# Reviewing the prospect of fermion triplets as dark matter and source of
baryon asymmetry in non-standard cosmology

Anirban Biswas  Mainak Chakraborty  Sarif Khan

###### Abstract

Indirect searches of Dark Matter (DM), in conjugation with ‘missing track
searches’ at the collider seem to confine SU(2)L fermion triplet DM (FTDM)
mass within a narrow range around 1 TeV. The canonical picture of the pure
FTDM is in tension since it is under-abundant for the said mass range. Several
preceding studies have reported that an extra species ($\phi$), redshifts
faster than the radiation ($\sim a^{-(4+n)}$ where $n>0$), leads to a faster
expanding early Universe by dominating in the energy density with an enhanced
Hubble parameter. This has the potential to revive the under-abundant FTDM
($\mathbb{Z}_{2}$ odd, lightest generation) by causing freeze-out earlier
without modifying the interaction strength between DM and thermal bath. On the
other hand, although the CP asymmetry produced due to the decay of
$\mathbb{Z}_{2}$ even heavier generations of the triplet remains unaffected,
its evolution is greatly affected by the non-standard cosmology. It has been
observed through numerical estimations that the minimum mass of the triplet,
required to produce sufficient baryon asymmetry of the Universe (BAU), can be
lowered up to two orders (compared to the standard cosmology) in this fast
expansion scenario. The non-standard parameters $n$ and $T_{r}$ (a reference
temperature below which radiation dominance prevails), which simultaneously
control DM abundance as well as the frozen value of BAU, are tightly
constrained from the observed experimental values. We have found that $n$ is
strictly bounded within the interval $0.4\lesssim n\lesssim 1.8$ where the
upper bound is imposed by the BAU constraint whereas the lower bound arises to
satisfy the correct DM abundance. It has been noticed that the restriction on
$T_{r}$ is not so stringent as it can vary from sub GeV to a few tens of GeV.

##  1 Introduction

The origin of dark matter(DM) and the explanation of the observed matter-
antimatter asymmetry are two most important and unresolved issues in particle
and astroparticle physics. The existence of dark matter has been suggested by
a plethora of observations at the galactic scale like galactic rotation curves
[1], gravitational lensing of distant celestial bodies [2], colliding galaxy
clusters (e.g. the Bullet cluster) [3] etc. Similarly, the observations at the
cosmological scale by WMAP [4] and Planck [5, 6] satellites have measured most
precisely the abundance of dark matter till date, $\Omega_{\rm
DM}h^{2}=0.120\pm 0.001$ at 68% C.L. [6], which is roughly around 25% energy
budget of our Universe. While Weakly Interacting Massive Particle (WIMP) [7,
8, 9, 10] characterised by the GeV–TeV scale mass window with weak scale
interaction cross-section is one of the most preferred dark matter candidates,
there exist various other possibilities like, non-thermal dark matter [11, 12,
13], asymmetric dark matter [14], primordial black hole as dark matter [15]
etc. On the other hand, there are overwhelming evidences in support of excess
baryonic matter over anti-matter at various cosmological epochs. The baryon-
to-photon ratio ($\eta_{B}$) predicted during the Big bang nucleosynthesis
(BBN) by measuring various light element abundances matches excellently with
that obtained from the baryon density at the time of recombination. The
current acceptable value of $\eta_{B}$ which carries the information about the
baryon asymmetry111It is to be noted that baryon asymmetry can also be
expressed as net baryon number density scaled by entropy density i.e
$Y_{B}=\frac{n_{B}}{s}$ and its experimental value lies within $(9.05\pm
0.150)\times 10^{-11}$.  of the Universe (BAU) is $\eta_{B}=(6.4\pm 0.1)\times
10^{-10}$ at 95% C.L. [6]. Non-zero neutrino masses and large leptonic mixing
[16, 17] have now become experimentally [18, 19, 20, 21, 22, 23, 24, 25]
verified fact and the experiments are now inching towards precise measurement
of the oscillation parameters [26]. Various seesaw mechanisms [27, 28, 29, 30,
31, 32], introduced to explain tiny mass of the active neutrinos, can act as a
bridge between DM phenomenology and mechanism of dynamical generation of
baryon asymmetry through lepton asymmetry(popularly known as Baryogenesis via
Leptogenesis [33]).

In this work, we have considered one of the minimal extensions of the SM to
address both the dark matter and the baryon asymmetry of the Universe. Here we
have added three SU$(2)_{\rm L}$ fermionic triplets with hypercharge $Y=0$ in
the particle spectrum of the SM. As a result, two of the three light neutrinos
have sub-eV Majorana masses through the Type-III seesaw mechanism and the
remaining one continues to be massless due to the imposition of an additional
$\mathbb{Z}_{2}$ odd parity on the lightest triplet fermion $\Sigma_{R\,1}$.
Accordingly, the triplet $\Sigma_{R\,1}$ gets decoupled from the neutrino
sector and its neutral component
$\Sigma^{0}_{1}(=\Sigma^{0}_{R\,1}+{\Sigma^{0}_{R\,1}}^{c})$ could be a
Majorana type dark matter candidate which satisfies the relic density bound
for $M_{\Sigma_{1}}\simeq 2.3$ TeV [34]. However, such a dark matter candidate
in the mass range $1.5\,\,{\rm TeV}\,\,\lesssim M_{\Sigma_{1}}\lesssim 2.7$
TeV is already ruled out by the indirect detection bound coming from dark
matter annihilation into $W^{+}W^{-}$ and $\gamma\gamma$ final states [35,
36]. There are some earlier attempts to generate correct relic abundance for
the low mass triplet fermionic dark matter by either introducing an additional
non-thermal contribution to the relic density from a decaying scalar [37] or
including a singlet fermion with a nonzero mixing with the triplet such that
the resulting dark matter candidate is predominantly a singlet-like state
[38]. In this work, we are considering another interesting way to bring relic
density in the correct ballpark value of Planck collaboration [6] in the low
mass regime of the triplet dark matter. Here, we modify the underlying
cosmology of the early222The word ‘early’ signifies an era which started
earlier than the thermal production scale of heavy fermion triplets and lasts
till the freeze-out epoch of the DM. Universe by adding extra species which
redshifts faster than the radiation content of the Universe. There is a large
ambiguity about the cosmology of the University prior to the era of light
nuclei (H, D, 3He, 4He, and 7Li) formation, popularly known as the Big bang
nucleosynthesis which began when the age of the Universe was $\sim 1$ second
and it continued for about three minutes. More importantly, the theoretical
predictions of BBN (e.g. light nuclei abundance), considering radiation
dominated Universe, are in good agreement with observational data. On the
contrary, very little information is available to us about the era before BBN
and in most of the literature a radiation dominated Universe has been
considered. However, the expansion rate of the Universe driven by the Hubble
parameter can be higher or lower compared to the case of radiation domination
depending on how the total energy density of the Universe redshift with
respect to expansion. The fast expansion results if there is an additional
source of energy on top of the radiation background composed of the SM
particles such that the total energy density $\rho_{\rm tot}>\rho_{\rm rad}$,
the energy density due to radiation. The early matter domination [39, 40, 41,
42], the kination [43, 44, 45, 46, 47, 48] etc. are a few examples of fast
expanding Universe. On the other hand, the opposite situation could also arise
particularly in some modified gravity theory [49, 50, 51, 52, 53] where a
negative contribution in the energy density leads to slow expansion [54].

In this work, we consider a fast expanding Universe by introducing a species
$\phi$ that dominates the energy density in the early Universe. Moreover, the
energy density of $\phi$ red-shifts at a faster rate with the cosmic scale
factor ($a$) compared to that of the radiation ($\rho_{\rm rad}\propto
a^{-4}$) i.e. $\rho_{\phi}\propto a^{-(4+n)}$ with $n>0$. Therefore, although
initially we have $\phi$ dominance, the radiation eventually takes over from
$\phi$ before the start of BBN ($T\sim 1$ MeV) since the latter decreases
rapidly due to expansion. The revival of radiation domination is characterised
by the temperature $T_{r}\geq 1$ MeV. Accordingly, we have two new parameters
$n$ and $T_{r}$ (both are not independent) controlling the cosmology of the
Universe for $T>T_{r}$. Depending upon the value of $n$ we can have different
expansion histories of the Universe. For example, the scenario with $n=2$ can
be realised if the kinetic energy of a scalar field dominates the energy
budget, which is known as the Kination era where $\rho_{\phi}\propto a^{-6}$
[45, 46, 47, 48]. This simple model for non-standard cosmology was first
proposed in [48] and thereafter a number of works have been done using this
model in the context of dark matter [55, 56, 57, 58, 59, 60] and leptogenesis
[61, 62, 63, 64, 65, 66, 67]. In this work, our prime motivation is to find an
allowed parameter space for the pure triplet dark matter which otherwise is
entirely ruled out by the combination of relic density and indirect detection
bounds for the standard cosmological scenario. In the radiation dominated
Universe, a triplet fermion of mass lower than 2.3 TeV remains under abundant
due to large annihilation and co-annihilation cross-sections into electroweak
gauge bosons and fermions, which results in a late freeze-out. However, this
situation can be changed in presence of a fast expanding Universe
characterised by a Hubble parameter larger than the standard scenario.
Moreover, the non-standard cosmology also affects the final baryon asymmetry
produced from lepton asymmetry through sphaleron involving processes before
the electroweak phase transition. In this model the other two heavier triplet
fermions, which are even under $\mathbb{Z}_{2}$ symmetry, have non-vanishing
Yukawa couplings with Lepton doublets and Higgs doublet. Accordingly, the
lepton asymmetry can be generated due to CP violating out of equilibrium
decays of these heavy triplet fermions at very high temperature. This has been
studied earlier considering a radiation dominated Universe before the BBN and
found [68, 69, 70] that the minimum mass of triplet fermion required to
generate the observed baryon asymmetry is $M_{\Sigma}\gtrsim 3\times 10^{10}$
GeV. In the present work, we explore the possibility of relaxing the above
mentioned lower bound333One can also lower the triplet mass bound by extending
the minimal model with additional fields as done in [71]. on $M_{\Sigma}$
obtained from BAU bound. Predominance of the extra scalar field $\phi$ in the
primordial Universe resulted in faster (than standard cosmology) expansion of
the Universe. Thus enhancement caused in the Hubble parameter pushes the rate
of decay/Inverse decay (of $\Sigma_{2}$ or $\Sigma_{3}$) and scattering
towards a lower value. Therefore the Boltzmann equations governing the
evolution of produced asymmetry will be altered with respect to its standard
counterpart. Although the modification of cosmology doesn’t affect the
asymmetry production, it has non-trivial effect in the evolution of the same.
Our aim in this part of the work is to identify the specific region of the
parameter space which can produce observed baryon asymmetry below the limit
(on the decaying triplet mass) set using standard cosmology.

The fascinating feature of this study is that the newly introduced non-
standard cosmology co-relates the apparently decoupled dark sector and the
Yukawa sector(responsible CP asymmetry generation) through the non-standard
parameters $T_{r},n$. We expect that the relic density bound together with the
observed BAU bound will allow a strongly constrained parameter space spanned
by the yet unbounded non-standard parameters $T_{r}-n$.

To address the issues raised in the preceding paragraphs in a comprehensive
and systematic manner, the rest of this paper is organized as follows. The
minimal extension scheme of SM adopted here, to account for the DM relic
abundance as well as the BAU, is discussed in Sec. 2. A qualitative overview
about the generation of active neutrino mass and mixing through Type-III
seesaw mechanism is presented in Sec. 3. Sec. 4 is solely devoted to the study
of DM-phenomenology. Its different subsections are arranged chronologically so
that the need for introduction of non-standard cosmology to save the fermion
triplet DM scenario can be clearly understood. So we start with a
comprehensive analysis of existing bounds (obtained from indirect searches and
collider experiments) on fermion triplet DM in sec. 4.1. The collective
results allow only a window for triplet DM mass, which is not enough to
produce adequate relic abundance in standard cosmological scenario. Thus non-
standard cosmology is introduced in Sec. 4.2 and its subsections discuss how
the freeze-out temperature of the DM can be modified (Sec. 4.2.1) in the newly
introduced cosmology, without harming the successful predictions of the BBN
(Sec. 4.2.3). Sec. 4.3 presents a detailed account of numerical analysis
related to the study of DM phenomenology in modified cosmological scenario.
Sec. 5 contains a rigorous analysis of Baryogenesis through Leptogenesis in
the present scenario. Discussions about Leptogensis, continue in its
constituent subsections, starting with generation of CP asymmetry due to the
decay of next to lightest triplet (Sec. 5.1), followed by a thorough
description of the Boltzmann equations in the non-standard (Sec. 5.2)
cosmology and ends with a meticulous analysis (Sec. 5.3) of numerical results
supplemented by plots of suitable model parameters. The noteworthy outcomes of
this exhaustive study are summarised in Sec. 6.

##  2 Model

In the present work, we have addressed a few well-recognised SM drawbacks
namely dark matter, neutrino mass and matter anti-matter asymmetry of the
Universe. To this end, we have extended the SM particle content by three
additional $SU(2)_{L}$ triplet fermions similar to the Type-III seesaw
mechanism. The complete particle spectrum with the associated charge
assignment is shown in Table 1. Among them, the lightest one is
$\mathbb{Z}_{2}$ odd and its neutral component serves as the suitable DM
candidate whereas the other two are $\mathbb{Z}_{2}$ even and contribute to
the neutrino mass through the Type-III seesaw mechanism. We have considered
its $(\Sigma_{1})$ mass in the TeV range. Fermion triplet DM heavier than
$2.4$ TeV is ruled out by Planck data due to the over-abundance. Moreover, for
mass beyond $1.5$ TeV, DM annihilation faces Sommerfeld enhancement [35, 36]
which is in direct contradiction with the indirect detection bound provided by
Fermi-LAT data [72]. Therefore, we have considered $\Sigma_{1}$ mass always
below $1.5$ TeV. This mass restriction forbids the lightest fermion triplet to
account for the full relic abundance of DM as reported by the experiments. We
also discuss in the result section, how the presence of an extra species
(denoted as $\phi$) in the early Universe assist us to obtain the total amount
of dark matter relic density even below $1$ TeV, which was not possible in
standard cosmological scenario. The other two triplet fermions, which are even
under $\mathbb{Z}_{2}$ symmetry, take part in the generation of neutrino mass,
lepton asymmetry, are kept at a very high scale $\mathcal{O}(10^{9})$ GeV. The
improvement of the lower bound on triplet mass, to obtain the correct value of
BAU, due to the presence of the extra species in the early Universe is
emphasized in the section dedicated to ‘Baryogenesis through Leptogenesis’.

| Gauge
---
Group
${\rm SU(2)}_{\rm L}$
${\rm U(1)}_{\rm Y}$
$\mathbb{Z}_{2}$
| Quarks
---
$Q_{L}^{i}=(u_{L}^{i},d_{L}^{i})^{T}$ | $u_{R}^{i}$ | $d_{R}^{i}$
$2$ | $1$ | $1$
$1/6$ | $2/3$ | $-1/3$
$1$ | $1$ | $1$
| Leptons
---
$L_{L}^{i}=(\nu_{L}^{i},e_{L}^{i})^{T}$ | $e_{R}^{i}$ | $\Sigma^{c}_{Rj}$ | $\Sigma^{c}_{R1}$
$2$ | $1$ | $3$ | $3$
$-1/2$ | $-1$ | $0$ | $0$
$1$ | $1$ | $1$ | $-1$
| Scalar
---
$\Phi$
$2$
$1/2$
$1$
Table 1: Particle contents and their corresponding charges under the SM gauge
group.

