The Cosmology of Ricci-Tensor-Squared Gravity in the Palatini Variational Approach

The Cosmology of Ricci-Tensor-Squared Gravity in the Palatini Variational Approach

Baojiu Li b.li@damtp.cam.ac.uk Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, Wilberforce Road, University of Cambridge, Cambridge CB3 0WA, United Kingdom    John D. Barrow j.d.barrow@damtp.cam.ac.uk Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, Wilberforce Road, University of Cambridge, Cambridge CB3 0WA, United Kingdom    David F. Mota d.mota@thphys.uni-heidelberg.de Institute of Theoretical Physics, University of Heidelberg, 69120 Heidelberg, Germany
September 27, 2019
Abstract

We consider the cosmology of the Ricci-tensor-squared gravity in the Palatini variational approach. The gravitational action of standard general relativity is modified by adding a function to the Einstein-Hilbert action, and the Palatini variation is used to derive the field equations. A general method of obtaining the background and first-order covariant and gauge-invariant perturbation equations is outlined. As an example, we consider the cosmological constraints on such theories arising from the supernova type Ia and cosmic microwave background observations. We find that the best fit to the data is a non-null leading-order correction to Einstein gravity, but the current data exhibit no significant preference over the concordance model. The growth of non-relativistic matter density perturbations at late times is also analyzed, and we find that a scale-dependent (positive or negative) sound-speed-squared term generally appears in the growth equation for small-scale density perturbations. We also estimate the observational bound imposed by the matter power spectrum for the model with to be roughly so long as the dark matter does not possess compensating anisotropic stresses.

pacs:
04.50.+h, 98.80.Jk, 04.80.Cc

I Introduction

The accumulating astronomical evidence for an accelerating cosmic expansion has stimulated many investigations into the nature of the dark energy, or other possible deviant gravitational effects, which might be responsible for this unexpected dynamics (for a review see, e.g., DEReview (2006)). Besides proposing to add some new (and purely theoretical) matter species into the energy budget of the universe, many investigators have also focused their attentions on modifying general relativity (GR) on the largest scales, so as to introduce significant modifications in the behaviour of gravity at late times when it is comparatively weak. One example of the latter sort is provided by the family of gravity models, which had also been considered before the discovery of cosmic acceleration (see for example Refs. Barrow (1983, 1988); Maeda (1989)) with reference to alternative forms of inflation and the existence of singularities. In Refs. Carroll (2005); Easson (2005), the authors discuss a specific model where the correction to GR is a polynomial function of the and quadratic curvature invariants (here, and are respectively the Ricci scalar, Ricci tensor and Riemann tensor calculated in the standard way from the physical metric ) and showed that there exist late-time accelerating attractors in Friedmann cosmological solutions to the theory. Barrow and Clifton established the general existence conditions for de Sitter, Einstein static, Gödel universes in theories where the Lagrangian is an arbitrary function of these three invariants Clifton (2005).

When the Ricci scalar in the Einstein-Hilbert action is replaced by some general functions of and , it becomes necessary to distinguish between two different variational approaches to deriving the field equations. In the metric approach, as in Refs. Carroll (2005); Easson (2005), the metric components are the only variational quantities and the field equations are generally of fourth-order, which makes the theories phenomenologically richer but more stringently constrained in most cases. Within the Palatini variational approach, on the other hand, we treat the metric and the connection as independent variables and extremize the action with respect to both of them; the resulting field equations are second order and easier to solve. The Palatini gravity is also proposed as an alternative to dark energy in a series of works Vollick (2003); Allemandi (2004, 2005). There has since been growing interest in these modified gravity theories: for the local tests of the Palatini and metric gravity models see PalatiniLGT (2007); MetricLGT (2007); for the cosmologies of these two classes of models see PalatiniCT (2006); MetricCT (2006); Koivisto (2006); Li (2006, 2006).

Both approaches to modifying gravity are far from problem-free. In the metric gravity models, the theory is conformally related to standard GR plus a self-interacting scalar field Barrow (1988), which generally introduces extra forces inconsistent with solar system tests MetricLGT (2007). The Palatini approach, on the other hand, generally leads to a large (or even negative) sound-speed-squared term in the growth equation of the non-relativistic matter perturbations on small scales. This induces effects on the cosmic microwave background (CMB) and the matter clustering power-spectra which deviate unacceptably from those which are observed Koivisto (2006); Li (2006, 2006). Again, these examples highlight the difficulties encountered when trying to make modifications to standard GR which are compatible with observations.

