# Power law coupling Higgs-Palatini inflation with a congruence between
physical and geometrical symmetries

1 José Edgar Madriz Aguilar, 2 Diego Allan Reyna E-mail address:
<EMAIL_ADDRESS>1 Departamento de Matemáticas, Centro Universitario
de Ciencias Exactas e ingenierías (CUCEI), Universidad de Guadalajara (UdG),
Av. Revolución 1500 S.R. 44430, Guadalajara, Jalisco, México,
and
2 Departamento de Física, Centro Universitario de Ciencias Exactas e
ingenierías (CUCEI), Universidad de Guadalajara (UdG), Av. Revolución 1500
S.R. 44430, Guadalajara, Jalisco, México.
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this paper we investigate a power law coupling Higgs inflationary model in
which the background geometry is determined by the Palatini’s variational
principle. The geometrical symmetries of the background geometry determine the
invariant form of the action of the model and the background geometry resulted
is of the Weyl-integrable type. The invariant action results also invariant
under the $U(1)$ group, which in general is not compatible with the Weyl group
of invariance of the background geometry. However, we found compatibility
conditions between the geometrical and physical symmetries of the action in
the strong coupling limit. We found that if we start with a non-minimally
coupled to gravity action, when we impose the congruence between the both
groups of symmetries we end with an invariant action of the scalar-tensor
type. We obtain a nearly scale invariant power spectrum for the inflaton
fluctuations for certain values of some parameters of the model. Also we
obtain values for the tensor to scalar ratio in agreement with PLANCK and
BICEP observational data: $r<0.032$.

###### pacs:

04.50. Kd, 04.20.Jb, 02.40k, 98.80k, 98.80.Jk, 98.80.Cq

Higgs-Palatini inflation, Weyl-Integrable geometry, Weyl group of symmetries,
$U(1)$ group of symmetries, Power spectrum of inflaton fluctuations.

##  I INTRODUCTION

Non-minimally coupled theories of gravity have been an object of interest in
recent years, particularly in the context of cosmological inflation, due to
the discovery of the Higgs scalar field Higgs1 ; Higgs2 . In the context of
the standard model (SM) of particles the unique scalar field that can play the
role of the inflaton by driving inflation is the Higgs scalar field. To
achieve a successful inflationary period the so-called non-minimal coupling
models of Higgs inflation have been proposed. A non-minimal coupling of the
Higgs with gravity of the form $|h|^{2}R$ is one of the most favored models by
observations Lambda1 ; Lambda2 ; Lambda3 . In certain models, these kind of
terms could arise from radiative corrections Lambda3 ; Lambda4 . Non-minimally
coupling Higgs inflationary models aligns excellently with current
observations of the cosmic microwave background (CMB) Lambda1 ; Lambda2 , and
the magnitude of the coupling constant $\epsilon$ between the Higgs field and
the Ricci scalar depends of the observed amplitude of perturbations in CMB,
resulting $\epsilon\gg 1$ Lambda5 . One of the problems in particle Higgs
models is related with the so called “hierarchy problem”. This consists in the
fact that the Higgs boson mass seems to be sensitive to quantum corrections
and the bare Higgs mass then needs to be fine-tuned to achieve a physical
Higgs boson mass many orders of magnitude smaller than the Planck scale
hierarchy . When the Higgs is coupled to gravity the hierarchy Higgs mass is
tackled through the use of the Palatini variation by several authors hp1 ; hp2
; hp3 ; hp4 ; hp5 ; hp6 ; hp7 ; hp8 .

A class of theories that naturally incorporates a scalar field is the scalar-
tensor theories of gravity. Nonetheless, it is not so clear, at least in the
Jordan frame, if the scalar field describes gravity or matter quirozaguilar .
This frame exhibits a non-minimal coupling with the scalar field, however in
the Einstein frame a minimal coupling is present faraoni . Usually one pass
from the Jordan to the Einstein frame by simply implementing a conformal
transformation of the metric. The main controversy relies in determining which
of the both frames is the physical one quirozaguilar . On the other hand, it
is a well-known fact that a geometry is characterized by the compatibility
condition between the connection and the metric:
$\nabla_{\mu}g_{\alpha\beta}=N_{\alpha\beta\mu}$. However, in general the
compatibility condition does not remain invariant only under conformal
transformations of the metric. Therefore, the usual manner in which we pass
from the Jordan to Einstein frame in standard scalar-tensor theories changes
the background geometry, and this could be why the physics in one or another
frame seems different. Particularly, under conformal transformations a
geodesic observer in one frame is not a geodesic observer on the other, and
viceversa, see for example quirozaguilar ; faraoni ; CR1 .

This controversy may be alleviated if the background geometry is not fixed
apriori as Riemannian. This is the main idea in a recently introduced new kind
of scalar-tensor theories known as geometrical scalar-tensor theories of
gravity CR1 ; CR2 . Geometrical scalar tensor theories of gravity are an
approach of scalar-tensor theories of gravity that arise as an attempt to
obtain a scalar invariant action under the group of symmetries of the
background geometry. Specifically, when a Platini’s variational principle is
adopted for a scalar-tensor theory the resulting background geometry is non-
Riemannian, and as a consequence the group of symmetries that leave invariant
the non-metricity condition is bigger than the diffeomorfism group. Thus, the
original action is not transforming as a scalar under this extended group.
Thus, in this geometrical approach a new action based on the original action
is proposed so that it becomes a scalar under the group of symmetries of the
background geometry II1 . In this context, several topics can be investigated
in the framework of this new approach, for example, Higgs inflation, the
formation of the seeds for cosmic magnetic fields and Dark energy scenarios
II1 ; II2 ; II3 .

