aainstitutetext: Institute of High Energy Physics, Austrian Academy of
Sciences, Nikolsdorfergasse 18, 1050 Vienna, Austriabbinstitutetext:
Department of Physics and Astronomy, University of California, Riverside, CA
92521, USAccinstitutetext: TRIUMF, 4004 Wesbrook Mall, Vancouver, BC V6T 2A3,
Canada

# Dark Freeze-out Cogenesis

Xiaoyong Chu<EMAIL_ADDRESS>b Yanou Cui<EMAIL_ADDRESS>a Josef
Pradler<EMAIL_ADDRESS>c Michael Shamma<EMAIL_ADDRESS>
###### Abstract

We propose a new mechanism where a multi-component dark sector generates the
observed dark matter abundance and baryon asymmetry and thus addresses the
coincidence between the two. The thermal freeze-out of dark matter
annihilating into meta-stable dark partners sets the dark matter relic
abundance while providing the out-of-equilibrium condition for baryogenesis.
The meta-stable state triggers baryon asymmetry production by its decay well
after the freeze-out and potentially induces a period of early matter
domination before its decay. The dark matter and baryon abundances are related
through number conservation within the dark sector (cogenesis). The
“coincidence” is a natural outcome with GeV- to TeV-scale symmetric dark
matter and the dark sector’s interactions with the Standard Model quarks. We
present a UV-complete model and explore its phenomenological predictions,
including dark matter direct detection signals, LHC signatures of new massive
particles with color charges and long-lived particles with displaced vertices,
dark matter-induced nucleon conversions, (exotic) dark matter indirect
detection signals, and effects on the cosmological matter power spectrum. As a
side result, we provide a novel analytical treatment for dark sector freeze-
out, which may prove useful in the study of related scenarios.

##  1 Introduction

The nature and dynamical origins of the baryon asymmetry and the observed dark
matter (DM) are two long-standing puzzles in particle cosmology. Meanwhile,
the observation that their abundances are strikingly similar,
$\Omega_{DM}/\Omega_{B}\approx 5$ planck , presents a coincidence problem
which is suggestive of a potential connection between the origins of DM and
baryons in the early Universe. Although there exists a cornucopia of
theoretical explanations which address each of these pieces separately, this
cosmic coincidence has inspired new directions in DM model-building (see e.g.,
Boucenna:2013wba ; Cui:2015eba  for reviews).

Weakly interacting massive particles (WIMPs) as DM candidates have been
leading the DM model-building efforts since the 1990s Primack:1988zm ;
Jungman:1995df . This paradigm is motivated largely by the observation that DM
with weak-scale interactions and masses can produce the correct DM abundance
through thermal freeze-out (WIMP “miracle”). However, conventional WIMPs have
become increasingly constrained by indirect/direct detection, and collider
experiments, see, e.g., Arcadi:2017kky ; Bertone:2018krk  and references
therein. This has led to the exploration of alternative DM candidates beyond
the WIMP paradigm. For example, asymmetric dark matter (ADM) Nussinov:1985xr ;
Barr:1990ca ; Kaplan:1991ah ; Kaplan:2009ag ; Davoudiasl:2010am ;
Zurek:2013wia ; adm  is an alternative inspired by the DM-baryon coincidence.
In this framework, the DM particle is distinct from its antiparticle, and an
asymmetry in their respective population densities is generated in the early
universe. The core idea of ADM is based on relating and generating DM and
baryon/lepton asymmetries through shared interactions. With the exception of a
few mechanisms (e.g. admfromlepto ; Buckley:2010ui ; Cui:2011qe ; Hall:2021zsk
) the observed coincidence is achieved with ADM masses of
$\mathcal{O}(\text{GeV})$.

Recently, there have been attempts at unifying WIMP DM and ADM mechanisms
wimpyBG ; McDonald:2011zza ; Davidson:2012fn ; wimpyBG2 ; Cui:2015eba ;
Farina:2016ndq ; Racker:2014uga ; Cui:2013bta ; Goudelis:2021lra ; Cui:2020dly
. Among the existing WIMP-related proposals, McDonald:2011zza  is highly
sensitive to various initial conditions, while both Davidson:2012fn  and WIMP
DM annihilation triggered “WIMPy baryogensis” wimpyBG  have sensitivity to
“washout”, i.e., a reduction in the baryon asymmetry due to inverse decays or
L-violating scatterings. The mechanism of “Baryogenesis from meta-stable
WIMPs” wimpyBG2  was then proposed as an alternative where the prediction is
more robust against model details: the baryon asymmetry is generated by a
long-lived WIMP that undergoes CP- and B-violating decays after its thermal
freeze-out. Such models also provide a strong cosmological motivation for
long-lived particle searches at collider experiments and have become a
benchmark for related studies Cui:2014twa ; Cui:2016rqt ; ATLAS:2019ems .
However, in its original form, it does not involve the specifics of DM, except
for assuming it is WIMP-like. The more recently proposed “WIMP
Cogenesis”Cui:2020dly  is a new realization of ADM with DM specifics
explicitly incorporated, and fully inherits the desirable qualities of both
WIMP and ADM production mechanisms. However, the permitted DM mass which
produces the observed DM-baryon coincidence is limited to a few GeV.

From the model building perspective it is desirable to further develop a
framework which incorporates the merits of WIMP-like and ADM-like mechanisms.
Specifically, a mechanism which maintains a symmetric DM candidate permitting
a wider range of DM masses, $\mathcal{O}({\rm GeV-TeV})$, while supplying a
predictive connection between $\Omega_{\rm DM}$ and $\Omega_{B}$ presents a
worthwhile theory target. In this work, we consider a realization of this goal
in the framework of an isolated dark sector which may have a thermal
temperature different from the Standard Model (SM) photon bath (see e.g.,
Ackerman:mha ; Feng:2008mu ; Chu:2011be ; Chacko:2015noa ; Dror:2016rxc ;
Adshead:2016xxj ; Asadi:2021bxp ). In particular, the dark sector is composed
of a stable state as DM and a meta-stable dark partner which is the “parent”
of the baryon asymmetry. Both states are initially in equilibrium with each
other through the efficient annihilation of DM into dark partners, before
freezing out. The dark partners are long-lived, and their decays into SM
particles trigger the generation of baryon asymmetry. The stable and meta-
stable states, and consequently, $\Omega_{\rm DM}$ and $\Omega_{B}$, are
connected through the overall number-conservation in the dark sector as
manifest in the annihilation process.

Independent of the baryogenesis motivation, we note that the particular dark
freeze-out scenario we consider in this work is new to the literature. While
there are existing studies on freeze-out in a decoupled dark sector
Carlson:1992fn ; Hochberg:2014dra ; Pappadopulo:2016pkp ; Dror:2016rxc ;
Asadi:2021bxp , some also involving meta-stable annihilation final states, the
present scenario distinguishes itself from known possibilities. For instance,
in the Co-Decaying dark matter case Dror:2016rxc , the meta-stable states
decay during the freeze-out process which exponentially depletes the DM
abundance, unlike in our case. We provide both numerical and analytical
approaches for tracking the dynamical evolution during the freeze-out going
beyond the instantaneous freeze-out approximation.

Figure 1: Schematic diagram outlining the key stages in the dark freeze-out
cogenesis mechanism.

The basic idea of Dark freeze-out Cogenesis is illustrated in the schematic of
Fig. 1. As we will elaborate later in the paper, a simple realization consists
of three dark states $\chi_{1,2,3}$, where $\chi_{1}$ is taken to be the
stable DM candidate and $\chi_{2}$ is the meta-stable component whose baryon
number- and CP-violating decays trigger baryogenesis. Departure from
equilibrium is achieved when $\chi_{1}$ and $\chi_{2}$ freeze-out in the
isolated sector. Finally, $\chi_{3}$ is required to induce CP-violation. When
the meta-stable $\chi_{2}$ component undergoes CP- and baryon number-violating
decays, its abundance is converted into the baryon asymmetry. The latter is
directly tied to the stable DM component’s abundance through number
conservation in the dark sector. We demonstrate our general ideas by
introducing a UV complete model, i.e., a particle content and their
interactions solely based on renormalizable interactions.

The remainder of the paper is organized as follows: in Sec. 2 we outline two
scenarios of dark sector freeze-out and describe the dynamics of the
cosmological evolution of matter abundances around the dark freeze-out. In
Sec. 3, we discuss the dynamics related to the subsequent $\chi_{2}$ decays
and the ensuing production of the baryon asymmetry. A UV-complete particle
physics model is introduced in Sec. 4. Various phenomenological implications
and observational signatures predicted by the model are detailed in Sec. 5
before concluding in Sec. 6. Further details on the freeze-out calculation are
relegated to App. A.

##  2 Dark Sector Freeze-out

In this section, we consider a dark sector with states $\chi_{1}$ and
$\chi_{2}$ that evolve independently from the thermal SM bath. This is
commensurate with initial conditions where $\chi_{1,2}$ decouple from SM while
still being relativistic. The state $\chi_{1}$ is stabilized by a
${\mathcal{Z}}_{2}$ symmetry, so that it plays the role of the DM candidate.
In contrast, a lighter state $\chi_{2}$ is meta-stable, and decays out-of-
equilibrium after $\chi_{1}$ freezes out, leading to the observed baryon
asymmetry (once interference effects with a state $\chi_{3}$, to be introduced
later, are taken into account). The dark sector is assumed to be self-
thermalized before freeze-out, characterised by a common dark temperature
$T^{\prime}$, which is set to be comparable to or lower than the photon
temperature $T$, $T^{\prime}\leq T$, so that the SM sector always dominates
the radiation energy of the Universe.

Depending on the mass hierarchy between $\chi_{1}$ and $\chi_{2}$, there are
various options for the details of DM freeze-out. In the following, we focus
on two principal possibilities, for each of which we present analytical
solutions: $m_{\chi_{1}}\gg m_{\chi_{2}}$ (“hierarchical scenario”) and
$m_{\chi_{1}}=m_{\chi_{2}}/(1-\delta)$ with $\delta\ll 1$ (“nearly degenerate
scenario”).111“Forbidden freeze-out” DAgnolo:2015ujb  with $\delta<0$ is
possible if $\chi_{2}$ is sufficiently short-lived; we will not discuss this
possibility here.  Throughout the paper, we use the subscripts, “$i$”, “f.o.”,
and “$f$” to represent the initial value, the value at $\chi_{1}$ freeze-out,
and the value well after $\chi_{1}$ freeze-out, respectively. The subscript
“$0$” is used for present values (except for $\sigma_{0}$ below), while the
superscript ′ denotes dark sector quantities that are measured in units of
$T^{\prime}$.

###  2.1 Hierarchical scenario: $\chi_{1}$ DM with $m_{\chi_{1}}\gg
m_{\chi_{2}}$

In the hierarchical scenario, we assume that DM $\chi_{1}$ is significantly
heavier than $\chi_{2}$, and reaches its final abundance via non-relativistic
freeze-out through $\chi_{1}\chi_{1}\to\chi_{2}\chi_{2}$ annihilation while
$\chi_{2}$ remains relativistic. For the parameters of our interest, this
generally requires $m_{\chi_{2}}\lesssim m_{\chi_{1}}/20$. We introduce a
time-dependent function, $\xi$, to characterize the temperature ratio between
the dark sector ($T^{\prime}$) and the SM ($T$), with its initial value
smaller than unity, $\xi_{i}\lesssim 1$. Given that the number of particle
species in the SM is much more than in the dark sector we consider, this
ensures that the Universe’s expansion is solely driven by the visible sector
particle content during the radiation-dominated epoch.

While the specific dark freeze-out scenario we consider was not studied in the
literature, the analysis of the Boltzmann equation evolution is analogous to
some of the existing work on freeze-out in a decoupled dark sector which are
useful references for our case, e.g., Ackerman:mha ; Feng:2008mu ; Chu:2011be
; Chacko:2015noa . In particular, here we use the approach in Chu:2011be  to
calculate the final abundance in terms of the yield variable, $Y_{j}\equiv
n_{j}/s$, where $n_{j}$ is the number density of species $i$ and
$s=(2\pi^{2}/45)g_{*S}T^{3}$ is the total (dark and visible sector) entropy
density, which, by assumption, essentially coincides with the SM one; $g_{*S}$
($g_{*}$) are the effective degrees of freedom in entropy (energy) and we take
$g_{*}=g_{*S}$ before neutrino decoupling.

