# Gravitational waves during Higgs inflation from complex geometrical scalar-
tensor theory of gravity

1 José Edgar Madriz Aguilar, 2 A. Bernal, 2 F. Aceves de la Cruz , 1 J. A.
Licea 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><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

In this paper we investigate tensor fluctuations of the metric at the end of a
Higgs inflationary period in the context of a recently introduced complex
geometrical scalar-tensor theory of gravity. In our model the Higgs field has
a geometrical origin and the affine connection is determined by the Palatini’s
principle. Additionally, we consider an extra contribution to the tensor-
fluctuations equation coming from the vacuum term in the energy momentum
tensor associated to the Higgs field. The Higgs potential is rescaled by the
non-canonicity function of the kinetic term of the field which is modified by
the symmetry group of the background geometry. We obtain a nearly scale
invariant spectrum and a scalar to tensor ratio in agreement with PLANCK 2018
cosmological results.

###### pacs:

04.50. Kd, 04.20.Jb, 02.40k, 98.80k, 98.80.Jk, 04.30w

Weyl-Integrable geometry, Higgs scalar field, geometrical scalar-tensor
gravity, inflation, gravitational waves.

##  I INTRODUCTION

Geometrical scalar tensor theories of gravity is an approach of scalar-tensor
theories of gravity that arises as an attempt to obtain an 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, and so the original action is not transforming
as an scalar under this extended group. Thus in this geometrical approach a
new action is proposed in order to be a scalar under the group of symmetries
of the background geometry II1 . A previous approach in which the Palatini’s
principle has been incorporated in scalar-tensor theories of gravity can be
found for example in CR1 ; CR2 . To achieve the invariance of the action under
the new group of geometrical symmetries it is incorporated a gauge vector
field in the covariant derivative that in certain scenarios can be identified
with the electromagnetic potential. Several topics have been investigated in
the framework of this approach, like for example, Higgs inflation, the
formation of the seeds for cosmic magnetic fields and Dark energy scenarios
II1 ; II2 ; II3 . Moreover, this geometrical approach can also be extended to
a new formulation of $f(R)$ theories obtained by broken the Weyl gauge
symmetry imposed by the background geometry. Cosmological backreaction
consequences and CMB imprints of the new contributions have been also
investigated II4 .

Inflationary models can be considered as a solution to the problems of the Big
Bang cosmology, whose main predictions have been verified by the acoustic
peaks of CMB primordial temperature anisotropies II5 . An important prediction
of inflationary models is the relic background of gravitational waves (GW).
Formally these GW are described as tensor perturbations of the metric
generated by the primordial density perturbations during inflation. The fact
that LIGO reported the detection of gravitational waves sourced by
astrophysical objects has motivated the search of primordial GW coming from
inflation. Several experiments for the detection of GW are in order, for
example, the Laser Interferometer Antenna (LISA) EX1 ; EX2  and the DECI-hertz
Interferometer Gravitational Wave Observatory (DECIGO) EX3 ; EX4 . Among the
zoo of inflationary models, Higgs inflationary models have attracted the
interest of cosmologists because the Higgs is the unique scalar particle that
has been detected. Minimal coupling Higgs inflationary models, in the context
of general relativity, have the problem of reproducing the amplitude of
density perturbations consistently with observational data, which in general
will depend on the quartic coupling of the Higgs potential. One way to sort
the problem is the proposal of non-minimally coupled models. However, those
models are not free of problems. For example, the tree-level unitarity problem
that appears when radiative corrections in the standard effective Higgs
potential are regarded HIN1 . The unitarity limit during inflation gives a
different energy scale than the one in which the electroweak vacuum tree-level
unitarity is violated, which motivates an ultraviolet extension of the model
that can leave to problems with the amplitude of primordial density
perturbations HIN2 ; HIN3 .

One characteristic of non-minimal coupling models of Higgs inflation is that
they work in two frames: the Jordan and the Einstein frames. The pass from the
Jordan to the Einstein frames is implemented by means of a conformal
transformation of the metric of the form
$\bar{g}_{\alpha\beta}=\Omega(h)g_{\alpha\beta}$, with $h$ being the physical
Higgs field and $g_{\alpha\beta}$ the metric in the Jordan frame. The
background geometry is assumed riemannian and therefore it holds that
$\nabla_{\mu}g_{\alpha\beta}=0$, which is the metricity condition for this
kind of geometry. However, due to the conformal transformation of the metric
it is not difficult to verify that in the Einstein frame
$\nabla_{\mu}\bar{g}_{\alpha\beta}\neq 0$. So in this frame the background
geometry is no longer riemannian. This fact, has not been considered in the
majority of non-minimal coupled Higgs inflationary models, where in spite of
taking the conformal transformation of the metric, the riemannian is still
being considered as the background geometry in the Einstein frame. But this is
in fact considered a very important issue in the geometrical scalar-tensor
theories of gravity. The appearance of a gauge vector field that can play the
role of an electromegnetic potential and the energy rescaling of the Higgs
potential suggested by the symmetry group of the background geometry are
examples of consequences of considering the change of the background geometry
when passing from one frame to another.

