# Energy Conditions in $f(P)$ Gravity

Snehasish Bhattacharjee 111Email<EMAIL_ADDRESS>Department of Physics, Indian Institute of Technology, Hyderabad 502285, India

###### Abstract

$f(P)$ gravity is a novel extension of ECG in which the Ricci scalar in the
action is replaced by a function of the curvature invariant $P$ which
represents the contractions of the Riemann tensor at the cubic order p . The
present work is concentrated on bounding some $f(P)$ gravity models using the
concept of energy conditions where the functional forms of $f(P)$ are
represented as a) $f(P)=\alpha\sqrt{P}$, and b) $f(P)=\alpha\exp(P)$, where
$\alpha$ is the sole model parameter. Energy conditions are interesting linear
relationships between pressure and density and have been extensively employed
to derive interesting results in Einstein’s gravity, and are also an excellent
tool to impose constraints on any cosmological model. To place the bounds, we
ensured that the energy density must remain positive, the pressure must remain
negative, and the EoS parameter must attain a value close to $-1$ to make sure
that the bounds respect the accelerated expansion of the Universe and are also
in harmony with the latest observational data. We report that for both the
models, suitable parameter spaces exist which satisfy the aforementioned
conditions and therefore posit the $f(P)$ theory of gravity to be a promising
modified theory of gravitation.

##  I Introduction

Modified gravity theories are being continuously developed to explain the
cosmic evolution from the primordial inflation just seconds after the big
bang, to the formation of large-scale structures, and finally to the presently
observed acceleration without citing dark matter and dark energy. Furthermore,
in addition to being observationally harmonious p1 ; p2 ; p3 ; p4 ; p5 ,
modified gravity theories are quantizable gravitational theories and transpire
naturally in the gravitational interactions of string theories p7 ; p6 ; p ,
and encompasses Einstein’s gravity as a distinct limit p8 . $f(R)$ gravity was
the first modified gravity introduced in literature where the Ricci scalar
($R$) is replaced by a suitable functional form of $R$, and is remarkable in
explaining numerous cosmological observations 2p4 ; 2p5 ; 2p6 ; 2p7 ; 2p8 ;
2p9  (see fr  for a review on $f(R)$ gravity). Motivated by the successes of
$f(R)$ gravity, several other gravitational theories emerged where the Ricci
scalar is replaced by or with other curvature invariants such as the Gauss-
Bonnet scalar, the Torsion scalar, and the Nonmetricity scalar (see rev  for a
recent review on various modified gravity theories).
Recent years have seen a rapid development of a class of modified gravity
theories employing cubic contractions of the Riemann tensor and is commonly
termed Einsteinian cubic gravity (ECG) p31 . Ref p  reported a novel extension
of ECG in which the Ricci scalar in the action is replaced by a function of
the curvature invariant $P$ which represents the contractions of the Riemann
tensor at the cubic order. This class of modified gravity theory have been
very successful in several cosmological domains such as inflation p35 ,
spherically symmetric black hole solutions p32 ; p33 ; p34 , and late-time
acceleration p ; 2p .
Energy conditions are interesting linear relationships between pressure and
density and have been extensively employed to derive interesting results in
Einstein’s gravity. For instance, the Hawking-Penrose singularity theorems
employ the strong energy condition (SEC) ft33 , whose violation warrants an
accelerated expansion of the cosmos. Another example is the invoking of the
null energy condition (NEC) to prove the second law of black hole
thermodynamics ft34 ; ft35 .
Energy conditions are an excellent tool to impose constraints on any
cosmological model ft36 . Energy conditions have been used to derive
constraints on several modified gravity theories ft45 ; ft46 ; ft47 ; ft48 ;
ft49 ; ft50 ; ft51 ; ft52 ; bhatta ; jamil ; garcia . They are also employed
to bound several cosmological observables such as the lookback time, curvature
parameter, distance modulus, and the deceleration parameter ft37 ; ft38 ; ft39
; ft40 ; ft41 ; ft42 ; ft43 ; ft44 .
In this work, we shall derive the expressions for the energy conditions in
$f(P)$ gravity to impose strict bounds on this class of modified gravity by
employing the observational constraints on the Hubble, the deceleration, and
the jerk parameters. The manuscript is organized as follows: In Section II, we
provide an overview of $f(P)$ gravity and obtain the expressions of the energy
density, pressure, and the EoS parameter. In Section III, we summarize the
concept of energy conditions and obtain the expressions for the energy
conditions in $f(P)$ gravity. In Section IV we impose bounds on some $f(P)$
gravity models using the concept of energy conditions and in Section V we
present the results and conclusions. Throughout the work we shall work with
natural units and use $H=0.692$, $q=-0.545$, and $j=0.776$ cap/2019 .

