Inflation in Energy-Momentum Squared Gravity in Light of Planck2018
Marzieh<EMAIL_ADDRESS>Narges<EMAIL_ADDRESS>and Kourosh<EMAIL_ADDRESS>(Corresponding Author)
Department of Theoretical Physics, Faculty of Basic Sciences, University of
Mazandaran,
P. O. Box 47416-95447, Babolsar, IRAN

###### Abstract

We study cosmological dynamics of the energy-momentum squared gravity. By
adding the squared of the matter field’s energy-momentum tensor
($\zeta\,\textbf{T}^{2}$) to the Einstein Hilbert action, we obtain the
Einstein’s field equations and study the conservation law. We show that the
presence of $\zeta\,\textbf{T}^{2}$ term, breaks the conservation of the
energy-momentum tensor of the matter fields. However, an effective energy-
momentum tensor in this model is conserved in time. By considering the FRW
metric as the background, we find the Friedmann equations and by which we
explore the cosmological inflation in $\zeta\,\textbf{T}^{2}$ model. We
perform numerical analysis on the perturbation parameters and compare the
results with Planck2018 different data sets at $68\%$ and $95\%$ CL, to obtain
some constraints on the coupling parameter $\zeta$. We show that  for
$0<\zeta\leq 2.1\times 10^{-5}$, the $\zeta\,\textbf{T}^{2}$ gravity is an
observationally viable model of inflation.
PACS: 98.80.Bp, 98.80.Cq, 98.80.Es
Key Words: Cosmological Inflation, Modified Gravity, Observational
Constraints.

##  1 Introduction

The standard model of cosmology despite all successes in the early years of
development of cosmology, failed to justify some observations of the universe.
Considering a single canonical scalar field (inflaton) with a flat potential,
leading to the slow-roll of the inflaton and causing the enough exponential
expansion of the early universe, is one simplest way to solve some main
problems of the standard model of cosmology. In the simple single field
inflation model, we get the adiabatic, scale invariant and gaussian dominant
modes of the primordial perturbations [1, 2, 3, 4, 5, 6, 7, 8]. However,
models with non-Gaussian distributed and not exactly scale invariant
perturbations have attracted a lot of attentions [8, 9, 10, 11, 12, 13, 14,
15, 16, 17, 18, 19, 20, 21].

One class of the interesting models in describing the early time inflation and
primordial perturbations is the one related to the modified gravity. Modified
gravity, in its simplest form, is a function of the Ricci scalar ($f(R)$) [22,
23, 24, 25, 26, 27, 28]. A lot of work on the inflation and perturbations
issue, have been done in the modified gravity and interesting results have
been obtained [29, 30, 31, 32, 33, 34]. Another interesting proposal in the
modified gravity is to consider an arbitrary coupling between the Ricci scalar
and Lagrangian density of the matter fields in the theory [35, 36, 37, 38, 39,
40, 41, 42, 43, 44, 45, 46]. In this regard, some authors have been attracted
to the models in which there is an arbitrary coupling between a function of
the Ricci scalar and the trace of the energy-momentum tensor of the matter
part [47, 48, 49, 50, 51].

In Ref. [52], the authors have considered an energy-momentum squared gravity
model in which they have added $\zeta\,\textbf{T}^{2}$ to the Einstein-Hilbert
action which $\textbf{T}^{2}=T_{\mu\nu}\,T^{\mu\nu}$. In this term,
$T_{\mu\nu}$ is the energy-momentum tensor of the matter part of the theory
and $\zeta$ is the coupling constant parameter which its positive values lead
to a viable cosmological scenario. They have shown that, the presence of this
term leads to a maximum energy density correspond to a minimum length. In this
way there is a bounce in the early universe, avoiding the early time
singularity. Besides, to constraint $\zeta$, one can find the scalar spectral
index, tensor spectral index and tensor-to-scalar ratio in the model and
compare the results with Planck2018 data set. The constraints on the
perturbation parameters $n_{s}$ and $r$, obtained from Planck2018 TTT, EEE,
TTE and EET data, is as $n_{s}=0.9658\pm 0.0038$ and $r<0.072$, respectively
[53, 54, 55]. Also, Planck2018 TT, TE, EE +lowE+lensing+BK14+BAO+LIGO and
Virgo2016 data implies the constraint $-0.62<n_{T}<0.53$ on the tensor
spectral index [53, 54, 55]. By using these released data, one can find some
constraints on the model’s parameter space.

