2024

[1]Siamak Akhshabi

[1]Department of Physics, Faculty of Sciences, Golestan University, Gorgan,
IRAN

# The Role of Torsion in Trans-Planckian Physics

Elham Arabahmadi<EMAIL_ADDRESS><EMAIL_ADDRESS>*

###### Abstract

The torsion of spacetime, if exists, plays an important role at the very early
universe when the spin density of particles was very high. It is generally
believed that in extremely high energies of the early universe a new physics
called trans-Planckian physics should be considered. Since the initial
conditions for inflation are probably a result of this new physics, here we
consider spin and torsion as trans-Planckian effects and analyze their
influence on the power spectrum of scalar and tensor perturbations at the end
of the inflationary era.

###### keywords:

Torsion, Einstein-Cartan Theory, Inflation, Trans-Planckian Physics,
Cosmological Perturbations, Gravitational Waves

##  1 Introduction

Despite the success of general relativity and the standard model of cosmology
on which it is based, it seems that at the very high energies of the early
universe close to the big bang, the effects of a new physics, generally
thought to be the illusive theory of quantum gravity should be considered.
These effects are conveniently called trans-Planckian as they become very
important near the Planck scales. Although there is still no clear agreement
on the exact nature of this new trans-Planckian physics, it is expected that
it can find an answer to the important problem of compatibility of gravity
with quantum mechanics. The effects of this Trans-Planckian physics can change
the power spectrum of initial density fluctuations produced at the end of the
inflationary period, because the initial conditions at the beginning of that
period are probably affected by the Trans-planckian effects Niemeyer ; Hui ;
Shiu ; Shankar ; Kempf ; Easther2 ; Kaloper ; Brandenberger1 . On the other
hand, recent analysis of the current inflationary models seem to suggest that
the initial energy scale of inflation should be significantly lower than the
Planck scale BrandennVafa ; YongCai . More specifically, it has previously
been realized that if the inflationary phase lasts long enough to successfully
produce the cosmological structures observed today, the present-day length
scales should originate from modes that are smaller than the Planck length
during inflation Martin ; BrandennMartin ; Niemeyer1 . This is the so-called
trans-Planckian problem of the inflationary cosmology Chalmers ; Easther1 ;
Easther3 ; Bozza ; MartinnBranden . It was generally expected that a complete
theory of quantum gravity would avoid this problem; however, it has been
argued in BedroyanVafa  that in a consistent quantum theory of gravity, sub-
Planckian quantum fluctuations should remain quantum and never become larger
than the Hubble horizon. This is called the Trans-Planckian Censorship
Conjecture (TCC), which forces the trans-Planckian perturbation modes to be
hidden by the Hubble horizon. The combination of the trans-Planckian problem
and the TCC, puts strict bounds on some inflationary parameters in models with
single or multiple scalar fields, most specifically the number of e-folds,
energy density of the inflation, slow-roll parameters and the tensor-to-scalar
ratio BrandennVafa . There has been considerable interest in the literature
about the implications of these issues and their resolution, see, for example,
Shojai1 ; Kadota ; Brahma ; Tenkanen ; YongCai2 ; Torabian ; KamalinBranden ;
WilsonnBranden .

Most of the analysis mentioned above is based on the assumption that the
theory of general relativity (GR) and the standard model of cosmology, which
is based on GR can correctly describe the evolution of the universe, at least
from a few seconds after the big bang to the present day. However various
issues of the standard model of cosmology, such as singularity problem and the
nature of dark matter and dark energy, led researchers to consider gravity
theories that modify the general relativity in some way. These modified
gravity theories have attracted considerable interest in recent times and
various aspects of them have been studied extensively in the literature
Faraoni ; Nojiri1 ; Cappozz1 ; Clifton ; Nojiri2 . The issue of trans-
Planckian problem in modified gravities is also of great interest as these
theories could potentially modify the prediction and constraints of the
inflationary era. As an example, trans-Planckian corrections to the power
spectrum of scalar perturbations in $f(R)$ gravity has been recently studied
in Shojai2 . Among the modified theories of gravity, the ones containing
torsion are some of the most natural extensions of GR Blag1 ; Hayashi .
Torsion, which is coupled to the spin of matter, appears naturally in many
theories of quantum gravity Scherk ; Hehl1974 ; Hehl1976 ; Shapiro . In
particular, it has been shown that at low energies, the effective Lagrangian
of string theory is mathematically equivalent to a metric theory of gravity
that includes torsion Hammond1 ; Hammond2 . Additionally, the strength of the
field associated with the Caleb-Ramond field in string theory behaves as a
torsion field within the context of field geometry Kalb . Penrose also has
demonstrated that torsion naturally emerges when spinors are permitted to
undergo rescaling through a complex conformal factor Penrose .

