# Perturbative approach to $f(R)$-gravitation in FLRW cosmology

Pham Van Ky1,2, Nguyen Thi Hong Van1, and Nguyen Anh Ky1
1 Mathematical-, high energy- and astro-physics group,
Institute of physics,
Vietnam academy of science and technology (VAST),
10 Dao Tan, Ba Dinh, Hanoi, Viet Nam.
2 Graduate university of science and technology, VAST,
18 Hoang Quoc Viet, Cau Giay, Hanoi, Viet Nam.
E-mail: pvky AT iop.vast.vn; nhvan AT iop.vast.vn; anhky AT iop.vast.vn

###### Abstract

We look for a perturbative solution of the Universe in the $f(R)$ theory of
gravitation developed perturbatively around the general theory of relativity
(with the cosmological constant included). As a result, we obtain a general
explicit solution that can be used for any model with an arbitrary (but well-
defined) $f(R)$ function (just satisfying given perturbation conditions). This
perturbative solution shows how the Hubble parameter $H(t)$ depends on time
(as well as on the cosmological constant and the mater density) to adapt the
evolution of the Universe. As an illustration, applying this approach to the
model with $f(R)=R-2\Lambda+\lambda R^{2}$ we see that it can describe three
consecutive epochs of the Universe evolution. This model at first glance looks
similar to the Starobinsky model but does not completely coincide with the
latter due to the presence of the cosmological constant $\Lambda$ and the
difference in the size and the nature of the coeficients of their $R^{2}$
term. Despite the fact that the perturbation is applied for a flat FLWR metric
indicates that the obtained solution can mainly describe the evolution of the
late Univesre, it may also fit an early Universe. Hopefully, the present
method when applied to the case with a more general FLRW metric can describe
more precisely different stages of the evolution of the Universe.

##  1 Introduction

Published by A. Einstein in 1915, the general theory of relativity (GR) [1, 2]
has become one of the pillars of the modern physics and is very well tested
both theoretically and experimentally. It can explain or predict different
astrophysical and cosmological phenomena, such as those in the radiation-
dominated and matter-dominated eras of the Universe (see, for example, [3, 4,
5, 6, 7], for some recent or new results supporting the GR) but it fails to
explain other phenomena, mainly in the acceleration periods of the Universe.
The GR is still facing difficulties in describing the issues such as quantum
gravity [8, 10], the cosmic inflation [9, 10, 11, 12, 13, 14], the origin of
dark matter [15] (see also, for example, [16] in a particle physics aspect),
the accelerated expansion of the Universe (called otherwise the dark energy
problem) [17, 18, 19], etc. These problems call for an extension or
modification of the GR but so far there has been no satisfactory theory
suggested. To name a few famous theories extending the GR. The string theory
(for a review, see, for instance, [20]) expected as a ”theory of everything”
is a very complicated ultra-high energy theory and weakly developed in the
phenomenological aspect, therefore, it is very difficult for an experimental
test. The similar situation is with a more special theory - supergravity [21,
22]. However, these theories are based on the concept of supersymmetry [23] of
which the LHC has not found any sign. In general, to explain the formation and
the evolution of the Universe is always a challenge in physics. One of the
first attempts was made by A. Einstein not long after the birth of the GR.

Einstein modified his original GR by adding a term called the cosmological
constant to the Lagrangian of the Einstein-Hilbert action. By introducing the
cosmological constant, Einstein tried to ”keep” the Universe static as it
could have been generally imagined in his epoch. Later, Einstein, however,
blamed himself that the introduction of the cosmological constant was a
blunder after knowing the theoretical works by A. Friedmann [24] and G.
Lemaître [25] and the observation by E. Hubble [26] proving the expansion of
the Universe. According to these works the Universe is not static at all but
dynamical, more precisely, as stated by Hubble’s law, different parts of the
Universe, e.g. galaxies, are receding from each other at a speed proportional
to their relative distance. Moreover, today we know that the Universe is not
only expanding [26] but is expanding at an increasing rate [17, 18, 19].
Despite this fact, the idea of the cosmological constant may not be completely
useless. It has been expected to help us in solving the dark energy problem by
treating the cosmological constant proportional (or equivalent) to the dark
energy- or vacuum energy density (corresponding to an equation of state with
negative pressure) as we can see by examining Einstein’s equation with
cosmological constant directly. However, another difficulty arising here is
this treatment gives rise to the so-called “cosmological constant problem” or
”vacuum catastrophe”: there is a huge discrepancy, of many orders of
magnitude, between the observed value of the vacuum energy density (generated
by the cosmological constant) and that estimated by quantum field theory [27,
28, 29]. It is the ever known biggest discrepancy between theory and
experiment/observation in physics. The next problem is, with the cosmological
constant in the Einstein equation the Hubble parameter $H$ is also a constant
(independent of time) and therefore, the GR does not accurately describe the
evolution of $H$ over time (very slowly but $H$ changes over time). Adding the
cosmological constant is just the first modification of the GR. There have
been so far many other attempts to extend or modify the GR in order to explain
phenomena beyond the GR. The so-called $f(R)$ theory of gravitation [30, 32,
33, 31, 32, 33], or just the $f(R)$ theory or $f(R)$-gravitation, for short,
is one of the simplest modified theories of the GR, suggested to solve the
problems emerging from the very geometry of the space-time.

Nowadays, the $f(R)$ theory is becoming a hot topical issue and attracting
much interest of a number of astrophysicists and cosmologists (see, for
instance, [10, 14, 30, 31, 32, 33] and references therein). The $f(R)$ theory,
via a transformation to an alternative form - the scalar-tensor theory [30,
32] (in particular, the Jordan-Brans-Dicke theory [1]), can make communication
between particle physics and cosmology. In this framework the scalar fields
may play a crucial role and they, e.g., the inflatons111This idea of Alan
Guth, however, was later rejected by himself [13] but still used elsewhere
[34] could be sometimes [12] treated as the Higgs bosons [35] believed now to
be discovered ten years ago by the LHC collaborations ATLAS and CMS [36, 37]
(see also [38] for a review on the introduction, the search and the discovery
of the Higgs boson). This conforms the fact that different cosmological
problems, such as those of cosmic inflation, dark matter, dark energy, etc.,
can be investigated in the viewpoints of both geometry and particle physics
compatible with each other. Instead of the particle physics approach [10, 30],
here, within the $f(R)$ theory, we will follow the geometrical path to study
some aspects of the evolution of the Universe.