In this paper, following Ref. [52], we consider the energy-momentum squared
gravity model and organize the paper as follows. In section 2, we obtain the
main equations in the $\zeta\,\textbf{T}^{2}$ gravity model. By considering an
additional term as $\zeta\,\textbf{T}^{2}$ in the Einstein-Hilbert action, we
obtain the Einstein’s field equations and study the conservation law for the
effective energy-momentum tensor. We show that the effective energy-momentum
tensor obeys the conservation law. In section 3, we explore the cosmological
dynamics in the $\zeta\,\textbf{T}^{2}$ gravity model. In this regard, we
obtain the Friedmann and the conservation equations of the model. In section
4, the cosmological inflation in this model is studied. By obtaining the slow-
roll parameters, we find $n_{s}$, $n_{T}$ and $r$ in terms of the model’s
parameters. Then, we seek for the observational viability of the model in
confrontation with Planck2018 data. In section 5, we summarize the model and
its results.

##  2 The Setup of $\zeta\,\textbf{T}^{2}$ Gravity

To study the energy-momentum squared gravity model, we start with the
following action

$S=\frac{1}{2\kappa}\int\sqrt{-g}\\\
\Big{[}R-2\Lambda-\zeta\,\textbf{T}^{2}\Big{]}\,d^{4}x+S_{M}\,.$ (1)

In this action, $g$ is determinant of metric, $R$ is Ricci scalar,
$\kappa=8\pi G$ is the gravitational constant and $\Lambda$ is the
cosmological constant. Note that, as mentioned in [52], the presence of the
cosmological constant leads to a positive acceleration in EMSG model in form
of $\ddot{a}=\frac{2\,\Lambda}{3}\,a\,,$ without considering any scalar field
in the theory. Also, the matter part of action is defined as follows

$S_{M}=\int\sqrt{-g}\,{\cal{L}}_{M}\,d^{4}x\,,$ (2)

where, ${\cal{L}}_{M}$ is the Lagrangian of the matter fields. By assuming
that the matter Lagrangian density is only a function of metric and not its
derivative, the energy-momentum tensor of the matter fields is given by

$T_{\mu\nu}=\frac{-2}{\sqrt{-g}}\frac{\delta(\sqrt{-g}{\cal{L}}_{M})}{\delta
g^{\mu\nu}}={\cal{L}}_{M}g_{\mu\nu}-2\frac{\delta{\cal{L}}_{M}}{\delta
g^{\mu\nu}}\,.$ (3)

By varying action (1) with respect to the metric $g^{\mu\nu}$, we obtain the
Einstein’s field equations in the $\zeta\,\textbf{T}^{2}$ gravity model as

$R_{\mu\nu}-\frac{1}{2}R\,g_{\mu\nu}+\Lambda\,g_{\mu\nu}+\frac{1}{2}\zeta\,\textbf{T}^{2}\,g_{\mu\nu}-\zeta\,\frac{\delta\textbf{T}^{2}}{\delta
g^{\mu\nu}}-\kappa\,T_{\mu\nu}=0\,.$ (4)

While [52, 57]

$\frac{\delta\textbf{T}^{2}}{\delta
g^{\mu\nu}}=2\,\Bigg{(}T_{\mu}^{\sigma}\,T_{\nu\sigma}+T^{\alpha\beta}\,\frac{\delta
T_{\alpha\beta}}{\delta g^{\mu\nu}}\Bigg{)},$ (5)

where $\textbf{T}^{2}=T_{\mu\nu}\,T^{\mu\nu}$. Now, we can rewrite equations
(4) as follows

$R_{\mu\nu}-\frac{1}{2}R\,g_{\mu\nu}+\Lambda\,g_{\mu\nu}=\kappa\,T_{\mu\nu}-\frac{1}{2}\zeta\,\textbf{T}^{2}\,g_{\mu\nu}+\zeta\,\Theta_{\mu\nu}+\zeta\,T_{\mu\nu}\,,$
(6)

In the above equation, $\Theta_{\mu\nu}$ is defined as

$\Theta_{\mu\nu}=-{\cal{L}}_{M}\,g_{\mu\nu}+\frac{1}{2}\,g_{\mu\nu}\,{\cal{L}}_{M}-\frac{1}{2}T_{\mu\nu}-2\,g^{\mu\nu}\,\frac{\partial^{2}{\cal{L}}_{M}}{\partial
g^{\mu\nu}\partial g^{\alpha\beta}}.$ (7)