Therefore, it seems that torsion effects, if they exists, are important in the
Trans-planckian era and their effects can be seen in the observations related
to the time close to the big bang, especially the inflation period. In this
article, considering the space-time torsion as a trans-planckian effect, we
investigate its influence on cosmic parameters, especially inflationary
parameters. To estimate the Trans-Planckian effects of torsion, we use the
Einstein-Cartan theory (ECT), which is the simplest generalization of general
relativity in the presence of torsion. In 1922, Cartan introduced his theory
to suggest that torsion represents the macroscopic expression of the intrinsic
angular momentum (spin) of matter Cartan . Several years later, the concept of
spin was reintroduced independently into general relativity by Kibble and
Sciama.Kibble . Subsequently, the EC theory has been studied extensively as it
became the subject of much research; (for example see Hehl1976 ; Trautman  and
references therein) as it serves as the most straightforward classical
extension of Einstein’s general relativity Blagojevic . Generalizations of
Einstein-Cartan theory also include the Poincaré gauge theory of gravity
Obukhov1  and the most general metric-affine theory of gravity Hehl1995 .
Indeed, there are several avenues for experimentally testing gravitational
theories that incorporate non-vanishing spacetime torsion (see for example
Hammond ; Mao ; Kostelecky ; March ; Hehl2013 ; Puetzfeld ; Lin ; Zamani1 ;
Pereira ; Akhshabi ; Ranjbar ). However, experimental or observational
evidence validating the unique predictions of theories with torsion or
confirming the existence of spacetime torsion remains elusive. The primary
reason for this absence of evidence lies in the fact that these theories
depart from classical general relativity only at exceptionally high energy
densities Kranas . Such extreme densities can only be found in extremely
compact objects like neutron stars and black holes, or during the initial
stages of the universe’s evolution, the latter of which we focus on this
article.

There are various ways to approach the trans-Planckian problem in inflationary
cosmology. One way, which we follow here is based on the choice of the vacuum
state and the time to impose the initial condition for inflation. If we had
been allowed to follow a given mode arbitrarily far back in time, we could
have argued for a unique vacuum, the Bunch-Davies vacuum, which also is the
vacuum typically used in calculations of the primordial spectrum. In the
presence of a fundamental scale, one cannot follow a given fluctuation mode to
scales smaller than the Planck length Danielsson1 ; Danielsson2 ; Danielsson3
. As a result, in order to incorporate the effects of trans-Planckian physics
which now include significantly large torsion and spin close to the initial
singularity, we assume that all effects due to unknown high energy quantum
gravity theory can be captured by a suitable choice of vacuum and then look
for signs of the new trans-Planckian physics in fluctuations of the cosmic
microwave background or in large scale structures Danielsson4 . This
assumption is particularly natural in an expanding universe, as has been
elaborated in Danielsson2 . In subsequent sections we use this procedure to
calculate various parameters at the end of the inflationary era. structure of
the paper is as follows: In section 2 we review the fundamental equations of
Einstein-Cartan theory of gravity and present the Friedmann equations used to
extract the cosmological dynamics in this theory. Section 3 is devoted to
studying the scalar perturbations generated at the end of inflation, taking
into account the effects of spin and torsion on their spectrum. In section 4,
by using the results of previous section, we examine the effects of the new
choice of vacuum in trans-Planckian physics to determine the modified power
spectrum in the presence of torsion. Section 5 briefly discusses the
generation and propagation of tensor perturbations (i.e. gravitational waves)
in Einstein-Cartan cosmology. Finally, section 6 is devoted to the discussion
and conclusions.

##  2 Einstein-Cartan theory of Gravity

In this section we mainly follow the conventions and notation of reference
mainEC . The Lagrangian in the Einstein-Cartan theory of gravity is assumed to
be the same as the Einstein-Hilbert action of general relativity.

$S=\frac{1}{2\kappa}\int d^{4}x\mathfrak{R}+\int d^{4}x\mathcal{L}_{m}$ (1)

Here $\mathfrak{R}$ is the Ricci scalar constructed with the full connection,
which now contains torsion and curvature. As a result, there are two
independent dynamical variables: the metric tensor as well as the new
independent torsion field (which can also be described by the independent part
of the spin connection), both of these fields contributes to the total
gravitational interactions. In the subsequent calculations, we assume the
metric signature to be (+,-,-,-). Torsion tensor is defined as the
antisymmetric part of the connection

$T_{\mu\nu}^{\alpha}=\Gamma_{[\mu\nu]}^{\alpha}=\frac{1}{2}(\Gamma_{\mu\nu}^{\alpha}-\Gamma_{\nu\mu}^{\alpha})$
(2)

We also defined the modified torsion tensor as

$Q\indices{{}_{\mu\nu}^{\alpha}}\equiv
T\indices{{}_{\mu\nu}^{\alpha}}+\delta_{\mu}^{\alpha}T\indices{{}_{\nu\lambda}^{\lambda}}-\delta_{\nu}^{\alpha}T\indices{{}_{\mu\lambda}^{\lambda}}$
(3)

In ECT, the connection can be divided into two parts

$\Gamma_{\mu\nu}^{\alpha}=\overset{\circ}{\Gamma}^{\alpha}_{\mu\nu}-K_{\mu\nu}^{\alpha}$
(4)

where $\overset{\circ}{\Gamma}\,^{\alpha}_{\mu\nu}$ is the Christoffel
connection and $K_{\mu\nu}\,^{\alpha}$ is called the contorsion tensor. The
relation between contorsion and torsion tensors can be expressed as

$K_{\mu\nu}\,^{\alpha}=-T_{\mu\nu}\,^{\alpha}+T_{\nu}\,^{\alpha}\,{}_{\mu}-T^{\alpha}\,_{\mu\nu}$
(5)

$T_{\mu\nu}\,^{\alpha}=-K_{[\mu\nu]}\,^{\alpha}=-\frac{1}{2}\,(K_{\mu\nu}\,^{\alpha}-K_{\nu\mu}\,^{\alpha})$
(6)