Various models based on the $f(R)$ theory have been suggested. Two of popular
versions of the theory are, for example, the model $f(R)=R+\alpha R^{2}$ (with
$\alpha$ being a coefficient independent of $R$ and other curvature values)
used to explain the accelerated expansion of the Universe during the
inflationary era [30, 39], and the model $f(R)=R+\beta/R$ (with the
coefficient $\beta$ independent of $R$ and other curvature values) used to
explain the accelerated expansion of the Universe in the late epochs including
the present one (the dark energy problem) [30], etc. Each $f(R)$ model can
explain some phenomena in some period of the Universe evolution, but none of
them is perfect and powerful enough to describe different phenomena beyond the
GR in different cosmic times. Regardless any model is considered, to solve the
corresponding (extended) Einstein equation occupies a central position.

In general, to find a solution, especially, an exact one, of an $f(R)$-theory
is very hard, even, sometimes, impossible. To simplify the situation, we can
use a perturbation method and impose some reasonable condition such as the
spherical symmetry which is a good approximation in many cases including that
of a homogeneous and isotropic Universe following the cosmological principles.
Perturbative solutions of the $f(R)$ theory in a central field of a distinct
gravitational source (such as a star, a black hole, etc.) were studied
recently in Refs. [40, 41, 42]. The latter, however, have not yet been applied
to the Universe as a whole. In this article, we go ahead to look for an
approximate solution of a general $f(R)$ theory (i.e., with arbitrary, but
well-defined, $f(R)$ functions) applied to a homogeneous and isotropic
Universe, adopting, thus, the Friedmann-Lemaître-Robertson-Walker (FLRW)
metric [43]. We make perturbation around the GR theory, which means that the
$f(R)$ theories used are slightly different from the GR (as the GR has been
very precisely tested, it is assumed that any deviation from this theory
should be small). As a result, we obtain an explicit perturbative solution for
an FLRW Universe. This perturbative solution improves the GR in the sense that
the Hubble parameter H is not a constant at all but depends on time to
accommodate the evolution of the Universe. As the observed Universe is almost
flat [44], it is reasonable to work here, in a perturbation approach, for the
late Universe (at least), with a flat FLRW metric. At the meantime, we are
working on the early Universe where the flat metric might not be a good
approximation but the same approach could be applied to a general (non-flat)
FLRW metric to describe an early Universe.

In this article the following conventions are used:

  * •

Signature of the Minkowski metric: $(+,-,-,-)$, that is, the infinitesimal
distance is given as

$ds^{2}=\eta_{\mu\nu}dx^{\mu}dx^{\nu}\equiv{dx^{0}}^{2}-dx^{2}-dy^{2}-dz^{2},~{}\text{with}~{}x^{0}=ct.$

  * •

Riemannian curvature tensor:

$\displaystyle R^{\alpha}_{~{}\mu\beta\nu}=$
$\displaystyle\frac{\partial\Gamma^{\alpha}_{\mu\beta}}{\partial
x^{\nu}}-\frac{\partial\Gamma^{\alpha}_{\mu\nu}}{\partial
x^{\beta}}+\Gamma^{\alpha}_{\sigma\nu}\Gamma^{\sigma}_{\mu\beta}-\Gamma^{\alpha}_{\sigma\beta}\Gamma^{\sigma}_{\mu\nu}.$

  * •

Ricci tensor: $R_{\mu\nu}=R^{\alpha}_{~{}\mu\alpha\nu}$.

  * •

Scalar curvature: $R=g^{\mu\nu}R_{\mu\nu},~{}g^{\mu\nu}=g^{\mu\nu}(x)$.

  * •

Energy-momentum tensor of a macroscopic object:

$T_{\mu\nu}=\frac{1}{c^{2}}(\varepsilon+P)u_{\mu}u_{\nu}-Pg_{\mu\nu},$

where $u^{\mu}=\displaystyle\frac{dx^{\mu}}{d\tau}=c\frac{dx^{\mu}}{ds}$,
while $\varepsilon$ and $P$ are the energy density and the pressure,
respectively.

##  2 Perturbative solutions of $f(R)$–theory of gravitation and FLRW
cosmology

The Einstein equation of the GR (or the GR equation for short) with the
cosmological constant $\Lambda$ included can be derived from the Lagrangian
${\cal L}_{\Lambda}=R-2\Lambda$. This (the GR with the cosmological constant)
is the theory around which our perturbative $f(R)$-theory will be developed.
The $f(R)$-theory is a more-general theory with the Lagrangian ${\cal
L}_{G}=f(R)$, where $f(R)$ is a scalar function of the scalar curvature $R$,
leading to the following equation generalizing the Einstein equation [30, 32,
33]:

$\displaystyle f^{\prime}(R)R_{\mu\nu}-g_{\mu\nu}\square
f^{\prime}(R)+\nabla_{\mu}\nabla_{\nu}f^{\prime}(R)-\frac{1}{2}f(R)g_{\mu\nu}=-kT_{\mu\nu},$
(1)

where $k=\displaystyle\frac{8\pi G}{c^{4}}$ ,
$\square=\nabla_{\mu}\nabla^{\mu}$ with $\nabla_{\mu}$ the covariant
derivative, and $f^{\prime}(R)=\displaystyle\frac{d}{dR}f(R)$. This theory in
an appropriate condition, as shown below, can describe the evolution of the
Universe in different stages. To this end we will work with those
$f(R)\equiv{\cal L}_{G}$ which can be developed perturbatively around ${\cal
L}_{\Lambda}$. Since our Universe in large scale is homogeneous, isotropic and
nearly flat, it makes sense to choose its geometry based on a flat (or almost
flat) FLRW metric.

