Singularity problem in f(R) model with non-minimal coupling
We consider the non-minimal coupling between matter and the geometry in the f(R) theory. In the new theory which we established, a new scalar has been defined and we give it a certain stability condition. We intend to take a closer look at the dark energy oscillating behavior in the de-Sitter universe and the matter era, from which we derive the oscillating frequency, and the oscillating condition. More importantly, we present the condition of coupling form that the singularity can be solved. We discuss several specific coupling forms, and find logarithmic coupling with an oscillating period in the matter era , can improve singularity in the early universe. The result of numerical calculation verifies our theoretic calculation about the oscillating frequency. Considering two toy models, we find the cosmic evolution in the coupling model is nearly the same as that in the normal f(R) theory when . We also discuss the local tests of the non-minimal coupling f(R) model, and show the constraint on the coupling form.
The rapid development of observational cosmology starting from 1990s shows that the expansion of our universe in the present epoch is accelerating. The most popular theory to explain the accelerating universe is dark energy (DE). One possible dark energy candidate accepted by most of researchers is the cosmological constant. However, the cosmological constant that remains a mystery, can not be explained clearly in any known theory. If the cosmological constant originates from the vacuum energy in quantum field theory, as many people believe, its energy scale is too large to be compatible with the observed dark energy density la (). Moreover, the observation indicates that the dark energy equation of the state may cross the phantom divide line . This suggests that the cosmological constant Hannestad:2002ur (); Cepa:2004bc (); cmb () may not be the only candidate for dark energy. There also exist some other dark energy models, ranging from quintessence, phantom, quintom to chaplygin gas models. For the nice reviews on dark energy, see Li:2011sd (); Copeland:2006wr (); Cai:2009zp (). An alternative scenario for dark energy is infrared(IR) modified gravity. Among many IR modified gravity models, gravity is of particular interest. One important feature in gravity is the intrinsic existence of an extra dynamical scalar degree of freedom, besides the massless graviton. Therefore it is possible to study both the early-time inflation and late-time acceleration of the universe in the framework of gravity, without introducing a scalar field by hand. More interestingly, the effective equation of state could be smaller than in dark energy models, indicating the scalar behaves like a phantom in the Jordan frame. The model with a Lagrangian density () was proposed for dark energy r-n1 (); ca0 (); r-n3 (); Pi:2009an (). However this model is plagued by matter instability rn4 (); rn5 () and difficulty to match local gravity constraints. Later on, researchers have proposed many viable models, seeing st (); hu (); tsu (); linder (); Elizalde:2011ds (); Elizalde:2010ts (); Cognola:2007zu (); Bamba:2010ws (); qx (). For nice reviews on f(R) theories, see Nojiri:2010wj (); tsu3 ().
Nonetheless, it has been pointed out in Refs. Elizalde:2011ds (); Appleby:2008tv (); Tsujikawa:2007xu (); Arbuzova:2010iu (); Bamba:2011sm (); Nojiri:2008fk (); Bamba:2008ut () that viable f(R) models generally suffer from a singularity problem. For most of the f(R) models, in order to evade the local gravity tests, is a common feature. This leads that the oscillating frequency becomes very large. In this case, the Ricci scalar, Hubble parameter and EOS parameter of DE have intense oscillations in the high redshift region. One possible way proposed by Lee:2012dk () to solve this problem is adding a term to the f(R) function. However, according to the constraint of f(R) inflation, . If we consider , then the adding term contributes to the oscillating frequency. So, this term may has some contribution in high red-shift region such as , but in the matter era , such term is of little help. Even recently, this problem is till be discussed in odin2012 ().
A generalization of the f(R) theories was proposed in allemandi (); inagaki (); bert () firstly by including the theory an explicit function of the Ricci scalar R with the matter Lagrangian density . As a result of the coupling, shown in Bertolami:2007gv (), the equation of motion which is the non-geodesic, and an extra force orthogonal to the four-velocity, arise. The implications of the non-minimal coupling on the stellar equilibrium were discussed in Bertolami:2007vu (), where the constraints on the coupling were obtained. The equivalence between a scalar theory and the model with the non-minimal coupling was considered in Bertolami:2008zh (); Bertolami:2008im (), where the authors showed that the non-minimal coupling f(R) theory corresponds to a two-field scalar theory. Especially in the non-minimal coupling model, the matter part was extended to a arbitrary function of the Lagrangian density of the matter in Harko:2008qz (). Later, the coupling model was analyzed in Nesseris:2008mq () to study the matter perturbation and gave a possible way to prevent the f(R) theory conflicting with the galaxies matter spectrum tests, in Paramos:2011rw () to study the accelerated expansion of the universe, and in Bertolami:2010ke () to discuss the reheating, which gives a constraint about the coupling form .