##  II $f(P)$ Gravity: An overview

The nontopological cubic term $P$ is defined as p

$P=\beta_{1}R_{i}{}^{\rho}{}_{j}{}^{\sigma}R_{\rho}{}^{\gamma}{}_{\sigma}{}^{\delta}R_{\gamma}{}^{i}{}_{\delta}{}^{j}+\beta_{2}RR_{ij\rho\sigma}R^{ij\rho\sigma}+\beta_{3}R^{j}_{i}R^{\rho}_{j}R^{i}_{\rho}+\beta_{4}R^{\rho\sigma}_{ij}R^{\gamma\delta}_{\rho\sigma}R^{ij}_{\gamma\delta}\\\
+\beta_{5}R^{i\rho}R^{j\sigma}R_{ij\rho\sigma}+\beta_{6}R^{\sigma\gamma}R_{ij\rho\sigma}R^{ij\rho\sigma}{}_{\gamma}+\beta_{7}RR^{ij}R_{ij}+\beta_{8}R^{3},$
(1)

where $\beta_{i}$ are free parameters. Now, to ensure that $f(P)$ theories
have a spectrum similar to that of Einstein’s gravity, the parameters
$\beta_{7}$, and $\beta_{8}$ need to take the following forms p31

$\beta_{7}=\frac{1}{12}\left[3\beta_{1}-48\beta_{2}-9\beta_{3}-24\beta_{4}-5\beta_{5}-16\beta_{6}\right],$
(2)
$\beta_{8}=\frac{1}{72}\left[-6\beta_{1}+64\beta_{2}+9\beta_{3}+36\beta_{4}+3\beta_{5}+22\beta_{6}\right].$
(3)

The action in $f(P)$ gravity reads p

$\mathcal{S}-\frac{1}{2\kappa}\int\sqrt{-g}\left(R+f(P)\right)d^{4}x,$ (4)

where $g$ represents the metric determinant, $R$ the Ricci scalar,
$\kappa=8\pi G$ the Newton’s gravitational constant, and $f(P)$ is a suitable
functional choice of $P$. Varying the action (Eq. 4) yields the following
field equation

$G_{ij}=\kappa(\bar{H}_{ij}+T_{ij}),$ (5)

where $T_{ij}$ is the energy-momentum tensor and is defined as

$T_{ij}=-\left(\frac{2}{\sqrt{-g}}\right)\left[\frac{\delta(\mathcal{L}_{m}\sqrt{-g})}{\delta
g^{ij}}\right],$ (6)

and

$\bar{H}_{ij}=R^{\epsilon\varepsilon\zeta}(_{i}\bar{K}_{j})_{\zeta\epsilon\varepsilon}+g_{ij}f(P)\nabla^{\epsilon}\nabla^{\varepsilon}\bar{K}_{\epsilon(ij)\varepsilon}.$
(7)

The tensor $\bar{K}_{\epsilon\varepsilon ij}$ is defined as
$K_{\epsilon\varepsilon ij}f^{{}^{\prime}}(P)$, where
$f^{{}^{\prime}}(P)=df/dP$, and $K_{\epsilon\varepsilon ij}$ can be expressed
as p

$K_{\epsilon\varepsilon
ij}=12\left[6R_{\epsilon}{}^{\zeta}(_{i}{}^{\sigma}R_{\nu})_{\sigma\varepsilon\zeta}+\frac{1}{2}R_{\epsilon\varepsilon}^{\zeta\sigma}R_{ij\zeta\sigma}+2g_{\varepsilon}(_{i}{}R_{\nu})_{\sigma\epsilon\zeta}R^{\zeta\sigma}-4R_{\zeta}(_{i}g_{j})(_{\epsilon}R_{\varepsilon})^{\zeta}-2g_{\epsilon}(_{i}R_{j})_{\sigma\varepsilon\zeta}R^{\zeta\sigma}-2R_{\epsilon}(_{\epsilon}R_{\varepsilon})_{j}\right].$
(8)

For a FLRW background, the field equations (Eq. 5) take the following forms p

$3H^{2}=\kappa\rho_{eff},$ (9) $2\dot{H}+3H^{2}=-\kappa p_{eff},$ (10)

where, $\rho_{eff}=\rho_{m}+\rho_{f(P)}$, and $p_{eff}=p_{m}+p_{f(P)}$.
Furthermore, $\rho_{f(P)}$, and $p_{f(P)}$ are defined respectively as p