The thermally averaged annihilation cross section
$\langle\sigma_{\chi_{1}}v\rangle$ is conventionally parametrized as

$\displaystyle\langle\sigma_{\chi_{1}}v\rangle\equiv\sigma_{0}(x^{\prime})^{-n}=\sigma_{0}x^{-n}\xi^{n},$
(1)

where $x^{(^{\prime})}=m_{\chi_{1}}/T^{(^{\prime})}$ and $\sigma_{0}$ is a
reduced “$n$-wave” annihilation cross section of $\chi_{1}$ with
$n=0,1,2,\dots$ corresponding to $s,p,d,\dots$-wave annihilation with a
respective relative velocity scaling as $v^{2n}$ in $(\sigma_{\chi_{1}}v)$
before taking the non-relativistic thermal average. Concretely, we obtain

$Y_{\chi_{1},f}={n+1\over\lambda}x_{\text{f.o.}}^{n+1}\xi^{-n}\,,$ (2)

where
$\lambda=\left[s\langle\sigma_{\chi_{1}}v\rangle/H\right]|_{x=1}=0.264g_{*S}/g_{*}^{1/2}m_{\text{Pl}}m_{\chi_{1}}\sigma_{0}$
with $H|_{x=1}=H(m_{\chi_{1}})$ being the Hubble expansion rate evaluated at a
photon temperature $T=m_{\chi_{1}}$; $m_{\text{Pl}}=1.2\times
10^{19}~{}\text{GeV}$ is the Planck mass. The freeze-out point,
$x_{\text{f.o.}}$, measured in terms of photon temperature, is given by

$\displaystyle x_{\text{f.o.}}\equiv{m_{\chi_{1}}\over T_{\text{f.o.}}}$
$\displaystyle=$
$\displaystyle\xi\ln\left[0.038\,\xi^{5/2}\langle\sigma_{\chi_{1}}v\rangle
m_{\rm Pl}m_{\chi_{1}}\frac{g_{\chi}}{\sqrt{g_{*}}}c(c+2)\right]$ (3)
$\displaystyle-\frac{\xi}{2}\ln\left\\{\xi\ln\left[0.038\,\xi^{5/2}\langle\sigma_{\chi_{1}}v\rangle
m_{\rm Pl}m_{\chi_{1}}\frac{g_{\chi}}{\sqrt{g_{*}}}c(c+2)\right]\right\\}\,,$

Here, $c=0.3$ yields good agreement with the exact numerical solution for
$\xi\geq 0.01$. It is worth mentioning that although the temperature ratio has
been treated as a constant function above, one should in practice take its
value at freeze-out (referred to as $\xi_{\text{f.o.}}$), which is slightly
larger than an initially attained temperature ratio $\xi_{i}$: $\chi_{1}$
annihilation heats up the dark sector, and increases this ratio by a factor
$2^{1/3}$ as can be obtained from the conservation of entropy in the self-
thermalized dark sector. In practice we adopt a step function for the
temperature ratio, $\xi$, w.r.t. $x^{\prime}$ as:
$\xi(x^{\prime})=\xi_{i}\text{~{}for }x^{\prime}\leq 4$ and
$2^{1/3}\xi_{i}\text{~{}for }x^{\prime}\geq 4$. Non-relativistic freeze-out
mostly happens at $x^{\prime}\geq 4$, where most of the entropy density in
dark sector is stored in $\chi_{2}$ kolbturner .

Since
$x_{\text{f.o.}}={m_{\chi_{1}}/(T^{\prime}_{\text{f.o.}}/\xi_{\text{f.o.}})}=(2^{1/3}\xi_{i})x^{\prime}_{\text{f.o.}}$,
one may re-write the solution to the DM abundance as

$Y_{\chi_{1},f}=(2^{1/3}\xi_{i})\,\left({n+1\over\lambda}x_{\text{f.o.}}^{\prime
n+1}\right)\propto{2^{1/3}\xi_{i}\over\langle\sigma_{\chi_{1}}v\rangle}\,,$
(4)

in analogy to the standard thermal freeze-out solution:
$Y_{\chi_{1},f}={(n+1)}\,x_{\text{f.o.}}^{n+1}/\lambda$. In addition, since
the freeze-out point $x^{\prime}_{\text{f.o.}}$, measured in units of the dark
temperature $T^{\prime}$ and broadly in the range 3-30, is not sensitive to
mild changes of parameters, an $s$-wave cross section
$\langle\sigma_{\chi_{1}}v\rangle/(2^{1/3}\xi_{i})\sim{\mathcal{O}}(1)$ pb is
needed to achieve the observed DM abundance in an otherwise standard
cosmological thermal history. In case of later co-moving entropy increase by a
factor $\Delta$ (to be discussed later), the annihilation cross section should
be comparatively smaller, satisfying
$\langle\sigma_{\chi_{1}}v\rangle\,\Delta/(2^{1/3}\xi_{i})\sim{\mathcal{O}}(1)$
pb, in order to generate the observed DM relic density. Compared to the
standard case $T^{\prime}=T$, dark freeze-out with $(2^{1/3}\xi_{i})/\Delta<1$
needs to happen at a relatively higher SM temperature $T$ in order to realize
the correct relic abundance.

For example, considering the interaction term
$ig_{j}\bar{\chi}_{j}\gamma^{5}\chi_{j}A$ ($j=1,\,2$) induced by a heavy
pseudoscalar state $A$ (this is one of the interactions realized in our UV
complete model presented in Sec. 4), the annihilation cross section is
dominantly $s$-wave and given by

$\sigma_{\chi_{1}}v={g_{1}^{2}g_{2}^{2}\sqrt{s(s-4m_{\chi_{2}}^{2})}\over
16\pi(s-m_{A}^{2})^{2}}\to{g_{1}^{2}g_{2}^{2}m_{\chi_{1}}^{2}\over 4\pi
m_{A}^{4}}\,,$ (5)

where the limit relates to non-relativistic relative initial motion with
$m_{A}\gg 2m_{\chi_{1}}$. Therefore, in order to generate the observed DM
abundance, this points to

$m_{\chi_{1}}\sim 0.18\,\text{TeV}\,\left({0.1\over
g_{1}g_{2}}\right)\,\left({m_{A}\over{\rm
TeV}}\right)^{2}\,\left({\xi_{\text{f.o.}}/\Delta\over 0.01}\right)^{1/2}\,.$
(6)

Since $\chi_{2}$ particles remain relativistic during the freeze-out (in the
case of $m_{\chi_{2}}/m_{\chi_{1}}\lesssim x^{\prime}_{\text{f.o.}}$), their
abundance is fixed in terms of the temperature ratio as
$Y_{\chi_{2},f}=0.42g_{\chi}\xi_{i}^{3}/g_{*}-Y_{\chi_{1},f}$. This is because
in our setup, by assumption, all interactions in the dark sector preserve the
total number of $\chi_{1}$ and $\chi_{2}$, such that
$Y_{\chi_{1}}+Y_{\chi_{2}}=\text{const.}$, up to changes in co-moving entropy.

In the end, the observed baryon asymmetry $Y_{B}$ produced from $\chi_{2}$
decays is proportional to the “would-be” $\chi_{2}$ abundance $Y_{\chi_{2},f}$
if it had not decayed: $Y_{B}\propto\epsilon_{\rm CP}Y_{\chi_{2},f}$, where
$\epsilon_{\rm CP}$ is the CP asymmetry of the decay and $\tau$ is the
$\chi_{2}$-lifetime at rest. A later entropy increase, which may be induced by
the late decay of $\chi_{2}$, would dilute the $\chi_{2}$ abundance, requiring
a larger $\epsilon_{\rm CP}$ to yield the observed baryon asymmetry today.
More explanation about the final yield of matter abundances will be given in
Sec. 4.

###  2.2 Nearly degenerate scenario: $\chi_{1}$ DM with
$m_{\chi_{1}}=m_{\chi_{2}}+\delta m$

In the second, nearly degenerate scenario, we assume that $\chi_{1}$ is
marginally heavier than $\chi_{2}$ and $\chi_{1}$ is still stabilized by a
${\mathcal{Z}}_{2}$ symmetry. The mass degeneracy is characterised by a
dimensionless parameter,
$\delta\equiv(m_{\chi_{1}}-m_{\chi_{2}})/m_{\chi_{1}}\ll 1$. The final
abundances of DM candidate $\chi_{1}$ and of meta-stable $\chi_{2}$ are
dominantly set by non-relativistic freeze-out of
$\chi_{1}\chi_{1}\leftrightarrow\chi_{2}\chi_{2}$. The crucial difference from
the previous scenario is that $\chi_{2}$ is now non-relativistic during the
freeze-out of $\chi_{1}$. This, in turn, leads to a faster decrease of dark
temperature. While dark freeze-out scenarios involving two near-degenerate
states have been discussed in literature (see e.g., Dror:2016rxc ;
Okawa:2016wrr ; Kopp:2016yji ; Bernal:2017mqb ; DAgnolo:2018wcn ;
Maity:2019hre ), in our scenario the decay of the meta-stable state occurs
well after the freeze-out era, resulting in different dynamics. Here we
provide an analytical solution to this problem, which to our knowledge is
novel.

Due to the mass degeneracy, the difference from the earlier discussed
hierarchical scenario is that, both $\chi_{1}$ and $\chi_{2}$ are non-
relativistic at DM freeze-out. At the same time, the total co-moving number of
$\chi_{1}$ and $\chi_{2}$ particles is still conserved, and the ratio of
$n_{\chi_{1}}$ and $n_{\chi_{2}}$ in equilibrium depends on their mass
splitting and the dark temperature. In this scenario, the temperature ratio
$T^{\prime}/T$ can deviate significantly from its initial value during
Universe expansion, so we must follow both number and energy density in the
dark sector. They are described by the respective Boltzmann equations,

$\displaystyle\dot{n}_{\chi_{1}}+3Hn_{\chi_{1}}=-(\dot{n}_{\chi_{2}}+3Hn_{\chi_{2}})=-\langle\sigma_{\chi_{1}}v\rangle
n_{\chi_{1}}^{2}+\langle\sigma_{\chi_{2}}v\rangle n_{\chi_{2}}^{2}\,,$ (7)

as well as

$\dot{\rho^{\prime}}+3H({\rho^{\prime}}+p^{\prime})=0\,.$ (8)

where the dot signifies the derivative with respect to cosmic time and
$\rho^{\prime}$ ($p^{\prime}$) is the energy density (pressure) of the dark
sector. For a dark sector in kinetic equilibrium, as assumed here, we are to
solve the set of three Boltzmann equations to obtain the dark temperature,
$T^{\prime}$, and two number densities, $n_{\chi_{1,2}}$ (or equivalently, two
chemical potentials $\mu_{\chi_{1,2}}$). Note, however, that
$\langle\sigma_{\chi_{1}}v\rangle$ and $\langle\sigma_{\chi_{2}}v\rangle$ are
related through the principle of detailed balance; see Eq. (37) in the
appendix.

For illustration, we again take the pseudoscalar mediated model with unaltered
annihilation cross section given by Eq. (5), and solve the set of equations
above for several benchmark parameters. The results are given in Fig. 2. The
right panel shows that freeze-out is associated with sizable $x^{\prime}$
since the Boltzmann suppression of $Y_{\chi_{1}}$ with respect to
$Y_{\chi_{2}}$ is determined by the mass splitting (orange dot-dashed
lines),222The function $e^{-\delta x^{\prime}}Y_{\rm tot}$ is an approximation
to the quasi-static equilibrium solution $Y^{\rm QSE}_{\chi_{1}}$, defined by
Eq. (39) in the appendix, in the limit $x^{\prime}\gtrsim 1/\delta$; the
detailed derivation is provided there. instead of the DM mass (red dotted
lines), when both DM annihilation and its inverse process are in equilibrium.
This is dictated through overall dark number conservation, and by the fact
that the inverse process, $\chi_{2}\chi_{2}\to\chi_{1}\chi_{1}$, only becomes
suppressed once $T^{\prime}\ll\delta m$. Nevertheless, when the same solutions
are shown as a function of $x$ (left panel of Fig. 2) one observes that
freeze-out happens with $x_{\text{f.o.}}\sim 3$, smaller than the typical
value, $x_{\text{f.o.}}\sim 20$, in the standard case ($\xi\equiv 1$) of TeV
DM freeze-out, to get the observed DM abundance; the reason for it is
explained below Eq. (4).

