HRI-RECAPP-2021-012

# Self-interacting freeze-in dark matter in a singlet doublet scenario

Purusottam Ghosh  Partha Konar  Abhijit Kumar Saha  and Sudipta Show

###### Abstract

We examine the non-thermal production of dark matter in a scalar extended
singlet doublet fermion model where the lightest admixture of the fermions
constitutes a suitable dark matter candidate. The dark sector is non-minimal
with the MeV scale singlet scalar, which is stable in the Universe lifetime
and can mediate the self-interaction for the multi-GeV fermion dark matter
mitigating the small scale structure anomalies of the Universe. If the dark
sector is strongly coupled, it undergoes internal dark thermal equilibrium
after freeze-in production, and we end up with suppressed relic abundance for
the fermion dark matter in a radiation dominated Universe. In contrast, the
presence of a modified cosmological phase in the early era drives the fermion
dark matter to satisfy nearly the whole amount of observed relic. It also
turns out that the assumption of an unconventional cosmological history can
allow the GeV scale dark matter to be probed at LHC from displaced vertex
signature with improved sensitivity.

##  1 Introduction

In the sea of apparent voidness at experimental fronts, some remarkable signs
of progress have recently been made in dark matter theory, with new elegant
ideas concerning dark matter production mechanisms and the nature of its
interaction. One of the exciting proposals is the non-thermal origin of dark
matter production known as an freeze-in [1]. The freeze-in process holds under
the assumption that the dark matter is feebly coupled to the visible sector
and initially having zero number density. At a later stage, it can be produced
non thermally from decay or scattering of visible sector particles.

On another front, the collisionless cold dark matter paradigm suffers from
small scale structure anomalies such as cusp vs core [2, 3, 4] and the too-
big-to-fail problems [5, 6, 4] etc. As a resolution it has been proposed that
the self-interaction [7, 8, 9, 10, 11, 12], a form of non-gravitational
interactions of Dark Matter (DM), is capable of addressing the problems
associated with the small scale structure. Tight constraints on dark matter
self-interactions come in terms of the ratio of such self-scattering cross
section over DM mass from observations at Bullet Cluster and N-body
simulations [13, 4].

Studies related to self-interacting dark matter in a typical WIMP (weakly
interacting massive particle) paradigm are prevalent in literature (see, for
example, [14, 15, 16, 17, 18] for some of the recent works). On the other
hand, proposals for self-interacting dark matter with freeze-in production are
somewhat limited and can be found in refs. [19, 20]. In ref. [20], the dark
matter is fermionic in nature, and it self-interacts via a light scalar
mediator. Similar analysis has been exercised in ref. [19], considering a
vector dark matter candidate. Irrespective of the production mechanisms, a
strong dark matter self-interaction craves for a light mediator with mass
ranging from keV to MeV scale. Another notable aspect is the dark sector
thermalisation [20] due to the strong interaction between the dark matter and
the light mediator. This reduces the relic abundance of DM after its freeze-in
production. To avoid that, restriction on the mediator mass turns severe and
lowers it towards $\mathcal{O}(1)$ KeV.

In this paper, we attempt to offer a freeze-in production [1] scenario of GeV
scale vector fermion DM that carries a bright prospect of testability at
collider experiments and is capable of alleviating small scale structure
problems by undergoing sufficient self-interaction. In view of this, we adopt
the singlet doublet framework where the dark matter ($\chi$) turns out to be
the lightest admixture of the singlet and doublet [21, 22, 23, 24, 25, 26, 27,
28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46,
47, 48] 111Earlier works [49, 50] on singlet doublet freeze-in dark matter
reveal that the dark matter is mostly singlet dominated with vanishing mixing
angle with the doublet.. We extend the set-up by adding one gauge singlet real
scalar ($\phi$) field. The new scalar mediates the self-interaction for the
fermion dark matter candidate. Additionally, it has no decay mode and is hence
stable in Universe lifetime. The singlet doublet dark matter model has rich
collider signatures as explored in [26, 49, 50]. The heavier charged fermion
or neutral partner is part of the additional $SU(2)_{L}$ doublet fermion.
Their copious production at colliders is expected due to the $SU(2)_{L}$ gauge
mediated interactions. The long-lived charged particle could give rise to
disappearing charge tracks, whereas the long-lived heavier neutral particle
can be traced through searching displaced vertices at LHC.

We focus on working in a scenario where the GeV scale fermion dark matter
predominantly contribute to the total relic abundance. Stronger interaction
strength between the fermionic dark matter and the mediator scalar is desired
for producing large self-interaction [14, 15, 51, 19, 20]. On the other hand,
a strongly coupled dark sector generally leads to the formation of internal
dark thermal equilibrium with a separate dark sector temperature $T_{D}$ [20,
52, 53, 54, 55, 56]. In our present scenario, after the freeze-in production
of $\chi$, they start annihilating into a pair of $\phi$ and vice versa,
maintaining the dark sector equilibrium before a freeze-out mechanism triggers
for $\chi$. In literature, such rich dynamics inside a hidden dark sector is
familiarly known as the reannihilation effect. Considering radiation dominated
Universe (RD), we find that after the dark sector freezes out, $\phi$ shares a
sizeable contribution to the total relic abundance. Thus, a $\chi$ dominated
scenario is very unlikely.

Since our aim is to provide a $\chi$ dominated scenario which may also give
rise to velocity-dependent fermionic self-interacting dark matter (see [20]
for a similar exercise in a different framework), the proposed set up as
discussed above in the radiation dominated Universe does not work. In the
present study, we apprehend that the presence of a non-standard epoch in the
early Universe could be useful to alleviate this. A faster expansion of the
Universe can prevent the dark sector from reaching thermal equilibrium or slow
down the dark sector interaction rate. These assist in suppressing the
$\chi\chi\to\phi\phi$ conversion rate to obtain a $\chi$ dominated dark matter
scenario. We have employed two particular kinds of modified cosmology [57, 45,
47] namely kination domination and faster than kination domination. We have
found that such non-standard epochs in the early Universe can work remarkably
to reach our defined aim of obtaining $\Omega_{\chi}h^{2}\sim 0.12$ [58] which
has the potential to resolve the cosmological problems as earlier stated.
Additionally, the presence of a non-standard epoch helps to provide a
complementary probe of the GeV scale dark matter in the singlet doublet
framework via displaced vertices which is not possible in a radiation
dominated scenario as shown in [49, 50].

The paper is organised as follows. In section 2, we furnish the model
structure and the corresponding Lagrangians. Next, in section 3, we discuss
the dark matter phenomenology considering the standard cosmology and two
different forms of non-standard cosmology. We describe the discovery prospects
of the proposed model at collider experiments in section 4. Estimate of the
self-interaction cross section for the singlet doublet dark matter are
presented in section 5.Finally, we summarise by pointing out the new findings
of our analysis and draw the conclusion in section 6.

##  2 The Model

We extend the SM particle content by one $SU(2)_{L}$ doublet fermion ($\Psi$)
with hypercharge $Y=-\frac{1}{2}$ and one hyperchargeless gauge singlet vector
fermion ($\chi$). In addition, we also include a real scalar singlet field
($\phi$) which assists the DM to yield strong dark matter self-interaction. We
impose a discrete $Z_{2}$ symmetry under which the SM particles and the $\phi$
field transform trivially while the BSM fermions are assigned odd $Z_{2}$
charges. The Lagrangian of the scalar sector is read as

$\displaystyle\mathcal{L}_{\rm
scalar}=|D^{\mu}H|^{2}+\frac{1}{2}(\partial^{\mu}\phi)(\partial_{\mu}\phi)-V(H,\phi),$
(2.1)

where the covariant derivative is defined as,

$\displaystyle
D^{\mu}=\partial^{\mu}-ig\frac{\sigma^{a}}{2}W^{a\mu}-ig^{\prime}\frac{Y}{2}B^{\mu},$
(2.2)

with $g$ and $g^{\prime}$ being the $SU(2)_{L}$ and the $U(1)_{Y}$ gauge
couplings respectively. The scalar potential $V=V(H)+V(\phi)+V(\phi,H)$ takes
the following form,