In the case of the majority of non-minimally coupled Higgs inflationary
models, it is also used a conformal transformation of the metric to pass to an
action free of the non-minimal coupling term, i.e. to the corresponding
Einstein frame. This leads to the same controversy of the scalar-tensor
theories generated by changing the background geometry, and just like in those
cases when the change of the geometry is considered, the original action is no
longer an invariant under the new geometrical symmetry group. In this manner,
in this paper we develop a Higgs inflationary model, with a power law non-
minimally coupled to gravity term in the action, in the Palatini formalism,
where we introduce an invariant action under the geometrical group of
symmetries of the background geometry. In other words, unlike traditional
models we use the group of symmetries of the background geometry to obtain the
correct action for the model. Thus in section I we give a brief introduction.
Section II is devoted to the construction of the invariant action of the
model. In section III we derive the field equations of the particular Higgs
inflationary model. In section IV we study the scalar fluctuations of the
metric at the end of inflation. Finally, section V is left for some
conclusions.

##  II basic formalism with A non-minimally coupled complex action

We start with an action for a complex scalar field $\phi$ non-minimally
coupled to gravity in the form

${\cal S}=\int
d^{4}\sqrt{-g}\Bigl{[}(1+\epsilon(\phi\phi^{{\dagger}})^{n})M_{pl}^{2}R+\frac{1}{2}g^{\alpha\beta}\phi_{,\alpha}\phi^{{\dagger}}_{,\beta}-V(\phi\phi^{{\dagger}})\Bigr{]},$
(1)

where $g$ denotes the determinant of the metric, $R$ is the Ricci scalar
curvature, $\epsilon$ is a coupling parameter, $M_{pl}=(16\pi G)^{-1/2}$ is
the reduced Planckian mass, the dagger $\dagger$ is denoting transposed
complex conjugate, and

$V(\phi\phi^{\dagger})=\frac{\lambda}{2}\left(\phi\phi^{\dagger}-\sigma^{2}\right)^{2}$
(2)

is the scalar Higgs potential, with $\lambda=0.129$ and $\sigma=246\,GeV$
being the Higgs vacuum expectation value III1; III2. Due to the explicit
$U(1)$ global symmetry from the complex scalar field, a gauge derivative
$\phi_{:\alpha}$ in terms of a vector field $B_{\mu}$ must be introduced in
order to obtain a locally invariant action

${\cal S}=\int
d^{4}\sqrt{-g}\Bigl{[}\left(1+\epsilon(\phi\phi^{{\dagger}})^{n}\right)M_{pl}^{2}R+\frac{1}{2}g^{\alpha\beta}\phi_{:\alpha}\phi^{{\dagger}}_{:\beta}-V(\phi\phi^{{\dagger}})-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\Bigr{]},$
(3)

where $\phi_{:\mu}=\phi_{,\mu}+i\alpha B_{\mu}\phi$ with $\alpha$ being a
gauge parameter of the derivative and
$F_{\mu\nu}=\partial_{\mu}B_{\nu}-\partial_{\nu}B_{\mu}$ is the strength field
describing the dynamics of the gauge boson field $B_{\mu}$. The action (3) is
invariant under the physical group of symmetries

$\displaystyle\tilde{\phi}$ $\displaystyle=$ $\displaystyle\phi
e^{i\gamma\theta(x)},$ (4) $\displaystyle\tilde{B}_{\lambda}$ $\displaystyle=$
$\displaystyle B_{\lambda}-\frac{\gamma}{\alpha}\theta(x)_{,\lambda}\,,$ (5)

where $\gamma$ and $\alpha$ are gauge parameters and the transformation (5) is
obtained when $\tilde{\phi}_{:\nu}=e^{i\gamma\theta}\phi_{:\nu}$ is imposed.
The expressions (4) and (5) are physical symmetries of the action (3). A
spontaneously symmetry breaking occurs when $\phi=\phi^{\dagger}$. Thus it is
not difficult to verify that after the breaking the minimum of the potential
is : $\phi=\pm\sigma$. Hence, it follows that the action (3) now reads

${\cal S}=\int
d^{4}x\,\sqrt{-g}\left[(1+\epsilon\phi^{2n})M_{pl}^{2}\,R+\frac{1}{2}g^{\alpha\beta}\phi_{:\alpha}\phi_{:\beta}-V(\phi)-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\right].$
(6)

The Palatini’s variational principle, i.e. the variation of the action with
respect to the affine connection reveals the background geometry associated to
(6) as a non-Riemannian geometry characterized by the compatibility condition:
$\nabla_{\lambda}g^{\mu\nu}=-[\ln(1+\epsilon\phi^{2n})]_{,\lambda}\,g^{\mu\nu}$,
which after the field rescaling $\varphi=-\ln(1+\epsilon\phi^{2n})$ becomes

$\nabla_{\lambda}g^{\mu\nu}=\varphi,_{\lambda}g^{\mu\nu}.$ (7)

Thus the background geometry is one of the Weyl-Integrable type with null
torsion, where $\varphi$ playing the role of the Weyl scalar field and
$\nabla_{\lambda}$ denotes the Weyl covariant derivative. Notice that there is
a geometric symmetry group that leaves invariant (7), this is the group of
Weyl transformations that reads

$\displaystyle\bar{g}$ $\displaystyle=e^{f}g,$ (8)
$\displaystyle\bar{\varphi}$ $\displaystyle=\varphi+f,$ (9)

being $f(x^{\gamma})$ a well behaved complex function of the space-time
coordinates.