Figure 2: The abundance evolution for the near-degenerate case with
$m_{\chi_{1}}=1$ TeV and $\delta=0.1$ as a function of $x$ (left panel) and
$x^{\prime}$ (right panel); we assume $\xi_{i}=0.1$, and
$g_{1}g_{2}/m_{A}^{2}=1/(4\,\text{TeV})^{2}$. The red (blue) solid lines show
the evolution of $Y_{\chi_{1}}$ ($Y_{\chi_{2}}$), and the dotted red lines
show the equilibrium value of $Y_{\chi_{1}}$ calculated from $T^{\prime}$ with
zero chemical potential. The orange dot-dashed lines show $e^{-\delta
x^{\prime}}Y_{\rm
tot}=e^{-(m_{\chi_{1}}-m_{\chi_{2}})/T^{\prime}}(Y_{\chi_{1},i}+Y_{\chi_{2},i})$,
which is an approximation of the exact quasi-static equilibrium solution,
$Y^{\rm QSE}_{\chi_{1}}$ (purple dashed lines); see Eq. (39) for more details.
Both show that the suppression of $Y_{\chi_{1}}$ is solely due to the mass
splitting, independent of the absolute mass of ${\chi_{1}}$. The observed DM
abundance is shown as gray dot-dashed lines.

While these equations can only be solved numerically, which is how we obtained
our final results, here we also provide analytical explanation/approximation
to better demonstrate the underlying physics. An empirical relation between
$T^{\prime}$ and $T$ can be obtained using the observation that a decoupled
non-relativistic species cools down adiabatically with the scaling
$T^{\prime}\propto T^{2}$; for a relativistic species $T^{\prime}\propto T$
instead (away from epochs of entropy transfer). The transition between these
two scaling laws happens approximately at $T^{\prime}\sim m_{\chi_{2}}/4$.
Using the insights we gained from our numerical study, a good approximation at
$T^{\prime}\ll m_{\chi_{2}}$ is

${\xi(T)\over 2^{1/3}\xi_{i}}\approx{2^{1/3}\xi_{i}T\over
m_{\chi_{2}}/4}={4\times 2^{1/3}\xi_{i}\over x(1-\delta)}\,$ (9)

at any temperature $T$, where $\xi_{i}$ is the initial temperature ratio,
$2^{1/3}$ is the reheating factor discussed earlier. That is, we obtain the
relation $x^{\prime}=\beta x^{2}$ at $T^{\prime}\ll m_{\chi_{2}}$, with the
dimensionless constant $\beta\equiv(1-\delta)/(4\times 2^{2/3}\xi_{i}^{2})$.
See Fig. 12 in the Appendix for a comparison with full numerical results.

Figure 3: Contours of $x_{\text{f.o.}}$ (left panel) and $Y_{\chi_{1},\,f}$
(right panel) obtained from the analytical solution of Sec. 2.2 as a function
of mass splitting, $\delta$, and the initial temperature ratio between the
dark and SM sectors, $\xi_{i}$. In both panels, we choose
$m_{\chi_{1}}=1\,$TeV, as well as
$g_{1}g_{2}/m_{A}^{2}=1/(4\,\text{TeV})^{2}$. The dashed black lines show the
observed DM abundance. The red (gray) stars depict the parameter choices used
in Fig. 2 (Fig. 13), where the Boltzmann equations are solved numerically for
$\delta=0.1$ and $\xi_{i}=0.1$.

Adopting Eq. (9) allows us to solve Eq. (7) analytically as detailed in App.
A. Following the notation of Eq. (2), we obtain an approximate solution of a
non-relativistic freeze-out,

$Y_{\chi_{1},f}\approx\frac{(2n+1)\beta^{n}}{\lambda}x_{\text{f.o.}}^{2n+1}\,,$
(10)

where the freeze-out time is given by

$x_{\text{f.o.}}^{2}={1\over\beta\delta}\ln\left[\frac{0.21c(2+c)\lambda\xi^{3}_{i}\beta^{1/2}\delta^{1/2}}{(1-\delta)^{3/2}}\,{g_{\chi}\over
g_{*}}\right]-{n+{3\over
2}\over\beta\delta}\ln\left\\{\ln\left[\frac{0.21c(2+c)\lambda\xi^{3}_{i}\beta^{1/2}\delta^{1/2}}{(1-\delta)^{3/2}}\,{g_{\chi}\over
g_{*}}\right]\right\\}\,.$ (11)

The solution is different from Eq. (2) of the previous scenario, mostly due to
$T^{\prime}\propto T^{2}$ at freeze-out here. Nevertheless, in the $s$-wave
limit, we have

$Y_{\chi_{1},f}\approx\frac{1}{\lambda}x_{\text{f.o.}}\propto{1\over
m_{\chi_{1}}\sigma_{0}}x_{\text{f.o.}}\,,$ (12)

in both scenarios. It in turn suggests that $\Omega_{\rm DM}$ is generally not
sensitive to the DM mass, as it cancels in the ratio
$x_{\text{f.o.}}/m_{\chi_{1}}$ and for as long as the $s$-wave annihilation
cross section $\sigma_{0}$ is held constant. Hence, the yield decreases for
increasing $\sigma_{0}$.

The analytical solutions are illustrated in Fig. 3 by scanning the parameter
space for $m_{\chi_{1}}=1$ TeV and
$g_{1}g_{2}/m_{A}^{2}=1/(4\,\text{TeV})^{2}$. For $\xi=0.1$ and $\delta=0.1$,
it gives the freeze-out point $x_{\text{f.o.}}\approx 3$, as well as the
observed DM abundance, in good agreement with the exact numerical results of
Fig. 2. For additional numerical results of $n_{\chi_{1,2}}$ evolution with
various parameter sets, see Fig. 13 in the appendix. For instance, reducing
the mass splitting by a factor of $10$ to $\delta=0.01$ gives
$x_{\text{f.o.}}\approx 8$ and a final abundance about ten times larger
(middle gray star in Fig. 3). This can be explained as follows: estimating the
freeze-out condition through
$(Y_{\chi_{1}}s)\langle\sigma_{\chi_{1}}v\rangle\sim H$, one obtains
$Y_{\chi_{1}}\sim H/(s\langle\sigma_{\chi_{1}}v\rangle)\propto 1/T$ at freeze-
out. That is, smaller mass splitting leads to a lower freeze-out temperature,
and thus larger value of the final ${\chi_{1}}$ yield. In contrast, reducing
the temperature ratio gives an less populated dark sector, and thus a smaller
final abundance of $\chi_{1}$.

Finally, we turn to the $\chi_{2}$ abundance. Number conservation of the sum
of $\chi_{1}$ and $\chi_{2}$ directly translates into $Y_{\chi_{2},f}\approx
0.42\xi^{3}_{i}g_{\chi}/g_{*}-Y_{\chi_{1},f}$ at the end of freeze-out.
Eventually, we demand $\chi_{2}$ to decay into the visible sector, in analogy
to the hierarchical scenario before.

###  2.3 Potential matter-domination caused by $\chi_{2}$

In both the hierarchical and near-degenerate scenarios described above, at the
end of DM freeze-out the $\chi_{2}$ abundance is given by

$Y_{\chi_{2},f}\approx{0.42g_{\chi}\xi_{i}^{3}\over g_{*}}-Y_{\chi_{1},f}\,.$
(13)

As we focus on $\xi_{i}\geq 10^{-2}$ and the DM abundance should satisfy
$Y_{\chi_{1},f}\approx 10^{-10}(4\,\text{GeV}/m_{\phi})<10^{-10}$ for
$m_{\chi_{1}}$ well above GeV, the equation above gives $Y_{\chi_{2},f}\gg
Y_{\chi_{1},f}$, thus the second term on the R.H.S. can be neglected in
practice. Note that the insensitivity of $Y_{\chi_{2},f}$ to the annihilation
cross section is a result of the dark number conservation and the Boltzmann
suppression of $Y_{\chi_{1},f}$ relative to $Y_{\chi_{2},f}$ around freeze-out
time due to the mass splitting. Interestingly, depending on the lifetime of
$\chi_{2}$, the $\chi_{2}$ abundance, now mainly decided by $\xi_{i}$, may
lead to an epoch of early matter-domination (EMD). This would modify the
prediction for current-day particle abundances, due to entropy injection at
the end of EMD.

Figure 4: Evolution of the dark sector with an early matter-domination epoch
as a function of the scale factor (normalized to a present value $a_{0}=1$)
for $m_{\chi_{2}}=1\,$TeV, $\tau=10^{-2}$ sec, and $\xi_{i}=0.3$. The left
panel shows the evolving ratio of matter-to-radiation energy densities,
$\rho_{\rm M}/\rho_{\rm R}$ (dashed line), and $\Delta$–the ratio of entropy
density defined by $a^{3}(\rho_{\rm M}+\rho_{\rm R})^{3/4}$ to its initial
value (solid line); see Eq. (14). In the right panel, the purple solid (black
dashed) line shows the evolution of the SM temperature $T$ in our model (in
standard cosmology). In both panels, the vertical gray lines depict the
beginning (end) of this epoch at $a_{\alpha}$ ($a_{\beta}$), while horizontal
gray lines correspond to unity in left panel, and $T=m_{\chi_{2}}$ in right
panel.

If $\chi_{2}$ decays while still being relativistic, the Universe’s expansion,
by assumption, is dominated by SM radiation, and the final baryon asymmetry
remains unaltered from Eq. (17). In contrast, the decay of a non-relativistic
species transfers additional entropy to the SM sector. The dilution factor,
measured in terms of the ratio of total co-moving entropies before
($S_{\alpha}$) and after ($S_{\beta}$) decay may be estimated as

$\Delta\equiv{S_{\beta}\over S_{\alpha}}={a_{\beta}^{3}\over
a_{\alpha}^{3}}{s_{\beta}\over s_{\alpha}}={a_{\beta}^{3}\over
a_{\alpha}^{3}}\left({\rho_{\beta}\over\rho_{\alpha}}\right)^{3/4}\,,$ (14)

where the subscripts, $\alpha$ and $\beta$, mark the beginning and end points
when the Universe exits and re-enters the radiation-domination, respectively.
Note that the radiative entropy ($S$) does not include the non-relativistic
matter contribution; the last quantity is thereby used during the EMD epoch.
Using the energy conservation law $\dot{\rho}=-3H(\rho+p)$ and the equation of
state $p=w\,\rho$, one obtains

${\rho_{\beta}\over\rho_{\alpha}}=\exp\left(-3\int^{a_{\beta}}_{a_{\alpha}}da\,{1+\omega\over
a}\right)\,,$ (15)

where $\omega=1/3$ ($0$) for a radiation-dominated (matter-dominated)
Universe. During the transition, $\omega$ varies and needs to be calculated
numerically. The results for one benchmark parameter set is shown in Fig. 4.

We may also obtain a simplified analytic estimate for the dilution factor
$\Delta$ by assuming a sudden transition between matter- and radiation-
dominated epochs. For this, one may take $\rho\propto a^{-3}$ during matter
domination, thus $\Delta\approx(a_{\beta}/a_{\alpha})^{3/4}$ from Eq. (14),
i.e., $\Delta$ is mostly decided by the duration of this period. Matter-
domination starts at the scale factor $a=a_{\alpha}$, when
$(Y_{\chi_{2},f}s)m_{\chi_{2}}\approx\rho_{\rm SM}$ with a photon temperature
$T(a_{\alpha})\approx 4Y_{\chi_{2},f}m_{\chi_{2}}/3$, as well as a matter (or
radiation) energy density $\rho_{M}(a_{\alpha})\approx
g_{*}(Y_{\chi_{2},f}m_{\chi_{2}})^{4}$ at this time. The matter-domination
epoch ends when the Hubble rate equals the $\chi_{2}$ decay rate, and
$\rho_{M}(a_{\beta})=3\Gamma_{\chi_{2}}^{2}/(8\pi G)$ using
$\Gamma_{\chi_{2}}=H(a_{\beta})$. Therefore, we obtain

$\Delta\approx\max\left\\{1,\,\left({\rho_{M}(a_{\alpha})\over\rho_{M}(a_{\beta})}\right)^{1/4}\right\\}\approx\max\left\\{1,\,0.7\xi_{i}^{3}\,{g_{\chi_{2}}\over
g_{*}^{3/4}}\left({m_{\chi_{2}}\over\sqrt{m_{\rm pl}\
\Gamma_{\chi_{2}}}}\right)\right\\}\,.$ (16)

Evidently, when $\Delta\approx 1$, the matter-domination epoch was either very
brief or never happened. The expression above can be tested against the
numerical solution of Fig. 4.333Note that the decoupling of heavy SM particles
contributes to the evolution of $T$ in the right panel of Fig. 4, but it does
not affect the entropy evolution in the left panel.  For example, while Eq.
(16) yields $\Delta\approx 41$, Fig. 4 points to $\Delta\approx 43$. For
better guidance, we point out that $\Delta\lesssim 10^{3}$ for $m_{\chi_{2}}$
smaller than 1 TeV, due to $g_{\chi_{2}}\ll g_{*}$ and the Big Bang
Nucleosynthesis (BBN) bound on the lifetime of ${\chi_{2}}$.