In this work we will focus on the Ricci-squared gravity models within the Palatini variational approach, which we also denote by the gravity. It turns out that the Ricci tensor, and Ricci scalar, , appearing in the field equations in the Palatini approach are not the ones calculated from the physical metric, , (we consider the metric as the physical one because it is this metric which the matter Lagrangian density depends on and the energy-momentum conservation law holds with respect to) as in GR, and we denote the GR equivalents by and respectively to distinguish them from the Palatini quantities). Such a modification of gravity has indeed been considered in Allemandi (2004) and shown to give an accelerating cosmology. However, our work differs from Allemandi (2004) in that we replace the Ricci scalar in the gravitational action with rather than simply and we concentrate more on the cosmology at the first-order perturbation level, especially the late-time cold-dark-matter (CDM) density perturbation growth. We emphasize the similarity to the Palatini gravity models also.

Our presentation is organized as follows. In Sec. II we briefly introduce the model and outline the methods used to derive the background and first-order covariant and gauge-invariant (CGI) perturbed field equations. In Sec. III, we present the modified Friedmann equation and apply it to a specific family of theories with ; the constraints on the parameter space from cosmological data are also given. Then, in Sec. IV, we analyse the growth of CDM density perturbations at late cosmological times for this model. Since this analysis shares some similarities with the Palatini gravity models, we also present a similar discussion of the latter for comparison. Our discussion and conclusions are presented in Sec. V.

Throughout this work our convention is chosen as where run over and ; the metric signature is and the universe is assumed to be spatially flat and filled with CDM and black body radiation.

Ii Field Equations in Gravity

In this section we briefly introduce the main ingredients of gravity and outline the strategy for deriving the general perturbation equations that govern the dynamics of small inhomogeneities in the cosmological models that arise in this theory.

ii.1 The Gravity Model

We will start our discussion with the modified Einstein-Hilbert action in the present model,

(1)

in which with the Newtonian gravitational constant. Here, is given by

(2)

and ; note that is a new and independent variable with respect to which we extremise the action, and is different from the Christoffel symbol calculated using the metric . is assumed to be a symmetric tensor (if it contains an antisymmetric part then the field equation will be spoiled as discussed in Borowiec (1998)) and could be used to raise or lower its indices. Varying the action Eq. (1) with respect to the metric (note that as they are independent) gives the modified Einstein equations:

(3)

where with and is the energy-momentum tensor of the fluid matter (CDM and radiation).

On the other hand, varying the action with respect to the new variable with the relation , one arrives at another field equation

(4)

where represents the covariant derivative compatible to (the covariant derivative compatible to is denoted, as conventionally, by ). Just like in the Palatini models, this equation implies some relation between the physical metric and the metric whose Christoffel symbol is . However, because of the presence of the second term in the parentheses this relation is nontrivial and some further algebra will be needed to explicate it. Before doing that, we will present some preliminary definitions and expressions, one of which is the notation of decomposition.

ii.2 The Decomposition

The main idea of decomposition Ellis (1989, 1998); Challinor (1999); Tsagas (2007) is to make spacetime splits of physical quantities with respect to the 4-velocity of an observer. The projection tensor is defined as and can be used to obtain covariant tensors perpendicular to . For example, the covariant spatial derivative of a tensor field is defined as

(5)

The energy-momentum tensor and covariant derivative of the 4-velocity are decomposed respectively as

(6)
(7)

In the above, is the projected symmetric trace-free (PSTF) anisotropic stress, the heat flux vector, the isotropic pressure, the PSTF shear tensor, , is the vorticity, ( is defined here as the mean expansion scale factor) the volume expansion rate scalar, and is the fluid acceleration; the overdots denote time derivatives expressed as , brackets mean antisymmetrisation, and parentheses symmetrization. The velocity normalization is chosen to be . The quantities are referred to as dynamical quantities and as kinematical quantities. Note that the dynamical quantities can be obtained from the energy-momentum tensor through the relations

(8)

Decomposing the Riemann tensor and making use the Einstein equations, we could obtain, after linearization, the perturbed (constraint and propagation) equations Ellis (1989, 1998); Challinor (1999); Tsagas (2007). Here, we shall not list all of them because most are irrelevant for the following discussion; rather we will use the linearised Raychaudhuri equation

(9)

the linearised conservation equations for the energy density:

(10)

and the linearised Friedmann equation

(11)

The above equations are derived and presented for standard general relativity, and so the variables describe imperfect fluid matter. For general modified gravity theories, such as those presented here, the modification to GR might be parameterized as an effective energy-momentum tensor. In this case the formalism of these equations is preserved and one just needs to replace by the total effective quantities of the same sort: Hwang (1990).

ii.3 The Field Equations in Gravity

In Eq. (4), we see that is a symmetric tensor density of weight 1, and so we can introduce a new metric by means of the following relation

(12)

where the Levi-Civita connection of the metric is just , as we referred to above.