$\rho_{f(P)}=-f(P)+18\beta H^{4}(\dot{H}-H\frac{\partial}{\partial
t}+H^{2})f^{{}^{\prime}}(P),$ (11) $p_{f(P)}=f(P)+6\beta
H^{3}\left[2(H^{2}+2\dot{H})\frac{\partial}{\partial
t}-5H\dot{H}+H\frac{\partial^{2}}{\partial
t^{2}}-3H^{3}\right]f^{{}^{\prime}}(P).$ (12)

The EoS parameter ($\omega_{f(P)}=p_{f(P)}/\rho_{f(P)}$) therefore reads

$\omega_{f(P)}=\left[\frac{f(P)+6\beta
H^{3}\left\\{2(H^{2}+2\dot{H})\frac{\partial}{\partial
t}-5H\dot{H}+H\frac{\partial^{2}}{\partial
t^{2}}-3H^{3}\right\\}f^{{}^{\prime}}(P)}{-f(P)+18\beta
H^{4}(\dot{H}-H\frac{\partial}{\partial t}+H^{2})f^{{}^{\prime}}(P)}\right].$
(13)

Additionally, for a FRW metric, $P=6\beta H^{4}(2H^{2}+3\dot{H})$, and $\beta$
is defined as

$\beta=-\beta_{1}+8\beta_{2}+2\beta_{3}+4\beta_{4}.$ (14)

##  III Energy Conditions in $f(P)$ Gravity

The energy conditions transpire from the Raychaudhuri equation for the cosmic
expansion defined as

$\frac{d\theta}{d\tau}=\omega_{ij}\omega^{ij}-\left[R_{ij}u^{i}u^{j}+\frac{1}{3}\theta^{2}+\sigma_{ij}\sigma^{ij}\right],$
(15)

where $\theta$, $\omega_{ij}$, and $\sigma^{ij}$ are the expansion, rotation
and shear coupled to the congruence described by the vector $u^{i}$, and
$R_{ij}$ denote the Ricci tensor.

Additionally, the evolutionary expansion for the congruence of null geodesics
expressed through a null vector $\kappa^{i}$ reads

$\frac{d\theta}{d\tau}=\omega_{ij}\omega^{ij}-\left[R_{ij}\kappa^{i}\kappa^{j}+\frac{1}{2}\theta^{2}+\sigma_{ij}\sigma^{ij}\right],$
(16)

where $\theta$, $\omega_{ij}$, and $\sigma^{ij}$ are now coupled to the null
geodesics.

One may note that the Eqs. 15 and 16 are solely geometric in nature and are
identical for all gravitational theories.
It is also evident that $\sigma_{ij}\sigma^{ij}\geq 0$ since the shear is a
spatial tensor and therefore, from Eqs. 15 and 16 it is clear that for any
hypersurface orthogonal congruences described by $\omega_{ij}=0$, the
necessary criteria required to establish gravity as an attractive force read
$R^{ij}u^{i}u^{j}\geq 0$ and $R^{ij}\kappa^{i}\kappa^{j}\geq 0$ ft48 . These
inequalities further corroborate the fact that the geodesic congruences pivot
within some finite values of the parameters on the geodesics ft48 . We shall
now try to derive the energy conditions in $f(P)$ gravity using the modified
gravitational equations 5. The energy conditions are self-governing and are
entirely geometrical in nature jamil ; ft33  and can be expressed as follows
garcia ; ft41 :

$\textbf{{Null Energy Condition} (NEC):}\hskip 7.22743pt\rho_{eff}+p_{eff}\geq
0,$ (17) $\textbf{{Strong Energy Condition} (SEC):}\hskip
7.22743pt\rho_{eff}+3p_{eff}\geq 0,\hskip 7.22743pt\text{and}\hskip
7.22743pt\rho_{eff}+p_{eff}\geq 0,$ (18) $\textbf{{Weak Energy Condition}
(WEC):}\hskip 7.22743pt\rho_{eff}\geq 0,\hskip 7.22743pt\text{and}\hskip
7.22743pt\rho_{eff}+p_{eff}\geq 0.$ (19)

Now, in order to proceed, it is useful to re-express the deceleration ($q$),
and jerk parameters ($j$) in terms of the Hubble parameter as follows:

$q=-1\left[\frac{1}{H^{2}}\frac{\dot{a}}{a}\right],$ (20)

and

$j=\left[\frac{1}{H^{3}}\frac{\ddot{a}}{a}\right].$ (21)

Using Eq. 20 and Eq. 21, we can now write the time derivatives of $H$ in terms
of $q$ and $j$ as follows:

$\dot{H}=-H^{2}(q+1),$ (22)

and

$\ddot{H}=H^{4}(j+2+3q).$ (23)

