COSMOS \(e'\) soft Higgsotic attractors
Eur. Phys. J. C
COSMOSe soft Higgsotic attractors
Sayantan Choudhury 0
0 Department of Theoretical Physics, Tata Institute of Fundamental Research , Colaba, Mumbai 400005 , India
In this work, we have developed an elegant algorithm to study the cosmological consequences from a huge class of quantum field theories (i.e. superstring theory, supergravity, extra dimensional theory, modified gravity, etc.), which are equivalently described by soft attractors in the effective field theory framework. In this description we have restricted our analysis for two scalar fields  dilaton and Higgsotic fields minimally coupled with Einstein gravity, which can be generalized for any arbitrary number of scalar field contents with generalized noncanonical and nonminimal interactions. We have explicitly used R2 gravity, from which we have studied the attractor and nonattractor phases by exactly computing two point, three point and four point correlation functions from scalar fluctuations using the InIn (SchwingerKeldysh) and the δN formalisms. We have also presented theoretical bounds on the amplitude, tilt and running of the primordial power spectrum, various shapes (equilateral, squeezed, folded kite or countercollinear) of the amplitude as obtained from three and four point scalar functions, which are consistent with observed data. Also the results from two point tensor fluctuations and the field excursion formula are explicitly presented for the attractor and nonattractor phase. Further, reheating constraints, scale dependent behavior of the couplings and the dynamical solution for the dilaton and Higgsotic fields are also presented. New sets of consistency relations between two, three and four point observables are also presented, which shows significant deviation from canonical slowroll models. Additionally, three possible theoretical proposals have presented to overcome the tachyonic instability at the time of late time acceleration. Finally, we have also provided the bulk interpretation from the three and four point scalar correlation functions for completeness.

S. Choudhury: Presently working as a Visiting (PostDoctoral) fellow
at DTP, TIFR, Mumbai.
Contents
10.2 Dynamical dilaton at late times . . . . . . . . .
10.3 Details of the δN formalism . . . . . . . . . .
10.3.1 Useful field derivatives of N . . . . . .
10.3.2 Secondorder perturbative solution with
various source . . . . . . . . . . . . . .
10.3.3 Expressions for perturbative solutions
in final hypersurface . . . . . . . . . . .
10.3.4 Shift in the inflaton field due to δN . . .
10.3.5 Various useful constants for δN . . . . .
10.4 Momentum dependent functions in four point
function . . . . . . . . . . . . . . . . . . . . .
References . . . . . . . . . . . . . . . . . . . . . . . . .
1 Introduction
The inflationary paradigm is a theoretical proposal which
attempts to solve various longstanding issues with standard
Big Bang cosmology and has been studied earlier in various
works [
1–12
]. But apart from the success of this theoretical
framework it is important to note that no single model exists
till now using which one can explain the complete evolution
history of the universe and also one is unable to break the
degeneracy between various cosmological parameters
computed from various models of inflation [
13–33
]. It is
important to note that we have the vacuum energy contribution
generated by the trapped Higgs field in a metastable vacuum
state which mimics the role of an effective cosmological
constant in effective theory. At the later stages of the universe
such a vacuum contribution dominates over other contents
and correspondingly the universe expands in an
exponential fashion. But using such metastable vacuum state it is
not possible to explain the tunneling phenomenon and also
impossible to explain the end of inflation. To serve both of the
purposes the effective potential for inflation should have a flat
structure. Due to such a specific structure the effective
potential for inflation satisfies the flatness or slowroll condition
using which one can easily determine the field value
corresponding to the end of inflation. There are various classes
of models in existence in the cosmological literature where
one has derived such a specific structure of inflation [
14,34–
39
]. For example, the Coleman–Weinberg effective potential
serves this purpose [
40,41
]. Now if we consider the finite
temperature contributions in the effective potential [
42,43
]
then such thermal effects need to localize the inflaton field
to small expectation values at the beginning of inflation. The
flat structure of the effective potential for inflation is such
that the scalar inflaton field slowly rolls down in the valley of
potential during which the scale factor varies exponentially
and then inflation ends when the scalar inflaton field goes to
the nonslowrolling region by violating the flatness
condition. At this epoch inflaton field evolves to the true minimum
very fast and then it couples to the matter content of the
universe and reheats our universe via subsequent oscillations
about the minimum of the slowly varying effective
potential for inflation. This class of models is a very successful
theoretical probe through which it is possible to explain the
characteristic and amplitude of the spectrum of density
fluctuations with high statistical accuracy (2σ CL from Planck
2015 data [
44–46
]) and at late times these perturbations act
as the seeds for the large scale structure formation, which we
observe at the present epoch. Apart from this huge success
of the inflationary paradigm in the slowly varying regime it
is important to mention that these density fluctuations
generated from various classes of successful models were
unfortunately large enough to explain the physics of standard Grand
Unified Theory (GUT) with wellknown theoretical
frameworks and also it is not possible to explain the observed
isotropy of the Cosmic Microwave Background Radiation
(CMBR) at small scales during the inflationary epoch. The
only physical possibility is that the self interactions of the
inflaton field and the associated couplings to other matter
field contents would be sufficiently small to possibly satisfy
these cosmological and particle physics constraints. But the
prime theoretical challenge at this point is that for such a
setup it is impossible to achieve thermal equilibrium at the
end of inflation. Consequently, it is not at all possible to
localize the scalar inflaton field near zero Vacuum Expectation
Value (VEV), φ = 0φ0 = 0, where 0 is the
corresponding vacuum state in quaside Sitter space time.
Therefore, a sufficient amount of expansion will not be obtained
from this prescribed setup. Here it is important to note that,
for a broad category of effective potentials, the inflaton field
evolves with time very slowly compared to the Hubble scale
following slowroll conditions and satisfies all of the
observational constraints [
44–46
] computable from various
inflationary observables from this setup. However, apart from the
success of the slowroll inflationary paradigm the density
fluctuations or more precisely the scalar component of the
metric perturbations restricts the coupling parameters to be
sufficiently small and allows huge finetuning in the
theoretical setup. This is obviously a not recommendable
prescription from a model builder’s point of view. Additionally, all
these classes of models are not ruled out completely by the
present observed data (Planck 2015 and other joint data sets
[
44–47
]), as they are degenerate in terms of the
determination of inflationary observables and associated cosmological
parameters in precision cosmology. There are various ideas
existing in the cosmological literature which can drive
inflation. These are:
• Category I: In this class of models, inflation is driven
through a field theory which involves a very high energy
physics phenomenon. Example: string theory and its
supergravity extensions [
13,15–17,19,22,23,25,48–82
],
various supersymmetric models [
14,34–39
], etc.
• Category II: In this case, inflation is driven by
changing the mathematical structure of the gravitational sector.
This can be done using the following ways:
1. Introducing higher derivative terms of the form of
f (R), where R is the Ricci scalar [
83–86
].
Example: the Starobinsky inflationary framework, which
is governed by the model [83] f (R) = a R + b R2,
where the coefficients a and b are given by a = M 2p
and b = 1/6M 2. If we set a = 0 and b = 1/6M 2 = α
then we can get back the theory of scale free gravity
in this context. In this paper we will explore the
cosmological consequences from the scale free theory of
gravity.
2. Introducing higher derivative terms of the form of
Gauss–Bonnet gravity coupled with a scalar field in
a nonminimal fashion, where the contribution in the
effective action can be expressed as [
87,88
]
(1.1)
SG B =
d4x √−g f (φ)[Rμναβ Rμναβ
− 4Rμν Rμν + R2],
where f (φ) is the inflaton dependent coupling which
can be treated as the nonminimal coupling in the
present context. This is also an interesting possibility
which we have not explored in this paper. Here one
cannot consider the Gauss–Bonnet term in the gravity
sector in 4D without coupling to other matter fields, as
in 4D the Gauss–Bonnet term is a topological surface
term.
3. Another possibility is to incorporate the effect of
nonminimal coupling of the inflaton field and the
gravity sector [
89–91
]. The simplest example is f (φ)R
gravity theory. For Higgs inflation [
89
], f (φ) =
(1 + ξ φ2), where ξ is the nonminimal coupling
of the Higgs field. Here one can consider a more
complicated possibility as well by considering a
noncanonical interaction between inflaton and f (R)
gravity by allowing an f (φ) f (R) term in the 4D
effective action [
92
]. For the construction of the
effective potential we have considered this possibility.
4. One can also consider the other possibility, where
higher derivative nonlocal terms can be incorporated
in the gravity sector [
93–102
]. For example one can
consider the possibilities R f1( )R, Rμν f2( )Rμν ,
Rμναβ f3( )Rμναβ , R f4( )∇μ∇ν ∇α∇β Rμναβ , Rμναβ
f5( )∇α∇β ∇ν ∇ρ ∇λ∇γ Rμρλγ , Rμναβ f6( )∇α
∇β ∇ν ∇μ∇λ∇γ ∇η∇ξ Rλγ ηξ , where is defined as
= √1−g ∂μ[√−g gμν ∂ν ]; it is the d’Alembertian
operator in 4D and the fi ( )∀i = 1, 2, . . . , 6 are
analytic entire functions containing higher derivatives up
to infinite order. This is itself a very complicated
possibility which we have not explored in this paper.
• Category III: In this case, inflation is driven by
changing the mathematical structure of both the gravitational
and the matter sector of the effective theory. One of the
examples is to use Jordan–Brans–Dicke (JBD) gravity
theory [
103,104
] along with extended inflationary
models which includes noncanonical interactions. By
adjusting the value of the Brans–Dicke parameters one can
study the observational consequences from this setup.
Instead of Jordan–Brans–Dicke (JBD) gravity theory one
can also use nonlocal gravity or many other complicated
possibilities.
In this paper, we consider the possibility of the soft
inflationary paradigm in an Einstein frame, where a chaotic
Higgsotic potential is coupled to a dilaton via exponential type
of potential, which is appearing through the conformal
transformation from Jordan to Einstein frame in the metric within
the framework of scale free α R2 gravity. Here it is
important to mention that, in the case of a soft inflationary model,
the dilaton exponential potential is multiplied by a coupling
constant of the Higgsotic theory which mimics the role of
an effective coupling constant and its value always decreases
with the field value. One can generalize this idea for any
arbitrary matter interactions which is also described by
generalized P(X, φ) theory [
105,106
] (see Appendix 10.1 for
more details). In this context also it is important to specify
that one can treat the field dependent couplings in the simple
effective potentials or maybe in generalized P(X, φ)
functionals, entailing a decaying behavior with dilaton field value
as it contains an overall exponential factor which is coming
from the dilaton potential itself in an Einstein frame. This
is a very interesting feature from the point of view of RG
flow in QFT as the field dependent coupling in an Einstein
frame captures the effect of field flow (energy flow). In this
context instead of solving directly the RGE for the effective
coupling, we solve the dynamical equations for the fields
and the effective coupling for powerlaw and exponential
attractors. Due to the similarities in the two techniques here
one can arrive at the conclusion that in cosmology solving a
dynamical attractor problem in the presence of effective
coupling in an Einstein frame mimics the role of solving RGE in
QFT. Thus due to the exponential suppression in the
effective coupling in an Einstein frame it is naturally expected
from the prescribed framework that for suitable choices of
the model parameters soft cosmological constraints can be
obtained [
107,108
]. As in this prescribed framework the
dilaton exponential coupling plays a very significant role, one
can ask the very crucial question of its theoretical origin.
Obviously there are various sources in existence from which
one can derive exponential effective couplings or more
precisely the effective potentials for dilaton. These possibilities
are:
Scale free gravity +
Higgso c scalar field
(in Jordan frame)
Two,three and four
point (using δN)
func on for infla on
+Rehea ng constraints
Consistent with
observa on (Planck
and other joint data)
Consistent with
rehea ng and late
me accelera on
Conformal Transforma on
Conformal Transforma on
Attractor phase
Power law
(stable) a ractor
Einstein gravity+
Higgso c dilaton
coupled twofield
theory
(in Einstein frame)
Dynamical
A ractor
solu ons
Exponen al
(unstable) a ractor
Stability from String Theory,
nonminimal coupling and
contribu on from mass
Scale free gravity +
Higgso c scalar field
(in Jordan frame)
Two, three and four
point (using Schwinger
Keldysh/InIn)
func on for infla on
+Rehea ng constraints
Consistent with
observa on (Planck
and other joint data)
Consistent with
rehea ng and late
me accelera on
Non
attractor
phase
Einstein gravity+
Higgso c dilaton
coupled twofield
theory
(in Einstein frame)
Assume
dilaton is
heavy field
Freezing dilaton
at UV cutoff of
Effec ve theory
Dynamical non a ractor
solu on
(a) Diagrammatic representation of attractor phase of soft (b) Diagrammatic representation of nonattractor phase of
Higgsotic inflation. In this representative diagram we have soft Higgsotic inflation. In this representative diagram we
shown the steps followed during the computation. have shown the steps followed during the computation.
Dynamical dilaton
Fixed dilaton
Infla on from so
Higgso c sector coupled with dilaton (in Einstein frame)
Attractor
phase
Non
attractor
phase
New consistency rela ons for NG+ PGW for two a ractors
Old consistency rela on for NG+ New consistency rela on for PGW
(c) Diagrammatic representation of attractor and nonattractor dynamical
phase of soft Higgsotic inflation which is coupled with dilaton in Einstein
frame.
• Source I: One of the sources for dilaton exponential
potential is string theory, appearing in the Category I.
Specifically, superstring theory and low energy
supergravity models are the theoretical possibilities in string
theory [
109–118
] where dilaton exponential potential
appears in the gravity part of the action in a Jordan
frame and after a conformal transformation in the
Einstein frame such dilaton effective potential is coupled
with the matter sector. The most important example is
the αattractor which mimics a class of inflationary
models in N = 1 supergravity in 4D. For details see Refs.
[
119–129
].
• Source II: Another possible source of the dilaton
exponential potential is coming from a modified gravity
theory framework such as f ( R) gravity [
83–86
], f (φ) f ( R)
gravity [
89–92
] and Jordan–Brans–Dicke theory [
103,
104
] in the Jordan frame, which appear in the
Category II (1 and 3) and Category III. After transforming
the theory in the Einstein frame via conformal
transformation one can derive the dilaton exponential potential.
In Fig 1a–c, we have shown the diagrammatic
representation of attractor and nonattractor phases of soft Higgsotic
inflation. In these representative diagrams we have shown the
steps followed during the computation. In this work we have
addressed the following important points through which it is
possible to understand the underlying cosmological
consequences from the proposed setup. These issues are:
• Transition from scale free gravity to scale dependent
gravity have discussed and its impact on the solutions
in the attractor and nonattractor regime of inflation have
also discussed.