The action (6) can be rewritten in terms of the previous field $\varphi$ as

${\cal S}=\int
d^{4}\sqrt{-g}\Bigl{[}e^{-\varphi}M_{pl}^{2}\,R+\frac{1}{2}g^{\alpha\beta}\omega(\varphi)D_{\alpha}\varphi
D_{\beta}\varphi-U(\varphi)-\frac{1}{4}F_{\alpha\beta}F^{\alpha\beta}\Bigr{]},$
(10)

where $U(\varphi)=V[\phi(\varphi)]$, the redefined covariant derivative
$D_{\lambda}=\varphi_{,\lambda}+i\alpha A_{\lambda}\varphi$ obeys the relation

$\phi_{:\lambda}=-\frac{e^{-\varphi}(e^{-\varphi}-1)^{\frac{1}{2n}-1}}{\epsilon^{1/2n}}\,D_{\lambda}\varphi\,,$
(11)

where the redefined gauge field is given by

$A_{\mu}=\frac{e^{\varphi}}{\varphi}\left(e^{-\varphi}-1\right)B_{\mu},$ (12)

and

$\omega(\varphi)=\frac{e^{-2\varphi}}{\epsilon^{1/n}}\left(e^{-\varphi}-1\right)^{\frac{1-2n}{n}}.$
(13)

The strength tensor can also be written in terms of $A_{\mu}$, and it has the
form

$F_{\alpha\beta}=\partial_{\alpha}\left[\left(\frac{e^{-\varphi}}{e^{-\varphi}-1}\right)\varphi
A_{\beta}\right]-\partial_{\beta}\left[\left(\frac{e^{-\varphi}}{e^{-\varphi}-1}\right)\varphi
A_{\alpha}\right].$ (14)

Now, in order to achieve a Weyl-invariance of the action (10), we need the
Weyl-invariance of the kinetic term in (10). Thus, we demand the
transformation of the covariant derivative
$D_{\lambda}\bar{\varphi}=D_{\lambda}\varphi$. Hence, it follows the rule

$\bar{A}_{\lambda}\bar{\varphi}=A_{\lambda}\varphi+i\alpha^{-1}f_{,\lambda}\,.$
(15)

The transformation rule (15) written in terms of $B_{\alpha}$ reads

$\frac{e^{-\bar{\varphi}}-1}{e^{-\bar{\varphi}}}\,\bar{B}_{\lambda}=\frac{e^{-\varphi}-1}{e^{-\varphi}}\,B_{\lambda}+i\alpha^{-1}f_{,\lambda}.$
(16)

As the equation (16) comes from demanding a Weyl-invariance then it can be
considered as a geometrical symmetry. Now, comparing equations (5) and (16) it
is not difficult to note that the physical and geometrical symmetries for the
gauge field $B_{\sigma}$ do not match each other.

So, in order to the geometric and physical symmetries for the gauge field to
be the same we will consider the strong coupling limit, which is obtained when
$\epsilon(\phi\phi^{\dagger})^{n}\gg 1$ in the action (3). Thus, we arrive to
the condition

$\epsilon\gg\frac{1}{(\phi\phi^{\dagger})^{n}}\gg\frac{1}{\min{\\{(\phi\phi^{\dagger})^{n}\\}}}=\frac{1}{\sigma^{2n}}=(0.004)^{2n}.$
(17)

In this limit the Weyl scalar field and the redefined gauge field can be
approximated by

$\displaystyle\varphi=-\ln(\epsilon\phi^{2n}),$ (18) $\displaystyle
A_{\mu}=-\frac{2n\alpha}{\varphi}\,B_{\mu}.$ (19)

In the same manner the expressión (11) reduces to

$\phi_{:\lambda}=-\frac{e^{-\varphi/2n}}{2n\epsilon^{1/2n}}\,D_{\lambda}\varphi.$
(20)

Hence, the equation (13) in this limit reduces to

$\omega(\varphi)=\frac{e^{-\varphi/n}}{(2n)^{2}\epsilon^{1/n}}.$ (21)

Therefore, using (19) in (15) we obtain

$\bar{B}_{\lambda}=B_{\lambda}-\frac{i}{2\pi\alpha^{2}}\,f_{,\lambda}.$ (22)

Now, by equationg (22) with (5) we obtain that the geometrical and physical
symmetries for $B_{\mu}$ match when

$f=\frac{2\pi\alpha^{2}}{i}\left(\frac{\gamma\theta}{\alpha}+\kappa\right),$
(23)

where $\kappa$ is an integration constant. Thus, (23) can be considered as a
compatibility relation between geometrical and physical symmetries for
$B_{\mu}$.

Getting back to the Weyl-invariance of the action (10), even if (15) holds,
the potential term (the third term) results not Weyl-invariant due to the fact
that the determinant of the metric under (8) transforms as
$\sqrt{-\bar{g}}=e^{2f}\sqrt{-g}$. Thus a complete Weyl invariant action
$\bar{{\cal S}}={\cal S}$ (in all of its terms) can be proposed as

${\cal S}_{inv}=\int
d^{4}\sqrt{-g}\Bigl{[}e^{-\varphi}M_{pl}^{2}\,R+\frac{1}{2}g^{\alpha\beta}\omega(\varphi)e^{-\varphi}D_{\alpha}\varphi
D_{\beta}\varphi-e^{-2\varphi}U(\varphi)-\frac{1}{4}F_{\alpha\beta}F^{\alpha\beta}\Bigr{]},$
(24)

where the aditional conditions

$\displaystyle\bar{\omega}(\bar{\varphi})$
$\displaystyle\equiv\omega(\bar{\varphi}-f)=\omega(\varphi),$ (25)
$\displaystyle\bar{U}(\bar{\varphi})$ $\displaystyle\equiv
U(\bar{\varphi}-f)=U(\varphi),$ (26)

must be valid.