It may be noted that the matter sector (i.e, $\rho_{m}$ and $p_{m}$) do not
violate the energy conditions and that the violation of energy conditions is
solely regulated by $\rho_{fp}$ and $p_{fp}$. Therefore, by substituting Eqs.
22, and 23 in Eqs. 11, and 12, we define the energy conditions in $f(P)$
theories of gravity as follows:

$\textbf{NEC:}\hskip 7.22743pt-f(P)-18\beta
H^{4}(H^{2}(q+1)-H\frac{\partial}{\partial t}+H^{2})f^{{}^{\prime}}(P)\\\
+f(P)+6\beta
H^{3}\left[2(H^{2}-2\left\\{H^{2}(q+1))\right\\}\frac{\partial}{\partial
t}+5H(H^{2}(q+1))+H\frac{\partial^{2}}{\partial
t^{2}}-3H^{3}\right]f^{{}^{\prime}}(P)\geq 0$ (24)

$\textbf{SEC:}\hskip 7.22743pt-f(P)-18\beta
H^{4}(H^{2}(q+1)-H\frac{\partial}{\partial t}+H^{2})f^{{}^{\prime}}(P)\\\
+3\left[f(P)+6\beta
H^{3}\left[2(H^{2}-2\left\\{H^{2}(q+1))\right\\}\frac{\partial}{\partial
t}+5H(H^{2}(q+1))+H\frac{\partial^{2}}{\partial
t^{2}}-3H^{3}\right]f^{{}^{\prime}}(P)\right]\geq 0,\\\ \text{and}\\\ \hskip
7.22743pt-f(P)-18\beta H^{4}(H^{2}(q+1)-H\frac{\partial}{\partial
t}+H^{2})f^{{}^{\prime}}(P)\\\ +f(P)+6\beta
H^{3}\left[2(H^{2}-2\left\\{H^{2}(q+1))\right\\}\frac{\partial}{\partial
t}+5H(H^{2}(q+1))+H\frac{\partial^{2}}{\partial
t^{2}}-3H^{3}\right]f^{{}^{\prime}}(P)\geq 0$ (25)

$\textbf{WEC:}\hskip 7.22743pt-f(P)-18\beta
H^{4}(H^{2}(q+1)-H\frac{\partial}{\partial t}+H^{2})f^{{}^{\prime}}(P)\geq
0,\\\ \text{and}\\\ \hskip 7.22743pt-f(P)-18\beta
H^{4}(H^{2}(q+1)-H\frac{\partial}{\partial t}+H^{2})f^{{}^{\prime}}(P)\\\
+f(P)+6\beta
H^{3}\left[2(H^{2}-2\left\\{H^{2}(q+1))\right\\}\frac{\partial}{\partial
t}+5H(H^{2}(q+1))+H\frac{\partial^{2}}{\partial
t^{2}}-3H^{3}\right]f^{{}^{\prime}}(P)\geq 0.$ (26)

##  IV EC bounds on $f(P)$ theories

We shall now use Eqs. 24, 25 and 26 to bound some suitable $f(P)$ gravity
models. To place the bounds, we shall ensure that the energy density (Eq. 11)
must remain positive and the pressure (Eq. 12) must be negative to assure that
the constraints on $f(P)$ gravity models are compatible with the current
accelerated expansion of the Universe. Next, we shall investigate to find
corners in parameter spaces for a chosen model which permits the EoS parameter
(Eq. 13) to attain a value close to $-1$ to ensure that the bounds are in
harmony with the current observational data. We note that for an accelerating
Universe, the SEC must violate and therefore offers a consistency check to the
constraints obtained for the models.
In the subsequent subsections, we shall try to place bounds for two specific
$f(P)$ gravity models where the functional forms of $f(P)$ are represented as
a) $f(P)=\alpha\sqrt{P}$, and b) $f(P)=\alpha\exp(P)$, where $\alpha$ is the
sole model parameter for these models.

###  IV.1 $f(P)=\alpha\sqrt{P}$

For the first case, let us assume the functional form of $f(P)$ is given by

$f(P)=\alpha\sqrt{P},$ (27)

where $\alpha$ is the model parameter.

