Inflation and dark matter in the inert doublet model
HJE
In ation and dark matter in the inert doublet model
Sandhya Choubey 0 1 2 3 4 5
Abhass Kumar 0 1 2 3 5
0 AlbaNova University Center , 106 91 Stockholm , Sweden
1 Anushaktinagar , Mumbai
2 Chhatnag Road, Jhunsi , Allahabad 211 019 , India
3 Homi Bhabha National Institute, Training School Complex
4 Department of Physics, School of Engineering Sciences, KTH Royal Institute of Technology
5 HarishChandra Research Institute
6 400094 , India
We discuss in ation and dark matter in the inert doublet model coupled nonminimally to gravity where the inert doublet is the in aton and the neutral scalar part of the doublet is the dark matter candidate. We calculate the various in ationary parameters like ns, r and Ps and then proceed to the reheating phase where the in aton decays into the Higgs and other gauge bosons which are nonrelativistic owing to high e ective masses. These bosons further decay or annihilate to give relativistic fermions which are
Cosmology of Theories beyond the SM; Beyond Standard Model

nally
responsible for reheating the universe. At the end of the reheating phase, the inert doublet
which was the in aton enters into thermal equilibrium with the rest of the plasma and its
neutral component later freezes out as cold dark matter with a mass of about 2 TeV.
1 Introduction
2
The standard model of particle physics has been very successful with highly accurate
predictions. However, it still has no answer for various problems like dark matter and in ation.
Both in ation and dark matter have been established very rmly through various
observations particularly of the cosmic microwave background (CMB) radiation. In ation [1]
has long been the most successful theory to answer cosmological problems like the horizon
problem and homogeneity. The most popular in ationary models are those that have an
extra scalar particle which acts as the in aton. Recent experiments like Planck [
2
] and
WMAP7 [3] have placed bounds with high accuracy on in ationary parameters like the
spectral index, the tensor to scalar ratio and the scalar power spectrum. There have been
a variety of in ation models over the years. The Higgs in ation [4, 5] models are the most
simple in the sense that they do not involve any extra eld and have just one more
parameter through which the eld couples to gravity but they come with their share of problems.
The quartic coupling
of Higgs eld at high energy scales (& 1010 GeV) becomes negative.
This can cause problems with the stablility of the vacuum [6]. Another problem comes in
the form of nonunitarity. The scalar power spectrum bounds require
104 [7] which
breaks unitarity at scales around mP l=
1013 GeV [8]. To avoid running into problems
in a Higgs in ation model, often an extra scalar stabilizing
eld is added and such
scenarios are called sin ation. These models have an extra gauge singlet scalar particle that
acts as the in aton while the Higgs eld acts as a portal to the standard model to reheat
the universe. There can be variations in this model and in [9] distinctions between the
variations is studied. The in ationary potential is usually taken to be either a chaotic
one or a Starobinsky one. Chaotic in ation [10] models include power law potentials like
m2 2 +
4. These were the rst type of potentials used to study in ation. On the other
{ 1 {
hand Starobinsky models have exponential potentials. We will discuss more about them
later. A good review for in ationary cosmology in the light of data can be had in [11].
Dark matter has been studied extensively over the years. Thanks to the many
experiments and observations, we now have a good estimate for dark matter distribution in and
around our galaxy and in the universe at large. Planck results [12] together with other
astronomical observations have put down the abundance of dark matter in the universe
to
dmh2 ' 0:12. The most commonly studied dark matter scenarios are the so called
WeaklyInteracting Massive Particles (WIMP). In recent years however, as dark matter
detection experiments have become better and colliders like LHC are probing higher
energies, the absence of any new particle at the weak scale has put the WIMP scenario in a
x and people have started looking at other options like axions, feebly interacting massive
particles (FIMP) and strongly interacting massive particles (SIMP) [13{16] among others.
One of the simplest models of dark matter  the scalar singlet dark matter model is still
being sustained and there have been updates to it [17]; see also [18].