Field equations of Einstein-Cartan gravity are obtained by varying the action
with respect to dynamical variables Hehl1976 ; Hehl1974 ; Hehl1973 ; Hehl19744
. For the matter part, the metric energy momentum tensor $\sigma_{\mu\nu}$ and
the spin tensor $S_{\,\,\nu\mu}^{\alpha}$ are defined as mainEC

$\sigma_{\mu\nu}:=-\frac{2}{\sqrt{-g}}\frac{\delta\mathcal{L}_{m}}{\delta
g^{\mu\nu}}\mbox{ }\mbox{ }\mbox{ , }\mbox{ }\mbox{
}S\indices{{}_{\alpha}^{\nu\mu}}:=\frac{1}{\sqrt{-g}}\frac{\delta\mathcal{L}_{m}}{\delta
K\indices{{}_{\mu\nu}^{\alpha}}}$ (7)

Using the above relations, the two field equations of Einstein-Cartan theory
will have the following form mainEC

$G_{\mu\nu}+\accentset{\star}{\nabla}_{\lambda}\left(-Q\indices{{}_{\mu\nu}^{\lambda}}+Q\indices{{}_{\nu}^{\lambda}{}_{\mu}}-Q\indices{{}^{\lambda}_{\mu\nu}}\right)=\kappa\sigma_{\mu\nu}$
(8)

$T\indices{{}_{\lambda\nu}^{\mu}}+\delta_{\lambda}^{\mu}T\indices{{}_{\nu\alpha}^{\alpha}}-\delta^{\mu}_{\nu}T\indices{{}_{\lambda\alpha}^{\alpha}}=\kappa
S\indices{{}_{\lambda\nu}^{\mu}}$ (9)

where we defined
$\accentset{\star}{\nabla}_{\lambda}=\nabla_{\lambda}+2T_{\lambda\alpha}^{~{}~{}\alpha}$.
The second field equation (9) is just an algebraic equation for torsion in
terms of its source $S\indices{{}_{\mu\nu}^{\alpha}}$ while by some
manipulations, the first equation, i.e. (8) can be brought to the following
form resembling the usual Einstein equation mainEC

$\accentset{\circ}{G}_{\mu\nu}\equiv\accentset{\circ}{R}_{\mu\nu}-\frac{1}{2}g_{\mu\nu}\accentset{\circ}{R}=\kappa\tilde{\sigma}_{\mu\nu}$
(10)

Here, $\tilde{\sigma}_{\mu\nu}$ will include the torsion source terms

$\begin{split}\tilde{\sigma}_{\mu\nu}:=\sigma_{\mu\nu}+\kappa&\left\\{-4S\indices{{}_{\mu\lambda}^{[\alpha}}S\indices{{}_{\nu\alpha}^{\lambda]}}-2S_{\mu\lambda\alpha}S\indices{{}_{\nu}^{\lambda\alpha}}+S_{\alpha\lambda\mu}S\indices{{}^{\alpha\lambda}_{\nu}}\right.\\\
&\left.+\frac{1}{2}g_{\mu\nu}\left(4S\indices{{}_{\lambda}^{\beta}{}_{[\alpha}}S\indices{{}^{\lambda\alpha}_{\beta]}}+S^{\alpha\lambda\beta}S_{\alpha\lambda\beta}\right)\right\\}\end{split}$
(11)

In order to examine the cosmological dynamics in Einstein-Cartan theory, we
need to examine spatially homogeneous and isotropic spacetimes with non-
vanishing torsion. In the cosmological context, one may associate torsion with
the intrinsic spin of the matter components in the universe, or it may also be
treated as a geometric entity intrinsic to the background spacetime
Capozziello ; Kranas . In the latter case, the symmetries of the Friedmann-
Robertson-Walker spacetime force the torsion tensor to have a very specific
form which preserves the spatial homogeneity and the spatial isotropy of the
background Tsamparlis ; Olmo ; Beltran . In the most general case, the torsion
is fully determined by two scalar functions depending only on time Goenner .
However, in the Einstein-Cartan theory where Weyssenhoff fluid is usually used
to describe spin fluid Weyssenhoff ; Obukhov , a single time dependent
function is sufficient to completely describe the torsion tensor. The
corresponding non-zero components of the spin tensor then can be found from
the algebraic field equation (9) as

$s(t)=S_{123}=S_{231}=S_{312}=-S_{[123]}=S_{0ii}=S_{022}=S_{033}=-S_{i0i}$
(12)

By making these choices, we can now derive the torsional counterparts of the
familiar Friedmann equations by substituting the Friedmann-Robertson-Walker
metric and the spin tensor (12) into the field equations (10) and assuming
that the usual energy-momentum tensor for matter in the universe
$\sigma_{\mu\nu}$ is described by a perfect fluid. The final result is given
in reference mainEC  as

$H^{2}=\frac{\kappa}{3}\,(\rho-\frac{1}{4}\kappa s^{2})$ (13)

$H^{2}+\dot{H}=-\frac{\kappa}{6}(\rho+3\,p-\kappa s^{2})$ (14)

In which $\kappa$ is the coupling constant, $s$ is the spin density and $\rho$
and $p$ are the energy density and isotropic pressure of the fluid,
respectively. Assuming that the spins of fermionic matter in the universe are
not polarized, the spin density $s$ can be related to the energy density of
matter $\rho$ and subsequently to the scale factor $a(t)$ by relation Gasp ;
Nurg ; mainEC ; Qorani