In this paper, in the framework of complex geometrical scalar-tensor theories
of gravity, we study primordial gravitational waves generated during a Higgs
inflationary stage. To achieve our goal the paper is organized as follows.
Section I is left for a brief introduction. Section II is devoted to the
construction of the invariant action of the model in the context of
geometrical scalar-tensor theories of gravity. In section III we derive the
field equations of the particular Higgs inflation model. In section IV we
study the tensor fluctuations of the metric in order to obtain the powwer
spectrum and the scalar to tensor ratio for primordial gravitational waves at
the end of inflation. Finally, section V is left for some conclusions.

##  II The action in the complex geometrical scalar-tensor theory

Let us start with a traditional complex scalar-tensor theory, whose action can
be written as II1 ; II2

${\cal S}=\int
d^{4}x\,\sqrt{-g}\,e^{-(\Phi+\Phi^{\dagger})}\left[\frac{M_{p}^{2}}{2}R+\Omega(\Phi+\Phi^{\dagger})\,\Phi^{,\mu}\Phi_{,\mu}^{\dagger}-V(\Phi+\Phi^{\dagger})\right],$
(1)

where $g$ denotes the determinant of the metric, $R$ is the Ricci scalar
curvature, $\Omega((\Phi+\Phi^{\dagger})$ is a well-behaved differentiable
function, the dagger $\dagger$ is denoting transposed complex conjugate,
$V(\Phi+\Phi^{\dagger})$ is the scalar potential and $M_{p}=(8\pi G)^{-1/2}$
is the reduced Planckian Mass. Adopting the Palatini’s variational principle
the backgroun geometry associated to (1) is one of the Weyl-Integrable type
characterized by the compatibility condition:
$\nabla_{\alpha}g_{\mu\nu}=(\Phi+\Phi^{\dagger})_{,\alpha}\,g_{\mu\nu}$ with
$\nabla_{\sigma}$ denoting the Weyl covariant derivative. The geometrical
symmetry group that leaves invariant this condition is the Weyl group of
transformations

$\displaystyle\bar{g}_{\lambda\sigma}$ $\displaystyle=$ $\displaystyle
e^{f+f^{\dagger}}g_{\lambda\sigma},$ (2) $\displaystyle\bar{\Phi}$
$\displaystyle=$ $\displaystyle\Phi+f,$ (3)

being $f(x^{\gamma})$ a well behaved complex function of the space-time
coordinates. As the action (1) is not an invariant under the Weyl group, an
invariant action results to be II1 ; II2

${\cal
S}_{inv}=\int\,d^{4}x\,\sqrt{-g}\,e^{-(\Phi+\Phi^{\dagger})}\left[\frac{M_{p}^{2}}{2}R+\Omega(\Phi+\Phi^{\dagger})\,\Phi^{:\mu}\Phi_{:\mu}^{\dagger}-e^{-(\Phi+\Phi^{\dagger})}V(\Phi+\Phi^{\dagger})-\frac{1}{4}e^{(\Phi+\Phi^{\dagger})}H_{\mu\nu}H^{\mu\nu}\right],$
(4)

where it was introduced the gauge covariant derivative
$\Phi_{:\alpha}=\nabla\Phi+i\epsilon B_{\alpha}\Phi$, with $B_{\alpha}$ being
a gauge field well-defined in the space-time manifold, $\epsilon$ is a
coupling constant and $H_{\mu\nu}=(\Phi B_{\nu})_{,\mu}-(\Phi B_{\mu})_{,\nu}$
is a field strength tensor. This is the action of a geometrical scalar-tensor
theory of gravity that is different from the traditional action (1). The
invariance of (4) is achieved only when the next transformations are valid