Using the flat FLRW metric (here the unit in which $c=1$ is used) [1, 30]

$\displaystyle ds^{2}=dt^{2}-a^{2}(t)\left(dx^{2}+dy^{2}+dz^{2}\right),$ (2)

we find the following non-zero Ricci tensor elements [1]

$\displaystyle R_{00}$ $\displaystyle=3\frac{\ddot{a}}{a},$ (3) $\displaystyle
R_{ij}$
$\displaystyle=\left(\frac{\ddot{a}}{a}+2\frac{\dot{a}^{2}}{a^{2}}\right)g_{ij},$
(4)

where $i,j=1,2,3$. Combining Eqs (2) – (4) we obtain

$\displaystyle R^{0}_{~{}0}$ $\displaystyle=3\frac{\ddot{a}}{a},$ (5)
$\displaystyle R^{i}_{~{}j}$
$\displaystyle=\left(\frac{\ddot{a}}{a}+2\frac{\dot{a}^{2}}{a^{2}}\right)\delta^{i}_{~{}j},$
(6)

therefore,

$\displaystyle R=6\left(\frac{\ddot{a}}{a}+\frac{\dot{a}^{2}}{a^{2}}\right).$
(7)

With denoting

$\displaystyle T^{0}_{~{}0}=\rho,$ (8) $\displaystyle
T^{1}_{~{}1}=T^{2}_{~{}2}=T^{3}_{~{}3}=-P.$ (9)

and using (2), the equation (1) leads to two independent equations of the
Universe

$3f^{\prime}(R)\frac{\ddot{a}}{a}-3\frac{\dot{a}}{a}\dot{f}^{\prime}(R)-\frac{1}{2}f(R)=-k\rho,$
(10)
$f^{\prime}(R)\left(\frac{\ddot{a}}{a}+2\frac{\dot{a}^{2}}{a^{2}}\right)-2\frac{\dot{a}}{a}\dot{f}^{\prime}(R)-\ddot{f}^{\prime}(R)-\frac{1}{2}f(R)=kP,$
(11)

where $\displaystyle\dot{f}^{\prime}(R)=\frac{\partial}{\partial
t}f^{\prime}(R)$ and
$\displaystyle\ddot{f}^{\prime}(R)=\frac{\partial^{2}}{\partial
t^{2}}f^{\prime}(R)$. Next, for later use in our perturbation approach we
write $f(R)$ in the form

$f(R)=R-2\Lambda+\lambda h(R),$ (12)

with $\Lambda$ and $\lambda$ being constants, and $h(R)$ being a scalar
function of $R$ such that $|\lambda h(R)|\ll|R-2\Lambda|$. Thus, taking (7)
and (12) into account we rewrite (10) and (11) in the following way

$\displaystyle 3\lambda
h^{\prime}(R)\frac{\ddot{a}}{a}-3\frac{\dot{a}^{2}}{a^{2}}-3\lambda\frac{\dot{a}}{a}\dot{h}^{\prime}(R)+\Lambda-\frac{\lambda}{2}h(R)=-k\rho,$
(13)

and

$\displaystyle[-2+\lambda h^{\prime}(R)]\frac{\ddot{a}}{a}+[-1+2\lambda
h^{\prime}(R)]\frac{\dot{a}^{2}}{a^{2}}-2\lambda\frac{\dot{a}}{a}\dot{h}^{\prime}(R)-\lambda\ddot{h}^{\prime}(R)+\Lambda-\frac{\lambda}{2}h(R)=kP,$
(14)

respectively, and then, combining the latter equations (13) and (14) we obtain

$\displaystyle 6[1-\lambda^{2}h^{\prime
2}(R)]\frac{\dot{a}^{2}}{a^{2}}+3\lambda\dot{h}^{\prime}(R)[2+\lambda
h^{\prime}(R)]\frac{\dot{a}}{a}-2\Lambda[1+\lambda h^{\prime}(R)]$
$\displaystyle+3\lambda^{2}\ddot{h}^{\prime}(R)h^{\prime}(R)+\lambda
h(R)[1+\lambda h^{\prime}(R)]-k\rho[2-\lambda h^{\prime}(R)]+3\lambda
kh^{\prime}(R)P=0.$ (15)

For the GR (when $\lambda=0$) these equations are reduced to

$\displaystyle
R^{\mu}_{~{}\nu}-\frac{1}{2}R\delta^{\mu}_{~{}\nu}+\Lambda\delta^{\mu}_{~{}\nu}=-kT^{\mu}_{~{}\nu},$
(16) $R_{GR}=4\Lambda+kT=4\Lambda+k(\rho-3P)=\overline{\Lambda},$ (17)
$H_{GR}=\frac{\dot{a}}{a}=\sqrt{\frac{\Lambda}{3}+\frac{k\rho}{3}},$ (18)

where the notations $T=T^{\mu}_{\mu}$ and
$\overline{\Lambda}=4\Lambda+k(\rho-3P)$ are used.

Now, suppose that in the formula (12) we consider the term $\lambda h(R)$ as a
perturbation term about the GR (when $f(R)=R-2\Lambda$). Thus in the equations
(15) we can treat the terms containing $\lambda$ as perturbation terms about
the GR equations. We solve the equation (15) by a perturbation method as
follows: substituting the solutions of the GR equations (17) and (18) into the
perturbative terms of (15), we get

$\displaystyle 6\frac{\dot{a}^{2}}{a^{2}}-6\lambda^{2}h^{\prime
2}(\overline{\Lambda})\left(\frac{\Lambda}{3}+\frac{k\rho}{3}\right)+3\lambda\dot{h}^{\prime}(\overline{\Lambda})[2+\lambda
h^{\prime}(\overline{\Lambda})]\sqrt{\frac{\Lambda}{3}+\frac{k\rho}{3}}-2\Lambda[1+\lambda
h^{\prime}(\overline{\Lambda})]$
$\displaystyle+3\lambda^{2}\ddot{h}^{\prime}(\overline{\Lambda})h^{\prime}(\overline{\Lambda})+\lambda
h(\overline{\Lambda})[1+\lambda
h^{\prime}(\overline{\Lambda})]-k\rho[2-\lambda
h^{\prime}(\overline{\Lambda})]+3\lambda kh^{\prime}(\overline{\Lambda})P=0.$
(19)