To go further, we need to express explicitly. This is easy to do in principle, because Eq. (3) is just an algebraic equation for . To see this, let us write the symmetric tensor in a general way as

(13)

where is the 4-velocity of the observer referred to above. Substituting Eqs. (6, 13) into Eq. (3), we get

which leads to the following four equations:

(14)
(15)
(16)
(17)

where are functions of . Thus given the specified form of and the values of , the quantities can be obtained from Eqs. (14, 15), at least numerically. Then, and can also be calculated from Eqs. (16, 17) provided the values of and are given. Note that and are nonzero only at first order in perturbation. Taking the time derivatives of Eqs. (14, 15), and using the background values of , we could easily obtain and by solving the two linear algebraic equations. Similarly, and could be worked out (here, is the spatial derivative). In what follows, we shall assume that and their derivatives have been calculated.

The next step is to find out the relation between and . We could rewrite Eq. (12) as

(18)

Taking the determinants of both sides and equating we get

(19)

with

(20)

Thus, we conclude from Eq. (18) that

(21)
(22)

Obviously, and need to be evaluated respectively. For , we have

To calculate this, let us write and where run over , and are first order metric variables of which is traceless, is the spatial component of , and is the metric of 3 dimensional flat space. As a result and . From these expressions the components of can also be obtained and one can substitute all these quantities into the above equation to get . Since and are only of first order and because is traceless, it is then not difficult to see that up to first order (note that the facts and indicate that and )

(23)

For , we know that it is symmetric as the inverse matrix of a symmetric matrix, and so could be written as

(24)

Using

and

it is then easy to obtain, to first order, that

(25)

As a result, we have now the relations between the two metrics and and their inverses as

(26)
(27)

where

(28)
(29)
(30)
(31)

A discussion of how the two Ricci tensors and are related to one another is given in the appendix, with the help of which the Einstein equation Eq. (3) can be rewritten as

(32)

where

(33)

and (see the appendix for a definition of the tensor )

(34)
(35)

With the aid of Eqs. (II.2, 33, 34, 35) one could identify and and express them in terms of and which are functions of and (c.f., Eqs. (14, 15, 28, 29)), and which are also functions of (c.f., Eqs. (16, 17)). However, from Eqs. (26 - 31, 69, 70) one can see that this process will involve a lot of calculation. In the present work we will not perform detailed numerical calculations of the perturbation equations of the model; instead, in the next two sections of the paper we will:

1. Study the background evolutions of general Ricci-squared gravity models. As an example, we will consider a specific family of theories with , and constrain the allowed parameter space using data sets supernovae (SNe) luminosity distances and the CMB shift parameter.

2. Present a simple argument to show that this class of modified gravity theory, like those arising in the Palatini theory, generally possesses a scale-dependent effective sound-speed-squared term which affects the growth of CDM density perturbations and thus influences the matter power spectrum Koivisto (2006); Li (2006, 2006) on small scales.

Iii The Cosmological Background Evolution

In order to analyse the cosmological background evolution we can neglect the and terms, and hence the quantities . As a result, the equations are greatly simplified.

We are interested in the modified Friedmann equation in the present model. From Eqs. (II.2, 11, 33) we have

(36)

in which is the Hubble expansion rate and is expressed as

(37)

with and given in Eqs. (34, 35). After a lengthy calculation, and using Eq. (9) to eliminate the term which appears in Eq. (37), we obtain the following simple result

(38)

There are two interesting points regarding Eq. (38). Firstly, we see that only , and not its time derivatives or enter the equation. Secondly, the second-order derivative of does not appear either; to see the consequence of this, note that since (with for matter and for radiation), the dependence of on could be expressed in terms of , e.g., in radiation-dominated era /3 and in matter-dominated era , and so we have . Consequently, Eq. (38) has the form

(39)

where is a complicated function of and (at late times and it becomes a function of alone).

Figure 1: (color online) The time evolution of the effective equation of state for the model. For clearness we only show the cases of (the dashed curve) and (, the solid curve). The other parameters defining the cosmology are and . Clearly an accelerating phase following the standard matter-dominated era can be realized within this model. Note that the redshift range corresponds to the future of the present epoch, where the Universe evolves into a de Sitter stage.

As we discussed above, knowing and means that we know and . Thus, given a specific form for Eq. (38) completely determines the background cosmological evolution of the model. As a particular example let us consider the case of

(40)

where and are the model parameters. Note that here is always non-negative, and corresponds to a picking the standard cosmology of GR.