$\displaystyle
V(H)=-{\mu_{H}^{2}}\,(H^{\dagger}H)+\lambda_{H}\,(H^{\dagger}H)^{2},$ (2.3)
$\displaystyle
V(\phi)=\frac{m_{\phi}^{2}}{2}\phi^{2}+\frac{\lambda_{\phi}}{4!}\phi^{4}+\frac{b_{3}}{3!}\phi^{3},$
(2.4) $\displaystyle V(\phi,H)=\frac{\lambda_{\phi
H}}{2}\phi^{2}(H^{\dagger}H)+a_{3}~{}\phi(H^{\dagger}H).$ (2.5)

Note that we do not write the liner term for $\phi$ in $V(\phi)$ since it can
always be absorbed through the redefinition of the other parameters. We assume
that all the mass scales and the coupling coefficients are real and positive.
In this limit, the vacuum expectation values (vev) of the scalars $H$ and
$\phi$ after minimising the potential $V$ are obtained as,

$\displaystyle\langle H\rangle=v~{},~{}~{}~{}\langle\phi\rangle=0.$ (2.6)

The Lagrangian for the fermionic sector is written as,

$\displaystyle\mathcal{L}=\mathcal{L}_{f}+\mathcal{L}_{Y},$ (2.7)

where,

$\displaystyle\mathcal{L}_{f}=$
$\displaystyle\;i\overline{\Psi}\gamma_{\mu}D^{\mu}\Psi+i\overline{\chi}\gamma_{\mu}\partial^{\mu}\chi-
m_{\Psi}\overline{\Psi}\Psi-m_{\chi}\overline{\chi}\chi$ (2.8)
$\displaystyle\mathcal{L}_{Y}=$
$\displaystyle~{}-Y\overline{\Psi}\tilde{H}\chi+h.c.-\lambda\phi\overline{\chi}\chi-\delta\phi\overline{\Psi}\Psi,$
(2.9)

where we have defined $\Psi^{T}=(\psi^{+}~{}~{}\psi^{0})$.

The Dirac mass matrix for the neutral fermions after the spontaneous breakdown
of the electroweak symmetry is obtained as,

$\displaystyle\mathcal{M}_{D}=\begin{pmatrix}m_{\Psi}&M_{D}\\\
M_{D}&m_{\chi}\end{pmatrix},$ (2.10)

where we define $M_{D}=\frac{Yv}{\sqrt{2}}$. After diagonalisation of
Eq.(2.10), we are left with two neutral Dirac particles which we identify as
$\xi_{1}$ and $\xi_{2}$. The mass eigenvalues of $\xi_{1}$ and $\xi_{2}$ are
given by,

$\displaystyle m_{\xi_{1}}\approx
m_{\chi}-\frac{M_{D}^{2}}{m_{\Psi}-m_{\chi}}$ (2.11) $\displaystyle
m_{\xi_{2}}\approx m_{\Psi}+\frac{M_{D}^{2}}{m_{\Psi}-m_{\chi}}$ (2.12)

Therefore, the lightest eigenstate is $\xi_{1}$, which is the stable DM
candidate of our framework. The stability of the DM is ensured by the unbroken
$\mathcal{Z}_{2}$ symmetry. The mixing between two flavor states, i.e. neutral
part of the doublet ($\psi^{0}$) and the singlet field ($\chi$) is
parameterised by,

$\displaystyle\sin 2\theta\simeq\frac{2Yv}{\Delta M},$ (2.13)

where $\Delta M=m_{\xi_{2}}-m_{\xi_{1}}\approx m_{\Psi}-m_{\chi}$ in the small
$Y$ limit. Here, $\xi_{1}$ can be identified with the singlet $\chi$. Since
the present analysis involves freeze-in dark matter production, the mixing
between the singlet and doublet fermions is always very tiny. Therefore,
henceforward we identify the dark matter candidate as $\chi$.

##  3 Dark matter relic

The present set-up has two stable particles $\phi$ and $\chi$, and both can
contribute to total DM relic abundance. The relative contribution of each
component to the DM relic is dependent on early Universe history, as we
discuss below. We have worked with $m_{\phi}\ll m_{\chi}$ and
$\lambda\sim\mathcal{O}(10^{-1})$ with an aim to obtain a large amount of
self-interaction for $\chi$ that can address the problems associated with
small scale structures of the Universe. Irrespective of the cosmological
history of the Universe, we assume $a_{3},\lambda_{\phi H},\delta,Y\ll 1$
always such that both $\phi$ and $\chi$ can never equilibrate with the SM
bath. Additionally, we have considered a specific choice
$a_{3},\delta,\lambda_{\phi H}\ll Y$ which implies $\phi$ can only be
populated at a non-negligible rate (depending on the expansion rate of the
Universe) from the $\chi\chi\rightarrow\phi\phi$ process.

###  3.1 Radiation dominated Universe

In the standard description of Big bang cosmology, the Universe is radiation
dominated prior to BBN. Initially both the components begin with zero number
density and first $\chi$ gets produced from the SM bath. To be specific,
production of $\chi$ at early Universe occurs primarily from $\Psi$ decay with
thermally averaged decay width $\langle\Gamma_{\Psi}\rangle^{T}$ where $T$ is
the SM temperature. Before EW symmetry breaking, the DM production takes place
only from the $Y\Psi\tilde{H}\chi$ vertex. After EW symmetry breaking, the
production channels for DM are namely, $\psi^{\pm}\rightarrow\chi W^{\pm}$,
$\xi_{2}\rightarrow\chi Z$ and $\xi_{2}\rightarrow\chi h$. In order to express
the $\chi$ production rate as function of temperature, we define
$\langle\Gamma_{\Psi}\rangle^{T}=\langle\Gamma_{\Psi}\rangle~{}\Theta(T-T_{\rm
EW})+\langle\Gamma_{\Psi^{\pm}}+\Gamma_{\xi_{2}}\rangle~{}\Theta(T_{\rm
EW}-T)$ where $T_{\rm EW}\sim 160$ GeV indicates the EW symmetry breaking
temperature. Subsequently, the $\chi\chi\rightarrow\phi\phi$ process yields
$\phi$ and when the number density of $\phi$ is sufficient, it can further
annihilate to $\chi$ and form a local dark sector thermal equilibrium with an
uniform temperature $T_{D}$ provided

$\displaystyle r\equiv\frac{n_{\chi}\langle\sigma
v\rangle_{\chi\chi\to\phi\phi}}{\mathcal{H}}\gg 1.$ (3.1)

Finally $\chi$ freezes out and both contribute non-negligibly to the total
relic abundance. The set of Boltzman equations that governs such dynamics with
$z(=\frac{m_{\chi}}{T})$ are given by [20, 52, 53, 54, 55, 56],

$\displaystyle\frac{dY_{\phi}}{dz}=\frac{s}{\mathcal{H}z}\langle\sigma
v\rangle_{\chi\chi\rightarrow\phi\phi}^{T_{D}}\left[Y_{\chi}^{2}-\left(\frac{Y_{\chi}^{\rm
eq}(T_{D})}{Y_{\phi}^{\rm eq}(T_{D})}\right)^{2}Y_{\phi}^{2}\right]$ (3.2)
$\displaystyle\frac{dY_{\chi}}{dz}=-\frac{s}{\mathcal{H}z}\langle\sigma
v\rangle_{\chi\chi\rightarrow\phi\phi}^{T_{D}}\left[Y_{\chi}^{2}-\left(\frac{Y_{\chi}^{\rm
eq}(T_{D})}{Y_{\phi}^{\rm
eq}(T_{D})}\right)^{2}Y_{\phi}^{2}\right]+\frac{45z^{2}}{2\pi^{2}m_{\chi}^{3}}\frac{s{\langle\Gamma_{\Psi}\rangle}^{T}}{\mathcal{H}}\big{(}Y_{\Psi}^{\rm
eq}(T)-Y_{\chi}\big{)}.$ (3.3)

| $\lambda$ | $m_{\Psi}$ | $m_{\chi}$ | $m_{\phi}$ | $Y$ | $\Omega_{\chi}h^{2}$ | $\Omega_{\phi}h^{2}$
---|---|---|---|---|---|---|---
RD-I | 0.13 | 1.5 TeV | 20 GeV | 10 MeV | $5.40\times 10^{-10}$ | 0.01 | $0.11$
RD-II | 0.085 | 1.5 TeV | 10 GeV | 10 MeV | $5.45\times 10^{-10}$ | 0.01 | 0.11
Table 1: Two representative benchmark points that describe the reannihilation
scenario in the present framework considering radiation dominated Universe.