$\displaystyle\bar{\Phi}\bar{B}_{\lambda}$ $\displaystyle=$ $\displaystyle\Phi
B_{\lambda}+i\epsilon^{-1}f_{,\lambda},$ (5)
$\displaystyle\bar{\Phi}^{\dagger}\bar{B}_{\lambda}$ $\displaystyle=$
$\displaystyle\Phi^{\dagger}B_{\lambda}-i\epsilon^{-1}f^{\dagger}_{,\lambda},$
(6) $\displaystyle\bar{\Omega}(\bar{\Phi}+\bar{\Phi}^{\dagger})$
$\displaystyle\equiv$
$\displaystyle\Omega(\bar{\Phi}+\bar{\Phi}^{\dagger}-f-f^{\dagger})=\Omega(\Phi+\Phi^{\dagger}),$
(7) $\displaystyle\bar{V}(\bar{\Phi}+\bar{\Phi}^{\dagger})$
$\displaystyle\equiv$ $\displaystyle
V(\bar{\Phi}+\bar{\Phi}^{\dagger}-f-f^{\dagger})=V(\Phi+\Phi^{\dagger}).$ (8)

In terms of the Weyl invariant metric:
$\gamma_{\alpha\beta}=e^{-(\Phi+\Phi^{\dagger})}g_{\alpha\beta}$ and the new
fields

$\displaystyle\varphi$ $\displaystyle=$ $\displaystyle\sqrt{\xi}\,e^{-\Phi},$
(9) $\displaystyle A_{\mu}$ $\displaystyle=$ $\displaystyle
B_{\mu}\ln\left(\frac{\varphi}{\sqrt{\xi}}\right),$ (10)

the action (4) can be put in the form II1 ; II2

${\cal S}_{inv}=\int d^{4}x\,\sqrt{-\gamma}\left[\frac{M_{p}^{2}}{2}{\cal
R}+\frac{1}{2}\omega(\varphi\varphi^{\dagger})D^{\mu}\varphi(D_{\mu}\varphi)^{\dagger}-\hat{V}(\varphi\varphi^{\dagger})-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\right],$
(11)

where $D_{\lambda}\varphi=\,\\!^{(R)}\nabla_{\lambda}\varphi+i\epsilon
A_{\lambda}\varphi$ is an effective Riemannian gauge covariant derivative,
$F_{\alpha\beta}=A_{\beta,\alpha}-A_{\alpha,\beta}=-H_{\alpha\beta}$ is the
Faraday tensor, ${\cal R}$ is the Riemannian Ricci scalar and where the next
relations are valid

$\displaystyle\Phi_{:\sigma}$ $\displaystyle=$
$\displaystyle-\frac{1}{\varphi}D_{\sigma}\varphi,$ (12)
$\displaystyle\omega(\varphi\varphi^{\dagger})$ $\displaystyle=$
$\displaystyle\frac{2\,\Omega\left[\ln(\varphi\varphi^{\dagger}/\xi)\right]}{\varphi\varphi^{\dagger}},$
(13) $\displaystyle\hat{V}(\varphi\varphi^{\dagger})$ $\displaystyle=$
$\displaystyle V\left(\ln\frac{\varphi\varphi^{\dagger}}{\xi}\right),$ (14)

with $\xi$ being a constant parameter introduced in order to the field
$\varphi$ has the correct physical units.

##  III The field equations of the Higgs inflationary model

In order to propose a Higgs inflationary model we start considering the Higgs
potential

$V(\Phi\Phi^{\dagger})=\frac{\lambda}{4}\left(\Phi\Phi^{\dagger}-\sigma^{2}\right)^{2},$
(15)

with $\lambda=0.129$ and the vacuum expectation value $\sigma=246\,GeV$ III1 ;
III2 . In terms of the field $\varphi$ the expression (15) reads

$V(\varphi\varphi^{\dagger})=\frac{\lambda}{4}\left(\frac{\varphi\varphi^{\dagger}}{\xi}-\sigma^{2}\right)^{2}.$
(16)

On the other hand, the action (11) has a Riemannian background geometry and
thus the fields $\varphi$ and $A_{\mu}$ respect the gauge transformations

$\displaystyle\bar{\varphi}=\varphi\,e^{i\epsilon\theta(x)},$ (17)
$\displaystyle\bar{A}_{\nu}=A_{\nu}-\theta_{,\mu},$ (18)

where $\theta(x)$ is a well-behaved gauge function. In this manner, breaking
the symetry by taking $\varphi=\varphi^{\dagger}$ and with the gauge election
$\theta_{,\nu}=A_{\nu}$, the action (11) acquires the form