• Explicit calculation of the δN formalism is presented
by considering the effect up to secondorder
perturbation in the solution of the field equation in attractor
regime. Additionally deviation in the consistency
relation between the nonGaussian amplitude for four point
and three point scalar correlation function a.k.a. Suyama–
Yamaguchi relation is presented to explicitly show the
consequences from attractor and nonattractor phase.
• Additionally, new sets of consistency relations are
presented in attractor and nonattractor phases of inflation
to explicitly show the deviation from the results obtained
from a canonical single field slowroll model.
• Detailed numerical estimations are given for all the
inflationary observables for attractor and nonattractor phases
of inflation which confronts well Planck 2015 data.
Additionally, constraints on reheating is also presented for
attractor and nonattractor phase.
• Bulk interpretation are given in terms of S, T and U
channel contribution for all the individual terms obtained
from three and four point correlation function.
• Scale dependent behaviors of the nonminimal coupling
between inflaton field and additional dilaton field are
given in an Einstein frame for powerlaw and exponential
types of attractor.
• Three possible theoretical proposals have presented to
overcome the tachyonic instability [
130–134
] at the time
of late time acceleration in a Jordan frame and due to this
fact the structure of the effective potentials changes in an
Einstein frame as well. These proposals are inspired by:
– I. NonBPS Dbrane in superstring theory [23,135–
140].
– II. An alternative situation where we switch on the
effects of additional quadratic mass term in the
effective potential.
– III. Also we have considered a third option where we
switch on the effect of nonminimal coupling between
scale free α R2 gravity and the inflaton field.
Now before going to the further technical details let us clearly
mention the underlying assumptions to understand the
background physical setup of this paper:
1. We have restricted our analysis up to monomial φ4
model and due to the structural similarity with Higgs
potential at the scale of inflation we have identified
monomial φ4 model as Higgsotic model in the present
context.
2. To investigate the role of scale free theory of gravity, as
an example we have used α R2 gravity. But the present
analysis can be generalized to any class of f (R) gravity
models.
3. In the matter sector we allow only simplest canonical
kinetic term which are minimally coupled with α R2
gravity sector. For such canonical slowroll models the
effective sound speed cS = 1. But for completeness one can
consider a most generalized version of P(X, φ) models,
where X = − 21 gμν ∂μφ∂ν φ and the effective sound speed
cS < 1 for such models. For example one can consider
the following structure [
56,105
]:
1
P(X, φ) = − f (φ)
1
1 − 2X f (φ) + f (φ) − V (φ),
(1.2)
which is exactly similar to the DBI model. But here
one can implement our effective Higgsotic models in
V (φ) instead of choosing the fixed structure of the DBI
potential in UV and IR regime. Here one can choose
[
56
] f (φ) ≈ φg4 , which is known as the throat factor in
string theory. In string theory g is the parameter which
depends on the flux number. But other choices for f (φ)
are also allowed for the general class of P(X, φ)
theories which follows the above structure. Similarly one can
consider the following structure of P(X, φ) for tachyon
and Gtachyon models given by [
23,141
]
For Tachyon: P(X, φ) = −V (φ)√1 − 2X α , (1.3)
For GTachyon: P(X, φ) = −V (φ)(1 − 2X α )q
(1/2 < q < 2), (1.4)
where α is the Regge slope. Here we consider the most
simple canonical form, P(X, φ) = X − V (φ), where
V (φ) is the effective potential for the monomial φ4 model
considered here for our computation.
4. As a choice of the initial condition or precisely as the
choice of vacuum state we restrict our analysis using
Bunch–Davies vacuum. If we relax this assumption, then
we can generalize the results for α vacua as well.
5. During our computation we have restricted ourselves up
to the minimal interaction between the α R2 gravity and
matter sector. Here one can consider the possibility of
nonminimal interaction between α R2 gravity and matter
sector.
6. During the implementation of the InIn formalism [
2
]
to compute three and four point correlation function we
have use the fact that the additional dilaton field is fixed
at Planckian field value to get the nonattractor behavior
of the present setup. One can relax this assumption and
can redo the analysis of the InIn formalism to compute
three and four point correlation function without freezing
the dilaton field and also use the attractor behavior of
the model to simplify the results.
7. During the computation of correlation functions using a
semi classical method, via the δN formalism [
23,142–
146
], we have restricted up to secondorder contributions
in the solution of the field equation in FLRW background
and also neglected the contributions from the back
reaction for all type of effective Higgsotic models derived in
an Einstein frame. For completeness, one can relax these
assumptions and redo the analysis by taking care of all
such contributions.
8. In this work we have neglected the contribution from the
loop effects (radiative corrections) in all of the effective
Higgsotic interactions (specifically in the self couplings)
derived in the Einstein frame. After switching on all such
effects one can investigate the numerical contribution of
such terms and comment on the effects of such terms in
precision cosmology measurement.
9. We have also neglected the interactions between gauge
fields and Higgsotic scalar field in this paper. One can
consider such interactions by breaking conformal
invariance of the U (1) gauge field in the presence of time
dependent coupling f (φ (η)) to study the features of
primordial magnetic field through inflationary
magnetogenesis [
147–149
].
The plan of this paper is as follows:
• In Sect. 2, we start our discussion with f (R) = α R2
gravity where a scalar field is minimally coupled with
the gravity sector and contains only canonical kinetic
term. Next in the matter sector we choose a very simple
monomial model of potential, V (φ) = λ4 φ4, which can
be treated as a Higgs like potential as at the scale of
inflation, the contribution from the VEV of Higgs is almost
negligible.
• Further, in Sect. 3, we provide the field equations in a
Jordan frame written in a spatially flat FLRW background.
Next, we perform a conformal transformation in the
metric to the Einstein frame and introduce a new dilaton field.
Further, we derive the field equations in an Einstein frame
and try to solve them for two dynamical attractor features:
a powerlaw solution, and exponential solution. However,
the second case give rise to tachyonic behavior which can
be resolved by consideringI. nonBPS Dbrane in
superstring theory, II. via switching on the effect of quadratic
term in the effective potential and III. by introducing a
nonminimal coupling between matter and α R2 gravity
sector.
• Next, in Sect. 4, using two dynamical attractors,
powerlaw and exponential solution, we study the
cosmological constraints in the presence of two fields. We study
the constraints from primordial density perturbation, by
deriving the expressions for two point function and the
present inflationary observables in Sect. 4.2. Further, we
repeat the analysis for tensor modes and also comment on
the future observables – the amplitude of the tensor
fluctuations and tensortoscalar ratio in Sect. 4.3.
Additionally, in Sect. 4.4, we study the constraint for the reheating
temperature. Finally, in Sects. 5.1 and 5.2, we derive the
expression for the inflaton and the nonminimal coupling
at horizon crossing, during reheating and at the onset of
inflation for the two above mentioned dynamical
cosmological attractors.
• Further, in Sect. 6, we have explored the cosmological
solutions beyond attractor regime. Here we restrict
ourselves at spatially flat FLRW background and made
cosmological predictions from this setup in Sect. 7.1. To
serve this purpose we have used the ADM formalism
using which we compute two point functions and
associated present observables using the Bunch–Davies initial
condition for scalar fluctuations in Sects. 7.2.1 and 7.2.2.
Further, in Sects. 7.3.1 and 7.3.2, we repeat the procedure
for tensor fluctuations as well where we have computed
two point function and the associated future observables.
We also derive a few sets of consistency relations in this
context which are different from the usual single field
slowroll models. Further, in Sect. 7.4, we derive the
constraints on reheating temperature in terms of observables
and the number of efoldings.
• Next, in Sects. 8.1.1 and 8.1.2, as a future probe, we
compute the expression for three point function and the
bispectrum of scalar fluctuations using the InIn formalism
for the nonattractor case and the δN formalism for the
attractor case. Further, we derive the result for a
nonGaussian amplitude fNloLc for equilateral and squeezed
limit triangular shape configuration. Also we give a bulk
interpretation of each of the momentum dependent terms
appearing in the expression for the three point scalar
correlation function in terms of S, T and U channel
contributions. Further, for the consistency check we freeze
the additional field in the Planck scale and redo the
analysis of the δN formalism. Here we show that the
expression for the three point nonGaussian amplitude
is slightly different as expected for the single field case.
Further, in Sects. 8.1.1 and 8.1.2, we compare the results
obtained from the InIn formalism and δN formalism for
the non attractor phase, where the additional field is
fixed in Planck scale. Finally, we give a theoretical bound
on the scalar three point nonGaussian amplitude.
• Finally, in Sects. 8.2.1 and 8.2.2, as an additional future
probe, we have also computed the expression for the four
point function and the trispectrum of scalar fluctuations
using the InIn formalism for the nonattractor case and
δN formalism for the attractor case. Further, we derive
the results for nonGaussian amplitude gNloLc and τ NloLc for
equilateral, countercollinear or folded kite and squeezed
limit shape configuration from the InIn formalism.
Further we give a bulk interpretation of each of the
momentum dependent terms appearing in the expression for the
four point scalar correlation function in terms of S, T and
U channel contributions. In the attractor phase
following the prescription of δN formalism we also derive the
expressions for the four point nonGaussian amplitude
gNloLc and τ NloLc. Next we have shown that the consistency
relation connecting three and four point nonGaussian
amplitude aka Suyama–Yamaguchi relation is modified
in the attractor phase and further given an estimate of the
amount of deviation. Further, for the consistency check
we freeze the additional field in Planck scale and redo
the analysis of the δN formalism. Here we show that the
four point nonGaussian amplitude is slightly different
as expected for the single field case. Finally, we give a
theoretical bound on the scalar four point nonGaussian
amplitude.
2 Model building from scale free gravity
To describe the theoretical setup let us start with the total
action of f (R) gravity coupled minimally along with a scalar
inflaton field φ:
S =
d4x √−g f (R) −
(∂μφ)(∂ν φ) − V (φ)
(2.1)
gμν
2
where in general f (R) can be an arbitrary function of the
Ricci scalar R. For example one can choose a generic form
given by [
150,151
]
mial powers of R. Here also one can treat the α Rn term as
an additional quantum correction to the Einstein gravity.
4. In our computation we set a1 = a = 0, a2 = b =
α, an = 0∀n > 2, which is known as scale free gravity
in a Jordan frame: f (R) = α R2, where α is a
dimensionless scale free coefficient. For this type of theory if
we perform the conformal transformation from Jordan
to Einstein frame then we will induce a constant term in
the effective potential and this can be interpreted as the
4D cosmological constant using which one can fix the
scale of the theory for early and late universe. But in our
computation we introduce an additional scalar field in the
action in a Jordan frame, which we identified to be the
inflaton. After a conformal transformation in an Einstein
frame we get an effective potential which is coming from
the interaction between the dilaton exponential potential
and the inflationary potential as appearing in a Jordan
frame. We will show that here the two fields, dilaton and
inflaton form dynamical attractors using which one can
very easily solve this twofield complicated model in the
context of cosmology.
Next we will discuss the structure of the inflational as
appearing in Eq. (2.1). Generically in 4D effective theory the
effective potential can be expressed as
f (R) =
an Rn,
∞
n=1
where an∀n are the expansion coefficients for the above
mentioned generic expansion. Here one can note down the
following features of this generic choice of the expansion:
1. If we set a1 = M 2p/2, an = 0∀n > 1, then one can
get back the wellknown Einstein–Hilbert action (GR) in
Jordan frame: f (R) = M 2p R/2. In this particular case
Jordan frame and Einstein frame are exactly the same
because the conformal factor for the frame
transformation is unity. This directly implies that no dilaton potential
appears due to the frame transformation from Jordan to
Einstein frame. But since in this paper we are
specifically interested in the effects of modified gravity sector,
the higher powers of R are more significant in the above
mentioned generic expansion of f (R) gravity.
2. If we set, a1 = a = M 2p/2, a2 = b = α, an = 0∀n > 2,
then one can get back the specific structure of the very
wellknown Starobinsky model: f (R) = a R + b R2 =
M 2p R/2 + α R2. Here one can treat the α R2 term as an
additional quantum correction to the Einstein gravity.
3. One can also set a1 = a = M 2p/2, an = α∀n ≥ 2,
then one can get back the specific structure f (R) =
M 2p R/2 + α Rn, which describes the situation where the
Einstein–Hilbert gravity action is modified by the
mono(2.2)
V (φ) =
Vren(φ)
Renormalizable part
+
∞
δ=5
Jδ(g)
φδ
M δp−4
Nonrenormalizable part
=
∞
δ=0
Cδ(g)
φδ
M δp−4
where Jδ(g) and Cδ(g) are the Wilson coefficients in effective
theory. Here g stands for the scale of theory and the
dependences of the Wilson coefficients on the scale can be exactly
computed for a full UV complete theory using
renormalization group equations. In this paper a similar scale
dependence on the couplings we will calculate using dynamical
attractor method in an Einstein frame, which exactly
mimics the role of solving renormalization group equations in
the context of cosmology. As written here, the total effective
potential is made by renormalizable (relevant operators) and
nonrenormalizable (irrelevant operators) part, which can be
obtained by heavy degrees of freedom from a known UV
complete theory. In our computation we just concentrate on
the renormalizable part of the action, which can be recast as
V (φ) =
Cδ(g)
4
δ=0
φδ
M δp−4
.
Next to get the Higgslike monomial structure of the potential
we set C3(g) = 0, as in this paper our prime motivation is
to look into only Higgsotic potentials. Consequently we get
(2.3)
(2.4)
V (φ) = C0 + C2(g)M 2pφ2 + C4(g)φ4.
To get the Higgsotic structure of the potential one should set
C0(g) = λ4 v4, C2(g) = − λ2 v2, C4(g) = λ4 .
Here v is the VEV of the field φ. Consequently, one can write
the potential in the following simplified form:
V (φ) = λ4 (φ2 − v2)2.
Now we consider a situation where scale of inflation as well
as the field value are very much larger than the VEV of the
field. This assumption is pretty consistent with inflation with
Higgs field. Consequently, in our case the final simplified
monomial form of the Higgsotic potential is given by
λ 4
V (φ) = 4 φ .
Further varying Eq. (2.1) with respect to the metric and using
Eqs. (2.2) and (2.8) the equation of motion (modified Einstein
equation) for the α R2 scale free gravity can be written as
G˜ μν := α[{Rμν + 2(gμν
− ∇μ∇ν )} + Gμν ]R = Tμν
wgαhβe∇reα∇thβe =D’Aglαeβm∇bαe∂rβtian= op√e1−ragt∂oαr( √is−dgegfiαnβe∂dβ )asand th=e
energymomentum stress tensor can be expressed as
Tμν = − √−g
2
δ(√−gLM )
δgμν
= ∂μφ∂ν φ − gμν
1 gαβ ∂αφ∂β φ + λ4 φ4 .
