An Inverse $f(R)$ Gravitation for Cosmic Speed up, and Dark Energy
  Equivalent
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Modern Physics Letters A
c© World Scientific Publishing Company
An Inverse f(R) Gravitation for Cosmic Speed up, and Dark Energy
Equivalent
Sohrab Rahvar
Department of Physics, Sharif University of Technology, P.O.Box 11365-9161, Tehran, Iran.∗
Yousef Sobouti
Institute for Advanced Studies in Basic Sciences,
P.O.Box 45195-1159, Zanjan, Iran
Received (Day Month Year)
Revised (Day Month Year)
To explain the cosmic speed up, brought to light by the recent SNIa and CMB ob-
servations, we propose the following: a) In a spacetime endowed with a FRW metric,
we choose an empirical scale factor that best explains the observations. b) We assume
a modified gravity, generated by an unspecified field lagrangian, f(R). c) We use the
adopted empirical scale factor to work back retroactively to obtain f(R), hence the term
‘Inverse f(R)’. d) Next we consider the classic GR and a conventional FRW universe
that, in addition to its known baryonic content, possesses a hypothetical ‘Dark Energy’
component. We compare the two scenarios, and find the density, the pressure, and the
equation of the state of the Dark Energy required to make up for the differences between
the conventional and the modified GR models.
Keywords: Cosmology; Dark Energy; Modified Gravity.
95.36.+x, 98.80.Jk, 98.80.Es
∗rahvar@sharif.edu
http://arxiv.org/abs/0704.0680v2
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2 Rahvar & Sobouti
As cosmological standard candles, supernovae type Ia (SNIa) appear dimmer
than what one expects from a Cold Dark Matter (CDM) model of the universe
1,2,3. This observation and other evidences from the Cosmic Microwave Background
(CMB) measurements indicate that the universe is in an acceleration phase of its
expansion 4,5,6. A conventional CDM scenario does not explain this speed up.
Some authors have stipulated a dark energy component to make up for whatever
dynamical effects that the known energy momentum content of the model leaves
unaccounted for 7,8,9,10,11,12,13,14,15,16,17,18,19,20,21. Others have entertained
alternatives to Einstein’s gravitation 22,23,24,25,26,27,28,29,30. Yet a third school
have resorted to inhomogeneous FRW universes to explain the dilemma 31. Since
all these approaches attempt to answer the same question, all should be equivalent,
and there should be a way to translate one language to the other.
Here, we are concerned with the ’dark energy’ and ’alternative gravitation’ sce-
narios. We suggest to begin with a Freidman- Robertson- Walker (FRW) universe,
to choose its scale factor, a(t), in a way that best explains the available observa-
tions, and to work out the dynamics of the spacetime. Next, to write down the
field equations for a modified f(R) gravitation, and knowing a(t), to solve for f(R).
Finally, to attribute whatever deviations from the conventional FRW results there
is, to a dark energy field, and to obtain its density, pressure, equation of state, etc.
The choice of the scale factor
In a FRW metric, ds2 = −dt2 + a(t)2dx2, single term scale factors of the form
a ∝ tβ lead to constant deceleration parameters, q = −äa/ȧ2 = (1 − β)/β, and do
not serve the purpose. We propose the following two-term ansatz
a(t) =
1 + p
(t/t0)
1 + p(t/t0)
, (1)
where t0 is the age of the universe, and α and p are the free parameters of the model,
to be adjusted to ensure compatibility of the emerging results with observations.
The factor (1+p)−1 is introduced to have a(t0) = 1. By letting either α or p tend to
zero, one recovers the standard CDM universe. Hereafter, for economy in writing,
we will use the time parameter τ = p(t/t0)
2α/3 instead of the conventional time t.
From Eq. (1) one finds
[1− (1 + α)(2α− 1)τ ] [1 + τ ] [1 + (1 + α)τ ]−2 (2)
)−3/2α [1 + (1 + α)τ ] [1 + τ ]
, (3)
R = t02R = 6H2(1− q) =
)−3/α (4)
× [1 + (2 + 5α+ 2α2)τ + (1 + 5α+ 4α2)τ2] [1 + τ ]−2 .
For α > 1/2, q can become negative and remain nonsingular throughout. Transition
from a decelerated phase of expansion to an accelerated one takes place at τtrans =
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An Inverse f(R) Gravitation for Cosmic Speed up, and Dark Energy Equivalent 3
[(α+ 1)(2α− 1)]−1 or ttrans = [(α+ 1)(2α− 1)p]
−3/2α
t0 < t0. For −1 < α < 1/2,
q is always positive and nonsingular. For α < −1, q can become negative but also
singular. The last two possibilities are discarded. For all values of α and p,
limit q(t → 0) =
limit q(t → ∞) = (1− 2α)/2(1 + α). (5)
In Fig.(1) we have plotted q(t) versus the redshift, z = a(t0)/a(t) − 1, for several
values of α and p = 1/3.
