∎

e1e-mail<EMAIL_ADDRESS>e2e-mail<EMAIL_ADDRESS>e3e-mail:
<EMAIL_ADDRESS>
11institutetext: UM-DAE Centre for Excellence in Basic Sciences, Mumbai
400098, India 22institutetext: Department of Physics, Indian Institute of
Technology Bombay, Mumbai 400076, India 11institutetext: UM-DAE Centre for
Excellence in Basic Sciences, Mumbai 400098, India 22institutetext: Department
of Physics, Indian Institute of Technology Bombay, Mumbai 400076, India

# Can we bypass no-go theorem for Ricci-inverse Gravity?

Indranil Dase1,addr1  Joseph P Johnsone2,addr2  S. Shankaranarayanane3,addr2

(Received: date / Accepted: date)

###### Abstract

Recently, Amendola et al. proposed a geometrical theory of gravity containing
higher-order derivative terms 2020-Amendola.etal-Phys.Lett.B . The authors
introduced anticurvature scalar $(A)$, which is the trace of the inverse of
the Ricci tensor ($A^{\mu\nu}=R_{\mu\nu}^{-1}$). In this work, we consider two
classes of Ricci-inverse — Class I and Class II — models. Class I models are
of the form $f(R,A)$ where $f$ is a function of Ricci and anticurvature
scalars. Class II models are of the form ${\cal F}(R,A^{\mu\nu}A_{\mu\nu})$
where ${\cal F}$ is a function of Ricci scalar and square of anticurvature
tensor. For both these classes of models, we numerically solve the modified
Friedmann equations in the redshift range $1500<z<0$. We show that the late-
time evolution of the Universe, i.e., evolution from matter-dominated epoch to
accelerated expansion epoch, _can not_ be explained by these two classes of
models. Using the reduced action approach, we show that we _can not bypass_
the no-go theorem for Ricci-inverse gravity models. Finally, we discuss the
implications of our analysis for the early-Universe cosmology.

††journal: Eur. Phys. J. Plus

##  1 Introduction

General Relativity (GR) has passed many observational tests from weak-gravity
regime to strong-gravity regime 2006-Will-LivingRev.Rel. ; 2016-Abbott.Others-
Phys.Rev.Lett. . However, GR with a Universe containing baryonic matter and
radiation can not explain some cosmological observations. The most important
among them are the cosmological inflation in the early universe
2000–Cosmologicalinflationlarge , and the late-time accelerated expansion of
the universe 1999-Perlmutter.Others-Astrophys.J. ; 1998-Riess.Others-Astron.J.
. Even though a solution for an expanding universe exists in the GR framework
with baryonic matter/radiation, to explain the accelerated expansion, one
needs to make modifications to the theory. One option is to introduce either
exotic forms of energy/matter with negative pressure, commonly known as dark
energy 2000-Sahni.Starobinsky-Int.J.Mod.Phys. ; 2006-Copeland.etal-
Int.J.Mod.Phys. ; 2012-Yoo.Watanabe-Int.J.Mod.Phys.D .

The most popular dark energy model, the $\Lambda$CDM model, can successfully
explain the observational results; however, it is not without problems. The
value of $\Lambda$ is many orders smaller than the value of vacuum energy
predicted by the standard model of particle physics 1968-Sakharov-
Sov.Phys.Dokl. ; 1989-Weinberg-Rev.Mod.Phys. . Recently, there also has been
disagreement between the value of the cosmological parameters estimated from
the high and low redshift observations using the $\Lambda$CDM model
2013-Marra.etal-Phys.Rev.Lett. ; 2013-Verde.etal-Phys.DarkUniv. .

Another possible way of explaining the accelerated expansion is by modifying
the geometric part of the Einstein Hilbert action, giving rise to the modified
gravity theories 2010-DeFelice.Tsujikawa-LivingRev.Rel. ; 2017-Nojiri.etal-
Phys.Rept. ; 2012-Bamba.etal-Astrophys.SpaceSci. . It has been shown that
modified gravity models with higher-order curvature terms like the $f(R)$
models can lead to the accelerated expansion of the Universe
2010-Sotiriou.Faraoni-Rev.Mod.Phys. ; 2019-Johnson.Shankaranarayanan-
Phys.Rev.D ; 2011-Nojiri.Odintsov-Phys.Rept. . However, we still do not have a
model which is in perfect agreement with _all_ the cosmological observations.
For instance, we still don’t have an understanding of the late-time
acceleration and the associated issues like the $H_{0}$ tension and
$\sigma_{8}$ tension. 2003-Peebles.Ratra-Rev.Mod.Phys. ;
2020-DiValentino.etal-Phys.Rev.D ; 2022-Johnson.etal-JCAP  Hence it is always
interesting to consider new possible mechanisms via geometric quantities that
can potentially explain the late-time acceleration without invoking exotic
physics.

In Ref. 2020-Amendola.etal-Phys.Lett.B  an alternate gravity theory has been
proposed in which the Einstein Hilbert action is modified by introducing the
anti-curvature tensor ($A^{\mu\nu}$) defined by
$A^{\mu\nu}R_{\nu\sigma}=\delta^{\mu}_{\sigma}$ and the anti-curvature scalar
$A\equiv g_{\mu\nu}A^{\mu\nu}$. It has to be noted that the anti-curvature
scalar $A$ is _not_ the inverse of the Ricci scalar $R$ i.e., $A\neq R^{-1}$.
Amandola et al. considered modified gravity action containing positive or
negative powers of the anti-curvature scalar. The authors proved a general no-
go theorem for these models and showed that cosmic trajectories from a
decelerated phase could not smoothly join an accelerated phase.

In this work, we consider generalized anti-curvature models to confirm/infirm
the no-go theorem for the Ricci-inverse gravity theories. We consider two
broad classes of Ricci-inverse — Class I and Class II — models. Class I models
are of the form $f(R,A)$ where $f$ is a function of Ricci and anticurvature
scalars. Class II models are of the form ${\cal F}(R,A^{\mu\nu}A_{\mu\nu})$
where ${\cal F}$ is a function of Ricci scalar and square of anticurvature
tensor. We derive the modified Friedmann equations for the two classes and
show that the models have an attractor solution describing the accelerated
expansion. Furthermore, we explicitly show the presence of singularities does
not allow a smooth transition from a matter-dominated Universe to the phase of
accelerated expansion. For a better understanding, we use the reduced action
approach for these models, identify these singularities, and show that they
are consistent with the results obtained from the study of the evolution
equations. To our knowledge, such an analysis has not been carried out for
these classes of anti-curvature gravity models.

In Section (2), we introduce the Ricci and anticurvature tensors and scalars
in the flat-space FLRW metric. In Section (3), we introduce the $f(R,A)$ model
where $f(R,A)$ is an arbitrary function of $R$ and $A$, and then after
subsequently analyzing various specific sub-models in this category, we show
that it cannot be a viable cosmological model as it fails to describe the
transition from the matter-dominated Universe to the late-time accelerated
expansion. In Section (4) we introduce ${\cal F}(R,A^{\mu\nu}A_{\mu\nu})$
model where ${\cal F}(R,A^{\mu\nu}A_{\mu\nu})$ is an arbitrary function of $R$
and $A^{\mu\nu}$ and derive the general evolution equations. Analyzing the
evolution for a few specific examples in this class of models, we show that
the problems that plagued the $f(R,A)$ models remains for the ${\cal
F}(R,A^{\mu\nu}A_{\mu\nu})$ models as well. In Section (5), we use the reduced
action approach to obtain a fundamental understanding of the reason why the
Ricci-inverse models can not explain the evolution of the Universe into the
late-time accelerated phase. Section 6 looks at the possibility of mapping the
modified gravity models with Ricci-inverse gravity models. In Section (7), we
conclude by briefly discussing the result and a possible model that might
succeed. Appendices A and B contain details of the calculations in the main
text.

Metric signature $(-,+,+,+)$ and natural units are used throughout the work
where $c=\hbar=8\pi G=1$. $H$ is the Hubble parameter defined as
$H=\dot{a}(t)/a(t)$, where $a(t)$ is the scale factor and is related to the
redshift $z$ as $a=1/(1+z)$. Unless otherwise specified dot stands for $d/dt$
and prime stands for $d/d\log a$, subscript $``R"$ denotes derivative with
respect to $R$, subscript $``A"$ denotes derivative with respect to $A$, and
subscript $``A^{2}"$ denotes derivative with respect to
$A_{\mu\nu}A^{\mu\nu}$.

##  2 Ricci-inverse gravity models

As mentioned above, we aim to confirm/infirm whether we can bypass the no-go
theorem for Ricci-inverse gravity. In other words, we want to identify a class
of Ricci-inverse models that can describe the evolution of the Universe from
the radiation-dominated epoch to the matter-dominated epoch and then to the
late-time accelerated expansion epoch. In this section, we list the two
classes of models that we consider and obtain the modified field equations.

###  2.1 Two classes of Ricci-inverse gravity models

In this work, we consider the following two classes of models:

  1. 1.

Class I models: In this class we consider the action of the following form:

$S=\int d^{4}x\sqrt{-g}\left[f(R,A)+\mathcal{L}_{m}\right]$ (1)

where $f(R,A)$ is any general, smooth function of Ricci scalar ($R$) and
anticurvature scalar $(A$), and $\mathcal{L}_{m}$ refers to the matter
Lagrangian that is minimally coupled to gravity. One constraint is that, in
the appropriate limit, these models should reduce to GR. Varying the above
action w.r.t the metric leads to the following equation 2020-Amendola.etal-
Phys.Lett.B :

$\begin{split}f_{R}R^{\mu\nu}-f_{A}A^{\mu\nu}-\frac{1}{2}fg^{\mu\nu}+g^{\rho\mu}\nabla_{\alpha}\nabla_{\rho}f_{A}A^{\alpha}_{\sigma}A^{\nu\sigma}-\frac{1}{2}\nabla^{2}\left(f_{A}A^{\mu}_{\sigma}A^{\nu\sigma}\right)\\\
-\frac{1}{2}g^{\mu\nu}\nabla_{\alpha}\nabla_{\beta}(f_{A}A^{\alpha}_{\sigma}A^{\beta\sigma})-\nabla^{\mu}\nabla^{\nu}f_{R}+g^{\mu\nu}\nabla^{2}f_{R}=T^{\mu\nu}\end{split}$
(2)

  2. 2.

Class II models: In this class we consider the action of the following form:

$S=\int d^{4}x\sqrt{-g}\left[{\cal
F}(R,A^{\mu\nu}A_{\mu\nu})+\mathcal{L}_{m}\right]$ (3)

where ${\cal F}(R,A^{\mu\nu}A_{\mu\nu})$ is any general function containing
Ricci scalar and the inverse-Ricci tensor square ($A^{\mu\nu}A_{\mu\nu}$). One
constraint is that, in the appropriate limit, these models should reduce to
GR. Varying the above action w.r.t the metric leads to the following equation
(A contains the detailed derivation):

$\displaystyle{\cal F}_{R}R^{\mu\nu}-2{\cal
F}_{A^{2}}A^{\mu\rho}A^{\nu}_{\rho}-\frac{1}{2}{\cal
F}g^{\mu\nu}-\nabla^{\mu}\nabla^{\nu}{\cal F}_{R}+g^{\mu\nu}\nabla^{2}{\cal
F}_{R}+g^{\rho\nu}\nabla_{\alpha}\nabla_{\rho}({\cal
F}_{A^{2}}A_{\sigma\kappa}A^{\sigma\alpha}A^{\mu\kappa})$
$\displaystyle-\nabla^{2}({\cal
F}_{A^{2}}A_{\sigma\kappa}A^{\sigma\mu}A^{\nu\kappa})-g^{\mu\nu}\nabla_{\alpha}\nabla_{\rho}({\cal
F}_{A^{2}}A_{\sigma\kappa}A^{\sigma\alpha}A^{\rho\kappa})+2g^{\rho\nu}\nabla_{\rho}\nabla_{\alpha}({\cal
F}_{A^{2}}A_{\sigma\kappa}A^{\sigma\mu}A^{\alpha\kappa})$
$\displaystyle-g^{\rho\nu}\nabla_{\alpha}\nabla_{\rho}({\cal
F}_{A^{2}}A_{\sigma\kappa}A^{\sigma\mu}A^{\alpha\kappa})=T^{\mu\nu}$ (4)

###  2.2 Modified Friedmann equations

A large amount of cosmological and astrophysical observations provide
conclusive evidence that Universe underwent different epochs, starting from
the radiation-dominated epoch, followed by the matter-dominated epoch, and
finally, the accelerated epoch with smooth transitions in between. However, to
explain the current accelerated expansion, the standard model of cosmology
requires modifications either by introducing exotic matter fields or modifying
the geometric part. In this work, we consider Ricci-inverse models and aim to
describe the evolution of the Universe.