2
(2.5)
(2.6)
(2.7)
(2.9)
(2.10)
(2.11)
(2.12)
Here it is important to note that the Einstein tensor is defined
gμν
as Gμν := Rμν − 2 R. Now after taking the trace of
Eq. (2.9) we get R R = 6Tα , where the trace of the
energymomentum tensor is characterized by the symbol
T = Tμμ. In this modified gravity picture we have ∇μG˜ μν =
4α[∇μ, ]R = 0 where we use ∇μ Rμν = g2μν ∇μ R, which
directly follows from the Bianchi identity ∇μGμν = 0. Now
varying Eq. (2.1) with respect to the field φ we get the
following equation of motion in curved spacetime:
φ = −V (φ) = −λφ3
= −λφ3.
1
⇒ √−g ∂α
√−ggαβ ∂β φ
Further assuming the flat (k = 0) FLRW background metric
the Friedmann equations can be written from Eq. (2.9) as
a 2
H 2 = a˙
2H˙ + 3H 2 = 2
a¨ a˙ 2
a + a
pφ
where we have assumed the energymomentum tensor can be
described by a perfect fluid as Tνμ = diag(−ρφ , pφ , pφ , pφ )
where the energy density ρφ and the pressure density pφ can
be expressed for the scalar field φ as
ρφ = φ˙22 + λ4 φ4, pφ = φ˙22 − λ4 φ4.
Similarly the field equation for the scalar field φ in the flat
(k = 0) FLRW background can be recast as
φ¨ + 3H φ˙ + λφ3 = 0.
(2.8)
In the flat (k = 0) FLRW background we have the following
expressions:
R = 6( H..˙. + 2H 2),
R¨ = 6(H + 4H˙ 2 + 4H H¨ ).
R˙ = 6(H¨ + 4H H˙ ),
Substituting these results in Eqs. (2.12) and (2.13) the
Friedmann equations can be recast in the Jordan frame as
2H (H¨ + 3H H˙ ) − H˙ 2 = 1ρ8φα ,
9H˙ (H˙ + H 2) + 6H H¨ + H... = − 6pαφ .
In the slowroll regime (φ˙ 2/2 λ4 φ4) the energy density ρφ
and the pressure density pφ can be approximated by ρφ ≈
λ φ4, pφ ≈ − 4
4 λ φ4. Consequently Eqs. (2.15), (2.17) and
(2.18) can be recast as
3H φ˙ + λφ3 ≈ 0,
V (φ)
2H (H¨ + 3H H˙ ) − H˙ 2 ≈ 18α ,
... V (φ)
9H˙ (H˙ + H 2) + 6H H¨ + H ≈ − 6α ,
λ φ4. Further combining Eqs. (2.20) and
where V (φ) =... 4
(2.21) we get H = 3H˙ (3H 2 − 4H˙ ). For further analysis
one can also define the following sets of slowroll
parameters in a Jordan frame:
H = − HH˙2 , δH = − HH¨3 = H˙H − 2 2H ,
...
γH = − HH4 = 3 H (3 + 4 H ) , ηH = − Hφ¨φ˙ .
Further using these new sets of parameters in Eqs. (2.20) and
(2.21) can be recast into the following simplified form:
γH 21
2δH + 12 + 4
V (φ) λφ4
H ≈ − 18α H 4 = − 72α H 4 .
(2.13)
(2.14)
(2.15)
(2.16)
(2.17)
(2.18)
(2.19)
(2.20)
(2.21)
(2.22)
(2.23)
However, solving this twofield problem in the presence of
scale free gravity is itself very complicated for the following
reasons:
• Complication I: First of all, for a given structure of
inflationary potential in a Jordan frame (here it is the Higgsotic
potential as mentioned earlier) it is impossible to solve
directly the dynamical equations (2.20), (2.21) and (2.23)
due its complicated coupled structural form.
• Complication II: One can use various solution Ansatzes
to get approximated numerical results, but this is also
dependent on the structure of the inflaton potential in a
Jordan frame and how one can able to implement initial
condition (starting point) of inflation for arbitrary
structure of the effective potential.
• Complication III: In connection with the implementation
of the initial condition and to check the sufficient
condition for inflation in this complicated field theoretical
setup one needs to define the expression for number of
efoldings in terms of effective potentials. But this cannot
be very easy in the present context as the field equations
are coupled.
Due to these huge number of difficulties in a Jordan frame
we transform the total action into the Einstein frame using
a conformal transformation. After transforming the Jordan
frame action into the Einstein frame in the present context we
need the solve a two interacting field problem in the presence
of Einstein gravity. There are several ways one can solve this
problem. These possibilities are:
• Solution I: the first solution to this problem is to follow
the wellknown approach to solving twofield models of
inflation by following the method of curvature and
isocurvature perturbation in the semiclassical δN formalism.
For more accurate results one can also solve directly the
Mukhanov–Sasaki equation for this twofield model and
directly treat fluctuations quantum mechanically. Since
this methodology has been discussed in various earlier
works, we will not discuss this issue in this paper. See
Refs. [
152–156
] fore more details.
• Solution II: a second way of solving this problem is to
use dynamical attractor mechanism in the present
context where the two fields are connected through specific
relations, which can be obtained by solving dynamical
field equations in cosmology. This is equivalent to solving
renormalization group equations in the context of
quantum field theory as the dynamical attractor solution of two
fields captures the effects of all the energy scale. In our
computation we explore the possibility of two dynamical
attractors:
1. Powerlaw attractor
2. Exponential attractor
Here they have different cosmological consequences. But
they originate from the Higgsotic structure of the
effective potential which we will discuss in the next section in
detail.
• Solution III: a final possibility is to freeze the dilaton
field in the Planck scale or in the vicinity, so that one
can absorb it in the effective couplings in the Higgsotic
theory. This is identified as the nonattractor phase in the
context of cosmology. The physical justification for such
possibilities can also be explained from the UV behavior
of the 4D effective theory, which is known as the UV
completion of the effective theory. According to this
proposal we have two sectors in the theory:
1. Hidden sector: the hidden sector is made up of a heavy
field (in our case dilaton) which lies around the UV
cutoff of the effective theory, which is the Planck
scale. We are not able to probe directly this sector.
But we can visualize how its imprints on the low
energy effective theory.
2. Visible sector: the visible sector is made up of a
light field (in our case inflaton) which one can probe
directly. For present discussion the visible sector is
important to explain the cosmological evolution.
Usually in such a prescription one integrates the heavy
fields and finally gets an effective theory in the
visible sector. Here we use the fact that such a
procedure mimics the role of freezing the heavy dilaton field
near the Planck scale. The only difference is that in the
case of freezing the dilaton field we only concentrate
on the Higgsotic potential. But the integration of the
heavy field allows for all relevant and irrelevant
operators. However, by applying a similar argument one can
look into only the renormalizable Higgsotic part of the
total effective potential. Additionally, it is important to
note that at late times the dynamical picture is
completely opposite where the inflaton field freezes in the
vicinity of the Planck scale and the dynamical
contribution for late time acceleration comes from the
dilaton field. In a simpler way one can interpret this
physical prescription as the competitive dynamical
description of the two fields. During inflation the Higgsotic
field wins the game and at late times the dilaton serves
the same purpose. More precisely, within this
prescription dynamic features transfer from dilaton to Higgsotic
field (or any scalar inflaton) during inflation and at late
times a completely opposite situation appears, where
a similar transfer takes place from inflaton to dilaton
field.
In this paper we explore the possibility of Solution II and
Solution III in detail in the next section. For completeness
we briefly review also Solution I in the appendix.
3 Soft attractor: a twofield approach
where after applying C.T. the total potential can be recast as
In the present context let us introduce a scale dependent mode
, which can be written in terms of a no scale dilaton mode
as
2
= f (R)M −p2 = 2α R M −p2 = e 3 Mp
which mimics the role of a Lagrange multiplier and arises
in the Jordan frame without spacetime derivatives. In terms
of the newly introduced no scale dilaton mode the total
action of the theory (see Eq. (2.1)) can be recast as
S =
− g2μν (∂μφ)(∂ν φ) − λ4 φ4 .
To study the behavior of the proposed R2 theory of gravity
here we introduce the following conformal transformation
(C.T.) in the metric from Jordan frame to the Einstein frame:
gμν −C−.→T. g˜μν =
2gμν , gμν −C−.→T. g˜μν =
−2gμν ,
√−g −C−.→T.
−g˜ =
4√−g,
which satisfies the condition gμν gνβ = g˜μκ g˜κβ = δμβ. In the
present context the conformal factor is given by
=
√
√2
= e 2√3 Mp .
With this proposed C.T. in the metric the Ricci curvature
scalar in the Jordan frame (R) is related to the Einstein frame
(R˜ ) as
R =
2[R˜ + 6 ln
− 6g˜μν ∂˜μln
∂˜ν ln
]
where ∂˜μ = ∂x∂˜μ and ln ≡ √1−g˜ ∂α( −g˜ g˜αβ ∂β ln ).
After doing C.T. the total action can be recast in the Einstein
frame as1:
1 Here we apply Gauss’ theorem to remove the following contribution
in the total effective action:
d4x −g˜ 23 Mp ˜
=
=
where V0 = M 4p/8α exactly mimics the role of the
cosmological constant and the effective matter coupling (λ( )) in
the potential sector is given by λ( ) = λ4 = λe− 2√√32 Mp .
Now varying Eq. (3.7) with respect to the metric the field
equations can be expressed as
G˜μν :=
R˜μν − g˜2μν R˜
= T˜μν (φ, )
where the energymomentum tensor T˜μν (φ, ) for the
dilaton–inflaton coupled theory can be expressed as
(3.8)
(3.9)
T˜μν (φ, ) = −
2 δ
−g˜
−g˜L(φ, )
δg˜μν
= ∂˜μφ∂˜ν φ + ∂˜μ ∂˜ν
− g˜μν
+ 21 g˜αβ ∂˜α ∂˜β
+ W˜ (φ, ) .
2 ˜
1 gαβ ∂˜αφ∂˜β φ
Here for the matter part of the action the following property
holds between the Einstein frame and Jordan frame
energy
2 δ(√−g˜LM )
momentum tensor: T˜μν (φ, ) ⊃ T˜μν = − √−g˜ δg˜μν
= Tμ2ν , which implies that using the perfect fluid
assumption one can write T˜νμ
= μdiag(−ρ˜φ , p˜φ , p˜φ , p˜φ ) =
14 diag(−ρφ , pφ , pφ , pφ ) = Tν4 . Assuming the flat (k = 0)
FLRW background metric in an Einstein frame the
Friedmann equations can be written from Eq. (3.9) as2:
Additionally, the Hubble parameter in the Einstein frame
(H˜ ) can be expressed as its Jordan frame (H ) counterpart
2 It is important to mention here that the time interval in an Einstein
frame dt˜ is related to the time interval in a Jordan frame dt as dt˜ = dt.
H˜ 2 =
d H˜
dt˜ + H˜ 2 =
d ln a 2
dt˜
d2a
dt˜2
ρ
˜ ,
= 3M 2p
= −
(ρ˜ + 3 p˜)
6M 2p
ρ˜ =
p˜ =
d
dt˜
d
dt˜
2
2
+
+
dφ
dt˜
dφ
dt˜
2
2
+ W˜ (φ, ),
− W˜ (φ, ).
where the effective energy density (ρ˜) and the effective
pressure ( p˜) can be written in an Einstein frame as
(3.10)
(3.11)
(3.12)
1
= e− √6 Mp
as H˜ = 1 H + 21 d lndt 2 H + √6˙Mp . Also
the Klein–Gordon field equations for the inflaton field φ and
the new field can be written in the flat (k = 0) FLRW
background as
d2φ dφ
dt˜2 + 3H˜ dt˜ + ∂φ W˜ (φ, ) = 0
d2 d
dt˜2 + 3H˜ dt˜ + ∂ W˜ (φ, ) = 0.
Now in the slowroll regime the field equations are
approximated by
dφ
3H˜ dt˜ + λ( )φ3 = 0
d λ( )φ4
3H˜ dt˜ − √6M p
= 0,
To study the behavior of the proposed model let us consider
two cases, where the dynamical features are characterized by
1. Case I: powerlaw solution,
2. Case II: exponential solution.
which we discuss in the next subsection.
3.1 Case I: Powerlaw solution
We consider here large α, small V0(≈ 0) with λ > 0 with
effective potential
W˜ (φ, ) ≈
λ(4 ) φ4 = λ4 e− 2√√32 Mp φ4 (for Case I).
Consequently the field equations can be recast as
dφ 2√2
3H˜ dt˜ + λe− √3 Mp φ3 = 0,
λ
˜ = 12M 2p
d λφ4
3H˜ dt˜ − √6M p
2√2
H 2 e− √3 Mp φ4.
This is the case where the cosmological constant V0 or more
precisely the parameter α will not appear in the final solution.
The cosmological solutions of Eqs. (3.19)–(3.21) are given
by3:
3 Throughout the paper the subscript ‘0’ is used to describe the
inflationary epoch.
Case I
−
,
(a) Case I : Powerlaw behavior.
(b) Case II : Tachyonic behaviour.
Fig. 2 Behavior of the inflationary potential for a V0 ≈ 0 and λ > 0
(Case I) and b V0 = 0 and λ < 0 (Case II). In a the inflaton rolls down
from a large field value and inflation ends at φ f ≈ 1.09 M p. On the
other hand in b the inflaton field rolls down from a small field value
= − 16αλ( )φ02 ⎣⎢ 1 −
M 4p
≈ 2αλ( )φ04 ln
This is the specific case where the cosmological constant
is explicitly appearing in the potential. To end inflation we
need to fulfill an extra requirement that λ < 0 and this will
finally led to massless tachyonic solution. In Fig. 2a, b we
have shown the behavior of the inflationary potential for the
two cases, 1. V0 ≈ 0 and λ > 0, 2. V0 = 0 and λ < 0.
Figure 2a implies that the inflaton rolls down from a large
field value and inflation ends at φ f ≈ 1.09 M p. Also the
potential has a global minimum at φ = 0, around which
field is start to oscillate and take part in reheating. On the
other hand in Fig. 2b the inflaton field rolls down from a
small field value and the inflation ends at the field value
φ f = 2.88 α1/8 M p, where the lower bound on the
parameter α is, α ≥ 2.51 × 107, which is consistent with Planck
2015 data [
44–46
]. Within this prescription it is possible to
completely destroy the effect of cosmological constant at the
end of inflationary epoch. But within this setup to explain the
particle production during reheating and also explain the late
time acceleration of our universe we need additional features
in the total effective potential in scale free α R2 gravity theory.