The Lagrangian for the particle spectrum shown in Table 1 is given by,

$\displaystyle\mathcal{L}$ $\displaystyle=$ $\displaystyle\mathcal{L}_{\rm
SM}+\mathcal{L}_{\Sigma}\,,$ (2.1)

where $\mathcal{L}_{\rm SM}$ is the Lagrangian associated with the SM
particles and $\mathcal{L}_{\Sigma}$ is the Lagrangian for the triplet
fermions which has the following form,

$\displaystyle\mathcal{L}_{\Sigma}$ $\displaystyle=$
$\displaystyle\frac{i}{2}\sum_{j=1,2,3}{\rm
Tr}\left[\overline{\Sigma_{R\,j}}\,\not{D}\Sigma_{R\,j}+\overline{{\Sigma_{R\,j}}^{c}}\,\not{D}{\Sigma_{R\,j}}^{c}\right]-\sum^{3}_{\alpha=1}\sum^{3}_{j=1}\left(\sqrt{2}\,y^{\alpha
j}_{\Sigma}\overline{L_{L\,\alpha}}\,\Sigma_{R\,j}\tilde{\Phi}+h.c.\right)$
(2.2) $\displaystyle-$ $\displaystyle\frac{1}{2}\sum^{3}_{j=1}{\rm
Tr}\left[\overline{{\Sigma_{R\,j}}^{c}}\left(M_{\Sigma}\right)_{jj}\Sigma_{R\,j}+h.c\right],$

where the above three terms are, respectively, the kinetic term, the Yukawa
coupling term and the bare mass term for the fermion triplets $\Sigma_{j}$
($j=1,2,3$). The Latin index $j=1\,$to 3 stands for three generations of
fermion triplets whereas the Greek index $\alpha$ is used to designate three
different lepton flavours. It is to be noted that although the summation on
$j$ runs from $1$ to $3$ in the second term, the species $\Sigma_{R\,1}$ does
not couple to the SM Higgs and the lepton doublets, which implies that the
first column of the Yukawa coupling matrix $y_{\Sigma}$ is null. This has been
implemented by imposing a $\mathbb{Z}_{2}$ odd parity to the lightest triplet
generation i.e. $\Sigma_{R1}$ whereas all the other species are
$\mathbb{Z}_{2}$ even including the SM particles. It should be noted that we
have constructed the physical basic using a linear combination of component
field and its CP conjugate of each triplet $\Sigma_{Rj}$. In the fundamental
representation of SU(2)L, the triplet $\Sigma_{Rj}$ can be expressed as

$\Sigma_{Rj}=\left(\begin{array}[]{cc}{\Sigma^{0}_{Rj}}/{\sqrt{2}}&\Sigma^{+}_{Rj}\\\
\Sigma^{-}_{Rj}&-{\Sigma^{0}_{Rj}}/{\sqrt{2}}\end{array}\right)~{}.$ (2.3)

The $2\times 2$ representation of $\Sigma_{Rj}$ transforms under SU(2)L as
$\Sigma^{\prime}_{Rj}=U\Sigma_{Rj}U^{\dagger}$ with
$U=\exp(-i\frac{g}{2}{\theta_{a}\,\sigma_{a}})$ and
$\sigma_{a}\,(a=1,\,2,\,3)$ are the Pauli matrices. The complex conjugate of
$\Sigma_{Rj}$, ${\Sigma_{Rj}}^{*}$, does not transform in the similar manner
as $\Sigma_{Rj}$ under SU(2)L. However, the combination
$i\sigma_{2}\,{\Sigma_{Rj}}^{*}\,i\sigma_{2}$ has the identical transformation
property as that of the triplet $\Sigma_{Rj}$. Therefore, in order to have an
SU(2) invariant Lagrangian, the CP conjugate of triplet $\Sigma_{Rj}$ is
defined as

${\Sigma_{Rj}}^{c}=i\sigma_{2}\,C\,\overline{\Sigma_{Rj}}^{T}\,i\sigma_{2}=\left(\begin{array}[]{cc}{\Sigma^{0}_{Rj}}^{c}/\sqrt{2}&{\Sigma^{-}_{Rj}}^{c}\\\
{\Sigma^{+}_{Rj}}^{c}&-{\Sigma^{0}_{Rj}}^{c}/\sqrt{2}\end{array}\right)~{}.$
(2.4)

Nevertheless, the CP conjugates of the component fields follow the usual
definition i.e. $\Sigma^{0}_{Rj}=C\,\overline{\Sigma^{0}_{Rj}}^{T}$ and
$\Sigma^{\pm}_{Rj}=C\,\overline{\Sigma^{\pm}_{Rj}}^{T}$. Now, using both
$\Sigma_{Rj}$ and ${\Sigma_{Rj}}^{c}$, we can construct a triplet for each
generation as

$\displaystyle\Sigma_{j}$ $\displaystyle=$
$\displaystyle\Sigma_{Rj}+{\Sigma_{Rj}}^{c}$ (2.7)
$\displaystyle=\left(\begin{array}[]{cc}\Sigma^{0}_{j}/\sqrt{2}&\Sigma^{+}_{j}\\\
\Sigma^{-}_{j}&-\Sigma^{0}_{j}/\sqrt{2}\end{array}\right)\,,$

where, the neutral parts
$\Sigma^{0}_{j}=\Sigma^{0}_{Rj}+{\Sigma^{0}_{Rj}}^{c}$ are the Majorana
fermions while the charged parts
$\Sigma^{\pm}_{j}=\Sigma^{\pm}_{Rj}+{\Sigma^{\mp}_{Rj}}^{c}$ are the Dirac
fermions. In terms of newly defined fields $\Sigma^{0}_{j}$ and
$\Sigma^{\pm}_{j}$, the Lagrangian given in Eq. (2.2) takes the following form
[37]

$\displaystyle\mathcal{L}_{\Sigma}$ $\displaystyle=$ $\displaystyle
i\,\overline{\Sigma^{-}_{j}}\,\not{\partial}\Sigma^{-}_{j}+\frac{i}{2}\overline{\Sigma^{0}_{j}}\,\not{\partial}\Sigma^{0}_{j}-M_{\Sigma_{j}}\overline{\Sigma^{-}_{j}}\Sigma^{-}_{j}-\frac{M_{\Sigma_{j}}}{2}\overline{\Sigma^{0}_{j}}{\Sigma^{0}_{j}}-g\left(\overline{\Sigma^{-}_{j}}\not{W}^{-}\Sigma^{0}_{j}+h.c.\right)$
(2.8)
$\displaystyle+g\cos\theta_{W}\overline{\Sigma^{-}_{j}}\not{Z}\Sigma^{-}_{j}+g\sin\theta_{W}\overline{\Sigma^{-}_{j}}\not{A}\Sigma^{-}_{j}-\sqrt{2}\left\\{y^{\alpha
j}_{\Sigma}\left(\frac{1}{\sqrt{2}}\overline{\nu_{L\alpha}}P_{R}\Sigma^{0}_{j}+\overline{\ell_{L\alpha}}P_{R}\Sigma^{-}_{j}\right){H^{0}}^{*}\right.$
$\displaystyle\left.-y^{\alpha
j}_{\Sigma}\overline{\nu_{L\alpha}}P_{R}\Sigma^{+}_{j}H^{-}+\frac{y^{\alpha
j}_{\Sigma}}{\sqrt{2}}\,\overline{\ell_{L\alpha}}P_{R}\Sigma^{0}_{j}H^{-}+h.c.\right\\}\,.$

Here, $H^{-}$ and $H^{0}$ are charged and neutral components of the Higgs
doublet $H$ respectively. As we mentioned above, all the elements in the first
column of the Yukawa coupling matrix are zero i.e. $y^{\alpha 1}_{\Sigma}=0$
for all values of $\alpha$. This implies $\Sigma_{1}$ does not have Yukawa
coupling with the SM lepton and Higgs. Thus, the neutral component of
$\Sigma_{1}$, $\Sigma^{0}_{1}$ can be a suitable DM candidate.

##  3 Type III Seesaw explaining neutrino mass

As discussed in Section 2, among the three triplet fermions, the
$\mathbb{Z}_{2}$ odd one contributes to the DM phenomenology and the other two
heavier generations take part in neutrino mass generation by the Type III
seesaw mechanism. After the electroweak symmetry breaking, the neutrino mass
matrix in the ($\nu_{L}$ $\Sigma_{R}^{~{}c}$ ) basis takes the following form,

$\displaystyle\mathcal{L}_{NM}=-\frac{1}{2}\begin{pmatrix}\overline{\nu_{L}}&\overline{\Sigma^{c}_{~{}L}}\end{pmatrix}\begin{pmatrix}0&M_{D}\\\
M^{T}_{D}&M_{\Sigma}\end{pmatrix}\begin{pmatrix}\nu^{c}_{~{}R}\\\
\Sigma_{R}\end{pmatrix}+{\it h.c.}$ (3.1)

It is evident that the mass matrix, written in ($\nu_{L}$ $\Sigma_{R}^{~{}c}$
) basis, is a six dimensional square matrix if we consider three generations
of light active neutrinos and three generations of fermion triplet. The
constituent matrices $M_{D},M_{\Sigma}$ are square matrices of order $3\times
3$. In contrast to the usual type-I seesaw mechanism, the first column of
$M_{D}$ is null since the Yukawa coupling of $\Sigma_{1}$ is forbidden by the
$\mathbb{Z}_{2}$ symmetry. The three dimensional bare mass matrix of the
triplet has been assumed to be diagonal with a hierarchical mass spectrum
$M_{\Sigma_{1}}\ll M_{\Sigma_{2}}<M_{\Sigma_{3}}$. With the seesaw assumption
i.e, $M_{\Sigma}\gg M_{D}$, we block diagonalize the mass matrix shown in Eq.
(3.1), and get the following two eigenvalues (matrices) as,

$\displaystyle m_{\nu}$ $\displaystyle\simeq$ $\displaystyle-
M_{D}M^{-1}_{\Sigma}M^{T}_{D}$ $\displaystyle M_{\Sigma}$
$\displaystyle\simeq$ $\displaystyle{\rm
diag}(M_{\Sigma_{1}},M_{\Sigma_{2}},M_{\Sigma_{3}})\,.$ (3.2)

In Eq. (3.2), $m_{\nu}$ is the active neutrino mass matrix and $M_{\Sigma}$ is
the heavy fermion triplet mass matrix. The effective neutrino mass matrix
$m_{\nu}$ has to be diagonalized further in order to get the light neutrino
mass eigenvalues and mixing angles. In the present work, we are not concerned
about the exact flavour structure of the neutrino mass matrix and thus we are
not going into the details of the diagonalization procedure. Existing studies
on Type-III seesaw leptogenesis have introduced a lower bound on the next to
lightest triplet mass from BAU bound, $M_{\Sigma_{2}}\gtrsim 3\times 10^{10}$
GeV, which ensures sub-eV scale active neutrinos.

##  4 Lightest triplet as DM candidate

The lightest triplet $\Sigma_{1}$ has been made stable by switching off its
coupling to the SM particles through imposition of $\mathbb{Z}_{2}$ discrete
symmetry. The three components
$(\Sigma^{+}_{1},\Sigma^{0}_{1},\Sigma^{-}_{1})$ of the triplet are degenerate
at the tree level. However, the degeneracy is lifted when we consider
radiative corrections [73]. In that case, the charged component becomes
slightly heavier than the neutral one which makes the decay channel
$\Sigma^{\pm}_{1}\rightarrow\Sigma^{0}_{1}+W^{\pm}(\rm virtual)$444These
virtual $W^{\pm}$ decay into pions or leptons allowed. Thus $\Sigma^{0}_{1}$
is the lightest stable particle in our particle spectrum, which has the
potential to serve as a viable dark matter candidate by reproducing the
correct relic abundance as predicted by the Planck collaboration [6].

###  4.1 Constraints on triplet DM

Figure 1: Collider bound on the mass of the triplet fermion when the neutral
component is a DM candidate.

The detailed bound on the triplet fermion DM has been addressed in Ref. [74].
Considering the neutral component of the triplet fermion as WIMP DM candidate,
the bound can come from direct detection, indirect detection and collider
searches. In the present model, the WIMP-nucleon scattering cross-section
occurs at one loop level and gets suppressed accordingly [73]. As a result,
the spin independent scattering cross-section ($\sigma_{\rm SI}$) lies at
least one order of magnitude below the present bound from LUX-ZEPLIN (LZ) [75]
for $M_{\Sigma_{1}}=1$ TeV [37]. Moreover, a further suppression on
$\sigma_{\rm SI}$ can arise due to the negative contribution of two loop gluon
mediated processes [76]. On the other hand, in the present model, our DM
candidate ($\Sigma^{0}_{1}$) can only annihilate into $W^{+}W^{-}$ final state
at the tree level555Coannihilations with charged partners, $\Sigma^{\pm}_{1}$
are not considered here as these are not relevant for DM indirect detection..
These $W^{\pm}$ bosons eventually produce $\gamma$-rays after undergoing
cascade processes involving pions. Accordingly, we have an existing bound on
the $\langle{\sigma v}\rangle_{W^{+}W^{-}}$ from gamma-ray telescopes like
Fermi-LAT which is sensitive to $\gamma$-rays in the range $\sim 20$ MeV to
300 GeV [72]. The recent bound on $\langle{\sigma v}\rangle_{W^{+}W^{-}}$ by
Fermi-LAT through observation of $\gamma$-rays coming from dwarf spheroidal
galaxies (dSph) [77] allows a narrow region of triplet DM mass between
$1000\,\,{\rm GeV}\leq M_{\Sigma_{1}}\lesssim 1500$ GeV. For $1500\,\,{\rm
GeV}\lesssim M_{\Sigma_{1}}\lesssim 2700$ GeV, there is a rapid growth in
$\langle{\sigma v}\rangle_{W^{+}W^{-}}$ due to the Sommerfeld enhancement [35,
36] and thus is ruled out from non-observation of excess gamma flux over the
known astrophysical background. However, there is a more recent analysis in
Ref. [78] where666In the said reference the authors have argued that DM in the
mass range $20-10^{5}$ GeV is fully safe from the indirect detection searches
(when DM particle under consideration is produced by the thermal freeze-out
mechanism.). Here, we mention this result just as a general observation.
However, it has not been used in our analysis. the authors have derived a
model independent bound on the total annihilation cross-section of dark matter
after taking into account the bounds from Planck, Fermi-LAT and AMS.

The bound on $M_{\Sigma_{1}}$ from collider searches has a similar search
strategy as the SUSY charginos. As the mass difference between charged and
neutral components, $\Delta M=M_{\Sigma^{\pm}_{1}}-M_{\Sigma_{1}}\simeq 166$
MeV [73], is greater than the mass of charged pion, $\Sigma^{\pm}_{1}$ decays
predominantly into $\Sigma^{0}_{1}$ and $\pi^{\pm}$ with approximately $98\%$
branching ratio. Due to the small mass gap $\Delta M$, the produced pion
leaves a very short track at the collider. This kind of missing track searches
put an upper bound on the triplet fermion mass and the current bound is
$M_{\Sigma_{1}}>580$ GeV from ATLAS data with a luminosity of 136 $fb^{-1}$
[79] (as shown in Fig. (1)). Moreover, there is a future projection of HL-LHC
from $3\,\,ab^{-1}$ luminosity data which will probe the triplet fermion mass
up to $870$ GeV [80]. In this context it should be mentioned that considering
the bound given in Ref. [78], the lower bound on the triplet DM mass shifts to
$M_{\Sigma_{1}}\gtrsim 350$ GeV and in that case the bound from the LHC will
be important. Most importantly, the collider bound is hard to evade with the
introduction of beyond SM physics for the present setup, but we can easily
evade the indirect detection bound by introducing an additional dark
annihilation channel of DM. For example, we can introduce a dark abelian gauge
group under which $\Sigma_{1}$ is charged and then DM can annihilate into dark
gauge bosons and lower the indirect detection bound. Of course, in this kind
of setup, we also need to take into account the gauge anomaly which can be
addressed by introducing additional triplet fermion. The additional fermion
mass can be kept beyond the inflation scale which will isolate its
participation in our analysis. On the other hand, we can choose its mass at
the lower scale which will have an impact on our DM analysis. Quantitatively,
we anticipate that this kind of addition to our setup will change our DM
phenomenology minimally but qualitatively there will be no change in our
conclusion. This is just an example in the context of the present work to
evade the indirect detection bound if the present model is in conflict with
the future measurement of $\gamma-$ray data. It is to be mentioned that we
have not implemented this type of extension in our present study and it is
left for future exploration. Moreover, our leptogenesis study will remain
unaffected because the triplet fermions which take part in leptogenesis are
neutral under the dark abelian gauge group. For the lower mass range of DM, we
have considered the collider bound on the DM mass to be stricter than the
indirect detection bound, since the indirect detection bound can be evaded
following the techniques discussed above. On the other hand for the higher
mass range of DM, the indirect detection bound is more effective (or in other
words hard to evade) due to the occurrence of the Sommerfeld enhancement
peaks.