To evaluate the dark temperature $(T_{D}$) one needs to solve the Boltzman
equation for the total dark sector energy density
($\rho_{D}=\rho_{\chi}+\rho_{\phi}$) where $\rho_{i}$ indicates the energy
density for an individual component [20, 52, 53, 56, 51, 59, 60].

$\displaystyle\frac{d\rho_{D}}{dt}+3H(\rho_{D}+p_{D})$
$\displaystyle=\mathcal{P}_{\Psi}n_{\Psi}^{\rm eq},$ (3.4)

where the quantity $\mathcal{P}_{\Psi}$ represents the thermally averaged
energy transfer rate from the SM to dark sector. The notation
$p_{D}=p_{\chi}+p_{\phi}$ stands for the sum of the pressures for individual
components. The standard form of $p_{D}$ is given by,

$\displaystyle
p_{D}=\frac{g_{\chi}}{6\pi^{2}}\int_{m_{\chi}}^{\infty}\frac{(E_{\chi}^{2}-m_{\chi}^{2})^{3/2}}{e^{E_{\chi}/T_{D}}+1}dE_{\chi}+\frac{g_{\chi}}{6\pi^{2}}\int_{m_{\phi}}^{\infty}\frac{(E_{\phi}^{2}-m_{\phi}^{2})^{3/2}}{e^{E_{\phi}/T_{D}}+1}dE_{\phi},$
(3.5)

The symbols $g_{\chi}$ and $g_{\phi}$ imply the internal degrees of freedom
for $\chi$ and $\phi$ species. Additionally, one can write the energy density
of the dark sector as,

$\displaystyle\rho_{D}=\frac{g_{\chi}}{2\pi^{2}}\int\frac{(E_{\chi}^{2}-m_{\chi}^{2})^{1/2}E_{\chi}^{2}dE_{\chi}}{e^{\frac{E_{\chi}}{T_{D}}}+1}+\frac{g_{\phi}}{2\pi^{2}}\int\frac{(E_{\phi}^{2}-m_{\phi}^{2})^{1/2}E_{\phi}^{2}dE_{\phi}}{e^{\frac{E_{\phi}}{T_{D}}}+1}.$
(3.6)

Analytical computation of the integrations in RHS of Eqs.(3.5-3.6) in presence
of Fermi-Dirac distribution function looks complicated. Instead one can
compute the same in the Maxwell Boltzman-approximation. We find 222We verified
a close match between the numerical estimation from the standard expression
with this approximation for our calculation.,

$\displaystyle
p_{D}=\frac{g_{\chi}}{2\pi^{2}}m_{\chi}^{2}T_{D}^{2}K_{2}\left(\frac{m_{\chi}}{T_{D}}\right)+\frac{g_{\phi}}{2\pi^{2}}m_{\phi}^{2}T_{D}^{2}K_{2}\left(\frac{m_{\phi}}{T_{D}}\right)$
(3.7)
$\displaystyle\rho_{D}=\frac{g_{\chi}m_{\chi}^{2}}{2\pi^{2}}T_{D}\left[m_{\chi}K_{1}\left(\frac{m_{\chi}}{T_{D}}\right)+3T_{D}K_{2}\left(\frac{m_{\chi}}{T_{D}}\right)\right]+\frac{g_{\phi}m_{\phi}^{2}}{2\pi^{2}}T_{D}\left[m_{\phi}K_{1}\left(\frac{m_{\phi}}{T_{D}}\right)+3T_{D}K_{2}\left(\frac{m_{\phi}}{T_{D}}\right)\right].$
(3.8)

Replacing Eq.(3.7) and Eq.(3.8) in Eq.( 3.4), it is simple obtain the
evolution equation for $T_{D}$ as function of SM temperature.

Figure 1: The left plot shows the variation of dark sector temperature $T_{D}$
as a function of inverse temperature $z$. Corresponding evolutions for the
number densities of dark sector particles are shown in the right plot. Here,
after initial non-thermal production of $\chi$ strong self-interaction
generates a dark sector thermal equilibrium. Further, freeze-out produces the
$\phi$ abundance. The figures are computed considering the benchmark point RD-
II in Table 1. Figure 2: The ratio of the interaction rate among the dark
sector particles and Hubble parameter of the Universe following Eq.(3.1) as
function of temperature for for the benchmark point RD-II in Table 1.

In Table 1, we provide two reference points that instigate dark freeze out of
$\chi$ after the non-thermal production from the SM bath for two choices of
dark matter masses having magnitude 10 GeV and 20 GeV, respectively. The
evolutions of dark sector temperature and $\chi$ and $\phi$ abundances for the
benchmark point RD-II (with DM mass 10 GeV) are shown in Figure 1. To generate
these figures, we solve the set of coupled Boltzmann equations as described in
Eqs.(3.3-3.6) and also used Eqs.(3.7-3.8) with the initial conditions
$(Y_{\chi},~{}Y_{\phi},~{}T_{D})=0$ and $Y_{\Psi}=Y_{\Psi}^{\rm eq}(T)$. In
Figure 2, we display the ratio $r$ as defined in Eq.(3.1) to measure the
strength of dark sector interaction rate as a function of SM temperature for
the same benchmark point. This figure confirms the formation of dark sector
equilibrium after the non-thermal production of both the components with $r\gg
1$ for a finite period. In the left of Figure 1, we see the evolution pattern
of the dark temperature as a function of SM bath temperature. Since the dark
matter annihilation to SM particles is negligible, it turns out that $T_{D}\ll
T$ always. Initially, the dark temperature increases from zero due to constant
entropy injection from the SM bath and reaches a maximum value. Then, due to
the expansion of the Universe, it keeps decreasing. Till $T\sim m_{\Psi}$, the
production of $\chi$ continues from $\Psi$ decay, and hence we see a slower
decreasing rate of $T_{D}$ initially, and after that, it becomes steep.

The most intriguing part is, here, due to large $\lambda$, the $\chi$ keeps
annihilating to $\phi$ till the decoupling and ends with suppressed abundance
(see right of Figure 1). Hence for the chosen benchmark points in the
reannihilating scenario, we notice $\Omega_{\chi}\ll\Omega_{\phi}$. An
enhancement of $\lambda$ for a fixed $m_{\chi}$ may slightly increase the
relic of $\phi$ further, while it will be reduced further in the opposite
limit. However, a small $\lambda$ is not desirable in view of generating
sufficient self-interaction of $\chi$ to solve the small scale structure
problems of the Universe. It is to note that for both the reference points in
Table 1, the total relic abundance
$\Omega_{T}h^{2}=\Omega_{\phi}h^{2}+\Omega_{\chi}h^{2}\sim 0.12$ [58] remains
within the observed limit by Planck requiring a similar order of $Y$ value.

Thus, radiation dominated Universe with a large $\lambda$ fails to provide a
pure $\chi$ dominated scenario due to the late time annihilation of the dark
matter $\chi$ to mediator $\phi$. We anticipate that a non-standard Universe
in the form of kination or faster than kination domination could enforce
significant suppression of $\phi$ production rate from
$\chi\chi\rightarrow\phi\phi$ process even in the presence of large $\lambda$.
This may occur if the ratio in Eq.(3.1) remains comparatively suppressed till
late time. Note that such faster expansion of the Universe may also slow down
the production rate of $\chi$ from the SM bath. However, that can be increased
by adjusting $Y$.

###  3.2 Non-standard Universe

In standard description of cosmology, we generally assume that the Universe is
radiation dominated in the pre big bang nucleosynthesis (BBN) era. However,
due to lack of evidences, possibilities remain open that before BBN, the
Universe could have been occupied by a nonstandard fluid, redshifting faster
or slower than the radiation component. Earlier works have emphasised that
consideration of such modified cosmology poses non-trivial impact on the dark
matter phenomenlogy.