In more recent works, people have started to look for scenarios where both in ation
and dark matter can be explained by the same eld. Gauge singlet scalar models in the
sin ation scenario is a case in point. In this paper, we have combined in ation and dark
matter in the inert doublet model coupled nonminimally to gravity. Such a uni cation
was rst shown to be possible in [19] in string theory landscape. In [20{22] a gauge singlet
scalar is used as in ation and later after freeze out as the dark matter candidate. [23] has a
situation similar to sin ation where the in aton is very light and interacts very feebly to
become FIMP dark matter later. In ation and dark matter in two Higgs doublet models
was studied in [24]. A scalar WIMP dark matter candidate with nonminimal coupling to
gravity acting as the in aton was studied in [25].
The motivation for using inert doublet model in our case is the fact that pure Higgs
in ation is problematic and yet it is the only scalar eld present in the standard model.
Another scalar doublet similar to Higgs doublet but stabilized by an extra Z2 symmetry
such that it does not interact with leptons and quarks via Yukawa couplings can present a
viable candidate for both in ation and dark matter. The components of the inert doublet
can all act as in aton via a particular eld rede nition. At the same time, its neutral
scalar component can later become the dark matter candidate. The inert doublet through
its interactions with the vector gauge bosons and Higgs can also reheat the universe at the
end of in ation to ensure that the universe gets populated by standard model particles.
Another motivation for using this model is that it is similar to sin ation models in that
the potential turns out to be of the Starobinsky kind which gives some of the best t to
in ationary parameters like the spectral index. We will also look at the reheating phase
in some detail. In aton during reheating behaves as nonrelativistic matter and decays via
gauge and Higgs bosons to relativistic particles. We will look at the interactions happening
during reheating and later when we discuss dark matter, we will point out the changes that
take place in the interactions of the inert doublet compared to the reheating phase. The
electroweak (EW) symmetry breaking will play a role in determining the type of interactions
that the inert doublet undergoes.
{ 2 {
This paper is organized in the following manner. We describe the model in the next
section. In section 3, we study in ation and
nd the value of the various in ationary
parameters like the slow roll parameters, the spectral index and the tensor to scalar ratio.
In section 4 we study reheating which progresses by the decay of the in aton into
nonrelativistic vector and Higgs bosons which further annihilate into relativistic fermions. In
this section, we calculate the energy density stored in the relativistic particles and nd some
bounds on some model parameters. The inert doublet as a cold dark matter candidate is
taken up in section 5 where we
x some parameter values like the mass of dark matter
through relic density calculations. We end in section 6 with conclusions.
HJEP1(207)8
1
2
MP2 lR
D
1D
y
1
D
2D
y
2
V ( 1; 2)
1 12R
2 22R ;
2
1. The extra doublet is inert in the sense
that it does not have any Yukawa like couplings because of an inherent Z2 symmetry under
which this doublet is odd ( 2 !
2) while the Higgs and other standard model particles
1
; , where
stands for SM particles other than Higgs). The action
(2.1)
(2.2)
(2.3)
where D stands for the covariant derivative containing couplings with the gauge bosons.
During in ation, there are no elds other than the in aton so that the covariant derivative
will reduce to the normal derivative D
The minus sign in the kinetic terms is in
keeping with the metric convention of ( ; +; +; +). MP l is the reduced Planck mass, R
is the Ricci scalar and 1 and 2 are dimensionless couplings of the doublets to gravity.
The motivation behind these couplings is that quantum e ects invariably give rise to such
couplings at Planck scales [26].
The potential is:
V = m12j 1j2 + m22j 2j2 + 1(j 1j2)2 + 2(j 2j )
2 2
+ 3j 1j2j 2j2 + 4( y1 2)( y2 1) +
1
2 5[( y1 2)2 + c:c:]:
The two doublets have the components:
1 = p
1
2
h
!
and
2 = p
1
2
q
x ei
!
:
2
1
2 automatically satis es this condition
2
1
Note that there is no nonzero vacuum expectation value of the Higgs eld as the
electroweak symmetry is intact at in ationary scales. We want the inert doublet to be the
in aton. This is ensured if 22
21 . A choice where 1 and 1 are of the same order while
{ 3 {
In ation
The action in eq. (2.1) is written in the physical or the Jordan frame [27, 28] and has terms
where the scalars
1;2 couple quadratically to gravity. This makes it di cult to derive
meaningful results from the usual processes of quantum
eld theory. We need to make
some transformations where we can get rid of such coupled terms. This can be done by
a conformal transformation to the so called Einstein frame. Einstein frame is useful as in
this frame the action looks like a regular eld theory action with no explicit couplings to
gravity. Results for physical observables remain the same independent of the frame chosen.