It is a general notion that the reheating phenomenon can only
be explained if the effective potential has a local minimum
and a remnant contribution (vacuum energy or equivalent to
and the inflation ends at the field value φ f = 2.88 α1/8 M p, where the
lower bound on the parameter α is α ≥ 2.51 × 107, which is consistent
with Planck 2015 data [
44–46
]
cosmological constant) in the total effective potential finally
produce the observed energy density at the present epoch as
given by4 ρnow ≈ 10−47 GeV4, which is necessarily required
to explain the late time acceleration of the universe. Now here
one can ask a very relevant question: if we include some
additional features to the effective Higgsotic potential, which also
can be treated as a massless tachyonic potential, then how
one can interpret the justifiability as well as the behavior
of effective field theory framework around the minimum of
the potential which will significantly control the dynamical
behavior in the context of cosmology? The most probable
answer to this very significant question can be described in
various ways. In the present context to get a stable minimum
(vacuum) of the derived effective Higgsotic potential in an
Einstein frame here we discuss a few physical possibilities:
• Choice I: The first possible solution of the mentioned
problem is motivated from nonBPS Dbrane in
superstring theory. In this prescription the effective
potential have a pair of global extrima at the field value,
φextrema = φ = ±φV for the nonBPS Dbrane within
the framework of superstring theory [
23, 135–140
].
Additionally, it is important to note that here a one parameter
(γ ) family of global extrima exists at the field value,
φ = φV eiγ for the brane–antibrane system. Here φV
is identified to be the field value where the reheating
phenomenon occurs. At this specified field value of the
minimum the brane tension of the Dbrane configuration
which is exactly canceled by the negative contribution as
4 For Einstein gravity one can write the observed energy density at the
present epoch in the following form: ρnow ≈ 3H02 M2p, where H0 is the
Hubble parameter at the present epoch.
appearing in the expression for effective potential in an
Einstein frame. Here for the sake of simplicity we relax
a little bit the constraints as appearing exactly in Case II.
To explore the behavior of the derived effective
potential here we have allowed both of the signatures of the
coupling parameter λ. This directly implies the following
constraint condition:
2√2
λ e− √3 Mp φV4 +
− 4
2√2
λ e− √3 Mp φV4 +
4
p = 0 (for λ < 0),
p = 0 (for λ > 0),
(3.32)
(3.33)
(3.42)
,
(3.43)
(3.44)
(3.45)
(3.47)
(3.48)
The solutions of Eqs. (3.37)–(3.42) are given by
Choice I(v1)
−
,
d
3H˜ dt˜ −
λ(φ4 − φV4 ) e− √3 Mp = 0, (3.41)
2√2
√6M p
H 2
˜
where p is the above mentioned additional contribution
and in the context of superstring theory this is given by
p =
√2(2π )− p gs−1 for nonBPS Dpbrane,
2(2π )− p gs−1 for nonBPS Dp−D¯ p brane pair,
(3.34)
with string coupling constant gs. This implies that the
inflaton energy density vanishes at the minimum of the
tachyon type of the derived effective potential and in this
connection the remnant energy contribution is given by
V0 = M 4p/8α, which serves the explicit role of
cosmological constant in the context of late time acceleration of
the universe. In this case considering the additional
contribution as mentioned above the total effective potential
can be modified as
M 4p 2√2 4
λ e− √3 Mp (φ4 − φV )
v1: W˜ (φ, ) = 8α − 4
(for λ < 0),
M 4p 2√2 4
λ e− √3 Mp (φ4 − φV )
v2: W˜ (φ, ) = 8α + 4
(for λ > 0).
Here to avoid any confusion we have taken out the
signature of the coupling λ outside in the expression for the
effective potential for the λ < 0 case.
In the present context the field equations can be expressed
as
dφ 2√2
For v1: 3H˜ dt˜ − λe− √3 Mp φ3 = 0,
M 2p λ
˜ = 24α − 12M 2p
2√2
e− √3 Mp (φ4 − φV4 ).
(3.35)
(3.36)
(3.37)
(3.39)
(3.40)
dφ 2√2
For v2: 3H˜ dt˜ + λe− √3 Mp φ3 = 0,
In Fig. 3a, b we have shown the variation of the potential
with respect to the inflaton field for both cases. For Fig. 3a
the inflaton can roll down in both ways. Firstly, it can roll
down to a global minimum at the field value φV = 0
from higher to lower field value and take part in
particle production procedure during reheating. On the other
hand, in the same picture the inflaton can also roll down
from higher to lower field value in an opposite fashion.
(a) Choice I(v1) : Modified potential from superstring the (b) Choice I(v2) : Modified potential from superstring
theory with λ < 0. ory with λ > 0.
Fig. 3 Behavior of the modified effective potential for case II with a Choice I(v1): V0 = 0, λ < 0, b Choice I(v2): V0 = 0, λ > 0, where
Mp = 2.43 × 1018 GeV
In that case the inflaton goes up to the zero energy level
of the effective potential and cannot explain the thermal
history of the early universe in a proper sense. It is also
important to note that in this picture the position of the
maximum of the effective potential in the Einstein frame
is around the field value, φV = 0.42 M p. Figure 3b is
the case where the signature of the coupling λ is positive.
Also the behavior of the effective potential is completely
opposite compared to the situation arising in Fig. 3a. In
this case the inflaton field can be able to roll down to
higher to lower field value or lower to higher field value.
But in both cases the inflaton field settles down to a local
minimum at, φmin = φV = 0.42 M p and within the
vicinity of this point it will produce particles via reheating. In
the two situations the lower bound on the parameter α is
fixed at, α ≥ 2.51 × 107, which is perfectly consistent
with Planck 2015 data [
44–46
].
• Choice II: It is possible to explain the reheating as well
as the light time cosmic acceleration once we switch on
the effect of mass like quadratic term in the effective
potential. In such a case the modified effective potential
in an Einstein frame can be written as
M 4p
v1: W˜ (φ, ) = 8α +
m2c2 φ2 − λ4 φ4 e− 2√√32 Mp
M 4p
v2: W˜ (φ, ) = 8α −
m2c2 φ2 − λ4 φ4 e− 2√√32 Mp
(for mc2 > 0, λ < 0),
(for mc2 < 0, λ > 0).
(3.49)
(3.50)
Here to avoid any confusion we have taken out the
signature of the coupling λ outside in the expression for
the effective potential for λ < 0 case. In this context
during inflation the inflaton field satisfies the constraint
field satisfies φ
λ2 mc. After inflation when reheating starts, the
λ2 mc. Finally at the field value
φ = λ2 mc the remnant energy V0 = M 4p/8α serves
the purpose of explaining the late time acceleration of the
universe. In the present context the field equations can be
expressed as
dφ 2√2
For v1: 3H˜ dt˜ + (mc2φ − λφ3)e− √3 Mp = 0, (3.51)
2√2
e− √3 Mp .
dφ 2√2
For v2: 3H˜ dt˜ − (mc2φ − λφ3)e− √3 Mp = 0, (3.54)
2√2
2√2
e− √3 Mp = 0,
The behavior of the effective potential in an Einstein
frame is plotted in Fig. 4a, b, where the inflaton field
is rolling down from a large field to lower value or the
lower to larger field value and after inflation take part in
(a) Choice I(v1) : Modified potential from superstring the (b) Choice I(v2) : Modified potential from superstring
theory with λ < 0. ory with λ > 0.
where ξ represents the nonminimal coupling parameter
and φV represents the VEV of the field φ in this context.
After performing conformal transformation, the effective
action in the Einstein frame can be written as
where after applying C.T. the total modified effective
action can be written as
In the present context the field equations can be expressed
as
2√2
e− √3 Mp = 0,
The solutions of Eqs. (3.66)–(3.68) are given by:
− 0 ≈ 2√√23Mp ln a0
a
= 3M√2pα (t − t0)
(φ2 − φ02) 1 + ξ2 φ2 + φ02 − 2φV2 + 2φV2 ln φφ0 ,
1 + ξφV2
In Fig. 5, we have shown the behavior of the
effective potential with respect to inflaton field in the
presence of nonminimal coupling parameter, ξ = M −p2 and
ξ = 10−8 M −p2 depicted by red and blue colored curves,
respectively. For both of the cases we have taken the self
interacting coupling parameter λ > 0. Also it is
important to mention here that if we decrease the strength of
the nonminimal coupling parameter then the effective
potential becomes steeper. For both situations the
inflaton field can roll down from higher to lower or lower
to higher field values and finally settle down to a local
minimum at φV = M p.
4 Constraints on inflation with soft attractors
Here we require the following constraints to study the
inflationary paradigm in the attractor regime:
4.1 Number of efoldings
To get a sufficient amount of inflation from the proposed
setup (for both Case I and Case II), necessarily
+
N (φ0) − N (φ f ) ≈ ++ln
+
a f +
+
a0 ++
which is a necessary quantity to be able to solve the horizon
problem associated with standard bigbang cosmology. The
subscripts ‘f’ and ‘0’ physically signify the final and initial
values of the inflationary epoch. Further using Eqs. (3.24) and
(3.31) the field value at the end of inflation can be explicitly
computed for the above mentioned two cases as
• For Case I the expression for the field associated with
the end of inflation φ f is completely fixed by the value
initial field value φ0. Here no information for the field
dependent coupling λ(ψ f ) = λ( = f ) is required
for this case as the expression for φ f is independent of
the dilaton field dependent coupling.
• For Case II the expression for the field associated with the
end of inflation φ f is fixed by the value initial field value
φ0 as well as by the field dependent coupling λ(ψ f ) =
λ( f ).
4.2 Primordial density perturbation
4.2.1 Two point function
The next observational constraint comes from the imprints of
density perturbations through scalar fluctuations. Such
fluctuations in CMB map directly implies that5:
δρ
ρ
δρ
ρ
cr
=
AS ∼ 10−5
measured on the horizon crossing scales, where δρ is the
perturbation in the density ρ. Additionally, it is important to
note that AS, represents the amplitude of the scalar power
spectrum. Also in the present context for both cases one can
write
σ
δρ
ρ t1
= σ
δρ
ρ t2
where the parameter σ is the parameter in the present context,
which can be expressed in terms of equation parameter as,
σ = 1 + 3(1+w) , w = ρp . It is important to note that (t1, t2)
2
represent the times when the perturbation first left and
reentered the horizon, respectively. At time t1, Eq. (3.12)
perfectly hold good in the present context. On the other hand
at time t = t2 the representative parameter σ take the value,
σ = 3/2 and σ = 5/3 during the radiation and
matterdominated epochs, respectively. For the potential dominated
inflationary epoch, w ≈ −1 and consequently one can write
the following constraint condition:
δρ = φ˙ δφ˙ + ˙ δ ˙ − 3H˜ φ˙ δφ + ˙ δ
≈ −2H˜ φ˙ δφ + ˙ δ
where we use the symbol as ˙ ≡ d/dt˜ and one can write
down, δφ˙ ≈ H˜ δφ, δ ˙ ≈ H˜ δ , δφ ≈ H˜ , δ ≈ H˜ , and
finally the fractional density contrast can be expressed as
δρ
ρ
t2
=
with the following constraint on the parameter C as given by,
C ∼ O(1) and it serves the purpose of a normalization
constant in this context. Then we get the two physically
acceptable situations for both of the cases which can be written
as
δρ
Region I: φ˙  <  ˙  ⇒ ρ ≈
δρ
Region II: φ˙  >  ˙  ⇒ ρ ≈
H˜ 2 W˜ h ,
 ˙  ≈ 2√2M 2p
H 2
˜
φ˙ 
≈
W˜ h3/2
M 3p(∂φ W˜ )h
.
δρ
ρ
t2
≈
1
1 − σ
δρ
ρ
t1
.
Further using Eq. (3.12) and approximated equation of
motion in slowroll regime of fluctuation in the total energy
density or equivalently in the scalar modes can be written as
5 Here one equivalent notation for the amplitude of the scalar
perturbation used as √Pcmb = √P(Ncmb), which we have used in the
nonattractor case.
(4.6)
(4.7)
(4.8)
(4.9)
(4.3)
(4.4)
(4.5)
Here one can interpret the results as
• In Region I, the amplitude of the density fluctuation at the
horizon crossing is only controlled by the scale of
inflation and the magnitude of the dilaton dependent effective
coupling parameter λ( h).
• In Region II, the amplitude of the density fluctuation at
the horizon crossing is given by
δρ
ρ
Region II
= √
2
W˜ h
δρ
ρ
Region I
.
(4.10)
This implies that contribution from the first slowroll
parameter, as given by W˜ = M22p ∂φW˜W˜ , controls the
magnitude of the amplitude of density perturbation apart
from the effect from the scale of inflation and the
magnitude of the dilaton dependent effective coupling
parameter λ( h).
4.2.2 Present observables
Further using the approximate equations of motion the
fractional density contrast for the above mentioned two cases can
be written as
δρ
Case I: ρ ∼
δρ
Case II: ρ ∼
⎧ φ02
⎪⎪⎪ 4M2p
⎨
⎧
⎪⎪⎪⎪⎪⎨ 8√1 α
• In Region I and Region II of Case I, the amplitudes of the
density fluctuation at the horizon crossing are related by
δρ
ρ
δρ
ρ
( h −
0)
Region I
.
Region I
1/2
(4.13)
δρ
ρ
δρ
ρ
≈
This implies that if we know the field value at the
starting point of inflation then one can directly quantify the
amplitude of density perturbation. Most importantly, if
inflation starts from the vicinity of the Planck scale
i.e. φ0 ∼ √2M p ∼ O(M p) then by evaluating the
amplitude of the density perturbation in Region I one
can easily quantify the amplitude of the density
perturbation in Region II. In this setup within the range
50 < N f / h < 70, we get
∼
δρ
ρ
which is consistent with Planck 2015 data. But if
inflation starts at the following field value, φ0 = √2 M p,
where the parameter ≷ 1 then one ca write the
following relationship between the amplitude of the density
perturbation in Region I and Region II as
δρ
ρ
δρ
ρ
δρ
ρ
δρ
ρ
δρ
ρ
(4.11)
(4.12)
(4.16)
(4.17)
(4.18)
(4.20)
This implies that for ≷ 1 in Region II we get tightly
constrained result for the amplitude for the density
perturbation.
• In Region I and Region II of Case II, the amplitude of the
density fluctuation at the horizon crossing are related by
Region II
M 3p
≈ √8αλ( h)φ03
δρ
ρ
Region I
. (4.19)
This implies that if we know the field value at the starting
point of inflation, the dilaton field dependent coupling at
the horizon crossing λ( h) and the coupling of scale free
gravity α, then one can directly quantify the amplitude of
density perturbation. Most importantly, if inflation starts
from the vicinity of the Planck scale i.e. φ0 ∼ O(M p)
and we have an additional constraint:
λ( h) ∼ √
1
8α
,
(4.15)
then by evaluating the amplitude of the density
perturbation in the Region I one can easily quantify the amplitude
of the density perturbation in Region II. Here one can also
consider an equivalent constraint:
φ0 ∼
1
√8αλ( h)
1/3
M p.