A non-standard Universe can be sketched by assuming the presence of an
additional species ($\eta$) along with the radiation component in the
Universe. We consider the equation of parameter ($\omega$) of the nonstandard
fluid is larger than that of radiation. We parameterize this by
$\rho_{\eta}\propto a^{-3(1+\omega)}$ which can be converted to
$\rho_{\eta}\propto a^{-(4+n)}$ with $\omega=\frac{1}{3}(n+1)$ and $n>0$. The
modified description of the Universe leads to the redefinition of the Hubble
parameter ($\mathcal{H}$) as given by [57, 45, 47],

$\displaystyle\mathcal{H}^{2}=\frac{\rho_{R}+\rho_{\eta}}{3M_{P}^{2}},$ (3.9)

where $M_{P}$ refers the reduced Planck scale, $\rho_{R}$ and $\rho_{\eta}$
correspond to the energy densities of radiation and $\eta$ component. The
total energy density of the Universe in presence of $\eta$ as function of
temperature can be expressed as,

$\displaystyle\rho(T)$ $\displaystyle=\rho_{\rm rad}(T)+\rho_{\eta}(T)$ (3.10)
$\displaystyle=\rho_{\rm
rad}(T)\left[1+\frac{g_{*}(T_{r})}{g_{*}(T)}\left(\frac{g_{*s}(T)}{g_{*s}(T_{r})}\right)^{(4+n)/3}\left(\frac{T}{T_{r}}\right)^{n}\right],$
(3.11)

where $\rho_{R}=\frac{\pi^{2}}{30}g_{*}(T)T^{4}$ with $g_{*}$ stands for the
number of relativistic degrees of freedom. The relativistic entropy degrees of
freedom is denoted by $g_{*s}$. Thus a faster expanding Universe is simply
parameterised by the set of $(n,T_{r})$. A larger $n$ or smaller $T_{r}$
prompts the energy density of the Universe to redshift more faster. The
temperature $T_{r}$ implies the end of modified expansion rate and we get back
the radiation dominated phase. In case of standard radiation dominated
Universe, $\eta$ field would be absent and we simply consider $\rho=\rho_{\rm
rad}$. The BBN observation on the number of relativistic degrees of freedom
imposes a lower bound on $T_{r}~{}(\gtrsim(15.4)^{1/n}$ MeV). A special case
$n=2$ (or $\omega=1)$ is familiar as kination domination phase. For $n>2$, one
has to consider scenarios faster than quintessence with negative potential. We
refer the readers to [57], for a detailed description of a fast-expanding
Universe. Below we separately discuss the cases for kination domination and
faster than kination domination.

| $\lambda$ | $m_{\Psi}$ | $m_{\chi}$ | $m_{\phi}$ | $Y$ | $n$ | $T_{r}$ | $\Omega_{\chi}h^{2}$
---|---|---|---|---|---|---|---|---
FKD-I | 0.13 | 1.5 TeV | 20 GeV | 10 MeV | $2\times 10^{-8}$ | 3 | 20 MeV | 0.119
FKD-II | 0.085 | 1.5 TeV | 10 GeV | 10 MeV | $2.44\times 10^{-8}$ | 3 | 22 MeV | 0.119
Table 2: Two representative benchmark points that demonstrate pure freeze-in
non-standard cosmology with faster than kination domination.

####  3.2.1 Faster than kination phase

As earlier mentioned a faster than kination phase is realised for $n>2$. Here
we consider $n=3$. A faster expansion implies enhanced Hubble rate which can
suppress the interaction rate among the dark sector particles. If the
interaction rate among the dark sector particles gets heavily suppressed, that
leads to the production of $\chi$ by pure freeze-in. In that case the
governing Boltzman equation for $\chi$ looks very simple as given by [1, 61],

$\displaystyle\frac{dY_{\chi}}{dz}=\frac{45z^{2}}{2\pi^{2}m_{\chi}^{3}}\frac{s{\langle\Gamma_{\Psi}\rangle}^{T}}{\mathcal{H}}\big{(}Y_{\Psi}^{\rm
eq}(T)-Y_{\chi}\big{)}.$ (3.12)

Since the rate of the conversion process $\chi\chi\to\phi\phi$ is negligibly
low in case of pure freeze-in, we do not estimate the relic of $\phi$ here,
which is expected to be barely abundant in the present Universe. For the same
reason, the computation of dark temperature is not essential here.

Figure 3: Left: The ratio of the interaction rate among the dark sector
particles and Hubble paramter of the Universe as function of temperature for
the benchmark point FKD-II in Table 2. Right: The evolution of DM comoving
number density is shown with temperature for the same reference point.

In Table 2, we note down two benchmark points with BSM model inputs
$\\{\lambda,m_{\Psi},m_{\chi},m_{\phi}\\}$ are same as used in section 3.1. We
have considered zero initial abundance of $\chi$ since it remains out of
equilibrium in the early Universe. We fix $n=3$ for this case and vary the
$T_{r}$ and $Y$ in order to obtain correct relic abundance (by pure freeze-in)
as allowed by Planck. Recall that these two reference points have portrayed
reannihilation patterns with $\Omega_{\chi}h^{2}\ll\Omega_{\phi}h^{2}$ for a
radiation dominated Universe. In the right of Figure 3, we have depicted the
evolution of comoving abundance of $\chi$ with temperature considering the
reference point FKD-II. We have used $n=3$ and fixed the parameters $T_{r}$
and $Y$ such that correct relic abundance for $\chi$ is attained. In left of
Figure 3, we estimate the parameter $r$ as earlier defined and found it to be
order of $\sim\mathcal{O}(10^{-2})$. This indeed ensures the validity of
labelling the present scenario as pure freeze-in. Here we do not mention the
abundance of the $\phi$ field. Owing to the faster expansion of the Universe,
the produced $\chi$ never creates its own bath, and it is expected that the
contribution of $\chi\chi\rightarrow\phi\phi$ would be negligible in yielding
$\phi$. Nevertheless, a conservative test can always be performed by solving
the Boltzmann equations for $Y_{\phi}$ and $Y_{\chi}$ while considering the
maximum 333Maximum possible value of a thermally averaged cross section can be
found by equating the temperature of the corresponding bath with the heavier
mass scale approximately associated with the interaction. possible value of
$\langle\sigma v\rangle_{\chi\chi_{\rightarrow}\phi\phi}$. By utilising this
conservative method, we have found similar results (maximum uncertainty is
2$\%$) with $Y_{\phi}$ remaining much lower than $Y_{\chi}$. We also notice
from Table 2, that the required order of the Yukawa coupling is relatively
larger than the one shown in section 3.1 considering standard RD Universe. The
reason is obvious as a faster expanding Universe not only suppresses the rate
of $\chi\chi\rightarrow\phi\phi$ process but also slows down the production
process of $\chi$ and therefore, one needs to raise BSM Yukawa coupling
parameter $Y$ appropriately. To investigate the $T_{r}$ dependence on $Y$
further, in Figure 4, we show the relic density satisfied contours in the
$T_{r}-Y$ plane for $m_{\chi}=10$ GeV and 20 GeV considering $n=3$. We keep
the corresponding values of $\lambda$ and $m_{\Psi}$ in accordance with Table
2. For a fixed DM mass, a smaller $T_{r}$ requires larger $Y$. This occurs
since a smaller $T_{r}$ implies an enhanced Hubble rate, and it requires a
larger $Y$ to obey the relic abundance bound. Moreover, a smaller DM mass also
requires larger $Y$ to obey the relic bound. It is to note that in Figure 4,
we have kept $T_{r}$ below 25 MeV. Beyond this value of $T_{r}$, the ratio $r$
would turn larger than $\mathcal{O}(10)^{-2}$ and therefore, the contribution
of $\chi\chi\to\phi\phi$ conversion in the final DM relic may turn important
at some extent.