After the end of in ation, the transformation parameter becomes almost 1, making the two
HJEP1(207)8
frames equivalent. Following [28], we make the following conformal transformation on the
metric and the elds to get the action in the Einstein frame: de ning
= f ; h; q; x; g
where:
g~
=
2
g ;
2 = 1 +
Gij =
V~ =
1
V
4
:
2 ij +
M1P2 l ( 2 + h2) +
M2P2 l (q2 + x2);
2
(3.1)
(3.2)
(3.3)
(3.4)
(3.5)
(3.6)
The above G gives mixed kinetic terms. All these
elds are always present in the
lagrangian but during in ation,
elds other than the inert doublet components give no
contribution.
2 can also be simpli ed to exclude the MP l
1 ( 2 + h2) term. This allows us
{ 4 {
Let us look at the kinetic terms. First, we expand the prefactor G in a matrix form:
G = 6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
6
4
2 2+6 12 42=MP2 l 6 MP21l 2 h
2
2
66 6 MP21l 2 h
2+6 12h2=MP2 l
4
2
to simplify the matrix G as:
2
2+6 22q2=MP2 l
4
B = MP l q :
MP l log
x
A further simpli cation to a completely diagonal kinetic form can be obtained by
rearranging the elds as follows:
Substituting this rede nition of elds into the kinetic part, we get a diagonal kinetic
term which is:
1
+
1
2
+
1
12 2F (A)
F (A)
2 2(1 + B2=MP2 l)2 (@ B)2 +
F (A) B2
2 2(1 + B2=MP2 l)
(3.10)
where F (A) = 1
exp
Eq. (3.10) is still apparently not canonical. However, at the scales relevant for in ation
F (A) is of the order of 1 and the change in F (A) while A drops from values many times
larger than MP l to MP l is very small. This can be seen in
gure 1. With large 2 this
which makes the kinetic term canonical.
12 and the other elds can have a constant rescaling
All such terms from the Einstein frame potential in eq. (3.5) which are not quartic in
q and x can be neglected owing to the largeness of these two elds. The only relevant term
that remains is the quartic term 14 2 q2 + x2 2 which using the rede ned elds becomes:
Ve
2MP4 l "1
exp
potentials [29]; see also [30]. In gure 2 we show the in ationary potential vs. the eld
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
F(A)
HJEP1(207)8
0
1
4
5
2
3
A/MPl
where it can be seen that the potential is almost at at high eld values ensuring slow roll.
The slow roll parameters and with this potential are: =
1
2
MP2 l
1 dVe
Ve dA
2
=
= MP2 l V1e dd2AV2e =
"
4
3
4
3 h
1 + exp
h
2
exp
1 + exp
1 and thus slow roll is satis ed. In ation ends
when
' 1.
We would now like to get estimates for the values of A at the beginning and end of
in ation which will be needed to get the power spectrum. This can be done by looking at
the number of times the universe expanded by e times its own size, also called the number
of efolds N . It is obtained as follows:
N =
1 Z Aini Ve dA
MP2 l Aend Ve0
3
4
"
=
exp
exp
where Ve0 = ddVAe , Aini is the value of A at the beginning of in ation and Aend is the value
of A at the end of the in ation.
{ 6 {
0.2
0
0
Inflationary Potential
1
2
To get Aend, we make use of the fact that slow roll in ation ends when
eq. (3.12), which gives:
Using eq. (3.15) in eq. (3.14) for N = 601 we get
exp
exp
Looking at gure 2, we see that eld values are consistent with slowroll and its end.
With N
xed at 60 and the eld value at the start of in ation
xed, we can get the
scalar power spectrum (PS), the tensor to scalar ratio (r) and the spectral index (ns) as
1In principle N could be any number greater than around 50 to solve
atness and horizon problems.
