# Cosmological constraints on the decay of heavy relics into neutrinos

Thomas Hambye<EMAIL_ADDRESS>Marco Hufnagel<EMAIL_ADDRESS>Matteo Lucca<EMAIL_ADDRESS>Service de Physique Théorique, Université
Libre de Bruxelles, C.P. 225, B-1050 Brussels, Belgium

###### Abstract

A massive particle decaying into neutrinos in the early Universe is known to
be less constrained than if it was decaying into other standard model
particles. However, even if the decay proceeds into neutrinos, the latter
still inevitably emit secondary particles undergoing electromagnetic
interactions that can be probed. We analyse in details how sensitive various
cosmological probes are to such secondary particles, namely CMB anisotropies,
CMB spectral distortions, and Big Bang Nucleosynthesis. For relics whose
lifetime is shorter than the age of the Universe, this leads to original and
stringent bounds on the particle’s lifetime as a function of its abundance and
mass.

##  I Introduction

Many beyond the standard model (BSM) scenarios imply the presence of new
unstable particles during the early Universe epoch. For instance, dark matter
(DM) setups introducing new dark sector interactions typically predict the
existence of additional mediator particles, whose fate can affect the whole
thermal history of the Universe, spanning from Big Bang Nucleosynthesis (BBN)
predictions and the shape of the Cosmic Microwave Background (CMB) anisotropy
power spectra to the cosmic rays and high-energy neutrino spectra observed
today. Indeed, this is precisely the case for a wide range of models, such as
for instance self-interacting DM scenarios involving the exchange of a light
mediator [1, 2, 3, 4, 5, 6] or hidden sector DM setups communicating with the
SM through a portal interaction.

If these unstable particles have still not disappeared by today, they can be
probed by direct and indirect detection and also have a cosmological impact,
such as having a too large relic abundance, thereby overclosing the Universe
(unless they have tiny sub-eV masses or their numbers are severely suppressed
at some point). If instead they do decay with a lifetime smaller than the age
of the Universe, their impact, which depends on their characteristics (such as
mass and primordial abundance), is mostly cosmological.

Most cosmological probes are basically only sensitive to the amount of
electromagnetic (EM) material that is effectively injected into the early
Universe photon-baryon fluid (see e.g. [7, 8]). Thus, one could at first sight
expect that if an extra particle, henceforth generically referred to as
“$\phi$”, decays dominantly into non-EM material, such as into neutrinos or
dark radiation, the decaying particle will be poorly constrained by these
probes. However, even if the decay proceeds into neutrinos, e.g. via
$\phi\rightarrow\nu\bar{\nu}$ or $\phi\rightarrow\nu{\nu}$, still, charged and
colored particles will be produced inevitably via the emission of additional
on-shell or off-shell $W/Z$-bosons. This ultimately results in the production
of EM and hadronic showers of particles that can affect electromagnetically
the CMB and BBN. This way, by carefully evaluating the amount of EM material
that the neutrinos release, the decay of massive particles into neutrinos can
be constrained by cosmological probes. To determine these limits constitutes
the main goal of this work.

In the following, we will focus our analysis on the cosmological constraints
coming from CMB anisotropies, CMB spectral distortions (SDs), and BBN
photodisintegration. We will do that for the case where the particles have a
lifetime shorter than the age of the Universe. This implicitly implies that
the massive particle we assume to be decaying cannot constitute the DM
observed today, but still it can play a role in the early Universe.
Complementary constraints that rely on the assumption that the decaying
particle is instead the DM, and hence constrain lifetimes larger than the age
of the Universe, can be found in e.g. [9, 10, 11, 12, 13, 14, 15, 16].

In light of these considerations, our analysis is organized as follows. In
Sec. II we will outline the method we employed to derive the fractional amount
of energy emitted in the form of EM radiation by the decay process and present
the resulting relation between this quantity and the relic’s mass. In Sec. III
we will briefly describe the cosmological probes we will use to constrain the
characteristics of the decay. In Sec. IV we will combine the two previous
sections to derive original bounds on the decay of massive relics into
neutrinos. We close in Sec. V with a summary of our results and other
concluding remarks.

##  II Energy injection

To calculate the amount of EM energy (i.e. carried by photons and charged
particles) that is produced by the decay of a massive relic into primary
neutrinos, in the following we have to (i) model the particle shower
production, and (ii) quantify the interactions of the shower products with the
background plasma. We thereby explicitly focus on decays into electron
neutrinos, but note that any other flavour would give similar results (due to
lepton universality of electroweak interactions).

###  II.1 Electroweak shower

The emission of secondary particles from neutrinos through radiative
corrections has already been considered in various contexts in the literature
(see e.g. [17, 18, 11, 19, 20]). These studies usually rely on Pythia8 [21] to
properly simulate the electroweak shower. This approach assumes a
factorization between the process producing the primary neutrinos and the
gauge-boson emission. This is justified for neutrino energies above the mass
of the $W/Z$ bosons, in which case the dominant contribution comes from the
emission of one (or multiple) on-shell $W/Z$-bosons from any of the final-
state neutrinos. For a decay, this applies when $m_{\phi}\gtrsim 2\,m_{W,Z}$.
For these masses, the amount of EM energy that gets injected into the plasma
becomes effectively independent of the underlying BSM model and we can follow
the established procedure to simulate the shower by (i) generating spherically
symmetric $\nu_{e}\bar{\nu}_{e}$ events with energy
$E_{\nu_{e}}=E_{\bar{\nu}_{e}}=m_{\phi}/2$, and (ii) using these events as an
input for Pythia8.3 [21] with the electroweak shower option
enabled.111Specifically, we use the two options PartonShowers:model=2 and
Vincia:EWmode=3.