For both situations in the present setup within the range
50 < N f / h < 70, we get
∼
δρ
ρ
which is also consistent with Planck 2015 data. But if
inflation starts at the field value φ0 = M p, where the
parameter ≷ 1, and we define
where the parameter ≷ 1, then one can write the
following relationship between the amplitude of the density
perturbation in Region I and Region II as
δρ
ρ
.
δρ
ρ
This implies that for ≷ 1 and ≷ 1 in Region II we
get a tightly constrained result for the amplitude for the
density perturbation.
(4.23)
(4.24)
(4.25)
(4.26)
(4.27)
Table 1 Inflationary
observables and model
constraints in the light of Planck
2015 data [
44–46
] for the
dynamical attractors considered
in Case I and Case II
N f/h
50
60
70
AS (×10−9)
δρ
ρ
=
δρ
ρ
δρ
ρ
δρ
ρ
δρ
ρ
(4.21)
In this context the scalar spectral tilt can be written at the
horizon crossing as6
nS − 1 =
d ln AS
d f
h
≈
⎧ 3
⎪⎨ (N f/h+1)2
3
⎪⎩ N 2f/h
⎧
⎪⎨
≈
6
− (N f/h+1)3
6
⎩⎪ − N 3f/h
for Case I,
for Case II,
for Case I,
for Case II.
(4.29)
(4.30)
(4.31)
Finally combining Eqs. (4.29), (4.30) and (4.31) we get the
following consistency relation for both Case I and Case II:
(nS − 1)2 κS 2/3
βS = 3 = 3 − 6 . (4.32)
This is obviously a new consistency relation for the present
Higgsotic model of inflation and it is also consistent with
Planck 2015 data [
44–46
]. In Table 1 we have shown the
numerical estimations of the inflationary observables for the
Higgsotic attractors depicted in Case I and Case II within the
range 50 < N f / h < 70.
In Fig. 6, we have plotted the running of the spectral tilt for
scalar perturbation (κS = d2nS/d2 ln k) vs. spectral tilt for
scalar perturbation (nS) in the light of Planck 2015 data along
with various joint constraints. Here it is important to note that
for Case I and Case II the Higgsotic models are shown by the
green and pink colored lines. Also the big circle,
intermediate size circle and small circle represent the representative
points in (κS, nS) 2D plane for the numbers of efoldings
N f / h = 70, N f / h = 60 and N f / h = 50, respectively.
6 Here we use a new symbol N f/h, which is defined as
+
N f/h = ++ln
+
aahf ++++ = N (φh) − N (φ f ) ∼ 50−70,
(4.28)
Fig. 6 Plot for running of the
running of spectral index
κS = d2nS/d2 ln k vs. running
of the spectral index
βS = dnS/d ln k for scalar
modes. Here for Case I and
Case II we have drawn green
and pink colored lines. We also
draw the background of the
confidence contours obtained
from various joint constraints
[
44–46
]
To represent the present status as well as statistical
significance of the Higgsotic model for the dynamical attractors as
depicted in Case I and Case II, we have drawn the 1σ and 2σ
confidence contours from Planck+WMAP+BAO 2015 joint
data sets [
44–46
]. It is clear from Fig. 6 that, for Case I we
cover the range 0.59 × 10−3 < βS = ddlnnSk < 1.16 × 10−3
and −1.65 × 10−5 > κS = dd22lnnSk > −4.56 × 10−5 in
the (κS, βS) 2D plane. Similarly for Case II we cover the
dnS
range 0.62 × 10−3 < βS = d ln k < 1.20 × 10−3 and
−1.78×10−5 > κS = dd22lnnSk > −4.80×10−5 in the (κS, βS)
2D plane.
4.3 Primordial tensor modes and future observables
In terms of the number of efoldings (N ) the most useful
parametrization of the primordial scalar and tensor power
spectrum or equivalently the tensortoscalar ratio can be
written near the horizon crossing Nh = N (φh) as
8
r (N ) = M 2p
dφ
dN
2
= r (Nh)e(N −Nh){Ah+Bh(N −Nh)}
where in the slowroll regime of inflation the tensortoscalar
ratio r (Nh) can be written in terms of the inflationary
potential as
r = r (Nh) ≈ 8M2p
Vh
Vh
2
=
⎧ 128M2p
⎪⎪⎨ φh2
⎪ 512α2λ2( h)φh6
⎪⎩ M6p
(4.33)
for Case I,
for Case II,
(4.34)
and the symbols Ah, Bh and Ch are expressed in terms of the
inflationary observables at horizon crossing as Ah = nT −
nS+1, Bh = 21 (βT−βS). In the above parametrization Ah
Bh i.e. βS − 2(nS − 1) βT − 2nT is always required for
convergence of the Taylor expansion. Using this assumption
the relationship between field excursion, φ = φh − φ f and
tensortoscalar ratio r (Nh) can be computed as
 φ
M p ≈
A2h 0 2π +
r (Nh) e− 2Bh +erfi
8 Bh ++
Ah
√2Bh
− erfi √A2hBh − B8h N f / h +++++ . (4.35)
Now the scale of inflation is connected with the
tensortoscalar ratio in the following fashion:
V 1/4
h
= 23 π 2 ASr ( fh)
1/4
M p ∼ 7.9 × 10−3 M p ×
r ( fh) 1/4
Substituting Eq. (4.36) in Eq. (4.35) we compute the
relationship between field excursion and the scale of inflation
as
 φ
M p ≈
0
Vh
6π M 4p AS Bh
A2
e− 2Bhh ++erfi
+
+
Ah
√2Bh
− erfi √A2hBh − B8h N f / h +++++ . (4.37)
Also using Eq. (4.36) the tensortoscalar ratio can be written
as
r = r (Nh) =
⎪⎨ (2×λ1(0−h2)φMh4p)4
⎧
Further using Eqs. (4.34) and (4.38) we get the following
constraints from the primordial tensor perturbation:
Table 5 Constraint on scalar
four point nonGaussian
amplitude from equilateral,
folded kite and squeezed
configuration with assuming
Suyama–Yamaguchi
consistency relation
Table 6 Constraint on scalar
four point nonGaussian
amplitude from equilateral
configuration without assuming
Suyama–Yamaguchi
consistency relation
Scanning region
I
II
III
I
II
III
I
II
III
IV
I + II + III + IV
Scanning region
I
II
III
IV
I + II + III + IV
Scanning region
IV
I + II + III + IV
Scanning region
IV
I + II + III + IV
τNeqLuil
In Table 5, we give the numerical estimates and constraints
on the four point nonGaussian amplitude from
equilateral configuration with assuming Suyama–Yamaguchi
consistency relation. Also in Table 6, we give the numerical
estimates and constraints on the four point nonGaussian
amplitude from equilateral configuration without
assuming Suyama–Yamaguchi consistency relation. Here all the
obtained results are consistent with the two point and three
point constraints as well as with the Planck 2015 data [
44–
46
].
8.2.2 Using δN formalism
In this section using the prescription of δN formalism in the
attractor regime of cosmological perturbation we derive the
expression for the nonGaussian amplitudes associated with
the four point function of scalar curvature fluctuation as
gNeqLuil
−0.004 < gNeqLuil < −0.023
Now we already know that in the attractor regime
cosmological perturbation, solution for the additional field can be
expressed in terms of the inflaton field φ and using this fact
the expression for the nonGaussian amplitudes associated
with the four point function of scalar curvature fluctuation
can be recast as
where the new functions X1(φ), . . . , X6(φ) are defined as
X1(φ) = f (φ) 1 + V22(φ) + V42(φ) + V61(φ)
,
X2(φ) = −3 f (φ)
VV3((φφ)) + VV5((φφ)) ,
X3(φ) = f (φ) V 2(φ) V 2(φ)
V6(φ) + V4(φ)
,
X4(φ) = f (φ) 1 + V23(φ) + V43(φ) + V61(φ) ,
X5(φ) = −3 f (φ) VV7((φφ)) + 2 VV5((φφ)) + VV3((φφ)) ,
X6(φ) = − f (φ) VV3((φφ)) − 2 VV42((φφ)) − 3 VV82((φφ)) − 5 V 2(φ) .
V6(φ)
−3
where f (φ) = 1 + V21(φ) . Further substituting the
explicit form of the function V(φ) and N,φ , N,φφ , N,φφφ
for all derived effective potentials at φ = φ∗ we get
τ NloLc = Y2 X1(φ∗) + X2(φ∗)φ∗ + X3(φ∗)φ∗2 ,
25 2 2X4(φ∗) + X5(φ∗)φ∗ + X6(φ∗)φ∗2 .
gNloLc = 54 Y
Now we comment on the consistency relation between the
nonGaussian parameters derived from four point and three
(8.177)
(8.178)
(8.179)
point scalar correlation function in the attractor regime
of inflation. To establish this connection we start with
Eqs. (8.64), (8.178) and (8.179) and finally get new set of
consistency relations:
36
τ NloLc = 25 ( fNloLc)2 X1(φ∗) + X2(φ∗)φ∗ + X3(φ∗)φ∗2 ,
10
gNloLc = 27 ( fNloLc)2 2X4(φ∗) + X5(φ∗)φ∗ + X6(φ∗)φ∗2 .
gNloLc = 418265 τ NloLc 2222XX44((φφ∗∗)) ++ XX55((φφ∗∗))φφ∗∗ ++ XX66((φφ∗∗))φφ∗∗2233 .
It is a very wellknown fact that in the nonattractor regime,
where the additional field is frozen in the Planck scale
Suyama–Yamaguchi consistency relation [
176–178
] holds
true, which states:
36
τ NloLc = 25 ( fNloLc)2. (8.183)
Further using this results one can estimate the deviation in
the Suyama–Yamaguchi consistency relation if we go from
attractor regime to nonattractor regime of cosmological
perturbation as
 τ NloLc = [τ NloLcnonattractor − τ NloLcattractor]
36
= 25 ( fNloLc)2nonattractor{1 − Qcorr},
where the correction factor Qcorr can be written as
( fNloLc)2attractor
Qcorr = ( fNloLc)2nonattractor
X1(φ∗) + X2(φ∗)φ∗ + X3(φ∗)φ∗2 .
(8.180)
(8.181)
(8.182)
(8.184)
(8.185)
Here we need to point out a few crucial issues:
• First of all, to estimate the magnitude of the deviation
factor Qcorr we need to concentrate on two physical
situations, I. Super Planckian field regime and II. Sub
Planckian field regime.
• In the super Planckian field regime the deviation factor
Qcorr can be expressed as
⎧
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪
⎪⎪⎪⎪ ⎛
⎪⎪
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪
⎪⎪⎪⎪ ⎛
⎪⎪
⎧
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪ 1 −
⎪⎪⎪⎪ ⎛
⎪⎪
⎪⎪⎪⎪⎪⎪ ⎜⎝ 1 −
⎪⎪⎪
⎪⎪⎪⎪ ⎛
⎨
⎜ 1 −
⎪⎪ ⎝
⎪⎪⎪
⎪⎪⎪⎪ ⎛
⎪⎪
⎪⎪⎪⎪⎪⎪ ⎜⎝ 1 −
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ −
18M2p 72M4p 3888M8p
81φ∗2 + 6561φ∗4 + 43046721φ∗8 + · · ·
18M2p
φ2 +
∗
72M4p
φ4 +
∗
3888M8p
φ8
∗
3888M8p
mc2
φ∗8 1− mc2−λφ∗2
72M4p
4 4 +
φ∗4 1− φφV4
∗
3888M8p
φ∗8 1− φφV44 8 + · · · ⎠⎟
∗
⎞
72M4p
mc2
φ∗4 1− mc2−λφ∗2
⎞
4
8 + · · · ⎠⎟
18M2p
72M4p
2 4
φ∗4 1+ξ(φ∗2−φV2 )+ φφV2
∗
where the factor
f is defined as
( fNloLc)2attractor
f = ( fNloLc)2nonattractor
.
Now to give a proper estimate of the deviation in the
magnitude of the amplitude of nonGaussian parameter computed
from four point function in terms of the three point
nonGaussian amplitude for the time being we assume that the
results obtained from the attractor and nonattractor
formalism is almost at the same order of magnitude. In that case we
(8.187)
where the correction factor Jcorr 1 is highly suppressed
in the super Planckian region of the perturbation theory, but
those small corrections are important as precision
measurement is concerned in the context of cosmology. In the case of
our derived effective potentials we get the following
approximate expressions for the correction factor:
⎞
8 + · · · ⎟⎠
⎞
for Case I
for Case II
for Case II + Choice I
for Case II + Choice II
for Case II + Choice III.
for Case II + Choice III.
(8.186)
(8.189)
Page 64 of 82
• In the sub Planckian field regime the deviation factor
Qcorr can be expressed as
1 + 198683 Mφ∗66p + · · ·
for Case I
for Case II
for Case II + Choice I
for Case II + Choice II
for Case II + Choice III.
f ×
So it is clear that  Jcorr captures the effect of the deviation in
the Suyama–Yamaguchi consistency relation which are very
small and highly suppressed in the super Planckian regime
of inflation. But as far as precision cosmology is concerned,
this small effect is also very useful to discriminate between
all derived effective models considered in this paper. If in the
near future Planck or any other observational probe detects
the signature of primordial nonGaussianity with high
statistical significance then one can also further comment on the
significance of attractors and nonattractors in the context of
early universe cosmology.
Further using this results one can estimate the deviation in
the Suyama–Yamaguchi consistency relation if we go from
attractor regime to nonattractor regime of the cosmological
perturbation as
36
 τ NloLc = 25 ( fNloLc)2nonattractor{1 − f (1 + Ccorr)}
36
∼ 25 ( fNloLc)2nonattractorCcorr.
(8.195)
where the factor f is defined earlier, which is f ∼ O(1).
Consequently the deviation factor can be recast as
where the correction factor Ccorr 1 is suppressed in the
sub Planckian region of the perturbation theory, but those
small corrections are important as precision measurement
is concerned in the context of cosmology. In the case of our
derived effective potentials we get the following approximate
expressions for the correction factor:
(8.192)
(8.194)
Also the fractional change can be expressed as
+++ τ NloLc +++
++ (τ NloLc)nonattractor ++φ∗ Mp
f (1 + Ccorr) ∼ Ccorr.
(8.196)
So it is clear that Ccorr captures the effects of the deviation in
the Suyama–Yamaguchi consistency relation which are very
small and suppressed in the sub Planckian regime of inflation.
• From the study of sub Planckian and super Planckian
regime it is evident that when f ∼ O(1) i.e. the
nonGaussian amplitude obtained from three point function
in attractor and nonattractor regime for all the derived
effective potentials are of the same order then the
deviation from Suyama–Yamaguchi consistency relation is
very small. The only difference is in the sub Planckian
case this correction is greater than unity and on the other
hand in the super Planckian case this correction factor is
less than unity. But since we are interested in the
precision cosmological measurement, such small but
distinctive corrections will play a significant role in
discriminating between the classes of effective models of inflation
derived in this paper.