Considering that all the changes appear in the right hand side of field
equations, we can introduce the following effective energy-momentum tensor

$T^{eff}_{\mu\nu}=T_{\mu\nu}+\frac{1}{\kappa}\,\zeta\,\Theta_{\mu\nu}+\zeta\,T_{\mu\nu}-\frac{1}{2}g_{\mu\nu}\,\zeta\,\textbf{T}^{2}.$
(8)

and rewrite the Einstein field equations as follows

$G_{\mu\nu}+\Lambda\,g_{\mu\nu}=\kappa\,T_{\mu\nu}^{eff}.$ (9)

Note that, these equations are also obtained in Ref. [56], where the authors
have considered $\zeta\,(T^{\mu\nu}\,T_{\mu\nu})^{n}$. In this work, we study
the case with $n=1$.

Now, we seek for the conservation law in this model. In this regard, we obtain
the covariant derivative of the energy-momentum tensor as

$\displaystyle\nabla^{\mu}\,T_{\mu\nu}=\frac{1}{(\kappa+\frac{1}{2}\,\zeta)}\Bigg{[}\frac{1}{2}\,\zeta\,\textbf{T}^{2}\,g_{\mu\nu}+{\cal{L}}_{M}\,\zeta\,g_{\mu\nu}$
$\displaystyle-\frac{1}{2}\,\zeta\,g_{\mu\nu}\,{\cal{L}}_{M}+2\,\zeta\,g^{\alpha\beta}\frac{\partial^{2}{\cal{L}}_{M}}{\partial
g^{\mu\nu}\partial g^{\alpha\beta}}\Bigg{]}.$ (10)

This equation shows that by adding $\zeta\,\textbf{T}^{2}$ to the action, the
conservation law breaks. This means that, due to the interaction between the
matter and curvature sectors, the energy-momentum tensor is not conserved in
time. However, we have introduced an effective energy-momentum tensor in
equation (9) which satisfies the conservation equation. In fact, we have

$\nabla^{\mu}\,T^{eff}_{\mu\nu}=0\,,$ (11)

which shows that the effective energy-momentum tensor is conserved in time.

Up to here, we have obtained the main equations of the energy-momentum squared
gravity model and studied the conservation law in this model. In the next
section, we explore the cosmological implications of this interesting model.

##  3 Cosmology in $\zeta\,\textbf{T}^{2}$ Gravity

Since our universe is homogenous and isotropic in large scales, it is
convenient to adopt the following Friedmann-Robertson-Walker metric as the
background

$ds^{2}=dx^{\mu}dx^{\nu}g_{\mu\nu}=-dt^{2}+a(t)^{2}\bigg{[}\frac{dr^{2}}{1-kr^{2}}+r^{2}(d\theta^{2}+sin^{2}\theta
d\phi^{2})\bigg{]}.$ (12)

Considering that the observational data confirms the flat universe, we
consider the FRW metric with $k=0$. To obtain the Friedmann equations, it is
necessary to adopt a suitable choice for the Lagrangian since the tensor
$\Theta_{tt}$ is related to the matter field’s Lagrangian via equation (7). It
has been shown that one can choose either ${\cal{L}}_{M}=p$ (where $p$ is the
pressure) or ${\cal{L}}_{M}=-\rho$ (where $\rho$ is the energy density) in
Refs. [58, 59]. Here, we adopt the case ${\cal{L}}_{M}=p$. In this regard, we
find

$H^{2}=\frac{\kappa}{3}\,\rho+\frac{\Lambda}{3}\,-\zeta\,(\frac{1}{2}\,p^{2}\,+\frac{4}{3}\,\rho\,p+\frac{1}{6}\,\rho^{2})$
(13)

$\frac{\ddot{a}}{a}=-\frac{\kappa}{6}\,(\rho+3\,p)+\frac{\Lambda}{3}+\zeta\,(p^{2}+\frac{2}{3}\,\rho\,p+\frac{1}{3}\rho^{2})$
(14)

As we mentioned $\zeta$ is the coupling constant and the important point about
it is that, against other coupling constants in higher order theories of
gravity which are dimensionless, dimension of $\zeta$ is
$\frac{s^{4}}{kg^{2}}$ (Note that, in this paper we adopt the light speed as
$c=1$). Further more as mentioned in Ref. [52] this coupling constant should
be both positive and small enough in order to give interesting cosmological
results and pass classical gravitational tests in low energy regimes
respectively. We see in the next section that both of these conditions are
satisfied in our observational analysis.