The expressions for the NEC (Eq. 24), SEC (Eq. 25), and WEC (Eq. 26) read
respectively as

$\textbf{NEC:}\hskip 7.22743pt-\left[\frac{\sqrt{\frac{3}{2}}\alpha\beta
H^{6}(6q+7)}{\sqrt{\beta\left(-H^{6}\right)(3q+1)}}\right]\\\ +\frac{\beta
H^{5}\left[60H(q+1)\sqrt{\beta\left(-H^{6}\right)(3q+1)}+\sqrt{6}\alpha\left\\{12(j+3q+2)-H(12q(q+2)+11)\right\\}\right]}{2\sqrt{\beta\left(-H^{6}\right)(3q+1)}}\geq
0,$ (28) $\textbf{SEC:}\hskip
7.22743pt-\left[\frac{\sqrt{\frac{3}{2}}\alpha\beta
H^{6}(6q+7)}{\sqrt{\beta\left(-H^{6}\right)(3q+1)}}\right]\\\
+3\left[\frac{\beta
H^{5}\left[60H(q+1)\sqrt{\beta\left(-H^{6}\right)(3q+1)}+\sqrt{6}\alpha\left\\{12(j+3q+2)-H(12q(q+2)+11)\right\\}\right]}{2\sqrt{\beta\left(-H^{6}\right)(3q+1)}}\right]\geq
0,\\\ \text{and}\\\ -\left[\frac{\sqrt{\frac{3}{2}}\alpha\beta
H^{6}(6q+7)}{\sqrt{\beta\left(-H^{6}\right)(3q+1)}}\right]\\\ +\frac{\beta
H^{5}\left[60H(q+1)\sqrt{\beta\left(-H^{6}\right)(3q+1)}+\sqrt{6}\alpha\left\\{12(j+3q+2)-H(12q(q+2)+11)\right\\}\right]}{2\sqrt{\beta\left(-H^{6}\right)(3q+1)}}\geq
0,$ (29) $\textbf{WEC:}\hskip
7.22743pt-\left[\frac{\sqrt{\frac{3}{2}}\alpha\beta
H^{6}(6q+7)}{\sqrt{\beta\left(-H^{6}\right)(3q+1)}}\right]\geq 0\\\
\text{and}\\\ -\left[\frac{\sqrt{\frac{3}{2}}\alpha\beta
H^{6}(6q+7)}{\sqrt{\beta\left(-H^{6}\right)(3q+1)}}\right]\\\ +3\frac{\beta
H^{5}\left[60H(q+1)\sqrt{\beta\left(-H^{6}\right)(3q+1)}+\sqrt{6}\alpha\left\\{12(j+3q+2)-H(12q(q+2)+11)\right\\}\right]}{2\sqrt{\beta\left(-H^{6}\right)(3q+1)}}\geq
0.$ (30)

The expressions for the energy density $(\rho_{f(P)})$ and pressure
($p_{f(P)}$) for this model read respectively as

$\rho_{f(P)}=-\left[\frac{\sqrt{\frac{3}{2}}\alpha\beta
H^{6}(6q+7)}{\sqrt{\beta\left(-H^{6}\right)(3q+1)}}\right],$ (31)

$p_{f(P)}=\frac{\beta
H^{5}\left[60H(q+1)\sqrt{\beta\left(-H^{6}\right)(3q+1)}+\sqrt{6}\alpha\left\\{12(j+3q+2)-H(12q(q+2)+11)\right\\}\right]}{2\sqrt{\beta\left(-H^{6}\right)(3q+1)}}.$
(32)

The EoS parameter ($\omega_{f(P)}$) reads

$\omega_{f(P)}=\frac{\sqrt{6}\alpha(H(12q(q+2)+11)-12(j+3q+2))-60H(q+1)\sqrt{\beta\left(-H^{6}\right)(3q+1)}}{\sqrt{6}\alpha
H(6q+7)}.$ (33)

Substituting the values of $H$, $q$ and $j$ in Eqs. 31, 32, and 33, we obtain

$\rho_{f(P)}\simeq-1.9\alpha\sqrt{\beta}$ (34) $p_{f(P)}\simeq
9.2\alpha\sqrt{\beta}+1.5\beta$ (35)

and

$\omega_{f(P)}\simeq-\frac{0.8\sqrt{\beta}}{\alpha}-5.0.$ (36)

From Eqs. 34, 35, and 36 it is clear that:

  * •

For $\rho_{f(P)}\geq 0$: $\alpha\leq 0$, and $\beta\geq 0$.

  * •

For $p_{f(P)}\leq 0$: $\alpha\leq 0$, $\beta\geq 0$ and
$\frac{\alpha}{\sqrt{\beta}}\lesssim-0.15$.

  * •

For $\omega_{f(P)}\simeq-1$: $\alpha\leq 0$, $\beta\geq 0$ and
$\frac{\alpha}{\sqrt{\beta}}\simeq-0.2$.