• Finally, if we relax the assumption that the deviation
factor, f = 1, then one can consider the following two
situations1. First we consider, f 1. In this case in the
super Planckian and sub Planckian regime we get the
following results for the deviation in the Suyama–
Yamaguchi consistency relation:
36
 τ NloLcφ∗ Mp = 25 ( fNloLc)2nonattractor f (1 − Jcorr).
36
 τ NloLcφ∗ Mp = 25 ( fNloLc)2nonattractor f (1 + Ccorr).
Also the fractional change in the Suyama–Yamaguchi
consistency relation can be expressed as
+++ τ NloLc +++
+ (τ NloLc)nonattractor ++φ∗ Mp
+
+++ τ NloLc +++
++ (τ NloLc)nonattractor ++φ∗ Mp
=  f (1 − Jcorr),
=  f (1 + Ccorr).
(8.197)
(8.198)
(8.199)
In this specific situation the deviation factor is
large and consequently one can achieve a maximum
amount of violation in the Suyama–Yamaguchi
consistency relation. Here the results of the super
Planckian and sub Planckian field values differ due to the
presence of the correction factors Jcorr and Ccorr. Here
both Jcorr < 1 and Ccorr < 1, but for model
discrimination such small corrects are significant as
mentioned earlier.
2. Next we consider, f 1. In this case in the
super Planckian and sub Planckian regime we get the
following results for the deviation in the Suyama–
Yamaguchi consistency relation:
36
 τ NloLcφ∗ Mp = 25 ( fNloLc)2nonattractor1 −
36
 τ NloLcφ∗ Mp = 25 ( fNloLc)2nonattractor1 −
f .
f .
(8.200)
(8.201)
(8.202)
(8.203)
(8.204)
Also the fractional change in the Suyama–Yamaguchi
consistency relation can be expressed as
+++ τ NloLc +++
+ (τ NloLc)nonattractor ++φ∗ Mp
+
+++ τ NloLc +++
++ (τ NloLc)nonattractor ++φ∗ Mp
In this specific situation deviation factor is small
and consequently one can achieve very small amount
of violation in Suyama–Yamaguchi consistency
relation. Here the results of the super Planckian and sub
Planckian field value are almost the same as we have
neglected the terms f Jcorr 1 and f Ccorr 1.
Now to derive the results of nonGaussian amplitudes in
the nonattractor regime using the δN formalism we need to
freeze the value of the additional field in the Planck scale.
If we do this job then the expression for the four point
nonGaussian amplitude computed from scalar fluctuation can be
expressed as
+++ τ NloLc +++ = ++++ 2257 W f − 1+++ . (8.207)
++ (τ NloLc)InIn ++ +
Now if we claim that at the horizon crossing nonGaussian
amplitudes obtained from the δN and InIn formalism are of
the same order then in that case we get W f ∼ O(1).
Consequently the deviation in the Suyama–Yamaguchi consistency
relation can be recast as
72
 τ NloLc = (τ NloLc)δN − (τ NloLc)InIn ∼ 625 (( fNloLc)2)InIn.
+ τ NloLc +
Consequently the fractional deviation is given by +++ (τ NloLc)InIn +++ ∼
225 .
9 Conclusion
 τ NloLc = (τ NloLc)δN − (τ NloLc)InIn
To summarize, in the present article, we have addressed the
following points:
• Firstly we have started our discussion with a specific class
of modified theory of gravity, aka f (R) gravity where a
single matter (scalar field) is minimally coupled with the
gravity sector. For simplicity we consider the case where
the matter field contains only canonical kinetic term. To
build effective potential from this toy setup of modified
gravity in 4D we choose f (R) = α R2 gravity.
• Next to start with in the matter sector we choose a very
simple model of potential, V (φ) = λ4 φ4, where φ is a
real scalar field and λ is a real parameter of the monomial
model. This type of potential can be treated as a Higgs like
potential as the structure of Higgs potential is given by
V (H ) = λ4 (H † H − V 2), where λ is Yukawa coupling, H
is the Higgs SU(2) doublet and 0H 0 = V ∼ 125 GeV
is the VEV of the Higgs field. Now one can write the
Higgs SU(2) doublet as H † = (φ 0) and the
corresponding Higgs potential can be recast as V (φ) = λ4 (φ2 −V2)2.
Now at the scale of inflation, which is at O(1016 GeV),
contribution from the VEV is almost negligible and
consequently one can recast the Higgs potential in the
mono
λ φ4. The only difference is in the
mial form, V (φ) ≈ 4
case of Higgs where λ is the Yukawa coupling and in the
(8.206)
(8.208)
case of a general monomial model λ is a free
parameter of the theory. Due to the similar structural form of
the potential we call the general φ4 monomial model as
Higgsotic potential.
• Further, we provide the field equations in a spatially flat
FLRW background, which are extremely complicated to
solve for this setup. To simplify, next we perform a
conformal transformation in the metric and write down the
model action in the transformed Einstein frame. Next, we
derive the field equations in a spatially flat FLRW
background and try to solve them for two dynamical attractor
features: I. a powerlaw solution and II. an exponential
solution. However, the second case gives rise to
tachyonic behavior which can be resolved by considering the
nonBPS Dbrane in superstring theory, considering the
effect of mass like quadratic term in the effective
potential and considering the effect of nonminimal coupling
between f (R) = α R2 scale free gravity sector and the
matter field sector.
• Next, using two dynamical attractors, a powerlaw and
an exponential solution, we have studied the
cosmological constraints in the presence of two fields in an Einstein
frame. We have studied the constraints from primordial
density perturbation, by deriving the expressions for two
point function and the present observablesamplitude of
power spectrum for density perturbations,
corresponding spectral tilt and associated running and running of
the running for inflation. We have repeated the analysis
for tensor modes and also comment on the future
observables – the amplitude of the tensor fluctuations,
associated tilt and running, and the tensortoscalar ratio. We
also provide a modified formula for the field excursion in
terms of the tensortoscalar ratio, scale of inflation and
the number of efoldings. Further, we have compared our
model with Planck 2015 data and constrain the
parameter α of the scale free gravity and nonminimal coupling
parameter λ( h). Additionally, we have studied the
constraint for the reheating temperature. Finally, we derive
the expression for the inflaton and the coupling parameter
at horizon crossing, during reheating and at the onset of
inflation which are very useful to study the scale
dependent behavior. Most importantly, in the present context
one can interpret such scale dependence as an outcome
of RG flow in the usual Quantum Field Theory.
• Further, we have explored the cosmological solutions
beyond attractor regime. We have shown that this
possibility can be achieved if we freeze the field value of
the dilaton field in Einstein frame. This possibility can
be treated as a single field model where an additional
field freezes at a certain field value, which we fix at
the reduced Planck scale. To serve this purpose we have
used the ADM formalism and computed the two point
function and associated present inflationary observables
using Bunch–Davies initial condition for scalar
fluctuations. We have repeated the procedure for tensor
fluctuations as well. In the nonattractor regime, we have also
derived a modified version of the field excursion formula
in terms of the tensortoscalar ratio, scale of inflation
and the number of efoldings. We have also derived few
sets of consistency relations in this context which are
different from the usual single field slowroll models.
For example, instead of getting r = −8nT here we get
sidered the contribution from contact interaction term,
scalar and graviton exchange. In the attractor phase
following the prescription of the δN formalism we also
derive the expressions for the four point nonGaussian
amplitude gNloLc and τ NloLc. Next we have shown that the
consistency relation connecting three and four point
nonGaussian amplitude aka Suyama–Yamaguchi relation is
modified in the attractor phase and further given an
estimate of the amount of deviation. Further, for the
consistency check we freeze the dilaton field in the Planck scale
and redo the analysis of the δN formalism. By doing this
we have found that the expression for the four point
nonGaussian amplitude is slightly different as expected for
the single field case. Next we have also shown that the
exact numerical deviation of the consistency relation is of
the order of 2/25 by assuming nonGaussian three point
amplitude for attractor and nonattractor phase are of the
same order of magnitude. Further, we compare the results
obtained from the InIn formalism and δN formalism for
the nonattractor phase, where the dilaton field is fixed
in the Planck scale. Here, finally, we give a theoretical
bound on the scalar four point nonGaussian amplitude
computed from equilateral, folded kite and squeezed limit
configurations. The obtained results are consistent with
the Planck 2015 data.
The future prospects of our work are appended below:
• We have restricted our analysis up to monomial φ4 model
and due to the structural similarity with Higgs potential
at the scale of inflation we have identified monomial φ4
model as Higgsotic model in the present context.
• To investigate the role of scale free theory of gravity, as
an example we have used α R2 gravity. But the present
analysis can be generalized to any class of f (R)
gravity models and other class of higher derivative gravity
models.
• In the matter sector for completeness one can consider
most generalized version of P(X, φ) models, where X =
− 21 gμν ∂μφ∂ν φ. DBI is one of the examples of P(X, φ)
model which can be implemented in the matter sector
instead of simple canonical kinetic contribution.
• In this work, we have not given any three point
computation and found point scalar correlation function and
representative nonGaussian amplitudes using the InIn
formalism in the attractor regime in the presence of both
fields, φ and , for all classes of Higgsotic models. In
near future we are planning to present the detailed
calculation on this important issue.
• Generation of primordial magnetic field through
inflationary magnetogenesis is one of the important issues in
the context of primordial cosmology, which we have not
explored yet from our setup. One can consider such
inter
Page 68 of 82
actions by breaking conformal invariance of the U (1)
gauge field in the presence of time dependent coupling
f (φ (η)) to study the features of primordial magnetic
field through inflationary magnetogenesis. We have also
a future plan to address this issue.
• In this work we have restricted our analysis within the
class of Higgsotic models. For completeness in the future
we will extend this idea to all class of potentials allowed
by the presently available observed Planck data. We
will also include the effects of various types of
nonminimal and noncanonical interactions in the present
setup.
• In the same direction one can also carry forward the
present analysis in the context of various types of higher
derivative gravity setup and comment on the constraints
on the primordial nonGaussianity, reheating and
generation of primordial magnetic field through inflationary
magnetogenesis for completeness. Also one can consider
the possibility of nonminimal interaction between α R2
gravity and matter sector. In future we will investigate the
possibility of appearing new consistency relations in the
presence of higher derivative gravity setup and will give
proper estimate of the amount of violation from Suyama–
Yamaguchi consistency relation.
• During the computation of correlation functions using
semi classical method, via the δN formalism, we have
restricted up to secondorder contributions in the
solution of the field equation in FLRW background and also
neglected the contributions from the back reaction for
all type of effective Higgsotic models derived in an
Einstein frame. For more completeness, one can relax these
assumptions and redo the analysis by taking care of all
such contributions. Additionally, we have a future plan
to extend the semi classical computation of the δN
formalism of cosmological perturbation theory in a more
sophisticated way and will redo the analysis in the present
context.
• In this work, we also have not investigated the possibility
of getting dark matter and dark energy constraints from
the present up. Most importantly the present structure
of interactions in the Einstein frame shows that the two
fields, φ and , are coupled and due to this fact if we want
to explain the possibility of dark matter and dark energy
together; from this setup it is very clear that they are
coupled. But this is not very clear at the level of analytics
and detailed calculations. Here one can also investigate
these possibilities from this setup.
• In this work we have not investigated the contribution
from the loop effects (radiative corrections) in all of the
effective Higgsotic interactions (specifically in the self
couplings) derived in the Einstein frame. After
switching on all such effects one can investigate the specific
numerical contribution of such terms and comment on
the effects of such terms in precision cosmology
measurement.
• Here one can generalize the results for α vacua and study
its cosmological consequences for all types of derived
potential in the present context.
• In the present context one can also study the quantum
entanglement between the Bell pairs, which can be
created through the Bell inequality violation in cosmology
[
179–181
].
Acknowledgements SC would like to thank Department of
Theoretical Physics, Tata Institute of Fundamental Research, Mumbai and
specially the Quantum Structure of the Spacetime Group for providing
a Visiting (PostDoctoral) Research Fellowship. The work of SC was
supported in part by Infosys Endowment for the study of the Quantum
Structure of Space Time. SC takes this opportunity to thank sincerely
to Ashok Das, Sudhakar Panda, Shiraz Minwalla, Sandip P. Trivedi,
Gautam Mandal and Varun Sahni for their constant support and
inspiration. SC also thanks the organizers of Indian String Meet 2016 and
Advanced String School 2017 for providing the local hospitality during
the work. SC also thanks Institute of Physics, Bhubaneswar for
providing the academic visit during the work. Last but not least, SC would like
to acknowledge debt to the people of India for their generous and steady
support for research in natural sciences, especially for theoretical high
energy physics, string theory and cosmology.
Open Access This article is distributed under the terms of the Creative
Commons Attribution 4.0 International License (http://creativecomm
ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
Funded by SCOAP3.
10 Appendix
10.1 Effective Higgsotic models for generalized P ( X, φ)
theory
In this section, to give a broad overview of the effective
Higgsotic models let us start with a general f ( R) theory in the
gravity sector and generalized P ( X, φ) theory in the matter
sector. The representative actions in a Jordan frame is given
by
S =
d4 x √−g [ f ( R) + P ( X, φ)] ,
where P ( X, φ) is a arbitrary function of single scalar field φ
and the kinetic term X = − 21 gμν ∂μφ∂ν φ. In general f ( R)
is any arbitrary function of R. But for our purpose we choose
f ( R) = α R2 to study the consequences from scale free
gravity. From this representative action one can write down
the field equations in a spatially flat FLRW background as
H 2 =
a 2
˙
a
Here the total potential can be recast as
where V0 = M 4p/8α, exactly mimics the role of cosmological
constant as mentioned earlier.
In the case of Higgsotic model we can rewrite the total
potential as
(10.7)
the specific form of P(X, φ) as stated in Eq. (10.6) after a
conformal transformation we get
(10.3)
G(X˜ , φ, ) =
1
4
2H˙ + 3H 2 = 2 aa¨ + aa˙ 2
pφ
where for generalized P(X, φ) theory pressure pφ and
density ρφ can be written as
pφ = P(X, φ), ρφ = 2X P,X (X, φ) − P(X, φ).
(10.4)
Here the effective speed of sound parameter cS is defined as
cS =
0
P,X (X, φ)
P,X (X, φ) + 2X P,X X (X, φ)
.
If we choose the following functional form of P(X, φ):
as pointed out earlier, then we get the following simplified
expression for pφ and density ρφ :
Also one can consider any arbitrary slowroll effective
potential but for our purpose we choose monomial Higgsotic
λ φ4 in the Jordan frame.
model, V (φ) = 4
In the present context let us introduce a scale dependent
mode , which can be written in terms of a no scale dilaton
mode as = f (R)M −p2 = 2α R M −p2 = e 23 Mp = 2,
which plays the role of a Lagrange multiplier and arises in
the Jordan frame without spacetime derivatives. Here is
the conformal factor of the conformal transformation that we
perform from Jordan frame to Einstein frame.