In our setup, the EMD epoch occurs after DM freeze-out. Therefore, while the
dilution reduces the values of $Y_{\chi_{1}}$ and $Y_{\chi_{2}}$ separately by
a common factor $\Delta$, it does not change their ratio. That is, in absence
of other baryon asymmetry-depleting processes we have,

$Y_{\chi_{1},0}={Y_{\chi_{1},f}\over\Delta}\text{~{}~{}~{} and
~{}~{}~{}~{}}Y_{B}=\epsilon_{\rm CP}{Y_{\chi_{2},f}\over\Delta}\,,$ (17)

and the predicted relation between the DM abundance and baryon asymmetry is
not affected.

##  3 $\chi_{2}$ Decays, Baryogenesis, and the DM-Baryon Coincidence

As outlined earlier, the $\chi_{2}$ abundance following the dark freeze-out is
meta-stable, and its CP- and B- violating, out-of-equilibrium decays trigger
baryogenesis. We require that $\chi_{2}$ decays after freeze-out of both
$\chi_{1}$ and $\chi_{2}$, but before primordial nucleosynthesis: $T_{\rm
BBN}<T_{\chi_{2},\text{dec}}<T_{\text{f.o.}}$. This way, freeze-out and
baryogenesis can be treated separately, and the wash-out effect from inverse
decay of $\chi_{2}$ is ineffective during the baryogenesis stage. Washout
processes such as $udd\rightarrow\bar{u}\bar{d}\bar{d}$ are suppressed by the
large mediator mass (required by collider constraints, see Secs. 4 and 5) in
the effective operator for the process. A more detailed discussion on
suppressing various washout processes can be found in wimpyBG2  which is in
analogy to the scenario considered here.

The initial condition for $Y_{\chi_{2}}$ of this stage of evolution (i.e.,
$\chi_{2}$ decay) is set by the would-be abundance of $\chi_{2}$ after the
freeze-out of $\chi_{1}$, which is an essential factor for predicting the
baryon asymmetry. Recall from Sec. 2.3, via the overall number conservation
within the dark sector, the $\chi_{2}$ abundance after the freeze-out of
$\chi_{1}\chi_{1}\leftrightarrow\chi_{2}\chi_{2}$ is given by
$Y_{\chi_{2},f}=(Y_{\chi_{1},i}+Y_{\chi_{2},i})-Y_{\chi_{1},f}$ where
$Y_{\chi_{1},i}\approx Y_{\chi_{2},i}=0.21g_{\chi}/g_{*}\xi_{i}^{3}$. Assuming
that it violates baryon number by one unit (which is realized in the model
shown in the next section), the co-moving density of baryon asymmetry
$Y_{B}\equiv(n_{b}-n_{\overline{b}})/s$ observed today is

$Y_{B}=\epsilon_{CP}\int_{0}^{T_{\text{f.o.}}}\frac{dY_{\chi_{2}}}{dT}\exp\left(-\int_{0}^{T}\frac{\Gamma_{W}(\tilde{T})}{H(\tilde{T})}\frac{d\tilde{T}}{\tilde{T}}\right)dT+\dots\,,$
(18)

where $\epsilon_{CP}$ is the CP asymmetry in $\chi_{2}$ decays, and
$\Gamma_{W}$ is the rate of washout processes. The ellipses stand for any
possible additional sources of the baryon asymmetry prior to
$\chi_{2}$-decay,444 An initial contribution $Y_{B}^{\text{initial}}$ to the
baryon asymmetry can be accounted for by the addition of a term
$Y_{B}^{\text{initial}}\exp\left(-\int_{0}^{T_{\text{initial}}}\frac{\Gamma_{W}(T)}{H(T)}\frac{dT}{T}\right)$
where the exponential suppression is due to “wash-out.”  which we simply
assume to be zero in the following.

As mentioned earlier, the rate of washout processes that can potentially
reduce the baryon asymmetry is weak by construction and the exponential
factors can be dropped. This gives a simple approximate solution for the co-
moving baryon asymmetry in either, hierarchical or near-degenerate, scenario,

$Y_{B}\approx\epsilon_{CP}{Y_{\chi_{2},f}\over\Delta}={\epsilon_{CP}\over\Delta}\left(0.42\frac{g_{\chi}}{g_{*}}\xi_{i}^{3}-Y_{\chi_{1},f}\right),$
(19)

where $Y_{\chi_{1},f}$ is given in Eqs. (4) and (10). The result in Eq. (19)
directly connects the co-moving baryon asymmetry to the relic abundance of
$\chi_{1}$. Assuming that $\chi_{1}$ composes all of DM observed today, we
obtain the ratio of the DM and baryon abundances observed today as

$\frac{\Omega_{B}}{\Omega_{\chi_{1}}}=\epsilon_{CP}\frac{m_{p}}{m_{\chi_{1}}}\frac{Y_{\chi_{2},f}}{Y_{\chi_{1},f}}=\epsilon_{CP}\frac{m_{p}}{m_{\chi_{1}}}\times\begin{cases}0.42\frac{g_{\chi}}{g_{*}}\frac{\xi_{i}^{3}\lambda}{(n+1)\xi_{\rm
f.o.}^{-n}}\frac{1}{x_{\text{f.o.}}^{n+1}}-1&\text{hierarchical}\\\
0.42\frac{g_{\chi}}{g_{*}}\frac{\xi_{i}^{3}\lambda}{(2n+1)\beta^{n}}\frac{1}{x_{\text{f.o.}}^{2n+1}}-1&\text{near-
degenerate}\end{cases}\,,$ (20)

where $m_{p}$ is the proton mass; the degrees of freedom are to be evaluated
at the time of DM freeze-out $g_{*}\approx g_{*S}\sim 100$.555In the numerical
evaluation we carry along the temperature dependence of $g_{*}$ and $g_{*S}$,
whereas in the analytical estimates we take them as constants. In the case of
$s$-wave annihilation ($n=0$), neglecting the second term allows to further
simplify the expression to

$\frac{\Omega_{B}}{\Omega_{\chi_{1}}}\approx\frac{\epsilon_{CP}}{10^{-9}}\left[{\xi_{i}^{3}\over
x_{\text{f.o.}}}\frac{g_{\chi}}{g_{*}^{1/2}}\left(\frac{\sigma_{0}}{\rm
0.3\,pb}\right)\right]\,.$ (21)

The result in Eq. (20) for $\Omega_{B}$ takes a similar form as the
conventional WIMP miracle, augmented by the CP-asymmetry factor
$\epsilon_{CP}$ and proton-to-$\chi_{1}$ mass ratio $m_{p}/m_{\chi_{1}}$.
Taking similar values for the annihilation cross section, splitting $\delta$,
and dark sector temperature $\xi_{i}$ to those in Fig. 2 and Fig. 13, the
observed coincidence between dark and baryonic matter abundances can be
achieved with $m_{\chi_{1}}\sim\mathcal{O}(10-10^{4})~{}\text{GeV}$ and
$\epsilon_{CP}\approx\mathcal{O}(10^{-8}-10^{-2})$. In the next section, we
implement the general dynamics of $\chi_{1}$-$\chi_{2}$ freeze-out and
$\chi_{2}$ decay-triggered baryogenesis in a specific model which achieves
exactly this with dedicated numerical analysis.

##  4 Exemplary UV-complete Model

In this section we present a concrete model with fermionic dark states, which
include interactions realizing the
$\chi_{1}\chi_{1}\rightarrow\chi_{2}\chi_{2}$ freeze-out and generation of the
baryon asymmetry. For the latter, the three Sakharov conditions Sakharov  are
met by the CP- and baryon number-violating out-of-equilibrium decay of
$\chi_{2}$ to SM quarks.

We start by considering dark sector interactions necessary for freeze-out.
There are a number of options to achieve this and our core mechanism is
insensitive to the detailed realization of it. Here, we consider interactions
mediated by a pseudoscalar, $A$, or a scalar, $S$. The associated Lagrangian
then reads,

$\displaystyle\mathcal{L}_{\text{dark f.o.}}$
$\displaystyle=-g_{j}\bar{\chi}_{j}i\gamma^{5}\chi_{j}A-g^{\prime}_{j}\bar{\chi}_{j}\chi_{j}S\,,$
(22)

where the SM singlet Majorana fermions $\chi_{j}$ ($j=1,2,3$) are the members
of the dark sector. The interactions mediated by $A$ allow for $s$-wave
annihilation, whereas the leading contribution in a velocity expansion of the
annihilation cross section mediated by $S$ is $p$-wave. In order to avoid
additional complications of a dynamical role that $\chi_{3}$ may play during
dark freeze-out, we may take $g^{(^{\prime})}_{3}\ll 1$. This allows us to
neglect $\chi_{1,2}\chi_{1,2}\leftrightarrow\chi_{3}\chi_{3}$ processes and
$\chi_{3}$-induced energy transfer between dark and SM sectors even for
$\beta=O(1)$ ($\beta$ is to be introduced in Eq. 24). In addition, for
$m_{A(S)}\geq 2m_{\chi_{1}}$ we may also neglect the processes
$\chi_{1,2}\chi_{1,2}\leftrightarrow AA(SS)$. Taken together, we then map onto
the analysis of dark sector freeze-out of the preceding section. In passing,
we note that a hierarchical coupling structure among $g_{j}^{(^{\prime})}$ can
be addressed by an underlying dark flavor physics model, but which is beyond
the scope of this work.

In order to predict the yields of DM and baryon asymmetry from this model, we
first analyze the freeze-out process due to the annihilation
$\chi_{1}\chi_{1}\leftrightarrow\chi_{2}\chi_{2}$ mediated by either the
pseudoscalar $A$ ($s$-wave) or the scalar $S$ ($p$-wave). The non-relativistic
thermally averaged annihilation cross-section (away from the resonances) reads

$\displaystyle\sigma_{\chi_{1}}v\approx\frac{g_{1}^{2}g_{2}^{2}\sqrt{\delta(2-\delta)}}{64\pi
m_{\chi_{1}}^{2}\left[{m_{A}^{2}}/{(4m_{\chi_{1}}^{2})}-1\right]^{2}}+\frac{g_{1}^{\prime
2}g_{2}^{\prime 2}\left[\delta(2-\delta)\right]^{3/2}}{128\pi
m_{\chi_{1}}^{2}\left[{m_{S}^{2}}/{(4m_{\chi_{1}}^{2})}-1\right]^{2}}v^{2}\,,$
(23)

where there is no interference between the two mediators, and the
proportionality to $\sqrt{\delta}$ indicates the available limited final state
phase space. Apparently, in the case of $g^{\prime}_{j}=g_{j}$ and
$m_{S}=m_{A}$, the $s$-wave component mediated by $A$ dominates the DM
annihilation in the non-relativistic regime.

Figure 5: Tree-level and one-loop diagrams contributing to CP violation in
$\chi_{2}$ decays.