That is, we solve Eq. (15) at its first order of perturbation (19). From (19)
we can immediately obtain the Hubble parameter $H(t)\equiv\dfrac{\dot{a}}{a}$,
with

$\displaystyle H^{2}(t)$ $\displaystyle=\dfrac{\Lambda}{3}[1+\lambda
h^{\prime}(\overline{\Lambda})]+\frac{1}{6}k\rho[2-\lambda
h^{\prime}(\overline{\Lambda})]+\lambda^{2}h^{\prime
2}(\overline{\Lambda})\left(\frac{\Lambda}{3}+\frac{k\rho}{3}\right)$
$\displaystyle-\frac{1}{2}\lambda\dot{h}^{\prime}(\overline{\Lambda})[2+\lambda
h^{\prime}(\overline{\Lambda})]\sqrt{\frac{\Lambda}{3}+\frac{k\rho}{3}}-\frac{1}{2}\lambda^{2}\ddot{h}^{\prime}(\overline{\Lambda})h^{\prime}(\overline{\Lambda})$
$\displaystyle-\dfrac{1}{6}\lambda h(\overline{\Lambda})[1+\lambda
h^{\prime}(\overline{\Lambda})]-\dfrac{1}{2}\lambda
kh^{\prime}(\overline{\Lambda})P.$ (20)

Note that from the equation of state $P=\omega\rho$ for matter ($\omega=0$) we
have $P=0$, formula (20) becomes

$\displaystyle H^{2}(t)$ $\displaystyle=\dfrac{\Lambda}{3}[1+\lambda
h^{\prime}(\overline{\Lambda})]+\frac{1}{6}k\rho[2-\lambda
h^{\prime}(\overline{\Lambda})]+\lambda^{2}h^{\prime
2}(\overline{\Lambda})\left(\frac{\Lambda}{3}+\frac{k\rho}{3}\right)$
$\displaystyle-\frac{1}{2}\lambda\dot{h}^{\prime}(\overline{\Lambda})[2+\lambda
h^{\prime}(\overline{\Lambda})]\sqrt{\frac{\Lambda}{3}+\frac{k\rho}{3}}-\frac{1}{2}\lambda^{2}\ddot{h}^{\prime}(\overline{\Lambda})h^{\prime}(\overline{\Lambda})-\dfrac{1}{6}\lambda
h(\overline{\Lambda})[1+\lambda h^{\prime}(\overline{\Lambda})],$ (21)

where $\overline{\Lambda}$ now takes the value

$\overline{\Lambda}=4\Lambda+k\rho,$ (22)

and

$\displaystyle\dot{h}^{\prime}(\overline{\Lambda})=\dot{\overline{\Lambda}}h^{\prime\prime}(\overline{\Lambda}),$
(23)
$\displaystyle\ddot{h}^{\prime}(\overline{\Lambda})={\dot{\overline{\Lambda}}}^{2}h^{\prime\prime\prime}(\overline{\Lambda})+\ddot{\overline{\Lambda}}h^{\prime\prime}(\overline{\Lambda}).$
(24)

It is easily to find

$a(t)=a_{0}\exp\left[\int^{t}_{t_{0}}H(t)dt\right],$ (25)

with the constant $a_{0}=a(t_{0})$ and $H(t)$ given in (21). Along with (19),
combining (13) and (14) we also have the equation

$\displaystyle[6+3\lambda h^{\prime}(R)]\frac{\ddot{a}}{a}-[3+6\lambda
h^{\prime}(R)]\frac{\dot{a}^{2}}{a^{2}}+3\lambda\ddot{h}^{\prime}(R)-\Lambda+\frac{\lambda}{2}h(R)+2k\rho+3kP=0$
(26)

which will be used later on. The perturbative solution (25) with $H(t)$ given
in (21) can be applied to any model of the $f(R)$-theory satisfying
perturbation conditions (see [40] and below). Now let us do more detailed
calculations for some models of the $f(R)$-theory.

##  3 Applications to some models

###  3.1 Model I: $f(R)=R-2\Lambda+\lambda R^{2}$

This model resembles the Starobinsky model [39] (but does not coincide with
the latter corresponding to the Lagrangian
$f(R)_{Starobinsky}=R+R^{2}/6M^{2}$). Here $h(R)=R^{2}$, $h^{\prime}(R)=2R$
and $h^{\prime\prime}(R)=2$, thus, the formula (21) now takes the form

$\displaystyle
H^{2}(t)=\frac{\Lambda+k\rho(t)}{3}-\frac{\lambda}{3}\left[4\Lambda+k\rho(t)\right]\left[16\lambda\Lambda^{2}+8\lambda\Lambda
k\rho(t)+\frac{3}{2}k\rho(t)+\lambda k^{2}\rho^{2}(t)+6\lambda
k\ddot{\rho}(t)\right]$
$\displaystyle+\frac{4\lambda^{2}}{3}\left[\Lambda+k\rho(t)\right]\left[16\Lambda^{2}+8\Lambda
k\rho(t)+k^{2}\rho^{2}(t)\right]-\lambda
k\dot{\rho}(t)\left[2+8\lambda\Lambda+2\lambda
k\rho(t)\right]\sqrt{\dfrac{\Lambda+k\rho(t)}{3}}.$ (27)

At $\lambda=0$, we obtain the GR value, namely,

$H^{2}_{GR}=\frac{\Lambda+k\rho(t)}{3},$ (28)

which is the first term of (27), while the rest terms are perturbative ones.
Since $\rho(t)\longrightarrow 0$ as $t\longrightarrow\infty$ we get

$\displaystyle H(\infty)=H_{GR}(\infty)=\sqrt{\frac{\Lambda}{3}}.$ (29)

Up to the first oder of $\lambda$, the equation (27) has a relatively simple
form