Figure 1: Top left panel shows the profile of energy density $\rho_{f(P)}$,
top right panel shows the profile of the EoS parameter $\omega_{f(P)}$, bottom
left panel shows the profile of the Strong Energy Condition and the bottom
right panel shows the profile of the Weak Energy Condition for the first
($f(P)=\alpha\sqrt{P}$) model. The figures are drawn for
$\beta\left\\{0.1,0.11\right\\}$, and $\alpha\left\\{-0.04,-0.09\right\\}$.

###  IV.2 $f(P)=\alpha\exp(P)$

Let us now set the functional form of $f(P)$ to the following

$f(P)=\alpha\exp(P),$ (37)

with $\alpha$ being the model parameter.

Similar to the previous case, we shall express the NEC (Eq. 24), SEC (Eq. 25),
and WEC (Eq. 26) for this model respectively as

$\textbf{NEC:}\hskip 7.22743pt\left[\alpha e^{-6\beta
H^{6}(3q+1)}\left\\{18\beta H^{6}\left(-36\beta
H^{6}(q+1)(3q+1)-q\right)-1\right\\}\right]\\\
+\left[\begin{multlined}\left[30\beta H^{6}(q+1)+\alpha e^{-6\beta
H^{6}(3q+1)}\right]\\\ \times\left[\left(18\beta H^{6}\left(12\beta
H^{5}(3q+1)\left(36\beta
H^{6}(3q+1)(j+3q+2)-2H(q+1)(2q+1)-5(j+3q+2)\right)-1\right)+1\right)\right]\end{multlined}\left[30\beta
H^{6}(q+1)+\alpha e^{-6\beta H^{6}(3q+1)}\right]\\\ \times\left[\left(18\beta
H^{6}\left(12\beta H^{5}(3q+1)\left(36\beta
H^{6}(3q+1)(j+3q+2)-2H(q+1)(2q+1)-5(j+3q+2)\right)-1\right)+1\right)\right]\right]\geq
0,$ (38) $\textbf{SEC:}\hskip 7.22743pt\left[\alpha e^{-6\beta
H^{6}(3q+1)}\left\\{18\beta H^{6}\left(-36\beta
H^{6}(q+1)(3q+1)-q\right)-1\right\\}\right]\\\
+3\left[\begin{multlined}\left[30\beta H^{6}(q+1)+\alpha e^{-6\beta
H^{6}(3q+1)}\right]\\\ \times\left[\left(18\beta H^{6}\left(12\beta
H^{5}(3q+1)\left(36\beta
H^{6}(3q+1)(j+3q+2)-2H(q+1)(2q+1)-5(j+3q+2)\right)-1\right)+1\right)\right]\end{multlined}\left[30\beta
H^{6}(q+1)+\alpha e^{-6\beta H^{6}(3q+1)}\right]\\\ \times\left[\left(18\beta
H^{6}\left(12\beta H^{5}(3q+1)\left(36\beta
H^{6}(3q+1)(j+3q+2)-2H(q+1)(2q+1)-5(j+3q+2)\right)-1\right)+1\right)\right]\right]\geq
0,\\\ \text{and}\\\ \left[\alpha e^{-6\beta H^{6}(3q+1)}\left\\{18\beta
H^{6}\left(-36\beta H^{6}(q+1)(3q+1)-q\right)-1\right\\}\right]\\\
+\left[\begin{multlined}\left[30\beta H^{6}(q+1)+\alpha e^{-6\beta
H^{6}(3q+1)}\right]\\\ \times\left[\left(18\beta H^{6}\left(12\beta
H^{5}(3q+1)\left(36\beta
H^{6}(3q+1)(j+3q+2)-2H(q+1)(2q+1)-5(j+3q+2)\right)-1\right)+1\right)\right]\end{multlined}\left[30\beta
H^{6}(q+1)+\alpha e^{-6\beta H^{6}(3q+1)}\right]\\\ \times\left[\left(18\beta
H^{6}\left(12\beta H^{5}(3q+1)\left(36\beta
H^{6}(3q+1)(j+3q+2)-2H(q+1)(2q+1)-5(j+3q+2)\right)-1\right)+1\right)\right]\right]\geq
0,$ (39) $\textbf{WEC:}\hskip 7.22743pt\left[\alpha e^{-6\beta
H^{6}(3q+1)}\left\\{18\beta H^{6}\left(-36\beta
H^{6}(q+1)(3q+1)-q\right)-1\right\\}\right]\geq 0\\\ \text{and}\\\
\left[\alpha e^{-6\beta H^{6}(3q+1)}\left\\{18\beta H^{6}\left(-36\beta
H^{6}(q+1)(3q+1)-q\right)-1\right\\}\right]\\\
+\left[\begin{multlined}\left[30\beta H^{6}(q+1)+\alpha e^{-6\beta
H^{6}(3q+1)}\right]\\\ \times\left[\left(18\beta H^{6}\left(12\beta
H^{5}(3q+1)\left(36\beta
H^{6}(3q+1)(j+3q+2)-2H(q+1)(2q+1)-5(j+3q+2)\right)-1\right)+1\right)\right]\end{multlined}\left[30\beta
H^{6}(q+1)+\alpha e^{-6\beta H^{6}(3q+1)}\right]\\\ \times\left[\left(18\beta
H^{6}\left(12\beta H^{5}(3q+1)\left(36\beta
H^{6}(3q+1)(j+3q+2)-2H(q+1)(2q+1)-5(j+3q+2)\right)-1\right)+1\right)\right]\right]\geq
0.$ (40)