We consider a spatially flat Friedmann-Lemaître
-Robertson-Walker (FLRW) metric:

$g_{\mu\nu}={\rm diag}[-1,a^{2}(t),a^{2}(t),a^{2}(t)]$ (5)

where $a(t)$ is the scale factor. Various geometric quantities — anticurvature
tensor ($A^{\mu\nu}$), Ricci tensor ($R^{\mu\nu}$), Ricci scalar ($R$) and
anticurvature scalar ($A$) and anticurvature tensor square
($A^{\mu\nu}A_{\mu\nu}$) — for the FLRW space-time are given below:

$\begin{split}A^{\mu\nu}={\rm
diag}\left[-\frac{a(t)}{3\ddot{a}(t)},\frac{1}{2\dot{a}^{2}(t)+a(t)\ddot{a}(t)},\frac{1}{2\dot{a}^{2}(t)+a(t)\ddot{a}(t)},\frac{1}{2\dot{a}^{2}(t)+a(t)\ddot{a}(t)}\right]\\\
={\rm
diag}\left[-\frac{1}{3H^{2}(1+\xi)},\frac{1}{H^{2}(3+\xi)a^{2}(t)},\frac{1}{H^{2}(3+\xi)a^{2}(t)},\frac{1}{H^{2}(3+\xi)a^{2}(t)}\right]\\\
\end{split}$ (6) $\begin{split}R_{\mu\nu}={\rm
diag}\left[-\frac{3\ddot{a}(t)}{a(t)},2\dot{a}^{2}(t)+a(t)\ddot{a}(t),2\dot{a}^{2}(t)+a(t)\ddot{a}(t),2\dot{a}^{2}(t)+a(t)\ddot{a}(t)\right]\\\
={\rm
diag}\left[-3H^{2}(1+\xi),H^{2}(3+\xi)a^{2}(t),H^{2}(3+\xi)a^{2}(t),H^{2}(3+\xi)a^{2}(t)\right]\end{split}$
(7)
$\begin{split}R=\frac{6\left(\dot{a}^{2}(t)+a(t)\ddot{a}(t)\right)}{a^{2}(t)}=6H^{2}(2+\xi)\end{split}$
(8)
$\begin{split}A=\frac{2a(t)\left(\dot{a}^{2}(t)+5a(t)\ddot{a}(t)\right)}{3\ddot{a}(t)\left(2\dot{a}^{2}(t)+a(t)\ddot{a}(t)\right)}=\frac{2(6+5\xi)}{3H^{2}(1+\xi)(3+\xi)}\end{split}$
(9)
$A^{\mu\nu}A_{\mu\nu}=\frac{4(7\xi^{2}+15\xi+9)}{9H^{4}(1+\xi)^{2}(\xi+3)^{2}}$
(10)

where, $H\equiv\dot{a}/a$ is the Hubble parameter, and

$\xi\equiv\frac{H^{\prime}}{H}=\frac{a\ddot{a}}{\dot{a}^{2}}-1\,,$ (11)

is the negative of the slow-roll parameter $\epsilon$. Thus, the two modified
Friedmann equations are:

$\displaystyle\left(\frac{\dot{a}}{a}\right)^{2}$ $\displaystyle=$
$\displaystyle\dfrac{\rho_{\rm eff}}{3}\hskip 61.0pt$
$\displaystyle\left(\frac{\ddot{a}}{a}\right)$ $\displaystyle=$
$\displaystyle-\dfrac{1}{6}\left(\rho_{\rm eff}+3p_{\rm eff}\right)$ (12)

where $\rho_{\rm eff}$ and $p_{\rm eff}$ are the density and pressure of the
effective fluid containing multiple components. Using the definition of $\xi$
in Eq (11), combining the two equations in (12) leads to:

$p_{\rm eff}=-\frac{\rho_{\rm eff}}{3}(3+2\xi)\,.$ (13)

Thus, one can define an equation of state parameter of the effective fluid as:

$w_{\rm eff}=-\frac{2\xi}{3}-1\,.$ (14)

The value of $\xi$ at any time implies whether the expansion of the Universe
is accelerated or decelerated. Any cosmological model which is consistent with
observations should describe a smooth transition from a decelerated phase
($\ddot{a}<0$) to the accelerated phase ($\ddot{a}>0$). This corresponds to
the transition from $\xi<-1$ to $\xi>-1$.

For the Ricci-inverse models considered here, the Ricci-inverse corrections
contribute to the effective fluid density and pressure. Thus, we have:

$\displaystyle\rho_{\rm eff}$ $\displaystyle=$
$\displaystyle\rho_{t}+\rho_{A}\hskip 10.0pt;\hskip
10.0pt\rho_{t}=\rho_{m}+\rho_{r}\hskip 52.0pt$ $\displaystyle p_{\rm eff}$
$\displaystyle=$ $\displaystyle p_{t}+p_{A}\hskip 10.0pt;\hskip
10.0ptp_{t}=w_{m}\rho_{m}+w_{r}\rho_{r}=p_{r}$ (15)

where $(\rho_{A},p_{A})$ refer to the density and pressure of the
anticurvature terms, $(\rho_{m},p_{m})$ refer to the density and pressure of
the matter (baryons and dark matter) content in the Universe and
$(\rho_{r},p_{r})$ refer to the density and pressure of the radiation. It has
to be noted that the complete set of background evolution equations can be
written in terms of variables $\xi$, $\Omega_{r}$, $\Omega_{m}$, and the model
parameters that remain constant.

Since we assume that the late-time acceleration is due to the Ricci-inverse
modifications to GR, we do not include cosmological constant or dark energy
components. Substituting the form of $\rho_{\rm eff}$ from Eq. (15) in the
first Friedmann equation (12), we have:

$\Omega_{m}+\Omega_{r}+\Omega_{A}=1$ (16)

where $\Omega_{i}=\rho_{i}/3H^{2}$, where the subscript $i$ refers to
$r$(radiation) or $m$(matter) or $A$ (anitcurvature).

In this work, for both classes of models considered, we numerically solve the
modified Friedmann equations (12) numerically subject to the above constraint
Eq.(16) in the redshift range $1500<z<0$ for a range of initial values
$-3\leq\xi_{i}\leq 1$ that lead to non-diverging solutions. We assume the
matter to be pressureless dust. Hence, $w_{m}=0$ and $w_{r}=1/3$. We fix the
initial values for the remaining parameters to be $\Omega_{r_{i}}=0.39$,
$\Omega_{m_{i}}=0.59$ and $\xi_{i}^{\prime}=0$. The above initial values of
$\Omega_{r}$ and $\Omega_{m}$ correspond to radiation matter equality at
$z\sim 2740$. $\xi^{\prime}=0$ corresponds to the power-law evolution of the
scale factor $a$, which is consistent with the radiation/matter-dominated era.

In the rest of the work, we consider the two classes of Ricci-inverse models
introduced in Section (2.1). We show that the late-time evolution of the
Universe, i.e., evolution from matter-dominated epoch to accelerated expansion
epoch, can’t be explained by this model. Using the reduced action approach, we
provide a reason why we can not bypass the no-go theorem for Ricci-inverse
gravity models.

The following table contains values of $\xi$ and $w_{\rm eff}$ for different
epochs of the Universe. In particular, as we evolve the modified Friedmann
equations, $\xi$ needs to evolve from matter-dominated epoch ($\xi=-1.5$) to
the current epoch ($\xi\simeq-0.5$). As we will show, this transition is not
possible for these two classes of Ricci-inverse models.

$\xi$ | $p=-\frac{\rho}{3}(3+2\xi)$ | $w_{\rm eff}$ | Effective matter fluid description
---|---|---|---
0 | $p=-\rho$ | -1 | Cosmological constant
-2 | $p=\rho/3$ | 1/3 | Radiation
-1.5 | $p=0$ | 0 | Matter(pure dust)
-0.495 or -0.5 | $p=-0.67\rho$ | -0.67 | Current Universe
Table 1: $\xi$ for different epochs of the Universe

##  3 Class I models

As mentioned earlier, $f(R,A)$ in Eq. (1) is an arbitrary, smooth function of
Ricci and anticurvature scalars. We further classify this class into three
subclasses — Class Ia, Class Ib, and Class Ic. Class Ia models are polynomials
in $R$ and $A$. This model has been discussed in the literature
2020-Amendola.etal-Phys.Lett.B ; 2021-Do-Eur.Phys.J.C ; 2022-Do-Eur.Phys.J.C ;
2021-Scomparin . For completeness, we briefly discuss the results of this
class of models in B. Class Ib models are of the form $\exp[(RA)^{n}]$. Class
Ic models are non-polynomial functions of $R$ and $A$.

###  3.1 Class Ia models

In Ref. 2020-Amendola.etal-Phys.Lett.B , the authors proved a no-go theorem
for the case when $f(R,A)$ is a polynomial function of $R$ and $A$.
Specifically, the authors showed that for polynomial function of $R$ and $A$,
cosmic trajectories around $\xi=-1.5$ will never evolve to $\xi=-0.5$. B
contains the plots for these classes of models. In Ref. 2021-Do-Eur.Phys.J.C
the author showed that one gets a no-go theorem in the Ricci-inverse model for
both the isotropic and anisotropic inflation. This was further extended by
including the second-order anticurvature scalar term in Ref. 2022-Do-
Eur.Phys.J.C  where the author showed that it is impossible to have stable
inflation within FLRW cosmology.

This naturally leads to the following question: Can we bypass the no-go
theorem for an arbitrary power of $R$ and $A$? To address this, we consider
class Ib models.

###  3.2 Class Ib models

In this class, we consider the following form of $f(R,A)$:

$f(R,A)=Re^{\alpha(RA)^{n}}\,,$ (17)

where $n$ is an integer, and $\alpha$ (a dimensionless constant) decides the
deviation from GR. We have made the above choice for the following reasons:
First, it is equivalent to an infinite polynomial involving a combination of
powers of the Ricci and anticurvature scalars. Second, in the limit of
$\alpha\to 0$, it reduces to Einstein-Hilbert action. The modified Friedman
equations for this class of models are:

$\begin{split}\rho_{t}=3H^{2}e^{4^{n}\alpha\left(\frac{(2+\xi)(6+5\xi)}{(1+\xi)(3+\xi)}\right)^{n}}\hskip
2.0pt\left[\hskip 2.0pt1+\frac{2\times(3+2\xi)4^{n}\alpha
n\xi}{(1+\xi)(3+\xi)(6+5\xi)}\left(\frac{(2+\xi)(6+5\xi)}{(1+\xi)(3+\xi)}\right)^{n}\right.\\\
\left.+\frac{2\times 4^{n}\alpha
n\left(\frac{(2+\xi)(6+5\xi)}{(1+\xi)(3+\xi)}\right)^{n}\xi^{\prime}}{(1+\xi)^{2}(2+\xi)(3+\xi)^{2}(6+5\xi)^{2}}\hskip
2.0pt\Bigg{(}\hskip 2.0pt3(36+84\xi+75\xi^{2}+32\xi^{3})\right.\\\
\left.+6n\xi^{2}(3+4\xi)\left(1+4^{n}\alpha\left(\frac{(2+\xi)(6+5\xi)}{(1+\xi)(3+\xi)}\right)^{n}\right)\right.\\\
\left.+\xi^{4}\left(17+8n+8n\times
4^{n}\alpha\left(\frac{(2+\xi)(6+5\xi)}{(1+\xi)(3+\xi)}\right)^{n}\right)\Bigg{)}\right]\\\
\end{split}$ (18)