For smaller masses, $m_{\phi}/2\lesssim m_{W,Z}$, the primary decay and the
emission process do no longer factorize, meaning that Pythia8.3 alone is no
longer able to simulate the production of secondary particles. In fact, using
the same procedure as before for these masses, Pythia8.3 predicts that no EM
material is injected at all (see Fig. 1 below).222Similarly, the PPPC results
[17], which have been obtained with Pythia8.1, are to be used only for
neutrino energies above $100-150\,\mathrm{GeV}$. Nevertheless, for
$m_{W,Z}<m_{\phi}<2\,m_{W,Z}$ an on-shell gauge boson emission is still
possible from a neutrino with energy $E_{\nu}>m_{\phi}/2$, and for
$m_{\phi}<m_{W,Z}$ the gauge boson can still be produced off-shell. In these
cases, the dominant process is $\phi\rightarrow\nu_{e}f_{1}f_{2}f_{3}$ with
any three SM fermions $f_{1,2,3}$. To take this process into account, we
instead have to consider the full transition amplitude for the emission
process, which ultimately makes the calculation dependent on the actual BSM
scenario.

In order to quantify this effect, we consider a simple scenario where the
decaying particle is a massive vector boson coupling to neutrinos in the
following simple way

$\displaystyle\mathcal{L}\supset\lambda_{e}\phi_{\mu}\bar{\nu}_{e}\gamma^{\mu}P_{L}\nu_{e}\,.$
(1)

The results obtained in this case should be rather general and reflect what
one can expect also in other scenarios.333Strictly speaking, in its simplest
version, this model also predicts, via $SU(2)_{L}$ invariance, the production
of a pair of charged leptons, which will dominate the production of EM
material. However, still the amount of EM energy obtained from neutrinos will
be generic of what we can get from other models not necessarily producing
charged leptons. We will not enter here in the details of specific models.
Note, however, that Eq. (1) can be obtained (i.e. without a $SU(2)_{L}$
charged lepton counterpart), from the decay of a vector boson into two heavy
sterile neutrinos, mixing with ordinary neutrinos, in the spirit of Ref. [15].
Note also that we could also consider frameworks based on the existence of a
scalar triplet. For instance, the neutral component of a scalar triplet feebly
coupling to doublets of leptons decays only into two neutrinos, whereas its
singly and doubly charged $SU(2)_{L}$ partners could have disappeared long
before by decaying more quickly via gauge interactions (such as through the
process $\Delta^{+}\rightarrow\Delta^{0}l\nu$). One can check that this
example of scalar particle decay leads to a production of particles that is
very similar to the one obtained from the decay of a vector boson, Eq. (1). By
implementing this model, which we call cn, in FeynRules [22] and afterwards
using it inside of MadGraph3.2.0 [23], we then fully simulate events for the
processes $\phi\rightarrow\nu_{e}f_{1}f_{2}f_{3}$ and
$\phi\rightarrow\bar{\nu}_{e}f_{1}f_{2}f_{3}$, where $f_{1,2,3}$ can be any of
the kinematically available SM fermions. Specifically, this is done by running
the commands ($\texttt{phi}\leftrightarrow\phi$, $\texttt{f}\leftrightarrow
f_{1,2,3}$)

⬇

import model cn

define f = l+ l- vl vl~ j

generate phi > ve ve~

add process phi > ve f f f

add process phi > ve~ f f f

output

launch

for different masses $m_{\phi}$. The so-generated events are then used as in
input for Pythia8.3, which takes care of the remaining parts of the shower
process, including photon radiation and hadronization.

For the full range of masses, we then end up with a set of shower products
including stable particles ($e^{\pm}$, $\gamma$, $\nu_{i}/\bar{\nu}_{i}$,
$p^{\pm}$) as well as particles which can decay through weak interactions
($\mu^{\pm}$, $\pi^{\pm}$, $K^{\pm}$, $n/\bar{n}$). Once produced, all of
these particles can still undergo interactions with the background plasma,
which we discuss in the following.

###  II.2 Interactions with the background plasma

The characteristic timescales for (i) an unstable particle to decay via weak
interactions and (ii) a particle of the shower to interact electromagnetically
with the background plasma are very similar. Consequently, these two effects
are intertwined, meaning that in order to obtain the actual fraction of energy
that is injected into the EM plasma, it is usually necessary to solve the full
set of Boltzmann equations for the different shower products. In the following
we show, however, that the scattering reactions do not lead to a significant
redistribution of energy, which is why they can essentially be neglected.

Photons and electrons from the shower interact with the background plasma
mainly via double-photon pair creation $\gamma\gamma_{\text{th}}\rightarrow
f\bar{f}$ and inverse Compton scattering $e^{\pm}\gamma_{\text{th}}\rightarrow
e^{\pm}\gamma$ [24]. Here, $f$ is a SM fermion and the index ’th’ indicates
particles from the thermal background. In particular, the pair production of
electrons and positrons will occur if $E_{0}\gtrsim
m_{e}^{2}/[22T(\tau_{\phi})]$ [24], i.e. if $m_{\phi}/2\gtrsim E_{0}$, which
we find to be true for all parameter combinations considered in this work. In
this case, the given processes conserve the energy in the EM sector. However,
for $m_{\phi}/2>m_{\mu}^{2}/[22T(\tau_{\phi})]$, the process
$\gamma\gamma_{\text{th}}\rightarrow\mu^{-}\mu^{+}$ becomes relevant. Unlike
the electron pair production process, this reaction will transfer back part of
the energy to the neutrino sector via the muon decay $\mu^{-}\rightarrow
e^{-}\bar{\nu}_{e}\nu_{\mu}$. By comparing the reaction rates for the pair
production of electrons with the one for muons for an average photon energy
$\bar{E}_{\gamma}\sim 10^{-2}m_{\phi}$ and at a temperature $T(\tau_{\phi})$,
we find that the energy loss via muon production leads to a smaller than $1\%$
correction for $\tau_{\phi}\gtrsim
10^{4}\,\mathrm{s}\times(m_{\phi}/\mathrm{PeV})^{2}$. This covers almost the
full parameter space considered in this work (except for the shaded region in
Fig. 2 below). Thus we neglect this effect for simplicity.