###  4.2 Evolution of comoving number density in non-standard cosmology

As discussed earlier the neutral component of the lightest triplet fermion,
$\Sigma^{0}_{1}$, can be a very good DM candidate when we consider the
corresponding Yukawa couplings are absent due to the imposition of
$\mathbb{Z}_{2}$ symmetry. Corresponding Boltzmann equation governing the
evolution of DM comoving number density (a ratio of number density over
entropy density s) is given by,

$\displaystyle\frac{dY}{dz}=-\frac{M_{\Sigma_{1}}}{z^{2}}\frac{1}{3H(T)}\frac{d\,{s}}{dT}\langle\sigma
v\rangle_{\rm eff}\left(Y^{2}-Y^{2}_{eq}\right)\,,$ (4.1)

where $z=\frac{M_{\Sigma_{1}}}{T}$ is a dimensionless quantity and
${s}(T)=g_{s}(T)\frac{2\pi^{2}}{45}T^{3}$ with $g_{s}(T)$ being the effective
degrees of freedom (d.o.f) related to the entropy density ${\rm s}$. The
comoving density can also be expressed as $Y=\sum_{i}Y_{i}$ where $i$ runs for
all the species in the dark sector i.e., $\Sigma^{0}_{1}$, $\Sigma^{+}_{1}$
and $\Sigma^{-}_{1}$. The thermal averaged effective annihilation cross-
section (denoted as $\langle\sigma v\rangle_{\rm eff}$) includes contributions
from annihilation as well as co-annihilations involving dark sector species.
The general expression of $\langle\sigma v\rangle_{\rm eff}$ is given by [81]

$\displaystyle\langle\sigma v\rangle_{\rm eff}$ $\displaystyle=$
$\displaystyle\sum_{i\,j}\langle\sigma
v\rangle_{ij}\dfrac{n^{eq}_{\Sigma^{i}_{1}}}{n^{eq}_{\Sigma_{1}}}\dfrac{n^{eq}_{\Sigma^{j}_{1}}}{n^{eq}_{\Sigma_{1}}}\,,$
(4.2) $\displaystyle=$ $\displaystyle\sum_{i\,j}\langle\sigma
v\rangle_{ij}\dfrac{g_{i}\,g_{j}}{g^{2}_{\rm
eff}}\left(1+\Delta_{i}\right)^{3/2}\left(1+\Delta_{j}\right)^{3/2}\exp\left[-x\,(\Delta_{i}+\Delta_{j})\right]\,,$

where, $n^{eq}_{\Sigma^{i}_{1}}$ is the equilibrium number density of the
species $i$ and thus the total equilibrium number density is given by
$n^{eq}_{\Sigma_{1}}=\sum_{i}n^{eq}_{\Sigma^{i}_{1}}$. The last step of the
above equation can be easily computed using the Maxwell-Boltzmann distribution
with $g_{i}$ being the internal degrees of freedom of species $i$ and
$\Delta_{i}=(M_{\Sigma^{i}_{1}}-M_{\Sigma_{1}})/M_{\Sigma_{1}}$ is the
fractional mass difference between the species $i$ and the lightest one which
is nothing but our DM candidate $\Sigma^{0}_{1}$. Therefore, $\Delta_{i}=0$
and $\Delta{M}/M_{\Sigma_{1}}$ for $i=\Sigma^{0}_{1}$ and $\Sigma^{\pm}_{1}$
respectively where $\Delta{M}=M_{\Sigma^{\pm}_{1}}-M_{\Sigma_{1}}\simeq 166$
MeV as mentioned earlier. The quantity $g_{\rm eff}$ in the denominator is
given by

$\displaystyle g_{\rm
eff}=\sum_{i}g_{i}\left(1+\Delta_{i}\right)^{3/2}\exp(-x\Delta_{i})\,.$ (4.3)

The thermal averaged cross-section when the species $i$ and $j$ are in the
initial state is denoted by $\langle\sigma v\rangle_{ij}$, where
$i=j=\Sigma^{0}_{1}$ represents DM annihilation while all other channels are
co-annihilations. In the model under consideration, DM can only annihilate
into $W^{+}W^{-}$ final state. On the other hand, there are several co-
annihilation channels like $\Sigma^{0}_{1}\,\Sigma^{\pm}_{1}\rightarrow
W^{\pm}Z$, $\Sigma^{+}_{1}\,\Sigma^{-}_{1}\rightarrow W^{+}W^{-}$,
$\Sigma^{+}_{1}\,\Sigma^{-}_{1}\rightarrow f\bar{f}$ ($f$ is the SM fermion),
$\Sigma^{\pm}_{1}\,\Sigma^{\pm}_{1}\rightarrow W^{\pm}W^{\pm}$ and
$\Sigma^{\pm}_{1}\,\Sigma^{\mp}_{1}\rightarrow W^{\pm}W^{\mp}$. These co-
annihilation channels have a great impact on the freeze-out dynamics of DM as
the mass gap between DM ($\Sigma^{0}_{1}$) and the next-to-lightest species
($\Sigma^{\pm}_{1}$) is around $166$ MeV only, which is less than $0.02\%$ of
$M_{\Sigma_{1}}=1$ TeV. The expressions of individual annihilation and co-
annihilation cross-sections along with $\langle{\sigma v}\rangle_{\rm eff}$ in
$s$-wave approximation are given in Appendix A.

In the Boltzmann equation, $H(T)$ is the Hubble parameter which depends on the
total energy content of the Universe. However, due to the unavailability of
experimental data before the BBN, the energy content of the Universe in that
epoch is not well known to us. In this work, we will exploit this freedom by
modifying the Hubble parameter before the BBN [48]. In particular, we consider
an extra species $\phi$ that increases the total energy density ($\rho$) of
the Universe by contributing dominantly before the BBN and $\phi$ red-shifts
faster than the radiation. As $\rho>\rho_{rad}$, the Hubble parameter is also
large compared to the standard case of the radiation domination and thus the
scenario is referred to as the fast expanding Universe in the literature [48].
The expansion rate of the Universe is quantified by the Hubble parameter which
is related to the total energy density of the Universe as,

$\displaystyle H(T)=\sqrt{\frac{8\pi\,G\,\rho(T)}{3}}\,,$ (4.4)

where $G$ is the Gravitational constant. The total energy density is now the
sum of two components, i.e,

$\displaystyle\rho(T)=\rho_{\rm rad}(T)+\rho_{\phi}(T)\,.$ (4.5)

The first term ($\rho_{\rm rad}$) is the radiation contribution to the energy
density, which dominates the total energy budget in the standard cosmology
before the era of matter-radiation equality ($T\simeq 0.8$ eV) and has the
following form,

$\displaystyle\rho_{\rm rad}(T)=\frac{\pi^{2}}{30}\,g_{*}(T)\,T^{4}\,,$ (4.6)

where, $g_{\star}(T)$ is the effective relativistic d.o.f related to the
radiation energy density at the concerned temperature. On the other hand, the
second term in Eq. (4.5), representing energy density of the species $\phi$,
red-shifts as [48],

$\displaystyle\rho_{\phi}\propto a^{-(4+n)}\,,$ (4.7)

with $n>0$. Since $\rho_{\phi}$ red-shifts faster ($\rho_{\phi}\propto
T^{4+n}$ as $a\propto T^{-1}$) than the radiation, we can have a Universe
dominated by the energy density of $\phi$ in the pre-BBN era and reduces to
the standard picture at $T\leq T_{r}$. Imposing the entropy conservation in a
unit co-moving volume, the energy density $\rho_{\phi}$ takes the following
form [48],

$\displaystyle\rho_{\phi}(T)=\rho_{\phi}(T_{r})\left(\frac{g_{s}(T)}{g_{s}(T_{r})}\right)^{\frac{4+n}{3}}\left(\frac{T}{T_{r}}\right)^{4+n}\,,$
(4.8)

where, the reference temperature $T_{r}$ is defined as the temperature where
contributions of the radiation and the species $\phi$ becomes equal and for
$T<T_{r}$ we have the usual radiation dominance. Finally, using Eqs. (4.6 and
4.8), the total energy density at any temperature can be written as

$\displaystyle\rho(T)$ $\displaystyle=$ $\displaystyle\rho_{\rm
rad}(T)+\rho_{\phi}(T)$ (4.9) $\displaystyle=$ $\displaystyle\rho_{\rm
rad}(T)\left[1+\dfrac{g_{*}(T_{r})}{g_{*}(T)}\left(\frac{g_{s}(T)}{g_{s}(T_{r})}\right)^{\frac{4+n}{3}}\left(\frac{T}{T_{r}}\right)^{n}\right]\,,$
$\displaystyle=$ $\displaystyle\rho_{\rm
rad}(T)\left[1+\left(\frac{g_{*}(T)}{g_{*}(T_{r})}\right)^{\frac{1+n}{3}}\left(\frac{T}{T_{r}}\right)^{n}\right]\,.$

In the last step, we consider $g_{*}(T)=g_{s}(T)$, which is valid before the
era of neutrino decoupling that takes place around $T\sim 1$ MeV. Therefore,
for $T>>T_{r}$, $\rho(T)>>\rho_{\rm rad}(T)$ and thus the Hubble parameter
gets enhanced according to Eq. (4.4), which has a great impact on the
evolution of particles in the early Universe. In this work, we will study the
impact of such non-standard cosmology on our DM candidate and also on the
matter anti-matter asymmetry of the Universe in the context of a typical BSM
model where the SM particle spectrum is extended by three additional SU(2)L
triplet fermions. As we mentioned earlier, this model potentially explains the
three important issues of the SM, namely the nature of dark matter, the origin
of tiny neutrino mass and the physics behind the observed asymmetry between
matter and anti-matter of the Universe.

####  4.2.1 Determining the freeze-out temperature $T_{f}$

As discussed in the preceding section, the presence of an extra species in the
early Universe modifies the Hubble parameter in the following manner,

$\displaystyle H(T)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{8\pi^{3}\,Gg_{*}(T)}{90}}\,T^{2}\left[1+\left(\frac{g_{*}(T)}{g_{*}(T_{r})}\right)^{\frac{1+n}{3}}\left(\frac{T}{T_{r}}\right)^{n}\right]^{1/2}\,,$
(4.10) $\displaystyle=$ $\displaystyle H_{\rm
rad}(T)\,\left[1+\left(\frac{g_{*}(T)}{g_{*}(T_{r})}\right)^{\frac{1+n}{3}}\left(\frac{T}{T_{r}}\right)^{n}\right]^{1/2}\,.$
(4.11)

In terms of $z\,\,(=\frac{M_{\Sigma_{1}}}{T})$, the Hubble parameter is given
by

$\displaystyle H(z)=H_{\rm rad}(z)f(z,z_{r},n)\,,$ (4.12)

where, the function $f(z,z_{r},n)$ (following definition of $z$,
$z_{r}=\frac{M_{\Sigma_{1}}}{T_{r}}$) is defined as

$\displaystyle
f(z,z_{r},n)=\left[1+\left(\frac{g_{*}(M_{\Sigma_{1}}/z)}{g_{*}(M_{\Sigma_{1}}/z_{r})}\right)^{\frac{1+n}{3}}\left(\frac{z_{r}}{z}\right)^{n}\right]^{1/2}\,,$
(4.13)

and $H_{\rm rad}$ is the Hubble parameter for the standard radiation dominated
Universe. Since the reference temperature $T_{r}<<T_{f}$, the function
$f(z,z_{r},n)$ can be significantly larger than unity and this results in an
enhanced Hubble parameter. The increase in the Hubble parameter will make the
freeze-out of dark matter happen earlier in time. In the thermal WIMP
paradigm, DM gets thermally decoupled from the cosmic fluid once the
interaction rate of DM falls below the corresponding expansion rate of the
Universe and we can determine the freeze-out temperature by solving the
following equation,

$\displaystyle H(T)=n^{eq}_{\Sigma^{0}_{1}}\langle\sigma v\rangle_{\rm
eff}\,\,{\rm at}\,\,T=T_{f}\,,$ (4.14)

where, $n^{eq}_{\Sigma^{0}_{1}}$ and $\langle{\sigma v}\rangle_{\rm eff}$ are
already defined earlier in Sec. 4.2. Upon simplifying777This simplification
can be found in appendix B ( Eq. (B.5) - Eq. (B.7) ). the above equality using
the expressions of Hubble parameter (Eq. (4.12)) in non-standard cosmology,
the equilibrium number density of the DM particle and effective annihilation
cross-section, we get the following transcendental equation for $z_{f}$,

$\displaystyle z_{f}\simeq 28-\ln\left[\frac{M_{\Sigma_{1}}}{1000\,\,{\rm
GeV}}\right]-\ln\left[f(z_{f},z_{r},n)\right]+\frac{1}{2}\ln\left[z_{f}\right]\,,$
(4.15)

which can be solved iteratively to obtain the freeze-out temperature
$T_{f}\,\,(=\frac{M_{\Sigma_{1}}}{z_{f}}\,)$. It is to be noted that Eq.
(4.15) contains an additional term (with negative sign), compared to its
standard counterpart, which depends on two non-standard parameters $n$,
$T_{r}$. In our study, we have focused on the regime where the non-standard
contribution is large, which implies a significant negative contribution due
to the $\ln\left[f(z_{f},z_{r},n)\right]$ term and hence, $z_{f}$ in our case
turns out to be much smaller than that of the standard radiation dominated
case. The lower value of $z_{f}$ clearly signifies that our DM candidate
freezes out at a higher temperature. So we can expect a higher relic abundance
than the standard cosmological scenario. It is worth mentioning here that in
the standard scenario, we could have lowered the $z_{f}$ value only by
increasing the DM mass $M_{\Sigma_{1}}$ (i.e. by reducing $\langle{\sigma
v}\rangle_{\rm eff}$). However, the introduction of non-standard cosmology,
where we have two additional parameters $n,\,T_{r}$, can result in a similar
kind of lowering in $z_{f}$. We will see later in this study that the
reduction in $z_{f}$ due to the non-standard parameters will help us obtaining
the correct value of relic density for the lower values of dark matter mass
which is not at all possible in the standard case of radiation domination.