$\begin{split}s^{2}&=\frac{1}{2}\langle
S_{ijk}S^{ijk}\rangle=\frac{1}{8}(\hbar
c)^{2}\frac{n_{0}^{2}}{\rho_{0}^{2/(1+\omega)}}\rho^{2/(1+\omega)}\\\
&=B_{\omega}\rho^{2/(1+\omega)}=B_{\omega}\rho_{0}^{2/(1+\omega)}a^{-6}\end{split}$
(15)

that $S_{ijk}$ is the spin tensor, $\omega$ is the equation of state for the
matter in the universe, $\rho_{0}$ and $n_{0}$ are the present day energy
density and particle number density respectively, and $B_{\omega}$ is a
dimensional constant depending on $\omega$. In the next section we will use
the Friedmann equations and the expression for the spin density mentioned
above to examine the effects of spin and torsion on the spectrum of scalar
perturbations at the end of inflation.

##  3 The Effects of Torsion On the Spectrum of Scalar Perturbations

To begin with, we consider a perturbed scalar field (inflaton) as follows

$\phi(\tau,x)\,=\,\bar{\phi}(\tau)\,+\,\delta\phi(\tau,x)\,=\,\bar{\phi}(\tau)\,+\,\frac{f(\tau,x)}{a(\tau)}$
(16)

where $\phi(\tau,x)$ represents the perturbed scalar field, $\bar{\phi}(\tau)$
is the unperturbed background field and $f(\tau,x)$ is the perturbation
function. In order to see how perturbations changes in presence of torsion, we
shall start with the action of the inflaton field

$S=\int d\tau
d^{3}x\sqrt{-g}\,\bigg{[}\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-V(\phi)\bigg{]}\,.$
(17)

where $g\equiv det(g_{\mu\nu})=-a^{8}$ and $\tau$ is the conformal time. We
consider the unperturbed FRW line element as
$ds^{2}=a^{2}(\tau)(d\tau^{2}-dr^{2})$. Using this metric and relation (16) in
action (17), we will have

$\begin{split}S&=\int d\tau
d^{3}x\sqrt{-g}\,\bigg{[}\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-V(\phi)\bigg{]}\\\
&=\int d\tau
d^{3}x\bigg{[}\frac{a^{2}}{2}\Big{(}(\phi^{\prime})^{2}-(\nabla\phi)^{2}\Big{)}-a^{4}V(\phi)\bigg{]}\end{split}$
(18)

where a ’prime’ denotes differentiation with respect to the conformal
time111In general in this article a ’prime’ denotes differentiation with
respect to the argument of the function. For the perturbed scalar field (16),
the action will be

$\begin{split}S&=\int d\tau
d^{3}x\bigg{[}\frac{1}{2}a^{2}\bigg{(}\Big{(}\frac{\partial(\bar{\phi}(\tau)\,+\,\frac{f(\tau,x)}{a(\tau)})}{\partial\tau}\Big{)}^{2}\\\
&-\,\Big{(}\nabla\big{(}\bar{\phi}(\tau)\,+\,\frac{f(\tau,x)}{a(\tau)}\big{)}\Big{)}^{2}\bigg{)}\,-\,a(\tau)^{4}\,V(\phi)\bigg{]}\\\
\end{split}$ (19)

To extract the linearized perturbation dynamics, we will need the quadratic
action Baumann . By performing the Taylor expansion for the potential
$V(\phi))$ in (19) and focusing on the terms which are second order in
perturbation function $f$, we get the quadratic action after some
straightforward algebra as

$\begin{split}S^{(2)}&=\int d\tau
d^{3}x\bigg{[}\frac{1}{2}a^{2}(\tau)\bigg{(}\big{(}\frac{f^{\prime}(\tau,x)}{a(\tau)}\big{)}^{2}-2\frac{f^{\prime}(\tau,x)a^{\prime}(\tau)f(\tau,x)}{a^{3}(\tau)}\\\
&+\big{(}\frac{a^{\prime}(\tau)f(\tau,x)}{a^{2}(\tau)}\big{)}^{2}-\big{(}\frac{\nabla
f(\tau,x)}{a(\tau)}\big{)}^{2}\bigg{)}-a^{4}(\tau)\frac{V^{\prime\prime}(\phi)}{2!}\frac{f^{2}(\tau,x)}{a^{2}(\tau)}\bigg{]}\\\
&=\int d\tau d^{3}x\bigg{(}\frac{1}{2}f^{\prime
2}(\tau,x)-\frac{a^{\prime}(\tau)}{a(\tau)}f(\tau,x)f^{\prime}(\tau,x)\\\
&+\frac{a^{\prime
2}(\tau)}{2a^{2}(\tau)}f^{2}(\tau,x)-\frac{1}{2}\big{(}\nabla
f(\tau,x)\big{)}^{2}-a^{2}(\tau)V^{\prime\prime}(\phi)f^{2}(\tau,x)\bigg{)}\\\
&=\int d\tau d^{3}x\frac{1}{2}\bigg{(}f^{\prime 2}(\tau,x)-\big{(}\nabla
f(\tau,x)\big{)}^{2}-2\mathcal{H}f(\tau,x)f^{\prime}(\tau,x)\\\
&+\mathcal{H}^{2}f^{2}(\tau,x)-a^{2}(\tau)V^{\prime\prime}(\phi)f^{2}(\tau,x)\bigg{)}\\\
\end{split}$ (20)

where $\mathcal{H}=\frac{a^{\prime}(\tau)}{a^{2}(\tau)}$ represents the Hubble
parameter in terms of conformal time and $V^{\prime\prime}(\phi)$ is the
second derivative of the potential $V(\phi)$ in terms of $\phi$.