A similar reaction can also be induced by neutrinos, i.e.
$\nu_{i}\bar{\nu}_{i,\text{th}}\rightarrow f\bar{f}$, which can instead
transfer energy from the neutrino sector to the EM sector [25]. However, this
process is only relevant if the intermediate $Z$ boson is produced close to
resonance. This ”Glashow-like resonance” happens roughly when $m_{\phi}/2\sim
E_{\text{GR}}\sim m_{Z}^{2}/[\mathcal{O}(1)T_{\nu}]$, which we deduce by
solving the full Boltzmann equation for this process. Hence, this process is
relevant only if $m_{\phi}>10^{7}\,\mathrm{GeV}$ for
$\tau_{\phi}>10^{3}\,\mathrm{s}$. Coincidentally, Pythia8.3 only works
reliably for $E_{0}<10^{7}\,\mathrm{GeV}$, which is why we only consider
energies below this (artificial) threshold, meaning that pair production from
neutrinos can also be neglected.444Other reactions like
$\nu_{i}e^{-}_{\text{th}}\rightarrow\dots$ are additionally suppressed by the
small number density of thermal electrons.

Finally, the particles
$x\\!\in\\!\\{\mu^{\pm},\pi^{\pm},K^{\pm},p^{\pm},n/\bar{n}\\}$ can induce
additional energy injection via (i) their decay and/or (ii) scattering
reactions [26, 27] such as Thompson scattering $x\gamma_{\text{th}}\rightarrow
x\gamma$. However, as already mentioned above, both of these effects are
potentially intertwined due to their similar timescales. For example, pions
mainly decay via $\pi^{-}\rightarrow\mu^{-}\bar{\nu}_{\mu}\rightarrow
e^{-}\bar{\nu}_{e}\nu_{\mu}\bar{\nu}_{\mu}$, and thus deploy most of their
energy into the neutrino sector. However, the scattering reactions prior to
the decay instead transfer energy from the pions into the EM plasma, which can
potentially distort the $1:3$ ratio implied by the decay alone. To quantify
this effect, we approximate the energy loss as continuous via a differential
equation of the form $\dot{E}_{x}=D_{x}(E,t)$ with the energy-loss rate
$D_{x}$ of $x$,555For a list of the relevant reactions and their corresponding
energy-loss rates, see e.g. App. B of [26]. which we solve from an initial
time $t_{i}=\tau_{\phi}$ until some final time $t_{f}$ satisfying
$t_{f}=\tau_{\phi}+\min\\{E_{x}(t_{f})\tau_{x}/m_{x},t_{\text{pdi}}\\}$, where
$t_{\text{pdi}}$ is the typical timescale at which photodisintegration
reactions become important. The transferred energy $\Delta
E_{x}=E_{x}(t_{i})-E_{x}(t_{f})$ is then attributed to the total amount of EM
energy, while the remaining decay of the particle $x$ with energy
$E_{x}(t_{f})$ is properly handled by running Pythia8.3 again. Overall we find
that the inclusion of this energy-loss mechanism does not significantly change
our results, as it merely leads to $\sim 1\%$ corrections for
$\tau_{\phi}>10^{4}\,\mathrm{s}$. Consequently, it is justified to neglect
this effect altogether and simply let all unstable particles decay
immediately, which can be handled in the same Pythia8.3 run as the shower
generation itself.

###  II.3 Results

Having described the different steps of our calculation, in Fig. 1 we show the
fraction $\zeta_{\text{EM}}$ of energy injected into the EM plasma as a
function of the mass $m_{\phi}$ (solid red). In addition, we also show the
results that we obtain when only using spherically symmetric
$\nu_{e}\bar{\nu}_{e}$ events as an input for Pythia8.3 (dash-dotted purple),
as well as a comparison with the previous results from the PPPC [17] (dashed
gray).

Figure 1: The fraction $\zeta_{\text{EM}}$ of EM energy that is injected into
the plasma as a function of the mass $m_{\phi}$ (solid red). For comparison,
we also show the results that we obtain when using only Pythia8.3 (dash-dotted
purple), as well as the results obtained with the PPPC (dashed gray).

We find that our results agree well with both other approaches for large
masses. However, for $m_{\phi}$ around and below $\sim 2\,m_{W,Z}$, the three
curves start to differ substantially. The deviation between the Pythia8.3 and
PPPC curves for $m_{\phi}\sim 2\,m_{W,Z}$ can be explained by noting that the
PPPC is based on Pythia8.1, which was subject to a bug preventing it from
producing correct results in this region of parameter space (see [28] for
further details) and which has since been fixed in Pythia8.3.666We explicitly
checked that Pythia8.3 can reproduce the results from [28], which used an
independent technique to properly simulate the shower. For even smaller
masses, below $m_{\phi}\sim 2\,m_{W,Z}$, Pythia8.3 itself is no longer able to
simulate the initial gauge boson emission from spherically symmetric events,
and instead it becomes essential to properly generate the initial events with
MadGraph3.2.0 as explained above. In fact, while $\zeta_{\text{EM}}$ would
simply drop to zero at $m_{\phi}\sim 2\,m_{W,Z}$ when relying solely on
Pythia8.3, a proper treatment instead leads to a sizeable production – albeit
slightly model dependent – even for values of $m_{\phi}$ several orders of
magnitude below the $W/Z$ thresholds. More precisely, we find that
$\zeta_{\text{EM}}$ does not simply drop to zero at $m_{\phi}\lesssim
2\,m_{W,Z}$, but instead falls off continuously with
$\zeta_{\text{EM}}\propto(m_{\phi}/m_{W})^{4}$ for $m_{\phi}<m_{W,Z}$. The
latter expression is indeed expected from the off-shell behavior of the gauge-
boson propagator in the transition amplitude.

Overall, the results presented in this work do not only correct the previous
literature regarding this matter for $m_{\phi}\sim 2\,m_{W,Z}$, but also
extend it to even smaller masses. We will see that, despite this Fermi
suppression, the resulting particle production can still have cosmological
effects that can be probed for masses as low as $\sim$ GeV. It is to be
anticipated that this could also have effects in other contexts, such as for
DM indirect detection, and modify at low energies various results accordingly.