##  III the Riemannian limit

Weyl transformations define an equivalence class between every set of Weyl
manifolds together with their metrics, their scalar fields, as well as their
gauge fields II1 ; CR1 ; CR2 ; II2 ; II3 . For each election of the function
$f(x^{\sigma})$ it corresponds a distinct metric which specifies a distinct
member ($M,g,\varphi,B$) of the aforementioned class of equivalence. On the
other hand, in a Weyl-integrable geometry the differential line element is not
preserved under the Weyl group of transformations i.e. $\bar{ds}^{2}\neq
ds^{2}$. However, the particular choice $f=-\varphi$, specifies a manifold
with some effective metric of the form
$h_{\alpha\beta}=e^{-\varphi}g_{\alpha\beta}$, such that
$\bar{h}_{\alpha\beta}=h_{\alpha\beta}$. It is not difficult to verify that
this effective metric has a metricity condition of the form

$\nabla_{\mu}h_{\alpha\beta}=0,$ (27)

therefore, one of the elements of the equivalence class is indeed a Riemannian
space-time. On this Riemannian limit the action (24) takes the following form

${\cal S}=\int
d^{4}\sqrt{-h}\Bigl{[}M_{pl}^{2}\,R+\frac{1}{2}h^{\alpha\beta}\omega(\varphi)D_{\alpha}\varphi
D_{\beta}\varphi-U(\varphi)-\frac{1}{4}F_{\alpha\beta}F^{\alpha\beta}\Bigr{]}.$
(28)

where in the strong coupling limit the strength tensor (14) is given by

$F_{\mu\nu}=-\frac{1}{2n\alpha}\left[\partial_{\mu}(\varphi
A_{\nu})-\partial_{\nu}(\varphi A_{\mu})\right].$ (29)

On the other hand, the cosmological principle does not allow the existence of
a vector field on cosmological large scales. Thus, it is convenient to use the
gauge freedom associated to $A_{\mu}$. With this in mind, by using (5) and
(19) we arrive to

$\tilde{\varphi}\tilde{A}_{\lambda}=\varphi
A_{\lambda}+\frac{\gamma}{2n\alpha^{2}}\,\theta_{,\lambda}.$ (30)

Therefore, by means of the gauge election

$\theta_{,\lambda}=-\frac{2n\alpha^{2}}{\gamma}\varphi A_{\lambda},$ (31)

it can be verified that $\tilde{\varphi}\tilde{A}_{\lambda}=0$. Thus it
follows from (29) that $\tilde{F}_{\alpha\beta}=0$. In this manner, Weyl
transforming the action (28) under the gauge election made, we obtain the
effective action

${\cal S}=\int
d^{4}\sqrt{-h}\Bigl{[}R+\frac{1}{2}\omega(\varphi)h^{\alpha\beta}\partial_{\alpha}\varphi\partial_{\beta}\varphi-U(\varphi)\Bigr{]},$
(32)

which is a minimally coupled theory that still is carrying the non-canonical
kinetic term.

##  IV A Higgs Inflationary model

Once that we are placed on the Riemannian limit of our theory, we are in
position to formulate a Higgs inflationary model. With his purpose we consider
the Higgs potential (2). As we mentioned before a spontaneous symmetry
breaking occurs when $\phi=\phi^{{\dagger}}$. Thus, excitations around the
ground state of $V(\phi)$ can be written as
$\phi(x^{\mu})=\sqrt{\xi}+\zeta(x^{\mu})$ where $\zeta(x^{\mu})$ is the Higgs
scalar field associated to $\phi$. Analogously, in terms of $\varphi$ we can
consider the expansion $\varphi(x^{\nu})=\varphi_{v}+\mathcal{Q}(x^{\nu})$,
with $\varphi_{v}=-\ln(\epsilon\sigma^{2n})$ being the vacuum value for
$\varphi$ and $\mathcal{Q}$ the Higgs field associated to $\varphi$. Employing
these expressions we can write $\omega(\varphi)$ and the potential
$U(\varphi)$ in terms of $\mathcal{Q}(x^{\mu})$ as

$\displaystyle\omega(Q)$ $\displaystyle=$
$\displaystyle\frac{e^{-\varphi_{v}/n}e^{-\mathcal{Q}/n}}{(2n)^{2}\epsilon^{1/n}},$
(33) $\displaystyle U(Q)$ $\displaystyle=$
$\displaystyle\frac{\lambda}{2}\Bigl{[}\sigma^{2}e^{-Q/n}-\sigma^{2}\Bigr{]}^{2}.$
(34)

Hence, the action (32) in terms of the Higgs field $\mathcal{Q}$ is then given
by

${\cal S}=\int
d^{4}\sqrt{-h}\Bigl{[}R+\frac{1}{2}h^{\alpha\beta}\omega(\mathcal{Q})\partial_{\alpha}Q\partial_{\beta}Q-U(Q)\Bigr{]},$
(35)

where using the value of $\varphi_{v}$ in (33) the function
$\omega(\mathcal{Q})$ reads