0 1 2
Fig. 1. Plot of q(t) versus the redshift, z = a(t0)/a(t) − 1, for α = 0, 1, 2, 3, 4; and p = 1/3. The
case α = 0 gives the classic value, q = 2/3. As α increases from 1 to 4, transition to the accelerated
phase of expansion moves from later to earlier epochs, from smaller z’s to larger one.
As to H and R, both remain positive for all times. Both tend to ∞ as τ → 0
and decease monotonically to 0 as τ → ∞. They exhibit a normal behavior in the
neighborhood of τtrans = [(1 + α)(2α − 1)]−1.
Equation (3), written for the present epoch, reveals a constraint on α and p,
that should be observed in the final design of the model. Thus,
[1 + (1 + α)p]/[1 + p] =
H0t0 ≈
. (6)
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4 Rahvar & Sobouti
0 0.5 1 1.5
Fig. 2. The distance modulus of the SNIa Gold’s sample versus redshifts, black circles; And the
plot of Eq. (7) with the scale factor of Eq. (1), solid line. Parameters of the model are α = 2 and
p = 1/3.
Empirical values of α & p
The distance modulus (corrected for the reddening) and the (dimensionless) lumi-
nosity distance, DL(z;α, p), of supernovae are related as
µ = m−M = 5 logDL(z;α, p) + 25, (7)
DL(z;α, p) = (1 + z)
dzH(z;α, p)−1.
In Fig. (2), the observed distance modulus of 157 SNIa of Gold’s sample are plot-
ted versus the redshifts. The solid curve is the plot of equation (7), in which, in
compliance with the constraint of Eq. (6), we have chosen
α = 2, p = 1/3. (8)
The fit to the data points is adequate for our purpose, though the parameters can
be refined to optimize the fit. These numbers give ttrans = 0.43t0 and ztrans = 1.14.
On adopting H0 ≈ 70km sec−1 Mpc−1, one obtains an age of t0 ≈ 12.4Gyr for the
universe.
An inverse f(R) way out
We begin with a modified field equation generated by an, as yet, unspecified field
lagrangian, f(R),
Rµν −
Rgµν =
(f −RF )gµν (9)
(∇µ∇ν − gµν∇λ∇λ)F −
T (M)µν ,
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An Inverse f(R) Gravitation for Cosmic Speed up, and Dark Energy Equivalent 5
where F (R) = df(R)/dR, and κ = 8πG. For a universe of FRW type, filled with a
perfect fluid of density ρm and pressure pm, Eq. (9) and the equation of continuity
reduce to
3HḞ + 3H2F −
(RF − f)− κρm = 0, (10)
F̈ −HḞ + 2ḢF + κ(ρm + pm) = 0, (11)
ρ̇m + 3H(ρm + pm) = 0, (12)
We further neglect the pressure and integrate Eq. (12) to obtain ρm(t) = ρ0a(t)
Next we substitute for ρm in Eq. (11), assume α = 2, change the time variable to
τ = p(t/t0)
4/3, and find
(1 + τ)3τ2F ′′ −
(1 + τ)2(1 + 5τ)τF ′ (13)
(1 + τ)(1 +
τ + 3τ2)F +
where the ′′ ′ now stands for d/dτ , and we have, arbitrarily, put the dimensionless
constant (3/4)3(1+p)3t0
2κρ0, that appear in the course of mathematical manipula-
tions equal to one. We will shortly discuss the numerical solution of Eq. (13). Some
general remarks on its asymptotic behavior, however, are instructive. As τ → 0 we
F (τ) =
τ2 + . . .
+ c1 τ
−0.44P1(τ) + c2 τ
1.7P2(τ), τ → 0.
where P1 and P2 are calculable polynomials in τ , and begin with term 1; c1 and c2 are
constants of integration to be obtained from boundary conditions. The exponents,
−0.44 and 1.7 are approximate solutions of the indicial equation, s2− 5
= 0. As
τ → 0, F diverges to ∞ or converge to 0 on account of one or the other term. This
feature makes the solutions sensitive to a CDM type boundary conditions of the
form, F (τinitial) = 1 and F
′(τinitial) = 0. Presently we have no basis, observational
or otherwise, to make an intelligent guess as to what the appropriate boundary
conditions are. For the sake of argument, however, we have adopt c1 = c2 = 0, and
kept only the proper solution of Eq. (12). With F (τ) known, Eq.(10) becomes an
algebraic equation to calculate f(τ). The numerically calculated solutions of F (τ),
f(τ) and R = t02R(τ) are plotted in Fig. (3). Elimination of τ in favor R provides
F (R) and f(R)
It is instructive to examine the asymptotic behavior of F (R) and f(R) analyti-
cally. In the limit of small and large τ ’s, corresponding to large and small R’s, one
finds
F (R) =
6pR−2/3 + · · ·
R → ∞ (15)
R → 0. (16)
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6 Rahvar & Sobouti
0.1 0.2 0.3 0.4 0.5
100* F( )
Fig. 3. F (τ), proper solution of Eq. (13) (×100), dot- dashed line; f(τ), dashed line; and R(τ),
solid line; α = 2, p = 1/3.