##  III Cosmological probes

Now that the properties of the energy injection caused by the decay of the
massive relic have been outlined, we move on to the description of the
cosmological probes we will use to constrain such a process. In particular, we
will discuss CMB anisotropy and SD constraints in Secs. III.1 and III.2,
respectively, as well as BBN bounds in Sec. III.3. For each case we will
briefly describe the main effects that non-standard decays have on that given
probe, review existing constraints together with their limitations, and
discuss how to combine them with the results obtained in Sec. II for the
specifics of the decay process considered here.

###  III.1 CMB anisotropies

As it is very well known in the literature (see for instance [29, 30] for
detailed reviews), the main impact that an energy injection into the CMB
photon fluid has on the CMB anisotropy power spectra is given by the
inevitable modification it causes to the recombination and reionization
evolution. This happens on the one hand because of the increased photon (and
baryon) temperature, which can alter the thermal history of the Universe (and
hence the evolution of the free electron fraction), and on the other hand
because the injection of additional high-energy particles can directly re-
ionize the plasma if recombination has already happened.

In the context of decaying massive relics, the relative contribution of these
effects and their overall impact on the CMB power spectra is mainly dictated
by the lifetime $\tau_{\phi}$ of the decaying particle and by the total amount
of energy that is effectively injected in the form of EM radiation,777In full
generality, with EM radiation here we mean electrons, positrons, and photons.
However, since the energy-loss mechanism for positrons is the same as for
electrons, with the additional possibility to annihilate with the thermal
electrons to produce two photons, in this setup a positron is nothing else
than an electron and two photons. Furthermore, for initial injection energies
above $\mathcal{O}$(GeV), which are the ones relevant in this work (see Fig.
1), the injections of electrons and photons quickly translate into one another
because of the very efficient scattering and pair production processes (see
e.g. Sec. 2 of [31]). Therefore, for this probe the injection of either of
these three particles, electrons, positrons and photons, is almost perfectly
interchangeable (as it becomes clear e.g. from Fig. 9 of [31] for such high
energies) and we will thus only generically refer to EM radiation. i.e. by the
particle’s mass $m_{\phi}$, its fractional abundance
$f_{\rm\phi}=\Omega_{\phi}/\Omega_{\text{DM}}$, and the amount
$\zeta_{\text{EM}}$ of that fraction that gets deposited into the plasma in
the form of EM radiation (see e.g. [32, 8] for more details). Here,
$\Omega_{\phi}$ corresponds to the abundance that the $\phi$ particle would
have today if it was not decaying. Therefore, given the particle’s lifetime,
it is possible to calculate the maximum amount of energy that can be injected
into the system without spoiling the goodness of the fit to CMB anisotropy
data. Typically, this will then result in exclusion bounds in the
$f_{\rm\phi}\zeta_{\text{EM}}-\tau_{\phi}$ plane, with a mild dependence on
the particle’s mass displayed in form of a width on the final limits [11, 8,
31].

Here we adopt the recent bounds presented in [31], based on Planck 2015 data
[33].888To our knowledge, the only similar bounds derived using Planck 2018
data [34] are those reported in [35]. However, they are based on a mock
likelihood and assume a more simplified injection history with respect to
[31]. We decided therefore to employ the latter, and in doing so we are in any
case conservative. They coincide with the previous literature [11, 8] for
lifetimes larger than recombination, but are improved for smaller lifetimes.
These improvements result from more carefully following the delay between
injection and deposition time of the energy, thus extending the constraints
down to redshifts of the order of $z\simeq 10^{4}$. In particular, since the
mass range where the CMB anisotropy bounds are relevant in our case is always
well above 1$\,\mathrm{GeV}$ (even assuming $f_{\phi}$ as large as $10^{4}$,
see Fig. 2), here we will make use of the bounds derived for 1 TeV particles
in Fig. 9 of the reference. This approximation is possible and quite accurate
as this bound is almost identical to the one presented for 1$\,\mathrm{GeV}$
injections in the same figure.

###  III.2 CMB spectral distortions

Because of the aforementioned effects, CMB anisotropy constraints play a
dominant role when a significant fraction of the baryons in the Universe is
ionized. However, before recombination, when the free electron fraction is
equal to one, CMB anisotropies quickly become insensitive to eventual energy
releases and other cosmological probes need to be considered. The first
example we will discuss is given by CMB SDs (see e.g. [36, 37, 35, 38] for
recent reviews of the topic).

CMB SDs are, generally speaking, any type of distortion of the CMB photon
spectrum from a pure black body shape. They are caused by the gradual
inefficiency of number changing and scattering processes (such as
Bremsstrahlung and Compton scattering) and the consequent inability to restore
the black body spectrum after an injection of additional EM energy. These are
predicted to exist even within the $\Lambda$CDM paradigm [39, 40], and are of
course generated whenever extra energy is injected into the photon plasma via,
for instance, the decay of massive relics [41, 42, 43, 35, 44, 45].

For this reason, SDs are per se a very powerful constraining tool, which is
however limited by the currently outdated observational status. Indeed,
although several SD missions have been proposed in the meanwhile [46, 47, 48,
37], the most stringent bounds date back to the ’90 and the COBE/FIRAS mission
[49, 50]. Nevertheless, despite this limitation, since – in the context of
decaying massive relics – SDs are sensitive to lifetimes in the range between
$\tau_{\phi}\simeq 10^{6}\,\mathrm{s}$ and recombination999In principle, they
are still constraining even today but for lifetimes larger than
$\tau_{\phi}\simeq 10^{13}\,\mathrm{s}$ CMB anisotropy constraints inevitably
start to dominate. (see Fig. 2), they can still be very important to bridge
the gap between BBN and CMB anisotropy constraints. Furthermore, looking
towards the future, it is to be expected that SDs will be able to surpass BBN
bounds by several orders of magnitude and dominate the constraints in the
aforementioned time range, as forecasted in e.g. [35, 44, 38].