60 efolds solves the baryon asymmetry problem if in ationary energy scales are O[1016] GeV [31]. Lower
in ationary energy scales would need more efolds and vice versa. However, the number of efolds cannot
be much larger than 60.
{ 7 {
follows:
0.966
0.965
0.964
n
Number of efolds (N)
scalar ratio lie within Planck bounds.
later for energy density calculations.
where Ve0 is the derivative of Ve with respect to A and both Ve and Ve0 are calculated at the
Aini. The values of r and ns are well within the Plank bounds [
2
] of ns = 0:9677
0:0060
at 1
level and r < 0:11 at 95% con dence level. Since there is no reason for N to be
precisely 60, we look at the in ationary parameters over a range of N from 55 to 65 (see
gure 3 and 4). We see that in the entire region of N , the spectral index and the tensor to
We can use WMAP7 constraint for Ps [3] to relate 2 and 2 which will be needed
Ps = (2:430
0:091)
Having multiple scalar elds can give rise to multi eld e ects which can cause signi cant
isocurvature
uctuations. The presence of isocurvature
uctuations has been studied in
detail in [32{34]. Following them, we expand the elds to rst order i = 'i(t) +
i(x )
{ 8 {
(3.19)
(3.20)
(3.21)
(3.22)
HJEP1(207)8
0.0036
0.0034
where " is the turnrate vector in the eld space: "i = ^_i + ijk ^j '_ k with ijk being the
connection in the eld space for the eld space metric Gij . We also de ne the masssquared
matrix for the gauge invariant linearized perturbations [32]:
Mji = Gij Dj Dk V e
Riklj '_ k'_ l
where Di is the covariant derivative in the eld space w.r.t. eld 'i and Riklj is the Riemann
tensor in
eld space. These together are used to get a parameter ss which is used to
calculate the masssquare of the isocurvature uctuations s2 as follows:
2
1
Since 22
21 , in ation occurs along the
h
0 direction, thereby making s^1 and
s^2 zero. We are left with remaining two scalars q and x which have symmetric couplings
2 and 2
. For such a case, ss
very suppressed isocurvature fraction of iso
with Planck data [
2
].
1 (
O(10 6)) which means [34] s2=H2
' 0 giving a
O(10 5). The results are hence consistent
ss =
s^is^j Mji MP2 l
V e
At the end of in ation, the universe is too dilute for anything to be present. Unless the
universe is somehow repopulated by particles, it remains empty. It is at this juncture that
the energy density till now stored in the in aton starts to disperse as the in aton particles
annihilate or decay into other particles including those of the standard model. This phase
of the universe after in ation where in aton annihilates into other relativistic particles is
called reheating [35]. If the in aton decays or annihilates into bosons, parametric resonance
production of bosons triggers e cient reheating [36, 37] (see also [38, 39]) and at the end
of it, the universe becomes radiation dominated.
allows us to identify two distinct regions [40] marked by Acr =
q 2 MP l :
The conformal transformation and the rede nition of elds done in the previous section
In ation occurs in the second region where A > Acr which can also be written as
(q2 + x2)1=2 > Acr.
Much below MP l, the in ationary potential in eq. (3.11) can be
approximated by a quadratic potential well:
A
Ve =
'
Ve =
(
(q2 + x2)1=2
for A < Acr;
q 32 MP l log
2
for A > Acr:
2 MP4 l "1
This is a simple harmonic potential in which the in aton oscillates rapidly with
frequency !. This makes the oscillations coherent, the phase being the same at all points
in space. Since the potential is a simple harmonic one near the minimum, the average
energy density obeys the relation A = hA_ 2i and thus obeys the equation _A + 3H A = 0
which yields a 1=a3 evolution for the average energy density. This means that during this
period the in aton behaves as nonrelativistic matter. A matter dominated universe has
the following characteristics with respect to the scale factor and Hubble's constant:
The equation of motion for A during this phase is which gives on solving for ! H: H(t) =
a / t2=3;
a_(t)
a(t)
=
2
3t
:
A + 3HA_ +
dA
dVe = 0;
A = A0(t) cos(!t);
(4.1)
(4.2)
(4.3)
(4.4)
(4.5)
(4.6)
(4.7)
where
The quadratic phase ends when the amplitude of the oscillations A0 crosses Acr which
gives us the crossing time as tcr = 2!2 . In [35] it was shown that reheating occurs when the
eld oscillates in a quadratic potential well. Therefore in the present scenario, reheating
starts when the potential gets approximated by eq. (4.3) and ends when the amplitude A0
crosses Acr at time tcr.