$\displaystyle H^{2}(t)=\frac{\Lambda+k\rho(t)}{3}-\frac{\lambda
k\rho(t)}{2}\left[4\Lambda+k\rho(t)\right]-2\lambda
k\dot{\rho}(t)\sqrt{\dfrac{\Lambda+k\rho(t)}{3}}.$ (30)

Let us now derive an approximate form of function $\dot{\rho}(t)$ in (30).
From the formula (1), we have the equation

$\nabla_{\mu}T^{\mu}_{~{}\nu}=0,$ (31)

which can be rewritten as

$\dot{\rho}+3H(\rho+P)=0.$ (32)

Taking $H\simeq\sqrt{\displaystyle\frac{\Lambda+k\rho(t)}{3}}$ (and the
equation of state $P=\omega\rho$ for matter $\omega=0$) into account we get

$\dot{\rho}(t)\simeq-\sqrt{3}\rho(t)\sqrt{\Lambda+k\rho(t)},$ (33)

or

$\ddot{\rho}(t)=\dfrac{3\rho(t)\left[2\Lambda+3k\rho(t)\right]}{2}.$ (34)

Inserting (33) in (30) we get (with the constant $c$ put back),

$H(t)=\frac{\dot{a}}{a}=\sqrt{\frac{1}{3}c^{2}\Lambda+\frac{1}{3}c^{4}k\rho(t)+\frac{3\lambda}{2}c^{6}k^{2}\rho^{2}(t)},$
(35)

where $H(t)$ is the Hubble parameter at time $t$. This time dependence $H(t)$
is depicted via $H(\rho)$ in Fig. 1.

Figure 1: Hubble parameter $H$ as a function of $\rho$ between the time
$t_{1}$ and a very far future according to Eq. (35) in the Model-I. The point
$(\rho(t_{2}),H(t_{2}))$ is at the moment $t_{2}$ when the Universe’s
expansion starts to accelerate. The point $\rho=0$ is a limit which might
never be reachable. The time direction is from right to left.

Solving the equation (35) for $a(t)$ we easily find

$a(t)=a_{0}\exp\left(\int^{t}_{t_{0}}\sqrt{\frac{1}{3}c^{2}\Lambda+\frac{1}{3}c^{4}k\rho(t)+\frac{3\lambda}{2}c^{6}k^{2}\rho^{2}(t)}dt\right),$
(36)

where $a_{0}=a(t_{0})$. We note that according to (12) (with (17) taken into
account) the perturbation condition is

$|\lambda h(R)|\ll 2\Lambda,$ (37)

which, for the model $f(R)=R-2\Lambda+\lambda R^{2}$ with $h(R)=R^{2}$,
becomes

$|\lambda|\ll\frac{1}{8\Lambda}.$ (38)

As $\Lambda\sim 10^{-52}m^{-2}$ [45] the perturbation upper bound of $\lambda$
is very large, $\lambda\ll 10^{51}m^{2}$. With (34) and (35) the equation (26)
gets the form

$\displaystyle\left[1+4\lambda\Lambda+\lambda
c^{2}k\rho(t)\right]\frac{\ddot{a}}{a}=\frac{c^{2}\Lambda}{3}-\frac{1+2\lambda\Lambda}{6}c^{4}k\rho(t)-\frac{19}{6}\lambda
c^{6}k^{2}\rho^{2}(t)+\frac{4c^{2}\lambda\Lambda^{2}}{3},$ (39)

This equation depicted in Fig. 2 shows how the Universe is expanding
acceleratedly (as from the moment when the expansion acceleration becomes
positive).

Figure 2: Formula (39) shows how the Univesre is acceleratedly exapanding in
the Model-I. The point $(\rho(t_{2}),0)$ is at the moment $t_{2}$ when the
acceleration of the Universe’s expansion changes the sign. The point $\rho=0$
is a limit which might be never reached. The time direction is from right to
left.

From here we find the following condition for the accelerated expansion of the
Universe (for $\ddot{a}>0$):

$c^{2}k\rho(t)<\frac{\sqrt{612\lambda^{2}\Lambda^{2}+156\lambda\Lambda+1}-1-2\lambda\Lambda}{38\lambda}.$
(40)

If $\lambda\Lambda\ll 1$, thus $\sqrt{1+156\lambda\Lambda}\approx
1+78\lambda\Lambda$, the condition (40) becomes

$c^{2}k\rho(t)<2\Lambda,$ (41)

consistently with the GR.

To finish this subsection, let us make some comments on (35). In a late era,
such as the today era, when the density $\rho$ is very small the third term is
dominated by the second term, while in an earlier era, such as the
inflationary era, the third term dominates, therefore,

$H_{early}(t)=\frac{\dot{a}}{a}=\sqrt{\frac{1}{3}c^{2}\Lambda+\frac{3\lambda}{2}c^{6}k^{2}\rho^{2}(t)},$
(42)

thus,

$a_{early}(t)=a_{0}\exp\left(\int^{t}_{t_{0}}\sqrt{\frac{1}{3}c^{2}\Lambda+\frac{3\lambda}{2}c^{6}k^{2}\rho^{2}(t)}dt\right).$
(43)

Further discussions will be made in the next session.