####  4.2.2 Analytical approximation for DM relic density

We can solve the Boltzmann equation given in Eq. (4.1) analytically in the
limit where effective degrees of freedom $g_{*}$ is not changing significantly
with temperature, which is valid as long as the freeze-out occurs well above
the era of QCD phase transition ($T\sim 100$ MeV). The comoving number density
after freeze-out ($z>z_{f}$ or $T<T_{f}$) can be obtained easily as

$\displaystyle
Y_{\Sigma_{1}}\left(z\right)\simeq\left[\frac{1}{Y_{\Sigma_{1}}(z_{f})}+\sqrt{\frac{\pi}{45}}M_{\Sigma_{1}}\,M_{pl}\,J\left(z,z_{r},n\right)\right]^{-1},$
(4.16)

where $z=\frac{M_{\Sigma_{1}}}{T}$ as defined before, the function
$J(z,z_{r},n)$ contains $\langle{\sigma v}\rangle_{\rm eff}$ and it also
depends on the non-standard cosmology parameters in the following way

$\displaystyle J(z,z_{r},n)$ $\displaystyle=$
$\displaystyle\int^{z}_{z_{f}}\frac{g^{1/2}_{*}\,\beta(z^{\prime})\,\langle{\sigma
v}\rangle_{\rm eff}}{f(z^{\prime},z_{r},n)\,{z^{\prime}}^{2}}dz^{\prime}\,,$
$\displaystyle{\rm and}$ $\displaystyle\beta(z^{\prime})$ $\displaystyle=$
$\displaystyle 1-\frac{1}{3}\frac{d\ln g_{*}(z^{\prime})}{d\ln z^{\prime}}\,.$

Assuming $g_{*}$ does not change vigorously over the region of integration
(which also implies $\beta\simeq 1$) and the enhancement factor $f>>1$, the
above integration can be simplified as

$\displaystyle
J(z,z_{r},n)\simeq\left(\frac{g_{*}(M_{\Sigma_{1}}/z_{r})}{\bar{g_{*}}}\right)^{\frac{1+n}{6}}\sqrt{\frac{\bar{g_{*}}}{z^{n}_{r}}}\int_{z_{f}}^{z}\dfrac{\langle{\sigma
v}\rangle_{\rm eff}}{{z^{\prime}}^{2-n/2}}dz^{\prime}\,.$ (4.18)

In the above expression for $J$, if we neglect the first term within the
brackets and set $z_{r}=1$ and $n=0$, we will recover the function $J$ for the
radiation dominated Universe. In the present model, we have $s$ wave
annihilations and the expression of $\langle{\sigma v}\rangle_{\rm eff}$ under
$s$ wave approximation is given in Appendix A. This allows us to compute the
function $J$ analytically as

$\displaystyle J(z,z_{r},n)$ $\displaystyle\simeq$
$\displaystyle\left(\frac{g_{*}(M_{\Sigma_{1}}/z_{r})}{\bar{g_{*}}}\right)^{\frac{1+n}{6}}\sqrt{\frac{\bar{g_{*}}}{z^{n}_{r}}}\,\langle{\sigma
v}\rangle_{\rm
eff}\,\times\left\\{\begin{array}[]{ccc}\frac{2}{2-n}\left(\frac{1}{z^{1-n/2}_{f}}-\frac{1}{z^{1-n/2}}\right)&{\rm
for}&n\neq 2\\\ \ln(z/z_{f})&{\rm for}&n=2\end{array}\right.$ (4.21)

The presence of $z_{r}^{n/2}$ ($z_{r}>>1$) in the denominator makes the
function $J$ suppressed compared to the case of radiation domination which
eventually increases the comoving number density through Eq. (4.16) for a
fixed value of $\langle{\sigma v}\rangle_{\rm eff}$. The Eq. (4.18) matches
with Ref. [48] except the factor within the brackets which the authors have
considered $\sim\mathcal{O}(1)$. Finally, once we have the co-moving number
density then we can determine the DM relic density using the following
expression,

$\displaystyle\Omega_{\rm DM}h^{2}=2.755\times
10^{8}\left(\frac{M_{\Sigma_{1}}}{\rm GeV}\right)Y_{\Sigma_{1}}\left(z\gg
z_{f}\right)\,.$ (4.22)

####  4.2.3 BBN Bound

The extra species $\phi$ can contribute adequately to the total energy density
and hence to the Hubble parameter ($H$) during the epoch of BBN depending on
the parameters $T_{r}$ and $n$. The extra energy888The extra energy should be
red-shifted at least equal to or faster than the radiation energy.  that can
be accommodated at the time of BBN over the standard radiation background is
estimated in terms of parameter $N_{\rm eff}$ as

$\displaystyle\Delta{\rho}=\Delta{N_{\rm eff}}\,\rho_{\nu_{L}}\,,$ (4.23)

where, $\rho_{\nu}=\frac{7}{8}\,\alpha_{\nu/\gamma}\,\rho_{\gamma}$ is the
energy of a single species of active neutrino at BBN and
$\alpha_{\nu/\gamma}=\left(\frac{4}{11}\right)^{1/3}$ is the ratio of active
neutrino to photon temperature after neutrino decoupling around $T\lesssim$ 1
MeV999In our numerical analysis, we consider $\alpha_{\nu/\gamma}=1$.. The
allowed value of $\Delta{N_{\rm eff}}$ based on standard cosmological history
around the BBN is given by [82]

$\displaystyle\Delta{N_{\rm eff}}=N^{\rm BBN}_{\rm eff}-N_{\rm
SM}=0.374\,\,\text{at\,\,95\%\,\,C.L.}\,,$ (4.24)

where, theoretical prediction of $N_{\rm eff}$ based on the SM is $N_{\rm
eff}=3.046$ [83, 84] and $N^{\rm BBN}_{\rm eff}=2.88\pm 0.27$ at 68% C.L.
[82]. Now, assuming the extra contribution from $\phi$ remains within the
allowed limit gives the lower limit of the transition temperature $T_{r}$ for
a particular value of index $n$

$\displaystyle T_{r}\geq\left(\frac{g_{s}(T_{\rm
BBN})}{g_{s}(T_{r})}\right)^{\frac{4+n}{3n}}\left(\frac{g_{s}(T_{r})}{0.65\,\alpha^{4}_{\nu/\gamma}}\right)^{1/n}\,\,{\rm
MeV}\,.$ (4.25)

The above bound has been used in all the scatter plots and in particular a
sharp effect of this bound can be observed in the $n-T_{r}$ plane of RP of
Fig. (7) and LP of Fig. (10).

###  4.3 Numerical results: viability of fermion triplet DM

In this section, we are going to discuss the numerical findings of our study
with special emphasis on evolution of DM relic density in non-standard
cosmological background. We will proceed through graphical representation of
numerical results followed by meticulous analysis of those figures in order to
realize the impact of non-standard cosmology towards generation of adequate
relic abundance.

Various line plots systematically analyze the possibility of earlier freeze-
out of the DM in the modified cosmological scenario. The extent of non-
standard contribution is quantified by the parameters $T_{r},n$ which play a
pivotal role in modifying evolution of the DM relic abundance and thus the
frozen value of the relic density can be altered significantly by tuning the
said non-standard parameters. Larger relic abundance (compared to the standard
case), in agreement with the current experimental data, can be obtained for a
lower mass, of the dark matter, which is allowed by the indirect detection
bound as well as the bound obtained from collider searches. This entire
phenomenon has been depicted through numerous line plots in Sec. 4.3.1.

Relic density depends on the model parameters ($M_{\Sigma_{1}},T_{r},n$) in a
bit complicated manner. To understand the nature of this dependence, when the
parameters are varied simultaneously, we show scatter plots of frozen value of
the relic density with mass of the triplet DM, where the non-standard
parameters are shown by colour variation. Limiting values of these parameters
($M_{\Sigma_{1}},T_{r},n$), required for successful prediction of DM relic
density, have been estimated from similar scatter plots presented in the
following Sec. 4.3.1.

####  4.3.1 Graphical analysis of numerical results

Figure 2: LP and RP show the out of equilibrium check of DM for three
different values $n$ and $T_{r}$, respectively.

In the left panel (LP) of Fig. (2), we have shown variation in
$z-\frac{n_{eq}\langle\sigma{}v\rangle_{\rm eff}}{H(T)}$ plane where
$n_{eq}=\frac{g_{\Sigma}M^{3}_{\Sigma}K_{2}(z)}{2\pi^{2}z}$. Here $K_{2}(z)$
is the modified Bessel function of second kind, $H(T)$ is Hubble parameter at
temperature $T$ (defined in Eq. (4.4)) and $\langle\sigma{}v\rangle_{\rm eff}$
is the effective annihilation cross-section (defined in Eq. (LABEL:effective-
CS)). It is to be noted that the proposed DM candidate gets decoupled from the
thermal plasma as soon as the corresponding reaction rate becomes less than
the Hubble parameter at that instant. In the above figure, we have plotted the
ratio of these two quantities, the out of equilibrium condition is satisfied
whenever the value of this ratio falls below unity. Thus it is clear from the
figure (LP) that, as we increase the value of $n$, the DM goes out of
equilibrium for a lower value of $z$ (or equivalently for a higher value of
temperature $T$). In other words, the DM freezes earlier for higher values of
$n$. Hence it can be inferred that with the increment of non-standard
contribution, our DM goes out of equilibrium earlier and thus we will get
higher DM abundance (as obtained from Eq. (4.15)). On the other hand in the
right panel (RP), we have shown variation of the same quantity (as in LP) for
three different values of $T_{r}$ while $n$ is kept fixed. From the figure
(RP), it is clear that lowering the value of $T_{r}$ causes earlier freeze-
out, which implies a greater relic abundance of the DM.

Figure 3: LP: Variation of DM relic density with the DM mass for standard
cosmology (red dashed line) and non-standard cosmology (green solid line). RP:
Variation of DM relic density with $z$ for three different values of $n$.

In the LP and RP of Fig. (3), we have shown line plots in the
$M_{\Sigma_{1}}-\Omega_{\rm DM}h^{2}$ and $z-\Omega_{\rm DM}h^{2}$ planes,
respectively. In the LP, we have shown the dependence of relic density on DM
mass where the red dashed line corresponds to the pure triplet model with
standard cosmology 101010The line will be a little lower if we include the
Sommerfeld factor in the relic density calculation and the correct value of DM
relic density can be obtained for $2.7\lesssim M_{\Sigma_{1}}\,({\rm
TeV})\lesssim 3$ [85]. and the green solid line represents the case with non-
standard cosmological background before the BBN. The green region is ruled out
by Fermi-LAT due to the presence of the Sommerfeld enhancement spike around
that specific mass range. The magenta region is also ruled excluded by Fermi-
LAT and the cyan region is ruled out by the missing track search at the LHC.
From the figure, it is clear that we have a very narrow window between
$1000-1500$ GeV which is allowed after considering all the existing bounds
mainly the indirect detection bound. As discussed in Section 4.1, the indirect
detection bound can be evaded by introducing an additional annihilation mode
of DM. Therefore from the lower mass end, the collider bound
($M_{\Sigma_{1}}>580$ GeV) can be regarded as stricter than the indirect
detection bound, whereas it is difficult to escape the indirect detection
bound between $1.5$ TeV and $2.7$ TeV due to the presence of a spike (coming
from the Sommerfeld enhancement factor) in the annihilation cross-section
around the above mentioned mass range. We can see that without non-standard
cosmology it is very difficult to satisfy the DM relic density in the correct
ballpark value. On the other hand, if we introduce non-standard cosmology
before BBN then it helps to decouple the DM from the thermal bath earlier (can
be seen from Eq. (4.15)) and hence the higher value of relic abundance can be
obtained. Therefore, we can clearly see that for $n=2$ and $T_{r}=1.5$ GeV, we
can easily satisfy DM relic density for mass $M_{\Sigma_{1}}=950$ GeV. On the
other hand, in the RP, we have shown the DM relic density variation with $z$
for three different values of $n$. It is clear from the said figure that as we
increase the value of $n$, the DM relic abundance is also enhanced. This is in
agreement with the conclusion obtained from Fig. (2). The most important
information obtained from Fig. (3) is, in the non-standard scenario we can get
the correct value of DM relic density for DM mass $M_{\Sigma_{1}}=950$ GeV
which is not achievable for the same DM mass within the framework of standard
cosmology.

Figure 4: LP and RP show the evolution of DM relic density with $z$ for three
different values of DM mass $M_{DM}$ and reference temperature $T_{r}$,
respectively.

In the LP and RP of Fig. (4), we have shown the variation of DM relic density
with $z$ for three different values of DM mass $M_{\Sigma_{1}}$ (while
$T_{r},n$ kept fixed) and $T_{r}$ (while $M_{\Sigma_{1}},n$ kept fixed ),
respectively. It can be seen from the LP of the figure that increment of DM
mass enhances its relic density. This can be clearly understood from Eq. (A.1)
that thermal average cross-sections are inversely proportional to the square
of the DM mass and the relic density is inversely proportional to the thermal
average cross-sections (Eq. (4.16)). Therefore the DM relic density behaviour
with mass is justified. The red solid line signifies that correct value of DM
relic density can be obtained for $M_{\Sigma_{1}}=950$ GeV while the non-
standard parameters are kept at a value $T_{r}=1.5$ GeV and $n=2$. On the
other hand, in the RP, we have shown the DM relic density variation for three
different values of $T_{r}$. We have already discussed in the context of Fig.
(2) that smaller values of $T_{r}$ implies more non-standard contribution in
the DM production. The same behaviour of relic density is reflected in the RP
of Fig. (4), i.e if we lower the value of $T_{r}$, we will have more non-
standard contribution in DM relic density and vice versa. The $T_{r}$ and $n$
dependence of DM production will be more vivid when we discuss the scatter
plots among the different parameters in the forthcoming figures.

In this part, we are going to show a few scatter plots among the model
parameters which affect the DM relic density significantly. The relevant
parameters are $M_{\Sigma_{1}}$, $n$ and $T_{r}$ which have been varied in the
following range (keeping in mind the upper bound on DM mass and BBN bound on
$T_{r}$),

$\displaystyle 500\,\,{\rm GeV}\leq M_{\Sigma_{1}}\leq 1500\,\,{\rm
GeV}\,,10^{-4}\leq n\leq 5\,,\,\,{\rm and}\,\,10^{-3}\,\,{\rm GeV}\leq
T_{r}\leq 10^{3}\,\,{\rm GeV}\,.$ (4.26)

Figure 5: LP: Scatter plot in the $M_{\Sigma_{1}}-\Omega_{\rm DM}h^{2}$ plane
with the colour variation in the parameter $n$. In the RP, the same scatter
plot but the color variation represent the parameter $T_{r}$. In both panels,
the DM relic density has been assumed in the range $0.01$ to $0.12$.

In the LP of Fig. (5), we have shown scatter plot in the
$M_{\Sigma_{1}}-\Omega_{\rm DM}h^{2}$ plane allowing only those points for
which relic density lies within the range of $0.01-0.12$. Moreover, It should
be noted that in the scatter plots we have chosen strictly those combinations
of $(T_{r},n)$ which respect the BBN bound given in Eq. (4.25). Basically, the
plots of LP and RP (in Fig. 5) both represent exactly the identical data-set
composed of $(T_{r},n,M_{\Sigma_{1}},\Omega_{\rm DM}h^{2})$. In LP variation
of $n$ is shown by colour bar whereas in RP same technique is used to
designate $T_{r}$. The pink shaded111111The pink shaded regions in the other
plots imply the same thing. region is already ruled out by the $136$ $fb^{-1}$
luminosity data of LHC. The blue solid line (in LP as well as RP of Fig. (5))
corresponds to the DM relic density variation with DM mass without any non-
standard contribution. The $n$ value of any particular point in the plot can
be retrieved from its colour. The horizontal colour bar shows continuous
variation of $n$ with colour. Before explaining this figure if we look at Eq.
(4.9), we see that for $n>1$ we can have a sizeable contribution, from the
non-standard part, comparable to the standard part. On the other hand, if we
take $T_{r}\gg 1$ the non-standard contribution, to the DM evolution, can be
reduced to a large extent. In the figure, the lower sharp line (coinciding
with the blue line) corresponds to the DM relic density for corresponding DM
masses when non-standard contribution is negligible i.e.