We now discuss the part that enables baryogenesis. The associated Lagrangian
is given by

$\displaystyle\mathcal{L}_{\text{baryog.}}$
$\displaystyle=-\alpha_{j}\bar{u}_{j}P_{L}\chi_{2}\phi-\beta_{j}\bar{u}_{j}P_{L}\chi_{3}\phi-\eta_{kl}\epsilon_{kl}\phi^{*}\bar{d}_{k}P_{L}d^{c}_{l}+\text{h.c.}\,,$
(24)

where the chiral projectors ensure that only the right-handed quarks $u_{j}$
and $d_{j}$ of generation $j$ participate, $\epsilon_{kl}$ is an antisymmetric
symbol for flavor indices $k,l$. As previewed in general discussions,
$\chi_{1}$ is charged under a $\mathcal{Z}_{2}$ symmetry. Hence, it does not
couple to quarks and plays the role of the DM. $\chi_{2}$ is the meta-stable
state whose decays trigger baryogenesis. We include a third member in the dark
sector, $\chi_{3}$, which enables the CP asymmetry-inducing interference
between tree and loop-level diagrams. A complex scalar, or “di-quark” $\phi$
is also introduced. This state transforms as an anti-triplet under
$SU(3)_{C}$, has the same SM charge as the right-handed up-type quarks, and
couples to the right-handed down-type quarks of the SM. All Yukawa couplings
$\alpha_{j},\ \beta_{j},\ \eta_{jk}$ are generic, complex numbers.

Because of stringent LHC constraints on color-charged particles $\phi$ and to
allow for a tractable exposition, we shall consider the hierarchy $m_{\phi}\gg
m_{\chi_{j}}$. In addition, we take $m_{\chi_{2}}>m_{\chi_{3}}$ so that
contributions to the baryon asymmetry solely arises from $\chi_{2}$ decays as
shown in Fig. 5. The analogous diagram for $\chi_{3}$ decay with $\chi_{2}$ in
the loop does not contribute to an analogous CP asymmetry from $\chi_{3}$,
since here the kinematic cut is forbidden as $m_{\chi_{2}}>m_{\chi_{3}}$.666If
$m_{\chi_{2}}<m_{\chi_{3}}$, it would be $\chi_{3}$, instead of $\chi_{2}$,
whose decay contributes to the CP-asymmetry by switching the subscripts
$2\leftrightarrow 3$. Alternatively, if $m_{\chi_{j}}\gtrsim m_{\phi}$, the
kinematic cut can be made through $\phi$ and up-quark propagators, where both
$\chi_{2}$ and $\chi_{3}$ decays would have non-vanishing $\epsilon_{CP}$,
and, in principle, contribute to the baryon asymmetry.  Finally, an ordering
$|\alpha|\ll|\beta|<|\eta|$ will be assumed to ensure the branching ratio
Br${}_{\phi\to d_{k}d_{l}}\approx 1$, and a sufficiently early decoupling of
dark particles from the SM. For instance and as shown below, the smallness of
$|\alpha|$ together with a heavy $\phi$ guarantees that $\chi_{1,2}$ decouple
from both SM particles and $\chi_{3}$ before DM freeze-out. The hierarchy
$|\alpha|\ll|\beta|$ further forces $\chi_{3}$ to have a lifetime shorter than
$\chi_{2}$, decaying into SM quarks well before BBN.

The Lagrangian (24) enables $\chi_{2}$ decay that triggers baryogenesis. With
the assumed mass hierarchy $m_{\phi}\gg m_{\chi_{j}}$, we may integrate out
the heavier di-quark scalar, yielding the effective operators

$\mathcal{L}_{|\Delta
B|=1}\supset\frac{\alpha_{j}\eta_{kl}}{m_{\phi}^{2}}\,(\bar{u}_{j}P_{L}\chi_{2})\,(\bar{d}_{k}P_{L}d_{l}^{c})+\frac{\beta_{j}\eta_{kl}}{m_{\phi}^{2}}\,(\bar{u}_{j}P_{L}\chi_{3})\,(\bar{d}_{k}P_{L}d_{l}^{c})+\text{h.c.}\
.$ (25)

They induce the three-body decay $\chi_{2}\rightarrow udd$ with $|\Delta
B|=1$. The associated decay rate at $T<m_{\chi_{2}}$ is given by

$\displaystyle\Gamma_{\chi_{2},\text{dec}}\approx\frac{3|\alpha_{j}|^{2}|\eta_{kl}|^{2}m_{\chi_{2}}^{5}}{1024\pi^{3}m_{\phi}^{4}}\,,$
(26)

where the factor of three in Eq. (26) accounts for color multiplicity. The CP-
asymmetry produced in decays arises through the interference between tree-
level and 1-loop diagrams shown in Fig. 5. The asymmetry as a function of
$m_{\chi_{2}}$, $m_{\phi}$, and Yukawa couplings is given by Cui:2013bta

$\epsilon_{CP}=\text{Im}[\alpha_{j}\alpha_{m}\beta_{j}^{*}\beta_{m}^{*}]\frac{m_{\chi_{2}}^{2}}{20\pi|\alpha_{j}|^{2}m_{\phi}^{2}}\,,$
(27)

in the process of $\chi_{2}\to u_{j}d_{k}d_{l}$ (and
$\bar{u}_{j}\bar{d}_{k}\bar{d}_{l}$) with an intermediate up-type quark
$u_{m}$. Since we are interested in a broad range of $m_{\chi_{1}}$ spanning
over $\mathcal{O}(10-10^{4})~{}\text{GeV}$, all generations may contribute
when $m_{\chi_{2}}\gtrsim m_{t}$. Nevertheless, the constraints from
(di-)nucleon decay and neutron-antineutron oscillation on the effective
operators in Eq. (25) are very strong if $(u_{j}d_{k}d_{l})$ that couples to
$\chi_{3}$ contains only light quarks, such as $(uds)$; see e.g.,
Aitken:2017wie . Thus, here we focus on the combination of $(cds)$, or
$(udb)$, where sizable coupling-combination $\beta\eta$ is allowed, and
$\chi_{2,3}$ decays into them are kinematically accessible. Finally, we note
that the model is exempt from constraints from the null measurements of the
neutron electric dipole moment (EDM). The reason is that the interference
diagrams in Fig. 5 leading to CP violation only involve right-handed quarks
He:1996hb ; Yamanaka:2017mef .

Figure 6: Contours of constant $\Omega_{B}/\Omega_{\chi_{1}}=0.2$ for varying
annihilation coupling $g$ and DM mass $m_{\chi_{1}}$ and four values of the
mass splitting $\delta$ in the hierarchical (left) and the nearly degenerate
(right) scenarios where annihilation is dominantly $s$-wave with
$g\equiv\sqrt{g_{1}g_{2}}$ and/or $m_{A}\lesssim m_{S}$. The other model
parameters are pseudoscalar mass $m_{A}=2.1~{}\text{TeV}$, $\phi$ mass
$m_{\phi}=10~{}\text{TeV}$, initial temperature ratio $\xi_{i}=0.1$, and
$\chi_{3}-\text{quark}$ Yukawa coupling $\beta=1$.

Having established all the ingredients and hierarchies of a UV complete model,
we now turn to the identification of prospective parameter ranges. As
discussed above, DM decoupling with $\xi_{i}\leq 1$ prefers an earlier freeze-
out in comparison with the standard case. Under the assumption that $\chi_{1}$
freezes out for $T\gtrsim m_{\chi_{2}}/10$, the requirement that $\chi_{2}$
decays after $\chi_{1}$ freeze-out but before BBN yields the preferred range
of allowed decay couplings as

$10^{-9}\left({1\,\text{TeV}\over
m_{\chi_{2}}}\right)^{5/2}\left({m_{\phi}\over
10\,\text{TeV}}\right)^{2}\leq|\alpha||\eta|\leq
10^{-4}\left({1\,\text{TeV}\over
m_{\chi_{2}}}\right)^{3/2}\left({m_{\phi}\over 10\,\text{TeV}}\right)^{2}\,.$
(28)

Note that the last inequality is also approximately the condition that
$\chi_{2}$ decouples from the SM bath during $\chi_{1}$ freeze-out: if the
bound were violated, the process $\chi_{2}\leftrightarrow u_{j}d_{k}d_{l}$
becomes efficient and dark and SM sectors thermalize. As a result, for
$m_{\phi}=10$ TeV, the hierarchy $|\alpha|\ll|\eta|$ introduced for
baryogenesis leads to $|\alpha|\ll 10^{-2}$ for $m_{\chi_{2}}=1\,$TeV, and
$|\alpha|\ll 0.14$ for $m_{\chi_{2}}=30\,$GeV. Similarly, requiring $\chi_{3}$
to decay fast, say at $T\sim m_{\chi_{3}}$, suggests $|\beta||\eta|\gtrsim
0.1(m_{\phi}/10\,\text{TeV})^{4}(10\,\text{GeV}/m_{\chi_{3}})^{3}$, which can
be easily satisfied in the parameter region of our interest here. So the
existence of $\chi_{3}$ neither leads to additional early matter domination,
nor affects the standard BBN processes.

Figure 7: Contours of constant $\Omega_{B}/\Omega_{\chi_{1}}\approx 0.2$ for
varying annihilation coupling $g$ and DM mass $m_{\chi_{1}}$ and four values
of the mass splitting $\delta$ in the hierarchical (left) and the nearly
degenerate (right) scenarios where annihilation is dominantly $p$-wave in the
limit of $m_{A}\gg m_{S}$. In analogy to the choice made in Fig. 6, we choose
$g^{{}^{\prime}}\equiv\sqrt{g^{{}^{\prime}}_{1}g^{{}^{\prime}}_{2}}$. The
other model parameters are $m_{S}=2.1~{}\text{TeV}$,
$m_{\phi}=10~{}\text{TeV}$, initial temperature ratio $\xi_{i}=0.1$, and
$\chi_{3}$-quark Yukawa coupling $\beta=1$.

Putting all together and plugging the specific results into Eq. (20), we
obtain the relic abundance of DM and the baryon asymmetry. Taking Eq. (21),
their ratio in the exemplary UV model for $s$-wave dominated freeze-out
becomes

$\frac{\Omega_{B}}{\Omega_{\chi_{1}}}\sim 0.2\left({|\beta|^{2}\sin
2\theta_{\beta}\over x_{\rm f.o.}}\right)\left({\xi_{i}\over
0.1}\right)^{3}\left({\sqrt{g_{1}g_{2}}\over 0.2}\right)^{4}\left({30\,{\rm
TeV}\over
m_{\phi}(m_{A}^{2}/4m_{\chi_{1}}^{2}-1)}\right)^{2}\left({(1-\delta)^{2}\sqrt{\delta}\over
0.1}\right)\,,$ (29)

for which, and for numerical results below, we have assumed the Yukawa
couplings between $\chi_{2}$ and the up-type quarks to be real such that the
CP-asymmetry in Eq. (27) can be written in terms of a complex phase
$\theta_{\beta}$ of
$\beta:~{}\text{Im}[(\alpha\alpha\beta^{*}\beta^{*})^{2}]\rightarrow|\alpha|^{2}|\beta|^{2}\sin(2\theta_{\beta})$.
We observe, that the final ratio of $\Omega_{B}$ to $\Omega_{\chi_{1}}$ is
sensitive to the mass splitting $\delta$ and $m_{A}/m_{\chi_{1}}$, instead of
$m_{\chi_{1}}$ alone. In Fig. 6 we show contours of
$\Omega_{B}/\Omega_{\chi_{1}}=0.2$ as a function of the $\chi_{1}$ mass and DM
annihilation coupling $g\equiv\sqrt{g_{1}g_{2}}$ in the hierarchical (left)
and the nearly degenerate (right) cases of our exemplary model. Note that the
contributions from the scalar $S$ is subleading and is thus not taken into
account. To obtain the numerical results, we plug Eqs. (23) and (27) into the
solution for the $\chi_{1}$ freeze-out abundance given by Eqs. (4) and (10),
which is then used together with
$Y_{\chi_{2},f}\approx\left(0.42\xi_{i}^{3}{g_{\chi}}/{g_{*}}-Y_{\chi_{1},f}\right)$
to find the abundance of $\chi_{2}$ just after DM freeze-out. This, in turn,
gives rise to the observed baryon asymmetry,
$Y_{B}\propto\epsilon_{CP}Y_{\chi_{2},f}$. The contours have fixed mass
splitting in the range of $0.97<\delta<0.997$ (or equivalently,
$0.003m_{\chi_{1}}<m_{\chi_{2}}<0.03m_{\chi_{1}}$) in the hierarchical case
and $0.05<\delta<0.5$ in the degenerate case. The pseudoscalar mass is taken
to be $m_{A}=2.1~{}\text{TeV}$ in Fig. 6. In the limit of $m_{A}\gg m_{S}$
and/or $g^{{}^{\prime}}\gg g$, the annihilation channel via the scalar $S$
dominates the DM freeze-out, which case is shown in Fig. 7. In our analysis,
we fix $\theta_{\beta}=\pi/4$ to obtain the maximal CP-asymmetry.