The inert doublet can decay into the W and Z bosons through the kinetic coupling terms
and into the Higgs boson through the potential in the Lagrangian. The resultant particles
don't have a physical mass at this time but an e ective mass arising due to the in aton
oscillations. If the oscillation frequency ! is much larger than the expansion rate H, the
amplitude can be taken to be constant over one oscillation period. This allows us to write
down e ective mass terms for the vector and scalar bosons. When A
q 32 MP l but still
above Acr, (q2 + x2)
in terms of q and x to get:
2
MP2 l . Using this we can expand the log term in the de nition of A
is 4gp26 M2P l A W 2. This gives an e ective mass for W bosons to be:
The coupling of the inert doublet to W bosons is g42 (q2 + x2)W 2 which in terms of A
2 MP l jAj:
2
(4.8)
(4.9)
(4.10)
(4.11)
The other vector boson e ective masses can be related by the Weinberg angle.
The coupling to Higgs is through 3;4;5. This gives us an e ective mass term for the
Higgs boson:
m2h = p
1
6
3 +
4
2
MP l jAj:
2
In writing the Higgs e ective mass, we have taken equal contributions of q and x in A.
Note that eq. (4.9) is not an equation for vacuum expectation value as it is not
calculated at the minimum of the potential. It just describes the transformation between q and
x on one hand and A on the other in a particular regime mentioned above the equation that
follows from eq. (3.8). The masses in eqs. (4.10) and (4.11) are therefore not usual masses
obtained from spontaneous symmetry breaking but are just e ective masses coming out of
their interactions with the in aton elds when written in terms of the transformed eld A.
The weak coupling g is large which makes the vector bosons nonrelativistic. They will
decay and annihilate to other relativistic fermions to reheat the universe. If either of 3 and
4 is large, the produced Higgs too will be nonrelativistic and it will decay into fermions
through Yukawa interactions which will add to the relativistic energy density. The inert
doublet gives a cold dark matter candidate which means the combination of its couplings
to Higgs becomes of the order of 1 [41]. We choose a case where 3
1 is the dominant
coupling when compared to 4 and 5 which are taken to be very small just for the sake of
brevity. This enables us to remove the 4 term from the e ective mass of Higgs in eq. (4.11).
At low number densities of the produced W and Higgs bosons (nW and nh respectively)
their decay to fermions is the dominant channel to produce relativistic particles. If the
number density of the bosons becomes large, their production rate will become exponential due
to parametric resonancne. During the resonance phase, the W bosons will mostly annihilate
to produce fermions. Their decays will become subdominant channels of fermion
production. Higgs on the other hand can only produce fermions through decays. Following [40]
(see also [42, 43]), the production of W bosons in the linear and resonance regions is:
weak coupling constant.
Making the corresponding changes for the production of Higgs, we have:
d(nha3)
dt
8
>
>
= <
2P3 !K23a3;
(linear);
>
>: 2a3!Qnh: (resonance):
K1 =
K2 =
"
" g2MP2 l r
K1 and K2 have dimensions of energy and are dependent on the respective mass terms
with:
where ti are instants when the in aton A = 0. In aton can decay into W and Higgs bosons
only in the vicinity of A = 0 when the e ective masses of these bosons are much less than
the in aton e ective mass !.
W bosons decay into fermions with a decay rate given by: while their annihilation cross section is given by:
W W
g4 2Nl + 2NqNc
16
8 hm2W i
10
g
4
16 2hm2W i :
W = 4g2 is the
(4.13)
(4.14)
(4.15)
(4.16)
(4.17)
Parametric resonance production of W bosons can start only when their decay rate
in eq. (4.16) falls below their production rate through parametric resonance in eq. (4.12).