Figure 1: Time evolution of the slow-roll parameter $\varepsilon$ defined in
(21) in the presence of torsion. The value of $\varepsilon$ at the end of
inflation is within the range permitted by trans-Planckian censorship
conjecture. Figure 2: Time evolution of the slow-roll parameter $\eta$ defined
in (22) in the presence of torsion.

The Hubble slow-roll parameters, which represent the condition for inflation,
are defined as

$\varepsilon=-\frac{\dot{H}}{H^{2}}=1-\frac{\mathcal{H}^{\prime}}{\mathcal{H}^{2}}$
(21)

$\eta=\frac{\dot{\varepsilon}}{H\varepsilon}=\frac{a\varepsilon^{\prime}}{\mathcal{H}\varepsilon}$
(22)

Using the Friedmann equations (13) and (14) and also the relation between the
spin parameter and the scale factor in the EC cosmology, i.e., equation (15),
one can derive the time evolution of these slow-roll parameters. Note that in
effect in our model described by equations (13) and (14), there are two fields
that contribute to the total energy density: The scalar field with
$\omega\approx-1$ and fermionic matter with its own equation of state that
acts as the source of spin and torsion in the universe. It should be noted
that spin and torsion alone are capable of inducing inflation without the need
for a scalar field, as demonstrated in Pereira ; Gasp ; Qorani . In Einstein-
Cartan theory, the spin induced inflation can only occur for a limited range
of the equation of state parameter Gasp , however such strict limitations are
not present in the more general Poincaré gauge theory of gravity Qorani . For
a slow-rolling inflaton scalar field with $\omega\approx-1$ and a simple
quadratic potential, the time evolution of $\varepsilon$ and $\eta$ in the
presence of torsion are depicted in figures 1 and 2. The figures show
$\varepsilon,|\eta|\ll 1$ which indicates that we are in the slow-roll regime.
In BrandennVafa , it has been shown that the TCC requires that, for quantum
fluctuations of inflation to be responsible for the origin of the structure,
there should be an upper bound to the value of the first slow-roll parameter
$\varepsilon$ as $\varepsilon<10^{-31}$ at the end of inflation. As can be
seen in the figure 1, in our model in EC theory, the value of $\varepsilon$
can actually comply with the limitations of TCC at the end of inflation.

In order to examine the effects of torsion and spin on the scalar field
perturbations, we assume that gravitational interactions at very high energies
of the early universe where there is a high density of fermions can be
described by Einstein-Cartan theory. As the spin density and its associated
torsion were probably very large at the time of inflaton, we will use the
Friedmann equation in Einstein-Cartan theory (13) to introduce the effects of
spin and torsion into the quadratic action (20). By substituting the energy
density of the inflaton field $\rho=\frac{1}{2}\dot{\phi}^{2}+V(\phi)$ into
the Friedmann equation (13) we get

$\begin{split}\mathcal{H}^{2}&=\frac{\kappa
a^{2}(\tau)}{3}\bigg{(}\big{(}\frac{1}{2}\dot{\phi}^{2}+V(\phi)\big{)}\,-\,\frac{1}{4}\kappa
s^{2}\bigg{)}\\\
&=\frac{\kappa}{6}a^{2}(\tau)\dot{\phi}^{2}+\frac{\kappa}{3}a^{2}(\tau)V(\phi)-\frac{\kappa}{12}\kappa
a^{2}(\tau)s^{2}\\\ \end{split}$ (23)

And as a result, the inflaton potential $V(\phi)$ can be expressed as

$V(\phi)=\frac{3\bigg{(}\mathcal{H}^{2}-\frac{\kappa}{6}a^{2}(\tau)\dot{\phi}^{2}+\frac{\kappa^{2}}{12}s^{2}a^{2}(\tau)\bigg{)}}{a^{2}(\tau)\kappa}\\\
$ (24)

From the definition of potential slow-roll parameters we have the following

$\eta_{V}=M_{Pl}^{2}\big{(}\frac{V^{\prime\prime}(\phi)}{V(\phi)}\big{)}$ (25)

$V^{\prime\prime}(\phi)=\frac{\eta_{V}V(\phi)}{M_{Pl}^{2}}$ (26)

Substituting (26) into (23), we finally get

$V^{\prime\prime}(\phi)=\frac{V(\phi)\eta_{V}}{M^{2}_{Pl}}=\frac{\eta_{\nu}\bigg{(}\frac{3\mathcal{H}^{2}}{\kappa}-\frac{\dot{\phi}^{2}}{2}+\frac{\kappa
s^{2}}{4}\bigg{)}}{M^{2}_{Pl}}$ (27)