###  3.2 Model II: $f(R)=R-2\Lambda+\lambda
R^{2}+\displaystyle\frac{\gamma}{R}$

For a late era of the Universe, $\lambda R^{2}$ is small in comparison with
$\displaystyle\frac{\gamma}{R}$ we have

$f(R)=R-2\Lambda+\frac{\gamma}{R}$ (44)

with $h(R)=\displaystyle\frac{1}{R}$. Using (33), (34), (22) and the
approximations

$\displaystyle\overline{\Lambda}\simeq 4\Lambda+c^{2}k\rho(t),$ (45)
$\displaystyle\overline{\Lambda}^{2}\simeq 16\Lambda^{2}+8\Lambda
c^{2}k\rho(t),$ (46)

we get the solution (21) in the form (neglecting those terms of order
$\gamma^{2}$)

$\displaystyle
H(t)=\sqrt{\frac{1}{3}c^{2}\Lambda+\frac{1}{3}c^{4}k\rho(t)-\frac{c^{2}\gamma}{16\Lambda}+\frac{c^{4}\gamma
k\rho(t)}{16\Lambda^{2}}+\frac{c^{6}\gamma
k^{2}\rho^{2}(t)}{384\Lambda^{3}}},$ (47)

hence,

$\displaystyle
a(t)=a_{0}\exp{\int^{t}_{t_{0}}H(t)dt}=a_{0}\exp\left(\int^{t}_{t_{0}}\sqrt{\frac{1}{3}c^{2}\Lambda+\frac{1}{3}c^{4}k\rho(t)-\frac{c^{2}\gamma}{16\Lambda}+\frac{c^{4}\gamma
k\rho(t)}{16\Lambda^{2}}+\frac{c^{6}\gamma
k^{2}\rho^{2}(t)}{384\Lambda^{3}}}dt\right).$ (48)

As $\rho(t)\longrightarrow 0$ when $t\longrightarrow\infty$ (see (47))

$H(\infty)=\sqrt{\frac{1}{3}c^{2}\Lambda-\frac{c^{2}\gamma}{16\Lambda}},$ (49)

we see that a perturbation term $\displaystyle\frac{c^{2}\gamma}{16\Lambda}$
still contributes to the GR.

##  4 Numerical discussions

Let us now consider perturbation conditions numerically. In the system of
units SI with $c=299792458m/s$, $G=6.67259\times 10^{-11}kg^{-1}m^{3}s^{-2}$
and $k=2.0761154\times 10^{-43}kg^{-1}m^{-1}s^{2}$, the observed value of the
Hubble parameter today $H(\tau)$ is

$H(\tau)=\sqrt{\displaystyle\frac{c^{2}\Lambda}{3}}\sim 0.18199\times
10^{-17}s^{-1},$ (50)

following the cosmological constant determined via the data given in 2018 by
the Planck collaboration [45] (see also wikipedia.org):

$\Lambda\sim 1.1056\times 10^{-52}m^{-2}.$ (51)

From the perturbation conditions (see (12))

$\lambda h(R)\ll 2\Lambda,$ (52)

we have

$\lambda h(R)\ll 2.21\times 10^{-52}m^{-2}.$ (53)

As an illustration, let’s consider the model $f(R)=R-2\Lambda+\lambda R^{2}$,
then the perturbation condition (52) will be $\lambda R^{2}\ll 2\Lambda$,
taking $R\sim 4\Lambda$ (see (17)), we get the perturbation condition for this
model

$\lambda\ll\frac{1}{8\Lambda}.$ (54)

Using (51), we have

$\lambda\ll 1.1306\times 10^{51}m^{2}.$ (55)

Now we do some estimate calculations on (35). Let $\lambda$ take the limit
value (55). Based on this value we estimate the Hubble parameter at the time,
say, $t_{1}$ (counted from the Big Bang), when the perturbation condition
starts to be applicable, that is,

$\frac{1}{3}c^{4}k\rho(t_{1})=\frac{3\lambda}{2}c^{6}k^{2}\rho^{2}(t_{1}),$
(56)

or

$\rho(t_{1})=\frac{2}{9\lambda c^{2}k}=1.05\times 10^{-26}kg/m^{3}.$ (57)

Then

$H(t_{1})\sim 3.8796\times 10^{-18}s^{-1},$ (58)

therefore,

$t_{1}\simeq\frac{1}{H(t_{1})}\sim 8.17~{}\mbox{billion years}.$ (59)

The contribution of the perturbation term to the GR is quite significant at an
earlier time before $t_{1}$ and neglegible at a very late time.

Next, we estimate the Hubble parameter at the time $t_{2}$ when the
accelerated expansion of the Universe gets start (40),

$c^{2}k\rho(t_{2})=\frac{\sqrt{612\lambda^{2}\Lambda^{2}+156\lambda\Lambda+1}-1-2\lambda\Lambda}{38\lambda},$
(60)

or

$\rho(t_{2})=0.5280225\times 10^{-26}kg/m^{3}.$ (61)

We calculate the GR value of the Hubble parameter at that time $t_{2}$, with
$c^{2}k\rho_{GR}(t_{2})=2\Lambda$ (see (41))

$H_{GR}(t_{2})=\sqrt{\frac{1}{3}c^{2}\Lambda+\frac{1}{3}c^{4}k\rho_{GR}(t_{2})}\sim
0.315216\times 10^{-17}s^{-1},$ (62)

corresponding to the Hubble time

$t_{2GR}\simeq\frac{1}{H_{GR}(t_{2})}\sim 10.05~{}\mbox{billion years}.$ (63)

The Hubble parameter and time calculated according to (35) are

$\displaystyle H(t_{2})$
$\displaystyle=\sqrt{\frac{1}{3}c^{2}\Lambda+\frac{1}{3}c^{4}k\rho(t_{2})+\frac{3\lambda}{2}c^{6}k^{2}\rho^{2}(t_{2})}$
$\displaystyle\sim 0.27827\times 10^{-17}s^{-1},$ (64)

and

$t_{2}\simeq\frac{1}{H(t_{2})}\sim 11.39~{}\mbox{billion years},$ (65)

respectively (cf. Figs. 1 and 2). The accelerated expansion of the Universe
begins about 10.05 billion years after the Big Bang (according to the GR) or
about 11.39 billion years after the Big Bang (according to the model-I,
$f(R)=R-2\Lambda+\lambda R^{2}$). The effect of the $f(R)$ theory is
noticeable. The perturbation condition became applicable to the Model-I about
8.17 billion years after the Big Bang (i.e., about 5.6 billion years ago).