$\begin{split}p_{t}=\Omega_{r}H^{2}=H^{2}e^{4^{n}\alpha\left(\frac{(2+\xi)(6+5\xi)}{(1+\xi)(3+\xi)}\right)^{n}}\hskip
2.0pt\Bigg{[}\hskip 2.0pt-(3+2\xi)-\frac{2\times
4^{n}n\alpha\xi(3+2\xi)^{2}}{(1+\xi)(6+5\xi)}\left(\frac{(2+\xi)(6+5\xi)}{(1+\xi)(3+\xi)}\right)^{n}\\\
-2\times
4^{n}n\alpha\left(\frac{(2+\xi)(6+5\xi)}{(1+\xi)(3+\xi)}\right)^{n}\hskip
2.0pt\Bigg{(}\hskip
2.0pt\frac{3\xi^{\prime}}{(1+\xi)^{2}(6+5\xi)^{2}(3+\xi)^{2}}\hskip
2.0pt\Bigg{(}\hskip 2.0pt108+252\xi+225\xi^{2}\\\
+96\xi^{3}+17\xi^{4}+2n(3+2\xi)^{2}+2n(3+2\xi)^{2}\xi^{2}4^{n}\alpha\left(\frac{(2+\xi)(6+5\xi)}{(1+\xi)(3+\xi)}\right)^{n}\Bigg{)}\\\
+\frac{(\xi^{\prime})^{2}}{(1+\xi)^{3}(2+\xi)^{2}(3+\xi)^{3}(6+5\xi)^{3}}\Bigg{(}-9720-388556\xi-65070\xi^{2}-61371\xi^{3}-35532\xi^{4}-12834\xi^{5}\\\
-2782\xi^{6}-255\xi^{7}-12n\xi(2+\xi)(3+2\xi)(-27-36\xi+15\xi^{3}+5\xi^{4})\left(1+4^{n}\alpha\left(\frac{(2+\xi)(6+5\xi)}{(1+\xi)(3+\xi)}\right)^{n}\right)\\\
+4n^{2}\xi^{3}(3+2\xi)^{3}\left(1+2^{n}\alpha\left(3\times
2^{n}\left(\frac{(2+\xi)(6+5\xi)}{(1+\xi)(3+\xi)}\right)^{n}+8^{n}\alpha\left(\frac{(2+\xi)(6+5\xi)}{(1+\xi)(3+\xi)}\right)^{2n}\right)\right)\Bigg{)}\\\
+\frac{\xi^{\prime\prime}(108+252\xi+225\xi^{2}+96\xi^{3}+17\xi^{4}+2n\xi(3+\xi)^{2}\left(1+4^{n}\alpha\left(\frac{(2+\xi)(6+5\xi)}{(1+\xi)(3+\xi)}\right)^{n}\right)}{(1+\xi)^{2}(2+\xi)(3+\xi)^{2}(6+5\xi)^{2}}\Bigg{)}\Bigg{]}\end{split}$
(19)

Since the matter and radiation are non-interacting, they individually satisfy
the following energy conservation equations:

$\dot{\Omega}_{r}=-H(t)\Omega_{r}\left[4+\frac{2\dot{H}}{H^{2}}\right],\quad\dot{\Omega}_{m}=-H(t)\Omega_{m}\left[3+\frac{2\dot{H}}{H^{2}}\right]$
(20)

In terms of $\ln(a)$, the above equations lead to:

$\Omega_{r}^{\prime}+\Omega_{r}(4+2\xi)=0,\quad\Omega_{m}^{\prime}+\Omega_{m}(3+2\xi)=0$
(21)

Setting $\alpha=0$ in the above equations leads to standard GR equations.
Since $n$ is an arbitrary parameter, different values of $n$ lead to varied
evolution. To keep calculations tractable, we consider the following four
values of $n$: $n=1,2,-1$ and $-2$. For each of these values of $n$, we choose
different values of $\alpha$ and a large range of initial conditions for
$\xi$.

We evolve the above equations (18, 19, 21) numerically in the redshift range
$1500<z<0$ for the initial values of the three parameters to be
$\Omega_{r_{i}}=0.39$, $\Omega_{m_{i}}=0.59$, $-3\leq\xi_{i}\leq 1$ and
$\xi_{i}^{\prime}=0$.

In Figs. (1) - (9), we have plotted $\xi(a),\Omega_{r}(a),\Omega_{m}(a),$ and
$\Omega_{A}(a)$ as functions of $\ln(a)$ (corresponding to the redshift range
$1500<z<0$) for different values of $n$. In each set of figures, we have
plotted for at least two distinct $\alpha$ values. For each value of $\alpha$
we have plotted for different initial values of $\xi$. At $z=1500$ (or
$\ln(a)=-7.31$), $\xi$ is in the range $[-5,0.6]$. This is because we want to
obtain a Universe that evolves from a radiation-dominated epoch to a matter-
dominated epoch and then to a late-time accelerated expansion epoch (cf. Table
1).

In the rest of this section, we discuss the features of each of these
scenarios and identify whether, for a range of initial values of $\xi$, we can
obtain a Universe that evolves from a radiation-dominated epoch to a matter-
dominated epoch and then to a late-time accelerated expansion epoch.

Fig. 1 contains the plots for $n=1$ (in Eq. (17)) for two values of $\alpha$.
Plots on the left are for $\alpha=23$ and the plots are the right are for
$\alpha=0.07$. Different colors in the plots refer to different initial values
of $\xi$ in the range $[-5,0.6]$. From these plots, we infer the following:

$\alpha=23$ | $\alpha=0.07$
---|---
$\xi_{i}$ | $\xi_{f}$ | $\Omega_{{A}_{f}}$ | $\Omega_{{m}_{f}}$ | $\xi_{i}$ | $\xi_{f}$ | $\Omega_{{A}_{f}}$ | $\Omega_{{m}_{f}}$
$\xi_{i}>-1$ | $\xi_{f}=0$ | $\Omega_{{A}_{f}}=1$ | $\Omega_{{m}_{f}}=0$ | $\xi_{i}>-0.9$ | $\xi_{f}\approx-0.5$ | $\Omega_{{A}_{f}}=1$ | $\Omega_{{m}_{f}}=0$
$\xi_{i}\in[-1.2,-1]$ | $\xi_{f}\rightarrow\pm\infty$ | $\Omega_{{A}_{f}}=1$ | $\Omega_{{m}_{f}}=0$ | $\xi_{i}\in[-1.1,-1.0]$ | $\xi_{f}\rightarrow\pm\infty$ | $\Omega_{{A}_{f}}\rightarrow\pm\infty$ | $\Omega_{{m}_{f}}\rightarrow\pm\infty$
$\xi_{i}<-1.2$ | $\approx\xi_{i}$ | $\Omega_{{A}_{f}}<1$ | $\Omega_{{m}_{f}}\neq 0$ | $\xi_{i}\in[-1.5,-1.1]$ | $\xi_{f}\approx-1.4$ | $\Omega_{{A}_{f}}\rightarrow 1$ | $\Omega_{{m}_{f}}\rightarrow 0$

  1. 1.

For $\alpha=23$ and initial values of $\xi>-1$, we get a de Sitter attractor
($\xi=0$). Corresponding to these initial values, $\Omega_{r}$ and
$\Omega_{m}$ starts at $0.39$ and $0.59$ respectively, and quickly converge to
$0$; while $\Omega_{A}$ starts at 0.02 and quickly converges to $1$.

  2. 2.

For $\alpha=23$ and for the initial values of $\xi$ in the range $[-1.2,-1]$,
the values ($\xi,\Omega_{m},\Omega_{r})$ diverge.

  3. 3.

For $\alpha=23$ and initial values of $\xi<-1.2$, $\xi$ is nearly constant and
$\Omega_{r}$ and $\Omega_{m}$ does not quickly converge. Importantly, for
these initial values, $\Omega_{m}$ always approaches a non-zero value. For
these initial values of $\xi$, $\Omega_{A}$ starts at $0.02$ and does not
quickly converge and always approaches a value less than $1$.

  4. 4.

For $\alpha=0.07$ and initial values of $\xi>-0.9$, we get $\xi\approx-0.5$ as
an attractor ($w_{\rm eff}\approx-0.67$). Corresponding to these initial
values $\Omega_{r}$ and $\Omega_{m}$ start at $0.39$ and $0.59$ respectively
and quickly converge to $0$. $\Omega_{A}$ starts at 0.02 and quickly converges
to 1.

  5. 5.

For $\alpha=0.07$ and for the initial values of $\xi$ in the range
$[-1.1,-1]$, the values ($\xi,\Omega_{m},\Omega_{r})$ diverge.

  6. 6.

For $\alpha=0.07$ and initial values of $\xi$ in the range $[-1.5,-1.1]$ lead
to an attractor at around $\xi=-1.4$ (very close to matter dominated
Universe).$\Omega_{r}$ and $\Omega_{m}$ starts at $0.39$ and $0.59$
respectively and do not quickly converge. Like in the case of $\alpha=23$, for
these initial values, $\Omega_{m}$ slowly approaches zero. For these initial
values of $\xi$, $\Omega_{A}$ starts at $0.02$ and slowly converges to $1$.

(a) $\xi(a)$ vs ln(a),$\alpha=23$ (b) $\xi(a)$ vs ln(a),$\alpha=0.07$

(c) $\Omega_{m}(a)$ vs ln(a),$\alpha=23$ (d) $\Omega_{m}(a)$ vs
ln(a),$\alpha=0.07$

(e) $\Omega_{A}(a)$ vs ln(a),$\alpha=23$ (f) $\Omega_{A}(a)$ vs
ln(a),$\alpha=0.07$

Figure 1: Plot of $\xi(a),\Omega_{m}(a),\Omega_{A}(a)$ as a function of
$\ln(a)$ for $f(R,A)=Re^{\alpha(RA)}$. Plots on the left are for $\alpha=23$
and the plots are the right are for $\alpha=0.07$.

Fig. 2 contains the plots of $n=2$ (in Eq. (17)) for $\alpha=0.0016$. Plots on
the left are for the initial values of $\xi$ in the range $[-2,0.6]$ and the
plots are the right are for the initial values of $\xi$ in the range
$[-2.9,-2.1]$. Different colors in the plots correspond to different initial
values of $\xi$ in these ranges. From these plots, we infer the following:

  1. 1.

For initial values of $\xi>-0.9$, $\xi\approx-0.5$ is an attractor ($w_{\rm
eff}\approx-0.67$ corresponding to the current Universe). Corresponding to
these initial values, $\Omega_{r}$ and $\Omega_{m}$ starts at $0.39$ and
$0.59$, respectively, and quickly converge to $0$; while $\Omega_{A}$ starts
at $0.02$ and quickly converges to 1.

  2. 2.

For the initial values of $\xi$ in the range $[-2,-1.1]$, the values
($\xi,\Omega_{m},\Omega_{r})$ diverge.

  3. 3.

For initial values of $\xi<-2$, $\xi=-2.8$ is an attractor. For these initial
values, $\Omega_{r}$ and $\Omega_{m}$ start at $0.39$ and $0.59$,
respectively, and evolve to unphysical values, which contradicts the observed
trend. Hence, these initial values are excluded.

(a) $\xi(a)$ vs ln(a) (b) $\Omega_{r}(a)$ vs ln(a)

(c) $\Omega_{m}(a)$ vs ln(a) (d) $\Omega_{A}(a)$ vs ln(a)

Figure 2: Plot of $\xi(a),\Omega_{r}(a),\Omega_{m}(a),\Omega_{A}(a)$ as a
function of $\ln(a)$ for $f(R,A)=Re^{\alpha(RA)^{2}}$ and $\alpha=0.0016$ .

Fig. 3 contains the plots of $n=2$ (in Eq. (17)) for two values of $\alpha$.
Plots on the left are for $\alpha=-0.015$ and the plots are the right are for
$\alpha=0.1$. Different colors in the plots correspond to different initial
values of $\xi$ in the range $[-5,0.6]$. From these plots, we infer the
following:

  1. 1.

For $\alpha=0.1$ and initial values of $\xi>-0.9$, we get a de Sitter
attractor ($\xi=0$). Corresponding to these initial values, $\Omega_{r}$ and
$\Omega_{m}$ starts at $0.39$ and $0.59$ respectively, and quickly converge to
$0$; while $\Omega_{A}$ starts at 0.02 and quickly converges to $1$.

  2. 2.

For $\alpha=0.1$ and initial values of $\xi$ in the range $[-2,-1.5]$, $\xi$
diverges.

  3. 3.

For $\alpha=0.1$ and other initial values of $\xi$ in the range $[-1.5,-1.1]$,
$\xi$ is almost a constant, and $\Omega_{r}$ and $\Omega_{m}$ does not quickly
converge. Importantly, for these initial values, $\Omega_{m}$ always
approaches a non-zero value. For these initial values of $\xi$, $\Omega_{A}$
starts at $0.02$ and does not quickly converge and always approaches a value
less than $1$.

  4. 4.

For $\alpha=0.1$ and initial values of $\xi<-2$, we observe something akin to
convergence, but $\xi$ doesn’t converge within the physical range of $\ln(a)$.
For these initial values, $\Omega_{r}$ and $\Omega_{m}$ start at $0.39$ and
$0.59$, respectively, and evolve to very large values. Hence, these initial
values are not physically relevant.

  5. 5.

