESP) under
Grant No. 2018/25225-9 and 2023/01197-4.

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