Concretely, to account for SDs we will consider a combination of the bounds
derived in [8] and [44] (see Fig. 5 and 19 therein, respectively). The latter
reference, which focused on accurately describing the thermalization process
in the case of high energy injections, provides the currently101010During the
final stages of this work, an updated analysis of these constraints has been
put forward [45]. We expect that the difference between the two limits does
not affect our results and conclusions significantly. most accurate SD bounds
on the decay of massive particles for small lifetimes
($\tau_{\phi}<10^{7}\,\mathrm{s}$). However, since in [44] the authors mainly
focused on early times (i.e. only $\mu$ distortions, as the contribution from
$y$ distortions is unaffected by their treatment with respect to the previous
literature), here we complement them with the $y$ distortions bounds from [8],
which cover instead larger $\tau_{\phi}$ values.

Note finally that, unlike in the case of the aforementioned CMB anisotropy
limits, the bounds obtained by [44] are expressed in terms of the yield so
that their mass dependence can be taken into account without the need for any
approximation. On the other hand, those by [8] are expressed in the same plane
as the CMB anisotropy constraints, which means that one cannot extract exactly
the mass dependence of the $y$ bounds in this case. However, as can be seen
from Fig. 5 of [43], for high particle masses (above 10$\,\mathrm{GeV}$) this
dependence significantly reduces, so that the constraints become almost
universal and we can adopt them as such.

###  III.3 BBN

In order to push our bounds to even smaller lifetimes, we also account for
constraints from BBN (see e.g. [51] for a review of the topic). In a nutshell,
BBN describes the process of light-element formation in the early Universe,
which took place at temperatures of the order of $\mathrm{keV}-\mathrm{MeV}$
(or equivalently redshifts around $z\simeq 10^{6}-10^{9}$). Theoretically, BBN
predictions follow from a system of Boltzmann equations that describe the
various abundances of the involved elements, which can then be probed against
observations of such abundances in the first galaxies. This comparison shows a
remarkable agreement between the values predicted in the SM and the ones
inferred from observations [52].

Figure 2: Left panel: CMB and BBN constraints (see Sec. III) on the mass
$m_{\phi}$ and lifetime $\tau_{\phi}$ of a massive particle decaying into
electron neutrinos, $\phi\to\nu_{e}\bar{\nu}_{e}$ (see Sec. II) with a
fractional abundance $f_{\phi}=\Omega_{\phi}/\Omega_{\rm DM}=1$. Right panel:
Same as left panel but keeping only the best bound for $f_{\phi}=1$, showing
in addition the corresponding best bounds for $f_{\phi}=10^{4,2,-2,-4}$. In
both panels the time of recombination and the age of the Universe are shown
for reference as black dashed and dashed-dotted horizontal lines,
respectively.

Consequently, any BSM scenario must not distort the SM abundances by too much.
This can in fact be challenging, since any BSM particle can change BBN in many
different ways [53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63], namely by (i)
increasing the Hubble rate and thus the expansion rate of the Universe (which
is similar to a bound on $\Delta N_{\text{eff}}$), (ii) changing the time-
temperature relation via the injection of EM energy, and (iii) inducing late-
time photodisintegration reactions that can destroy the light nuclei
previously created during BBN. In this work, we focus on rather small
abundances of the decaying relics (for larger abundances see [63]), in which
case the first two effects are negligible. However, photodisintegration is
known to constrain even very small abundances, ten orders of magnitude below
the photon abundance [56, 57, 58, 59, 60, 61]. Therefore, this effect needs to
be properly taken into account for our analysis.

To this end, we deploy the public code ACROPOLIS [64], which allows to
calculate the evolution of the light-element abundances due to
photodisintegration reactions for any BSM model. In our scenario, these
disintegration reactions are specifically induced by the EM particles
originating from the electroweak shower, and its associated EM cascade
creating a set of non-thermal photons.111111For the energy considered in this
work, the non-thermal photon spectrum always takes the universal form [24],
with a normalisation proportional to the faction of EM energy that is injected
by the shower. These photons afterwards destroy part of the previously created
nuclei, thus potentially leading to conflicts with observations. We then
calculate the corresponding constraints by comparing the light-element
abundances after disintegration with the most recent measurements for
deuterium ($\text{D}/{}^{1}\text{H}=(2.547\pm 0.035)\times 10^{-5}$ [52]),
helium-4 (${}^{4}\text{He}/{}^{1}\text{H}=(2.45\pm 0.03)\times 10^{-1}$ [52]),
and helium-3 (${}^{3}\text{He}/\text{D}=(8.3\pm 1.5)\times 10^{-1}$ [65]),
which we only use to set an upper bound (c.f. e.g. [26, 65, 66]).

Finally, let us note that the hadronic particles from the electroweak shower
can also induce hadrodisintegration reactions, which can potentially lead to
even stronger constraints (see e.g. [26, 27, 67]). However, we do not consider
this mechanism here, meaning that our bounds are ultimately conservative, and
could potentially be improved in future work.

##  IV Results and discussion

Now that both the decay process and the cosmological probes used to constrain
it have been outlined, we move on to the discussion of the resulting bounds.
These are displayed in the $\tau_{\phi}-m_{\phi}$ plane in Fig. 2.

In the left panel of the figure, we show the combination of the constraints
resulting from CMB and BBN observations. These constraints have been obtained
from the general bounds in the $f_{\rm\phi}\zeta_{\rm EM}-\tau_{\phi}$ plane
discussed in Sec. III, by using the relation presented in Fig. 1 and assuming
the arbitrary value of $f_{\phi}=1$ (i.e. for a $\phi$ abundance, which – if
it was not decaying – would be equal today to the DM abundance). In this plot,
the relatively small hatched area indicates the region where muon pairs can
efficiently be produced and then partly decay back into neutrinos. As
explained above, we neglect this effect and consequently the results are only
approximate in this region, with the overall bound changing by at most 20%.