For $\alpha=-0.015$ and initial values of $\xi>-0.6$, we get a nearly de
Sitter attractor ($\xi\simeq 0.2$). Corresponding to these initial values,
$\Omega_{r}$ and $\Omega_{m}$ starts at $0.39$ and $0.59$ respectively, and
quickly converge to $0$; while $\Omega_{A}$ starts at $0.02$ and quickly
converges to $1$.

  6. 6.

For $\alpha=-0.015$ and initial values of $\xi$ in the range $[-2,-1.6]$,
($\xi,\Omega_{m},\Omega_{r})$ diverge.

(a) $\xi(a)$ vs ln(a),$\alpha=-0.015$ (b) $\xi(a)$ vs ln(a),$\alpha=0.1$

(c) $\Omega_{m}(a)$ vs ln(a),$\alpha=-0.015$ (d) $\Omega_{m}(a)$ vs
ln(a),$\alpha=0.1$

(e) $\Omega_{A}(a)$ vs ln(a),$\alpha=-0.015$ (f) $\Omega_{A}(a)$ vs
ln(a),$\alpha=0.1$

Figure 3: Plot of $\xi(a),\Omega_{m}(a),\Omega_{A}(a)$ as a function of
$\ln(a)$ for $f(R,A)=Re^{\alpha(RA)^{2}}$. Plots on the left are for
$\alpha=-0.015$ and the plots are the right are for $\alpha=0.1$.

Fig. 5 contains the plots of $n=-1$ (in Eq. (17)) for two values of $\alpha$.
Plots on the left are for $\alpha=-15$ and the plots are the right are for
$\alpha=700$. Different colors in the plots refer to different initial values
of $\xi$ in the range $[-5,0.6]$. From these plots, we infer the following:

  1. 1.

For $\alpha=-15$ and initial values of $\xi>-1.2$, we get $\xi=-0.5$ as an
attractor ($w_{\rm eff}\simeq-0.67$ corresponding to the current Universe).
For these initial values, $\Omega_{r}$ and $\Omega_{m}$ start at $0.39$ and
$0.59$ respectively, and quickly converge to $0$; while $\Omega_{A}$ starts at
0.02 and quickly converges to $1$.

  2. 2.

For $\alpha=-15$ and initial values of $\xi$ in the range $[-2,-1.2]$,
($\xi,\Omega_{m},\Omega_{r})$ diverge.

  3. 3.

For $\alpha=-15$ and initial values of $\xi<-2$, we get another attractor at
$\xi=-2$ (radiation dominated epoch). For these initial values, $\Omega_{r}$
and $\Omega_{m}$ starts at $0.39$ and $0.59$, respectively and diverge. Hence,
these initial values are not physically relevant.

  4. 4.

For $\alpha=700$ and initial values of $\xi$ in the range $[-0.4,0.4]$, we get
a de Sitter attractor ($\xi=0$). Corresponding to these initial values,
$\Omega_{r}$ and $\Omega_{m}$ starts at $0.39$ and $0.59$ respectively, and
quickly converge to $0$; while $\Omega_{A}$ starts at 0.02 and quickly
converges to $1$.

  5. 5.

For $\alpha=700$ and initial values of $\xi$ in the range $[-1.2,-1]$,
($\xi,\Omega_{m},\Omega_{r})$ diverge.

  6. 6.

For $\alpha=700$ and initial value of $\xi=-1.2$, $\xi$ is almost a constant.
For the initial values in the range $[-1,-0.5]$, $\Omega_{r}$ and $\Omega_{m}$
starts at 0.39 and 0.59 respectively and converges to 0; while, $\Omega_{A}$
starts at 0.02 and converges to 1.

(a) $\xi(a)$ vs ln(a),$\alpha=-15$ (b) $\xi(a)$ vs ln(a),$\alpha=700$ (c)
$\Omega_{m}(a)$ vs ln(a),$\alpha=-15$ (d) $\Omega_{m}(a)$ vs
ln(a),$\alpha=700$

(a) $\Omega_{A}(a)$ vs ln(a),$\alpha=-15$ (b) $\Omega_{A}(a)$ vs
ln(a),$\alpha=700$

Figure 5: Plot of $\xi(a),\Omega_{m}(a),\Omega_{A}(a)$ as a function of
$\ln(a)$ for $f(R,A)=Re^{\alpha(RA)^{-1}}$. Plots on the left are for
$\alpha=-15$ and the plots are the right are for $\alpha=700$.

Fig. 7 contains the plots of $n=-2$ (in Eq. (17)) for two values of $\alpha$.
Plots on the left are for $\alpha=-1100$ and the plots are the right are for
$\alpha=-120$. Different colors in the plots refer to different initial values
of $\xi$ in the range $[-5,0.6]$. From these plots, we infer the following:

  1. 1.

For $\alpha=-1100$ and initial value of $\xi>-0.9$, we get $\xi=-0.02$ (pure
dust) as a attractor. For initial value $\xi=-1$, $\xi$ is a constant. For any
other initial value of $\xi<-1$ (like $\xi=-1.1$), $\xi$ diverges.

  2. 2.

For $\alpha=-120$ and initial value of $\xi>-0.9$, we get $\xi=-0.5$
(corresponding to the current Universe) as a attractor. For initial values of
$\xi=-1$ and $-1.1$, $\xi$ is a constant. For any other initial value of
$\xi<-1.2$ (like $\xi=-1.2$), $\xi$ diverges.

  3. 3.

For both values of $\alpha$ and initial values of $\xi>-1$, $\Omega_{r}$ and
$\Omega_{m}$ starts at $0.39$ and $0.59$ respectively, and converge to 0;
while, $\Omega_{A}$ starts at $0.02$ and converges to $1$.

(a) $\xi(a)$ vs ln(a),$\alpha=-1100$ (b) $\xi(a)$ vs ln(a),$\alpha=-120$

(a) $\Omega_{m}(a)$ vs ln(a),$\alpha=-1100$ (b) $\Omega_{m}(a)$ vs
ln(a),$\alpha=-120$

(c) $\Omega_{A}(a)$ vs ln(a),$\alpha=-1100$ (d) $\Omega_{A}(a)$ vs
ln(a),$\alpha=-120$

Figure 7: Plot of $\xi(a),\Omega_{m}(a),\Omega_{A}(a)$ as a function of
$\ln(a)$ for $f(R,A)=Re^{\alpha(RA)^{-2}}$. Plots on the left are for
$\alpha=-1100$ and the plots are the right are for $\alpha=-120$.

(a) $\xi(a)$ vs ln(a),$\alpha=0.9$ (b) $\xi(a)$ vs ln(a),$\alpha=4250$

(a) $\Omega_{m}(a)$ vs ln(a),$\alpha=0.9$ (b) $\Omega_{m}(a)$ vs
ln(a),$\alpha=4250$

(c) $\Omega_{A}(a)$ vs ln(a),$\alpha=0.9$ (d) $\Omega_{A}(a)$ vs
ln(a),$\alpha=4250$

Figure 9: Plot of $\xi(a),\Omega_{m}(a),\Omega_{A}(a)$ as a function of
$\ln(a)$ for $f(R,A)=Re^{\alpha(RA)^{-2}}$. Plots on the left are for
$\alpha=0.9$ and the plots are the right are for $\alpha=4250$.

Fig. 9 contains the plots for two different values of $\alpha$. From the
plots, it is clear that the features are very similar. From these plots, we
notice a universal feature of class Ib models: There exists _at least one_
singularity point in the $\xi$ range $[-1.2,-1]$. Thus, a Universe from a
matter-dominated epoch ($\xi=-1.5$) to an accelerating Universe ($\xi=-0.5$)
will not be possible as one will encounter at least one singularity point in
the range $\xi$ range $[-1.2,-1]$. Thus, our analysis shows that class Ib
models _can not bypass_ the no-go theorem.

###  3.3 Class Ic models

In this class, $f(R,A)$ is a combination of Class Ia and Class Ib models. In
Ref. 2020-Amendola.etal-Phys.Lett.B , the authors conjectured that Lagrangian
with some function of $A$ multiplied with $e^{\alpha(RA)^{n}}$ might smoothly
evolve from a matter-dominated Universe to a late-time accelerated universe.

Before we proceed with the detailed analysis of such models, let us understand
the existence of singularity points in the $\xi$ range $[-1.2,-1]$. From Eq.
(9), we see that $\xi=-1.2,-1$ and $-3$ are special points of the
anticurvature scalar. If the function $f(R,A)$ is a proportional to $A$, then
the action will diverge at $\xi=-1$ and $-3$. If the function $f(R,A)$ is
inversely proportional to $A$, then the action will diverge at $\xi=-1.2$.
This implies that as the Universe evolves from the matter-dominated epoch
($\xi=-1.5$) to the current accelerated epoch ($\xi=-0.5$), we will have to
encounter a singularity either at $\xi=-1.2$ or at $\xi=-1$. As we show, this
feature will be present irrespective of the form of $f(R,A)$.

While the models in this class are extensive, we only consider a class of
models that will reduce to Einstein-Hilbert action $(f(R,A)\sim R)$. For
example, to confirm/infirm the no-go theorem, we consider the following form
of $f(R,A)$:

$f(R,A)=R+\alpha(R^{m+1}A^{m})e^{\beta(RA)^{n}}$ (22)

Note that in the limit of $\alpha\to 0$ and $\beta\to 0$, the above function
reduces to GR.

Applying the chain rule of differentiation on the exponential term, we have:

$\frac{d}{dg_{\mu\nu}}(e^{\alpha(RA)^{n}})=\alpha
e^{\alpha(RA)^{n}}n(RA)^{n-1}\left(\frac{dR}{dg_{\mu\nu}}+\frac{dA}{dg_{\mu\nu}}\right)\,.$
(23)

The singularities in $A$ (as can be seen in Eq. (9)) will propagate to the
terms including $dA/dg_{\mu\nu}$. Depending on the value of $n$ (i. e., $n>0$
or $n<0$), the modified Friedmann equations in the flat-space FLRW metric will
always have $(\xi+1)$ or $(5\xi+6)$ as a factor in the denominator leading to
a singularity either at $-1$ or at $-1.2$.

Including $R^{m+1}A^{m}$, we see that the singularities in this can not be
canceled by $A^{n-1}$ and $dA/dg_{\mu\nu}$. In other words, like in Class Ia
and Ib models, here again, we can remove only one singularity. To demonstrate
this, in Fig. 11, we have plotted $\xi,\Omega_{r},\Omega_{m}$ and $\Omega_{A}$
as a function of $\ln(a)$ for $\alpha=\beta=0.04,m=n=1$ in the action (22).

(a) $\xi(a)$ vs ln(a) (b) $\Omega_{r}(a)$ vs ln(a)

(a) $\Omega_{m}(a)$ vs ln(a) (b) $\Omega_{A}(a)$ vs ln(a)

Figure 11: Plot of $\xi(a),\Omega_{r}(a),\Omega_{m}(a),\Omega_{A}(a)$ as a
function of $\ln(a)$ for $f(R,A)=R+\alpha(R^{2}A)\exp[\alpha(RA)]$ and
$\alpha=0.04$.

Extending this model by considering a polynomial of $R$ instead of
$(R^{m+1}A^{m})$, multiplied with $e^{\alpha(RA)^{n}}$, the above problem will
still exist for some terms. Hence, we will always get either one or both
singularities ($\xi=-1,-1.2$) in these kinds of models.

Note that extending the above model by including a polynomial of $R$ and $A$
in the exponential will lead to singularities in the resulting modified
Friedmann equations. This is because of the presence of $A$ in the polynomial.
The same analysis and argument apply to models with polynomials of $R$ and $A$
in both the numerator and denominator.

###  3.4 Can Class I models bypass the no-go theorem?

For certain initial conditions, Class I models lead to $\xi=-1.5(w_{\rm
eff}=0)$ (matter-dominated universe) or $\xi=0(w_{\rm eff}=-1)$ (de Sitter
universe) or $\xi=-0.5(w_{\rm eff}=-0.67)$ (current Universe) as attractors.
However, we can not circumvent the no-go theorem as it is impossible to
smoothly evolve from $\xi=-1.5$ to $\xi=-0.5$. Hence Class I models can not
explain the late-time acceleration of the universe.