${\cal S}=\int d^{4}x\sqrt{-\gamma}\,\left[\frac{M_{p}^{2}}{2}{\cal
R}+\frac{1}{2}\omega_{eff}({\cal H}){\cal H}^{,\mu}{\cal
H}_{,\mu}-V_{eff}({\cal H})\right],$ (19)

where ${\cal H}$ is the Higgs field that obeys:
$\varphi(x^{\alpha})=\sqrt{\xi}\,\sigma+{\cal H}(x^{\alpha})$, and
$V_{eff}({\cal H})=V[\sqrt{\xi}\,\sigma+{\cal H}(x^{\alpha})]$. Unitarizing
the kinetic term in (19) we arrive to

${\cal S}=\int d^{4}x\sqrt{-\gamma}\,\left[\frac{M_{p}^{2}}{2}{\cal
R}+\frac{1}{2}\phi^{,\mu}\phi_{,\mu}-U(\phi)\right],$ (20)

where

$\displaystyle\phi(x^{\alpha})$ $\displaystyle=$
$\displaystyle\int\sqrt{\omega_{eff}({\cal H})}\,d{\cal H},$ (21)
$\displaystyle U(\phi)$ $\displaystyle=$ $\displaystyle V_{eff}({\cal
H}(\phi))=\frac{\lambda}{4}\left[\frac{(\sqrt{\xi}\,\sigma+{\cal
H}(\phi))^{2}}{\xi}-\sigma^{2}\right]^{2}.$ (22)

The field equations resulting from (20) are then

$\displaystyle{\cal R}_{\mu\nu}-\frac{1}{2}{\cal R}\,\gamma_{\mu\nu}$
$\displaystyle=$ $\displaystyle M_{p}^{-2}T_{\mu\nu},$ (23)
$\displaystyle\Box\phi+U^{\prime}(\phi)$ $\displaystyle=$ $\displaystyle 0,$
(24)

where the energy-momentum tensor for the scalar field $\phi$ is given by

$T_{\mu\nu}=\phi_{,\mu}\phi_{,\nu}-\frac{1}{2}\gamma_{\mu\nu}\left(\phi^{,\alpha}\phi_{,\alpha}-2U(\phi)\right),$
(25)

and $\Box$ is denoting the D’Alambertian operator.

Now, in order to allow the potential (22) to show a plateu for large enough
field values, suitable to describe a period of inflation, we consider the
anzats

$\omega_{eff}({\cal H})=\frac{1}{\left[1-\beta^{2}(\sqrt{\xi}\,\sigma+{\cal
H}^{4})\right]^{5/2}},$ (26)

with $\beta$ being a constant parameter with $M_{p}^{-2}$ units. By using
(21), the equation (26) implies a relation between the inflaton field $\phi$
and the Higgs field ${\cal H}$ of the form

$\phi=\frac{\sqrt{\xi}\,\sigma+{\cal H}}{[1-\beta^{2}(\sqrt{\xi}\sigma+{\cal
H})^{4}]^{1/4}}.$ (27)

Thus, the inflationary potential (22) reads

$U(\phi)=\frac{\lambda}{4\xi^{2}}\left(\frac{\phi^{4}}{1+\beta^{2}\phi^{4}}\right).$
(28)

A similar potential is obtianed for example in III3 . Once inflation starts it
is verified that $\beta^{2}\phi^{4}\ll 1$ and hence the potential can be
approximated by $U(\phi)\simeq(\lambda/4\xi^{2})\phi^{4}$.

##  IV Tensor fluctuations of the metric

The background of gravitational waves generated at the end of inflation are
due to sourceless tensor fluctuations of the metric. However, the energy-
momentum tensor (25) can be formally decomposed as a pressureless material and
a vacuum parts IV1 . The vacuum component is given by

$T_{\mu\nu}^{(vac)}=\left(U(\phi)-\frac{1}{2}\phi^{,\alpha}\phi_{,\alpha}\right)\gamma_{\mu\nu}.$
(29)

The perturbed line element has the form

$ds^{2}=dt^{2}-a^{2}(t)\left(\delta_{ij}+h_{ij}\right)dx^{i}dx^{j}\quad,$ (30)

where the tensor fluctuations of the metric are describe by
$h_{ij}(x^{\alpha})$, that satisfies $tr(h_{ij})=0$ and $h_{ij}^{\,,i}=0$.
Hence, it follows from (23) that tensor fluctuations of the metric obey the
dynamical equation