Equation (27) can be substituted into the last term in the quadratic action
(20) and due to the presence of spin density parameter $s$, it carries the
effects of torsion. We should note that the Hubble parameter is approximately
constant during inflation and, also in the slow-roll regime, the kinetic term
$\dot{\phi}^{2}$ is negligible. On the other hand, as stated earlier, the spin
density $s$ was probably very high in the very early universe. As a result of
these arguments, we can safely assume that during the inflationary period, the
last term in (27) was dominant and we can safely neglect the other two terms.
With these consideration the quadratic action will be

$\begin{split}S^{(2)}\,&=\,\frac{1}{2}\int\,d\tau\,d^{3}x\,\bigg{[}f^{\prime
2}(\tau,x)\,-\,\big{(}\nabla f(\tau,x)\big{)}^{2}\\\
&+\,\bigg{(}\frac{a^{\prime\prime}(\tau)}{a(\tau)}\,-\,a^{2}(\tau)V^{\prime\prime}(\phi)\bigg{)}f^{2}(\tau,x)\,\bigg{]}\\\
&=\,\frac{1}{2}\int\,d\tau\,d^{3}x\,\bigg{[}f^{\prime
2}(\tau,x)\,-\,\big{(}\nabla f(\tau,x)\big{)}^{2}\\\
&+\,\bigg{(}\frac{a^{\prime\prime}(\tau)}{a(\tau)}\,-\,a^{2}(\tau)\big{(}\frac{\eta_{V}\kappa
s^{2}}{4M^{2}_{Pl}}\big{)}\bigg{)}\,f^{2}(\tau,x)\bigg{]}\\\ \end{split}$ (28)

Now we can use the Euler-Lagrange equation

$\frac{\partial\mathcal{L}}{\partial
f(\tau,x)}\,-\,\frac{d}{d\tau}\,\big{(}\frac{\partial\mathcal{L}}{\partial
f^{\prime}(\tau,x)}\big{)}\,=\,0,$ (29)

to derive the equation of motion for perturbation field. We have

$\begin{split}&\frac{\partial\mathcal{L}}{\partial
f(\tau,x)}=2f(\tau,x)\bigg{(}\frac{a^{\prime\prime}(\tau)}{a(\tau)}-a^{2}(\tau)\big{(}\frac{\eta_{V}\kappa
s^{2}}{4M^{2}_{Pl}}\big{)}\bigg{)}\\\ &\frac{\partial\mathcal{L}}{\partial
f^{\prime}(\tau,x)}=2f^{\prime}(\tau,x)\end{split}$ (30)

Combining the above, we get the equivalent of the famous Mukhanov-Sasaki
equation in Einstein-Cartan cosmology

$f^{\prime\prime}(\tau,x)-\nabla^{2}f(\tau,x)-\bigg{(}\frac{a^{\prime\prime}(\tau)}{a(\tau)}-a^{2}(\tau)\big{(}\frac{\eta_{V}\kappa
s^{2}}{4M^{2}_{Pl}}\big{)}\bigg{)}f(\tau,x)=0$ (31)

where the effects of torsion are contained in the second term inside
parentheses. In order to extract the time evolution of scalar perturbations,
as usual, we perform a Fourier expansion on the perturbation parameter
$f(\tau,x)$ as

$f(\tau,x)=\int\frac{d^{3}k}{(2\pi)^{3}}\exp^{ik.x}f_{k}(\tau)$ (32)

Substituting into (31) and simplifying, we have

$\begin{split}&f^{\prime\prime}_{k}(\tau)\
+\bigg{(}k^{2}-\frac{a^{\prime\prime}(\tau)}{a(\tau)}-a^{2}(\tau)\big{(}\frac{\eta_{V}\kappa
s^{2}}{4M^{2}_{Pl}}\big{)}\bigg{)}f_{k}(\tau)=0\end{split}$ (33)

In order to solve the above equation, we relate the spin density parameter $s$
to the scale factor with the help of the relation (15), by doing so (LABEL:F)
will become

$f_{k}^{\prime\prime}(\tau)+\Big{(}k^{2}-\frac{a^{\prime\prime}}{a}-a^{2}\big{(}\frac{\eta_{V}\kappa}{4M^{2}_{pl}}(B_{\omega}\rho_{0}^{\frac{2}{1+\omega}}a^{-6})\big{)}\Big{)}f_{k}(\tau)=0\,.$
(34)

To solve the equation (34) and to obtain the time evolution of scalar field
perturbation $f(\tau)$, we first examine the limiting regimes of this
equation. At late times ($\tau\rightarrow 0$), the third term inside the
parentheses of equation (34) is negligible compared to the other two terms, as
it is proportional to $a^{-4}$. In this case, the equation is as follows

$f_{k}(\tau)+\big{(}k^{2}-\frac{a^{\prime\prime}}{a}\big{)}f_{k}(\tau)=0$ (35)

if as usual we assume that the scale factor and conformal time have a relation
as $a\sim\tau^{\frac{1}{2}-\mu}$, the solution to the above equation will be

$f(\tau)=\sqrt{-\tau}\big{(}A\,J(\mu,-k\tau)+B\,Y(\mu,-k\tau)\big{)}$ (36)

where $J$ and $Y$ are Bessel functions of the first and second kind,
respectively (the solution can also be written using Hankel functions) and $A$
and $B$ are constants which should be determined by initial conditions. On the
other hand, in the very early universe ($\tau\rightarrow-\infty$) the third
term in (34) is dominant compared to the other two terms. In this case
equation (34) becomes