##  5 Experimental Signatures

The concrete model outlined in the previous section lends itself to
observational signatures and experimental tests, in particular in relation to
the stable DM candidate $\chi_{1}$ and its unstable dark sector partners
$\chi_{2,3}$. These include conventional and novel direct and indirect
detection signals, as well as modifications to primordial density
fluctuations. We will sketch the prospects of these searches below, leaving a
more-in-detail analysis for future work.

###  5.1 New particle searches at colliders

Figure 8: Diagrams relevant for $\phi$ (left) searches and $\chi_{2,3}$
production at the LHC.

To produce the baryon asymmetry with renormalizable operators, the color anti-
triplet, di-quark scalar $\phi$ has been introduced. As a result of the color
charge it carries, LHC bounds on its mass are quite strong. As outlined in
Sec. 4, the di-quark scalar $\phi$ decays to two down-type quarks. At the LHC,
$\phi\phi^{*}$ are pair-produced dominantly via gluon fusion, and subsequently
decay to 4 jets: $pp\rightarrow\phi\phi^{*}\rightarrow 4j$, as shown in Fig.
8. LHC searches for di-quark scalars constrain the mass of $\phi$. For
instance, the recent CMS search CMS:2018mgb  constrains the production cross
section of di-quark scalar resonances to be below fb at $\sqrt{s}=13$ TeV,
excluding $m_{\phi}\lesssim 7~{}\text{TeV}$.

Figure 9: Diagrams showing dominant contributions to $\chi_{1}$, $\chi_{2}$
and $\chi_{3}$ production at the LHC.

The $\chi_{j}$ ($j=1,2,3$) particles can also potentially be produced at
colliders. The production of $\chi_{1}$ is shown on the left panel of Fig. 9,
and would lead to a missing transverse energy (MET) signal. However, due to
the combination of loop factor, the required large mass of the di-quark scalar
$\phi$ in the loop diagram, and the additional initial state radiation (ISR)
photon or jet required for tagging the event, the $\chi_{1}$ production rate
is found to be too small to be detectable. The production of $\chi_{2}$ and
$\chi_{3}$ can proceed via t-channel $\phi$-exchange as shown in the right
panel of Fig. 9. The subsequent $\chi_{2,3}$ decays can lead to prompt jets,
displaced vertices or MET, depending on their decay length. Again,
$m_{\phi}\gtrsim 7$ TeV suppresses the production rates of $\chi_{2}$ and
$\chi_{3}$. Since successful co-genesis requires $|\alpha|\ll 10^{-2}$ for
$m_{\chi_{2}}=1$ TeV, the production cross section of $\chi_{2}$ falls
significantly below fb, well beyond the LHC reach. The prospect for $\chi_{3}$
production is better, as the coupling $|\beta|$ can be $O(1)$, leading to fb-
sized LHC production cross sections for $\sqrt{s}=13\,$TeV and
$m_{\phi}=10\,$TeV Kilic:2015vka . The prospect of detecting $\chi_{3}$
depends on its decay length following production. It may vary over a wide
range,

$c\gamma\tau_{\chi_{3}}=4\text{\,cm}\left({\gamma\over
10}\right)\left({10^{-4}\over\beta\eta}\right)^{2}\left({1\,\text{TeV}\over
m_{\chi_{3}}}\right)^{5}\left({m_{\phi}\over 10\,\text{TeV}}\right)^{4}\,,$
(30)

where $\gamma=E_{\chi_{3}}/m_{\chi_{3}}$ is the Lorentz factor; the same
expression also applies to $\chi_{2}$ after replacing $m_{\chi_{3}}$ and
$\beta$ with $m_{\chi_{2}}$ and $\alpha$. $\chi_{3}$ decay can hence lead to
prompt multijet events, MET, or displaced jet signals. For the size of the
production cross section, the displaced vertex channel is the most promising
one, with the potential of detection at the High-Luminosity LHC Cui:2014twa ,
due to the generally low background CMS:2019zmd ; CMS:2018tuo ;
CidVidal:2018eel ; ATLAS:2018yii .

We would like to note that although with the LHC it is generally challenging
to detect $\chi_{j}$ particles in the model as outlined above, the detection
prospects are expected to improve notably with the proposed future higher
energy colliders Mangano:2017tke ; ILC:2019gyn ; CEPCStudyGroup:2018ghi ;
Roloff:2018dqu ; FCC:2018byv . Furthermore, model-variations can increase the
production rate of $\chi_{j}$. For instance, interaction terms, such as a
trilinear $A$-$A$-$H$, may open up new channels at higher energy colliders for
both $\chi_{1}$ and $\chi_{2}$ production with potentially appreciable rate,
independently of $m_{\phi}$. In addition, while we consider $\phi$ dominantly
decays into a pair of down-type quarks, a sizable branching ratio of
$\phi\to\chi_{3}u$ is possible, depending on the ratio $|\beta/\eta|^{2}$,
which provides a new, potentially efficient channel for $\chi_{3}$ production
as shown in the right panel of Fig. 8; for $\chi_{2}$ the corresponding ratio
is much smaller and less prospective.

In summary, the di-quark and $\chi_{j\,}$’s in the co-genesis framework that
utilizes quark-couplings in its UV-complete representation could leave
observable signals in various search channels at high luminosity run of the
LHC, yet are potentially challenging. On the other hand, the proposed future
colliders at the high energy frontier have more promising capacity to reveal
detectable signatures from this framework.

###  5.2 Direct detection of $\chi_{1}$

Figure 10: Diagrams contributing to DM-nucleus scattering
$\chi_{1}N\rightarrow\chi_{1}N$. The diagram on the left is the dominant
contribution in the minimal model while the diagram on the right may dominate
if the $S$-Higgs mixing is present.

In the example model detailed in Sec. 4, there are no interactions between the
stable DM candidate $\chi_{1}$ and quarks at tree-level. However, DM-nucleus
interactions can be induced at 1-loop as shown in the left panel of Fig. 10.
Integrating out the heavier mediating states generates low-energy effective
operators which induce interactions between $\chi_{1}$ and nucleons $N=n,\ p$.
They are either spin-independent, $\mathcal{L}\propto\bar{N}NS$, or spin-
dependent, $\mathcal{L}\propto\bar{N}(i\gamma^{5})NA$. The effective
interactions lead to DM-nucleon elastic scattering cross sections which can be
estimated using dimensional analysis:

$\sigma^{\rm SI}_{\chi_{1}N}\sim{f_{N}^{2}\over
256\pi^{5}}\left[\frac{\alpha^{2}g^{\prime}_{1}g^{\prime}_{2}m_{\chi_{2}}m_{\chi_{1}}m_{N}^{2}}{m_{\phi}^{2}m_{S}^{2}(m_{\chi_{1}}+m_{N})}\right]^{2}\text{~{}and~{}}\,\sigma^{\rm
SD}_{\chi_{1}N}\sim\frac{v^{2}f_{N}^{2}}{512\pi^{5}}\left[\frac{\alpha^{2}g_{1}g_{2}m_{\chi_{2}}^{2}m_{\chi_{1}}m_{N}}{m_{\phi}^{2}m_{A}^{2}(m_{\chi_{1}}+m_{N})}\right]^{2},$
(31)

which applies in the limit $m_{\phi,S}\gg m_{\chi_{2}}\gg m_{N}$ Kumar:2013hfa
. Here the first component, mediated by $S$, is a spin-independent cross
section, while the latter, mediated by $A$, is both velocity suppressed and
spin-dependent; the coefficient $f_{N}\approx 0.3$, is counting for the
valence quark content of the nucleon COMPASS:2007esq . For instance, under the
assumption that
$g_{1}g_{2}/m_{A}^{2}=g^{\prime}_{1}g^{\prime}_{2}/m_{S}^{2}=1/(4\,\text{TeV}^{2})$,
$m_{\chi_{1}}=3\,$TeV, and $m_{\chi_{2}}=0.1\,$TeV, which gives the observed
DM abundance with $\xi\sim 1$, setting $m_{\phi}=10\,$TeV together with Eq.
(28) require $|\alpha|\ll 1$, resulting in the spin-independent part
$\sigma^{\rm SI}_{\chi_{1}N}\ll 10^{-56}\,\text{cm}^{2}\,$; the spin-dependent
component is even smaller due to the velocity suppression. We conclude that in
the current setup, there are no direct detection prospects. Finally, we point
out that there can be contributions to direct detection at tree-level when the
scalar $S$ mixes with the SM Higgs. In order for the dark sector and SM to
remain decoupled as per our working hypotheses, the $S$-Higgs mixing angle
must be small, $\theta_{S}\lesssim 10^{-7}$. In the case of $m_{S}\geq m_{h}$,
the scattering cross section between $\chi_{1}$ and proton through mixing can
be estimated as $10^{-42}({g^{\prime}_{1}\theta_{S}})^{2}\,\text{cm}^{2}$
Lopez-Honorez:2012tov , and thus will not be detectable in the foreseeable
future.

An interesting possibility arises through $t$-channel scattering of protons
(or neutrons) with $\chi_{j}$ at one loop via off-shell
$A\,(S),~{}\chi_{2},~{}\text{and}~{}\phi$. An example of this process with a
proton initial state is shown in Fig. 11.

Figure 11: Example of exotic nucleon conversion arising in dark sector
baryogenesis.

Therefore, it is important to note that the relative kinetic energy of the
galactic $\chi_{1}$ DM flux at earth interacting with nuclei falls short to
kinematically access this channel, but rather requires a “boosted” $\chi_{1}$
component. A particularly interesting environment may involve neutron stars in
regions of high $\chi_{1}$ density accelerating $\chi_{1}$ to sufficiently
large energy Guver:2012ba , such that this nucleon conversion process occurs
and leads to observable effects such as anomalous heating. Note that the
energy spectrum of the incoming and outgoing states can differ significantly
from the elastic scattering usually considered Acevedo:2019agu ; Bell:2018pkk
. We leave a detailed investigation of nucleon conversion via capture in
neutron stars, and other stellar objects, to future work.

###  5.3 (Exotic) indirect detection through $\chi_{2}$

The low-redshift $\chi_{1}$ annihilation into possibly energetic $\chi_{2}$
particles shares some features with the boosted dark matter scenario
Agashe:2014yua ; Berger:2014sqa . However, the $\chi_{2}$ final states decay
electromagnetically within a second, leading to box-shaped energy spectra of
final SM quarks Ibarra:2012dw , which in turn can be experimentally
constrained. Approximately, the annihilation cross section is given by
$\langle\sigma_{\chi_{1}}v\rangle\approx(\xi_{\text{f.o.}}/\Delta)$ pb, in
order to yield the observed DM abundance. This leads to lower bounds on DM
mass around $m_{\chi_{1}}\geq 50\,(\xi_{\text{f.o.}}/\Delta)$ GeV from Planck
and indirect search experiments (see e.g., Elor:2015bho ; Slatyer:2015jla ;
Fermi-LAT:2019lyf ; Kahlhoefer:2021sha )777Extremely small mass splittings,
below the typical kinetic energy of galactic DM, can lead to a velocity-
dependent cross section. This is a fine-tuned situation that does not happen
in the generic parameter space of interest here. .

In addition, DM can be captured by celestial objects, and consequently
annihilate inside. Given the short lifetime of $\chi_{2}$, one particularly
interesting signature is that the boosted $\chi_{2}$ produced from DM
annihilation in the centre of Earth decays inside a large-volume terrestrial
detector, for which we take IceCube IceCube:2016aga  as an example. The signal
requires the in-flight decay length of $\chi_{2}$,
$l_{\chi_{2}}\equiv\tau\,c\sqrt{m_{\chi_{1}}^{2}/m_{\chi_{2}}^{2}-1}$, to be
on the order of or longer than the Earth’s radius, $r_{\rm Earth}$. If
satisfied, the probability for $\chi_{2}$ to decay inside its detector of
fiducial volume $1\,$km3 is given by

$\left.P_{\chi_{2}\to X}\right|_{\text{detector}}={V_{\text{detector}}\over
l_{\chi_{2}}\cdot 4\pi r_{\rm Earth}^{2}}\approx 0.8\times
10^{-13}\left({\text{0.02\,sec}\over\tau\sqrt{m_{\chi_{1}}^{2}/m_{\chi_{2}}^{2}-1}}\right)\,.$
(32)

Assuming a sensitivity threshold of several events per year in the detector
same as Feng:2015hja  and $l_{\chi_{2}}/c=\text{0.02\,sec}$, the probability
given by Eq. (32) implies that the DM annihilation rate, $\Gamma_{A}$, in the
Earth’s centre should reach $10^{6}$ particles per second, in order to be
detectable in IceCube, which is in agreement with Schuster:2009au .