In this work, we study the oscillating behavior of the dark energy in the non-minimal coupling f(R) theory. We consider the non-pressure dust as the main contribution of the matter Lagrangian and the coupling form is arbitrary function of Ricci scalar. So the energy-momentum tensor of the matter is generally conserved and the matter density is proportional to . Similar to the f(R) theories, the non-minimal coupling f(R) model also has a scalar freedom which consists of the f(R) part , and the coupling part, . Through adding the coupling part, the oscillating frequency can be modified, so that the singularity problem may be solved.
This paper is organized as follows. In Section II, we present the definitions in the non-minimal coupling f(R) model and derive the equation of motion. Then we study the stability condition. In Section III and IV, we solve the EOM of dark energy in de-Sitter universe and the matter era theoretically. And we mainly focus on the study of oscillating behavior of the EOS parameter of dark energy. In Section V, we show that a logarithmic coupling can solve the singularity problem. In Section VI, we give the numerical result, and discuss some toy models. In Section VI, we discuss the local tests of the non-minimal coupling f(R) model. Finally, in section VIII, we make the conclusions.
Ii definitions and the stability condition
ii.1 definitions and the conservation equations
The action allemandi () we consider is
The parameter , and we set it to be unit in the following sections. L denotes matter Lagrangian. By varying the action with respect to the metric , following Bertolami:2007gv (); Harko:2008qz (); Nesseris:2008mq ()with some modifications, we get the modified Einstein equation
where . The matter energy-momentum tensor is defined as
Using the Bianchi identity, , and the identity
which indicates the non-minimal coupling between curvature and matter yields a exchange between matter and the geometry. In the absence of the coupling, , one recovers the covariant conservation of the energy-momentum tensor. A simple choice of the matter Lagrangian is , if we consider the non-pressure dust as the main contribution. Considering the FRW metric, and , we find the matter density is conserved
which yields the conserved equation of the matter density
So, the evolution of the matter density is the same as that in CDM, .
In a flat FRW metric with a scale factor , according to Eq. (2), we get the modified Friedmann equations,
where , , and the dot denotes a derivative with respect to the cosmic time .
By rewriting Eq. (8), we can define the effective energy density,
and the effective pressure,
Here is the mass scale and is the matter density at the present time.
The EOS-parameter for dark energy can be expressed by as
When tends to be a constant, the EOS-parameter tends to be -1.
ii.2 field equations and the stability condition
In this section, we discuss the stability in the local gravity. Taking the trace of Eq. (2), we get
First, we decompose the quantities R, and into the background part with a constant curvature and the perturbed part: , and . We consider R close to the mean-field value , and the metric still very close to the minkowski case. The linear expansion of Eq. (14) in a time-independent background gives
where is the mass of the scalar
and . The stability condition is given by
Note that, to respect the solar system constraints, we must let , and . So this condition (17) can be rewritten approximately as
Especially in the late time de-Sitter universe, , and , then the stability condition (17) recovers to that in the normal f(R) gravity.
Iii Oscillations in the de-Sitter universe
The trace equation (14) of the field can be recast in the form
Similar to the dynamic of normal f(R) gravity, there is a new scalar freedom which decides the cosmic evolution, and the effective potential has a minimum at
In the de-Sitter universe, neglecting the contribution of the matter, the effective potential has a minimum at
where is a constant. Using Eq. (12), the Ricci scalar can be expressed as
and we have used .
When , we consider the perturbations around the de-Sitter solution of the dark energy density , hence
In this case, we can rewrite the coefficients as
The solution of Eq.(23) is
where B is a constant depending on the initial condition and . Depending on the sign of the discriminant in the square root of Eq. (35), there are two possible behaviors for this model. If , the solution approaches the de-Sitter point as a power function of (1+z). Otherwise if , the dark energy shows an oscillating behavior as
Now we write out the discriminant,
Combining the stability condition Eq. (17), and we know, when
the dark energy has an oscillating behavior near the de-Sitter point. In this case, according to the definition in Eq. (13), one has
where . The EOS parameter of dark energy also has an oscillating form. We have noticed that, on condition , the oscillating amplitude becomes smaller and smaller. Therefore, as time goes on, tends to be -1. If we take the e-folding number as the variable, the oscillating frequency is
Iv Oscillating in the matter era
In the matter dominated era, and , in order to match the local gravity tests, the following conditions
must be satisfied. And the minimum point of the effective potential in the Eq. (19) is
So, we neglect the dark energy contribution, and we write out the expression of Ricci scalar
In this case, according to Eq. (23), the EOM in the matter era is
The solution of Eq. (45) is
where is a constant depending on initial conditions. Generally speaking, for different forms of , for example, corresponds to and corresponds to . Note that . Therefore, , , and which mean we can neglect the third term in Eq. (50). In this case, the discriminant in the square root of Eq. (50) must be negative, and the solution (50) becomes an oscillating form
where . Using Eq.(13) again, we give EOS parameter of dark energy near the redshift ,
Since we care about the oscillating, we present the oscillating frequency,
which corresponds to that in normal f(R) theory Elizalde:2011ds (), where we have used the condition .