In summary, the above discussion reveals that a faster than kination dominated
early Universe ($n=3$) is able to provide a pure freeze-in yield for $\chi$
with absolutely dominant share to the total relic abundance even in the
presence of $\lambda\sim\mathcal{O}(10^{-1})$. This is in sharp contrast to
the RD Universe where we found $\Omega_{\chi}h^{2}\ll\Omega_{\phi}h^{2}$ with
the same values of $\lambda$ and mass scales of the dark sector particles. In
the upcoming section, we discuss how the scenario evolves in the case of
kination dominated ($n=2$) early Universe.

Figure 4: Relic satisfied contours in $T_{r}-Y$ plane considering $n=3$ for
$m_{\chi}=10$ GeV and 20 GeV.

####  3.2.2 Kination phase

Earlier, we have seen, faster than kination domination has led to pure freeze-
in for the parameters that show reannihilation in the case of an RD Universe
by suppressing the interaction rate among dark sector particles. Now, we
consider the kination domination case ($n=2$), which leads to a relatively
slower expansion rate of the Universe than $n=3$ and hence may enhance the
interaction rate inside the dark sector. Thus to realise pure freeze-in with
kination domination era, a relatively smaller value of $Y$ for a fixed $T_{r}$
is expected to obey the relic bound compared to the $n=3$ case. Indeed such
prediction emerges to be correct as evident from Figure 5. Using the
approximate method as commented earlier, we have found that for the respective
ranges for $Y$ and $T_{r}$ in Figure 5, the ratio $r$ always remains
$\lesssim\mathcal{O}(10^{-2})$ and thus relic $\phi$ is always suppressed with
an uncertainity $5\%$ atmost.

Figure 5: Relic satisfied contours in $T_{r}-Y$ plane for $m_{\chi}=10$ GeV
and 20 GeV considering $n=2$.

One may also wonder whether a reannihilating scenario for the dark matter
$\chi$ with final correct relic abundance is possible in non-standard
cosmological models. To investigate this, let us stick to the $n=2$ case. This
would obviously require a larger $T_{r}$ compared to the pure freeze-in case
with $n=2$. Assuming internal equilibrium of dark sector is established for
some period in the early Universe, the set of Boltzman equations remain same
as in Eq.(3.2-3.4). We would like to reuse the same set of benchmark values
for ($\lambda,m_{\Psi},m_{\chi},m_{\phi}$) as originally introduced in section
3.1. We would like to emphasise that, these points depict a reannihilation
pattern in the radiation dominated Universe, while the presence of a kination
or faster than kination dominated epoch transform it to pure freeze-in with
almost full $\chi$ occupancy in the total relic abundance provided suitable
choices for $T_{r}$ are made. Here, we find out the estimates of $(Y,T_{r})$
such that it shows a reannihilation pattern with
$\Omega_{\chi}h^{2}\gg\Omega_{\phi}h^{2}$. We consider zero initial abundance
for the dark sector particles as well as $T_{D}^{\rm ini}=0$. In Table 3, we
note down the required order of $T_{r}$ and the Yukawa coupling that predicts
such scenario considering the same set of
($\lambda,m_{\Psi},m_{\chi},m_{\phi}$) as used in section 3.1.

| $\lambda$ | $m_{\Psi}$ | $m_{\chi}$ | $m_{\phi}$ | $Y$ | $n$ | $T_{r}$ | $\Omega_{\rm\chi}h^{2}$ | $\Omega_{\phi}h^{2}$
---|---|---|---|---|---|---|---|---|---
KD-I | 0.13 | 1.5 TeV | 20 GeV | 10 MeV | $2.4\times 10^{-9}$ | 2 | 320 MeV | 0.115 | 0.003
KD-II | 0.085 | 1.5 TeV | 10 GeV | 10 MeV | $3.75\times 10^{-9}$ | 2 | 180 MeV | 0.118 | 0.001
Table 3: The representative benchmark points that lead to reannhilation after
freeze-in production of $\chi$ in presence of kination dominated epoch with
$n=2$.

The evolution patterns of $Y_{\chi}$ and $Y_{\phi}$ for the benchmark point
KD-II in Table 3 are similar to the case of RD Universe as shown in Figure 6.
In the left of Figure 6, the ratio $r$ is plotted as a function of the
temperature of the SM bath. This figure shows $r$ crosses unity for a brief
period and dark sector equilibrates. This feature enables $\chi$ to annihilate
at a late time to $\phi$ after production from the standard model bath.
However, the annihilation rate is suppressed due to faster expansion of the
Universe, which results in reduced relic for $\phi$ as compared to the one in
RD Universe. We also notice such an enhanced expansion rate of the Hubble
parameter slows down the $\chi$ production process itself, but that can be
adjusted by tuning the Yukawa coupling $Y$ appropriately (see Table 3). In
right of Figure 6, the temperature $T_{D}$ is plotted as a function of SM
temperature $T$. We see a nontrivial pattern for the temperature evolution of
the dark sector here. The $T_{D}$ remains constant up to $z\sim 0.01$ then
reduces with $z$. This has occurred specifically due to an accidental
cancellation between the second term of LHS in Eq.(3.4) and the term in RHS
while solving the Boltzman equation for dark sector energy density. Such
cancellation is triggered by the same temperature dependence of the Hubble
parameter and $n_{\Psi}^{\rm eq}$ when $\Psi$ is relativistic. In Figure 7 the
evolution patterns for the number densities of $\chi$ and $\phi$ considering
the benchmark point KD-II are shown as function of SM temperature. The
obtained patterns are similar as in case of RD Universe. The comoving number
density of $\chi$ drops from its freeze-in value due to late time dark freeze
out. A similar reannihilation pattern for $m_{\chi}=20$ GeV can be obtained as
well with the proper assignment of other relevant parameters as pointed out in
Table 3. For both the benchmark points of Table 3, $\chi$ occupies the maximum
share of total relic abundance, whereas $\phi$ can contribute up to 2$\%$ of
the total relic.

Figure 6: Left: The variation of parameter $r$ is shown against the SM
temperature for the reference point KD-II. Right: The variation of $T_{D}$ on
SM temperature $T$ is shown for the same reference point.

Before we close this section, let us draw a clear comparison between the
impacts of standard and non-standard cosmology within the present setup. We
begin by fixing the mediator mass $m_{\phi}=10$ MeV and dark matter at 10 GeV
and 20 GeV, respectively. In an RD Universe, a larger value of $\lambda$
(motivated from generating velocity dependent large self interaction) leads to
the annihilation of $\chi$ to $\phi$ after its production by forming dark
sector equilibrium. Consequently, we obtain $\phi$ as a dominant DM component
rather than $\chi$. We then anticipate that presence of a non-standard epoch
(kination or faster than kination) in the early Universe can assist in
realising a dominant share of $\chi$ in total DM relic abundance with or
without reaching dark sector thermal equilibrium for the same dark sector
parameters as used in the RD scenario. We have utilised the same benchmark
points as used in the RD case and show that, indeed, a fast-expanding Universe
changes the DM dynamics completely and helps in realising $\chi$ as the main
dark matter component. This occurs since the presence of a modified cosmology
enhance the expansion rate of the Universe and suppresses the conversion rate
of the $\chi\chi\to\phi\phi$ process. One can also spot that it requires a
larger amount of Yukawa coupling $Y$ to obey the relic density bound in a non-
standard era compared to the case in RD Universe. This, in turn, improves the
collider search prospects of the present set up as we will talk about shortly.

Figure 7: The evolution of $\chi$ and $\phi$ number densities are shown with
SM temperature for the benchmark point KD-II.

##  4 Discussion on Collider searches

We shall now briefly discuss the detection prospects of the proposed singlet
scalar extended singlet doublet freeze-in DM at colliders. The possible
collider signatures of the singlet doublet fermion DM model have been
discussed in detail in the context of WIMP [62, 41] and FIMP [50, 49] DM
scenarios. The collider sensitivity of the singlet doublet dark matter model
is mainly based on the $Y\overline{\Psi}\tilde{H}\chi$ vertex. Note that the
same vertex determines the production efficiency of the dark matter in the
early Universe. Depending on the cosmological history and dark sector
dynamics, we found that the value of $Y$ keeps changing for fixed DM mass and
$m_{\Psi}$ or $m_{\xi_{2}}$. For example, a fast expanding Universe prefers
larger $Y$ as compared to the RD Universe in each of the pure freeze-in or
reannihilation scenarios to obtain a constant dark matter relic. Therefore in
the present framework, the colliders can perhaps be utilised to test the
cosmological history at the early Universe and the hidden dark sector dynamics
in the context of the singlet doublet model.