From Eq. (9), we see that $\xi=-1.2,-1$ and $-3$ are special points of the
anticurvature scalar. If the function $f(R,A)$ has a term which is
proportional to $A$, then the action will diverge at $\xi=-1$ and $-3$. If the
function $f(R,A)$ has a term which is inversely proportional to $A$, then the
action will diverge at $\xi=-1.2$. This implies that as the Universe evolves
from the matter-dominated epoch ($\xi=-1.5$) to the current accelerated epoch
($\xi=-0.5$), we will have to encounter a singularity either at $\xi=-1.2$ or
at $\xi=-1$. Choosing the form of $f(R,A)$ can remove one singularity, but not
both the singularities. Since, we need to cross both the points to evolve from
from the matter-dominated epoch ($\xi=-1.5$) to the current accelerated epoch
($\xi=-0.5$), class I models _can not_ bypass the no-go theorem.

##  4 Class II models

In this section, we consider the following general action:

$S=\int d^{4}x\sqrt{-g}\,{\cal F}(R,A^{\mu\nu}A_{\mu\nu})$ (24)

where ${\cal F}(R,A^{\mu\nu}A_{\mu\nu})$ is an arbitrary, smooth function of
Ricci scalar and anticurvature tensor square. We further classify this class
into two subclasses (Class IIa, and Class IIb). Class IIa models are
polynomials in $R$ and $A^{\mu\nu}A_{\mu\nu}$. Class IIb models are of the
form $\exp[\alpha(R^{2}A^{\mu\nu}A_{\mu\nu})^{n}]$.

###  4.1 Class IIa models

As mentioned above, in this class, we consider the cases where ${\cal
F}(R,A^{\mu\nu}A_{\mu\nu})$ is polynomial. Moreover, to keep calculations
tractable, we consider the following two forms:

$\displaystyle{\cal F}_{1}(R,A^{\mu\nu}A_{\mu\nu})$ $\displaystyle=$
$\displaystyle R\left(1+\frac{\alpha}{R^{2}A^{\mu\nu}A_{\mu\nu}}\right)\,;$
$\displaystyle{\cal F}_{2}(R,A^{\mu\nu}A_{\mu\nu})$ $\displaystyle=$
$\displaystyle
R\left(1+\frac{\alpha}{R^{2}A^{\mu\nu}A_{\mu\nu}}+\frac{\alpha}{R^{4}(A^{\mu\nu}A_{\mu\nu})^{2}}\right)$
(25)

where $\alpha$ is a constant. For simplicity, we have fixed the same
coefficients for ${\cal F}_{2}$. In the limit of $\alpha\to 0$, the above
models reduce to GR.

Fig. 12 contains the plots for ${\cal F}_{1}$ for two values of $\alpha$.
Plots on the left are for $\alpha=-65$ and the plots are the right are for
$\alpha=-17$. Different colors in the plots refer to different initial values
of $\xi$ in the range $[-5,0.6]$. From these plots, we infer the following:

  1. 1.

For $\alpha=-65$ and initial values of $\xi>-0.7$, we get a de Sitter
attractor ($\xi=0$). For initial values of $\xi$ in the range $[-1.3,-0.8]$,
we get another attractor at $\xi=-1$. Corresponding to these initial values,
$\Omega_{r}$ and $\Omega_{m}$ starts at $0.39$ and $0.59$ respectively, and
quickly converge to $0$; while $\Omega_{A}$ starts at 0.02 and quickly
converges to $1$.

  2. 2.

For $\alpha=-65$ and for the initial values of $\xi$ in the range
$[-1.6,-1.3]$, the values ($\xi,\Omega_{m},\Omega_{r})$ diverge. For initial
values of $\xi<-1.6$, we get attractor at $\xi=-2$ (corresponding to
radiation-dominated epoch). Note that this is different from that of Class I
models. However, for these models, $\Omega_{r}$ settles to a non-zero value.
Hence, this is not consistent with cosmological observations.

  3. 3.

For $\alpha=-17$, for different initial values of $\xi$ we observe the same
feature. The only difference is that for initial values of $\xi>-0.7$ we get
an attractor at $\xi=-0.5$.

(a) $\xi(a)$ vs ln(a),$\alpha=-65$ (b) $\xi(a)$ vs ln(a),$\alpha=-17$ (c)
$\Omega_{m}(a)$ vs ln(a),$\alpha=-65$ (d) $\Omega_{m}(a)$ vs
ln(a),$\alpha=-17$

(e) $\Omega_{A}(a)$ vs ln(a),$\alpha=-65$ (f) $\Omega_{A}(a)$ vs
ln(a),$\alpha=-17$

Figure 12: Plot of $\xi(a),\Omega_{m}(a),\Omega_{A}(a)$ as a function of
$\ln(a)$ for ${\cal F}_{1}$. Plots on the left are for $\alpha=-65$ and the
plots are the right are for $\alpha=-17$.

Fig. 13 contains the plots for ${\cal F}_{2}$ for two values of $\alpha$.
Plots on the left are for $\alpha=-16$ and the plots are the right are for
$\alpha=65$. For $\alpha=-16$, we observe the same results as in the case of
${\cal F}_{1}$ for $\alpha=-17$. For $\alpha=65$, we get the same trend, with
the only difference being there are no attractors for initial values greater
than $\xi=-0.8$.

(a) $\xi(a)$ vs ln(a),$\alpha=-16$ (b) $\xi(a)$ vs ln(a),$\alpha=65$

(c) $\Omega_{m}(a)$ vs ln(a),$\alpha=-16$ (d) $\Omega_{m}(a)$ vs
ln(a),$\alpha=65$

(e) $\Omega_{A}(a)$ vs ln(a),$\alpha=-16$ (f) $\Omega_{A}(a)$ vs
ln(a),$\alpha=65$

Figure 13: Plot of $\xi(a),\Omega_{m}(a),\Omega_{A}(a)$ as a function of
$\ln(a)$ for ${\cal F}_{2}$. Plots on the left are for $\alpha=-16$ and the
plots are the right are for $\alpha=65$.

###  4.2 Class IIb models

In this class, we consider the following exponential forms of ${\cal F}$:

${\cal
F}_{3}(R,A^{\mu\nu}A_{\mu\nu})=Re^{\alpha/(R^{2}A^{\mu\nu}A_{\mu\nu})}\qquad{\cal
F}_{4}(R,A^{\mu\nu}A_{\mu\nu})=Re^{\alpha(R^{2}A^{\mu\nu}A_{\mu\nu})}$ (26)

where $\alpha$ (a dimensionless constant) decides the deviation from GR.

Fig. 14 contains the plots for ${\cal F}_{3}$ and $\alpha=700$. Different
colors in the plots refer to different initial values of $\xi$ in the range
$[-5,0.6]$. From these plots we infer the following:

  1. 1.

For initial values of $\xi>-0.4$, we get a de Sitter attractor ($\xi=0$).

  2. 2.

For initial values of $\xi$ in the range $[-2,-0.4]$, $\xi$ is almost a
constant. Corresponding to these initial values, $\Omega_{r}$ and $\Omega_{m}$
starts at $0.39$ and $0.59$ respectively, and quickly converge to $0$; while,
$\Omega_{A}$ starts at 0.02 and quickly converges to $1$.

  3. 3.

For the initial value of $\xi=-2$, the values ($\xi,\Omega_{m},\Omega_{r})$
diverge. For initial values of $\xi<-2$, we get attractor at $\xi=-2$
(corresponding to radiation-dominated epoch). However, for these models,
$\Omega_{r}$ and $\Omega_{m}$ evolve to very large values. Hence, this is not
consistent with cosmological observations.

(a) $\xi(a)$ vs ln(a) (b) $\Omega_{r}(a)$ vs ln(a)

(c) $\Omega_{m}(a)$ vs ln(a) (d) $\Omega_{A}(a)$ vs ln(a)

Figure 14: Plot of $\xi(a),\Omega_{r}(a),\Omega_{m}(a),\Omega_{A}(a)$ as a
function of $\ln(a)$ for ${\cal F}_{3}$ and $\alpha=700$.

Fig. 15 contains the plots for ${\cal F}_{4}$ and $\alpha=0.005$. Different
colors in the plots refer to different initial values of $\xi$ in the range
$[-5,0.6]$. From these plots, we infer the following:

  1. 1.

For initial values of $\xi=-1$, the values ($\xi,\Omega_{m},\Omega_{r})$
diverge.

  2. 2.

The attractors are at $\xi\approx-0.5$, $\xi=-1.5$ and $\xi\approx-2.7$. For
the attractor point $\xi\approx-2.7$, $\Omega_{r}$ and $\Omega_{m}$ diverge or
evolve to unphysical values.

  3. 3.

For all initial values that lead to an attractor $\xi\approx-0.5$,
$\Omega_{r}$ and $\Omega_{m}$ start at $0.39$ and $0.59$ respectively, and
quickly converge to $0$; while $\Omega_{A}$ starts at 0.02 and quickly
converges to $1$.

  4. 4.

For all initial values that lead to an attractor $\xi\approx-1.5$,
$\Omega_{r}$ and $\Omega_{m}$ starts at $0.39$ and $0.59$ respectively, and do
not converge to $0$; while, $\Omega_{A}$ starts at 0.02 and do not converge to
$1$.

(a) $\xi(a)$ vs ln(a) (b) $\Omega_{r}(a)$ vs ln(a)

(c) $\Omega_{m}(a)$ vs ln(a) (d) $\Omega_{A}(a)$ vs ln(a)

Figure 15: Plot of $\xi(a),\Omega_{r}(a),\Omega_{m}(a),\Omega_{A}(a)$ as a
function of $\ln(a)$ for ${\cal F}_{4}$ and $\alpha=0.005$.

###  4.3 Can Class II models bypass the no-go theorem?

From the analysis of this section, we see that the singularity at $\xi=-1$ is
removed in the majority of the cases, and the singularity at $\xi=-1.2$ is
removed for all cases. However, we can not bypass the no-go theorem. In other
words, points close to $\xi=-1.5$ can not evolve to $\xi=-0.5$. Unlike the
Class I models, this is not because of the singularities at $\xi=-1.2$ and
$\xi=-1$. In these models, $\xi$ evolves to the attractor $\xi=-2$. In other
words, $\xi=-2$ ($w_{\rm eff}=1/3$, corresponding to radiation dominated) is
the late-time attractor (in most cases). More specifically, this model also
fails to explain the late-time evolution of the universe. Thus, Class II
models _can not bypass_ the no-go theorem.

##  5 Understanding the No-Go theorem from the reduced action

The action principle for the gravity sector reproduces the field equations
when all metric components are varied independently. If space-time has a
particular symmetry, it is possible to impose this symmetry in the full action
and integrate the redundant coordinates leading to the reduced action. Varying
the reduced action with respect to the remaining dependent variables may not
be equivalent to the reduced gravity equations, i.e., the equation resulting
from the imposition of the symmetry to the full gravity field equations. When
they are equivalent, the corresponding reduced Lagrangian is said to be valid.
A well-known example of a non-valid case is the class B family of Bianchi
models 1972-Maccallum.Taub-CMP . The class of spatially homogeneous Bianchi
cosmologies, including the FLRW space-time and Bianchi IX, are valid cases
1972-Maccallum.Taub-CMP ; 2010-Appignani.etal-JCAP ;
2015-Nandi.Shankaranarayanan-JCAP . Also, in the case of FLRW space-time, this
is valid up to second-order in perturbations 2015-Nandi.Shankaranarayanan-JCAP
. This section shows that the reduced action approach of Ricci-inverse models
provides an understanding of the no-go theorem.

Numerical analysis of the modified Friedmann equations of Class I and Class II
models shows that these models can lead to a late-time accelerated Universe.
However, neither of these models can have a smooth transition from a
decelerated universe to a late-time accelerated universe. One possible reason
is the appearance of the singularities in the anticurvature scalar either at
$\xi=-1.2$ or at $\xi=-1$. With this as a cue, we use the reduced action
approach to understand the no-go theorem.

Class | $\mathcal{L}$ | Reduced form
---|---|---
Class I | $\frac{1}{A}$ | $\frac{3(1+\xi)(3+\xi)}{2(6+5\xi)}$
Class I | $\exp(RA)$ | $\exp\left(\frac{4(2+\xi)(6+5\xi)}{(1+\xi)(3+\xi)}\right)$
Class II | $\frac{1}{R\,A^{\mu\nu}A_{\mu\nu}}$ | $\frac{3(1+\xi)^{2}(3+\xi)^{2}}{8(2+\xi)(7\xi^{2}+15\xi+9)}$
Table 3: Lagrangian corresponding to reduced action of Class I and Class II
models

Figure 16: Plot of the reduced Lagrangian in FLRW background vs. $\xi$

For both Class I (Eq. 1) and Class II (Eq. 3) models action, we obtain the
corresponding reduced action by applying the symmetry of the FLRW space-time.
The reduced action is a function of $\xi$ (excluding the $H^{2}$ factor).
Table 3 contains the Lagrangian corresponding to the reduced action for some
Class I and Class II models. We do not include the Lagrangian corresponding to
the Einstein-Hilbert action as it does not contribute to any singularities
during the evolution of the Universe.