If we hope the non-minimal coupling f(R) model improve the singularity problem, according to the expression (53),it is required that must not be the decreasing function of Ricci scalar. In the matter dominated era, we ignore the normal part. On the other hand, by considering think the scalar field stays at the minimal point of the effective potential, or equivalently , we get
Recalling the stability condition (17), we get the constraint of
On the other hand, to evade the singularity problem, considering , must not be a decreasing function of Ricci scalar. Therefore we get the condition that improve the singularity,
V specific couplings
In the normal f(R) gravity, there is a singularity problem that the oscillating frequency is too large due to . We expect to reduce the oscillating frequency by adding a coupling term. Here, we consider some specific models.
v.1 logarithmic coupling
Basing on the f(R) theory, we consider a logarithmic coupling
where is the cosmological constant sharing the same unit with the cosmological constant , and the parameter is a small non-dimensional constant. In the matter era, for most of f(R) theory, to evade the solar system tests, the is set to be very small. Here, we set reasonably. Such a coupling has a virtue that it has little influence on the evolution of the early universe, for example, inflation and reheating era, as long as we take a proper coupling constant . In the inflation and reheating era, considering c is a cosmological constant , , we get , especially when , we get approximately. This recovers the normal f(R) theory. As for the local gravity, , however, the detailed discussion, which we put in the Sec. VII, may be more complex. Here, we just set to be a small constant without dimension. Therefore, the scalar is
where we have used the approximation . According to the definition, can be rewritten as
So, we have the approximation and . In this case, we can get the oscillating frequency in the matter era,
which indicates that, the period is proportional to the red-shift z if we consider a small logarithmic coupling. This solves the singularity problem in the matter era.
v.2 power-law coupling
We also consider the power-law coupling, assuming that
where is a constant with the same unit as .
Firstly, we discuss the stability condition in power-law coupling, and we get
In order to improve the large oscillating frequency, assuming in the matter era, according to Eq. (17), one has
Then we have two condition
Therefor, we do not consider the inverse power-law coupling which is used to mimic the dark matter in Paramos:2011rw (), either the case . Using condition (56), we get , which is always satisfied, if we consider stability condition.
v.3 exponential coupling
Ones assume a power-law exponential coupling of the form
where the must be very small in the matter era to recover the normal f(R) theory. We get the ,
Therefor the discussion is the same as the power-law coupling.
There are also some other exponential coupling, such as
where the c is a constant which sharing the same unit and order with the cosmological constant . In this model, decreases rapidly that means is still very small in the matter era. So, the exponential coupling is unavailable to improve the singularity problem either. Using condition (56), we get
which is bigger than 7/3.
Vi numerical result and the late time evolution
In this section, using numerical calculation, we examine and certify the theoretical calculation about the oscillating frequency in the matter era. Then, using several possible coupling forms, we discuss the behavior EOS parameter in the de-Sitter universe.
vi.1 non-singularity with a small logarithmic coupling
We set the coupling constant , and the initial condition , , . Therefore, in the matter era, seeing Eq. (58) and Eq. (59), when the contribution of is insignificant comparing with the coupling term, the coupling constant will dominate. This will suppress the oscillating in the early universe.