$\lim_{n>1,T_{r}\gg
1}\left(\frac{g_{*}(T)}{g_{*}(T_{r})}\right)^{\frac{1+n}{3}}\left(\frac{T}{T_{r}}\right)^{n}\ll
1\,.$

It can also be seen from the figure that the green points correspond to $n<1$
which gives more contribution than the standard contribution because of less
suppression due to the aforementioned effect. But as we have seen in Fig. (2),
the larger values of $n$ imply more contribution from the non-standard
cosmology and this is also consistent with the present figure because if we
take a fixed value of $M_{\Sigma}$ and move along the ordinate then the DM
relic density increases with the increase of $n$. In the RP of the same figure
we have used similar kind of colour variation to bring out the nature $T_{r}$
dependence of relic density. The dark-pink points, represent higher values of
$T_{r}$, give a smaller contribution in DM relic density which is reflected in
the region of negligible non-standard contribution. In the RP, for a fixed
value of $M_{\Sigma}$, if we move along the y coordinate we get an increment
in DM relic density which corresponds to lower values of $T_{r}$ as
represented by the colour variation. In both the figures, It has been checked
explicitly that, for $n<0.4$ we can never achieve the correct value of DM
relic density for the allowed DM mass range. We can’t go below $n=0.4$ since
it would result in a large value of $T_{r}$ (obtained from the bound of Eq.
(4.25)) which again tends to nullify the non-standard effect.

Figure 6: LP and RP show the scatter plots in the $M_{\Sigma_{1}}-n$ and
$M_{\Sigma_{1}}-T_{r}$ plane after demanding DM relic density in the range
$0.05$ to $0.12$. In both the panel the color variation shows the variation of
DM relic density.

In the LP of Fig. (6), we have shown the scatter plot in $M_{\Sigma_{1}}-n$
plane where the colour bar corresponds to the DM relic density in the range
$0.05-0.12$ and like Fig. (5), we have also demanded that all the points
satisfy BBN bound represented by Eq. (4.25). We know from the LP of Fig. (3)
that without non-standard cosmology $M_{\Sigma_{1}}=1200$ GeV corresponds to
DM relic density $\Omega_{\rm DM}h^{2}\sim 0.03$. Thus from Fig. (6), it is
clear that if the DM mass is in the range of $500-1200$ GeV and the non-
standard parameter in the range $0.1\leq n\leq 0.4$, we can not have DM relic
density around 0.05 for the whole range of $T_{r}$ considered in our work and
therefore this region is not allowed. The correct value of DM relic density
can be achieved for the whole range of DM mass if we are in the region $n>1$
whereas the value of $n$ can be as low as $\sim 0.4$ if we take
$M_{\Sigma_{1}}\sim 1500$ GeV. For the range $1200<M_{\Sigma_{1}}({\rm
GeV})<1500$ and $0.1<n<0.4$, the non-standard is not sufficient to produce the
correct value of DM relic density. On the other hand in the RP we have shown
the variation of the parameters $M_{\Sigma}-T_{r}$ which produce DM relic
density in the range of $0.05-0.12$. It can be noticed from the figure that
for the range $10<T_{r}({\rm GeV})<1000$ and $500<M_{\Sigma_{1}}({\rm
GeV})<1200$, the non-standard contribution is not sufficient to raise the DM
relic density up to $\Omega_{\rm DM}h^{2}=0.05$. On the other hand, if we take
$T_{r}$ below $10$ GeV, we can always obtain the correct value of DM relic
density. For the range $10<T_{r}({\rm GeV})<1000$ and
$1200<M_{\Sigma_{1}}({\rm GeV})<1500$, the non-standard contribution is not
sufficient to have the correct value of DM relic density.

Figure 7: Scatter plot in the $T_{r}-n$ plane after demanding DM relic density
in the $0.05-0.12$. The colour variation represents DM relic density.

In the LP of Fig. (7), we have shown variation in the $T_{r}-n$ plane upon
imposing the constraint that DM relic density (which is depicted in the colour
bar) should lie within the range $0.05-0.12$. The top-left corner region
represents an overabundant region, i.e. the DM relic density $\Omega_{\rm
DM}h^{2}>0.12$. This can be easily understood since the region corresponds to
a large non-standard contribution which drives the relic abundance towards a
higher value. On the contrary, we do not have any points in the top-right
corner because that region has less contribution from non-standard cosmology
due to the high value of $T_{r}$. Thus our DM is severely underabundant in
that region ($\Omega_{\rm DM}h^{2}<0.05$). We also have discussed before that
we can never get the correct value of relic density ($0.117\leq\Omega_{\rm
DM}h^{2}\leq 0.123$) for $M_{\Sigma_{1}}\leq 1.5$ TeV if we are below $n=0.4$.
This figure also confirms the fact that the region below $T_{r}\lesssim 10$
GeV and $n>0.4$ can reproduce the correct value of DM relic density. In the
RP, we have shown the allowed region in $n-T_{r}$ plane that satisfies DM
relic density in the $3\sigma$ range only. While doing this scanning, we have
varied all three variables $n$, $T_{r}$ and $M_{\Sigma_{1}}$ simultaneously.
The corresponding variation of DM mass has been indicated by the colour bar in
the range $500\,\,{\rm GeV}\leq M_{\Sigma_{1}}\leq 1500$ GeV. The region
disallowed by BBN for producing excess $\Delta{N}_{\rm eff}$ (as given in Eq.
(4.25)) is shown by the dark cyan colour. From this plot, one can easily
notice that a larger DM mass is required to satisfy $\Omega_{\rm DM}h^{2}$ in
$3\sigma$ range when the effect of non-standard cosmology is small i.e. for
the lower values of $n$ and the higher values of $T_{r}$. Similarly, as we
increase the DM mass, the correct value of DM density can be achieved for a
bit higher value of $T_{r}$. However, once we go beyond $T_{r}\simeq 20$ GeV
then one can never achieve the correct DM relic abundance for any value of $n$
when $M_{\Sigma_{1}}\leq 1.5$ TeV. In the later part of the article, we will
further constrain the $n-T_{r}$ parameter space once we impose the BAU bound.

##  5 Baryogenesis through Type-III seesaw leptogenesis

The popular Type-I seesaw [27, 28, 29, 30] mechanism (SM + three generations
of a gauge singlet heavy fermion), is primarily introduced to explain the non-
zero mass of the active neutrinos, it has the potential to account for the
observed baryon asymmetry of the Universe. Presence of Majorana type mass term
in the Lagrangian ensures the lepton number violation by two units. The CP
violating out of equilibrium decay of the singlet fermions (right handed
neutrinos) to lepton Higgs pair gives rise to lepton asymmetry which gets
converted into baryon asymmetry by sphaleron interactions [86].

If the said fermion singlet is replaced by SU(2)L triplet fermion, the
Majorana mass term and the Yukawa coupling term seems to be identical to that
of the singlet except the fact that the Yukawa term now consists of three fold
extra terms (which signifies the three components of the triplet) compared to
its singlet counterpart. Unlike the singlets, the fermion triplets have gauge
interactions. At high temperature these gauge interactions are strong enough
to promptly drive the triplet abundance to its equilibrium value. Thus in this
case the final asymmetry is almost independent of the initial condition (i.e.
whether we start from vanishing /equilibrium initial abundance).

The preliminary objective of this work is to assess the viability of fermion
triplet DM (which is very tightly constrained by indirect detection searches
of DM and collider searches) in a non-standard cosmological background. The
neutral component of the lightest triplet acts as a DM candidate since it is
odd under $\mathbb{Z}_{2}$ discrete symmetry, whereas the heavier generations
of the triplet as well as the whole SM particle content are even under
$\mathbb{Z}_{2}$. The next to lightest triplet, having normal Yukawa coupling
can decay to lepton, Higgs pair to produce the CP asymmetry required to
explain the BAU. Existing studies [68, 69, 70] in the literature dealing with
hierarchical fermion triplet leptogenesis have already set a lower bound on
the mass of the lightest decaying triplet to be $M_{\Sigma}\gtrsim 3\times
10^{10}$ GeV121212 This is similar to the Davidson-Ibarra bound [87] obtained
in case of singlet, i.e $M_{N_{1}}\gtrsim 10^{9}$ GeV.. It is to be noted that
the said lower bound also depends on the effective neutrino mass parameter
$(\tilde{m})$ and its specific value mentioned above can be obtained for
$\tilde{m}\sim 0.001$ eV. It can be found from the concerned plot [68, 70] of
allowed (by BAU bound) parameters in $M_{\Sigma_{2}}-\tilde{m}$ plane, that
the lowest allowed value of $M_{\Sigma_{2}}$ lies near $10^{11}$ GeV over a
wide range of $\tilde{m}$. Thus in the remaining portion of our analysis
$10^{11}$ GeV is referred to as the lower limit of $M_{\Sigma_{2}}$ for
standard cosmology.

Now in context of non-standard cosmology, it will be an intriguing study to
examine whether the above mentioned bound can be lowered further in current
scenario. Implications of the baryon asymmetry bound on the free parameters
$(T_{R},n)$ of non-standard cosmology is also worth studying.

###  5.1 CP asymmetry

The fermion triplet $\Sigma_{i}=\Sigma_{R\,i}+{\Sigma_{R\,i}}^{c}$ is actually
composed of three components and in two by two notation it is expressed as

$\Sigma_{i}=\left(\begin{array}[]{cc}\Sigma^{0}_{i}/\sqrt{2}&\Sigma^{+}_{i}\\\
\Sigma^{-}_{i}&-\Sigma^{0}_{i}/\sqrt{2}\end{array}\right)~{}.$ (5.1)

Among these three components
$\left(\Sigma^{+}_{i},\Sigma^{0}_{i},\Sigma^{-}_{i}\right)$, the neutral one,
which is a Majorana fermion, mimics the behaviour of right handed neutrino as
in the case of Type-I seesaw. The charged states,
$\Sigma^{\pm}_{i}=\Sigma^{\pm}_{R\,i}+{\Sigma^{\mp}_{R\,i}}^{c}$, are the
Dirac fermions. Now, due to the SU(2)L symmetry, it can be argued that decay
widths (to lepton Higgs pair) of all three components should be equal and are
given by

$\Gamma_{\Sigma^{0}_{i}}=\Gamma_{\Sigma^{+}_{i}}=\Gamma_{\Sigma_{i}^{-}}=\frac{M_{\Sigma_{i}}}{8\pi}|y^{\dagger}_{\Sigma}y_{\Sigma}|_{ii},$
(5.2)

where ‘$i$’ is the generation index of the triplet. Thus, for our future
calculations we will use the notation $\Gamma_{\Sigma_{i}}$ (omitting the
‘charge’ index) for the decay width of a specific generation of the triplet
irrespective of its charge. It is evident that the thermally averaged decay
rate (Eq. (C.2)) becomes three times to that of the well known singlet right
handed neutrino, since the equilibrium number density of the decaying triplet
(considering the neutral and charged components) is simply three times of the
right handed neutrino. The CP asymmetry parameter [88], quantified by the
difference between decay rates of ‘heavy triplet to lepton Higgs pair’ and its
CP conjugate process, is expressed mathematically as

$\epsilon_{\Sigma_{i}}=\frac{\Gamma\left(\Sigma_{i}\longrightarrow
L~{}H\right)-\Gamma\left(\overline{\Sigma_{i}}\longrightarrow\bar{L}~{}\bar{H}\right)}{\Gamma\left(\Sigma_{i}\longrightarrow
L~{}H\right)+\Gamma\left(\overline{\Sigma_{i}}\longrightarrow\bar{L}~{}\bar{H}\right)}~{},$
(5.3)

where the asymmetry parameter includes contribution from all three components
of the triplet. The generic expression for CP asymmetry produced due to the
decay of the next-to-lightest triplet ($\Sigma_{2}$) is given by [68, 69, 70]

$\epsilon_{\Sigma_{2}}=\frac{3}{2}\left(\frac{M_{\Sigma_{2}}}{M_{\Sigma_{3}}}\right)\left(\frac{\Gamma_{\Sigma_{3}}}{M_{\Sigma_{3}}}\right)I_{\Sigma}\left(\frac{f_{v}-2f_{s}}{3}\right)~{},$
(5.4)

where

$\displaystyle
I_{\Sigma}=\frac{\mathcal{I}m\left[\left(y^{\dagger}_{\Sigma}y_{\Sigma}\right)_{23}^{2}\right]}{|y^{\dagger}_{\Sigma}y_{\Sigma}|_{22}|y^{\dagger}_{\Sigma}y_{\Sigma}|_{33}}$
$\displaystyle{\rm and}$
$\displaystyle\frac{\Gamma_{\Sigma_{3}}}{M_{\Sigma_{3}}}=\frac{|y^{\dagger}_{\Sigma}y_{\Sigma}|_{33}}{8\pi}~{}.$
(5.5)

$f_{v}$ and $f_{s}$ are vertex and self energy correction functions given by

$\displaystyle
f_{v}=\frac{{M_{\Sigma_{3}}}^{2}({M_{\Sigma_{3}}}^{2}-{M_{\Sigma_{2}}}^{2})}{({M_{\Sigma_{3}}}^{2}-{M_{\Sigma_{2}}}^{2})^{2}+{M_{\Sigma_{2}}}^{2}{\Gamma_{\Sigma_{3}}}^{2}}~{},$
(5.6) $\displaystyle
f_{s}=2\frac{{M_{\Sigma_{3}}}^{2}}{{M_{\Sigma_{2}}}^{2}}\left[\left(1+\frac{{M_{\Sigma_{3}}}^{2}}{{M_{\Sigma_{2}}}^{2}}\right)\ln\left(1+\frac{{M_{\Sigma_{3}}}^{2}}{{M_{\Sigma_{2}}}^{2}}\right)-1\right]~{}.$
(5.7)

When the triplet mass spectrum is strongly hierarchical, effectively the
asymmetry is produced (and survives till present epoch) solely due to the
decay of lightest generation of $\mathbb{Z}_{2}$ even triplet, which is
$\Sigma_{2}$ in this case. So we have to deal with $\epsilon_{\Sigma_{2}}$
only. For the sake of notational simplicity, from now on we will omit the
subscript ‘$2$’ in $\epsilon_{\Sigma_{2}}$. In strongly hierarchical limit
both the above mentioned loop functions reduce to unity. Therefore the
expression for CP asymmetry (Eq. (5.4)) simplifies to

$\displaystyle\epsilon_{\Sigma}$ $\displaystyle=$
$\displaystyle\frac{3}{2}\frac{M_{\Sigma_{2}}}{M_{\Sigma_{3}}}\frac{\Gamma_{\Sigma_{3}}}{M_{\Sigma_{3}}}I_{\Sigma}\left(-\frac{1}{3}\right)$
(5.8) $\displaystyle=$
$\displaystyle-\frac{1}{16\pi}\frac{M_{\Sigma_{2}}}{M_{\Sigma_{3}}}\frac{\mathcal{I}m\left[\left(y^{\dagger}_{\Sigma}y_{\Sigma}\right)_{23}^{2}\right]}{|y^{\dagger}_{\Sigma}y_{\Sigma}|_{22}}~{},$

which is exactly $1/3$ of the CP asymmetry obtained from the decay of
hierarchical RH neutrinos in case of Type-I seesaw mechanism. Following the
same arguments as that of the fermion singlet, the theoretically allowed
maximum value of CP asymmetry turns out to be [68, 69, 70]

$\displaystyle\epsilon^{\rm max}_{\Sigma}$ $\displaystyle=$
$\displaystyle\frac{M_{\Sigma_{2}}}{8\pi v^{2}}(m_{3}-m_{1})$ (5.9)
$\displaystyle=$ $\displaystyle\frac{M_{\Sigma_{2}}}{8\pi
v^{2}}\left(\frac{\Delta{m^{2}}_{31}}{m_{3}+m_{1}}\right)~{},$

where $m_{i}(i=1,2,3)$ are the light neutrino mass eigenvalues131313Since two
triplet fermions take part in the active neutrino mass generation so the mass
of the lightest component among the active neutrinos would be zero, i.e.,
$m_{1}=0$. and $v(=246{\rm GeV})$ is the SM VEV. To get an estimate for the
lowest value of $M_{\Sigma_{2}}$ that can produce adequate asymmetry to meet
the observed value of baryon asymmetry, this above mentioned upper limit of CP
asymmetry has to be used in the Boltzmann equations. A few earlier studies
[68, 69, 70] dealing with fermion triplet leptogenesis have shown that the
concerned lower limit on the mass of the lightest triplet lies near $10^{11}$
GeV.

