CC BY
51
UDC 530.1; 539.1
The universe evolution in exponential F(R)-gravity
K. Bamba1, A. Lopez-Revelles2,\ R. Myrzakulov6, S. D. Odintsov2,3,4,6 and L. Sebastiani5,6
1KMi! Nagova University, Nagoya, Japan 2ICE/CSIC-IEEC, Bellaterra (Barcelona),'Spain. ‘3 ICREA, Barcelona, Spain 4 Tomsk State Pedagogical University, Tomsk, Russia 5 Dipartimento di Fisica, University di Trento, Italia 6 Department of General & Theoretical Physics, Eurasian National University, Astana, Kazakhstan
E-mail: odintsov@tspu.edu.ru
A generic feature of viable exponential F(R)-gravity is investigated. An additional modification to stabilize the effective dark energy oscillations during matter era is proposed and applied to two viable models. An analysis on the future evolution of the universe is performed. Furthermore, a unified model for early and late-time acceleration is proposed and studied.
Keywords: gravity, dark energy, universe, cosmology.
1 Introduction
The current acceleration of our universe is supported by various observations and cosmological data [1]. There exist several descriptions of this acceleration and, among them, the most important are given by the ACDM Model of General Relativity (GR) plus Cosmological Constant and by the modified gravitational theories, like F(R)-gravity [2]. Here, we study some aspects and generic feature of viable F(R)-gravity models
A
power-law and exponential gravity. We show that the behavior of higher derivatives of the Hubble parameter may be influenced by large frequency oscillations of effective dark energy during matter era, which makes solutions singular and unphysical at a high redshift. As a consequence, we examine an additional correction term to the models in order to remove any instability with keeping the viability properties. Some comments about the growth of matter perturbations of corrected models are given and future universe evolution is analyzed. We also propose a way to describe the early time acceleration of universe by applying exponential gravity to inflationary cosmology and a unified description between inflation and current acceleration is presented. This work is based on Refs. [3,4]. We denote the gravitational constant 8nGN by k2-
2 Realistic F(R)-models in the FLRW uni-
verse
The action describing F(R)-gravity is given by
I = / d4x^—g
Jm
L(matter) ^.jje matter Lagrangian. In realistic models
reproducing the standard cosmology of GR with a suitable correction to realize current acceleration and/or inflation one has F (R) = R + f (R) and the modification of gravity is encoded in the function f (R). In this case, we may use an effective fluid-like representation of modified gravity and in FLRW space-time described by the metric ds2 = —dt2 + a(t)2dx2, the equations of Motion (EOM) assume the form
3
1
PDE + Pm — —2H i PDE + Pm —---------------2 ( 2H + 3H
K2 K2 V
(2)
Here, pm and Pm are the energy density and pressure of standard matter and pDE and PDE are the effective dark energy density and pressure given by modified gravity. In order to study the dynamics of F(R) gravity models, we may introduce the variable yH(z) — pDE/pm(o) as a function of the red shift z, where pm(0) — 3m2/k2 is the matter energy density at the present time, being m2 a mass scale. From the EOM one derives an equation on the type [5]
yH(z) + Ji (z)yH(z) + J2(z)yH(z) + J3(z) = 0, (3)
where Ji(z), J2 (z) and J3(z) depend on the F(R)-model. In what follows, we will consider viable models representing a realistic scenario to account for dark energy, in particular a suitable reparameterization of the Hu-Sawicki model [61,
F (R) = R — 2A < 1 —
1
(R/A)4 + 1
(1)
and a simple example of exponential gravity [7] F (R) = R — 2A |"l — e-R/A
(4)
(5)
F(R) R
only, g is the determinant of the metric tensor and
This models are very carefully constructed such that in the high curvature regime F(R ^ A) — R — 2A. Here,
Figure 1: Evolution of the jerk parameter in pure exponential model (a) and in exponential model with correction term (b) overlapped with the ones in ACDM Model.
A plays the role of the Cosmological Constant which decreases in the flat limit (F(R ^ 0) ^ 0) where the Minkowski solution is recovered.