Fig. 1 shows the EOS parameter of dark energy as a function of red-shift. In the late time, with a small coupling, we can not distinguish the result from that in normal f(R) theory without coupling. However, in the matter era, there are some differences. Fig. 2 shows the oscillating at different red-shift. Note that, as the red-shift gets bigger, the oscillating period getting bigger. It reads as:
redshift numerical calculating period theoretic predict z=4 T= 0.00042 0.00044 z=5 T= 0.00053 0.00053 z=6 T= 0.00062 0.00062 z=7 T= 0.00071 0.00071
Comparing with the result in the literatureElizalde:2011ds (), there is no more singularity in this model. Form the Table. 1, we get the relationship
which can be also got from the theoretical result Eq. (60). Note that, when , there is some differences between the numerical and the theoretical calculation. This is because our approximation and is no longer valid in the low redshift region. However when , our theoretical calculation is still valid.
vi.2 late time evolution in toy models
If we ignore the singularity problem in the early universe and care more about the late time universe, we want to check how the non-minimal coupling wound influence the late time universe. To match the solar system tests, when Ricci scalar is large, must tend to be 1 while considering the coupling form. We still consider a exponential model, and with a exponential coupling:
When , in the matter era, which assure . Fig. 3 shows the detail. when , we can distinguish the results from that in normal f(R) theory. When , the result is nearly the same. In this model, we set and the initial red-shift is .
We also consider the model
proposed in qx () recently. Therefore we add new exponential coupling form
when , in the matter era, that assure . where the parameters we set is , and . the initial red-shift is .
Fig. 4 shows the details. In this coupling model, the oscillating amplitude is reduced comparing with the normal f(R) theory.
Vii local tests
In this section, we consider local tests of the non-minimal coupling f(R) model. We will use the similar method by Hu and Sawicki hu (). We consider a spherically symmetric isotropic metric,
where we assume and near the source. especially, in the GR limit, since the solar system tests provide the strongest limits on .
Generally, according to the definition of the Ricci tensor, we get
And form the modified Einstein equation (2), we get the time-time component of the field equation
If we consider the static solution, combining this equation with the trace equation (14), we get the time-time component of Ricci tensor
Then we get
In the high curvature region, for general f(R) model, we have and . On the other hand, taking proper form of , which has the limit and in the high curvature region, we have . So we consider
In the same limit, from the trace equation of the field equation (14), we have
Therefore, together with has a solution
We assume the solution remains finite at , , and when we assume . So, we have
It is difficult to find an accurate solution of the Ricci scalar. However, we can consider the field stays at the minimal point approximately, and the deviation from the minimum is
where the last three terms we know is small, so we care the field gradients . And we define . A sufficient condition for is the compton condition
If this condition is always satisfied, we consider . If this condition is violated beyond some outer radius, such as the sun radius, this approximation is not valid until the compton condition is locally satisfied again outside the sun. Then the the field stays at the minimum again.
Anyway, the curvature near the sun is high, so we think comparing with . On the other hand, we use the galaxy radius instead of , considering the field stay at the minimum point . So we get
Finally the deviation from the GR metric is given by
Generally, is the constraint of normal f(R) component. If we require
the coupling model may evade the solar system test. For logarithmic coupling, , the constraint is .
Viii Conclusions and discussion
We consider a non-minimal coupling between geometry and matter in the f(R) theory. We find this theory also has a independent scalar similar to the scalar in the normal f(R) theory. We give its stability condition (17), which is also similar to normal f(R) theory.
We discuss the evolution of dark energy in the de-Sitter universe, when the condition (38) is satisfied, the dark energy density will have oscillating behavior. Meanwhile, we find the dark energy is always oscillating in the matter era, with an oscillating frequency . Adding the coupling part into the f(R) theory, under the condition that , the oscillation is suppressed successfully by the coupling term in the matter era.
Three specific coupling forms have been studied, and it turns out that the logarithmic coupling, with a oscillating period , is a good choice. The power law coupling also can suppress the oscillating becoming strong. However, the reheating constraint gives a overlarge energy scale. And this will disappointedly lead to very small , which makes it difficult to reduce the oscillating frequency in the matter era. The exponential coupling can not satisfy the increasing function condition (69).
Through numerical calculation, we verified our theoretical calculation of the oscillating frequency in the matter era, and found only when the numerical calculation has some differences with the theoretical calculation. This is because our approximation and is no longer valid in the low redshift region. However when , our theoretical calculation is still valid. We take two toy models as examples, and show that the non-minimal coupling does not change the cosmic evolution of f(R) model in the late universe. Especially when , we can not distinguish between coupling model and the f(R) model.
In the end, we discuss the local test constraint on the non-minimal f(R) model. Generally, we obtain the constraint . in the logarithmic coupling model, this constraint forces us to set the coupling parameter to , which leads a small oscillating period , seeing Fig. 1. However, its oscillating frequency is decreasing as the redshift increasing and the amplitude is not too large comparing with the normal f(R) theories. So, the singularity problem in normal f(R) model is no longer so serious when we consider the non-minimal coupling.
The work was in part supported by NSFC Grant No. 10975005. We gratefully acknowledge Bin Chen for a careful reading of the manuscript and insightful suggestions.
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