$\omega(Q)=\frac{\sigma^{2}e^{-\mathcal{Q}/n}}{(2n)^{2}}.$ (36)

Now, by means of the field transformation

$\Phi(x)=\int dQ\sqrt{\omega(Q)},$ (37)

we can unitarize the kinetic term in (35). Thus we obtain for the rescaled
field $\Phi$ and its corresponding potential

$\Phi(x)=-\sigma e^{-Q/2n},$ (38)

$U(\Phi)=\frac{\lambda\sigma^{4}}{2}\Bigl{[}\Bigl{(}\frac{\Phi}{\sigma}\Bigr{)}^{2}-1\Bigr{]}^{2}.$
(39)

Therefore, the action (35) can be written as

${\cal S}=\int
d^{4}\sqrt{-h}\Bigl{[}R+\frac{1}{2}h^{\alpha\beta}\partial_{\alpha}\Phi\partial_{\beta}\Phi-U(\Phi)\Bigr{]},$
(40)

where it is important to emphasize that due to the unitarization of the
kinetic term the potential energy has been rescaled. This fact makes it
possible for the Higgs to have enough energy to inflate the universe.

Straight forward calculations show that the action (40) leads to the field
equations

$\displaystyle G_{\alpha\beta}$ $\displaystyle=8\pi
G\\{\Phi,_{\alpha}\Phi,_{\beta}+\frac{1}{2}h_{\alpha\beta}(\Phi^{\prime\mu}\Phi,_{\mu}-2U(\Phi))\\}$
(41) $\displaystyle\Box\Phi$ $\displaystyle+U^{\prime}(\Phi)=0,$ (42)

where $U^{\prime}(\Phi)$ denotes the derivative with respect to $\Phi$. Now we
can consider the 3D spatially flat Friedmann-Lemaitre-Robertson-Walker metric

$ds^{2}=dt^{2}-a(t)(dx^{2}+dy^{2}+dz^{2}),$ (43)

where t $a(t)$ is the cosmological scale factor. Due to the cosmological
principle we can adopt the semi-classical approximation

$\Phi(x,t)=\Phi_{b}(t)+\delta\Phi(x,t),$ (44)

where $\Phi_{b}$ is the homogeneous classical solution to the field equations
and $\delta\Phi$ denotes the quantum fluctuations of $\Phi$. As with any other
field, the mean fluctuations are null,
$\langle\delta\Phi\rangle=\langle\delta\dot{\Phi}\rangle=0$. Here we work in
the linearized theory, assuming the perturbations to be small
$\delta\Phi(x,t)$ so that higher order perturbations are discarded. It follows
from (42) and (43) that the classical and quantum parts for the inflaton can
be splitted into different regimes

$\displaystyle\ddot{\Phi}_{b}+3H_{b}\dot{\Phi}_{b}+U^{\prime}(\Phi_{b})=0$
(45)
$\displaystyle\delta\ddot{\Phi}+3H_{b}\delta\dot{\Phi}+\frac{1}{a^{2}}(\nabla^{2}\delta\Phi)+U^{\prime\prime}(\Phi_{b})\delta\Phi=0,$
(46)

where $H=\dot{a}/a$ is the Hubble parameter. On the classical side, besides
having the relation (45), we also have the following Friedmann equation

$H_{b}=\frac{8}{3}\pi G(\frac{1}{2}\dot{\Phi}^{2}_{b}+U(\Phi_{b})),$ (47)

we can take the time derivative of the previous equation and solve for
$\dot{\Phi}_{b}$, $\dot{\Phi}^{2}_{b}=-\dot{H}_{b}/(4\pi G)$, also observe
that since $\dot{H}_{b}=H^{\prime}_{b}\dot{\Phi}_{b}$, then
$\dot{\Phi}_{b}=-H^{\prime}_{b}/(4\pi G)$. Now, under the slow roll condition,
we have $|\dot{\Phi}_{b}|\ll|U(\Phi_{b})|$, and therefore $\dot{\Phi}_{b}$
becomes

$\dot{\Phi}_{b}=-\sqrt{\frac{\lambda}{3\pi G}}\Phi_{b}.$ (48)

By integrating both sides of (48) we can obtain the explicit form for the
background field

$\Phi_{b}=\Phi_{be}e^{-\sqrt{\frac{\lambda}{3\pi G}}(t-t_{e})}$ (49)

where $\Phi_{be}=\Phi(t_{e})$ with $t_{e}$ denoting the time when inflation
ends. The previous expression for $\Phi_{b}x$ near the end of inflation, i.e.
$t\approx t_{e}$, can be approximated by a Taylor expansion as

$\Phi_{b}(t)\approx\Phi_{be}\Bigl{[}1+\frac{\lambda}{3\pi
G}(t_{e}-t)\Bigr{]}.$ (50)

From (47) and (49) the explicit form for $H(\Phi_{b}(t))$ reads

$H_{b}(t)=\sqrt{\frac{4\pi
G\lambda\sigma^{4}}{3}}\Bigl{[}\frac{\Phi_{be}^{2}}{\sigma^{2}}e^{-\sqrt{\frac{\lambda}{3\pi
G}}(t-t_{e})}-1\Bigr{]}.$ (51)

Then by substituting the Hubble parameter in terms of $H(a(t))$ we can solve
for the scale factor $a(t)$ right after performing an integration from $a$ to
$a_{e}$ and from $t$ to $t_{e}$ in order to obtain

$a=a_{e}e^{\pi G\sigma^{2}(1-e^{2\sqrt{\frac{\lambda}{3\pi
G}}(t-t_{e})})}e^{\sqrt{\frac{4\pi G\lambda\sigma^{4}}{3}}(t-t_{e})}.$ (52)

Thus, near the end of inflation the scale factor can be approximated by

$a(t)\approx a_{e}\exp{\sqrt{\frac{4\pi G\lambda\sigma^{4}}{3}}(t-t_{e})},$
(53)

leaving an approximated constant Hubble parameter at the end of this stage.