3 DE-oscillations in the matter dominated era
Since in matter dominated era R = 3m2(z + 1)3, one may locally solve Eq. (3) around z + Sz, where |Sz| C z. The solution reads
yu(z+Sz) = ao+boSz+Coe2(z0+1) ( a±V“”-4^)5z , ^
where C0 is a generic constant and a0, b0, a and P are constants depending on the model. It turns out that the dark energy perturbations remain small z
diverge in expanding universe, when Sz < 0. The explicit conditions (1-F,(R))F,,,(R)/(2F"(R)2) > -7/2 and 1/(RF"(R)) > 12 are found for stable matter era [3]. In the case of realistic gravity described by the models (4)-(5), where a0 = A/(3m2), it is easy to see that such conditions are well satisfied. Furthermore, since in both of the models F"(R) ^ 0+, we understand that the discriminant of Eq. (6) is negative and dark energy oscillates with frequency F(z) ~ (R * F"(R))-1/2 /(z + 1). Since by increasing the curvature F"(R) tends to zero, for large values of the red-shift the dark energy density oscillates with a high frequency and also its derivatives become large and maybe dominant in some derivatives of the Hubble parameter which approach an effective singularity making unphysical the solution. Moreover, it may be stated that
A
F,,(R)
frequency of dark energy becomes. As a consequence, despite the fact that the dynamics of the universe depends on the matter and the dark energy density remains very small, some divergences in the derivatives of the Hubble parameter can occur, showing a different
1The form of the jerk parameter as a function of time is j(t)
2We have used Mathematica 7 ©.
feature of this kind of models with respect to the case of GR with an undynamic Cosmological Constant. Since in the models (4)-(5) the approaches to the Cosmological Constant are different from each other, we may say that dark energy oscillations in matter era are a generic feature of realistic F(R) gravity, in which the cosmological evolution is similar to those in ACDM Model. In order to remove the divergences in the derivatives of the Hubble parameter, we can introduce a function g(R) for which the oscillation frequency of the dark energy density acquires a constant value F(z) = 1/\/S, S > 0, during matter era, stabilizing the oscillations with the use of a correction term. This term has been derived as
/ R \ 1/3
g(R) = , <7>
where Y = 1.702 S, so that (R * g"(R))-1/2 /(z + 1) when R = 3m2(z + 1)3. We explore the models
(4)-(5) with adding these correction. The effects of the compensating term vanish in the de Sitter epoch, when R = 4L and the models resemble to a model with an effective cosmological constant, provided that 7 C (m2/A)1/3. For example, a reasonable choice is to put Y = 1/1000, which corresponds to the oscillations frequency of the dark energy during matter era F = 41.255 with a period T — 0.152. In order to appreciate the role of the proposed correction, let us consider the jerk parameter j(z), which depends on the second derivative of H(z)1, for the exponential model (5), whose numerical extrapolation2 is plotted
A
the left side of Fig. 1. Despite the fact that the dark energy density pDE — A/k2 is negligible at high red shift, the effects of its oscillations become relevant in the jerk parameter, which grows up with an oscillatory behavior and diverges at z — 2.8. On the right side of the figure, we plot the graphic of the jerk parameter for the same model with the addition of the correction term g(R) previously discussed. Also in this case the
= a'"(t)/(a(t)H(t)), a(t) being the scale factor of the universe.
jerk parameter oscillates with the same frequency of the dark energy, but here such oscillations have a constant frequency and do not diverge at high red shift.
It is also important to stress that the term added to stabilize the dark energy oscillations in the matter dominated epoch does not cause any problem to the good proprieties of the models (4)-(5), which continue to satisfy all the cosmological and local gravity constraints. It has been demonstrated in several papers that in order to evade the solar system tests [8] or avoid large corrections to Newton Law at large cosmological scales [9], a modified gravity theory must satisfy the following relation
1
3R
F '(R)
F ''(R)
- R » 1,
(8)
R
constant background around a matter source. For the models (4)-(5) without correction terms, this condition is always well satisfied, due to the fact that F"(R ^ A) ~ 0+. Next, if we consider the addition of g(R), whose second derivative is dominant, the left side
of the letter equation reads 34/3m2/(2A7) (R/m2)2/3. Despite the fact that this quantity is smaller than the one obtained for the pure models (4)-(5), it still remains sufficiently large and the correction to the Newton Law is very small. For example, at the typical value of the curvature in the solar system, namely R ~ 10-61eV2, it corresponds to 2 x 106, which is very large effectively. Concerning the matter instability [10], this might also occur when the curvature is rather large, as on a planet (R ~ 10-38eV2), as compared with the average curvature of the universe today (R ~ 10-66eV2). This instability is avoided as soon as 1/(RF"(R)) is much larger than one on the planet surfaces. For our correction term g(R), this quantity-evaluated at R ~ 10-38eV2 corresponds to 4 x 1021 and also in this case we do not have any particular problem.