$\delta{\cal R}_{\mu\nu}-\frac{1}{2}{\cal
R}^{(b)}\delta\gamma_{\mu\nu}-\frac{1}{2}\delta{\cal
R}\,\gamma_{\mu\nu}^{(b)}=M_{p}^{-2}\,T_{\mu\nu}^{(vac)}(\phi_{b}),$ (31)

where we have employed a semiclassical aproximation for the inflaton field
that reads

$\phi(x^{\lambda})=\phi_{b}(t)+\delta\phi(x^{\lambda}),$ (32)

where the espectation values $<\phi>=\phi_{b}$ and $<\dot{\delta\phi}>=0$,
being $\phi_{b}(t)$ the background inflaton field defined on cosmological
scales and $\delta\phi$ describing the quantum fluctuations of the inflaton on
small scales. Thus $\gamma_{\mu\nu}^{(b)}$ is the background metric, ${\cal
R}^{(b)}$ accounts for the Ricci scalar evaluated on the background metric and
$\delta{\cal R}$ represents the fluctuations of the Ricci scalar generated by
the perturbed metric in (30).

Thus, with the use of (30) the energy density for the vacuum scalar field
coming from (29) results

$\rho_{vac}=-\left[\frac{1}{2}\dot{\phi}_{b}^{2}-U(\phi_{b})\right].$ (33)

In order to obtain a positive $\rho_{vac}$ necessarily
$U(\phi_{b})>(1/2)\dot{\phi}_{b}^{2}$, so the slow-roll condition on the
inflaton field must be valid. The Ricci scalar has no contributions of the
first order of tensor metric fluctuations and its background value is given by

${\cal R}^{(b)}=-6\left(\dot{H}+2H^{2}\right),$ (34)

with $H(t)$ being the Hubble parameter. Thus, in the traceless-transverse (TT)
gauge and in the slow-roll regime, it follows from (31) that the dynamics of
the tensor modes is given by the linearized equations

$\delta{\cal R}_{ij}-\frac{1}{2}{\cal
R}^{(b)}\,\delta\gamma_{ij}=M_{p}^{-2}\,U(\phi_{b})\,\delta\gamma_{ij}.$ (35)

With the help of (30) the expression (35) reduces to

$\ddot{h}^{i}_{j}+3H\dot{h}^{i}_{j}-\frac{1}{a^{2}}\nabla^{2}h^{i}_{j}-2(2\dot{H}+3H^{2})h^{i}_{j}+\frac{2}{M_{p}^{2}}U(\phi_{b})h^{i}_{j}=0.$
(36)

On the other hand, it follows from (23) and (24) that the background dynamics
is given by

$\displaystyle 3H^{2}=M_{p}^{-2}\,U(\phi_{b}),$ (37)
$\displaystyle\ddot{\phi}_{b}+3H\dot{\phi}_{b}+U^{\prime}(\phi_{b})=0.$ (38)

By using (28) for $\beta^{2}\phi_{b}^{4}\ll 1$ in (37) we obtain a scale
factor of the form II1

$a(t)=a_{e}\exp\left[\frac{\phi_{e}^{2}}{8M_{p}^{2}}\left(1-\exp\left(4M_{p}\sqrt{\frac{\lambda}{3\xi^{2}}}(t_{e}-t)\right)\right)\right],$
(39)

which at the end of inflation becomes

$a(t)\simeq
a_{e}\exp\left(-\frac{\phi_{e}^{2}}{2M_{p}}\sqrt{\frac{\lambda}{3\xi^{2}}}t_{e}\right)\exp\left(\frac{\phi_{e}^{2}}{2M_{p}}\sqrt{\frac{\lambda}{3\xi^{2}}}t\right),$
(40)

where $t_{e}$ denotes the time at the end of inflation, $a_{e}=a(t_{e})$ and
$\phi_{e}=\phi_{b}(t_{e})$. Following the canonical quantization procedure we
implement the Fourier expansion

$h^{i}_{j}(t,\bar{r})=\frac{e^{-\frac{3}{2}\int H(t)dt}}{(2\pi)^{3/2}}\int
d^{3}k\,\sum_{\alpha=+,\times}\,\\!^{(\alpha)}e^{i}_{j}\left[a_{k}^{(\alpha)}e^{i\bar{k}\cdot\bar{r}}\xi_{k}(t)+a_{k}^{(\alpha)\,\dagger}e^{-i\bar{k}\cdot\bar{r}}\xi_{k}^{*}(t)\right]$
(41)

with the creation $a_{k}^{(\alpha)\,\dagger}$ and annihilation
$a_{k}^{(\alpha)}$ operators obeying the algebra