Comparing them, we nd that resonance production of W bosons can start only when
A0 <
2
0:5625
Production of relativistic particles through decay of W takes place very slowly and would
reheat the universe long after the resonance period would have ended [40] while production
of relativistic particles through annihilation is a much faster process and can yield enough
relativistic particles to reheat the universe. Annihilation can occur only when the number
density of W bosons is large. This makes the occurrence of parametric resonance necessary
allowing us to put a lower bound on 2
When W is produced through resonance, its number density increases exponentially
and the dominant channel for production of fermions is by annihilations of W bosons
following eq. (4.17).
We need to check these conditions for fermion production via Higgs as well. The decay
rate of Higgs into fermions is given by the Yukawa couplings:
In eq. (4.20), only the coupling to top is important as it is large while the coupling for other
fermions is very small. The top quark can later decay or annihilate into other fermions.
Comparing eq. (4.20) to resonance production rate of Higgs in eq. (4.13), we nd that
Higgs production enters the resonance regime only after:
A0 <
Comparing eq. (4.21)2 to eq. (4.18), we see that if 3 . 0:006, Higgs production will
enter the resonance regime around the same time as W boson. For even a small amount
of resonance production in Higgs to occur, 3 cannot be greater than 0:41 2
. Since the
inert doublet is a dark matter candidate with large couplings, Higgs production will not
enter resonance regime till long after the end of the quadratic phase of the potential. The
production rate of Higgs remains small and its decay to fermions is at a much lower rate
than the annihilation of gauge bosons.
During parametric resonance production of gauge bosons, almost all the W bosons
get converted to fermions giving a complete transfer of energy density from W bosons to
relativisitc fermions which can be obtained by solving the following equation [40]
d( a4)
dt
= 2a4q
hm2W i
4Q2!2
W W
;
(4.22)
2The eq. (4.21) contains only
3 in the denominator instead of full 3 + 4=2 because of our choice of
large 3. The fact that Higgs won't be produced via resonance stands even if 4 is used.
2 &
1
60
h =
y
2
16 hmhi:
which after integration gives
=
10
where tp is the time when the parametric resonance starts given by the condition in
eq. (4.18) and tcr is the end of reheating. During this conversion, almost all the W bosons
get converted to fermions so that the only remaining particles by the time reheating ends
are the fermions apart from the inert doublet particles. Putting in the numbers, we get:
At this time, energy density in A is: (4.24)
In the nonrelativistic limit where vrel
1, we can rewrite eq. (5.1) as
jCM vrel =
1
We can now obtain the reheating temperature from the energy density in relativistic
particles:
5
Dark matter
where g is the number of degrees of freedom in the relativistic plasma.
The end of reheating marks the end of the quadratic oscillations phase of the rearranged
eld A. Since now, A is the same as (q2 + x2)1=2 and the Jordan and Einstein frames have
become equivalent, we can come back to using the physical Jordan frame. The in aton
eld no longer has an e ective mass !. Rather things go back to the original inert doublet
potential given in eq. (2.2) with the inert doublet having a mass of m2. In the beginning,
the inert doublet obtains a thermal equilibrium with the rest of the relativistic plasma
and evolves as radiation. Later, as the temperatures fall and the inert doublet becomes
nonrelativistic, its evolution is given by the Boltzmann equation. It freezesout as a cold
relic and thus becomes a candidate for cold dark matter.
We will use the observed relic abundance of dark matter
dm h2 = 0:12 [12] to calculate
certain parameters in the model. The interactions of the neutral scalar part of the inert
doublet are its annihilations into the vector bosons and Higgs. There are no decays of any of the
inert doublet components as they are prevented by the Z2 symmetry. At the tree level, there
are 4point interactions (see gure 5). The scattering crosssection for these processes is:
jCM vrel =
32 m22 (1 + vr2el=4) processes
X
jMj2:
The amplitude Pprocesses jMj2 for the neutral scalar component of
actual dark matter candidate is:
where g and g0 are the weak couplings to the vector bosons.
Apart from these 4point interactions, there are trilinear couplings as well which include
the annihilation of a pair of inert doublet particles via the gauge boson or the Higgs channel
into SM particles as shown in gure 6.