One of the possible signatures at LHC could be the disappearing charge track
signature induced by $\psi^{\pm}\rightarrow\pi^{\pm}\xi_{2}$ decay with decay
length $c\tau_{\psi^{+}}\sim\mathcal{O}(1)$ cm. Independent analyses by ATLAS
and CMS collaborations [63, 64] have inferred strong restriction on the
chargino mass considering a supersymmetric framework as a benchmark model.
Since the decay $\psi^{+}\rightarrow\pi^{\pm}\xi_{2}$ is equivalent to
chargino to Higgsino production, in the present analysis, the bounds provided
by ATLAS and CMS can be employed in our analysis as well. In Figure 8, the
purple shaded region is disfavored due to non-observation of disappearing
charge track assuming $\text{Br}(\psi^{\pm}\to\pi\xi_{2})\simeq 1$. In our
analysis, we have considered $m_{\Psi}$ to be of the TeV scale. Thus the bound
arising from the disappearing charge track signature is not important for our
case.

Figure 8: Discovery prospects of the chosen benchmark points (both in standard
and non standard cosmology) in $Y-m_{\xi_{2}}$ plane. The blue region is
disfavored by the search of disappearing charge track signature. We have also
shown the present and proposed LHC exclusion sensitivities by the dashed red
and green contours respectively as taken from [49].

On the other hand, due to gauge mediated interactions, the heavy charged
($\psi^{\pm}$) and neutral (approximately $\xi_{2}$) components of dark
fermion doublet in this model can be produced at a hadronic collider. The
relevant processes are $p~{}p\to\psi^{+}~{}\psi^{-}$, $\xi_{2}~{}\xi_{2}$ and
$\psi^{\pm}~{}\xi_{2}$ which further decay to DM ($\chi~{}{\rm or}~{}\xi_{1}$)
along with jets in final states via $hh$, $hZ$, $ZZ$, $W^{+}W^{-}$ modes. The
presence of feeble interaction between heavy doublet and the DM
($Y\overline{\Psi}\tilde{H}\chi$) which is the one of requirements of freeze-
in scenario makes the heavy neutral state $\xi_{2}$ longlived
($c\tau_{\xi_{2}}>1$ mm) at the typical scale of detector length. The heavy
charged state, $\psi^{\pm}$ promptly decays to a heavier neutral state,
$\xi_{2}$ with a soft pion which is difficult to probe. The charge fermion,
$\psi^{\pm}$ can also decay directly into DM with $W$ ($\psi^{\pm}\to
W^{\pm}\chi$), which is suppressed when the singlet doublet mixing is small.
In the limit of small singlet doublet mixing, the direct and associate
productions of $\xi_{2}~{}\xi_{2}$ are one of the promising modes for our
analysis which can give rise to displaced vertices with jets plus missing
energy signature (DV+MET) before reaching the end of the tracker. Similar kind
of search has been performed by ATLAS with $\sqrt{s}=13$ TeV and
$\mathcal{L}=32.8$ fb-1 in the context of split supersymmetric models (i.e.
gluino-neutralino) [65]. The displaced events will be analysed by looking at
the individual jet tracks which are originating from a displaced vertex. The
ATLAS DV+$\not{E_{T}}$ search estimates the final events targeting at least
one displaced vertex with jets and large missing transverse energy. So the
signal processes which can give rise to the DV+$\not{E_{T}}$ signature at LHC
in this framework are given as:

$\displaystyle pp$ $\displaystyle\to$
$\displaystyle\big{(}\psi^{+}\psi^{-}\big{)}\to\overline{\xi}_{2}~{}\xi_{2}+{\rm~{}soft~{}pions}\to
hh/hZ/ZZ+\overline{\chi}\chi\to\text{jets}+\overline{\chi}\chi;$ (4.1)
$\displaystyle\to$
$\displaystyle\big{(}\psi^{\pm}\xi_{2}\big{)}\to\overline{\xi}_{2}~{}\xi_{2}+{\rm~{}soft~{}pion}\to
hh/hZ/ZZ+\overline{\chi}\chi\to\text{jets}+\overline{\chi}\chi;$
$\displaystyle\to$ $\displaystyle\overline{\xi}_{2}~{}\xi_{2}\to
hh/hZ/ZZ+\overline{\chi}\chi\to\text{jets}+\overline{\chi}\chi.$

Note that here we have only considered hadronic decay channels of $h$ and $Z$,
since large hadronic branching yields a sizeable cross-section of the
$pp\to\text{jets}+\overline{\chi}\chi$. The above signal cross-section mainly
depends on the LLP (long-lived particle) mass, $m_{\xi_{2}}(\simeq m_{\Psi})$
and the Yukawa coupling, $Y$. The heavy neutral state, $\xi_{2}$ can decay
inside ($1\text{~{}mm}-100~{}\text{m}$) or outside the detector that crucially
depends on the strength of $Y$.

An accurate prediction of discovery prospects of our proposed scenario
requires proper recasting with the exact limits from CMS and ATLAS. One has to
perform a careful reconstruction and selection of events employing suitable
cuts, and by taking into account the generator-level efficiency along with
background estimation. The details of the recasting strategy of this singlet
doublet model have been performed in ref.[49]. In Figure 8 we extract the
present and proposed exclusion bounds obtained on $m_{\xi_{2}}-Y$ plane from
[49] and check the sensitivities of the benchmark points we used so far,
implying different kinds of dark matter dynamics both in standard and non-
standard cosmology. The benchmark points FKD-I and FKD-II, which signify pure
freeze-in in non-standard cosmology, are inside the 300 fb-1 exclusion limit
and can be traced in the future runs of ATLAS. One of two benchmark points
(KD-II) is also likely to be probed in the next run of ATLAS. If tested, these
would probably indicate the presence of modified cosmology with the values of
non-standard cosmological parameters as listed in Table 2 and Table 3. The
benchmark points labelled as RD-I and RD-II are far outside the reach of ATLAS
300 fb-1 run. For these reference points, the decay lengths of $\xi_{2}$ turns
out to be very large with $c\tau_{\xi_{2}}\sim\mathcal{O}(1)$ km. Thus it is
challenging to search $\xi_{2}$ with such long lifetimes at ATLAS and CMS with
current and future sensitivities. The proposed MATHUSLA surface detector
experiment [66, 67] could be capable of probing such a long-lived particle. We
do not discuss this in detail and refer the readers to [50].

##  5 Dark matter self-interaction

The present set-up resembles a two-component dark matter framework. We have
seen in sections 3.2.1 and 3.2.2, that $\chi$ could be the main component with
almost 100$\%$ relic share owing to the presence of modified cosmology. Hence
$\chi$, being adequately abundant in the present Universe, can be an ideal
candidate for self-interacting dark matter. As earlier mentioned, few long-
standing tensions between astrophysical observations and N-body stimulations
for cold DM suggests the DM to be self-interacting with
$1~{}\text{cm}^{2}/\text{gm}\lesssim\sigma/m_{\chi}\lesssim
10~{}\text{cm}^{2}/\text{gm}$ for DM relative velocity $(30\lesssim
v_{d}\lesssim 200)$ km/s [68]. In the present framework, the self interaction
can take place through $\phi$ mediation. We assume the $\chi$ to be symmetric
in nature. The DM self-interaction processes are
$\chi\chi\rightarrow\chi\chi$,
$\overline{\chi}\overline{\chi}\rightarrow\overline{\chi}\overline{\chi}$ and
$\overline{\chi}\chi\rightarrow\overline{\chi}\chi$. The nonrelativistic self-
scattering in DM halos is conventionally described by a Yukawa potential,