$f_{k}^{\prime\prime}(\tau)+\Big{(}k^{2}-\big{(}\frac{\eta_{V}\kappa}{4M^{2}_{pl}}(B_{\omega}\rho_{0}^{\frac{2}{1+\omega}}a^{-4})\big{)}\Big{)}f_{k}(\tau)=0$
(37)

where the solution will be

$\begin{split}f(\tau)&=C\sqrt{-\tau}\,J\Bigg{(}\frac{1}{4\mu},\frac{\sqrt{\frac{-\rho_{0}^{\frac{2}{1+\omega}}B_{\omega}\kappa\eta_{V}}{M_{pl}^{2}}}\,k\,\tau^{2\mu}}{4\mu}\Bigg{)}\\\
&+\,D\sqrt{-\tau}\,Y\Bigg{(}\frac{1}{4\mu},\frac{\sqrt{\frac{-\rho_{0}^{\frac{2}{1+\omega}}B_{\omega}\kappa\eta_{V}}{M_{pl}^{2}}}\,k\,\tau^{2\mu}}{4\mu}\Bigg{)}\end{split}$
(38)

where again $C$ and $D$ must be determined by initial conditions. Note that
the physical interval of the conformal time is in the form of
$-\infty<\tau<0$. The inside of the parentheses in (38) contains the
coefficient of $B_{\omega}$ and therefore shows the effects of the spin tensor
and thus the torsion on high energies.

##  4 Torsion and trans-Planckian physics

Now we examine the issue of applying the initial conditions to the above
solutions. Here we are interested in the effects of torsion at the very early
universe where quantum effects were important. At early times during the
inflation, all of the perturbation modes we are interested in were inside the
horizon. On these scales, the fluctuations of the inflaton field are described
by the use of quantum harmonic oscillators. The standard procedure in dealing
with these oscillators involves expressing the perturbation field $f$ and its
conjugate momentum $\pi=\frac{\partial\mathcal{L}}{\partial f^{\prime}}$ where
$\mathcal{L}$ is the Lagrangian in the quadratic action (28) in terms of the
ladder (or creation and annihilation) operators which obey standard
commutation relations. The ladder operators here are time dependent, but they
can be expressed in terms of operators at a fixed time by the help of
Bogolubov transformations. The most important point in the procedure is the
choice of the vacuum state, which is defined as the state annihilated by the
lowering operator. This choice can be ambiguous in a time-dependent
background. The standard choice is to impose the initial condition at
$\tau\rightarrow-\infty$ which will lead to the unique Bunch-Davis vacuum
Bunch . However, as stated previously, due to our ignorance of the physics
beyond the Planck scale, we cannot follow a mode that far back in conformal
time. Instead to obtain the effects of torsion on trans-Planckian physics we
use the method first introduced in Danielsson2  and impose the initial
conditions in high energies at $\tau\rightarrow\frac{\Lambda}{Hk}$ instead of
the usual $\tau\rightarrow-\infty$. Here $\Lambda$ is the energy scale
(usually the Planck scale) in which the effects of trans-Plankian physics
become important. Since the relationship between conformal time and the scale
factor is in the form of $\tau=-\frac{1}{aH}$, the relationship between the
comoving momentum $k$ and the physical momentum $p$ will be
$k=ap=-\frac{p}{H}$.

Figure 3: Power spectrum of scalar perturbations in terms of the wave number
$k$ from numerically solving equation (34) in the absence of torsion (blue
line) and in the presence of Torsion (green line). In both cases the new
trans-Planckian initial conditions were imposed. In the cases with and without
torsion we set $B_{\omega}=1$ and $B_{\omega}=0$ in the numerical analysis of
equation (34) respectively.

Now we try to solve the equation (37) with the initial condition imposed at
$\tau\rightarrow\frac{\Lambda}{Hk}$. In order to obtain the conformal time in
which the initial conditions should be applied, we set the physical momentum
$p$ equal to $\Lambda$. The solution to equation (34), with the new time for
imposing initial conditions, is also a suitable combination of Bessel
functions where the constant coefficients should be determined by the new
initial conditions. Here, in order to show the trans-Planckian effects of
torsion, we shall numerically solve equation (34) with the said initial
conditions and determine the power spectrum for scalar perturbations in the
presence of torsion. The scalar power spectrum is related to the perturbation
function $f$ via the relation $P_{k}=\frac{k^{3}}{2\pi^{2}}|f_{k}(\tau)|^{2}$.
In general relativity with the standard Bunch-Davis vacuum, imposing the
initial conditions at $\tau\rightarrow-\infty$ leads to the standard relation
$P_{k}=\Big{(}\frac{H}{2\pi}\Big{)}^{2}$, however, as found in Danielsson2 ,
imposing the initial condition at $\tau\rightarrow\frac{\Lambda}{Hk}$, causes
the power spectrum $P_{k}$ to contain an oscillating term
$P_{k}=\Big{(}\frac{H}{2\pi}\Big{)}^{2}\Bigg{(}1-\frac{H}{\Lambda}\sin{\frac{2\Lambda}{H}}\Bigg{)}$.
In the presence of torsion, the result for the trans-Planckian power spectrum
is presented in figure 3 after numerically solving equation (34) with the new
initial conditions. The results show that the presence of torsion increases
the amplitude of scalar perturbations for each given mode compared to the
general relativistic result. As can be seen from the figure, when the initial
conditions are imposed not at infinite past but at some other conformal time,
the resulting power spectrum is no longer scale invariant. This and also the
fluctuating behavior of the spectrum of scalar perturbations from the
inflationary period due to the trans-Planckian effects is a phenomenon that
has already been observed in previous work Danielsson2 ; Easther1 .