##  V  The inflaton fluctuations

Now, in the semi-classical approach we are able to study the inflaton quantum
fluctuations $\delta\Phi$. The dynamics of these fluctuations is determined by
(46). With this in mind, we adopt the standard canonical quantization
procedure, and we impose the canonical commutation relations at every Cauchy
surface. According to the action (35) the Lagrangian has the form

$L=\sqrt{-h}\Bigr{[}R+\frac{1}{2}\dot{\Phi}^{2}+\frac{1}{2a^{2}}(\nabla\Phi)^{2}-U(\Phi)\Bigr{]}.$
(54)

The canonical conjugate momentum is given by $\Pi=\frac{\partial
L}{\partial\dot{\Phi}}=\sqrt{h}\dot{\Phi}$. Therefore the commutation
relations are of the form

$[\delta\Phi(t,\vec{r}),\delta\dot{\Phi}(t,\vec{r^{\prime}})]=\frac{i}{\sqrt{-h}}\delta^{(3)}(\vec{r}-\vec{r^{\prime}}).$
(55)

Here we can introduce an auxiliary field $\delta\chi(t,\vec{r})$ such that

$\delta\Phi(t,\vec{r})=\exp\Bigl{[}-\frac{3}{2}H_{b}(t)dt\Bigr{]}\delta\chi(t,\vec{r}).$
(56)

In terms of the previous auxiliary field, the dynamical equation for the
fluctuations (46) becomes

$\delta\ddot{\chi}+\frac{1}{2a^{2}}\nabla^{2}\delta\chi-\Bigl{[}\frac{9}{4}H^{2}+\frac{3}{2}\dot{H}-U^{\prime\prime}(\Phi_{b})\Bigr{]}\delta\chi=0.$
(57)

Now, we can consider the Fourier expansion of $\delta\chi$ in terms of the
annihilation and creation operators as

$\delta\chi(t,\vec{r})=\frac{1}{(2\pi)^{3/2}}\int
d^{3}k\Bigl{[}\hat{a}_{k}e^{i\vec{k}\cdot\vec{r}}\xi_{k}(t)+\hat{a}^{{\dagger}}_{k}e^{-i\vec{k}\cdot\vec{r}}\xi^{*}_{k}(t)\Bigr{]}.$
(58)

With the help of (57) and (58) the dynamics of the modes $\xi_{k}$ is governed
by

$\ddot{\xi}_{k}+\Bigl{[}\frac{k^{2}}{a^{2}}-\frac{9}{4}H^{2}+\frac{3}{2}\dot{H}-U^{\prime\prime}(\Phi_{b})\Bigr{]}\xi_{k}=0.$
(59)

It follows from (50) that near the end of inflation
$U^{\prime\prime}(\Phi_{e})$ reads

$U^{\prime\prime}(\Phi_{e})=2\lambda(3\Phi_{e}^{2}-\sigma^{2}).$ (60)

Therefore the equation (59) near to the end of inflation is given by

$\ddot{\xi}_{k}+\Bigl{[}\frac{k^{2}}{a^{2}}-\frac{9}{4}H_{e}^{2}-2\lambda(3\Phi_{e}^{2}-\sigma^{2})\Bigr{]}\xi_{k}=0.$
(61)

Employing (56) and (58) the normalization condition results

$\dot{\xi}_{k}^{*}\xi_{k}-\dot{\xi}_{k}\xi_{k}^{*}=\frac{i}{a_{e}^{3}}.$ (62)

Hence, adopting the Bunch-Davies vacuum, the normalized solution of (61) can
be written as

$\displaystyle\xi_{k}(t)=\frac{1}{2}\sqrt{\frac{\pi}{\alpha
a_{e}^{3}}}\cdot\mathcal{H}_{\nu}^{(1)}[z(t)].$ (63)

where
$\nu=\frac{1}{2\alpha}\sqrt{9H_{e}^{2}+24\lambda\Phi_{e}^{2}-8\lambda\sigma^{2}}$
and $z(t)=\frac{k}{\alpha a_{e}}e^{-\alpha(t_{e}-t)}$, where
$\alpha=\sqrt{4\pi G\lambda\sigma^{4}}/\sqrt{3}$.