Overall, we find that CMB anisotropy constraints (green) from Planck 2015 can
probe mediator masses down to approximately 10$\,\mathrm{GeV}$ for lifetimes
around the epoch of recombination (which we show as a dashed black line for
reference). Note that these bounds are expected to improve [35] with a Planck
2018 analysis and in the next decade with data from CMB-S4 [68, 69], possibly
reaching masses of the order of a fraction of a GeV. For lifetimes larger than
roughly $10^{14}\,\mathrm{s}$, CMB anisotropy constraints on $m_{\phi}$ reduce
linearly with $\tau_{\phi}$, and as soon as we consider lifetimes larger than
the age of the Universe (indicated by the dashed-dotted black line) the
cosmic-ray and high-energy neutrino experiments come into play. These latter
bounds are summarised in e.g. Fig. 2 of [9].

For lifetimes smaller than $10^{11}\,\mathrm{s}$ (or correspondingly redshifts
of the order of $10^{4}$) these CMB limits drop very sharply and BBN
constraints (purple) begin to dominate. In this case, masses above $\sim
100\,\mathrm{GeV}$ are excluded for lifetime larger than $\sim
10^{5}\,\mathrm{s}$. However, BBN photodisintegration does not constraint
smaller lifetimes. Finally, although less constraining than BBN bounds, for
the sake of generality we also include bounds from CMB SDs as (not) observed
by FIRAS (blue and red). As already mentioned above, and as argued in a number
of references (see e.g. [35, 44, 38]), these limits could be largely improved
in the future. Already with PIXIE-like sensitivities [46] future measurements
of CMB SDs would improve on BBN bounds by orders of magnitude, potentially
probing the $10-100\,\mathrm{GeV}$ range also for lifetimes smaller than
recombination.

The right panel of Fig. 2 shows how the bounds change when one varies the
initial abundance of the decaying particle. For a value of $f_{\phi}$ equal to
$10^{4}$, the bounds on $m_{\phi}$ improve by approximately one order of
magnitude with respect to the $f_{\phi}=1$ case of the left panel. This stems
from the fact that, as explained in Sec. III, the bounds are on the product
$f_{\phi}\zeta_{\rm EM}$. Thus, if one increases $f_{\phi}$, the amount of EM
material produced, $\zeta_{\rm EM}$, has to decrease accordingly, and
$\zeta_{\rm EM}$ scales as $m_{\phi}^{-4}$ for the masses of $\phi$ that apply
here (that is to say below the $W/Z$ thresholds, see Fig. 1). Instead, as Fig.
2 shows, for values of $f_{\phi}$ below unity the scaling is less trivial, as
a result of the fact that in this case larger values of $\zeta_{\rm EM}$ are
allowed, so that in Fig. 1 masses of $\phi$ above the $W/Z$ threshold apply
for the bounds (where the scaling of $\zeta_{\rm EM}$ in $m_{\phi}$ is
different than below these thresholds). Note that as a result of the fact that
above the thresholds the dependence of $\zeta_{\rm EM}$ on $m_{\phi}$ becomes
gradually very weak, for a value of $f_{\phi}$ as small as $10^{-4}$, all
masses below approximately 200$\,\mathrm{GeV}$ are allowed.

Note that, for $f_{\phi}\gtrsim 10^{5}$, other bounds that we do not consider
here will apply, especially for lifetimes shorter than recombination time, in
particular from the number of relativistic degrees of freedom that are allowed
by BBN (already mentioned above) and CMB, see e.g. [63]. Note also that the
larger $f_{\phi}$ the more this particle, which becomes matter once non-
relativistic, will dominate the energy density of the Universe until it
decays. This can modify in particular the radiation/matter equality time if
the lifetime is larger than this time. Thus we limit ourselves to values of
$f_{\phi}$ up to $10^{4}$, so that we can safely neglect these additional
constraints.

As a final remark, the values of $f_{\phi}$ we consider are smaller than the
ones one would obtain if, for instance, the decaying particle would thermalize
with the SM thermal bath and decouple from it relativistically. For this
extremely large abundance case we obtain limits on $m_{\phi}$ which are one
order of magnitude more stringent than for the $f_{\phi}=10^{4}$ line in Fig.
2, but this possibility is anyway excluded by the $\Delta N_{\rm eff}$
constraints, unless the lifetime is quite short [56, 54, 6]. Values of
$f_{\phi}$ such as those considered in Fig. 2 can be obtained e.g. for a non-
relativistic decoupling or from a relativistic decoupling in a hidden sector
thermal bath which would be colder than the SM thermal bath [70, 71, 72]. The
latter situation can apply to various self-interacting DM models with light
mediators, see e.g. [73], but for values of the light mediator mass below
$\sim 100\,\mathrm{MeV}$ that are slightly smaller than the values of the
masses probed in Fig. 2.

##  V Summary and conclusions

Many BSM setups involve new particles that could play a role in the early
Universe. This includes DM scenarios involving dark sectors, where e.g. the DM
has its own new interactions mediated by new particles. The decay of such
massive relics can significantly alter the thermal history of the Universe,
especially if the lifetime of these particles is larger than the age of the
Universe at the time of BBN, i.e. $t\sim 1$ s. In particular, if a sizeable
amount of the decay products ultimately end up in the form of EM radiation,
these decays can have an effect on cosmological probes, such as on CMB
anisotropies and SDs and on the BBN process.

Therefore, in order to avoid such stringent constraints, these unstable
particles are often assumed to decay into neutrinos, as they do not interact
as strongly with the thermal bath. However, even when this decay channel is
assumed, i.e. $\phi\rightarrow\nu_{e}\bar{\nu}_{e}$, this process can have an
effect on the aforementioned cosmological probes because they inevitably lead
to the emission of on-shell or off-shell $W/Z$ gauge bosons, producing showers
of electroweak and hadronic components, ultimately leading to an EM energy
injection.

In this work, we determined the CMB and BBN constraints that this implies in
terms of mass and lifetime of the decaying particle. We did that for lifetimes
shorter than the age of the Universe and find new, stringent constraints that
are presented in Fig. 2. They exclude any potential scenario where a decaying
particle would have a lifetime larger than $\sim 10^{4}-10^{7}$ s and a mass
above a few GeV to a few hundred GeV (depending on the lifetime and abundance
of the decaying particle ).