Note that F (R) is a dimensionless scalar as it should be. With F (R) = df/dR
known, it is a matter of simple integration to obtain f(R). Thus,
f(R) =
6pR−2/3 + · · ·
, R → ∞, (17)
, R → 0. (18)
At early epochs (large R’s), the spacetime approaches the conventional FRW uni-
verse with the classical GR while for the later times we have a positive acceleration
universe. Another point about this action is that for R → 0 in the solar system
scales, f(0) = 0 and f ′(0) = 0. In this case we will have standard GR equation
and f(R) evades from the solar system test. Recently this type of models have been
studied by introducing action in the form of f(R) = R+ f1(R), where for R = 0 in
the solar system, f(0) = 0 and for larger R’s, in cosmological scales and inside the
large scale structures the action reduces to f(R) = R− Λ 32,33.
In the remaining range of R, integration is done numerically and the results are
plotted in Fig. (4).
Dark Energy equivalent
Instead of the modified gravitation considered above, let us assume a classic FRW
universe. That is, let f(R) = R and F = 1. Let this universe, however, have a
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An Inverse f(R) Gravitation for Cosmic Speed up, and Dark Energy Equivalent 7
10 20 30 40
Fig. 4. Numerical plot of f(R) versus R. τ is eliminated between f(τ) and R(τ); α = 2, p = 1/3.
’Dark Energy’ component, in addition to its conventional baryonic content. The
counterparts of Eqs. (10), and (11) will be
3H2 − κ (ρde + ρm) = 0, (19)
2Ḣ + κ (ρde + ρm) + κ (pde + pm) = 0. (20)
Subtracting Eq. (10) from (19), and Eq. (11) from (20) gives
κρde = 3H
2(1− F )− 3HḞ +
(RF − f), (21)
κ(ρde + pde) = F̈ −HḞ − 2Ḣ(1− F ), or
κpde = F̈ + 2HḞ −H2(1− 2q)(1− F )
(RF − f). (22)
The equation of state for the dark energy is obtained by eliminating τ , implicit in
F and H , between Eqs. (21) and (22). This is done numerically and w = pde/ρde
as a function of the redshift is plotted in Fig. (5).
Concluding remarks
The algorithm of Fig. 6 summarizes the path we have followed in this letter. We
have resorted to the SNIa observations to design an empirical FRW metric that
allows the model universe to transit from a phase of decelerated expansion at early
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8 Rahvar & Sobouti
0 1 2 3
-0.685
-0.68
-0.675
-0.67
Fig. 5. Equation of State of the Dark Energy: Numerical plot of w = pde/ρde versus the redshift
epochs to an accelerated one at later times. The spacetime is almost a CDM model
and the gravity is almost the classic GR at very early times, but evolves away in
course of time. Next we have maintained that the so-designed spacetime is deducible
from a modified non- Hilbert- Einstein field lagrangian, f(R) . Knowing the metric,
we have solved the modified field equations retroactively for the sought-after f(R).
Finally, we have compared our results with those of a conventional FRW model
and have attributed the differences between the two to a dark energy component.
Eventually, we have extracted the density, the pressure, and the equation of state
of this stipulated energy.
We note that our choice of the scale factor and the adjustment of its free parameters,
to comply with the available cosmological observations, is, by no means, unique.
The goal is simply to demonstrate that the use of the observations at the outset,
to deduce the rudiments of what seems reasonable, facilitates the access to possible
formal underlying theories, the action based f(R) gravity in our case. With the
availability of more extensive and more accurate data in future one may come back
and revise the model. See also 34 for a similar emphasis.
One of us (YS)35 has followed the same path to propose a modified gravitation for
galactic environments and to explain the flat rotation curves and the Tully-Fisher
relation in spiral galaxies without recourse to hypothetical dark matters.
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An Inverse f(R) Gravitation for Cosmic Speed up, and Dark Energy Equivalent 9
boundary
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equation of
f=f(F, F’, )
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from observation
Fig. 6. Algorithm of inverse f(R): Choose a scale factor; Solve field equations first for F = df/dR
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