The expressions for the energy density $(\rho_{f(P)})$ and pressure
($p_{f(P)}$) for the second model read respectively as

$\rho_{f(P)}=\left[\alpha e^{-6\beta H^{6}(3q+1)}\left\\{18\beta
H^{6}\left(-36\beta H^{6}(q+1)(3q+1)-q\right)-1\right\\}\right],$ (41)

$p_{f(P)}=\begin{multlined}\left[30\beta H^{6}(q+1)+\alpha e^{-6\beta
H^{6}(3q+1)}\right]\\\ \times\left[\left(18\beta H^{6}\left(12\beta
H^{5}(3q+1)\left(36\beta
H^{6}(3q+1)(j+3q+2)-2H(q+1)(2q+1)-5(j+3q+2)\right)-1\right)+1\right)\right]\end{multlined}\left[30\beta
H^{6}(q+1)+\alpha e^{-6\beta H^{6}(3q+1)}\right]\\\ \times\left[\left(18\beta
H^{6}\left(12\beta H^{5}(3q+1)\left(36\beta
H^{6}(3q+1)(j+3q+2)-2H(q+1)(2q+1)-5(j+3q+2)\right)-1\right)+1\right)\right].$
(42)

The EoS parameter ($\omega_{f(P)}$) for this model reads

$\omega_{f(P)}=\frac{\begin{multlined}\left[30\beta H^{6}(q+1)+\alpha
e^{-6\beta H^{6}(3q+1)}\right]\\\ \times\left[\left(18\beta H^{6}\left(12\beta
H^{5}(3q+1)\left(36\beta
H^{6}(3q+1)(j+3q+2)-2H(q+1)(2q+1)-5(j+3q+2)\right)-1\right)+1\right)\right]\end{multlined}\left[30\beta
H^{6}(q+1)+\alpha e^{-6\beta H^{6}(3q+1)}\right]\\\ \times\left[\left(18\beta
H^{6}\left(12\beta H^{5}(3q+1)\left(36\beta
H^{6}(3q+1)(j+3q+2)-2H(q+1)(2q+1)-5(j+3q+2)\right)-1\right)+1\right)\right]}{\alpha
e^{-6\beta H^{6}(3q+1)}\left(18\beta H^{6}\left(-36\beta
H^{6}(q+1)(3q+1)-q\right)-1\right)}.$ (43)

We now substitute the respective values of $H$, $q$ and $j$ into 41, 42, and
43 to obtain

$\rho_{f(P)}\simeq\alpha e^{0.42\beta}(\beta(2.3\beta+1.1)-1)$ (44)
$p_{f(P)}\simeq\alpha
e^{0.3\beta}(\beta(\beta(6.9\beta+13.5)-2.0)+1.)+1.5\beta$ (45)

and

$\omega_{f(P)}\simeq-\frac{\alpha(\beta(\beta(6.9\beta+13.5)-2.0)+1.)+1.5e^{-0.4\beta}\beta}{\alpha(\beta(-2.3\beta-1.1)+1)}.$
(46)

Given the complexity of Eqs. 44, 45, and 46, it is therefore difficult to
constrain $\beta$ and $\alpha$. Let us therefore proceed by assuming first
$\beta=+1$, and then $\beta=-1$ and check for which of these two cases, we get
$\rho_{f(P)}\geq 0$, and $p_{f(P)}\leq 0$.

Case 1: $\beta=+1$ gives

$\rho_{f(P)}\simeq 3.4\alpha,$ (47)

and

$p_{f(P)}\simeq 29.3\alpha+1.5.$ (48)

From Eq. 47, and 48 we get the following constraints for $\alpha$:

  * •

For $\rho_{f(P)}\geq 0:\alpha\geq 0$,

  * •

For $p_{f(P)}\leq 0$: $\alpha\leq 0$.