The explicit mode functions in the IR sector or superhorizon sector ($-kt<<1$)
are dictated by the Hankel function behavior on this regime, which is given by
$\mathcal{H}_{\nu}^{(1)}\approx\frac{i}{\pi}\Gamma(\nu)\Bigl{(}\frac{z(t)}{2}\Bigr{)}^{\nu}$.
Therefore the modes $\xi_{IR}$ have the form

$\xi_{IR}=\frac{1}{2}\sqrt{\frac{\pi}{\alpha
a_{e}^{3}}}\frac{i}{\pi}\Gamma(\nu)\Bigl{(}\frac{z(t)}{2}\Bigr{)}^{\nu}.$ (64)

The mean square fluctuations of the field $\langle 0|\delta\chi^{2}|0\rangle$
are given by

$\langle\delta\chi^{2}\rangle=\frac{1}{2\pi^{2}}\int^{\epsilon_{kh}}_{0}\frac{dk}{k}(\xi_{k}\xi_{k}^{*}),$
(65)

where $\epsilon_{kh}$ is the upper limit for the modes inside the horizon. In
terms of our original inflaton field $\delta\Phi$, the mean squared
fluctuations are given by

$\displaystyle\langle\delta\Phi^{2}\rangle=\frac{e^{\frac{3}{2}\int
Hdt}}{2\pi^{2}}\int^{\epsilon_{kh}}_{0}\frac{dk}{k}k^{3}(\xi_{k}\xi_{k}^{*})_{IR}.$
(66)

Now, with the help of (64) we arrive to

$\langle\delta\Phi^{2}\rangle=\Bigl{(}\frac{a}{a_{e}}\Bigr{)}^{3-2\nu}\frac{2^{2\nu}}{8\pi^{3}}\frac{\Gamma^{2}(\nu)}{(H_{e}a_{e})^{3}}\frac{1}{(H_{e}a_{e})^{-2\nu}}\int^{\epsilon_{kh}}_{0}\frac{dk}{k}k^{3-2\nu},$
(67)

Hence, it follows that the corresponding power spectrum reads

$\mathcal{P}_{s}(k)=\Bigl{(}\frac{a}{a_{e}}\Bigr{)}^{3-2\nu}\frac{2^{2\nu}}{8\pi^{3}}\frac{\Gamma^{2}(\nu)}{(H_{e}a_{e})^{3}}\frac{1}{(H_{e}a_{e})^{-2\nu}}\,k^{3-2\nu}.$
(68)

Now we are in position to determine the spectral index and the scalar to
tensor ratio associated to the inflaton fluctuations in this model.

###  V.1 The spectral index and the scalar to tensor ratio

Once we have obtained the spectral index for inflaton fluctuations we are in
position to calculate the scalar to tensor ratio r. In order to do so, we will
use the fact without the strong coupling limit condition, the geometrical
scalar fiel $\varphi$ is given by $\varphi=-\ln(1+\epsilon\phi^{2n})$. Thus,
considering that $\varphi=\varphi_{v}+Q$, with
$\varphi_{v}=\ln(1+\epsilon\sigma^{2n})^{-1}$, it is not difficult to verify
that with the help of (2) and (13) we arrive to

$\displaystyle U(Q)$ $\displaystyle=$
$\displaystyle\frac{\lambda}{2}\left[\frac{1}{\epsilon^{1/n}}(1+\epsilon\sigma^{2n})e^{-Q/n}-\sigma^{2}\right]^{2},$
(69) $\displaystyle\omega(Q)$ $\displaystyle=$
$\displaystyle\frac{(1+\epsilon\sigma^{2n})^{2}e^{-2Q}}{\epsilon^{1/n}}\left[(1+\epsilon\sigma^{2n})e^{-Q}-1\right]^{\frac{1-2n}{n}}.$
(70)

Now, using (37) we obtain

$\Phi=-\frac{2n}{\epsilon^{1/2n}}\left[(1+\epsilon\sigma^{2n})e^{-Q}-1\right]^{\frac{1}{2n}}$
(71)

In this manner, the potential (69) for $n=2$ can be written as

$U(\Phi)=\frac{\lambda}{2}\left[\left(\frac{\epsilon\Phi^{4}+b_{0}}{b_{0}\epsilon}\right)^{1/2}-\sigma^{2}\right]^{2},$
(72)

where $b_{0}=2^{8}$. The spectral index, is given by

$n_{s}=4-\frac{1}{\alpha}\sqrt{9H_{e}^{2}+24\lambda\Phi_{e}^{2}-8\lambda\sigma^{2}}.$
(73)

In terms of the number of e-foldings

$N(\Phi)=M_{p}^{-2}\int^{\Phi}_{\Phi_{e}}\frac{U(\Phi)}{U^{\prime}(\Phi)}\,d\Phi,$
(74)

where $M_{p}=(8\pi G)^{-1/2}$, the equation (73) is given approximately by

$n_{s}\simeq
4-\frac{1}{\alpha}\sqrt{9H_{e}^{2}+\frac{24\lambda\Theta^{2}}{N}-8\lambda\sigma^{2}},$
(75)

where

$\Theta=\frac{\sigma\sqrt{2b_{0}}}{4M_{p}\epsilon^{1/4}}.$ (76)

Hence, considering a typical value $H_{e}\simeq 10^{-5}M_{p}$ and $N=63$ we
obtain that for $\epsilon\in[1.6697\times 10^{-7},1.6435\times 10^{-7}]$ it is
achieved the observational bound $n_{s}=0.968\pm 0.006$ given by PLANCK 2018
cosmological constraints PRIC . In the figure [1] there is a plot showing the
increasing behavior of $n_{s}$ with respect to $N$ for $\epsilon=1.6435\times
10^{-7}$.