From equations $(13)$ and $(14)$, we can define the following effective energy
density and pressure

$\rho_{eff}=\rho-\frac{1}{2}\zeta\,\Big{(}\rho^{2}+3p^{2}+8\rho\,p\Big{)}\,,$
(15)

$p_{eff}=p-\frac{1}{2}\zeta\,\Big{(}\rho^{2}+3p^{2}\Big{)}\,,$ (16)

leading to the following continuity equation

${\dot{\rho}}_{eff}+3\,H(\rho_{eff}+p_{eff})\,=0$ (17)

Which dot means derivatives with respect to cosmological time. This equation
means that the effective energy density is conserved in time.

After obtaining the main cosmological equation in the $\zeta\,\textbf{T}^{2}$
gravity model, we study the inflation and observational viability of this
model in the next section.

##  4 Inflation and Observational Viability of $\zeta\,\textbf{T}^{2}$ Gravity

By using the definition of a constant and positive equation of state as
$\omega=\frac{p}{\rho}$ , we can rewrite the Friedmann equations (13) and (14)
as follows respectively

$H^{2}=\frac{\kappa}{3}\,\rho+\frac{\Lambda}{3}-\frac{1}{6}\,\lambda\,\rho^{2}\,,$
(18)
$\dot{H}+H^{2}=-\frac{\kappa}{6}\,\xi\,\rho+\frac{\Lambda}{3}+\beta\,\rho^{2}$
(19)

where constants are defined as

$\lambda=\zeta(3\,\omega^{2}+8\,\omega+1)\,.$ (20) $\xi=1+3\,\omega$ (21)
$\beta=\zeta\,(\omega^{2}\,+\frac{2}{3}\,\omega+\frac{1}{3})$ (22)

By differentiating the equation $(19)$ with respect to cosmological time, we
obtain

$\ddot{H}+2H\dot{H}=-\frac{\kappa}{6}\xi\,\dot{\rho}+2\beta\,\rho\,\dot{\rho}\,.$
(23)

Which $\rho$ and $\dot{\rho}$ must be substitute. By rearranging the equation
$(18)$ in order to express the energy density, we have

$\rho=\frac{\kappa+\sqrt{-6\,H^{2}\,\lambda+2\,\Lambda\,\lambda+\kappa^{2}}}{\lambda}$
(24)

Also, the time derivative of the energy density by using equations $(15)$,
$(16)$ and $(17)$ is given by

$\displaystyle\dot{\rho}=\frac{H}{\lambda^{2}\,(-1+\kappa+\sqrt{-6\,H^{2}\,\lambda+2\,\Lambda\,\lambda+\kappa^{2}})}\bigg{[}9H^{2}\,\zeta\,\xi\,\lambda+9H^{2}\,\lambda^{2}$
$\displaystyle-3\,\zeta\,\xi\,\kappa\sqrt{-6\,H^{2}\,\lambda+2\,\Lambda\,\lambda+\kappa^{2}}\,-3\,\xi\,\zeta\,\kappa^{2}-3\,\zeta\,\xi\,\lambda\,\Lambda-3\,\kappa\,\lambda\sqrt{-6\,H^{2}\,\lambda+2\,\Lambda\,\lambda+\kappa^{2}}$
$\displaystyle+3\,\chi\,\lambda\,\sqrt{-6\,H^{2}\,\lambda+2\,\Lambda\,\lambda+\kappa^{2}}-3\,\kappa^{2}\,\lambda+3\,\chi\,\lambda\,\kappa-2\,\Lambda\,\lambda^{2}\bigg{]}$
(25)

Where we have defined

$\chi=\zeta\,\xi\,,$ (26)

By substituting equations $(24)$ and $(25)$ into equation $(23)$, we obtain a
Li$\acute{e}$nard equation as

$\ddot{H}+AH\dot{H}+BH^{3}=0\,.$ (27)

Where the parameters $A$ and $B$ are

$A=\frac{4\,\zeta}{\kappa}\,\,,\quad\quad
B=\frac{-27\,\zeta}{\kappa}+\frac{2\,\Lambda}{\kappa^{2}}\,.$ (28)

In Ref. [52] it has been shown that, to resolve the singularity of the early
universe and achieve the positive acceleration expansion in the energy-
momentum squared gravity model the universe should be dominated with the
radiation component. In this regard, to obtain the coefficients (28), we have
adopted $\omega=\frac{1}{3}$. The solution of the differential equation (27)
in the general form is