In terms of the newly introduced no scale dilaton mode
the total action of the theory (see Eq. (2.1)) can be recast as
S =
After doing C.T. the total action can be recast in the Einstein
frame as
M 2p
2
R˜ + G(X˜ , φ, )
(10.9)
where after applying C.T. the functional G(X˜ , φ, ) is
defined in an Einstein frame as
G(X˜ , φ, ) =
1
4
P(X˜ , φ) − M8α4p e2 23 Mp .
(10.10)
Here X˜ is the kinetic term after a conformal transformation,
which is defined as X˜ = − 21 g˜μν ∂˜μφ∂˜ν φ. Now in the case of
The rest of the computation is exactly similar to what we
have performed earlier, only the structure of the total effective
potential changes.
Here it is important to note that apart from f (R) gravity
one can consider various other possibilities. To give a clear
picture about various classes of twofield attractor models
one can consider the following 4D effective action in an
Einstein frame:
where J (X, Y, φ, ) is the general functional of the two
fields φ and : the following specific mathematical
structure:
c
J (X, Y, φ, ) = e− M1 p X + Y − W (φ, ).
(10.16)
Here c1 and c2 characterize the effective coupling constant in
4D, which are different for various types of source theories. In
the EFT setup these are identified as the Wilson coefficients.
Additionally, it is important to note that the kinetic terms for
(10.11)
(10.12)
(10.13)
(10.14)
gμν
the φ and field are defined as X = − 2 ∂μφ∂ν φ and Y =
gμν
− 2 ∂μ ∂ν . Here W (φ, ) is the 4D effective potential,
which is given by the following expression:
This is a nonseparable form of the twofield effective
poten
c
tial where one can treat V (φ) as usual inflaton field and e− M2 p
as the dilaton exponential coupling.
This type of effective theory can be derived from the
following class of models:
1. Type I: Consider an action in a Jordan frame where the
scalar field is nonminimally coupled with the gravity
sector as given by
(10.17)
S =
d4x √−g 2 f1( )R − f2( )gμν ∂μ ∂ν
− U ( ) + X − V (φ)] .
(10.18)
Here f1( ) is the nonminimal coupling and f2( ) is the
noncanonical interaction. This type of theories include
the following subclass of models:
• Jordan Brans Dicke (JBD) theory: In this case we
have
, U ( ) = 0,
Here ω is the JBD parameter and for powerlaw
inflation ω > 1/2.
• Induced gravity theory: In this case we have
ω
f1( ) = 16π , f2( ) = 16π
M p
.
Here g1 and g2 are coupling constants. For powerlaw
inflation g1 < 1/2.
• Nonminimally coupled theory: In this case we have
f1( ) = M22p − ξ2 2, f2( ) = 21 , U ( ) = 0,
(10.20)
(10.21)
(10.22)
c1 = c22 , c2 = =<>= 2(6ξ3MξM2p 2p+ 1) ,
(10.23)
for ξ = 1/6
for ξ = 1/6.
(10.24)
After doing the conformal transformation in an Einstein
frame one can derive the required form of the effective action
from all these models.
2. Type II: Consider an action in a Jordan frame where the
scalar field is minimally coupled with the f (R) gravity
sector as given by
S =
d4x √−g [ f (R) + X − V (φ)] .
(10.25)
Here f (R) is an arbitrary functional of the Ricci scalar R.
After doing the conformal transformation in an Einstein
frame one can derive the required form of the effective
action.
3. Type III: Consider a 4 + D dimensional Kaluza–Klein
theory with an additional scalar field. This type of theories
includes the following subclass of models:
• Extra dimensional theoryI: In this case the inflaton
is introduced in the 4D effective action in a Jordan
frame:
S =
1
2 R + 4 1 − D
× gμν ∂μ ∂ν ; − U ( ) + X − V (φ) .
(10.26)
In this case we have
c2
c1 = 2 , c2 =
8D
D + 2
0 2
= M p 2 1 + D
ln( ).
(10.27)
But from this type of model no powerlaw inflationary
solutions are possible.
• Extra dimensional theoryII: In this case the inflaton
is introduced in the 4 + D dimensional action in a
Jordan frame as given by
S =
d4+D x
R + X − V (φ) .
1
2κ42+D
(10.28)
Here g4+D is the determinant of the 4 + D
dimensional metric and κ42+D is the 4 + D dimensional
gravitational coupling constant. In this case also we
have
c1 = 0, c2 =
2D
D + 2
,
From this type of model powerlaw inflationary
solutions are possible for all extra D dimensions.
4. Type IV: Consider an action in a Jordan frame from
superstring theory in 10 dimension with fixed Kalb–
Ramond background. In this case the scalar field is
nonminimally coupled with the gravity sector:
S =
d4x√−g e−2 R + 4gμν ∂μ ∂ν
+ X − V (φ) .
In this case is known as the dilaton field. But from
this type of model no powerlaw inflationary solutions
are possible. Here additionally we have two classes of
the solutions:
Class I:
After the completion of the phase of reheating, the total
system enters the radiation dominated stage, at the beginning of
which the total energy density is governed by Eq. (7.47). At
that stage, the scalar inflaton fields have almost settled down
in one of the potential valleys of the derived EFT potentials
and get its VEV for the proposed model in R2 gravity setup
in an Einstein frame. To make the computation simpler we
also assume that at the level of perturbations the dilaton field
is almost decoupled from the Standard Model fields and
the only dynamical field present in the model at late times.
Henceforth, we will treat as a dynamical field minimally
coupled to the R2 gravity in a conformally transformed
Einstein frame and also assume that the field is noninteracting
with other matter degrees of freedom and radiation content of
the universe at late times. During this epoch the total potential
is characterized by the following expression:
W˜ (φˆ , ) = V0 1 +
,
Dominant at late time
= V0 + λˆ exp
(10.33)
where V0 is defined as V0 = M8α4p , and the VEV of the inflaton
field φ is denoted by the symbol φˆ . Here one can set φˆ ∼
O(M p) for the proposed model at late time scale.
Once the contribution of the inflaton scalar field φ gets
its VEV the corresponding energy density ρm ≡ ρφ =
Constant. Now in the present context to characterize the
features of late time acceleration of the universe let us introduce
equation of state parameter wX(= w ), which is defined as
pX
wX = ρX =
d
dt˜
d
dt˜
2
2
− W˜ (φˆ , )
+ W˜ (φˆ , )
and the continuity equation in the present context can be
written as
dρX + 3H˜ (1 + wX)ρX = 0. (10.35)
dt˜
For the qualitative analysis of the prescribed system in the
Einstein frame and in order to compare with present day
observations, we introduce the following sets of
dimensionless density parameters and shifted equation of state
parameter:
ρX
= 3H˜ 2 M 2p
,
m ≡
ρm
φ = 3H˜ 2 M 2p
X ≡
X ≡
m ≡
ρr
r ≡ 3H˜ 2 M 2p
,
= 1 + w
= 1 + wX,
φ = 1 + wφ = 1 + wm .
(10.34)
(10.36)
(10.37)
In order to transform the cosmological equations into a
simplified autonomous system, we define the following
dimensionless auxiliary variables for the study of present dynamical
system at late time scale:
x ≡ √
6H˜ M p
W˙˜ (φˆ , )
, y ≡ √
3H˜ M p
≡ −M p∂ ln W˜ (φˆ , ),
Case I: δ
φV4
1 − φ4
1 − (mc2m−c2λφ2)
2 2 φV2
1 + ξ2 (φ2 + φ0 − 2φV2 ) + ξ2 (φ2 − φ0 ) + φ2
V(φ) = − √
φ
6M p ×
for Case I
for Case II
for Case II + Choice I(v1&v2)
for Case II + Choice II(v1&v2)
for Case II + Choice III.
dx
dN
dy
dN
d
dN
d r
dN
d m
which can be recast in the autonomous form as
W˜ (φˆ , )∂ W˜ (φˆ , )
(∂ W˜ (φˆ , ))2
r ) −
r ),
dN
together with an additional constraint condition, X + r +
m = x 2 + y2 + m + r = 1. Also using these
dimensionless variables Eqs. (10.34) and (10.36) can be recast as
(10.39)
weff − 3r ,
X
One can also define the total effective equation of state as
peff pX + pm + pr
weff ≡ ρeff = ρX + ρm + ρr
p + pφ + pr
= ρ + ρφ + ρr
= x 2 − y2 + 3r .
For an accelerated expansion effective equation of state
satisfy the following constraint, weff < −1/3. Using this
methodology mentioned in this section one can study the
constraints on the model from late time acceleration which
is beyond the scope of our discussion in this paper.
10.3 Details of the δN formalism
10.3.1 Useful field derivatives of N
To simplify the calculation for δN let us consider all these
possibilities to write down the infinitesimal change in field
in terms of the inflaton field φ:
(10.40)
(10.41)
∂φ ∂
∂φ ∂φ ∂
∂φ ∂ ∂φ =
9φ
= − √6M p
φ
δφ,
Case II: δ
= − √6M p
Case II + Choice I(v1&v2):
δφ,
δ
δ
φ
,
= − √6M p
Case II + Choice II(v1&v2):
φ
mc2
δ = − √6M p δφ 1 − (mc2 − λφ2) ,
Case II + Choice III:
φ
δφ 1 + ξ2 (φ2 + φ02 − 2φV2 )
= − √6M p
(10.42)
(10.43)
(10.44)
(10.45)
(10.46)
(10.47)
(10.48)
(10.49)
(10.50)
(10.51)
This additionally implies that one can write down the
following differential operator for the field:
1
∂ = V(φ) ∂φ , ∂2 =
1 2 V (φ)
V2(φ) ∂φ − V3(φ) ∂φ ,
∂3 =
V31(φ) ∂φ3 − 3 VV4((φφ)) ∂φ2 + 3 V 2(φ) ∂φ ,
V5(φ)
V(1φ) ∂φ2 − VV2((φφ)) ∂φ , ∂ ∂φ = V(1φ) ∂φ2,
V(1φ) ∂φ3 − 2 VV2((φφ)) ∂φ2
V2(φ) − 2 V 2(φ)
V (φ)
V3(φ)
V(1φ) ∂φ3 − VV2((φφ)) ∂φ2 ,
∂φ ,
∂ ∂φ∂φ = V(1φ) ∂φ,
3
∂φ∂ ∂ = V2(φ) ∂φ3 − 3VV3((φφ)) ∂φ2 + 3 VV42((φφ)) ∂φ ,
1
∂ ∂φ∂ = V21(φ) ∂φ3 − 2VV3((φφ)) ∂φ2 + 2 VV42((φφ)) ∂φ ,
(10.52)
(10.53)
+3 VV42((φφ)) N,φ ,
N, φ = V2(φ) ∂φ3 − 2VV3((φφ)) ∂φ2 + 2 VV42((φφ)) ∂φ N
1
1
= V2(φ) N,φφφ − 2VV3((φφ)) N,φφ
+2 VV42((φφ)) N,φ ,
fiwehlCedroeφnsii.eseq.duee=finntl∂eyφdo.ansethcaenpawrrtiiatel derivative with respect to the N, φ = V211(φ) ∂φ3 − VV3((φφV)) (∂φφ2)
N = V2(φ) N,φφφ − V3(φ) N,φφ .
N, = V(1φ) ∂φN = V(φ) N,φ,
1
1 2 V (φ)
N, = V2(φ) ∂φ − V3(φ)∂φ N
1 V (φ)
= V2(φ) N,φφ − V3(φ) N,φ ,
1 2
N,φ = V(φ) ∂φ − VV2((φφ)) ∂φ N
(10.54) If we neglect the quadratic slowroll corrections then the
solutionofEq.(8.40)takesthefollowingformforalldifferent
cases considered here:
10.3.2 Secondorder perturbative solution with various
source
(10.60)
(10.61)
(10.62)
1
= V(φ) N,φφ − VV2((φφ)) N,φ ,
1 2 1
N, φ = V(φ) ∂φ N = V(φ) N,φφ,
N, = V3(φ) ∂φ3 − 3VV4((φφ)) ∂φ2 + 3VV52((φφ)) ∂φ N
1
(10.55)
= V3(φ) N,φφφ − 3VV4((φφ)) N,φφ + 3VV52((φφ)) N,φ ,
1
N,φφ = V(1φ) ∂φ3 − 2VV2((φφ)) ∂φ2
− VV2((φφ)) − 2VV32((φφ)) ∂φ N,
1
= V(φ) N,φφφ − 2VV2((φφ)) N,φφ
− VV2((φφ)) − 2VV32((φφ)) N,φ ,
N,φ φ = V(1φ) ∂φ − VV2((φφ)) ∂φ2 N
3
1
= V(φ) N,φφφ − VV2((φφ)) N,φφ ,
1 3 1
N, φφ = V(φ) ∂φ N = V(φ) N,φφφ,
N,φ = V21(φ) ∂φ3 − 3VV3((φφ)) ∂φ2 + 3 VV42((φφ)) ∂φ N
1
= V2(φ) N,φφφ − 3VV3((φφ)) N,φφ
(10.57)
(10.58)
(10.59)
(10.56)
+e−3Ht(4 cφL3(1 + 3Ht)D1 − 9H2D3) .
(10.63)
For Case I:
1 27φ∗HeHYt
2 = D4 + 27H3 Y2(3 + Y)3 {−Y(3 + Y)2
×(4 cφL3 + H2Y(3 + Y))
+ H(4 cφL3(−18 + Y(3 + Y)(−6 + Ht(3 + 2Y)))
+ H2Y(3 + Y)(−9 + Y(3 + Y)(−2 + Ht(3 + 2Y))))}
+9H2 cφL3t(φL + 4D2)
For Case II:
1 27φ∗HeHYt
2 = D4 + 27H3 Y2(3 + Y)3
× 4−Y(3 + Y)2 −4 cφL3 + H2Y(3 + Y)
+ H −4 cφL3(−18 + Y(3 + Y)(−6 + Ht(3 + 2Y)))
+H2Y(3 + Y)(−9 + Y(3 + Y)(−2 + Ht(3 + 2Y))) 5
+9H2t β − cφL3 (φL + 4D2)
−e−3Ht 4 cφL3(1 + 3Ht)D1 + 9H2D3 .