The fact that these constraints can go down to a few GeV masses stems from a
proper calculation of the amount of the EM material that is produced just
above the $W/Z$ threshold and below (from the emission of an off-shell gauge
boson). We anticipate that these results for low masses (where Pythia gives
vanishing results) could be relevant in other contexts too.

## Acknowledgements

We thank Marco Cirelli, Paul Frederik Depta, Nadege Iovine, and Filippo Sala
for helpful discussions. MH and ML are supported by an F.R.S.-FNRS fellowship.
This work is further supported by the “Probing dark matter with neutrinos”
ULB-ARC convention, by the IISN convention 4.4503.15, and by the Excellence of
Science (EoS) project No. 30820817 - be.h “The H boson gateway to physics
beyond the Standard Model”. TH further acknowledges the support of the
Institut Pascal at Université Paris-Saclay during the Paris-Saclay
Astroparticle Symposium 2021, where part of this work has been done, and the
related IN2P3 master project UCMN.

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# The Separate Universe approach for multifield inflation models

Hugo Holland  Institut d’Astrophysique Spatiale, rue Jean Teillac
91405 ORSAY CEDEX, France

###### Abstract

Primordial black holes could constitute part or all of dark matter but they
require large inhomogeneities to form in the early universe. These
inhomogeneities can strongly backreact on the large scale dynamics of the
universe. Stochastic inflation provides a way of studying this backreaction
and getting an estimation of the abundance of primordial black holes. Because
stochastic inflation focuses on large scale dynamics, it rests on the separate
universe approach. However, the validity of this approach has only been
checked in single field models, but not in multifield models in which we
expect strong boosts in the power spectrum, leading to the formation of
primordial black holes. We will check the validity of a separate universe
approach in multifield models by matching it with a complete cosmological
perturbation theory approach at large scales. In particular, we wish to
compare these two paradigms and their differences in the adiabatic and
entropic directions of the phase space. This will give us a range of validity
and conditions one needs to verify in order to apply the separate universe
approach and stochastic inflation in multifield models.

##  1 Stochastic inflation & Separate Universe

The search for dark matter candidates has given rise to many suitable
candidates. One of these are primordial black holes (PBH). These black holes
are different than astrophysical black holes as they did not form after the
collapse of an object such as a star, but formed from the collapse of overly
dense regions in the primordial universe. We now have many constraints on the
proportion of dark matter that PBH could account for depending on their masses
[1]. However some mass regions are still completely open and there is still
hope to detect a gravitational wave background coming from PBH with LISA [2].
There are several ways of ensuring the creation of PBH in inflationary models,
but all of them boil down to the same issue : creating a massive boost in the
power spectrum. These boost can appear in both single field models of
inflation and multifield models [3] [4]. In any model, these boosts will
produce very large fluctuations and will impact the dynamics of large scale
fluctuations, which in turn will impact the abundance of collapsed structures
such as PBHs or dark matter haloes [5]. The way to study these dynamics is the
Stochastic Inflation formalism [6].

###  1.1 Stochastic inflation

We know that during inflation the Hubble radius $(aH)^{-1}$ decreases. This
means that for a given field $\phi$, each mode $\phi_{k}$ is either sub
horizon or super horizon. During inflation there will be a continuous flow of
sub horizon modes crossing into the super horizon regime as seen in 1(left).

Figure 1: Solution to the horizon problem via inflation. Bauman TASI
lectures(left) And Quantum to classical transition of fluctuations Artigas
Grav&Cosmo 2024 (right)

This continuous flow of subhorizon modes crossing the horizon, is treated as a
white noise in the dynamics of the super horizon modes, giving a Langevin
equation. We will denote the super horizon modes as $\phi_{IR}$.

$\dot{\phi}_{IR}=-\frac{\partial V}{\partial\phi}+\xi_{\phi}$ (1)

Where $\xi_{\phi}$ is the noise that depends on the sub horizon scales. In
this equation we can immediately notice that there are no gradients, so no
space dependence. That is because we have assumed that each Hubble patch is
homogeneous and isotropic and that they all evolve independently from one
another. This is the separate universe approach [7] [8].

###  1.2 The separate universe approach

We will now have two different approaches in order to treat the different
regimes. Having lost the inhomogeneity for the super horizon modes, we can
study them at a non linear level. On the contrary the sub horizon modes evolve
in a inhomogeneous background so we will restrict ourselves to a linear
evolution. The evolution of a given mode compared to the background is given
in 1 (right). In stochastic inflation we allow a abrupt transition from one
regime to the other as the mode crosses the horizon. This abrupt transition is
exctly what the separate universe approach aims to justify.

##  2 Separate universe & multifield inflation

In this section we will present a very generic model of multifield inflation,
often referred to as ”non linear sigma models” and how the separate universe
approach affects such models.

###  2.1 Multifield inflation

We start from a very generic Lagrangian for multifield models of inflation
with $I{,}J\in\llbracket 1,n\rrbracket$ is

$S=\int d^{4}x\sqrt{-g}\big{[}\frac{1}{2}M_{pl}^{2}{\cal
R}-\frac{1}{2}g^{\mu\nu}G_{IJ}\partial_{\mu}\phi^{I}\partial_{\nu}\phi^{J}-V(\phi^{I})\big{]}$
(2)

Where $\phi^{I}$ are our fields and $G^{IJ}$ is a coupling metric that depends
on all of the fields. It is interesting to note that at the classical level
these models are equivalent to non minimally coupled models in which
$G^{IJ}=\delta^{IJ}$ and ${\cal R}$ is replaced by an $f({\cal R})$ function.
This model doesn’t only encompass non minimal couplings to gravity, it can
also represent non linear couplings between the fields or derivative couplings
like those that appear in EFT of inflation. Many models of multifield high
energy physics can be generalised with this metric coupling. We then make use
of the ADM formalism and perturb all of our objects in two ways. Firstly we do
a complete cosmological perturbation theory development, then we do the same
thing but with the separate universe assumption.