We may consider the prospects of such signature in the concrete UV model. As
discussed in Sec. 5.2, the non-relativistic scattering between $\chi_{1}$ and
nucleons is dominated by the $S$-mediated interaction, leading to a spin-
independent cross section $\sigma^{\rm SI}_{\chi_{1}N}$. In the parameter
region of interest, where the annihilation cross section is
$\langle\sigma_{\chi_{1}}v\rangle\leq 1\,$pb, equilibrium between capture and
annihilation in Earth is not reached Lundberg:2004dn ; IceCube:2016aga . That
is, the actual DM annihilation rate in Earth scales as $(\sigma^{\rm
SI}_{\chi_{1}N})^{2}\,\langle\sigma_{\chi_{1}}v\rangle$, up to subleading
effects IceCube:2016aga . Moreover, a comparison of Figs. 6 and 8 in
IceCube:2016aga  allows us to estimate the coefficient of this scaling for
$m_{\chi_{1}}\geq 100\,$GeV, leading to

$\Gamma_{A}\sim 10^{9}\,\text{s}^{-1}\left({10\,\text{TeV}\over
m_{\chi_{1}}}\right)^{2}\left({\sigma^{\rm SI}_{\chi_{1}N}\over
10^{-41}\text{cm}^{2}}\right)^{2}\left({\langle\sigma_{\chi_{1}}v\rangle\over
1\,\text{pb}}\right)\,.$ (33)

As an example, for 100 GeV DM with $\langle\sigma_{\chi_{1}}v\rangle=0.1\,$pb,
detecting $O(1)$ events per year requires $\Gamma_{A}\sim 10^{6}$ s-1,
implying a sensitivity of $\sigma^{\rm SI}_{\chi_{1}N}\sim
10^{-45}-10^{-46}\rm cm^{2}$. This is comparable to current direct detection
constraints, but still needs to be improved by several orders of magnitude to
eventually probe our model. Such improvement may be achieved if the DM
annihilation is significantly enhanced at low velocities (e.g., through
Sommerfeld enhancement with a light mediator), bringing the Earth capture and
annihilation into equilibrium. Note that this simple re-scaling does not apply
for $m_{\chi_{1}}\leq 100\,$GeV, where the bound gradually gets weaker, except
for DM masses that trigger resonant capture of abundant elements in Earth
IceCube:2016aga . A more detailed investigation of captured DM abundances is
deferred to future work.

###  5.4 Modifications to primordial density fluctuations

New particles and their associated dynamics in our framework may also leave an
imprint on cosmological observables. One aspect is structure formation. For
instance, the metastable $\chi_{2}$ can induce an EMD epoch, during which the
subhorizon density perturbations grow linearly. In contrast, the motion of DM
particles leads to damping of density perturbations at small scales. As
studied in the literature (e.g., Erickcek:2011us  and most recently in
Delos:2021rqs ), the total effect on density perturbations is mainly decided
by two scales: the horizon size at the end of the EMD epoch and the scale
below which primordial fluctuations are suppressed by DM streaming out of
over- and under-dense regions.

The scale associated with perturbation growth is decided by the horizon size
at cosmic time $t=\tau$, where $\chi_{2}$ decays and the EMD epoch ends. Its
value can be expressed in terms of co-moving distance as

$L_{\rm end}={\tau}\,\left({s(T_{\rm end})\over s(T_{0})}\right)^{1/3}\approx
10^{-5}\,\sqrt{\tau\over 0.02\,{\rm s}}\,{\rm Mpc}\,,$ (34)

where $s(T)$ is the radiation entropy density at temperature $T$, $T_{end}$ is
the reheating temperature post the EMD epoch; the photon temperature at
present is $T_{0}=2.35\times 10^{-4}$ eV planck . Perturbations at scales
below $L_{\rm end}$ thus grow linearly w.r.t. the scale factor $a$ during this
EMD epoch Dodelson:2003ft ; Erickcek:2011us .

On the other hand, density perturbations of DM can be suppressed due to random
energy transfers either via sound waves or traveling particles, referred to as
acoustic or collision(-less) damping Green:2003un ; Loeb:2005pm ;
Jeong:2014gna . Since $\chi_{1}$ does not significantly scatter after its
freeze-out, collisionless damping dominates in the suppression of structure.
This free-streaming scale is estimated by

$L_{\rm fs}=a_{0}\int^{t_{\rm eq}}_{0}dt{v(t)\over a(t)}\approx 10^{-9}\,{{\rm
TeV}\over m_{\chi}/\xi_{i}}\,{\rm Mpc}\,\,$ (35)

where we have neglected the logarithmic dependence on the exact time when DM
kinetically decouples. Finally, we note that the last equality of Eq. (35)
assumes a standard cosmology. In presence of a matter-dominated epoch, the
ratio of $a_{0}/a(t)$ becomes larger than in standard cosmology at $t<\tau$
(compare solid and dashed lines in right panel of Fig. 4), thus a “stretching
factor” $\Delta^{1/3}$ needs to be multiplied in this case.

Although there exist two competing effects as outlined above (enhancement and
suppression), observationally, the most relevant scale is the larger one of
$L_{\rm end}$ and $L_{\rm fs}$. In our framework, the scale governing
additional perturbation growth, $L_{\rm end}$, is larger than the suppression
scale, $L_{\rm fs}$, as the former is the horizon size when DM is already non-
relativistic. As a result, we generally expect a peak in density perturbations
between the two scales, leading to a potentially observable enhancement in the
local subhalo abundance. The ensuing altered predictions of the ionization
history as well as of the halo-mass function in the late Universe can be
probed with various strategies Erkal2016TheNA ; Chluba:2019kpb ;
Furugori:2020jqn ; Lee:2020wfn ; Vikaeus2021ConditionsFD ; Delos:2021rqs . In
particular, future Pulsar Timing Arrays (PTA) will be able to constrain the
abundance of local subhalos down to a halo mass of $10^{-10}\,M_{\odot}$, and
thus have the sensitivity to probe effects on primordial perturbations at co-
moving scales as small as $10^{-7}$ Mpc Ramani:2020hdo ; Lee:2020wfn . Such a
scale corresponds to $\tau\sim 10^{-6}$ s, or, conversely, a reheating
temperature of $T_{\rm end}\sim 300$ MeV following the end of the EMD era.

##  6 Conclusions

In this work, we identify a new mechanism for the joint generation of the
baryon and DM abundances. DM $\chi_{1}$ makes a thermal freeze-out via
annihilation into a lighter metastable dark partner $\chi_{2}$ through an
overall dark number conserving process $\chi_{1}\chi_{1}\to\chi_{2}\chi_{2}$.
The lighter state $\chi_{2}$ subsequently decays to SM quarks. Its chemical
decoupling and CP- and B-violating interactions with SM ensure the fulfillment
of the Sakharov conditions, while its interactions with $\chi_{1}$ ensure
fulfillment of the relic density requirement by observations. The lifetime of
$\chi_{2}$ is assumed such that its decay happens after $\chi_{1}$-$\chi_{2}$
freeze-out yet before primordial nucleosynthesis. By itself, this only
requires small couplings to SM, and, in general the dark and observable sector
temperatures, $T^{\prime}$ and $T$, may differ.

We present a novel analytical treatment for a two-state dark sector freeze-out
and subsequent baryogenesis assuming $T^{\prime}\leq T$ and for which
$T^{\prime}=T$ is contained as a special case. There are then two principal
options. First, in the hierarchical scenario there is a mass hierarchy
$m_{\chi_{1}}/m_{\chi_{2}}\gtrsim 10$ such that $\chi_{2}$ is relativistic
during $\chi_{1}$ freeze-out. The relic abundance of DM, $\chi_{1}$, is
established following the dark freeze-out, and depends on the annihilation
cross-section and the ratio $T^{\prime}/T$, resembling the prediction familiar
from WIMPs. The $\chi_{2}$ abundance prior to its decay is also fixed by the
freeze-out, yet is insensitive to the annihilation cross-section.
Consequently, the ensuing prediction of the baryon asymmetry is predominantly
determined by the CP asymmetry $\epsilon_{\rm CP}$ and the initial value of
$T^{\prime}/T$. In the nearly degenerate scenario ($\delta\ll 1$) both
$\chi_{1}$ and $\chi_{2}$ freeze out non-relativistically. However, because of
overall $\chi_{1}+\chi_{2}$ number conservation in the dark sector, its
freeze-out yield again essentially coincides with that of radiation such as in
the previous case. Therefore, the baryon asymmetry likewise only depends on
$\epsilon_{\rm CP}$ and the temperature ratio.

In either scenario, $\chi_{2}$ may dominate the energy budget of the Universe
before its decay, leading to an early matter dominated era. The associated
entropy injection dilutes both baryon asymmetry and DM abundance. However, it
leaves their relative proportion unchanged. In the case of $s$-wave
annihilation, the prediction for $\Omega_{B}/\Omega_{\rm DM}$ only depends on
the DM annihilation cross section—which may take on its usual thermal value
$O({\rm pb})$—and on $\epsilon_{\rm CP}$, once the initial temperature ratio
is fixed; see Eq. (21). We verify our analytical estimates by numerically
solving the Boltzmann equations, and find excellent agreement for benchmark
parameter points.

We then realize our general ideas by introducing a UV complete model. Here, a
heavy pseudoscalar $A$ mediates the dominant, $s$-wave annihilation of the
fermionic DM state $\chi_{1}$, while a massive CP-even scalar mediator $S$ may
play a more favorable role for detection prospects in DM direct detection
experiments. The connection to the SM is made through Yukawa interactions
between fermionic states $\chi_{2,3}$, an $SU(3)_{c}$ charged scalar $\phi$,
and SM quarks. These interactions mediate the tree-level B-violating decay of
$\chi_{2}$ which—through its interference with the loop-induced decay by the
intermediate state $\chi_{3}$—becomes CP-violating. The colored state $\phi$
can, e.g., be pair-produced through gluons and decay to four jets, and is
currently constrained by the LHC data to be at least of multi-TeV mass. Pair
production of $\chi_{1,2,3}$ would lead to missing energy signals and/or in
the form of displaced vertices. Some of these collider signatures can be
within reach of the High-Luminosity LHC, while future high energy colliders
are more promising for detection. The conventional DM direct detection signal
from this model is strongly suppressed due to the decoupled nature of the dark
sector. Model-specific indirect signals of baryogenesis from DM capture and
annihilation in the Earth and other stellar objects, exotic signatures at
large volume neutrino experiments, and departures of a standard matter power
spectrum warrant further study.

In summary, we identify a new generic mechanism connecting the DM and baryon
abundances. We term it “Dark freeze-out Cogenesis” on account of a separate
thermal dark sector evolution in the early Universe, and that both DM
abundance and baryon asymmetry are seeded during the same freeze-out dynamics
in a dark sector. Our mechanism works over a vast range of dark sector masses.
The price to pay is restrictions on the size of the interactions with the SM
that ensure certain decoupling conditions to hold. This renders direct and
indirect tests more challenging, while these restrictions may be lifted with
future model-building efforts in this direction.

## Acknowledgments

YC and JP thank KITP (supported by National Science Foundation under Grant No.
NSF PHY-1748958) for hospitality. XC is supported by the Austrian Science Fund
FWF under Grant No. FG1. YC is supported in part by the US Department of
Energy under award number DE-SC0008541. MS is supported by TRIUMF who receives
federal funding via a contribution agreement with the National Research
Council of Canada.