(10.64)
For Case II + Choice I(v1):
1 27φ∗HeHYt
2 = D4 + 27H3 Y2(3 + Y)3
4−Y(3 + Y)2 −4 cφL3 + H2Y(3 + Y)
+ H −4 cφL3(−18 + Y(3 + Y)(−6 + Ht(3 + 2Y)))
+H2Y(3 + Y)(−9 + Y(3 + Y)(−2 + Ht(3 + 2Y))) 5
For Case II + Choice II(v2):
1
2 = D4 + 54H 3
54φ∗ H eHYt
Y2(3 + Y)3
:−Y(3 + Y)2 −McφL + 4 cφL3 + H 2Y(3 + Y)
+ H (−4 cφL3 + McφL)(18 − Y(3 + Y)(−6 + H t (3 + 2Y)))
+H 2Y(3 + Y)(−9 + Y(3 + Y)(−2 + H t (3 + 2Y))) ;
+9H 2t 2β + φL(−Mc(φL + 2D2) + cφL2 (φL + 4D2))
−e−3Ht 2φL(Mc − 4 cφL2 )(1 + 3H t)D1 + 18H 2D3
For Case III:
1
2 = D4 + 27H 3φL
27φ∗ H eHYt
Y2(3 + Y)3
× :−Y(3 + Y)2 ξ ξ + H 2φLY(3 + Y)
ξ ξ (−18 + Y(3 + Y)(−6 + H t (3 + 2Y)))
+ H
+H 2φLY(3 + Y)(−9 + Y(3 + Y)(−2 + H t (3 + 2Y))) ;
+9H 2t φL(β + ξ ) + ξ ξ D2
(10.69)
(10.67)
.
.
(10.68)
+9H 2t β + cφV4 − cφL3 (φL + 4D2)
−e−3Ht 4 cφL3 (1 + 3H t)D1 + 9H 2D3
.
(10.65)
27φ∗ H eHYt
Y2(3 + Y)3
× :−Y(3 + Y)2 4 cφL3 + H 2Y(3 + Y)
+ H 4 cφL3 (−18 + Y(3 + Y)(−6 + H t (3 + 2Y)))
+H 2Y(3 + Y)(−9 + Y(3 + Y)(−2 + H t (3 + 2Y))) ;
+9H 2t β − cφV4 + cφL3 (φL + 4D2)
+e−3Ht 4 cφL3 (1 + 3H t)D1 − 9H 2D3
.
(10.66)
54φ∗ H eHYt
Y2(3 + Y)3
× :−Y(3 + Y)2 McφL − 4 cφL3 + H 2Y(3 + Y)
+ H (4 cφL3 − McφL)(18 − Y(3 + Y)(−6 + H t (3 + 2Y)))
+H 2Y(3 + Y)(−9 + Y(3 + Y)(−2 + H t (3 + 2Y))) ;
+9H 2t 2β + φL(Mc(φL + 2D2) − cφL2 (φL + 4D2))
+e−3Ht 2φL(Mc − 4 cφL2 )(1 + 3H t)D1 − 18H 2D3
× 4−Y(3 + Y)2 4 cφL3 + H 2Y(3 + Y)
+ H
−4 cφL3 (18 + 6Y(3 + Y))
10.3.3 Expressions for perturbative solutions in final
hypersurface
Neglecting the contribution from the quadratic slowroll term
and taking up to linearorder term in slowroll we get the
following result:
1
1(N = 0) = D2 − 3H D1 +
(10.70)
Similarly if we neglect the quadratic slowroll corrections
then the solution of 2(N = 0) takes the following form for
all different cases considered here:
For Case I:
1
2(N = 0) = D4 + 27H 3
27φ∗ H
Y2(3 + Y)3
× 4−Y(3 + Y)2 4 cφL3 + H 2Y(3 + Y)
+ H
−4 cφL3 (18 + 6Y(3 + Y))
−H 2Y(3 + Y)(9 + 2Y(3 + Y)) 5
+ 4 cφL3 D1 − 9H 2D3
For Case II:
1
2(N = 0) = D4 + 27H 3
27φ∗ H
Y2(3 + Y)3
× 4−Y(3 + Y)2
−4 cφL3 + H 2Y(3 + Y)
+ H 4 cφL3 (18 + 6Y(3 + Y))
−H 2Y(3 + Y)(9 + 2Y(3 + Y)) 5
− 4 cφL3 D1 + 9H 2D3
For Case II + Choice I(v1):
1
2(N = 0) = D4 + 27H 3
× 4−Y(3 + Y)2
−4 cφL3 + H 2Y(3 + Y)
+ H 4 cφL3 (18 + 6Y(3 + Y))
−H 2Y(3 + Y)(9 + 2Y(3 + Y)) 5
.
− 4 cφL3 D1 + 9H 2D3
For Case II + Choice I(v2):
1
2(N = 0) = D4 + 27H 3
27φ∗ H
Y2(3 + Y)3
27φ∗ H
Y2(3 + Y)3
(10.71)
(10.72)
(10.73)
−H2Y(3 + Y)(9 + 2Y(3 + Y)) 5
+ 4 cφL3D1 − 9H2D3 .
× 4−Y(3 + Y)2 McφL − 4 cφL3 + H2Y(3 + Y)
+ H (4 cφL3 − McφL)(18 + 6Y(3 + Y))
−H2Y(3 + Y)(9 + 2Y(3 + Y))) 5
(10.74)
× 4−Y(3 + Y)2 −McφL + 4 cφL3 + H2Y(3 + Y)
+ H (−4 cφL3 + McφL)(18 + 6Y(3 + Y))
− 2φL(Mc − 4 cφL2)D1 + 18H2D3 . (10.76)
Analytical expression for the shift in the inflaton field from
linearorderandsecondordercosmologicalperturbationtheory can be written up to considering the contributions from
the firstorder slowroll contribution as
δφ1(N = 0) = δφ1∗ = φ∗ ˆ 1(N = 0) = φ∗ Dˆ2
− 3φH∗ Dˆ 1 + Y(3φ+∗Y)2 −Y(3 + Y)2
+ H (−9 + Y(3 + Y){−2 + H(3 + 2Y)t})]. (10.78)
For Case I:
δφ2(N = 0) = δφ2∗ = φ∗ ˆ 2(N = 0)
φ∗ 27H
= φ∗ Dˆ4 + 27H3 Y2(3 + Y)3
× 4−Y(3 + Y)2 4 cφL3 + H2Y(3 + Y)
+ H −4 cφL3(18 + 6Y(3 + Y))
−H2Y(3 + Y)(9 + 2Y(3 + Y)) 5
+ 4 cφL3 Dˆ1 − 9H2 Dˆ3 .
For Case II:
+ H 4 cφL3(18 + 6Y(3 + Y))
−H2Y(3 + Y)(9 + 2Y(3 + Y)) 5
− 4 cφL3 Dˆ1 + 9H2 Dˆ3 .
× 4−Y(3 + Y)2 −4 cφL3 + H2Y(3 + Y)
For Case II + Choice I(v1):
δφ2(N = 0) = δφ2∗ = φ∗ ˆ 2(N = 0)
φ∗ 27H
= φ∗ Dˆ4 + 27H3 Y2(3 + Y)3
+ H 4 cφL3(18 + 6Y(3 + Y))
−H2Y(3 + Y)(9 + 2Y(3 + Y)) 5
− 4 cφL3 Dˆ1 + 9H2 Dˆ3 .
For Case II + Choice I(v2):
δφ2(N = 0) = δφ2∗ = φ∗ ˆ 2(N = 0)
φ∗ 27H
= φ∗ Dˆ4 + 27H3 Y2(3 + Y)3
× 4−Y(3 + Y)2 4 cφL3 + H2Y(3 + Y)
+ H −4 cφL3(18 + 6Y(3 + Y))
+ 4 cφL3 Dˆ1 − 9H2 Dˆ3 .
(10.79)
(10.80)
(10.81)
(10.82)
For Case II + Choice II(v1):
2(N = 0) = φ∗Dˆ 4 +
φ
∗
54H 3
54H
Y2(3 + Y)
ξ ξ Dˆ1 − 9H 2φLDˆ 3
.
10.3.5 Various useful constants for δN
For the derived effective potentials (φ ) and (φ ) can be recast as
B ∗ C ∗
φ
∗
54H 3
54H
Y2(3 + Y)
φ
∗
27H
27H 3φL Y2(3 + Y)
4
3 −Y(3 + Y)
−McφL + 4 cφL + H 2Y(3 + Y)
3
4
3 −Y(3 + Y)
ξ ξ + H 2φLY(3 + Y)
4
3 −Y(3 + Y)
McφL − 4 cφL + H 2Y(3 + Y)
3
for Case II + Choice I(v1&v2)
for Case II + Choice II(v1&v2)
for Case II + Choice III.
(10.84)
(10.85)
(10.86)
(10.87)
for Case I & II
for Case II + Choice I(v1&v2)
for Case II + Choice II(v1&v2)
for Case II + Choice III,
3
−
2
+
2λmc2φ∗2
mc2
(mc2−λφ2)2 1− (mc2−λφ∗2) ⎭
∗
2 1− mc2−λφ∗2 − (mc2−λφ∗2)2
2λmc2φ∗2
4 2 ⎫
φ
1+3 V4 ⎪
φ∗ ⎬
4 2
φV
1− φ4 ⎪⎭
∗
1− mc2−λφ∗2
2λmc2φ∗2
2
2
∗
mc2
1− mc2−λφ∗2
1 (mc2−λφ∗2)2 + (mc2−λφ∗2)3
6mc2λφ∗
8λ2mc2φ∗3 ⎞ ⎫
mc2
1− mc2−λφ∗2
⎟
⎠
⎪
⎬
⎪
⎭
2 2
φV
1+ξ(3φ∗2−φV2 )− φ2
φ2
6ξφ∗2+2 V
φ2
∗
2 ⎟
φV ⎠
1+ξ(φ∗2−φV2 )+ φ2
⎞ ⎫
⎪
⎬
⎪
⎭
3
⎪ Yφ∗2
⎪
⎪
⎪
⎪
⎪⎪⎪ φV4
⎪⎪⎪⎪⎪⎪⎪ Y1φ∗2 13−+ φφφV4∗44
⎪
⎪
⎪⎨ ∗
1 ⎨
1 ⎨ 8
3 +
3 +
1 ⎨ 8
3 +
B(φ∗) =
C(φ∗) ≈
Additionally the constants G1(φ∗) and G2(φ∗), as appearing in the expression for fNloLc, are defined as
6M2p −2
6M2p −2
4 2 ⎟
φV ⎠
φ∗2 1− φ4
6M2p
mc2
12M2p
12M2p
36M4p
36M4p
⎞ −2 ⎛
2 ⎟
⎠
6M2p
2 2 ⎟
φV ⎠
φ∗2 1+ξ(φ∗2−φV2 )+ φ2
+ 81φ2
∗
+ φ2
∗
6M2p −1 6M2p
6M2p −1 6M2p
φ3
∗
G1(φ∗) =
and
G2(φ∗) =
1
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪⎪ 1
⎪
⎪
⎪
⎪
⎪
⎪⎪ ⎛
⎪
⎪
⎪
⎪
⎪⎪ ⎜ 1 +
⎪⎪ ⎝
⎨
for Case I
for Case II
for Case II + Choice I(v1&v2)
for Case II + Choice II(v1&v2)
for Case II + Choice III
36M4p
for Case I
for Case II
Choice I(v1&v2)
+
Choice II(v1&v2)
+
Choice III.
+
φ4
6M2p 1+3 V
φ4
∗
4 3
φ
V
φ3 1− φ4
∗
6M2p
4 2 ⎟
6M2p
mc2
2 ⎟
6M2p
2 2 ⎟
6M2p 1− mc2−λφ2 − (mc2−λφ2)2
∗ ∗
2λmc2φ∗2
mc2
∗ mc2−λφ∗2
3
φ
φ3 1+ξ(φ∗2−φV2 )+ φ2
V
∗
∗
2 3
∗
10.4 Momentum dependent functions in four point function
Momentum dependent functions Gˆ S(k1, k2, k3, k4), Wˆ S
(k1, k2, k3, k4) and Rˆ S(k1, k2, k3, k4) as appearing in four
point function are defined as
S
Gˆ (k1, k2, k3, k4) =
S(k˜ , k1, k2)S(k˜ , k3, k4)
k1 + k23
× k2.k4 +
× k2.k3 +
× k3.k4 +
[k2.(k1 + k2)] [k4.(k3 + k4)]
[k2.(k1 + k2)] [k3.(k3 + k4)]
[k3.(k3 − k4)] [k4.(k3 − k4)]
k1 + k22
k1 + k22
k1 + k22
k1.k3 +
+ k1.k4 +
− k1.k2 +
k1 + k22
k1 + k22
k1 + k22
[k1.(k1 + k2)] [k4.(k3 + k4)]
[k1.(k1 + k2)] [k2.(k3 + k4)]
12M2p
36M4p
⎞ −2 ⎛
2 +
36M4p
φ∗4 1− mc2−λφ∗2
12M2p
4 ⎟
⎠
2 2 +
φV
φ∗2 1+ξ(φ∗2−φV2 )+ φ2
2 4 ⎟
φV ⎠
φ∗4 1+ξ(φ∗2−φV2 )+ φ2
(10.88)
(10.89)
(10.90)
with
and
where
and the momentum dependent functions A1(k1,k2,k3,k4),
A2(k1,k2,k3,k4) and A3(k1,k2,k3,k4) are defined as
(k3.k4)((k1.k2)(k12 + k22) + 2k12k22)
8k1 + k22
A1(k1,k2,k3,k4) =
+(1,2 ↔ 3,4)
(10.94)
A2(k1,k2,k3,k4) = −
2k3k4(k3 + k4)((k1.k2)(k12 + k22) + 2k12k22)(k3k4 + k3.k4) + (3,4 ↔ 1,2)3
8k1 + k24
1
−2k1 + k22 k12k42(k2.k3)(k2 + k3) + k12k32(k2.k4)(k2 + k4) + k22k42(k1.k3)(k1 + k3) + k22k32(k1.k4)(k1 + k4)
(k1.k2)
+ 8k1 + k22 ((k1 + k2)((k3.k4)(k32 + k42) + 2k32k42) + k3k4(k3 + k4)(k3k4 + k3.k4)) + (1,2 ↔ 3,4) , (10.95)
2k1 + k22
3k1k2k3k4(k1k2 + k1.k2)(k3k4 + k3.k4)
.
4k1 + k22
2k1 + k23Gˆ S(k1,k3,k2,k4) k1k2(k1 + k2)2((k1 + k2)2 − k32 − k42 − k3k4)
(Kˆ − 2(k3 + k4))2Kˆ 2((k1 + k2)2 − k1 + k22)
k1 + k2 k1 + k2 k1 + k2
× − 2k1k2 − k32 + k42 + 4k3k4 − (k1 + k2)2 + k1 + k22 − (k1 + k2)2
1 1 3
Kˆ − 2(k1 + k2) − Kˆ + 2(k1 + k2) + (1,2 ↔ 3,4)
k1 + k23(k1 + k22 − k12 − k22 − 4k1k2)(k1 + k22 − k32 − k42 − 4k3k4) .
− 2(k1 + k22 − k12 − k22 − 2k1k2)(k1 + k22 − k32 − k42 − 2k3k4)
(10.96)
(10.97)
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