The gauge boson mediated diagrams are momentum dependent. Their contribution is
small compared to the one shown in gure 5. Most of the thermal equilibrium evolution of
the dark matter particles occur above the EW symmetry breaking scale where the Higgs
mediated diagram of gure 6 are not present.
To calculate the freezeout temperature and the relic abundance, we need to solve the
Boltzmann equation. Assuming only swave annihilations, one can obtain the xf = mTf2 3
3We use m2 as the mass of the neutral scalar component of 2 because there are no mass corrections
which occur only after EW symmetry breaking.
0.2
0.1
2h 0.15
)
Ω
(
e
c
n
a
d
n
u
b
a
c
i
l
e
R 0.05
0
0
λ3=1
λ3=0.9
λ3=0.8
λ3=0.7
λ3=0.6
λ3=0.5
g
g1s=2
g
dmh2 to
;
(5.4)
(5.5)
(5.6)
(5.7)
HJEP1(207)8
500
1000
1500
2000
2500
Mass of dark matter (m)
Tf =
xf
m2 = 1:89 TeV;
m2 = 65:7 GeV:
This calculation has been done using a xed set of values for 3
, 4 and 5 with 3
1
and 4; 5
1. In principle, 3 can vary between 0:5 to 1 while still keeping
4 and 5
very small. The e ect of varying
3 on the mass of the dark matter candidate is shown
in gure 7 where the solid horizontal line shows the value of the relic abundance obtained
from Planck 2015 [12] and is equal to 0.12. Note that m2 is the dark matter mass till EW
symmetry breaks which occurs around the same time as freezeout. After the symmetry
black band is the Planck 2015 result.
at freezeout where Tf is the freezeout temperature, and the relic abundance
where mP is the Planck mass (not the reduced Planck mass which we have denoted as MP l),
g and g are the number of degrees of freedom in the plasma and the entropic number of
degrees of freedom respectively and h vi is taken from eq. (5.2).
Planck 2015 data for the relic abundance can now be used to get estimates for the
mass of the dark matter and its freezeout temperature. We obtain:
1
0.9
breaks, dark matter mass will get a small correction of order 100 GeV. The table 1 gives
the values of dark matter mass satisfying the relic abundance constraint for various values
of 3
. The corresponding freezeout temperatures are a little below the EW symmetry
breaking scale suggesting that we include the Higgs mediated diagrams in gure 6 in the
calculations. However, their contribution to the calculations are very small and any changes
that they bring about in dark matter masses are beyond the second decimal place.
6
Conclusion
Explaining in ation and dark matter remain two challenges for any theory beyond the
standard model of particle physics. The inert doublet model has been studied extensively
in the literature in the context of generating neutrino masses and mixing as well as dark
matter. The doublet is called inert because of a Z2 charge assignment which forbids all
Yukawa couplings of this doublet with the standard model fermions. This is done to avoid
all undesirable avor violations in the model. In this work we showed that the inert doublet
coupled nonminimally to gravity could act both as the in aton driving slowroll in ation as
well as the cold dark matter of the universe. We obtained a Starobinsky like potential from
the model and showed that both slowroll parameters ;
1. We calculated the scalar
power spectrum, the tensor to scalar ratio and the spectral index in our model and showed
them to be well within the observed limits from Planck. After successfully reheating the
universe, the inert doublet attains thermal equilibrium and eventually freezesout as a cold
dark matter. We obtained bounds on the couplings of the scalar potential from reheating
and dark matter constraints and showed that the Planck bound on relic abundance can be
satis ed for neutral scalar component mass of the inert doublet of around 1.3 to 2 TeV.
Acknowledgments
The authors would like to thank the Department of Atomic Energy (DAE) Neutrino Project
under the XII plan of HarishChandra Research Institute. This project has received
funding from the European Union's Horizon 2020 research and innovation programme
InvisiblesPlus RISE under the Marie SklodowskaCurie grant agreement No 690575. This project
has received funding from the European Union's Horizon 2020 research and innovation
programme Elusives ITN under the Marie Sklodowska Curie grant agreement No 674896.
Open Access.
This article is distributed under the terms of the Creative Commons Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
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