###  5.2 Boltzmann equations for leptogenesis in standard cosmology

The set of Boltzmann equations are used to track the evolution of lepton
asymmetry with temperature as the Universe gradually cools down. Let us define
few important parameters, necessary for writing the Boltzmann equations.
Instantaneous abundance of a particle species is conventionally expressed in
terms of scaled number density $Y_{a}(z)(=\frac{n_{a}(z)}{s(z)})$, where
$n_{a}(z)$ is the number density of the particle species ‘$a$’, $s(z)$ is the
entropy density and $z$ is a dimensionless parameter
($z=M_{\Sigma_{2}}/T$)141414We have used this definition of $z$ only for
leptogenesis analysis. It should not be confused with the $z$ used in analysis
of DM. varying inversely with the temperature of the Universe. Since all three
components of the triplet contribute equally to the asymmetry, it would be
more convenient to handle the set of Boltzmann equations in terms of total
number density of a specific triplet generation, i.e
$n_{\Sigma_{i}}=n_{\Sigma^{0}_{i}}+n_{\Sigma^{+}_{i}}+n_{\Sigma^{-}_{i}}$. In
the present case, the asymmetry is assumed to be generated solely due to the
decay of $\Sigma_{2}$. Therefore we are concerned with the number density
$n_{\Sigma_{2}}$ only151515Expressions for equilibrium number densities of
different particle species are given in Appendix B.. Since there is no other
generation of the triplet involved in the Boltzmann equations, we can safely
use the notation $n_{\Sigma}$ for $n_{\Sigma_{2}}$. The evolution of the
triplet abundance and the corresponding $(B-L)$ asymmetry are governed by the
following set of Boltzmann equations [70]

$\displaystyle
sHz\frac{dY_{\Sigma}}{dz}=-\gamma_{D}\left(\frac{Y_{\Sigma}}{Y^{eq}_{\Sigma}}-1\right)-2\gamma_{A}\left(\frac{{Y_{\Sigma}}^{2}}{{Y^{eq}_{\Sigma}}^{2}}-1\right),$
(5.10) $\displaystyle
sHz\frac{dY_{B-L}}{dz}=-\gamma_{D}\epsilon_{\Sigma}\left(\frac{Y_{\Sigma}}{Y^{eq}_{\Sigma}}-1\right)-\frac{Y_{B-L}}{Y^{eq}_{l}}\left(\frac{\gamma_{D}}{2}+\gamma^{sub}_{\Sigma}\right),$
(5.11)

where $\gamma$ generically designates the reaction density161616Detailed
expression is given in Appendix C. of the process under consideration and the
subscript ‘$D$’ denotes the decay process, whereas ‘$A$’ stands for $2$
$\rightarrow$ $2$ gauge boson mediated scattering processes. The
$\gamma^{sub}_{\Sigma}$ term accounts for the contribution from $\Delta L=2$
scattering processes where the resonant contribution has already been
subtracted. It is clear from the above set of Boltzmann equations that there
is an extra term (compared to that of Type-I seesaw) involving $\gamma_{A}$ in
the first equation. It appears since the fermion triplet can couple to the
gauge bosons unlike the gauge singlet fermions. The above set of Boltzmann
equations can be rewritten in a convenient form as

$\displaystyle\frac{dY_{\Sigma}}{dz}=-D(z)\left(Y_{\Sigma}(z)-Y^{eq}_{\Sigma}(z)\right)-{S_{A}}(z)\left({Y_{\Sigma}(z)}^{2}-{Y^{eq}_{\Sigma}(z)}^{2}\right),$
(5.12)
$\displaystyle\frac{dY_{B-L}}{dz}=-\epsilon_{\Sigma}D(z)\left(Y_{\Sigma}(z)-Y^{eq}_{\Sigma}(z)\right)-\left(W(z)+W^{sub}(z)\right)Y_{B-L}(z),$
(5.13)

where different decay and scattering terms are restructured and can be
expressed in terms of some fundamental constants and parameters of the model
as follows. Decay $(D)$ term and gauge boson mediated scattering ($S_{A}$)
terms are simplified as

$\displaystyle D(z)$ $\displaystyle=$
$\displaystyle\frac{\gamma_{D}(z)}{s(z)H(z)zY^{eq}_{\Sigma}(z)}$ (5.14)
$\displaystyle=$
$\displaystyle\frac{n^{eq}_{\Sigma}(z)\Gamma_{\Sigma}\frac{K_{1}(z)}{K_{2}(z)}}{s(z)H(z)z\frac{n^{eq}_{\Sigma}(z)}{s(z)}}$
$\displaystyle=$ $\displaystyle\left(\frac{M_{pl}}{8\pi
v^{2}1.66{g_{\ast}}^{1/2}}\right)\tilde{m}z\frac{K_{1}(z)}{K_{2}(z)}$

where $\tilde{m}=8\pi\Gamma_{\Sigma}\frac{v^{2}}{{M_{\Sigma_{2}}}^{2}}$. The
thermal reaction rate of gauge boson mediated two body scattering process
$\gamma_{A}$, involves integral of the reduced cross-section ($\hat{\sigma}$),
of the relevant process, over the centre of mass energy of the initial
particles, is computed as

$\displaystyle\gamma_{A}(z)$ $\displaystyle=$
$\displaystyle\frac{{M_{\Sigma_{2}}}^{4}}{64\pi^{4}z}\int\limits^{\infty}_{x_{min}}\sqrt{x}K_{1}(z\sqrt{x})\hat{\sigma_{A}}(x)dx$
$\displaystyle=$ $\displaystyle\frac{{M_{\Sigma_{2}}}^{4}}{64\pi^{4}z}I(z)\,,$
$\displaystyle{\rm where,}$ $\displaystyle I(z)$ $\displaystyle=$
$\displaystyle\int\limits^{\infty}_{x_{min}}\sqrt{x}K_{1}(z\sqrt{x})\hat{\sigma_{A}}(x)dx\,,$

and the integration variable $x$ is defined as the ratio of centre of mass
energy ($s$) to the square of decaying particle’s mass $(M_{\Sigma_{2}})$. The
minimum value of centre of mass energy for a generic two body process
$\left(a,b\rightarrow i,j,....\right)$ is given by $s_{min}={\rm
max}\left[(m_{a}+m_{b})^{2},(m_{i}+m_{j}+...)^{2}\right]$. The analytical
expression for the reduced cross-section is given in Appendix D. The gauge
mediated scattering term ($S_{A}$) of Eq. (5.12) can be written as a function
of the above mentioned integral as

$\displaystyle S_{A}(z)$ $\displaystyle=$
$\displaystyle\frac{2\gamma_{A}(z)}{s(z)H(z)z{Y^{eq}_{\Sigma}(z)}^{2}}$ (5.16)
$\displaystyle=$
$\displaystyle\left(\frac{\pi^{2}g^{1/2}_{\ast}M_{pl}}{1.66\times
180g^{2}_{\Sigma}}\right)\frac{1}{M_{\Sigma_{2}}}\left(\frac{I(z)}{z{K_{2}(z)}^{2}}\right)\,,$

where $g_{\Sigma}$ is the internal degrees of freedom of the Majorana type
fermion triplet $\Sigma$. Similarly the washout terms ( $W$ and $W^{sub}$ of
Eq. (5.13) ) due to inverse decay and $\Delta L=2$ scattering (off-shell)
processes are given by

$\displaystyle W(z)$ $\displaystyle=$
$\displaystyle\left(\frac{3M_{pl}}{1.66\times 32\pi
v^{2}g^{1/2}_{\ast}}\right)\tilde{m}z^{3}K_{1}(z)\,,$ (5.17) $\displaystyle
W^{sub}(z)$ $\displaystyle=$ $\displaystyle\left(\frac{M_{pl}}{1.66\times
64\pi^{2}g^{1/2}_{\ast}}\right)\frac{z^{3}}{{M_{\Sigma_{2}}}}\left\\{\int\limits^{\infty}_{x_{min}}\sqrt{x}K_{1}(z\sqrt{x})\left(\hat{\sigma_{s}}(x)+\hat{\sigma_{t}}(x)\right)dx\right\\}$
(5.18) $\displaystyle=$ $\displaystyle\left(\frac{M_{pl}}{1.66\times
64\pi^{2}g^{1/2}_{\ast}}\right)\frac{z^{3}}{{M_{\Sigma_{2}}}}\left(I_{s}(z)+I_{t}(z)\right)$

with $I_{s},I_{t}$ being integral of the $s$ channel171717The resonance
contribution has been properly subtracted for the $s$ channel process.
Analytical expressions for $\hat{\sigma_{s}}(x),\hat{\sigma_{t}}(x)$ are given
in Appendix D.  and $t$ channel (reduced) cross-sections.

####  5.2.1 Baryon asymmetry and efficiency factor at present epoch

At relatively lower (than the leptogenesis scale, i.e $T\ll M_{\Sigma_{2}}$)
temperature (or equivalently $z\gg 1$ ) when the strength of lepton/baryon
number violating interactions become negligibly small, $Y_{B-L}$ freezes to a
certain value and remains constant till the present epoch. Assuming vanishing
preexisting asymmetry (i.e $Y_{B-L}(z_{i})=0$) the set of Boltzmann equations
(Eqs. (5.12 and 5.13)) can be solved up to a arbitrary value of $z$ to obtain

$\displaystyle Y_{B-L}(z)$ $\displaystyle=$
$\displaystyle-\epsilon_{\Sigma}Y^{eq}_{\Sigma}(z\ll
1)\left\\{-\frac{1}{Y^{eq}_{\Sigma}(z\ll
1)}\int\limits^{z}_{z_{i}}dz^{\prime}\frac{dY_{\Sigma}}{dz^{\prime}}e^{-\int\limits^{z}_{z^{\prime}}\left(W(z^{\prime\prime})+W^{sub}(z^{\prime\prime})\right)dz^{\prime\prime}}\right\\}$
(5.19) $\displaystyle=$ $\displaystyle-\epsilon_{\Sigma}Y^{eq}_{\Sigma}(z\ll
1)\kappa(z)~{},$

where the integral inside the curly bracket gives efficiency factor evaluated
at any arbitrary $z$. If we assume $z_{f}$ to be the value of $z$ where $B-L$
asymmetry gets frozen, i.e

$\displaystyle Y_{B-L}(z_{f})$ $\displaystyle=$
$\displaystyle-\epsilon_{\Sigma}Y^{eq}_{\Sigma}(z\ll 1)\kappa(z_{f})$
$\displaystyle{\rm or},~{}Y^{f}_{B-L}$ $\displaystyle=$
$\displaystyle-\epsilon_{\Sigma}Y^{eq}_{\Sigma}(z\ll 1)\kappa^{f}~{}.$ (5.20)

The so called sphaleron process converts this frozen value of $B-L$ asymmetry
to baryon asymmetry which is connected to $Y^{f}_{B-L}$ through the
multiplicative sphaleron transition factor $a_{sph}=28/79$, i.e,

$\displaystyle Y_{B}(z)$ $\displaystyle=$ $\displaystyle
a_{sph}Y_{B-L}(z)=-a_{sph}\epsilon_{\Sigma}Y^{eq}_{\Sigma}(z\ll
1)\kappa(z)~{}~{}~{}{\rm and}$ (5.21) $\displaystyle Y^{f}_{B}$
$\displaystyle=$ $\displaystyle
a_{sph}Y_{B-L}(z_{f})=-a_{sph}\epsilon_{\Sigma}Y^{eq}_{\Sigma}(z\ll
1)\kappa^{f}~{}.$ (5.22)

The viability of our model can be tested through a comparison between the
theoretically predicted value of baryon asymmetry ($Y^{f}_{B}$) with the
observed one ($Y^{obs}_{B}$) which ranges between $8.9\times
10^{-11}<Y^{obs}_{B}<9.2\times 10^{-11}$ at $95\%$ confidence level [6].

####  5.2.2 Modification of the Boltzmann equations in non-standard cosmology

It has already been mentioned in Sec. 4.2 that, in non-standard cosmological
scenario the Hubble parameter is modified due to the non-trivial contribution
of the $\phi$ field in total energy density. It is well known that above the
scale of electroweak (EW) symmetry breaking all the SM particles can be
treated to be relativistic and thereby allows us to assume $g_{\ast}(T>T_{\rm
EW})=g_{SM}\simeq 106.75$. Since the processes relevant to generation and
washout of lepton asymmetry take place much above the EW scale, $g_{\ast}(T)$
in that temperature regime behaves as a constant with a value $g_{SM}$. In
present scenario the total energy density (Eq. (4.9)) includes additional
(apart from the well known radiation energy) temperature dependent terms
involving non-standard parameters $T_{r},n$. It affects the Hubble parameter
directly by introducing a steeper temperature dependence as already shown in
Eq. (4.12), i.e $H(z)=H_{\rm rad}(z)f(z,z_{r},n)$ (with $f(z,z_{r},n)>1$). The
particle physics inputs (i.e decay widths and scattering cross-sections) in
the Boltzmann equations are totally unaffected by the changes incorporated in
the cosmology. Therefore it can be stated unambiguously that in spite of
introduction of modified cosmology, the basic structure of the Boltzmann
equations (Eqs. (5.12 and 5.13)) remains unaltered, except the decay and
scattering terms which are modulated by non-standard effects through a $z$
dependent term. Lets us first inspect the non-trivial modification in the
decay term, i.e.

$\displaystyle\tilde{D}(z)$ $\displaystyle=$
$\displaystyle\frac{\gamma_{D}(z)}{s(z)H(z)zY^{eq}_{\Sigma}(z)}$ (5.23)
$\displaystyle=$
$\displaystyle\left(\frac{\gamma_{D}(z)}{s(z)H_{rad}(z)zY^{eq}_{\Sigma}(z)}\right)\frac{1}{f(z,z_{r},n)}\,,$
$\displaystyle=$ $\displaystyle\frac{D(z)}{f(z,z_{r},n)}~{}.$

Analogy between the structure of various decay and scattering terms guide us
to realize that all the concerned terms ($D,S_{A},W,W^{sub}$) will be modified
(to $\tilde{D},\tilde{S}_{A},\tilde{W},\tilde{W}^{sub}$ ) exactly in a similar
manner through the common multiplicative factor $1/f(z,z_{r},n)$, where

$f(z,z_{r},n)=\left[1+\left(\dfrac{g_{*}(M_{\Sigma_{2}}/{z})}{g_{*}(M_{\Sigma_{2}}/{z_{r}})}\right)^{\frac{1+n}{3}}\left(\frac{z_{r}}{z}\right)^{n}\right]^{1/2}.$
(5.24)

Thus, in order to find the baryon (precisely $B-L$) asymmetry produced due to
the decay of a heavy fermionic triplet in a non-standard cosmological
scenario, we have to solve the same set of Boltzmann equations (Eqs. (5.12 and
5.13)) with modified decay and scattering terms
($\tilde{D},\tilde{S}_{A},\tilde{W},\tilde{W}^{sub}$).