The scalar to tensor ratio defined by

$r=\frac{16M_{p}^{2}}{2}\left(\frac{U^{\prime}(\Phi)}{U(\Phi)}\right)^{2},$
(77)

results then

$r=\frac{128M_{p}^{2}\epsilon\Phi^{6}}{b_{0}(\epsilon\Phi^{4}+b_{0})\left(\sqrt{\frac{\epsilon\Phi^{4}+b_{0}}{\epsilon
b_{0}}}-\sigma^{2}\right)^{2}}.$ (78)

Hence, in terms of the number of e-foldings the tensor to scalar ratio can be
approximated by

$r\simeq\frac{64\,\sigma^{6}}{\sqrt{\epsilon}\,N(4\sigma^{4}+M_{p}^{4}N^{2})\left(\sqrt{\frac{4\sigma^{4}}{\epsilon\,M_{p}^{4}\,N^{2}}+1}-\sigma^{2}\right)^{2}}$
(79)

According to Tristan ; BPDATA1  observational data corresponding to PLANCK and
BICEP the scalar to tensor ratio has the improved limit $r<0.032$. The value
$r=0.032$ is achieved for $N=63$ when $\epsilon=2.730576184\times 10^{-10}$.
The same value for r can be obtained for $N=60$ when
$\epsilon=2.730573666\times 10^{-10}$. In general, for values of
$\epsilon>2.730576184\times 10^{-10}$ the observational limit $r<0.032$ is
achieved. In the figure [2] it is shown a plot exhibing the behavior of the
scalar to tensor ratio $r$ as a function of $N$ and $\epsilon$ where it can be
seen that $r$ is decreasing respect to $N$ and for values
$\epsilon=2.730573666\times 10^{-10}$ the values of $r<0.032$. The figure [3]
shows the decreacing behavior of $r$ as the e-foldings increase for
$\epsilon=2.730576184\times 10^{-10}$.

##  VI Final Comments

In this paper we have studied a Higgs-Palatini inflationary model with a power
law coupling. The Higgs scalar field is geometrized by means of the Palatini’s
principle which in this particular model ensures that the background geometry
is a non-Riemannian whose metric compatibility condition can be written in the
form of a Weyl-integrable one, where the Weyl geometrical scalar field is the
same scalar field that appears in the Higgs potential. This field has a
dynamics given by an action that is invariant under the $U(1)$ group of
symmetries, which is considered the physical group of symmetries of the
action. However, the Weyl-Integrable background geometry also imposes a
geometrical group of symmetries: the Weyl group of transformations. The
original action is not an scalar under the Weyl group so we propose a new
scalar action obeying this requirement but at the same maintaining the $U(1)$
symmetry. In the new action appears a gauge vector field that allows the
action to be an scalar under the Weyl group. Unfortunately, in general, the
$U(1)$ requirements on this gauge field are not compatible with the
geometrical conditions imposed by the Weyl group. However, in the strong
coupling limit of the Higgs field with gravity appears a compatibility between
the geometrical and physical groups of symmetries for the gauge field. This
compatibility is one of the main differences of this model with respect other
similar ones.

Even when we do not start with a scalar-tensor theory of gravity, the new
action invariant under the geometrical and physical groups of symmetries can
be seen as one of the scalar-tensor gravity type. Thus, in the Riemannian
limit of the theory, which corresponds to the Einstein’s frame in traditional
scalar-tensor theories of gravity, we have calculated the spectrum of inflaton
fluctuations, its spectral index and the tensor to scalar ratio. Considering a
typical value $H_{e}\simeq 10^{-5}M_{p}$ and $N=63$ we have obtained that for
$\epsilon\in[1.6697\times 10^{-7},1.6435\times 10^{-7}]$ it is achieved the
observational bound $n_{s}=0.968\pm 0.006$ given by PLANCK 2018 cosmological
constraints PRIC . In the figure [1] the plot shows the increasing behavior of
$n_{s}$ with respect to $N$ for $\epsilon=1.6435\times 10^{-7}$. In agreement
with the observational upper bound provided by using BICEP and PLANCK data:
$r<0.032$ Tristan ; BPDATA1 . In our case the value $r=0.032$ is achieved for
$N=63$ when $\epsilon=2.730576184\times 10^{-10}$. The same value for r can be
obtained for $N=60$ when $\epsilon=2.730573666\times 10^{-10}$. In general,
for values of $\epsilon>2.730576184\times 10^{-10}$ the observational limit
$r<0.032$ is achieved. In the figure [2] it is plotted the behavior of the
scalar to tensor ratio $r$ as a function of $N$ and $\epsilon$. It shows that
$r$ is decreasing respect to $N$ and for values $\epsilon=2.730573666\times
10^{-10}$ the values of $r<0.032$. Also for values of $\epsilon$ ranging in
the interval $[1.6697\times 10^{-7},1.6435\times 10^{-7}]$ the condition
$r<0.032$ is valid, given values of $r$ too small. The figure [3] shows the
decreacing behavior of $r$ as the e-foldings increase for
$\epsilon=2.730576184\times 10^{-10}$.

## Acknowledgements

J. E. Madriz-Aguilar, and Diego Allan Reyna acknowledge CONAHCYT México and
Centro Universitario de Ciencias Exactas e Ingenierias of Guadalajara
University for financial support.

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Figure 1: This plot shows the behavior of the spectral index $n_{s}$ as a
function of $N$ for $\epsilon=1.6435\times 10^{-7}$ and $H_{e}=10^{-5}M_{p}$.
It can be seen that $n$ is increasing respect to $N$ reaching the
observational values $n_{s}\simeq 1$. Figure 2: This plot shows the behavior
of the scalar to tensor ratio $r$ as a function of $N$ and $\epsilon$. It can
be seen that $r$ is decreasing respect to $N$ and for values
$\epsilon=2.730576184\times 10^{-10}$ the values $r<0.032$ are reached Figure
3: This plot shows the relation between the scalar to tensor ratio $r$ and the
umber of e-foldings $N$ for $\epsilon=2.730576184\times 10^{-10}$.