##  Appendix A Freeze-out Solution in the Nearly Degenerate Scenario

The Boltzmann equation for the $\chi_{1}$ evolution is determined by
$\chi_{1}$ annihilating into $\chi_{2}$. “Inverse” annihilation of the lighter
$\chi_{2}$ to heavier $\chi_{1}$ is forbidden at zero temperature but proceeds
off the exponential tail of the $\chi_{2}$ velocity distribution. Assuming
only this process for $\chi_{1}$ and $\chi_{2}$, the evolution of $\chi_{1}$
follows

$\dot{n}_{\chi_{1}}+3Hn_{\chi_{1}}=-\langle\sigma_{\chi_{1}}v\rangle
n_{\chi_{1}}^{2}+\langle\sigma_{\chi_{2}}v\rangle n_{\chi_{2}}^{2}.$ (36)

The principle of detailed balance dictates that the right hand side of Eq.
(36) vanishes in equilibrium, which gives the thermally averaged cross section
for $\chi_{2}$ annihilation, $\langle\sigma_{\chi_{2}}v\rangle$, in terms of
$\langle\sigma_{\chi_{1}}v\rangle$,

$\displaystyle\langle\sigma_{\chi_{2}}v\rangle=\bigg{(}\frac{n_{\chi_{1}}^{\text{eq}}}{n_{\chi_{2}}^{\text{eq}}}\bigg{)}^{2}\langle\sigma_{\chi_{1}}v\rangle=\bigg{(}\frac{g_{\chi_{1}}}{g_{\chi_{2}}}\bigg{)}^{2}(1-\delta)^{-3}e^{-2\delta
x^{\prime}}\langle\sigma_{\chi_{1}}v\rangle\,,$ (37)

where $\delta\equiv(m_{\chi_{1}}-m_{\chi_{2}})/m_{\chi_{1}}$ is the
dimensionless mass splitting, $g_{\chi_{1},\chi_{2}}$ are the number of
internal degrees of freedom in $\chi_{1}$/$\chi_{2}$, and
$x^{\prime}={m_{\chi_{1}}}/{T^{\prime}}$. The exponential factor arises from
the equilibrium number density with a vanishing chemical potential,
$n^{\text{eq}}=g\big{(}\frac{mT^{\prime}}{2\pi}\big{)}^{3/2}e^{-m/T^{\prime}}$.
Plugging Eq. (37) into Eq. (36) and replacing the number densities
$n_{\chi_{1,2}}$ with the co-moving densities $Y_{\chi_{1,2}}$ and cosmic time
$t$ in favor of $x=m_{\chi_{1}}/T=x^{\prime}\xi$, the $\chi_{1}$ abundance,
$Y_{\chi_{1}}\equiv n_{\chi_{1}}(T^{\prime})/s(T)$, evolves as

$\frac{dY_{\chi_{1}}}{dx}=-\frac{\lambda\xi^{n}}{x^{2+n}}\bigg{[}Y_{\chi_{1}}^{2}-(1-\delta)^{-3}e^{-2\delta
x^{\prime}}Y_{\chi_{2}}^{2}\bigg{]}\,,$ (38)

with other quantities explained in the main text.

Figure 12: Evolution of dark temperature $T^{\prime}$ in the nearly degenerate
scenario, where the couplings are fixed to
$g_{1}g_{2}/m_{A}^{2}=1/(4\,\text{TeV})^{2}$ and $m_{\chi_{1}}=1$ TeV, while
the left panel takes $\delta=0.1$, $\xi_{i}=0.01$, and the right takes
$\delta=0.01$, $\xi_{i}=0.1$. The red lines give exact values of $T^{\prime}$,
normalized to $m_{\chi_{2}}$, from numerically solving the Boltzmann
equations. In contrast, gray dotted lines give a simple $T^{\prime}\propto T$
scaling from its initial value at $T^{\prime}\gg m_{\chi_{2}}$ , while gray
dashed lines show the values from
$(m_{\chi_{1}}/T^{\prime})=\beta(m_{\chi_{1}}/T)^{2}$ of Eq. (9). It verifies
that Eq. (9), with the scaling $T^{\prime}\propto T^{2}$, works well in the
regime of $T^{\prime}\ll m_{\chi_{2}}$.

At $T^{\prime}\ll m_{\chi_{1}}$, number conservation requires
$Y_{\chi_{1}}+Y_{\chi_{2}}\equiv Y_{\chi_{1},i}+Y_{\chi_{2},i}\approx
0.42g_{\chi}/g_{*S}\xi^{3}_{i}$, if setting initially the temperature ratio
$\xi=\xi_{i}$ at $T^{\prime}\geq m_{\chi_{1}}$. When the dark sector is in
thermal equilibrium, the quasi-static condition, ${dY_{\chi_{1}}}/{dx}\approx
0$, needs to be satisfied for Eq. (38), leading to
$Y_{\chi_{1}}\approx(1-\delta)^{-3/2}e^{-\delta
x^{\prime}}Y_{\chi_{2}}=(1-\delta)^{-3/2}e^{-\delta
x^{\prime}}(Y_{\chi_{1},i}+Y_{\chi_{2},i}-Y_{\chi_{1}})$. In other words, the
quasi-static equilibrium (QSE) solution of $Y_{\chi_{1}}$, defined as $Y_{\rm
QSE}$, is expressed in terms of

$Y_{\chi_{1}}\approx Y_{\rm QSE}\equiv\frac{(1-\delta)^{-3/2}e^{-\delta
x^{\prime}}}{1+(1-\delta)^{-3/2}e^{-\delta
x^{\prime}}}\times(Y_{\chi_{1},i}+Y_{\chi_{2},i})\,.$ (39)

In general, the denominator is approximately one for cold freeze-out, i.e., if
$x^{\prime}_{\text{f.o.}}\gg 1/\delta$. We emphasize that our QSE solution is
significantly different from that of a WIMP freeze-out. In the WIMP case, DM
pair annihilate to final states that remain in chemical equilibrium with the
thermal bath so that the associated Boltzmann equation is directly $dY_{\rm
WIMP}/dx=-\lambda/x^{2+n}[Y_{\rm WIMP}^{2}-(Y^{\rm eq}_{\rm
WIMP}(\mu=0))^{2}]$, leading to a QSE solution $Y_{\rm QSE}=Y^{\rm eq}_{\rm
WIMP}(\mu=0)$, when all relevant processes are sufficient. In our framework,
non-vanishing chemical potentials arise for both dark particles due to the
total number conservation of $\chi_{1}$ and $\chi_{2}$, which is why now its
QSE solution has a dependence on the initial total dark abundance.

An implicit assumption here is that both particles are non-relativistic when
freeze-out happens, i.e., our results below are not valid for $\delta\to 1$.
The latter case is instead studied in the main text in the hierarchical
scenario of Sec. 2.1.

To solve the equation (38) at later time when the annihilation process
gradually becomes out of equilibrium, we recast it in terms of $\Delta\equiv
Y_{\chi_{1}}-Y_{\rm QSE}$ as

$\frac{d\Delta}{dx}\approx-\frac{dY_{\rm
QSE}}{dx}-\frac{\xi^{n}\lambda}{x^{2+n}}\Delta\bigg{[}\Delta+2Y_{\rm
QSE}\bigg{]}\,,$ (40)

neglecting sub-leading terms under the assumption $\delta x^{\prime}\gg 1$.
Note that $\xi=T^{\prime}/T$ is not a constant due to the fact that for a free
non-relativistic particle $T^{\prime}\propto 1/a^{2}$ while $T\propto 1/a$ for
the radiative thermal bath. That is, one can set $\xi=1/(\beta x)$, and thus
$x^{\prime}=\beta x^{2}$ in the non-relativistic limit $T^{\prime}\ll
m_{\chi_{2}}$. As argued in the main text, a reasonable choice for the
coefficient is $\beta=(1-\delta)/(4\times 2^{2/3}\xi_{i}^{2})$, as confirmed
by the numerical results in Fig. 12. Using this, we may write

$\frac{dY_{\rm QSE}}{dx}=-\frac{(Y_{\rm
QSE})^{2}}{Y_{\chi_{1},i}+Y_{\chi_{2},i}}\,\bigg{[}2\delta\beta
x\,(1-\delta)^{3/2}e^{\delta\beta x^{2}}\bigg{]}\,.$ (41)

Following standard procedure, we assume the L.H.S. of Eq. (40) to vanish at
freeze-out $x_{\text{f.o.}}$, and take ${\Delta}(x_{\text{f.o.}})=cY_{\rm
QSE}(x_{\text{f.o.}})$, to solve for $x_{\text{f.o.}}$. As fiducial value, we
choose $c=0.3$ with a rather mild dependence on the variation of that number.
It then follows that at $x=x_{\text{f.o.}}$ we may write,

$\frac{\xi^{n}\lambda}{x^{2+n}}c(2+c)={-\frac{1}{(Y_{\rm
QSE})^{2}}\frac{dY_{\rm
QSE}}{dx}}=\frac{1}{Y_{\chi_{1},i}+Y_{\chi_{2},i}}\,\bigg{[}2\delta\beta
x\,(1-\delta)^{3/2}e^{\delta\beta x^{2}}\bigg{]}\,.$ (42)

Substituting $Y_{\chi_{1},i}+Y_{\chi_{2},i}\approx
0.42g_{\chi}\xi^{3}_{i}/g_{*S}$ and $\xi=1/(\beta x)$, the exponential above
is expressed as

$e^{\delta\beta
x_{\text{f.o.}}^{2}}=\frac{\lambda}{x^{2+2n}\beta^{n}}c(2+c)\,{0.42g_{\chi}\xi^{3}_{i}\over
g_{*S}}{1\over 2\delta\beta
x\,(1-\delta)^{3/2}}=\frac{0.21c(2+c)\lambda\xi^{3}_{i}\beta^{1/2}\delta^{1/2}}{(1-\delta)^{3/2}}\,{g_{\chi}\over
g_{*S}}\left({1\over\delta\beta x_{\text{f.o.}}^{2}}\right)^{n+3/2}\,,$

where the L.H.S. increases and the R.H.S. decreases with growing
$x_{\text{f.o.}}$, so that there is a unique solution to $x_{\text{f.o.}}$.
Taking the logarithm of both sides, and iteratively replacing
$x_{\text{f.o.}}$ on the R.H.S. give us

$\delta\beta
x_{\text{f.o.}}^{2}=\ln\left[\frac{0.21c(2+c)\lambda\xi^{3}_{i}\beta^{1/2}\delta^{1/2}}{(1-\delta)^{3/2}}\,{g_{\chi}\over
g_{*S}}\right]-\left(n+{3\over
2}\right)\ln\left\\{\ln\left[\frac{0.21c(2+c)\lambda\xi^{3}_{i}\beta^{1/2}\delta^{1/2}}{(1-\delta)^{3/2}}\,{g_{\chi}\over
g_{*S}}\right]\right\\}\,,$ (43)

assuming the first term dominates, to yield the observed DM relic abundance.

Figure 13: The abundance evolution of $\chi_{1}$ and $\chi_{2}$ for
$m_{\chi_{1}}=1$ TeV and $g_{1}g_{2}/m_{A}^{2}=1/(4\,\text{TeV})^{2}$ as a
function of $x$ (left) and $x^{\prime}$ (right), same as Fig. 2, except that
the approximation lines, $e^{-\delta x^{\prime}}Y_{\rm tot}$, are omitted
here. The additional parameters are modified to $\delta=0.01$, $\xi_{i}=0.1$
(top), $\delta=0.01$, $\xi_{i}=0.01$ (middle), and $\delta=0.1$,
$\xi_{i}=0.01$ (bottom).

Finally, under the assumption that $Y_{\rm QSE}(x_{\text{f.o.}})\gg
Y_{\chi_{1},f}\gg Y_{\rm QSE}|_{x\to\infty}$, we obtain

$\frac{d\Delta}{dx}\approx-\frac{\xi^{n}\lambda}{x^{n+2}}\Delta^{2}=-\frac{\lambda}{\beta^{n}x^{2n+2}}\Delta^{2}\,,$
(44)

and integrating it over $x$ from $x_{\text{f.o.}}$ to $\infty$ results in

$Y_{\chi_{1},f}\approx\Delta|_{x\to\infty}\approx\frac{(2n+1)\beta^{n}}{\lambda}x_{\text{f.o.}}^{2n+1}=\frac{(2n+1)}{\lambda\beta^{1/2}\delta^{n+1/2}}\left(\delta\beta
x_{\text{f.o.}}^{2}\right)^{n+1/2}\,,$ (45)

where the last term is obtained from Eq. (43). We may verify our analytical
results above by comparing to the numerical results shown in Fig. 13, where
one finds good agreement. As expected, the evolution of $T^{\prime}$ is
insensitive to the detailed processes. This is because it is essentially
governed by the dominant number abundance $Y_{\chi_{2}}$, and, thus, by the
lighter state $m_{\chi_{2}}$ and $\xi_{i}$.

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