$\displaystyle V(r)=\pm\frac{\alpha_{D}^{2}}{4\pi r}e^{-m_{\phi}r},$ (5.1)

where “$\pm$” stands for repulsive and attractive potentials. The parameter
$\alpha_{D}$ is the analog of fine structure constant defined by
$\alpha_{D}=\frac{\lambda^{2}}{4\pi}$. In general, a fermion dark matter can
self-interacts via both scalar and vector portal. Note that the scalar
interactions are purely attractive, while a vector interaction could be both
attractive or repulsive [68]. In literature, uses of two kind of cross
sections can be noticed namely (i) transfer cross section ($\sigma_{T}$) and
(ii) viscosity cross section ($\sigma_{V}$), which are defined as follows [4,
68, 69, 70, 71, 72]:

$\displaystyle\sigma_{T}=\int
d\Omega(1-\cos\theta)\frac{d\sigma}{d\Omega}~{},~{}~{}~{}~{}\sigma_{V}=\int
d\Omega\sin^{2}\theta\frac{d\sigma}{d\Omega}~{}~{},$ (5.2)

where $\theta$ is the scattering angle. The viscosity cross section has
certain merits over the transfer one. For example, the $\sigma_{V}$ takes care
of the divergences in both forward and backward scatterings in the DM halo. In
addition, for self-interaction between identical particles, transfer cross
sector fails. In view of this, we calculate the $\sigma_{V}$ in the present
set up.

The description of the self-scattering could be of different natures
(classical, semi-classical or quantum), parameterized by two dimensionless
parameters $\kappa=\frac{m_{\chi}v_{d}}{m_{\phi}}$ and
$\beta=\frac{2\alpha_{\chi}m_{\phi}}{m_{\chi}v_{d}^{2}}$ correlated to the
momentum and strength of the potential relative to the kinetic energy. The
system is known to be in the classical, semi-classical, and quantum regime for
$\kappa\gg 1$, $\kappa\gtrsim 1$ and $\kappa\lesssim 1$ respectively. The
analytical form of $\sigma_{V}$ in classical and semi-classical regimes can be
found in ref. [73].

Figure 9: Self-interaction cross section as function of velocity for the two
different sets of $(m_{\chi},\lambda)$ that have been used to describe the
dark matter phenomenology in sections 3.2.1 and 3.2.2. The black arrow
indicates the constraint from the Bullet Cluster which is
$\frac{\sigma}{m_{\chi}}<0.7$ cm2/g for $v=4000$ km/s [13, 4]. We fix
$m_{\phi}$ at 10 MeV. We also show the observational data from dwarfs (blue),
low surface brightness (LSB) galaxies (orange), and clusters (green) as taken
from [74].

For the quantum case (excluding the Born approximation for
$2\beta\kappa^{2}\ll 1)$, in principle one needs to solve the Schrodinger
equation by partial wave analysis in order to compute the differential cross
section. The differential scattering cross section can be computed by,

$\displaystyle\frac{d\sigma}{d\Omega}=\frac{1}{k^{2}}\left|(2l+1)e^{i\delta_{l}}P_{l}(\cos\theta)\sin\delta_{l}\right|^{2}.$
(5.3)

where the phase shift for the partial wave is indicated by $\delta_{l}$. The
Schrodinger equation for the radial wave function $R_{l}(r)$ of the reduced DM
two-particle system is written as,

$\displaystyle\frac{1}{r^{2}}\frac{\partial}{\partial
r}\left(r^{2}\frac{dR_{l}}{dr}\right)+\left(k^{2}-\frac{l(l+1)}{r^{2}}-2\mu
V(r)\right)R_{l}=0,$ (5.4)

where $\mu=m_{\chi}/2$ is the reduced mass and $k=\mu v_{d}$. Then by
obtaining $\delta_{l}$ from the asymptotic form of the radial function
$R_{l}(r)$, one can reach at the approximated expression for the viscosity
cross section:

$\displaystyle\sigma_{V}m_{\phi}^{2}\simeq\frac{4\pi}{\kappa^{2}}\int_{0}^{\infty}dl\frac{(l+\frac{1}{2})(l+\frac{3}{2})}{2l+2}~{}\sin^{2}2\delta^{\prime}\left(l+\frac{1}{2}\right),$
(5.5)

where $\delta^{\prime}\left(l+1/2\right)\simeq\delta_{l+1}-\delta_{l}$. The
analytical computation of $\sigma_{V}$ looks unlikely in the above case, where
both quantum and non-resonant effects are important. However, for $\kappa\ll
1$, the S-wave scattering dominates, and the Hulthen potential can be
implemented to find the $\sigma_{V}$ analytically as indicated in [73].

In Figure 9, we estimate the velocity dependence of the viscosity cross
section $\sigma_{V}$ considering two sets of $(m_{\chi},\lambda)$ as used for
the DM analysis earlier in section 3.2.1 and section 3.2.2 with $m_{\phi}$= 10
MeV. Note that for both the benchmark points, we could obtain nearly same
percent relic abundance for the dark matter candidate $\chi$ in considering
non-standard cosmology namely kination domination or faster than kination
domination. For both the $(m_{\chi},\lambda)$ sets, the parameter
$\frac{\sigma_{v}}{m_{\chi}}$ remains in the range (1-10) cm2/g at
$v=(30-200)$ km/s. We also found that these points also satisfy the bound
$\frac{\sigma_{v}}{m_{\chi}}<0.7$ cm2/g at $v=4000$ km/s, arising from the
observations in Bullet Cluster galaxies. In the case of RD dominated Universe,
for the two benchmark points under our discussion, $\chi$ is not adequately
abundant in the present scenario, and hence the solution of small structure
anomalies remains unlikely. However, the assumption of a modified cosmology
before BBN turns instrumental in the simultaneous realisation of adequately
abundant $\chi$ component dark matter and sufficient self-interaction to
alleviate the small scale anomalies of the Universe.

##  6 Summary and Conclusion

The singlet doublet model is a simple particle extension of SM, providing a
viable dark matter candidate from mixing of doublet and singlet fermions after
the electroweak symmetry breaking. Different variants of this scenario are
extensively studied for its enriched dark sector and collider signatures. In
this work, we have examined whether a non-thermally produced and adequately
abundant GeV scale fermion doublet dark matter candidate has the potential to
be probed at colliders. We also discuss the prospect of alleviating the small
scale structure anomalies of the Universe.

We have minimally extended the singlet doublet dark matter model with a MeV
scale singlet scalar, which mediates the dark matter self-interaction. The
singlet scalar is stable in Universe lifetime since it has no decay mode. The
fermion dark matter has a non-thermal origin and can be produced adequately in
the early phase of the Universe. A strong self-interaction, in general,
prefers a sizable non-gravitational interaction strength between the dark
matter and the mediator particle. With such a notion, our computation
indicates the formation of internal dark thermal equilibrium when the
conversion process from the fermion dark matter to scalar mediator turns so
efficient that it significantly suppresses relic abundance for our dark matter
candidate. In fact, the mediator particle emerges to be way more abundant in
the present Universe. The process as mentioned above is prominent in a
standard radiation-dominated Universe, and we remain unsuccessful in obtaining
the singlet doublet fermion dark matter as the main component.

We adopt two specific non-standard cosmological scenarios such as kination and
faster than kination-dominated early Universe to circumvent this issue. The
motivation for such choices is to suppress the conversion process inside the
dark sector. We have found that proper tunings of non-standard cosmological
parameters completely alter the evolution patterns of the number densities for
the dark sector particles. In fact, the benchmark points which had earlier
manifested a reannihilating pattern in the RD dominated Universe depict a pure
freeze-in scenario in the non-standard Universe with negligible mediator
abundance in the present Universe. In addition, for some choices of the non-
standard cosmological parameters, the dark sector still goes through internal
thermal equilibrium. However, that does not cause colossal depletion to the
dark matter relic, unlike the case considering radiation domination.

In short, the presence of a modified cosmology helps realise the fermion-dark
matter as the main component that is adequately abundant in the present
Universe. An exciting consequence comes in terms of collider constraints,
where we found that the displaced Vertex signature can provide the robust
exclusion bound on the GeV scale DM parameter space. Some of our benchmark
points are already within the projected exclusion limit that can be tested by
LHC in the next run. We have further demonstrated that the realised parameter
space can indeed generate velocity-dependent sufficient self-interactions
(consistent with the bounds from observations at bullet cluster galaxies) with
a MeV scale mediator.