$H=\bigg{(}\frac{-2z}{A}\bigg{)}^{\frac{1}{2}}\,,$ (29)

which is given in the parametric form. Also, the parameter $z$ is given by

$z={\cal{C}}\exp\bigg{(}-\int{\frac{\sigma
d\sigma}{\sigma^{2}-\sigma-D}}\bigg{)}\,,$ (30)

where ${\cal{C}}$ is a constant and we have

$D=\frac{-2B}{A^{2}}\,.$ (31)

To see more details about obtaining the solution of the differential equation
(27), see the Appendix and also Ref. [60]. Now, we can obtain the derivatives
of $H$ as follows

$\dot{H}=\sigma z$ (32) $\ddot{H}=-A\,(\frac{\sigma+D}{\sigma})\,H\dot{H}$
(33)

and

$\dddot{H}=\dot{H}\Bigg{[}A^{2}H^{2}\bigg{(}\frac{\sigma+D}{\sigma}\bigg{)}^{2}-A\dot{H}\bigg{(}\frac{\sigma+D}{\sigma}\bigg{)}\Bigg{]}\,.$
(34)

We can use the equations (29) and (32)-(34) to study the inflation in the
$\zeta\,\textbf{T}^{2}$ gravity model. To this end, we define the following
slow-roll parameters [61]

$\epsilon_{1}\equiv\frac{-\dot{H}}{H^{2}}\,,$ (35)
$\epsilon_{2}\equiv\frac{\ddot{H}}{H\dot{H}}-\frac{2\dot{H}}{H^{2}}\,,$ (36)

and

$\epsilon_{3}\equiv\bigg{(}\ddot{H}H-2\dot{H}^{2}\bigg{)}^{-1}\,\Bigg{[}\frac{H\dot{H}\dddot{H}-\ddot{H}(\dot{H}^{2}+H\ddot{H})}{H\dot{H}}-\frac{2\dot{H}}{H^{2}}\Big{(}H\ddot{H}-2\dot{H}^{2}\Big{)}\Bigg{]}\,.$
(37)

It is possible to write the slow-roll parameters in terms of the e-folds
number. The e-folds number is defined as

$N\equiv\ln\bigg{(}\frac{a_{f}}{a_{i}}\bigg{)}=-\int_{a_{i}}^{a_{f}}H(t)\,dt\,,$
(38)

where $a_{i}$ and $a_{f}$ show the values of the scale factor at the beginning
and end of the inflation era, respectively. The e-folds number in
$\zeta\,\textbf{T}^{2}$ gravity model takes the following form

$N=\frac{2}{A\sqrt{4D+1}}\tanh^{-1}\bigg{(}\frac{2\sigma+1}{\sqrt{4D+1}}\bigg{)}.$
(39)

By using the definition of the e-folds number given in equation (38), we
obtain the following expressions for the slow-roll parameters

$\epsilon_{1}(N)\equiv\frac{-H^{\prime}(N)}{H(N)}\,,$ (40)
$\epsilon_{2}(N)\equiv\frac{H^{\prime\prime}(N_{)}}{H^{\prime}(N)}-\frac{H^{\prime}(N)}{H(N)}\,,$
(41)

and

$\epsilon_{3}(N)\equiv\Bigg{[}\frac{H(N)H^{\prime}(N)}{H^{\prime\prime}(N)H(N)-H^{\prime
2}(N)}\Bigg{]}\Bigg{[}\frac{H^{\prime\prime\prime}(N)}{H^{\prime}(N)}-\frac{H^{\prime\prime
2}(N)}{H^{\prime 2}(N)}-\frac{H^{\prime\prime}(N)}{H(N)}+\frac{H^{\prime
2}(N)}{H^{2}(N)}\Bigg{]}\,.$ (42)

These parameters in our $\zeta\,\textbf{T}^{2}$ model, take the following form

$\epsilon_{1}=\frac{\sigma A}{2}\,,$ (43)
$\epsilon_{2}=A\bigg{(}\frac{\sigma^{2}-\sigma-D}{\sigma}\bigg{)}\,,$ (44)
$\epsilon_{3}=A\bigg{(}\frac{-\sigma^{3}+\sigma+D}{-\sigma^{2}+\sigma+D}\bigg{)}\,.$
(45)

Now, by considering the equations (39) and (43)-(45), we can seek for graceful
exit of the model from inflation era. In the inflation era we have
$\epsilon_{1},\epsilon_{2},\epsilon_{3}\ll 1$. To have graceful exit from
inflation, one of the slow-roll parameters should reach unity. In this regard,
we plot the parameters $\epsilon_{1}$ and $\epsilon_{2}$ versus the e-folds
number for two sample values of $\zeta$. The results are shown in figure 1. As
figure shows, the slow-roll parameter $\epsilon_{2}$ meet unity at $N=60$.
This means that in our model inflation ends after 60 e-folding.