$\displaystyle\left[a_{k}^{(\alpha)},a_{k^{\prime}}^{(\alpha^{\prime})\,\dagger}\right]=\gamma^{\alpha\alpha^{\prime}}\delta^{(3)}\left(\bar{k}-\bar{k}^{\prime}\right),$
(42)
$\displaystyle\left[a_{k}^{(\alpha)},a_{k^{\prime}}^{(\alpha^{\prime})}\right]=\left[a_{k}^{(\alpha)\,\dagger},a_{k^{\prime}}^{(\alpha^{\prime})\,\dagger}\right]=0,$
(43)

and where the polarization tensor $e_{ij}$ satisfies the properties

${}^{(\alpha)e_{ij}}=\,\\!^{(\alpha)}e_{ji},\quad
k^{i}\,\\!{}^{(\alpha)}e_{ij}=0,$ (45)
${}^{(\alpha)}e_{ii}=0,\quad^{(\alpha)}e_{ij}(-\bar{k})=\,\\!^{(\alpha)}e_{ij}^{*}(\bar{k}).$

Now, following the canonical quantization procedure we impose the commutation
relation

$\left[h^{i}_{j}(t,\bar{r}),\Pi_{i}^{i}(t,\bar{r}^{\prime})\right]=i\delta^{(3)}\left(\bar{r}-\bar{r}^{\prime}\right),$
(46)

where $\Pi_{ij}=\partial L/\partial\dot{h}^{ij}$ is the canonical conjugate
momentum. The lagrangian for gravitational tensor modes has the form

$L=\frac{M_{p}^{2}a^{3}}{8}\left[\dot{h}_{ij}^{2}-\frac{1}{a^{2}}h_{ij,l}h^{ij,l}+\left(2(2\dot{H}+3H^{2})-\frac{2}{M_{p}^{2}}\right)h_{ij}h^{ij}\right].$
(47)

Thus (47) reduces to

$\left[h_{j}^{i}(t,\bar{r}),\dot{h}^{i}_{j}(t,\bar{r}^{\prime})\right]=\frac{4i}{a^{3}M_{p}^{2}}\delta^{(3)}(\bar{r}-\bar{r}^{\prime}).$
(48)

Now, inserting (41) in (48) we obtain

$\xi_{k}\dot{\xi}^{*}-\xi_{k}^{*}\dot{\xi}_{k}=\frac{4i}{M_{p}^{2}a_{o}^{3}},$
(49)

which is the normalization condition for the modes. With the help of (36) and
(41) the modes at the end of inflation are governed by the dynamical equation

$\ddot{\xi_{k}}+\left[\frac{{k^{2}}}{a_{e}^{2}e^{-2H_{e}t_{e}}}e^{-2H_{e}t}-\frac{33}{4}H_{e}^{2}+\frac{2}{M_{p}^{2}}U_{e}\right]\xi_{k}=0,$
(50)

where

$U_{e}=\frac{\lambda}{4\xi^{2}}\left(\frac{\phi_{e}^{4}}{1+\beta^{2}\phi_{e}^{2}}\right),$
(51)

and where we have employed (40), with

$H_{e}=\frac{\phi_{e}^{2}}{2M_{p}}\sqrt{\frac{\lambda}{3\xi^{2}}}\,.$ (52)

By means of (49) and considering the Bunch-Davies vacuum, the normalized
solution of (50) is given by

$\xi_{k}(t)=\frac{1}{M_{p}}\sqrt{\frac{\pi}{\tilde{a_{e}}^{3}H_{e}}}\mathcal{H}_{\nu}^{(2)}\left[Z(t)\right],$
(53)

where ${\mathcal{H}}_{\nu}^{(2)}[Z(t)]$ denotes the second kind Hankel
function and

$\displaystyle\nu$ $\displaystyle=$
$\displaystyle\frac{1}{H_{e}}\sqrt{\frac{33}{4}H_{e}^{2}-\frac{2U_{e}}{M_{p}^{2}}},$
(54) $\displaystyle Z(t)$ $\displaystyle=$
$\displaystyle\frac{k}{\tilde{a_{e}}H_{e}}e^{-H_{e}t},$ (55)

with $\tilde{a}_{e}=a_{e}\exp\left(-H_{e}t_{e}\right)$.