###  5.3 Analysis of numerical results: constraining the non-standard
parameters

The present study of fermion triplet leptogenesis in non-standard cosmology is
mainly motivated to inspect the following intriguing issues: (i) possibility
of producing observed asymmetry at some lower mass scale, (ii) whether baryon
asymmetry bound can impose a more stringent constraint on the non-standard
parameters which are already restricted by the DM relic density bound. It is
worth mentioning that, since we are interested to find the lower limit (using
BAU) on $M_{\Sigma_{2}}$, we should use theoretically possible maximum value
of the CP asymmetry parameter. So we have used $\epsilon^{\rm max}_{\Sigma}$
(Eq. (5.9)) as the CP asymmetry throughout the analysis (for both standard and
non-standard case).

Let us proceed with a brief discussion on the scheme of solution of Boltzmann
equations (Eq. (5.12), Eq. (5.13)) in non-standard cosmology (i.e, using
$\tilde{D},\tilde{S}_{A},\tilde{W},\tilde{W}^{sub}$). Although in standard
cosmology scenario the decay/inverse decay and the scattering terms, have been
expressed as explicit functions of $z$, they depend on other parameters like
$\tilde{m}$ and $M_{\Sigma_{2}}$ too. However in modified cosmological
scenario (along with the standard ones) the non-standard parameters play a
pivotal role in determining the decay/inverse decay and scattering terms.
Therefore they ($\tilde{D},\tilde{S}_{A},\tilde{W},\tilde{W}^{sub}$) become
functions of $(z;\tilde{m},M_{\Sigma_{2}},T_{r},n)$ where $z$ is the
integration variable and others are parameters of the model. It is worthwhile
to mention that we have kept $\tilde{m}$ fixed at a benchmark value $0.05$ eV
throughout the analysis, whereas the other parameters are varied within
suitable range to answer the questions raised in the first paragraph of this
section.

At first we solve the Boltzmann equations for leptogenesis assuming standard
radiation dominated cosmology for fixed value of triplet mass
$M_{\Sigma_{2}}$. We show the variation of $Y_{B}$ with $z$ (in left panel of
Fig. (8)) to get an estimate of the specific value of $z$ above which the
asymmetry freezes. The same exercise is repeated taking into account the
presence of non-standard terms in the Boltzmann equations. The evolution of
$Y_{B}$ is shown in the same plot together with its standard counter part.

Figure 8: Left: Evolution of baryon asymmetry $Y_{B}$ (red line) and triplet
abundance $Y_{\Sigma_{2}}$ (blue line) with $z$ assuming standard (solid line)
as well as non-standard (dashed line) cosmology for fixed benchmark value of
set of parameters as $(M_{\Sigma_{2}}=5\times 10^{9}{\rm
GeV},\tilde{m}=0.05~{}{\rm eV},T_{r}=1~{}{\rm GeV},n=1)$. Horizontal dotted
line represents the experimental value of baryon asymmetry. Right: Variation
of final (or frozen) value of baryon asymmetry ($Y^{f}_{B}$) with mass of the
decaying triplet ($M_{\Sigma_{2}}$) for fixed values of the set of parameters
$(\tilde{m}=0.05~{}{\rm eV},T_{r}=1~{}{\rm GeV},n=1)$. The particular points,
where the vertical dotted lines have touched the abscissa, denote the lowest
value of $M_{\Sigma_{2}}$ for which the BAU bound can be satisfied.

The figure (left panel, Fig. (8)) vividly shows that, for chosen set of
parameters the final asymmetry indeed freezes to a higher value in non-
standard case. To investigate about the nature of dependence of final
asymmetry ($Y^{f}_{B}$) on the decaying triplet mass ($M_{\Sigma_{2}}$), we
solve the Boltzmann equations for a continuous range of $M_{\Sigma_{2}}$. The
final asymmetry (for standard as well as non-standard) is plotted (right panel
Fig. (8)) against $M_{\Sigma_{2}}$, while the other parameters
($\tilde{m},T_{r},n$) are kept fixed at a bench mark value
$(\tilde{m}=0.05~{}{\rm eV},T_{r}=1~{}{\rm GeV},n=1)$. As anticipated,
$Y^{f}_{B}$ for both the plots shows monotonically increasing behaviour.
However, the non-standard curve (denoted by dashed line) shows a steeper
dependence, thereby satisfying the baryon asymmetry bound at a much lower
(approximately $2$ orders) value of $M_{\Sigma_{2}}$ compared to the standard
one. So we have got the answer of our first question, i.e the lower bound on
$M_{\Sigma_{2}}$ indeed decreases in presence of non-standard effect.

Let us recall the relation between final value of baryon asymmetry and
efficiency factor, i.e

$Y^{f}_{B}(M_{\Sigma_{2}},\tilde{m},T_{r},n)=-a_{sph}\epsilon_{\Sigma}(M_{\Sigma_{2}})Y^{eq}_{\Sigma}(z\simeq
0)\kappa^{f}(M_{\Sigma_{2}},\tilde{m},T_{r},n)$ (5.25)

where we have shown explicit dependence on model parameters. In principle this
functional dependence indicates that final asymmetry expected to vary with the
non-standard parameters $(T_{r},n)$ even if $M_{\Sigma_{2}}$ is kept fixed.
Therefore the conclusion drawn towards the end of last paragraph may not be
true for all combinations of $(T_{r},n)$. To examine its validity for other
(than those used to draw the Fig. (8)) values of ‘$n$’ we redraw the same plot
(i.e $Y^{f}_{B}$ vs $M_{\Sigma_{2}}$) for three sample values of $n$ and show
them in the left panel of Fig. (9).

Figure 9: Left: Variation of final (or frozen) value of baryon asymmetry
($Y^{f}_{B}$) with mass of the decaying triplet ($M_{\Sigma_{2}}$) for three
benchmark values of non-standard parameters $n$ while $\tilde{m}$ and $T_{r}$
kept fixed throughout. The lower limit on $M_{\Sigma_{2}}$ decreases up to
$n=1.5$. Right: Dependence of efficiency factor $(\kappa^{f})$ on the non-
standard parameter $n$ for three sample values of $M_{\Sigma_{2}}$. Horizontal
dashed lines represent the corresponding efficiency factor in standard
cosmology (obviously it does not depend on $n$). $M_{\Sigma_{2}}$ and
$\tilde{m}$ are in GeV unit.

It is observed that the said lower limit on $M_{\Sigma_{2}}$ decreases with
$n$ only up to a certain value of $n$ (which is $n=1.5$ in this case), further
increment in $n$ results in a drastic fall of final asymmetry which implies
increment of the lower bound of $M_{\Sigma_{2}}$. This observation allows us
to state unambiguously that the said lower bound can not be lowered
indefinitely by increasing $n$. There exists a critical value of $n$ (for each
fixed value of $M_{\Sigma_{2}}$) beyond which the final asymmetry declines
monotonically. A deeper understanding about the nature of efficiency factor is
required to shed some light on this issue. Eq. (5.25) clearly designates that
the non-standard parameters only affect the efficiency factor, whereas the CP
asymmetry parameter being solely dictated by the underlying particle physics
model remains unaffected. Variation of efficiency factor ($\kappa^{f}$) with
$n$ for three different fixed values of $M_{\Sigma_{2}}$ is depicted in the
right panel of Fig. (9). The efficiency factors obtained using standard
cosmology (for same three benchmark values of $M_{\Sigma_{2}}$) have been
shown in the same figure with horizontal dashed lines. It is to be noted that
the efficiency factor is indeed enhanced in non-standard case, but above a
certain value of ‘$n$’ it again starts to decrease and may become even lesser
than its standard counterpart. This peculiar behaviour of $\kappa^{f}$ can be
explained by studying the effect of non-standard parameters (specially $n$) on
the terms responsible for the creation and washout of asymmetry. The integral
representation (Eq. (5.19)) of efficiency factor $\kappa$ clearly indicates
its dependence on ‘rate181818This rate is again proportional to Decay and
scattering terms (Eq. (5.12)) of change of $Y_{\Sigma}$’ and ‘the integral
over washout terms’. It can be understood from Eq. (5.23), that the decay term
$\tilde{D}$ (and also $\tilde{S}_{A},\tilde{W},\tilde{W}^{sub}$) decreases
with increase in $n$. When $n$ is increased from zero, initially the effect of
decrease in washout (effectively increase in the integral) surpluses decrease
in the Decay and scattering terms. Thus the resultant effect becomes positive
which implies increment in the efficiency factor. However, beyond a critical
value of $n$ the decrement in the $\tilde{D}$ and $\tilde{S}_{A}$ dominates
overwhelmingly driving the efficiency factor towards a lower value.

The remarkable feature of this study is the interrelation, between DM
phenomenology and leptogenesis, which is brought about by the non-standard
parameters. These parameters have taken a crucial role in producing the
correct relic abundance within the experimentally allowed range of
$M_{\Sigma_{1}}$ as well as playing a vital role in bringing down the lower
limit on $M_{\Sigma_{2}}$ (required for successful prediction of BAU). In this
context, it should be mentioned that during the computation of DM relic
density, the non-standard parameters are varied within the range $10^{-3}\leq
T_{r}({\rm GeV})\leq 10$ and $0\leq n\leq 5$ while the mass of DM
$(M_{\Sigma_{1}})$ is taken between $500$ and $1500$ GeV. Moreover, we have
also imposed the bound on DM relic density in $3\sigma$ range as obtained by
the Planck collaboration i.e., $0.1172\leq\Omega_{DM}h^{2}\leq 0.1226$. For
leptogenesis, we have solved the Boltzmann equations for the entire range of
$(T_{r}-n)$ values previously allowed by relic density bound, while
$M_{\Sigma_{2}}$ has been varied independently over a wide range
$(10^{6}~{}-~{}10^{13})$ GeV.

Figure 10:  Left: Non-standard parameter space $(T_{r}~{}-~{}n)$ allowed by
relic density bound is denoted by deep green points whereas deep red points
are those which satisfy both relic density as well as baryon asymmetry bound.
The narrow magenta patch (which has actually covered few points in the dark
red region) denotes those typical combinations of $(T_{r},n,M_{\Sigma_{2}})$
which produce total energy density greater than the Planck limit. The area
painted in deep cayan stands for the region excluded by BBN bound. The plot
shows that the imposition of BAU bound reduces the allowed parameter space
considerably. Right: Allowed (by BAU bound) values of $M_{\Sigma_{2}}$ are
represented by different colours in $(T_{r}~{}-~{}n)$ plane. The vertical
colour bar stands for specific numerical values of $M_{\Sigma_{2}}$ as denoted
by different colours. It can be seen that the magenta patch has overshadowed
the allowed high mass region to some extent. This narrow region, although
allowed by the baryon asymmetry bound, is in tension with the Planck
constraint on the total energy density ($\rho_{\rm max}\sim 10^{66}$ GeV4).
However, the low mass region is completely safe from this bound. The deep
brown patch signifies the lowest achievable value of $M_{\Sigma_{2}}$, using
non-standard cosmology which is $\lesssim 10^{9}$ GeV.

It can be inferred from the relevant plot (left panel, Fig. (10)) that the
parameter space (in $(T_{r}-n)$ plane) allowed by the relic density bound,
shown by deep green points, has shrunk significantly to a smaller region
consisting of deep red points upon imposition of second round of constraint,
i.e. the BAU bound. For the sake of completeness we scan the whole allowed
parameter space (by BBN bound + DM relic bound + Baryon asymmetry bound) and
compute the maximum possible value of total energy density ($\rho_{\rm tot}$)
(using Eq. (4.9)) in our model. The typical combinations of
$(T_{r},\,\,n,\,\,M_{\Sigma_{2}})$ which result in $\rho_{\rm tot}>\rho_{\rm
max}$191919The maximum allowed energy density[89] of the Universe (as
suggested by Planck data [6]) after the inflation era is given by $\rho_{\rm
max}\sim 10^{66}$ GeV4., where $\rho_{\rm max}$ is the maximum allowed energy
density of the Universe after inflation as dictated by the Planck data [6],
are indicated by the magenta patch. Further in-depth studies related to
leptogenesis, like mass $(M_{\Sigma_{2}})$ distribution in $(T_{r}-n)$ plane,
have to be done using this common parameter space designated by the deep red
patch in the above mentioned plot.

The meticulous analysis of numerical results, done so far conveys a clear
message that dependence of final baryon asymmetry $Y^{f}_{B}$ on
$M_{\Sigma_{2}}$ and other non-standard parameters is quite convoluted. Even
the nature of dependence on a particular parameter (while others are kept
fixed) is not always monotonic. Nevertheless, to get a comprehensive picture
of the three dimensional parameter space (spanned by
$M_{\Sigma_{2}},\,\,T_{r},\,\,n$) allowed by recent experimental bound on BAU
(as well as relic density bound), we depict202020Only those combinations of
$M_{\Sigma_{2}},\,\,T_{r},\,\,n$ are considered which produces $Y^{f}_{B}$
within the experimental range. ${\log_{10}}\left(M_{\Sigma_{2}}\right)$ with
continuously varying colours in ${\log_{10}}\left(T_{r}\right)-n$ plane (right
panel, Fig. (10)). The vertical colour bar contains all the required numerical
information about the representative colours. The right most allowed region
(deep cyan) corresponding to $M_{\Sigma_{2}}\sim 10^{12}$ GeV has been covered
by the magenta patch which indicates that strict adherence with the Planck
bound on total energy density will lead to exclusion of this narrow region
from the allowed parameter space. However, the region associated with the
lower values of $M_{\Sigma_{2}}$ remains absolutely unaffected (by the
$\rho_{\rm max}$ bound). The deep brown patch in the figure signifies that the
lowest value of the decaying triplet mass, for which BAU bound can be
satisfied, is less than $10^{9}$ GeV, while $1.0\lesssim n\lesssim 1.6$ and
$T_{r}$ is less restricted, can take any value between $(1~{}-~{}10)$ GeV.
Further increment/decrement in $n$ will shift lower bound on $M_{\Sigma_{2}}$
towards higher value.

##  6 Summary and Conclusion

In this work, we have considered the Type-III seesaw model where the SM is
extended by three SU$(2)_{\rm L}$ fermion triplets to generate tiny Majorana
masses for the SM neutrinos. However, here we have imposed a $\mathbb{Z}_{2}$
symmetry on one of the triplet fermions $\Sigma_{R1}$ and accordingly, the
neutral component of $\Sigma_{R1}$ becomes a stable dark matter candidate and
one active neutrino remains massless. On the other hand, the heavier
generations of the newly introduced triplet fermions can couple to the SM
Higgs and leptons through Yukawa coupling, which in turn opens up the
possibility to account for the observed baryon asymmetry of the Universe
through CP violating out of equilibrium decay of triplet fermions to lepton
Higgs pair.

It is worthwhile to mention that although the stability of the proposed DM
candidate ($\Sigma^{0}_{1}$) has been ensured by the imposed $\mathbb{Z}_{2}$
symmetry, it remains under-abundant throughout the entire mass range (of
$M_{\Sigma_{1}}$) allowed by the indirect and collider searches of DM.
Assuming the evolution of the relic abundance through standard radiation
dominated cosmology, it can be shown that the correct relic density can be
obtained for a fermion triplet DM of mass $\sim 2.3$ TeV which is ruled out
unambiguously by the indirect searches of DM. To alleviate this problem or in
other words to achieve the correct relic abundance for a relatively lower mass
(allowed by experimental results) of the triplet, we resort to a typical form
of non-standard cosmology where primordial (well above the BBN temperature)
Universe expands faster than the standard picture. In the present work we have
shown that, indeed it is possible to produce the correct relic abundance for a
triplet mass as low as a few hundreds of GeV while the values of the non-
standard parameters ($T_{r}-n$) have been chosen judiciously.