Figure 1: The evolution of the slow-roll parameters $\epsilon_{1}$ and
$\epsilon_{2}$ versus the e-folds number during the inflation for the two
sample values of $\zeta$.

Another way to seek for inflation and its a graceful exit is the study of the
evolution of the Hubble parameter versus the e-folds number. The result is
shown in figure 2, which has been plotted for $\zeta=2.1\times 10^{-5}$. As
this figure shows, the Hubble parameter during the inflation changes very
slowly until inflation ends.

Figure 2: The evolution of the Hubble parameter versus the e-folds number
during inflation.

Also, the perturbation parameters are defined in terms of the slow-roll
parameters. In this regard, the scalar spectral index and its running are
given by [27, 47, 61, 63, 64]

$n_{s}\approx 1-2\epsilon_{1}-2\epsilon_{2}\,,$ (46)

and

$\alpha_{s}\approx-2\epsilon_{1}\epsilon_{2}-\epsilon_{2}\epsilon_{3}\,.$ (47)

respectively. The tensor spectral index, in terms of the slow-roll parameters,
is defined as

$n_{T}\approx-2\epsilon_{1}.$ (48)

Finally, the tenor-to-scalar ratio is given by

$r\approx 16\epsilon_{1}\,.$ (49)

By substituting equations (43)-(45) in equations (46)-(49) we obtain the
following expressions for the perturbation parameters

$n_{s}=1-A\sigma-2\,A\,\bigg{(}\frac{\sigma^{2}-\sigma-D}{\sigma}\bigg{)}\,,$
(50)
$\alpha_{s}=\Bigg{[}-A^{2}\,\bigg{(}\frac{\sigma^{2}-\sigma-D}{\sigma}\bigg{)}\Bigg{]}\Bigg{[}\sigma+\frac{-\sigma^{3}+\sigma+D}{-\sigma^{2}+\sigma+D}\Bigg{]}\,,$
(51) $n_{T}=-A\,\sigma\,,$ (52)

and

$r=8\,A\,\sigma\,.$ (53)

Finally, by using equation (53) to eliminate the parameter $\sigma$, we get

$n_{s}=\frac{3}{4}-\frac{r}{8}+2A-\frac{32B}{r}.$ (54)
$\alpha_{s}=\left(\frac{r}{8A}-1+\frac{16B}{AR}\right)\left[\frac{-rA}{8}-A^{2}(\frac{-r^{3}+64rA^{2}-1024AB}{-r^{2}+8Ar-128B})\right].$
(55)

and

$n_{T}=\frac{-r}{8}.$ (56)

After obtaining the main perturbations parameters, now we explore the model
numerically and compare the results with the observational data. In this
regard, we can examine the observational viability of our setup and obtain
some constraints on the coupling parameter $\zeta$. Note that, in Ref. [52] it
has been shown that only positive values of $\zeta$ lead to the viable
cosmology. Therefore, in our analysis, we consider only the positive values of
this parameter. Figure 3 shows the behavior of the running of the scalar
spectral index versus the scalar spectral index, in the background of the
Planck2018 TT, TE, EE+lowE+lensing data [53]. To plot this figure, we have
used equations (54) and (55), where the parameters $A$ and $B$ are given by
equation (28). This figure and forthcoming figures have been plotted for
$0<\zeta\leq 10^{-5}$. We can also study the behavior of the tensor-to-scalar
ratio versus the scalar spectral index by using equation (54). The result is
shown in figure 4, in the background of the Planck2018 TT, TE,
EE+lowE+lensing+BK14+BAO data set [54]. Also, figure 5 shows the tensor-to-
scalar ratio versus the tensor spectral index (see equation (56)) in the
background of the Planck2018 TT, TE, EE+lowE+lensing+BK14+BAO+LIGO and
Virgo2016 data [54]. To plot figures 3-5, we have borrowed the contour plots
released by Planck 208 team [53, 54, 55]. This is because, in this paper, we
compare the results of the numerical analysis in our model with the Planck
observational data. However, the blue regions are the numerical results of our
setup which have been obtained from equations (54)-(56). As these figures
show, the energy-momentum squared gravity model in some ranges of the model’s
parameter space is consistent with observational data. By performing the
numerical analysis, we have obtained some ranges of the parameter $\zeta$
which cause the viability of the model in confrontation with different data
sets. The constraints are summarized in table 1.