##  5 Gravitational Waves from inflation in the presence of torsion

We now turn our attention to the tensor perturbations generated after
inflation. Here we follow the notation of reference Sasaki . To study the
evolution of tensor perturbations, we consider a perturbed FRW metric in the
form

$\bar{g}_{\mu\nu}=g_{\mu\nu}+\delta g_{\mu\nu}$ (39)

and expand the tensor part of the metric perturbations in terms of the
harmonic functions as

$\begin{split}\bar{g}_{00}&=a^{2}(\tau)\\\
\bar{g}_{ij}&=-a^{2}(\tau)\Big{(}\gamma_{ij}+2H_{\tau}Y_{ij}\Big{)}\end{split}$
(40)

Here, $\gamma_{ij}$ is the three-dimensional spatial metric for constant
curvature hypersurfaces and $Y_{ij}$ is the harmonic function. By perturbing
the Einstein-Cartan field equation (10) and substituting this perturbed
metric, we will have for the left-hand side

$\bar{\accentset{\circ}{G}}_{ij}=\accentset{\circ}{G}_{ij}+\delta\accentset{\circ}{G}_{ij}$
(41)

where $\accentset{\circ}{G}_{ij}$ is constructed fully by the Christoffel
connection and so its perturbed components are well known (for example see
appendix D in Sasaki ). For the right-hand side of equation (10), the relevant
part of the perturbed standard energy-momentum tensor $\sigma_{\mu\nu}$ is
given by

$\bar{\sigma}_{ij}=p\Big{[}\delta_{ij}+\Pi_{T}Y_{ij}\Big{]}$ (42)

where $p$ is the isotropic pressure and $\Pi_{T}$ is the anisotropic stress.
The rest of the terms on the right-hand side consist of the spin tensor (or
equivalently torsion). Perturbing these term, we get

$\delta\tilde{\sigma}_{ij}=p\Pi_{T}Y_{ij}+\kappa\Big{[}4a^{-4}s\delta
s-6a^{-2}s^{2}H_{\tau}Y^{ij}\Big{]}$ (43)

where $\delta s$ is the perturbation in the spin density. As one can see from
the above relations, the system involves perturbations of multiple fields and,
in general, is quite complicated. In order to simplify the system of equations
and estimate the effects of torsion on the spectrum of gravitational waves
generated after inflation, we set $\delta s=\Pi_{T}=0$. In this case, after
doing a Fourier expansion of the perturbation field similar to the one in
(32), the equation governing the time evolution of the tensor perturbation
parameter $H_{\tau}$ can be derived from the perturbed field equation as

$\begin{split}H_{\tau}^{\prime\prime}+2\frac{a^{\prime}}{a}H^{\prime}_{\tau}+k^{2}H_{\tau}&-4\frac{a^{\prime\prime}}{a}H_{\tau}+2(\frac{a^{\prime}}{a})^{2}H_{\tau}\\\
&=-6\,\kappa^{2}\,a^{-2}\,s^{2}H_{\tau}\end{split}$ (44)

Figure 4: Gravitational wave signal from the numerical solution of equation
(44) for a radiation dominated universe with $a(t)\propto t^{\frac{1}{2}}$ and
for different k numbers ($k=5,15$). Figure 5: Gravitational wave signal from
the numerical solution of equation (44) for a matter dominated universe with
$a(t)\propto t^{\frac{2}{3}}$ and for different wave numbers ($k=5,15$).
Figure 6: Gravitational wave signal for both radiation and matter dominated
universes in the presence of torsion for different wave numbers ($k=5,15$).

where again a prime denotes differentiation with respect to the conformal
time. If, as before, we write the spin density in this equation in terms of
the scale factor with the help of equation (15), we can then solve the
equation numerically for different values of the equation of state parameter
$\omega$. The result of the numerical analysis for two interesting case of a
radiation dominated universe ($\omega=\frac{1}{3}$) and a matter dominated
universe ($\omega=0$) are given in figures 4 and 5 respectively. Figure 6
shows the gravitational wave signal for both matter and radiation dominated
universes for different values of the wave number $k$.

##  6 Conclusion

In this paper, considering torsion as a trans-Planckian effect, we studied the
generation and propagation of both scalar and tensor perturbations during
inflation. The key to account for the unknown trans-Planckian physics is to
apply the initial conditions not at the infinite past, but a finite conformal
time in which quantum effects become dominant. This amounts to a choice of
vacuum different from the usual Bunch-Davis vacuum used in standard
inflationary scenarios. The presence of torsion causes the amplitude of trans-
Planckian power spectrum to be greater than its general relativistic
counterpart. Moreover, trans-Planckian effects cause the power spectrum to no
longer be scale-invariant. As a further result, for a scalar inflaton field
with a simple quadratic potential in the presence of torsion, the value of the
first slow-roll parameter $\varepsilon$ lies within the range permitted by the
TCC. Here we considered Einstein-Cartan theory, the simplest of torsion
theories, to study inflationary dynamics. A more comprehensive analysis can be
done by using more general torsion theories of gravity with dynamical torsion
which can be considered in a future work.

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