4 Growth index
After the discovery of cosmic acceleration, a lot of theories for the dark energy have been proposed, and it has become very important to find a way to discriminating among these theories, most of them exhibit the same expansion history. In this context, the characterization of growth of the matter density perturbations is very significant in modified gravity, since models with the same cosmological evolution have a different cosmic growth history. In order to execute it, the so-called growth index y(z) is useful [11]. Under the subhorizon approximation, the matter density perturbation S = Spm/pm satisfies the equation S + 2HS — 4nGeff (a, k)pmS = 0 with k being the co-moving wavenumber and Geff (a, k) being the effective
gravitational “constant” depending on the model and
k
such that k2/a2 ^ H. The equation for matter perturbation has an equivalent description in terms of the growth rate fg = dln S/d ln a. The growth index 7(z) is defined as is defined as the quantity satisfying the relation fg(z) = Qm(z)7(z) with Om(z) = 8nGpm/3H2 being the matter density parameter. The growth index cannot be observed directly, but it can be determined from the observational data of both the growth factor and the matter density parameter at the same redshift and this is the reason for which in the last period its calculation in different theories has become of a great interest. In Ref. [4] various parameterizations for the growth index y(z) have been studied for the models (4)-(5) with correction term g(R). It has been found that the Ansatz which better fits the (numerical) solution of the growth rate for the considered models is given by 7 = 70 + 71^, where 707 are constants.
5 Future universe evolution
The confrontation of the models (4)-(5) with SNIa, BAO, CMB radiation and gravitational lensing has been executed in the past in several works and all the numerical analysis have demonstrated that they are consistent with the last observational data coming from our universe. In particular, they reproduce the correct amount of dark energy today QDE = 0.721 ± 0.015 [5,12]. Moreover, the addition of correction term (7) does not have any influence at the present day range of detection. Starting from Eq. (3), we are able to study perturbations around the de Sitter solution RdS = 4A of our universe today in the considered models. Performing the variation with respect to yH(z) = y0 + y1(z) with y0 = RdS/12m2 and |y1(z)| C 1 and assuming the contributions of radia-
y0
y1 (z)
L(z) = Co(z +1)2(3±V9-4?) + 4Z(z + 1)3 , (9)
P
where C0 is a generic constant and Z = 1 + (1 — F'(R))/(RF"(R)) and p = —4 + 4F'(R)/(RF"(R)), the both evaluated at R = RdS. The de Sitter solution is stable provided by condition F'(RdS)/((RdS)F"(RdS)) > 1, which is a well know result. It has also been demonstrated that since in real-F(R) 0 <
RdSF"(RdS) C 1, one has F'(RdS)/(RdSF"(RdS)) > 25/16, giving the negative discriminant of Eq. (9) and an oscillatory behavior to the dark energy density during this phase. Thus, in this case the dark energy Equation of State (EoS) parameter wDE = PDE/pDE oscillates infinitely often around the line of the phantom divide wDE = —1 [13]. These models possess one
crossing in the recent past, after the end of the matter dominated era, and infinite crossings in the future, but the amplitude of such crossings decreases as (z + 1)3/2 and it does not cause any serious problem to the accuracy of the cosmological evolution during the de Sitter epoch which is in general the final attractor of the system. However, since the existence of a phantom phase can give some undesirable effects such as the possibility to have the Big Rip [14] or Big Rip scenario with the disintegration of bound structures [15], it is important to observe that in the models (4)-(5) with the adding modification (7), the effective EoS parameter of the universe
2(z + 1) dH(z)
Weff =
PDE + Pm PDE + Pm
-1 +
dz
(10)
3H (z)
never crosses the phantom divide line in the past, and
z
— 1, it coincides with wde and the crossings occur. Let us consider for example the exponential case (5) plus term (7). The numerical evaluation of Eq. (3) leads to Hubble parameter
H(z) = Jm2 [yH (z) + (z + 1)3] , (11)
and therefore weff (z). We depict the corresponding cosmological evolution of weff (z) for —1 < z < 2 in Fig. 2. We can see that weff (z) starts from zero in the matter dominated era and asymptotically approaches -1 without any appreciable deviation. Only when z — —0.90, weff (z) crosses the line of phantom divide, how it is shown on the right. By considering the scale factor a(t) = exp (Hot), where H0 — 6.3 x 10—34eV— 1 is the Hubble parameter of the de Sitter universe, we may conclude that the crossings appear at 1026 years.