To finish we do a comparison of the perturbative solution of Model-I above
with the solution of the Starobinsky model
$f(R)_{Starobinsky}=R+\displaystyle\frac{1}{6M^{2}}R^{2}$, with a constant $M$
[30, 39]. Let us note again that the Model-I has a similar but not the same
form with the Starobinsky model as the Lagrangian $f(R)_{Starobinsky}$ does
not have ${\cal L}_{\Lambda}=R-2\Lambda$ as a perturbative limit, unlike the
Lagrangian $f(R)=R-2\Lambda+\lambda R^{2}$ of the Model-I perurbatively
developed around ${\cal L}_{\Lambda}$. We see first that when considering the
Universe in the late epoch (including the present time), as the term $R^{2}$
is too small (because $R$ is small), the Starobinsky model gives almost no
contribution to the Universe evolution in the present epoch, while the
perturbative solution in the model-I gives a significant contribution for an
appropriate value of $\lambda$. Therefore, the Starobinsky model, unlike the
model-I, cannot describe well the late Universe but the early one. More
precisely, the Starobinsky model contributes only during the era of the
inflationary Universe, when $R$, that is, $R^{2}$ was very big. In the
standard inflation model there must be $\dot{H}<0$, otherwise, the Universe
could be hyperinflationary [30]. The Starobinsky model is consistent with the
standard inflation condition for $\dot{H}\propto\displaystyle\frac{-M^{2}}{6}$
[30]. However, in the Starobinsky model the matter term (e.g., $\rho(t)$) has
no contribution to the inflation rate. The perturbative solution (42) of the
model-I, when applied to the inflationary Universe, despite that the flat FLRW
metric is used, as $\dot{H}<0$ due to $\dot{\rho}(t)<0$, is consistent with
the standard inflation, moreover, the matter term $\rho(t)$ also contributes
to the inflation process of the Universe. In other words, the model-I under
the current perturbation approach, is more ”flexible” than the Starobinsky
model in accommodating different periods of the Universe’s evolution.

##  5 Conclusions

A general perturbative solution of the $f(R)$ theory in the FLRW cosmology is
(25) with $H(t)$ given in (21). This solution describes an acceleratedly
expanding Universe ($\dot{a}>0$ and $\ddot{a}>0$). For the model
$f(R)=R-2\Lambda+\lambda R^{2}$ the solution becomes (35) and (36) showing
that this model can describe three consecutive eras of the Universe, while for
the model $f(R)=R-2\Lambda+\displaystyle\frac{\gamma}{R}$ the solution is (47)
and (48). We see that the Hubble parameter $H$ depends on time (while in the
GR theory it is a constant). These perturbative solutions improves the GR
solution in the sense that they show the evolution of H over time. We see that
when $t\longrightarrow\infty$, the Hubble parameter of the
$f(R)=R-2\Lambda+\lambda R^{2}$ model will approach that for the GR theory
(with $H=\sqrt{\displaystyle\frac{c^{2}\Lambda}{3}}$), but in the
$f(R)=R-2\Lambda+\displaystyle\frac{\gamma}{R}$ model it will not.

We would like to stress that it is reasonable to work in the perturbative
approach to the $f(R)$ theory, that means, we work with only those $f(R)$
satisfying the perturbation condition which could be present in some periods
of the Universe evolution and have shown that a perturbative $f(R)$-theory
could be a good theory of the Universe evolution in different stages.

We could consider applying the present work to an arbitrary (non flat in
general) FLRW metric and the problem would become more complicated but we hope
the method would work.

## Acknowledgement

This work is funded by the National Foundation for Science and Technology
Development (NAFOSTED) of Vietnam under Grant No. 103.99-2020.50.