Figure 3: Running of the scalar spectral index versus the scalar spectral index for $0<\zeta\leq 10^{-5}$ (blue region), in the background of Planck2018 TT, TE, EE+lowE+lensing data (red regions). The yellow arrow shows the direction in which the parameter $\zeta$ increases. Figure 4: Tensor-to-scalar ratio versus the scalar spectral index for $0<\zeta\leq 10^{-5}$ (blue region), in the background of the Planck2018 TT, TE, EE+lowE+lensing+BK14+BAO data set (red regions). The yellow arrow shows the direction in which the parameter $\zeta$ increases. Figure 5: Tensor spectral index versus the tensor spectral index for $0<\zeta\leq 10^{-5}$ (blue region), in the background of the Planck2018 TT, TE, EE+lowE+lensing+BK14+BAO+LIGO and Virgo2016 data (red regions). The yellow arrow shows the direction in which the parameter $\zeta$ increases. Table 1:  The ranges of the parameter $\zeta$ $\left(\frac{s^{4}}{kg^{2}}\right)$ in which the tensor-to-scalar ratio, the scalar spectral index, its running and the tensor spectral index of the $\zeta\,\textbf{T}^{2}$ gravity model are consistent with different data sets. |  | Planck2018 TT, TE, EE+lowE | Planck2018 TT, TE, EE+lowE | Planck2018 TT, TE, EE+lowE |
---|---|---|---|---|---
|  | +lensing | +lensing+BK14+BAO | lensing+BK14+BAO |
|  |  |  |  +LIGO$\&$Virgo2016 |
|  $68\%$ CL | $\zeta\leq 1.6\times 10^{-3}$ | $0<\zeta\leq 1.4\times 10^{-5}$ | $0<\zeta$ |
|  $95\%$ CL | $0<\zeta\leq 1.6\times 10^{-3}$ | $0<\zeta\leq 2.1\times 10^{-5}$ | $0<\zeta$ |

After studying the perturbation’s parameters numerically and obtaining some
constraints on the model from the observational data, it seems interesting to
seek the abundance of the the fluid $\rho$ with $\omega=\frac{1}{3}$
(corresponding to radiation component). For this purpose, we rewrite equation
(18) in terms of the density parameters $\Omega$ as

$1=\Omega_{{rad}}+\Omega_{{\Lambda}}-{\Omega^{2}_{{rad}}}\Omega_{{\zeta}}\,,$
(57)

Where

$\Omega_{{rad}}={\frac{\kappa\,\rho}{3{H}^{2}}}\,,\quad\Omega_{{\Lambda}}={\frac{\Lambda}{3{H}^{2}}}\,,\quad\Omega_{{\zeta}}={\frac{3\lambda\,{H}^{2}}{2{\kappa}^{2}}}\,,$
(58)

With $rad$ presenting the radiation component. As our numerical analysis has
shown, the strength of the energy-momentum squared gravity in our model is
small ($0<\zeta\leq 2.1\times 10^{-5}$). This means that even for small
strength of the energy-momentum squared gravity, it is possible to get the
observationally viable inflationary model. In this sense, to study the
abundance of the the fluids in our model, we adopt small value of
$\Omega_{\zeta}$ as $\Omega_{\zeta}=0.001$. Then, we find the abundance of
$\Omega_{r}$ and $\Omega_{\Lambda}$ at $68\%$ CL and $98\%$ CL, for this
adopted value of $\Omega_{\zeta}$. The result is shown in figure 6. According
to our analysis at $68\%$ CL, we have $\Omega_{r}=0.908\pm 0.003$ and
$\Omega_{\Lambda}=0.091\pm 0.003$.

Figure 6: The abundance of $\Omega_{r}$ and $\Omega_{\Lambda}$ at $68\%$ CL
(red region) and $98\%$ CL (plum region), for $\Omega_{\zeta}=0.001$.