6 Exponential gravity for early and late-time cosmic acceleration
One of the most important goals for modified gravity theories is that all the processes in the expansion history of the universe are realized, from inflation to the late cosmic acceleration epoch, without invoking the presence of dark components. Models of the type (5) may be combined in a natural way to obtain the phenomenological description of the inflationary epoch, mimicking a smooth version of F(R) = R—2A [1 — e—R/A] — A; 0(R—R;), where R is the transition scalar curvature for which a suitable cosmological constant Aj switches on producing inflation. Following the first proposal of Ref. [16], we consider the form of F(R)
of our universe
F(R)
R — 2A (1 — e—R) — A;
a—1
Ra
where R; and A; are the typical values of transition curvature and expected cosmological constant during inflation, respectively, and n is a natural number larger than unity (here, we do not write the correction term for the stability of oscillations in the matter dominated era). In Eq. (12), the last term 7(1/R“—1)Ra, where 7 is a positive dimensional constant and a is a real number, works at the inflation scale R; and is actually necessary in order to realize an exit from inflation. An accurate analysis of this model shows that if a > 1 mid n > 1, we avoid the effects of inflation at small curvatures and it does not influence the stability of the matter dominated era. The de Sitter point Rds which describes the early time acceleration exists if R; = RdS and it reads as RdS = 2/(7(2 — a) + 1) under the condition (RdS/R;)n ^ 1- Thus, the inflation is unstable if a7(a — 2) > 1. We may evaluate the characteristic number of e-folds during inflation N = log[(z; + 1)/(ze + 1)], where z; and ze are the redshifts at the beginning and at the end of early time cosmic acceleration. Given a small cosmological perturbation y1(zi) at the redshift z; to the effective dark energy density of inflation y0 = RdS/(12m2), so that y(z) = y0 + y1(z), we have from Eq. (3) avoiding the matter contribution
y1(z;) = Co(z; + 1)x , with
2 3—
16F '(RdS) RdSF "(RdS)
/25 -
(13)
(14)
Here, x < 0 if the de Sitter point is unstable. As a consequence, the perturbation yi(z) grows up in expanding universe as y1(z) = y1(zi) [(z + 1)/(z; + 1)]x. When y1(z) is on the same order of the effective modified gravity energy density y0(z) of the de Sitter solution describing inflation, the model exits from inflation. In this way, we can estimate the number of e-folds during inflation as
N — W
x V yo
(15)
(12)
A value demanded in most inflationary scenarios is at least N = 50-60. A classical perturbation on the (vacuum) de Sitter solution may be given by the presence of ultrarelativistic matter in the early universe.
e
x
on model parameters. Let us consider an example by supposing that ultrarelativistic matter/radiation is 106 times smaller than that of effective modified gravity energy density(~ A;) at the beginning of inflation. By choosing the viable model parameters n = 4,
x
1
e
Figure 2: Effective EoS parameter for —1 < z < 2 in exponential model. On the right, the region —1 < z < -0.9 is stressed.
Ri = 2Aj, Y = 1 and a = 8/3, such that the curvature at the inflation results to be RdS = 6A;, one has x = — 0.218 and, % using Eq. (15), we obtain N ~ 64 and ze = e-64(z; + 1) — 1. The numerical extrapolation of Eq. (3) gives us yH(ze) = 0.88yH(z;) and R(ze) = 0.853Rds, confirming that the effective modified gravity energy density and the curvature decrease at the end of inflation. After that, it is reasonable to expect that the physical processes described by the A
ification to gravity.
7 Conclusions
In this work, we have analyzed some feature of realistic F(R)-gravity mimicing the physics of ACDM Model. Two viable models have been considered. We have seen that since the corrections to GR at small curvatures may lead to undesired effects at high curva-
tures, some adding modifications are required. In particular, we have reconstructed a compensating term in order to stabilize the oscillations of the effective dark energy at high red shifts with retaining the viability proprieties. Furthermore, we have briefly discussed about the growth index and an investigation on the cosmological evolution of the universe has been executed showing that the effective crossing of the phantom divide, which characterizes the de Sitter epoch, occurs in the very far future. We also have considered the inflationary cosmology in exponential gravity. It has been
e
flation and the model parameters. Since at the end of inflation the effective modified gravity energy density and the curvature decrease, we may conclude that it is possible to acquire a gravitational alternative scenario for a unified description of inflation in the early universe with the late-time cosmic acceleration.
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Received 01.10.2012
К. Вамба, А. Лопез-Ревеллес, P. Мырзакулов, С. Д. Одинцов, JI. Себастиани ЭВОЛЮЦИЯ ВСЕЛЕННОЙ В ЭКСПОНЕНЦИАЛЬНОЙ F(Д)-ГРАВИТАЦИИ
Исследована общая особенность экспоненциальной F(Я)-гравитации. Предложена дополнительная модификация, чтобы стабилизировать колебания эффективной темной энергии в эпоху доминирования материи. Приведен анализ будущей эволюции Вселенной в рассматриваемой модели. Предложена и изучена единая модель для раннего и позднего времени ускорения.
Ключевые слова: тёмная энергия, F(R)-гравитация.
Вамба, К.
Нагойя университет,
Япония.
Лопез-Ревеллес, А.
ICE/CSIC-IEEC,
Bellaterra (Барселона), Испания.
Мырзакулов, Р.
Евразийский национальный университет,
Астана, Казахстан.
Одинцов, С. Д.
ICREA,
Барселона, Испания;
Томский государственный педагогический университет,
ул. Киевская 60, Томск, Россия;
Евразийский национальный университет,
Астана, Казахстан.
E-mail: odintsov@tspu.edu.ru
Себастиани, Л.
Университет Тренто,
Италия;
Евразийский национальный университет,
Астана, Казахстан.