In this manner, the amplitude of gravitational waves defined by
$<h^{2}>_{IR}=<0|h^{i}_{j}h_{i}^{j}|0>$, on the IR-sector, i.e. on
cosmological scales, is given by

$\langle h^{2}\rangle_{IR}=\frac{e^{-\int 3Hdt}}{2\pi^{2}}\int_{0}^{\epsilon
k_{H}}\frac{dk}{k}k^{3}\left(\xi_{k}\xi_{k}^{*}\right)\Bigr{|}_{IR},$ (56)

where $\epsilon=k_{max}^{IR}/K_{p}\ll 1$ is a dimensionless parameter, being
$k_{max}^{IR}=k_{H}(t_{r})$ the wave number associated with the Hubble radius
at the time $t_{r}$ when the modes re-enter to the horizon near the end of
inflation, $k_{p}$ is the Planckian wave number. For example, $\epsilon$
varies from $10^{-5}$ to $10^{-8}$ for a typical Hubble parameter during
inflation of the order $H\simeq 0.5\times 10^{-9}\,M_{p}$, which corresponds
to a number of e-foldings $N\simeq 63$.

Now, on cosmological scales and at the end of the inflationary period we can
employ the IR-asymptotic aproximation formula

${\cal
H}^{(1)}_{\nu}[Z]\simeq\frac{i}{\pi}\Gamma(\nu)\left(\frac{Z}{2}\right)^{-\nu}.$
(57)

Hence it follows from (53) to (56) that

$\langle
h^{2}\rangle_{IR}=\frac{2^{2\nu}}{\pi^{3}}\frac{\Gamma^{2}(\nu)}{M_{p}^{2}}\frac{H_{e}^{2}}{(\tilde{a_{e}}H_{e})^{3-2\nu}}e^{(2\nu-3)H_{e}t}\int_{0}^{\epsilon
k_{H}}\frac{dk}{k}k^{3-2\nu},$ (58)

where according to the modes equation (50) the wave number associated to the
horizon is given by

$k_{H}=\tilde{a_{e}}\sqrt{\frac{11}{2}\dot{H}+\frac{33}{4}H_{e}^{2}+\frac{2}{M_{p}^{2}}U_{e}}.$
(59)

In this manner we obtain a power spectrum derived from (58) of the form

$P_{h}(k)=\frac{2^{2\nu+2}}{\pi}\frac{\Gamma^{2}(\nu)}{M_{p}^{2}}\left(\frac{H_{e}}{2\pi}\right)^{2}e^{(2\nu-3)H_{e}t}\left(\frac{k}{\tilde{a_{e}}H_{e}}\right)^{3-2\nu}.$
(60)

It is not difficult to verify that a nearly scale invariant power spectrum of
the Harrison Zeldovick type can be achieved from (54) and (60) when
$U_{e}\simeq 3M_{p}^{2}H_{e}^{2}$. In this particular case the formula (60)
reduces to

$P_{h}(k)|_{\nu\simeq
3/2}\simeq\frac{2^{3}}{M_{p}^{2}}\left(\frac{H_{e}}{2\pi}\right)^{2}.$ (61)

The spectral index is given by

$n_{s}=4-\frac{2}{H_{e}}\sqrt{\frac{33}{4}H_{e}^{2}-\frac{2U_{e}}{M_{p}^{2}}}.$
(62)

The Planck 2018 results indicate that for the spectral index the limits are:
$n_{s}=0.9649\pm 0.0042$ PRIC . In the figure [1a] we show the behavior of the
spectral index $n_{s}$ versus $U_{e}$. It can be seen that the observational
values for $n_{s}$ can be obtained for $U_{e}\in\left[7.4258\times
10^{-19},7.4417\times^{-19}\right]M_{p}^{4}$.

On the other hand, as it was shown in II1 , the power spectrum associated to
the quantum fluctuations of the inflaton is given by

$P_{\delta\phi}(k)=\frac{2^{2\nu-1}}{\pi}\Gamma^{2}(\nu)\left(\frac{H_{e}}{2\pi}\right)^{2}e^{(2\nu-3)H_{e}t_{e}}\left(\frac{k}{\tilde{a_{e}}H_{e}}\right)^{3-2\nu}.$
(63)

Thus the power spectrum for curvature perturbations
$P_{\mathcal{R}}(k)=\frac{1}{2\epsilon}\frac{P_{\delta\phi}}{M_{p}^{2}}$
results

$P_{\mathcal{R}}(k)=\frac{2^{2\nu}}{4\pi\epsilon}\frac{\Gamma^{2}(\nu)}{M_{p}^{2}}\left(\frac{He}{2\pi}\right)^{2}e^{(2\nu-3)H_{e}t}\left(\frac{k}{\tilde{a_{e}}H_{e}}\right)^{3-2\nu},$
(64)

where we have employed de slow-roll parameter
$\epsilon=\frac{M_{p}^{2}}{2}\left(\frac{U^{\prime}}{U}\right)^{2}$. Hence the
scalar to tensor ratio $r=\frac{P_{h}}{P_{\mathcal{R}}}$ in terms of the
background inflaton field has the form