For convenience, we shall define the following dimensionless quantities

(41)

then Eqs. (14, 15) could be rewritten as

(42)
(43)

where

(44)
(45)

and than Eq. (38) reduces to

(46)

where .

In this paper we will set , so today we have and there are 3 equations (Eqs. (42, 43, 46)) for the 5 parameters and . Therefore, we are able to express all the other quantities in terms of and , which can therefore be treated as the two independent degrees of freedom of our model. Note that is a constant, and once evaluated at the present day, it could be used all through the cosmic history, which helps determine at arbitrary times.

In Figure 1 we have plotted the effective equation of state, defined by (where a star-superscript denotes the derivative with respect to ), as a function of the redshift. The values of are indicated beside the curves. At early times, when the corrections are negligible, the models all mimic the evolution of , and the same thing happens in the future. This is because during this era the matter (relativistic and non-relativistic) is greatly diluted so that the right-hand sides of Eqs. (42, 43) both vanish; consequently, we can solve them to show that and so is also constant. The deviation from occurs mainly at intermediate times, that is, in the recent past and future.

We now use the observational data on the background cosmology to constrain the parameter space (in the plane) of the present model. For this we jointly use the 157 measurements on SNe luminosity distance in the Gold data sets of Riess et al. Riess (2004) and the CMB shift (R) parameter. The SNe luminosity distance is expressed as

(47)

where . The measurements supply the extinction-corrected distance modulus (with in units of Mega-parsecs) and its uncertainty, , for individual SNe, so that the standard minimization, defined by

(48)

is easy to implement, where is the theoretically predicted distance modulus. As appears only as it does in Eq. (47), we could marginalize over it by integrating the probability density for all values of . For the CMB R-parameter, defined as

(49)

we adopt the observational value at from Wang (2006). Note that this does not depend on the specified value of .

Our constraining result is shown in Figure 2, where we have shown the 68% and 95% confidence regions respectively. The constrained intervals are roughly and at the 95% confidence level, with the best fitting values being with . Also note that the concordance model (the white star) lies within the 68% confidence region of our constraints.

Figure 2: The constraints on the parameter space of and in the present model from joint SNe and CMB shift parameter data sets. The grey and light grey regions represent the 68.3% and 95.4% confidence contours respectively. The white circle () is the best-fitting parameter of our model, and the star is the concordance model.

Thus, we see that the background cosmological data is able to constrain to be of order . In the next section we will briefly investigate the possible constraint from the growth of dark-matter density perturbations, and show that this may provide a potentially more stringent restriction on . However, considering that this latter limit depends on the properties of the dark matter, our background constraints given in this section are less model dependent.

Iv Effects on Late-time CDM Density Perturbation Growth

In this section we study the effects of the corrections to GR on the CDM density perturbation growth. We start by recalling the case of gravity because it shares some similarities with the one, while being technically simpler than the latter, and because a similar analysis for the former is still missing from Refs. Li (2006, 2006) (see however Koivisto (2006) for a slightly different treatment).

iv.1 The Case of Palatini Gravity

Recall that in our simplified model the universe is filled with CDM and radiation, and at later times the radiation energy density is negligible, so .

Taking the spatial derivative of the Raychaudhuri equation Eq. (9) (with the there being replaced by ), and working in the CDM frame (where the observer is comoving with CDM particles and thus ) Lewis (2000), we have

(50)

where is the CDM density perturbation contrast that is defined through , and is the Hubble expansion rate with respect to conformal time (note that a prime denotes the conformal time derivative, and a dot the cosmic comoving proper-time derivative); are respectively the harmonic expansion coefficients for and (defined via and Commet1 (2007)). Clearly we need to know about and which arise from the modifications to GR (c.f. Eq. (33)).

In the Palatini model, in which the Ricci scalar in the gravitational action is replaced with , Eqs. (26, 27) still hold, but with (see for example Li (2006))

(51)

Then, with the help of the calculations in the appendix, it is straightforward to show that

(52)

where , and the modified Einstein equation Li (2006),

can be rewritten as

in which the effective total energy-momentum tensor is given by

(53)

Using Eq. (II.2), we can now identify

(54)
(55)

Thus

(56)

in which represent the terms not involving second order (time and spatial) derivatives.

The reason why we keep only two second derivative terms explicitly on the right-hand side of Eq. (56) is that, after taking the spatial covariant derivative, the first term contributes a piece to Eq. (50) while the second term contributes a piece. None of the remaining terms in contribute these two pieces to Eq. (50). To be more explicit, recall that in the model, so

(57)

where we have used Eq. (10) to background order. As a result,

and Eq. (50) can be recast into the form