Since to obtain $\rho_{f(P)}\geq 0$, and $p_{f(P)}\leq 0$, $\alpha$ needs to
be positive and negative at the same time which is unphysical, it is therefore
clear that positive values for $\beta$ are not permissible for this model.

Case 2: $\beta=-1$ gives

$\rho_{f(P)}\simeq 0.2\alpha,$ (49)

and

$p_{f(P)}\simeq 6.2\alpha-1.5.$ (50)

Eqs. 49, and 50 result in the following constraints for $\alpha$:

  * •

For $\rho_{f(P)}\geq 0:\alpha\geq 0$,

  * •

For $p_{f(P)}\leq 0$: $0\leq\alpha\lesssim 0.24$.

Therefore, it is evident that there exist suitable values for $\alpha$ which
permit an accelerating Universe for this $f(P)$ gravity model given that
$\beta$ remain negative. Finally $\omega_{f(P)}$ for this model reads

$\omega_{f(P)}\simeq 53.3-\frac{12.5}{\alpha},$ (51)

where we report that for Eq. 51 $\simeq-1$, $\alpha\simeq 0.23$.

Figure 2: Top left panel shows the profile of energy density $\rho_{f(P)}$,
top right panel shows the profile of the EoS parameter $\omega_{f(P)}$, bottom
left panel shows the profile of the Strong Energy Condition and the bottom
right panel shows the profile of the Weak Energy Condition for the second
($f(P)=\alpha\exp(P)$) model. The figures are drawn for
$\beta\left\\{-1.1,-1\right\\}$, and $\alpha\left\\{0.23,0.24\right\\}$.

##  V Conclusions

Einsteinian cubic gravity (ECG) p31  is a class of modified gravity theories
employing cubic contractions of the Riemann tensor and have undergone rapid
and significant development in recent years. $f(P)$ gravity is a novel
extension of ECG in which the Ricci scalar in the action is replaced by a
function of the curvature invariant $P$ which represents the contractions of
the Riemann tensor at the cubic order p . This class of modified gravity
theory have been very successful in several cosmological domains such as
inflation p35 , spherically symmetric black hole solutions p32 ; p33 ; p34 ,
and late-time acceleration p ; 2p .
The present work is concentrated on bounding some $f(P)$ gravity models to
understand their viability in cosmology. Energy conditions are interesting
linear relationships between pressure and density and have been extensively
employed to derive interesting results in Einstein’s gravity, and are also an
excellent tool to impose constraints on any cosmological model. To place the
bounds, we have ensured that the energy density (Eq. 11) must remain positive
and the pressure (Eq. 12) must be negative to assure that the constraints on
$f(P)$ gravity models are compatible with the current accelerated expansion of
the Universe, and have also investigated to find corners in parameter spaces
for a chosen model which permit the EoS parameter (Eq. 13) to attain a value
close to $-1$ to ensure that the bounds are in harmony with the current
observational data.
In this work, we report bounds for two specific $f(P)$ gravity models where
the functional forms of $f(P)$ are represented as a) $f(P)=\alpha\sqrt{P}$,
and b) $f(P)=\alpha\exp(P)$, where $\alpha$ is the sole model parameter for
these models. We report that for the first model, in order to obtain
$\rho_{f(P)}\geq 0$, $p_{f(P)}\leq 0$, and $\omega_{f(P)}\simeq-1$, the model
parameter $\alpha$ must be negative ($\alpha\leq 0$), and $\beta$ must be
positive ($\beta\geq 0$), and $\frac{\alpha}{\sqrt{\beta}}\simeq-0.2$, while
for the second model, we could not report any conclusive constraints on
$\beta$ and $\alpha$ owing to the complexity of the Eqs. 44, 45, and 46.
Therefore, in order to obtain some qualitative understanding of the parameters
$\alpha$ and $\beta$, we proceeded by assuming first $\beta=+1$, and then
$\beta=-1$ and checked for which of these two cases, we get $\rho_{f(P)}\geq
0$, and $p_{f(P)}\leq 0$. We found that when $\beta=+1$, to obtain
$\rho_{f(P)}\geq 0$, and $p_{f(P)}\leq 0$, $\alpha$ needs to be positive and
negative at the same time which is unphysical and therefore makes it clear-cut
that positive values of $\beta$ are not permissible for this model. However,
for $\beta=-1$, there breathes suitable values for $\alpha$ which satisfy the
aforementioned conditions and also assures that $\omega_{f(P)}\simeq-1$ and
reads $\alpha\simeq 0.23$. Furthermore, we report from Figs. 1, and 2 that for
both the models, the SEC get violated and therefore offered a consistency
check to the constraints obtained for both the models.

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