Fig. 16 contains the plots for the different Class I and Class II models
discussed in Secs. (3, 4) as a function of $\xi$. Since these are plotted as
functions of $\xi$, the plots are _independent_ of the epoch of the evolution
of the Universe. In other words, we can apply the analysis to early-time or
late-time. From the plots, we infer the following:

  1. 1.

From the plot of $\exp[RA]$ in Fig. 16 (second row, first column), we notice
two features: First, minima occurs at $\xi=-0.5$. Second, there’s a
discontinuity at $\xi=-1$. For $\alpha=0.07$, these features are seen in Fig.
1. Specifically, $\xi=-0.5$ acts as an attractor. For the initial value of
$\xi=-1$, the values $(\xi,\Omega_{m},\Omega_{r})$ diverge.

  2. 2.

From the plot of $1/A$ in Fig. 16 (first row, second column), we see that the
anticurvature term has a discontinuity at $\xi\approx-1.2$ and two turning
points at $\xi\approx-0.7$ and $\xi\approx-1.5$. For $\alpha=-4$, these
features can be seen in Fig 23(a). We notice from the plot that
$\xi\simeq-0.7$ and $\xi\simeq-1.5$ are attractor. For the initial value of
$\xi=-1.2$, the values $(\xi,\Omega_{m},\Omega_{r})$ diverge.

  3. 3.

Thus, we can conclude that we can map the effective action approach to the
features obtained by solving the Eqs. (18, 19, 21) numerically in the redshift
range $1500<z<0$. For different values of $\alpha$, the attractor point or
initial values where the physical quantities diverge might change slightly;
however, the overall nature remains the same.

  4. 4.

The above conclusion is also true for the Class II models. From the plot of
$\exp[1/(R^{2}A^{\mu\nu}A_{\mu\nu})]$ in Fig. 16 (fourth row, second column),
we see that the anticurvature tensor term has three turning points
$\xi\approx-0.5,-1,-2.7$ and has two discontinuities at $\xi=-1.5,-2.5$. For
$\alpha=0.005$, these features are seen in Fig 15.

  5. 5.

From the plots in Fig. 16 of the reduced action, we notice that there exists
discontinuity while going from $\xi=-1.5~{}\text{to}~{}\xi=-0.5$. This
confirms from the analysis (in Secs. (3) and (4)) that it is not possible to
smoothly evolve from the matter-dominated epoch ($\xi=-1.5$) to the current
accelerated epoch ($\xi=-0.5$). Thus, the reduced action approach provides an
alternative understanding of the no-go theorem.

##  6 Mapping Modified gravity theories and Ricci-inverse models

The above results lead us to the following question: Can we map Ricci-inverse
gravity to modified theories of gravity? In other words, can modified gravity
theories mimic the features of Ricci-Inverse gravity? While the analysis is
not possible for a generic space-time, it is possible for FRW space-time.
Using the reduced action approach, we identify which modified theories of
gravity mimic Ricci-inverse models.

We consider six modified gravity models that can be written as a function of
the Ricci scalar, Ricci tensor, and Riemann tensor. In that sense, these
models can be the closest representations of Ricci-inverse models.

###  $\mathcal{L}=R+\frac{\alpha}{R^{2}}$, $[\alpha]=[L^{-6}]$

Figure 17: Reduced action for $R^{-2}$ vs $\xi(a)$

Fig. (17) is the plot of reduced action for the above Lagrangian versus $\xi$.
This model leads to a discontinuity at $\xi=-2$ and no minima around
$\xi=-1.5$ or $\xi=-0.5$. Thus, this will neither have a radiation-dominated
attractor nor explain the evolution from radiation to matter-dominated epoch
or matter-dominated epoch to the current accelerated epoch.

###  $\mathcal{L}=R+\alpha R^{2}$, $[\alpha]=[L^{6}]$

Figure 18: Reduced action for $R^{2}$ vs $\xi(a)$

Fig. 18 is the plot of reduced action for the Starobinsky model versus $\xi$.
From the plot, we see that this modified gravity model leads to a minimum at
$\xi=-2$ corresponding to radiation dominated phase. This modified gravity
model can possibly explain the evolution from the early Universe to radiation
dominated. Note that $\xi=-\epsilon$, where $\epsilon$ is the slow roll
parameter.

### Inverse Gauss-Bonnet models

We consider the following two such models that include the inverse of the
Gauss-Bonnet term:

$\displaystyle\mathcal{L}_{\rm GB}^{(1)}$ $\displaystyle=$ $\displaystyle
R+\frac{\alpha
R}{(R^{2}-4R^{\rho\sigma}R_{\rho\sigma}+R^{\mu\nu\alpha\beta}R_{\mu\nu\alpha\beta})}\qquad[\alpha]=\text{dimensionless}$
(27) $\displaystyle\mathcal{L}_{\rm GB}^{(2)}$ $\displaystyle=$ $\displaystyle
R+\frac{\alpha}{(R^{2}-4R^{\rho\sigma}R_{\rho\sigma}+R^{\mu\nu\alpha\beta}R_{\mu\nu\alpha\beta})}\qquad[\alpha]=[L^{-2}]$
(28)

Figure 19: Reduced action for $\mathcal{L}_{\rm GB}^{(1)}$ vs $\xi(a)$ Figure
20: Reduced action for $\mathcal{L}_{\rm GB}^{(2)}$ vs $\xi(a)$

Fig. 19 and Fig. 20 are the plots of reduced action for the inverse Gauss-
Bonnet models versus $\xi$. From the figures, we see that the inverse of
Gauss-Bonnet terms leads to a discontinuity at approximately $\xi\simeq-1$.
Thus, it is not possible to evolve smoothly from the matter-dominated epoch
($\xi=-1.5$) to the current accelerated epoch ($\xi=-0.5$) through this model.

### Generalized curvature models

Let us now consider the case where the coefficients of the Riemann, Ricci
tensor and Ricci scalar are different from that of Gauss-Bonnet —
$(aR^{2}+bR^{\rho\sigma}R_{\rho\sigma}+cR^{\mu\nu\alpha\beta}R_{\mu\nu\alpha\beta})$
where $a,b$ and $c$ are not equal to $1,-4$ and $1$, respectively.
Specifically, we consider the following two models:

$\displaystyle\mathcal{L}_{\rm Gen}^{(1)}$ $\displaystyle=$ $\displaystyle
R+\frac{\alpha
R}{(R^{2}-3\,R^{\rho\sigma}R_{\rho\sigma}-10\,R^{\mu\nu\alpha\beta}R_{\mu\nu\alpha\beta})}\qquad[\alpha]=\text{dimensionless}$
(29) $\displaystyle\mathcal{L}_{\rm Gen}^{(2)}$ $\displaystyle=$
$\displaystyle
R+\frac{\alpha}{(R^{2}-2.35R^{\rho\sigma}R_{\rho\sigma}-7.55R^{\mu\nu\alpha\beta}R_{\mu\nu\alpha\beta})}\qquad[\alpha]=[L^{-2}]$
(30)

Figure 21: Reduced action for $\mathcal{L}_{\rm Gen}^{(1)}$ with $a=1$,
$b=-3$, $c=-10$ Figure 22: Reduced action for $\mathcal{L}_{\rm Gen}^{(2)}$ vs
$\xi(a)$ with $a=1$, $b=-2.35$, $c=-7.55$

Fig. 21 and Fig. 22 are the plots of reduced action for the generalized
curvature models versus $\xi$. From the figures, we see that for some specific
values of $a,b$, and $c$, we get a minimum at $\xi=-0.5$. Hence, the evolution
from the matter-dominated epoch to the current accelerated epoch may likely be
explained through this model. We infer that the generalized curvature models
can provide similar features as that of Ricci-inverse gravity models. The
analysis needs to be for a generic space-time to do the complete mapping. We
plan to do this in future work.

However, this does _not_ imply that the inverse Ricci tensor can be expressed
as a combination of the Polynomials of Ricci tensor or Riemann tensor.
Similarly, it is not possible to generally express inverse Ricci scalar as a
function of Ricci scalar. For instance, as we have seen in section (2), for
the FLRW metric, the inverse Ricci scalar and Ricci scalar are given by

$A=\dfrac{2a(t)(\dot{a}^{2}(t)+5a(t)\ddot{a}(t))}{3\ddot{a}(t)(2\dot{a}^{2}(t)+a(t)\ddot{a}(t))},\quad
R=\dfrac{6}{a^{2}(t)}(\dot{a}(t)^{2}+\ddot{a}(t)a(t))$ (31)

As one can see, it is not possible to express $A$ as a function of $R$.

For the FLRW metric, the $00$ component of the anticurvature tensor is
obtained by calculating the matrix inverse of the Ricci tensor:

$A^{00}=\frac{R_{11}R_{22}R_{33}}{R_{00}R_{11}R_{22}R_{33}}$ (32)

We get this simple expression since the Ricci tensor is diagonal for the FLRW
metric. However, we see that the expression is a combination of the individual
components of the Ricci tensor. Even in this case, we cannot express the
anticurvature tensor as a function of the Ricci tensor. Hence an action that
is a function of anticurvature scalar $A$ cannot be expressed as a function of
Riemann and Ricci tensors and Ricci scalar. Thus, Ricci-inverse gravity is a
novel modified theory of gravity.

##  7 Conclusions

In this work, we considered two classes of Ricci-inverse models — Class I and
Class II models. Class I models ($f(R,A)$) are an arbitrary, smooth function
of Ricci and anticurvature scalars. Class II models ${\cal
F}(R,A^{\mu\nu}A_{\mu\nu})$ are arbitrary, smooth functions of Ricci scalar
and square of anticurvature tensor. Class I models are classified into three
subclasses — Class Ia, Class Ib, and Class Ic. Class Ia models are polynomials
in $R$, and $A$ 2020-Amendola.etal-Phys.Lett.B ; 2021-Do-Eur.Phys.J.C ;
2022-Do-Eur.Phys.J.C . Class Ib models are of the form $\exp[(RA)^{n}]$. Class
Ic models are non-polynomial functions of $R$ and $A$. Class II models are
further classified into two subclasses (Class IIa, and Class IIb). Class II
models are further classified into two subclasses — Class IIa, and Class IIb.
Class IIa models are polynomials in $R$ and $A^{\mu\nu}A_{\mu\nu}$. Class IIb
models are of the form $\exp[\alpha(R^{2}A^{\mu\nu}A_{\mu\nu})^{n}]$.

We evolved the modified Friedmann equations (18, 19, 21) numerically in the
redshift range $1500<z<0$ and obtained
$\xi(a),\Omega_{r}(a),\Omega_{m}(a),\Omega_{A}(a)$ as a function of $\ln(a)$
(corresponding to the redshift range $1500<z<0$) for different values of
$\alpha$. We showed that although these models can lead to the late-time
accelerated Universe, it can not evolve from a decelerated expansion to an
accelerated expansion. Hence, these two classes of Ricci-inverse models are
_not_ consistent with the observed background evolution of the Universe.

In the case of Class I models we found that as the Universe evolves from the
matter-dominated epoch ($\xi=-1.5$) to the current accelerated epoch
($\xi=-0.5$), we encounter a singularity either at $\xi=-1.2$ or at $\xi=-1$.
Choosing the form of $f(R,A)$ can remove one singularity but not both
singularities. Since, we need to cross both the points to evolve from from the
matter-dominated epoch ($\xi=-1.5$) to the current accelerated epoch
($\xi=-0.5$), class I models _can not_ bypass the no-go theorem. Unlike the
Class I models, Class II models do not have singularities at $\xi=-1.2$ and
$\xi=-1$. However, in these models, $\xi$ evolves to the attractor $\xi=-2$
and fails to explain the late-time acceleration of the Universe.

Using the reduced action approach, we obtained an alternative way of
understanding the no-go theorem. Specifically, we showed that the features
that the effective action approach exhibits can be mapped to the features
obtained by solving the Eqs. (18, 19, 21) numerically in the redshift range
$1500<z<0$. For different values of $\alpha$, the attractor point or initial
values where the physical quantities diverge might change slightly; however,
the overall nature remains the same. Hence, we can not bypass the no-go
theorem for Ricci-inverse gravity models.