##  5 Conclusion

In this paper, we have studied the cosmological dynamics of the energy-
momentum squared gravity. In this regard, we have considered an additional
term in the Einstein-Hilbert action as $\zeta\,\textbf{T}^{2}$, where $\zeta$
is a positive constant coupling and $\textbf{T}^{2}=T_{\mu\nu}\,T^{\mu\nu}$.
We have presented the Einstein’s field equations in $\zeta\,\textbf{T}^{2}$
gravity model and also studied the conservation law via the energy-momentum
tensor. We have shown that, by adding a $\zeta\,\textbf{T}^{2}$ term to the
action, the energy momentum of the matter fields breaks. However, if we
consider an effective energy-momentum tensor, the conservation law would be
satisfied. After that, by assuming the FRW metric as the background, we have
obtained the Friedmann equations in this setup. In this regard, we have
introduced the effective energy density and the effective pressure, by which
we have shown the conservation of the effective energy density in the energy-
momentum squared gravity model.

Then, we have studied the inflation phase in this model. By obtaining the
slow-roll parameters in this model, we have expressed the perturbation
parameters in terms of the model’s parameters. By performing a numerical
analysis on the scalar spectral index, its running, tensor spectral index and
tensor-to-scalar ratio, we have studied the viability of the
$\zeta\,\textbf{T}^{2}$ model in the context of the inflation. according to
our analysis, Planck2018 TT, TE, EE+lowE+lensing+BK14+BAO+LIGO and Virgo2016
data doesn’t give any constraint on the coupling parameter $\zeta$. However,
it is possible to set constraints on $\zeta$ by considering the Planck2018 TT,
TE, EE+lowE+lensing and also, Planck2018 TT, TE, EE+lowE+lensing+BK14+BAO data
sets.

In summary, the $\zeta\,\textbf{T}^{2}$ gravity model, which lead to the
bounce at early universe, is an observationally viable inflation model with
$0<\zeta\leq 2.1\times 10^{-5}$.

Appendix

We have the following Lienard differential equation

$\ddot{H}+AH\dot{H}+BH^{3}=0\,,$

where

$A=\frac{4\,\zeta}{\kappa}$

and

$B=\frac{-27\,\zeta}{\kappa}+\frac{2\,\Lambda}{\kappa^{2}}\,$

By assuming $H\equiv y$, we can rewrite the above Lienard equation as follows

$\ddot{y}+f(y)\dot{y}+g(y)=0\,,$

where

$f(y)=A\,y\,,$

and

$g(y)=B\,y^{3}\,.$

Now, we define $w(y)=\dot{y}$ and by which we convert our Lienard equation to
the Abel differential equation of the second kind as

$ww^{\prime}+f(y)w+g(y)=0\,,$

with $ww^{\prime}=\ddot{y}$ and $w^{\prime}\equiv\frac{dw}{dy}$. By
introducing $z=\int F(y)dy$, with $F(y)=-Ay$ and $G(y)=-By^{3}$, the Abel
equation takes the following canonical form

$ww_{,z}=w+\phi(z)\,,$

where $``,z$” demonstrates derivative with respect to $z$, and

$\phi(z)=\frac{G(y)}{F(y)}\,.$

Considering that

$z=-\frac{Ay^{2}}{2}\,,$

we find the following expression for $\phi(z)$

$\phi(z)=\frac{B}{A}\,y^{2}\,,$

leading to

$ww_{,z}=w+\frac{B}{A}\,y^{2}=w+D\,z\,.$

By defining $k(z)=Dz$, we get

$ww_{,z}=w+k(z)\,,$

which has the following solution

$z={\cal{C}}\exp\bigg{(}-\int{\frac{\sigma
d\sigma}{\sigma^{2}-\sigma-D}}\bigg{)}\,,\quad
w={\cal{C}}\sigma\exp\bigg{(}-\int{\frac{\sigma
d\sigma}{\sigma^{2}-\sigma-D}}\bigg{)}\,.$

Now, we can obtain the Hubble parameter and its derivatives. From $y\equiv H$
and $z=-\frac{Ay^{2}}{2}$, we find

$H=\left(\frac{-2z}{A}\right)^{\frac{1}{2}}\,.$

Using $\dot{H}=\dot{y}=w$, we obtain

$\dot{H}=\sigma\,z\,.$

Also, from $\ddot{H}=ww^{\prime}$ and considering that
$\frac{dw}{dy}\equiv\frac{dw}{dz}\frac{dz}{dy}$, we get

$\ddot{H}=-A\left(\frac{\sigma+D}{\sigma}\right)H\dot{H}\,.$

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