We conclude with the comment that, in general, a freeze-in dark matter being
feebly coupled to the visible sector is extremely hard to track at
experiments. Interestingly, the proposed framework of singlet doublet freeze-
in GeV scale DM has ample scopes to be indirectly probed e.g. in the
astrophysical experiments at galaxy scales due to the strong self-interaction
of the fermion dark matter as well as in the collider experiments by virtue of
modified cosmological theory.

##  Appendix A Cross section

The analytical expression for the cross section of
$\chi\chi\rightarrow\phi\phi$ process is given by,

$\displaystyle\sigma_{\chi\chi\rightarrow\phi\phi}$ $\displaystyle=$
$\displaystyle\frac{1}{32\pi s(s-4m_{\chi}^{2})}\times\Bigg{\\{}\frac{\ln
B}{s-2m_{\phi}^{2}}\Big{[}\lambda^{4}s^{2}-4\lambda^{4}\left(m_{\phi}^{2}-4m_{\chi}^{2}\right)s+6\lambda^{4}m_{\phi}^{4}-16\lambda^{4}m_{\phi}^{2}m_{\chi}^{2}$

$\displaystyle-32\lambda^{4}m_{\chi}^{2}\Big{]}-\frac{\sqrt{(s-4m_{\chi}^{2})(s-4m_{\phi}^{2})}}{m_{\chi}^{2}s-4m_{\phi}^{2}m_{\chi}^{2}+m_{\phi}^{4}}\Big{[}\lambda^{4}\left(2m_{\chi}^{2}(s+8m_{\chi}^{2})-16m_{\phi}^{2}m_{\chi}^{2}+3m_{\phi}^{4}\right)\Big{]}\Bigg{\\}}\,,$

where

$B~{}\equiv~{}\frac{s-2m_{\phi}^{2}+\sqrt{(s-4m_{\chi}^{2})(s-4m_{\phi}^{2})}}{s-2m_{\phi}^{2}-\sqrt{(s-4m_{\chi})(s-4m_{\phi}^{2})}}\,.$
(A.2)

##  Appendix B Decay widths

Before EW symmetry breaking the production of dark matter occurs from
$\Psi\rightarrow H\chi$ decay while in the post EW symmetry breaking era,
decays such as $\xi_{2}\rightarrow h\chi$, $\xi_{2}\rightarrow Z\chi$ and
$\psi^{\pm}\rightarrow W^{\pm}\chi$ turn active. The analytical expression for
the decay widths are read as,

$\displaystyle\Gamma_{\xi_{2}\rightarrow h\chi}=\frac{Y^{2}}{32\pi
m_{\xi_{2}}^{3}}\left\\{(m_{\xi_{2}}+m_{\chi})^{2}-m_{h}^{2}\right\\}F(m_{\xi_{2}},m_{\chi},m_{h}),$
(B.1) $\displaystyle\Gamma(\xi_{2}\rightarrow
Z\chi)=\frac{Y^{2}}{32\pi}\frac{\left\\{(m_{\xi_{2}}-m_{\chi})^{2}-m_{Z}^{2}\right\\}\left\\{(m_{\xi_{2}}-m_{\chi})^{2}+2m_{Z}^{2}\right\\}}{m_{\xi_{2}}^{3}(m_{\xi_{2}}-m_{\chi})^{2}}F(m_{\xi_{2}},m_{\chi},m_{Z}),$
(B.2) $\displaystyle\Gamma(\psi^{\pm}\rightarrow
W^{\pm}\chi)=\frac{Y^{2}}{32\pi}\frac{\\{(m_{\Psi}-m_{\chi})^{2}-m_{W}^{2}\\}\\{(m_{\Psi}+m_{\chi})^{2}+2m_{W}^{2}\\}}{m_{\Psi}^{3}(m_{\Psi}-m_{\chi})^{2}}F(m_{\Psi},m_{\chi},m_{W}),$
(B.3)

where $F(x,y,z)=\sqrt{x^{4}+y^{4}+z^{4}-2x^{2}y^{2}-2x^{2}z^{2}-2y^{2}z^{2}}$.
We also note that

$\displaystyle\Gamma_{\Psi\rightarrow H\chi}\simeq
4~{}\Gamma(\xi_{2}\rightarrow h\chi).$ (B.4)

##  Appendix C Energy Transfer rate from SM to dark sector

The aim of this appendix is to determine an analytical form of the energy
transfer rate for two body decay of $\Psi_{i}$ appearing in R.H.S of Eq.(3.4).
We start with the Boltzmann equation for the phase-space distribution of a
particle species $\Psi_{i}$:

$\frac{\partial f_{\Psi_{i}}(p_{\Psi_{i}},t)}{\partial
t}~{}=~{}H(t)p_{\Psi_{i}}\frac{\partial f_{\Psi_{i}}(p_{\Psi_{i}},t)}{\partial
p_{\Psi_{i}}}+C[f]\,,,$ (C.1)

where $C[f]$ is the collision operator for tree level $\Psi_{i}$ decay. For a
specific process $\Psi_{i}\rightarrow a+b$ process, the collision term takes
the following form,

$\displaystyle C[f]$ $\displaystyle=$
$\displaystyle-\frac{1}{2E_{\Psi_{i}}}\int
d\Pi_{a}d\Pi_{b}(2\pi)^{4}\delta^{(4)}(p_{\Psi_{i}}-p_{a}-p_{b})\times\bigg{[}\left|\mathcal{M}_{\Psi_{i}\rightarrow
a+b}\right|^{2}f_{\Psi_{i}}(1\pm f_{a})(1\pm f_{b})$ (C.2)
$\displaystyle~{}~{}~{}~{}-\left|\mathcal{M}_{a+b\rightarrow\Psi}\right|^{2}f_{a}f_{b}(1\pm
f_{\Psi_{i}})\bigg{]}\,,$

where $d\Pi_{i}=\frac{d^{3}p_{i}}{(2\pi)^{3}2E_{i}}$, and “$\pm$” symbolises
the Bose-enhancement and Pauli-blocking factors. We have used the Maxwell-
Boltzmann distribution approximation and use $f_{i}\approx e^{-E_{i}/T_{i}}$
and $1\pm f_{i}\approx 1$ to simplify the calculation. The energy-transfer
rate from the SM sector to the dark sector is obtained by

$\frac{d\rho_{\Psi_{i}}}{dt}~{}=~{}-3H(P_{\Psi_{i}}+\rho_{\Psi_{i}})+\int\frac{d^{3}p_{\Psi_{i}}}{(2\pi)^{3}}E_{\Psi_{i}}C[f]\,.$
(C.3)

In case for a decay, we write

$\displaystyle\int\frac{d^{3}p_{\psi}}{(2\pi)^{3}}E_{\psi}C[f]=n_{\Psi_{i}}\mathcal{P}_{\Psi_{i}},$
(C.4)

where we obtain
$n_{\Psi_{i}}\mathcal{P}_{\Psi_{i}}=n_{\Psi_{i}}m_{\Psi_{i}}\Gamma_{\Psi_{i}}$.

## Acknowledgments

PK and SS gratefully acknowledge the HPC resources (Vikram-100 HPC) and TDP
project at PRL for the computations. AKS appreciates Sougata Ganguly for
several discussions during the course of work. AKS is supported by NPDF grant
PDF/2020/000797 from Sci- ence and Engineering Research Board, Government of
India. PG would like to acknowledge the support from DAE, India for the
Regional Centre for Accelerator based Particle Physics (RECAPP), Harish
Chandra Research Institute.

##### Note added:

Shortly before submitting this manuscript, the preprint [75] appeared on the
arXiv, which also considers the self-interacting dark matter in an enlarged
variant of the singlet doublet framework. The dark sector is purely thermal in
their case under the assumption of a standard radiation dominated Universe. In
contrast, we have explored a freeze-in production of a hidden dark sector
considering a few different variants of cosmological scenarios.

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