From the point of view of particle physics, it seems that the dark sector and
the Yukawa sector (responsible for asymmetry generation) are disconnected by
the $\mathbb{Z}_{2}$ symmetry whereas the modified cosmology is bridging this
gap through its parameters $T_{r}-n$. These two parameters play a key role in
getting correct relic abundance of the DM as well as producing BAU in
agreement with recent data. It has been found in our analysis that although
the DM relic abundance constraint can be satisfied for high values of $n$ (up
to the maximum value used for scanning the parameter space), the baryon
asymmetry bound restricts it to be $\lesssim 1.8$. On the other hand, the
relic density constraint bounds $n$ from below with a limiting value of $0.4$.
Thus it can be concluded that simultaneous satisfaction of DM relic density
bound and BAU bound imposes a stringent restriction on the common parameter
space spanned by the non-standard parameters $T_{r}-n$. The exhustive analysis
of leptogenesis leads us to another noteworthy result, i.e, the lower bound on
triplet mass (obtained from the observed BAU) can be brought down further by
two orders of magnitude in comparison to the bound estimated using standard
cosmology.

## Acknowledgements

The research of AB is supported by the National Research Foundation of Korea
(NRF) funded by the Ministry of Education through the Center for Quantum Space
Time (CQUeST) with grant number 2020R1A6A1A03047877 and by the Ministry of
Science and ICT with grant number 2021R1F1A1057119. He also acknowledges the
support from National Research Foundation grants funded by the Korean
government (NRF-2021R1A4A2001897). M.C would like to acknowledge the financial
support provided by SERB-DST, Govt. of India through the NPDF project
PDF/2019/000622. This work used the Scientific Compute Cluster at GWDG, the
joint data center of Max Planck Society for the Advancement of Science (MPG)
and University of Göttingen. The authors would like to acknowledge the useful
discussion with Laura Covi.

##  Appendix A Thermal averaged cross-section

In this Appendix, we have listed the expressions of all annihilation and co-
annihilation cross sections in the $s$-wave approximation as given in Ref.
[34],

$\displaystyle{\sigma
v}_{\Sigma^{0}_{1}\Sigma^{0}_{1}}\simeq\frac{2\pi\alpha^{2}_{L}}{M^{2}_{\Sigma_{1}}}\,,\,\,\,{\sigma
v}_{\Sigma^{0}_{1}\Sigma_{1}^{\pm}}\simeq\frac{29\pi\alpha^{2}_{L}}{8M^{2}_{\Sigma_{1}}}\,,\,\,\,{\sigma
v}_{\Sigma_{1}^{+}\Sigma_{1}^{-}}\simeq\frac{37\pi\alpha^{2}_{L}}{8M^{2}_{\Sigma_{1}}}\,,\,\,\,{\sigma
v}_{\Sigma_{1}^{\pm}\Sigma_{1}^{\pm}}\simeq\frac{\pi\alpha^{2}_{L}}{M^{2}_{\Sigma_{1}}}\,.$
(A.1)

The effective thermal averaged cross-section can be found easily using Eq.
(4.2) as

$\displaystyle{\langle{\sigma v}\rangle}_{\rm eff}$ $\displaystyle=$
$\displaystyle\frac{g^{2}_{\Sigma^{0}_{1}}}{g^{2}_{\rm eff}}{\sigma
v}_{\Sigma^{0}_{1}\Sigma^{0}_{1}}+2\,\frac{g^{2}_{\Sigma_{1}^{\pm}}}{g^{2}_{\rm
eff}}\left[{\sigma v}_{\Sigma^{\pm}_{1}\Sigma^{\pm}_{1}}+{\sigma
v}_{\Sigma^{+}_{1}\Sigma^{-}_{1}}\right](1+\epsilon)^{3}\exp(-2\epsilon x)$
$\displaystyle+4\,\frac{g_{\Sigma^{0}_{1}}g_{\Sigma^{\pm}_{1}}}{g^{2}_{\rm
eff}}{\sigma
v}_{\Sigma^{0}_{1}\Sigma^{\pm}_{1}}(1+\epsilon)^{3/2}\exp(-\epsilon x)\,,$

where $g_{\Sigma_{1}^{0}}=g_{\Sigma^{\pm}_{1}}=2$,
$\epsilon=\frac{\Delta}{M_{\Sigma_{1}}}$, $g_{\rm
eff}=g_{\Sigma^{0}_{1}}+2\,g_{\Sigma_{1}^{\pm}}(1+\epsilon)^{3/2}\exp(-\epsilon
x)$ and $\Delta=M_{\Sigma^{\pm}_{1}}-M_{\Sigma_{1}}\simeq 166$ MeV for
$M_{\Sigma_{1}}\sim$ 1 TeV. So $\epsilon$ turns out to be negligibly small
($\sim 10^{-4}$). Thus in the limit $\epsilon\rightarrow 0$, the effective
annihilation cross-section (Eq. (LABEL:effective-CS)) can be approximated to
be

${\langle{\sigma v}\rangle}_{\rm
eff}\simeq\frac{111}{36}\frac{\pi{\alpha_{L}}^{2}}{{M_{\Sigma_{1}}}^{2}}~{}.$
(A.3)

##  Appendix B Equilibrium number density and freeze-out temperature

##### Number density of different species:

The number density of fermion triplet (for each generation) in terms of
temperature (or equivalently $z=M_{\Sigma}/T$) is given by

$n^{eq}_{\Sigma}(T)=\frac{3}{4}\zeta(3)\frac{g_{\Sigma}T^{3}}{2\pi^{2}}\left(\frac{M_{\Sigma}}{T}\right)^{2}K_{2}\left(\frac{M_{\Sigma}}{T}\right)~{},$
(B.1)

where $\zeta$ is the Riemann zeta function and $g_{\Sigma}$ is the internal
degrees of freedom for the triplet fermion. In this case $g_{\Sigma}=2+4=6$,
where the factor $2$ is for the Majorana fermion (neutral component) and the
factor $4$ is for the Dirac fermion (charged component) respectively.
Similarly number density of SM doublet leptons is

$n^{eq}_{l}(T)=\frac{3}{4}\zeta(3)\frac{g_{l}T^{3}}{\pi^{2}}~{},$ (B.2)

and that of photons is

$n^{eq}_{\gamma}(T)=\zeta(3)\frac{g_{\gamma}T^{3}}{\pi^{2}}~{},$ (B.3)

where $g_{\gamma}=g_{l}=2$ (since leptons can be treated to be massless above
EW symmetry breaking scale).

Co-moving entropy density as a function of temperature is expressed as

$s(T)=\frac{2\pi^{2}}{45}g_{\ast}T^{3}~{},$ (B.4)

where $g_{\ast}$ is number of effective relativistic degrees of freedom which
turns out to be $\simeq 106.75$ if all Standard Model particles are assumed to
be relativistic.

##### Freeze-out temperature in non-standard cosmology:

The dark matter gets thermally decoupled from the plasma at a specific
temperature $T_{f}$ (or equivalently $z_{f}(=M_{\Sigma_{1}}/T_{f})$ ) whenever
the equality

$H(z_{f})=n^{eq}_{\Sigma^{0}_{1}}(z_{f})\langle\sigma v\rangle_{\rm eff}$
(B.5)

is reached. Using the modified Hubble (due to the presence of non-standard
effects), i.e, $H(z_{f})=H_{\rm rad}(z_{f})f(z_{f},z_{r},n)$, effective
annihilation cross-section from Eq. ((A.3)) and approximated form of the
number density of the dark matter particle in the above equation we get

$\displaystyle\left(\frac{g_{\Sigma^{0}_{1}}}{\pi^{2}}{M_{\Sigma_{1}}}^{3}{z_{f}}^{-3/2}e^{-z_{f}}\right)\left(\frac{111}{36}\frac{\pi{\alpha_{L}}^{2}}{{M_{\Sigma_{1}}}^{2}}\right)=\left(1.66\sqrt{g_{\ast}(T_{f})}\frac{1}{M_{pl}}\frac{{M_{\Sigma_{1}}}^{2}}{{z_{f}}^{2}}\right)f(z_{f},z_{r},n).$
(B.6)

First we put numerical values of the constants
$(g_{\Sigma^{0}_{1}},\alpha_{L},M_{pl},g_{\ast}(T_{f}))$ in the above
equation. Then a few steps of algebraic manipulations lead us to a much
simplified form of the transcendental equation,

$\displaystyle{z_{f}}^{1/2}e^{-z_{f}}$ $\displaystyle\simeq$
$\displaystyle\left(6300\times
10^{-19}\right){M_{\Sigma_{1}}}f(z_{f},z_{r},n)$ $\displaystyle{\rm
or,~{}}z_{f}$ $\displaystyle\simeq$ $\displaystyle
35+\frac{1}{2}\ln{z_{f}}-\ln{{M_{\Sigma_{1}}}}-\ln{f(z_{f},z_{r},n)}$
$\displaystyle{\rm or,~{}}z_{f}$ $\displaystyle\simeq$ $\displaystyle
28+\frac{1}{2}\ln{z_{f}}-\ln{\left({M_{\Sigma_{1}}}/1000\right)}-\ln{f(z_{f},z_{r},n)},$
(B.7)

solving which we can get the freeze-out temperature
$T_{f}=M_{\Sigma_{1}}/{z_{f}}$.

##  Appendix C Reaction densities of various decay, inverse decay, scattering
processes

Let us consider a generic decay process $(\psi\rightarrow a+b+.....)$ and a
scattering process $(\psi+\chi\rightarrow a+b+.....)$ where $\psi$ is the
particle whose abundance is to be tracked. The space-time density of the
generic scattering process in thermal equilibrium is expressed as

$\gamma(\psi+\chi\rightarrow a+b+.....)=\int
d\Pi_{\psi}d\Pi_{\chi}f_{\psi}f_{\chi}\int
d\Pi_{a}d\Pi_{b}.....\left(2\pi\right)^{4}\delta^{4}(p_{\psi}+p_{\chi}-p_{a}-p_{b}-........)|\mathcal{M}|^{2},$
(C.1)

where $d\Pi_{i}=\frac{d^{3}p_{i}}{(2\pi)^{3}2E_{i}}$ is the phase space
factor, $|\mathcal{M}|^{2}$ is the squared transition amplitude and $f_{i}$ is
the distribution function of the concerned particle species. For a decay
process the above integral can be simplified to express the reaction density
as

$\gamma(\psi\rightarrow
a+b)=n^{eq}_{\psi}\frac{K_{1}(z)}{K_{2}(z)}\Gamma_{\psi},$ (C.2)

where $\Gamma_{\psi}$ is the decay width of $\psi$ to $a,b$ in the rest frame
of $\psi$ and $n^{eq}_{\psi}$ is the equilibrium number density of the $\psi$
particle. For the scattering process. For the scattering process the thermally
averaged reaction rate can be reduced to

$\gamma(\psi+\chi\rightarrow
a+b)=\frac{T}{64\pi^{4}}\int\limits^{\infty}_{s_{min}}ds\sqrt{s}K_{1}\left(\frac{\sqrt{s}}{T}\right)\hat{\sigma}(s),$
(C.3)

where the integration is evaluated over the centre of mass energy $(s)$, whose
minimum admissible value is either square of sum of the initial particle
masses or final particle masses (which ever is greater), i.e
$s_{min}=max\left[(m_{\psi}+m_{\chi})^{2},(m_{a}+m_{b})^{2}\right]$. Here
$\hat{\sigma}(s)$ is the reduced cross-section which is related to the actual
cross-section $(\sigma)$ of the concerned process through the
function212121$\lambda[x_{1},x_{2},x_{3}]=(x_{1}-x_{2}-x_{3})^{2}-4x_{2}x_{3}$
$\lambda$ as

$\hat{\sigma}=2s\lambda[1,\frac{{m_{\psi}}^{2}}{s},\frac{{m_{\chi}}^{2}}{s}]\sigma~{}.$
(C.4)

The integration variable $s$ in Eq. (C.3) can be replaced by a dimension-less
one $(x=s/{m_{\psi}}^{2})$ such that the reaction density of scattering can be
expressed as

$\displaystyle\gamma(\psi+\chi\rightarrow
a+b)=\frac{{M_{\psi}}^{4}}{64\pi^{4}z}\int\limits^{\infty}_{x_{min}}\sqrt{x}K_{1}(z\sqrt{x})\hat{\sigma}(x)dx.$
(C.5)

##  Appendix D Scattering cross-section of various processes

##### Gauge boson mediated scattering process:

Reduced scattering cross-section for the gauge boson mediated
processes222222It includes $\Sigma,\Sigma\rightarrow$ all possible fermion
doublets, Higgs, gauge bosons  is given by

$\hat{\sigma}_{A}=\frac{6{g_{2L}}^{4}}{72\pi}\left[\frac{45}{2}r(x)-\frac{27}{2}{r(x)}^{3}-\left\\{9\left({r(x)}^{2}-2\right)+18\left({r(x)}^{2}-1\right)^{2}\right\\}\ln\left(\frac{1+r(x)}{1-r(x)}\right)\right],$
(D.1)

where $g_{2L}$ is the $SU(2)_{L}$ gauge coupling constant and
$r(x)=\sqrt{1-4/x}$.

#####  $\Delta L=2$ scattering processes:

The off-shell part of the $\Delta L=2$ scattering cross-section (for
$s$-channel process) is given by

$\displaystyle\hat{\sigma}_{s}(LH\rightarrow\bar{L}H^{\ast})$
$\displaystyle=\frac{\left({y_{\Sigma_{2}}}^{\dagger}{y_{\Sigma_{2}}}\right)^{2}_{22}}{4\pi}\left[2+x{D^{2}_{s}}^{sub}+\left(2-3x\frac{m_{3}}{\tilde{m}_{1}}\right)Re\left\\{D_{s}\right\\}+3\frac{m_{3}}{\tilde{m}_{1}}\left(x\frac{m_{3}}{\tilde{m}_{1}}-2\right)\right.$
$\displaystyle-\left.\frac{2}{x}\ln(1+x)\left\\{1+\left(Re\left\\{D_{s}\right\\}+3\frac{m_{3}}{\tilde{m}_{1}}\right)(1+x)\right\\}\right]$
(D.2)

where $D_{s}$ is the $s$ channel propagator given by

$D_{s}=\frac{1}{s-{M_{\Sigma_{2}}}^{2}+i\Gamma_{\Sigma_{2}}M_{\Sigma_{2}}}$
(D.3)

and ${D^{2}_{s}}^{sub}$ is the modulus square of ‘resonance contribution
subtracted’ $s$ channel propagator, expressed in terms of $D_{s}$ as

${D^{2}_{s}}^{sub}=1-\frac{\pi}{\Gamma_{\Sigma_{2}}M_{\Sigma_{2}}|D_{s}|^{2}}\delta\left(s-{M_{\Sigma_{2}}}^{2}\right)~{}.$
(D.4)

Cross-section for the $t$ channel process is given by

$\displaystyle\hat{\sigma}_{t}(LL\rightarrow
H^{\ast}H^{\ast})=\frac{\left({y_{\Sigma_{2}}}^{\dagger}{y_{\Sigma_{2}}}\right)^{2}_{22}}{2\pi}\left[\frac{3x}{2}\left(\frac{{m_{3}}^{2}}{{\tilde{m}_{1}}^{2}}+\frac{2}{1+x}\right)+\left(3\frac{m_{3}}{\tilde{m}_{1}}-\frac{3}{2+x}\right)\ln\left(1+x\right)\right]~{}.$
(D.5)

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