We considered several other modified theories of gravity and applied the
reduced action approach to see if they could mimic Ricci-Inverse Gravity. We
found models of the form
$(aR^{2}+bR^{\rho\sigma}R_{\rho\sigma}+cR^{\mu\nu\alpha\beta}R_{\mu\nu\alpha\beta})^{-1}$
with values of $a,b$ and $c$ different than that of Gauss-Bonnet can
potentially explain the evolution of the Universe from the matter-dominated
phase to the accelerated expansion phase for some specific values of $a,b$ and
$c$ 2005-Carroll.etal-Phys.Rev.D .

Many modified gravity models lead to extra degrees of freedom which cause
instabilities. From our analysis for FLRW space-time, the extra degree of
freedom is not apparent. To understand the properties of the plausible extra
degree of freedom, one must do perturbation theory. The perturbation theory is
essential if the background analysis provides a possible explanation for the
accelerated expansion. In this work, we have explicitly shown that the inverse
Ricci model cannot explain the cosmic expansion history starting from the
radiation-dominated epoch to the matter-dominated epoch to the dark energy-
dominated epoch. Hence we have not done the stability analysis for Ricci-
inverse gravity.

We have not done a detailed analysis for the early Universe — Universe
evolving from an inflationary epoch to a radiation-dominated epoch. However,
the reduced action approach in section (5) is independent of the epoch. Hence,
our analysis can be extended and leads to the _same conclusion_ — Ricci-
inverse gravity cannot smoothly evolve from an inflationary epoch to a
radiation-dominated epoch in the flat FLRW space-time. In other words, Ricci-
inverse gravity can not provide a satisfactory explanation for a smooth exit
from inflation.

Thus, from these results, we can conclude that Ricci-inverse gravity can not
explain the late-time acceleration of the Universe in the FLRW metric.
However, if one can construct a scalar quantity with no singularities or no
singularities in the range $\xi\in[-2,0]$, then it is possible to construct a
Lagrangian such that it has minima around $\xi=-0.5$, which can act as an
attractor leading to the late-time acceleration of the Universe. One
possibility is to consider Riemann-inverse gravity. In this gravity, we can
define an “Anti-Riemann” tensor as:

$A_{\nu}^{\alpha\rho\gamma}R^{\mu}_{\alpha\rho\beta}=\delta^{\mu}_{\nu}\delta^{\gamma}_{\beta}$
(33)

This is beyond the scope of this work but is currently being investigated.

##  8 Acknowledgements

ID acknowledges the financial support provided by DST, Govt. of India through
INSPIRE scheme. JPJ is supported by CSIR Senior Research Fellowship, India.
The work is supported by ISRO-Respond Grant, India.

## Data availability

Data sharing is not applicable to this article as no datasets were generated
or analysed during the current study.

##  Appendix A Derivation of Eq. (4) for Class II models

Consider the action:

$S=\int\sqrt{-g}\hskip 2.0ptd^{4}x\left({\cal
F}(R,A^{\mu\nu}A_{\mu\nu})\right)$ (34)

Varying the action w.r.t the metric leads to:

$\begin{split}\delta S=\int d^{4}x\left[{\cal
F}\delta\sqrt{-g}+\sqrt{-g}\hskip 2.0pt\delta{\cal F}\right]\hskip 140.0pt\\\
=\int\sqrt{-g}\hskip 2.0ptd^{4}x\left[\frac{-1}{2}{\cal F}g_{\mu\nu}\delta
g^{\mu\nu}+{\cal F}_{R}R_{\mu\nu}\delta g^{\mu\nu}+{\cal
F}_{R}g^{\mu\nu}\delta R_{\mu\nu}+{\cal F}_{A^{2}}A^{\mu\nu}\delta
A_{\mu\nu}+{\cal F}_{A^{2}}A_{\mu\nu}\delta A^{\mu\nu}\right]\end{split}$ (35)

where ${\cal F}_{R}=\partial{\cal F}/\partial R$ and ${\cal
F}_{A^{2}}=\partial{\cal F}/\partial(A^{\mu\nu}A_{\mu\nu})$. We will calculate
each term in the RHS separately:

$\textbf{I term :}\qquad\qquad\frac{-1}{2}{\cal F}g_{\mu\nu}\delta
g^{\mu\nu}=\frac{1}{2}{\cal F}g^{\mu\nu}\delta g_{\mu\nu}$

$\textbf{II term :}\qquad\qquad{\cal F}_{R}R_{\mu\nu}\delta g^{\mu\nu}=-{\cal
F}_{R}R^{\mu\nu}\delta g_{\mu\nu}$

III term : $\qquad\qquad{\cal F}_{R}g^{\mu\nu}\delta R_{\mu\nu}$

$\displaystyle\delta
R^{a}_{bcd}=\nabla_{c}(\delta\Gamma^{a}_{db})-\nabla_{d}(\delta\Gamma^{a}_{cb})$
$\displaystyle=\frac{1}{2}\nabla_{c}[g^{ai}(-\nabla_{i}\delta
g_{db}+\nabla_{d}\delta g_{bi}+\nabla_{b}\delta g_{di})+\delta
g^{ai}\Gamma_{idb}]-\frac{1}{2}\nabla_{d}[g^{ai}(-\nabla_{i}\delta
g_{cb}+\nabla_{c}\delta g_{bi}+\nabla_{b}\delta g_{ci})+\delta
g^{ai}\Gamma_{icb}]$
$\displaystyle=\frac{g^{ai}}{2}\nabla_{c}[-\nabla_{i}\delta
g_{db}+\nabla_{d}\delta g_{bi}+\nabla_{b}\delta
g_{di}]-\frac{g^{ai}}{2}\nabla_{d}[-\nabla_{i}\delta g_{cb}+\nabla_{c}\delta
g_{bi}+\nabla_{b}\delta g_{ci}]$

We have:

$\displaystyle\delta R_{ab}=\frac{1}{2}[-\nabla^{d}\nabla_{d}\delta
g_{ab}+\nabla^{d}\nabla_{a}\delta g_{bd}+\nabla^{d}\nabla_{b}\delta
g_{ad}-\nabla_{a}\nabla_{b}(g^{cd}\delta g_{cd})]$

After contracting with metric, we get:

$\displaystyle g^{ab}\delta R_{ab}=\nabla^{b}\nabla^{a}\delta
g_{ab}-\nabla^{b}\nabla_{b}(g^{cd}\delta g_{cd})$ $\displaystyle{\cal
F}_{R}g^{\mu\nu}\delta R_{\mu\nu}={\cal F}_{R}\nabla^{\nu}\nabla^{\mu}\delta
g_{\mu\nu}-{\cal F}_{R}\nabla^{\mu}\nabla_{\mu}(g^{\beta\alpha}\delta
g_{\beta\alpha})$

Rewriting, we get:

$\displaystyle{\cal F}_{R}\nabla^{\nu}\nabla^{\mu}\delta
g_{\mu\nu}=\nabla^{\nu}[{\cal F}_{R}\nabla^{\mu}\delta
g_{\mu\nu}]-\nabla^{\mu}[\nabla^{\nu}{\cal F}_{R}\hskip 2.0pt\delta
g_{\mu\nu}]+\delta g_{\mu\nu}\nabla^{\mu}\nabla^{\nu}{\cal F}_{R}$
$\displaystyle-{\cal F}_{R}\nabla^{\mu}\nabla_{\mu}(g^{\beta\alpha}\delta
g_{\beta\alpha})=-\nabla^{\mu}[{\cal F}_{R}\nabla_{\mu}(g^{\beta\alpha}\delta
g_{\beta\alpha})]+\nabla_{\mu}[\nabla^{\mu}{\cal F}_{R}\hskip
2.0ptg^{\beta\alpha}\delta g_{\beta\alpha}]-g^{\beta\alpha}\delta
g_{\beta\alpha}\nabla_{\mu}\nabla^{\mu}{\cal F}_{R}$

The first two terms in both of the above formulas, when multiplied with
$\sqrt{-g}$, become a total derivative and hence won’t contribute to the
resulting equations (boundary terms). We then have:

$\displaystyle{\cal F}_{R}g^{\mu\nu}\delta
R_{\mu\nu}=[\nabla^{\mu}\nabla^{\nu}{\cal F}_{R}-g^{\mu\nu}\nabla^{2}{\cal
F}_{R}]\delta g_{\mu\nu}$

IV term : ${\cal F}_{A^{2}}A^{\mu\nu}\delta A_{\mu\nu}$

$\displaystyle
A_{\mu\nu}=g_{\nu\beta}g_{\mu\alpha}A^{\alpha\beta}\,;~{}~{}~{}~{}\delta
A_{\mu\nu}=g_{\nu\beta}g_{\mu\alpha}\delta A^{\alpha\beta}+g_{\nu\beta}\delta
g_{\mu\alpha}A^{\alpha\beta}+\delta g_{\nu\beta}g_{\mu\alpha}A^{\alpha\beta}$
$\displaystyle{\cal F}_{A^{2}}A^{\mu\nu}\delta A_{\mu\nu}={\cal
F}_{A^{2}}[A^{\mu\nu}g_{\nu\beta}g_{\mu\alpha}\delta
A^{\alpha\beta}+A^{\mu\nu}g_{\mu\alpha}A^{\alpha\beta}\delta
g_{\nu\beta}+A^{\mu\nu}g_{\nu\beta}A^{\alpha\beta}\delta g_{\mu\alpha}]$
$\displaystyle={\cal F}_{A^{2}}A_{\mu\nu}\delta A^{\mu\nu}+{\cal
F}_{A^{2}}(A^{\rho\nu}A^{\mu}_{\rho}\delta
g_{\mu\nu}+A^{\mu\rho}A^{\nu}_{\rho}\delta g_{\mu\nu})$

Interchanging $\mu$, $\nu$ in the last term and using the properties that
$\delta g_{\mu\nu}=\delta g_{\nu\mu}$ and $A^{\alpha\beta}=A^{\beta\alpha}$.
Then the last and second terms are identical. We then finally have:

$\displaystyle{\cal F}_{A^{2}}A^{\mu\nu}\delta A_{\mu\nu}={\cal
F}_{A^{2}}A_{\mu\nu}\delta A^{\mu\nu}+2{\cal
F}_{A^{2}}A^{\rho\nu}A^{\mu}_{\rho}\delta g_{\mu\nu}$

V term : ${\cal F}_{A^{2}}A_{\mu\nu}\delta A^{\mu\nu}$

$\displaystyle\delta
A^{\mu\nu}=\frac{-1}{2}A^{\mu\alpha}g^{\rho\lambda}(\nabla_{\rho}\nabla_{\alpha}\delta
g_{\beta\lambda}-\nabla_{\rho}\nabla_{\lambda}\delta
g_{\alpha\beta}-\nabla_{\beta}\nabla_{\alpha}\delta
g_{\rho\lambda}+\nabla_{\beta}\nabla_{\lambda}\delta
g_{\alpha\rho}+[\nabla_{\beta},\nabla_{\rho}]\delta
g_{\lambda\alpha})A^{\beta\nu}$

Let’s look at the first term here: $-\frac{1}{2}{\cal
F}_{A^{2}}A_{\mu\nu}A^{\mu\alpha}g^{\rho\lambda}\nabla_{\rho}\nabla_{\alpha}\delta
g_{\beta\lambda}A^{\beta\nu}$. Just like we did for the third term (write it
as the sum of three terms, two of them become total derivatives, and only the
third term contributes), we can do the same here. The final result is:

$\displaystyle-\frac{1}{2}{\cal
F}_{A^{2}}A_{\mu\nu}A^{\mu\alpha}g^{\rho\lambda}\nabla_{\rho}\nabla_{\alpha}\delta
g_{\beta\lambda}=-\frac{1}{2}\delta
g_{\beta\lambda}g^{\rho\lambda}\nabla_{\rho}\nabla_{\alpha}({\cal
F}_{A^{2}}A_{\mu\nu}A^{\mu\alpha}A^{\beta\nu})$

In arriving, we have used $\nabla_{c}g^{ab}=0$. We repeat the same for the
other terms:

$\displaystyle{\cal F}_{A^{2}}A_{\mu\nu}\delta A^{\mu\nu}=\frac{-1}{2}\delta
g_{\mu\nu}[g^{\rho\nu}\nabla_{\alpha}\nabla_{\rho}({\cal
F}_{A^{2}}A_{\sigma\kappa}A^{\sigma\alpha}A^{\mu\kappa})-\nabla^{2}({\cal
F}_{A^{2}}A_{\sigma\kappa}A^{\sigma\mu}A^{\nu\kappa})$
$\displaystyle-g^{\mu\nu}\nabla_{\alpha}\nabla_{\rho}({\cal
F}_{A^{2}}A_{\sigma\kappa}A^{\sigma\alpha}A^{\rho\kappa})+g^{\rho\nu}\nabla_{\rho}\nabla_{\alpha}({\cal
F}_{A^{2}}A_{\sigma\kappa}A^{\sigma\mu}A^{\alpha\kappa})$
$\displaystyle+g^{\rho\nu}\nabla_{\rho}\nabla_{\alpha}({\cal
F}_{A^{2}}A_{\sigma\kappa}A^{\sigma\mu}A^{\alpha\kappa})-g^{\rho\nu}\nabla_{\alpha}\nabla_{\rho}({\cal
F}_{A^{2}}A_{\sigma\kappa}A^{\sigma\mu}A^{\alpha\kappa})]$

We get:

$\displaystyle{\cal F}_{A^{2}}A_{\mu\nu}\delta A^{\mu\nu}=\frac{-1}{2}\delta
g_{\mu\nu}[g^{\rho\nu}\nabla_{\alpha}\nabla_{\rho}({\cal
F}_{A^{2}}A_{\sigma\kappa}A^{\sigma\alpha}A^{\mu\kappa})-\nabla^{2}({\cal
F}_{A^{2}}A_{\sigma\kappa}A^{\sigma\mu}A^{\nu\kappa})$
$\displaystyle-g^{\mu\nu}\nabla_{\alpha}\nabla_{\rho}({\cal
F}_{A^{2}}A_{\sigma\kappa}A^{\sigma\alpha}A^{\rho\kappa})+2g^{\rho\nu}\nabla_{\rho}\nabla_{\alpha}({\cal
F}_{A^{2}}A_{\sigma\kappa}A^{\sigma\mu}A^{\alpha\kappa})-g^{\rho\nu}\nabla_{\alpha}\nabla_{\rho}({\cal
F}_{A^{2}}A_{\sigma\kappa}A^{\sigma\mu}A^{\alpha\kappa})]$

Combining all the terms, we have:

$\begin{split}{\cal F}_{R}R^{\mu\nu}-2{\cal
F}_{A^{2}}A^{\mu\rho}A^{\nu}_{\rho}-\frac{1}{2}fg^{\mu\nu}-\nabla^{\mu}\nabla^{\nu}{\cal
F}_{R}+g^{\mu\nu}\nabla^{2}{\cal
F}_{R}+g^{\rho\nu}\nabla_{\alpha}\nabla_{\rho}({\cal
F}_{A^{2}}A_{\sigma\kappa}A^{\sigma\alpha}A^{\mu\kappa})\\\ -\nabla^{2}({\cal
F}_{A^{2}}A_{\sigma\kappa}A^{\sigma\mu}A^{\nu\kappa})-g^{\mu\nu}\nabla_{\alpha}\nabla_{\rho}({\cal
F}_{A^{2}}A_{\sigma\kappa}A^{\sigma\alpha}A^{\rho\kappa})+2g^{\rho\nu}\nabla_{\rho}\nabla_{\alpha}({\cal
F}_{A^{2}}A_{\sigma\kappa}A^{\sigma\mu}A^{\alpha\kappa})\\\
-g^{\rho\nu}\nabla_{\alpha}\nabla_{\rho}({\cal
F}_{A^{2}}A_{\sigma\kappa}A^{\sigma\mu}A^{\alpha\kappa})=0\end{split}$ (36)

##  Appendix B Class Ia models

Substituting the following form 2020-Amendola.etal-Phys.Lett.B :

$f(R,A)=R+\frac{\alpha}{A}$ (37)

in Eq. (2), we obtain the following modified Friedmann equations:

$\displaystyle\rho_{t}$ $\displaystyle=$ $\displaystyle 3\alpha
H^{2}\frac{(\xi+3)^{2}(5\xi+6)-18\xi^{\prime}}{4(5\xi+6)^{3}}+3H^{2}$ (38)
$\displaystyle p_{t}$ $\displaystyle=$ $\displaystyle-\frac{\alpha
H^{2}}{4(5\xi+6)^{4}}\left[(5\xi+6)\left((\xi+3)^{2}(2\xi+3)(5\xi+6)-18\xi^{\prime\prime}\right)+270(\xi^{\prime})^{2}-54(\xi+2)(5\xi+6)\xi^{\prime}\right]$
(39) $\displaystyle-2H^{2}\xi-3H^{2}$

In Fig. (23), we have plotted $\xi(a)$ as a function of $\ln(a)$. The
behaviour is identical to that derived in Ref. 2020-Amendola.etal-Phys.Lett.B
. We see from the plots that the initial values of $\xi$ close to $\xi=-1.5$
will never smoothly evolve to $\xi=-0.5$. This leads to the no-go theorem
2020-Amendola.etal-Phys.Lett.B .

(a) $\Omega_{m}+\Omega_{A}=1$ (b) $\Omega_{m}+\Omega_{A}+\Omega_{\Lambda}=1$

Figure 23: Plot of $\xi~{}{\rm vs}~{}\ln(a)$ for $f(R,A)=R+\alpha/A$ and
$\alpha=-4$. The left plot is for $\Omega_{m}+\Omega_{A}=1$ and the right plot
is for $\Omega_{m}+\Omega_{A}+\Omega_{\Lambda}=1$.

## References

  * (1) Luca Amendola, Leonardo Giani, and Giorgio Laverda.  Ricci-inverse gravity: a novel alternative gravity, its flaws, and how to cure them.  Phys. Lett. B, 811:135923, 2020.  arXiv:2006.04209, doi:10.1016/j.physletb.2020.135923.
  * (2) Clifford M Will.  The Confrontation between General Relativity and Experiment.  Living Rev. Rel., 17(3):4, 2006.  URL: http://www.livingreviews.org/lrr-2006-3, arXiv:1403.7377, doi:10.12942/lrr-2014-4.
  * (3) B P Abbott and Others.  Observation of Gravitational Waves from a Binary Black Hole Merger.  Phys. Rev. Lett., 116(6):61102, 2016.  arXiv:1602.03837, doi:10.1103/PhysRevLett.116.061102.
  * (4) David H Lyth Andrew R. Liddle.  Cosmological inflation and large-scale structure.  Cambridge University Press, 2000.
  * (5) S Perlmutter and Others.  Measurements of Omega and Lambda from 42 high redshift supernovae.  Astrophys. J., 517:565–586, 1999.  arXiv:astro-ph/9812133, doi:10.1086/307221.
  * (6) Adam G Riess and Others.  Observational evidence from supernovae for an accelerating universe and a cosmological constant.  Astron. J., 116:1009–1038, 1998.  arXiv:astro-ph/9805201, doi:10.1086/300499.
  * (7) Varun Sahni and Alexei A Starobinsky.  The Case for a positive cosmological Lambda term.  Int. J. Mod. Phys., D9:373–444, 2000.  arXiv:astro-ph/9904398, doi:10.1142/S0218271800000542.
  * (8) Edmund J Copeland, M Sami, and Shinji Tsujikawa.  Dynamics of dark energy.  Int. J. Mod. Phys., D15:1753–1936, 2006.  arXiv:hep-th/0603057, doi:10.1142/S021827180600942X.
  * (9) Jaewon Yoo and Yuki Watanabe.  Theoretical Models of Dark Energy.  Int. J. Mod. Phys. D, 21:1230002, 2012.  arXiv:1212.4726, doi:10.1142/S0218271812300029.
  * (10) A D Sakharov.  Vacuum quantum fluctuations in curved space and the theory of gravitation.  Sov. Phys. Dokl., 12:1040–1041, 1968.  doi:10.1023/a:1001947813563.
  * (11) Steven Weinberg.  The Cosmological Constant Problem.  Rev. Mod. Phys., 61:1–23, 1989.  doi:10.1103/RevModPhys.61.1.
  * (12) Valerio Marra, Luca Amendola, Ignacy Sawicki, and Wessel Valkenburg.  Cosmic variance and the measurement of the local hubble parameter.  Phys. Rev. Lett., 110(24):241305, 2013.  arXiv:1303.3121, doi:10.1103/PhysRevLett.110.241305.
  * (13) Licia Verde, Pavlos Protopapas, and Raul Jimenez.  Planck and the local universe: Quantifying the tension.  Phys. Dark Univ., 2:166–175, 2013.  arXiv:1306.6766, doi:10.1016/j.dark.2013.09.002.
  * (14) Antonio De Felice and Shinji Tsujikawa.  f(R) theories.  Living Rev. Rel., 13:3, 2010.  arXiv:1002.4928, doi:10.12942/lrr-2010-3.
  * (15) S Nojiri, S D Odintsov, and V K Oikonomou.  Modified Gravity Theories on a Nutshell: Inflation, Bounce and Late-time Evolution.  Phys. Rept., 692:1–104, 2017.  arXiv:1705.11098, doi:10.1016/j.physrep.2017.06.001.
  * (16) Kazuharu Bamba, Salvatore Capozziello, Shin’ichi Nojiri, and Sergei D. Odintsov.  Dark energy cosmology: the equivalent description via different theoretical models and cosmography tests.  Astrophys. Space Sci., 342:155–228, 2012.  arXiv:1205.3421, doi:10.1007/s10509-012-1181-8.
  * (17) Thomas P Sotiriou and Valerio Faraoni.  f(R) Theories Of Gravity.  Rev. Mod. Phys., 82(1):451–497, mar 2010.  URL: https://link.aps.org/doi/10.1103/RevModPhys.82.451, arXiv:0805.1726, doi:10.1103/RevModPhys.82.451.
  * (18) Joseph P. Johnson and S. Shankaranarayanan.  Low-energy modified gravity signatures on the large-scale structures.  Phys. Rev. D, 100(8):083526, 2019.  arXiv:1904.07608, doi:10.1103/PhysRevD.100.083526.
  * (19) Shin’ichi Nojiri and Sergei D. Odintsov.  Unified cosmic history in modified gravity: from f(r) theory to lorentz non-invariant models.  Phys. Rept., 505:59–144, 2011.  arXiv:1011.0544, doi:10.1016/j.physrep.2011.04.001.
  * (20) P.J.E. Peebles and Bharat Ratra.  The Cosmological Constant and Dark Energy.  Rev. Mod. Phys., 75:559–606, 2003.  arXiv:astro-ph/0207347, doi:10.1103/RevModPhys.75.559.
  * (21) Eleonora Di Valentino, Alessandro Melchiorri, Olga Mena, and Sunny Vagnozzi.  Nonminimal dark sector physics and cosmological tensions.  Phys. Rev. D, 101(6):063502, 2020.  arXiv:1910.09853, doi:10.1103/PhysRevD.101.063502.
  * (22) Joseph P. Johnson, Archana Sangwan, and S. Shankaranarayanan.  Observational constraints and predictions of the interacting dark sector with field-fluid mapping.  JCAP, 01(01):024, 2022.  arXiv:2102.12367, doi:10.1088/1475-7516/2022/01/024.
  * (23) Tuan Q. Do.  No-go theorem for inflation in Ricci-inverse gravity.  Eur. Phys. J. C, 81(5):431, 2021.  arXiv:2009.06306, doi:10.1140/epjc/s10052-021-09223-4.
  * (24) Tuan Q. Do.  No-go theorem for inflation in an extended ricci-inverse gravity model.  Eur. Phys. J. C, 82(1):15, 2022.  arXiv:2101.08538, doi:10.1140/epjc/s10052-021-09974-0.
  * (25) Mattia Scomparin.  Cosmic structures in ricci-inverse theories of gravity.  2 2021.  arXiv:2102.04676.
  * (26) Malcolm A. H. Maccallum and A. H. Taub.  Variational principles and spatially-homogeneous universes, including rotation.  Commun. Math. Phys., 25:173–189, 1972.  doi:10.1007/BF01877589.
  * (27) Corrado Appignani, Roberto Casadio, and S. Shankaranarayanan.  A Note on second-order perturbations of non-canonical scalar fields.  JCAP, 03:010, 2010.  arXiv:0905.4184, doi:10.1088/1475-7516/2010/03/010.
  * (28) Debottam Nandi and S. Shankaranarayanan.  ‘Constraint consistency’ at all orders in cosmological perturbation theory.  JCAP, 08:050, 2015.  arXiv:1502.04036, doi:10.1088/1475-7516/2015/08/050.
  * (29) Sean M. Carroll, Antonio De Felice, Vikram Duvvuri, Damien A. Easson, Mark Trodden, and Michael S. Turner.  The cosmology of generalized modified gravity models.  Phys. Rev. D, 71:063513, 2005.  arXiv:astro-ph/0410031, doi:10.1103/PhysRevD.71.063513.