## References

  * [1] S. Weinberg “Gravitation and cosmology: Principles and applications of the general theory of relativity”, John Wiley $\&$ Son, New York, 1972.
  * [2] L. D. Landau and E.M. Lifshitz, “The classical theory of fields”, vol. 2, Elsevier, Oxford, 1994.
  * [3] R. Penrose, “Gravitational collapse and space-time singularities” Phys. Rev. Lett. 14, 57-59 (1965)
  * [4] S. Gillessen, F. Eisenhauer, S. Trippe, T. Alexander, R. Genzel, F. Martins and T. Ott, “Monitoring stellar orbits around the massive black hole in the Galactic center”, Astrophys. J. 692, 1075-1109 (2009); [arXiv:0810.4674 [astro-ph]].
  * [5] B. P. Abbott et al. [LIGO Scientific and Virgo], “GW170817: Observation of gravitational waves from a binary neutron star inspiral”, Phys. Rev. Lett. 119 (2017) no.16, 161101; [arXiv:1710.05832 [gr-qc]].
  * [6] B. P. Abbott et al. [LIGO Scientific and Virgo], “Observation of gravitational waves from a binary black hole merger”, Phys. Rev. Lett. 116, no.6, 061102 (2016); [arXiv:1602.03837 [gr-qc]].
  * [7] M. Kramer, I. H. Stairs, R. N. Manchester, N. Wex, A. T. Deller, W. A. Coles, M. Ali, M. Burgay, F. Camilo and I. Cognard, et al. “Strong-field gravity tests with the double pulsar”, Phys. Rev. X 11, no.4, 041050 (2021); [arXiv:2112.06795 [astro-ph.HE]].
  * [8] C. Rovelli, Quantum gravity, Cambridge unversity press, Cambridge, 2010.
  * [9] A. A. Starobinsky, “Spectrum of relict gravitational radiation and the early state of the universe”, JETP Lett. 30, 682-685 (1979).
  * [10] A. D. Linde, “Particle physics and inflationary cosmology”, Harwood academic publishers, Chur, Switzerland, 1990; Contemp. Concepts Phys. 5, 1-362 (1990); [arXiv:hep-th/0503203 [hep-th]].
  * [11] A. R. Liddle and D. H. Lyth, “Cosmological inflation and large scale structure”, Cambridge university press, Cambridge 2000 (published online 2012).
  * [12] A. H. Guth, “The Inflationary Universe: A possible solution to the horizon and flatness problems”, Phys. Rev. D 23, 347-356 (1981).
  * [13] A. H. Guth, “The inflationary universe: The quest for a new theory of cosmic origins”, AddisonWesley, Reading, MA, 1997.
  * [14] V. Mukhanov, “Physical foundations of cosmology”, Cambridge unversity press, Oxford, 2005.
  * [15] A. Arbey and F. Mahmoudi, “Dark matter and the early Universe: a review”, Prog. Part. Nucl. Phys. 119, 103865 (2021); [arXiv:2104.11488 [hep-ph]].
  * [16] M. Drewes, T. Lasserre, A. Merle, S. Mertens, R. Adhikari, M. Agostini, N. Anh Ky, T. Araki, M. Archidiacono and M. Bahr, et al. “A white paper on keV sterile neutrino dark matter”, JCAP 01, 025 (2017); [arXiv:1602.04816 [hep-ph]].
  * [17] A. G. Riess et al. [Supernova Search Team], “Observational evidence from supernovae for an accelerating universe and a cosmological constant”, Astron. J. 116, 1009-1038 (1998); [arXiv:astro-ph/9805201 [astro-ph]].
  * [18] S. Perlmutter et al. [Supernova Cosmology Project], “Measurements of $\Omega$ and $\Lambda$ from 42 high redshift supernovae”, Astrophys. J. 517, 565-586 (1999); [arXiv:astro-ph/9812133 [astro-ph]].
  * [19] R. P. Crease, “The discovery of dark energy: Historical reflections” AIP Conf. Proc. 1166, no.1, 87-94 (2009).
  * [20] J. Polchinski, “String theory”. Vol. 1 & 2, Cambridge university press, Cambridge 2007.
  * [21] P. Van Nieuwenhuizen, “Supergravity”, Phys. Rept. 68, 189-398 (1981).
  * [22] Y. Tanii, “Introduction to supergravity” (ebook: doi:10.1007/978-4-431-54828-7), published by Springer in Tokyo, Japan, 2014.
  * [23] P. C. West, “Introduction to supersymmetry and supergravity” (2nd edition), World scientific, 1990.
  * [24] A. Friedman, “On the curvature of space” (translted from German) Z. Phys. 10, 377-386 (1922)
  * [25] G. Lemaitre, “A homogeneous universe of constant mass and increasing radius accounting for the radial velocity of extra-galactic nebulae” (translted from French) Mon. Not. Roy. Astron. Soc. 91, 483-490 (1931)
  * [26] E. Hubble, “A relation between distance and radial velocity among extra-galactic nebulae”, Proc. Nat. Acad. Sci. 15, 168-173 (1929).
  * [27] J. Martin, “Everything you always wanted to know about the cosmological constant problem (but were afraid to ask)”, Comptes Rendus Physique 13, 566-665 (2012); [arXiv:1205.3365 [astro-ph.CO]].
  * [28] R. J. Adler, B. Casey and O. C. Jacob, “Vacuum catastrophe: An elementary exposition of the cosmological constant problem”, Am. J. Phys. 63, 620-626 (1995)
  * [29] S. Weinberg, “The cosmological constant problem”, Rev. Mod. Phys. 61, 1-23 (1989)
  * [30] A. De Felice and S. Tsujikawa, “f(R) theories,” Living Rev. Rel. 13, 3 (2010); [arXiv:1002.4928 [gr-qc]].
  * [31] S. Nojiri and S. 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 [gr-qc]].
  * [32] Thomas P. Sotiriou and Valerio Faraoni, “$f(R)$ theories of gravity,” Rev. Mod. Phys. 82, 451 (2010).
  * [33] S. Capozziello and M. De Laurentis, “Extended theories of gravity,” Phys. Rept. 509, 167 (2011); [arXiv:1108.6266 [gr-qc]].
  * [34] F. Jegerlehner, “Higgs inflation and the cosmological constant”, Acta Phys. Polon. B 45, no.6, 1215 (2014); [arXiv:1402.3738 [hep-ph]].
  * [35] P. W. Higgs, “Broken symmetries and the masses of gauge bosons” Phys. Rev. Lett. 13, 508-509 (1964)
  * [36] G. Aad et al. [ATLAS], “Observation of a new particle in the search for the Standard Model Higgs boson with the ATLAS detector at the LHC” Phys. Lett. B 716, 1-29 (2012) [arXiv:1207.7214 [hep-ex]].
  * [37] S. Chatrchyan et al. [CMS], ‘‘Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC” Phys. Lett. B 716, 30-61 (2012); [arXiv:1207.7235 [hep-ex]].
  * [38] Nguyen Anh Ky and Nguyen Thi Hong Van, “Was the Higgs boson discovered?” Commun. in Phys. 25, no.1, 1-20 (2015); [arXiv:1503.08630 [hep-ph]].
  * [39] A. A. Starobinsky, “A new type of isotropic cosmological models without singularity”, Phys. Lett. B 91, 99-102 (1980).
  * [40] N. Anh Ky, P. V. Ky and N. T. Hong Van, “Perturbative solutions of the $f(R)$-theory of gravity in a central gravitational field and some applications,” Eur. Phys. J. C 78, no.7, 539 (2018) [erratum: Eur. Phys. J. C 78, no.8, 664 (2018)] [arXiv:1807.04628 [gr-qc]].
  * [41] Nguyen Anh Ky, Pham Van Ky and Nguyen Thi Hong Van, “Testing the $f(R)$-theory of gravity”, Commun. Phys. 29, no. 1, 35 (2019) [arXiv:1904.04013 [physics.gen-ph]].
  * [42] P. Van Ky, N. T. Hong Van and N. Anh Ky, “New phenomena with the f(R)-theory of gravitation in a central gravitational field,” J. Phys. Conf. Ser. 1506, no.1, 012001 (2020)
  * [43] S. Weinberg, “Cosmology”, Oxford university press, Oxford, 2008.
  * [44] R. J. Adler and J. M. Overduin, “The nearly flat universe”, Gen. Rel. Grav. 37, 1491-1503 (2005); [arXiv:gr-qc/0501061 [gr-qc]].
  * [45] N. Aghanim et al. [Planck], “Planck 2018 results. VI. Cosmological parameters”, Astron. Astrophys. 641, A6 (2020) [erratum: Astron. Astrophys. 652, C4 (2021)] [arXiv:1807.06209 [astro-ph.CO]].