$r=\frac{128M_{p}^{2}}{\phi_{b}^{2}(1+\beta^{2}\phi_{b}^{4})^{2}}.$ (65)

According to the Planck 2018 results $r<0.056$ PRIC . In the figure [1b] we
show a plot of $r$ vs $\phi_{b}$ for different values for $\beta$. In general
depending of the $\beta$ parameter the observational range of values for $r$
is achieved for an interval of values for $\phi_{b}$. For example, when
$\beta=40\,M_{p}^{-2}$ the scalar to tensor ratio results $r=0.04$ when
$\phi_{b}=0.511\,M_{p}$. Thus for this value of $\beta$ the observational
range $r<0.056$ is achived when $\phi_{b}>0.4945\,M_{p}$. Hence, as it is
shown in figure [1b] for increasing values of $\beta$ the observational range
for $r$ is reached for decreasing values of $\phi_{b}$.

(a) This is a plot of the variation of $n_{s}$ with respect $U_{e}$.

(b) This is a plot exhibiting the behavior of $r$ versus $\phi_{b}$ for
different values of $\beta$.

Figure 1: These plots shows the behaviour of the spectral index $n_{s}$ and
the scalar to tensor ratio $r$, respectively. In figure (a) the observational
value for $n_{s}$ is achieved for $7.4258\times
10^{-19}M_{p}^{4}<U_{e}<7.4417\times^{-19}M_{p}^{4}$. In figure (b) when
$\beta$ increases the value of $\phi_{b}$ for which $r$ enters in the
observational range $r<0.056$, decreases.

##  V Conclusions

In this paper we have investigated the background relic of gravitational waves
generated during a Higgs inflationary stage in the context of a geometrical
scalar-tensor theory of gravity. In this model the Higgs scalar field has a
geometrical origin because the background geometry is of the Weyl-integrable
type where the Weyl scalar field is related to the Higgs field. The background
geometry is asigned by the Palatini’s variational principle. The physical
field equations for the inflaton Higgs scalar field are obtained by recasting
the original action of the theory in terms of the so called invariant action.

The primordial gravitational waves are described by tensor fluctuations of the
metric. In general these kind of fluctuations are considered sourceless. One
important difference with respect other approaches is that in our model we
have decomposed the energy momentum-tensor into a pressureless matter plus a
vacuum components, and hence the vacuum part has been considered in the
formulation of the dynamical equation that governs the tensor fluctuations of
the metric.

As a consequence of taking into account the symmetry group of the Weyl-
Integrable background geometry of the original action, the standard Higgs
potential is rescaled by means of a function that makes the kinetic term non-
canonical, which is determined by two factors: the Weyl group of symmetries
and the requirement that such function must create an enough plateu in the
effective potential to achieve an initial Hubble parameter of the order
$H_{0}\simeq 10^{11}-10^{12}$ GeV, which allows to have the enough quantity of
inflation in agreement with PLANCK data PRIC ; UL1 . Aditionally, the slow-
roll conditions that are typically imposed, here are obtained through the
requirement that the energy density of the background inflaton field to be
positive. Hence, the rescaled potential ends up depending on the parameter
$\beta$ (see eq.(28)). We obtain a power spectrum for gravitational waves
nearly scale invariant in agreement with PLANCK data for a range of values for
the effective potential at the end of inflation $U_{e}$ given for
$[7.4528\times 10^{-19},7.4417\times 10^{-19}]M_{p}^{4}$. The amplitude of the
spectrum results proportional to $(H/2\pi)^{2}$ with $H$ evaluated at the end
of inflation. The scalar to tensor ratio $r$ resulted to be depending on the
$\beta$ parameter. Thus for increasing values of $\beta$ the observational
values of $r$ ($r<0.056$ according to PLANCK data ) are reached for decreasing
values of $\phi_{b}$ (see figure [1b]). For example, for
$\beta=40\,M_{p}^{-2}$ we obtain $r=0.04$.

## Acknowledgements

J. E. Madriz-Aguilar, A. Bernal, F. Aceves and J. A. Licea acknowledge CONACYT
México and Centro Universitario de Ciencias Exactas e Ingenierias of
Guadalajara University for financial support.

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