###  2.2 Validity conditions of the Separate Universe Approach

Once everything has been perturbed at the second order we can compare the
equations of motions found in the full CPT approach at large scales to those
found in the SU approach and check when they are consistent. This is
equivalent to comparing the two Hamiltonians that we get :

$\displaystyle H_{CPT}$ $\displaystyle=\int d^{3}\vec{x}\big{[}\delta
N\mathcal{S}^{(1)}+\delta N^{i}\mathcal{D}_{i}+N\mathcal{H}^{(2)}\ \big{]}$
(3) $\displaystyle H_{SU}$ $\displaystyle=\int d^{3}\vec{x}\big{[}\bar{\delta
N}\bar{\mathcal{S}}^{(1)}+\bar{N}\bar{\mathcal{H}}^{(2)}\big{]}$ (4)

Where a bar means that we are working in the separate universe formalism, $N$
and $N_{i}$ are the lagrange multipliers that come from the ADM formalism,
$S^{(1)}$ and $H^{(2)}$ are the scalar constraint at first and at second order
in perturbation, and $D_{i}$ is the diffemorphism contraint expressed at first
order in perturbation. We finally compare $S^{(1)}$ with $\bar{S}^{(1)}$ and
$H^{(2)}$ with $\bar{H}^{(2)}$ and we get three inequalities that need to be
verified in order to justify the separate universe approach. The results we
present here are twofold. On the left column we give the inequalities as we
found them in a fully covariant expression of the hamiltonians expressed in
the original field space. In the column on the right we give the same
inequalities but expressed in the adiabatic and entropic directions of our
field space [9].

$\displaystyle M_{pl}^{2}\big{(}\frac{k}{aH}\big{)}^{2}$
$\displaystyle\ll\frac{1}{H^{2}}\big{|}\frac{\pi_{I}\pi^{I}}{a^{6}}-V(\phi^{I})\big{|}\quad=\quad\frac{1}{H^{2}}\big{|}N^{2}\dot{\sigma}^{2}-V(\phi^{I})\big{|}$
(5) $\displaystyle M_{pl}^{2}\big{(}\frac{k}{aH}\big{)}^{2}$
$\displaystyle\ll\frac{1}{H^{2}}\big{|}\frac{\pi_{I}\pi^{I}}{a^{6}}+\frac{V(\phi^{I})}{2}\big{|}\quad=\quad\frac{1}{H^{2}}\big{|}N^{2}\dot{\sigma}^{2}+\frac{V(\phi^{I})}{2}\big{|}$
(6) $\displaystyle\big{|}\big{(}\frac{k}{aH}\big{)}^{2}G_{IJ}\big{|}$
$\displaystyle\ll\frac{1}{H^{2}}\big{|}V;_{IJ}\big{|}\quad\quad\quad\quad\Leftrightarrow\quad\quad\big{(}\frac{k}{aH}\big{)}^{2}\ll\frac{1}{H^{2}}\big{|}V;_{cc}\big{|}$
(7)

The first two inequalities tell us that we have to be at big enough scales for
the separate universe approach to work, as expected because we were aiming for
scales at which the hubble patches can be seen as homogeneous and isotropic.
This isn’t quite obvious to see but the right hand side is essentially
Friedmann’s equations which once divided by $M_{pl}^{2}$ gives one. The third
inequality is slightly more subtle as it tell us that the effective mass of
our fields have to be large enough. The adiabatic and entropic versions of the
third inequality show that the condition on the mass has to be valid for all
the directions. However, we expect the adiabatic direction to be the hardest
to constraint as it is the flatest and lightest directions. We are still
studying these directions in this approach as for more than three fields it is
very easy to constrain the first derivatives of the potential with respect to
the entropic ones [10].

##  3 Gauge fixing and Mukhanov Sasaki variables

One point that we have not yet discussed here is the issue of gauge fixing. It
has been shown that there are issues in some gauges when going from the
Separate universe framework to the full CPT framework as some gauge degrees of
freedom don’t exist in the former so cannot be easily fixed. This is fairly
simple to see when computing the equations of motions for the Mukhanov-Sasaki
gauge invariant variables in the separate universe approach, in particular for
the adiabatic direction :

$\displaystyle\ddot{\bar{Q}}_{\sigma}+3H\dot{\bar{Q}}_{\sigma}+\big{[}\mathcal{M}_{\sigma\sigma}-\omega_{1}^{2}-\frac{1}{M_{pl}^{2}a^{3}}$
$\displaystyle\partial_{t}(\frac{a^{3}\dot{\sigma}^{2}}{H})\big{]}\bar{Q}_{\sigma}=\mathcal{F}(\bar{Q}_{1},\dot{\bar{Q}}_{1})+\frac{3}{2}{M_{pl}^{2}a^{2}}\dot{\sigma}\bar{\mathcal{D}_{i}}.$
(8)

Where $\omega_{1}$ is the covariant rate turn, $\mathcal{M}_{\sigma\sigma}$ is
the effective mass of the adiabatic Mukhanov-Sasaki variable and
$\mathcal{F}(Q_{1},\dot{Q}_{1})$ gives the coupling between the adiabatic
direction and the first entropic direction. In the full CPT computation we get
an extra $k^{2}\bar{Q}_{\sigma}$ term in the left hand side and no
$\bar{D}_{i}$ term in the right hand side, which corresponds to an additional
source term coming from a diffeomorphism constraint even though such a
constraint doesn’t naturally appear in the hamiltonian formalism in the SU
approach [7].

##  4 Conclusion

The separate universe approach works as long as we are being careful with it.
We need to work at large enough scales or guaranty that any potential $k^{2}$
terms are suppressed in front of the effective mass of our fields. The gauge
fixing issues that appear here are the same than in single field models, as
long as we start from the CPT framework we won’t have any problems, but one
needs to be careful when fixing a gauge in the SU framework and trying to get
back to a more general framework as new degrees of freedom might still be
unfixed.

## Acknowledgments

I would like to thank Julien Grain and Lucas Pinol for the many enlightening
conversations I regularly have with them.

## References

## References

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