Contents
Abstract

We consider five dimensional conformal gravity theory which describes an anisotropic extra dimension. Reducing the theory to four dimensions yields Brans-Dicke theory with a potential and a hidden parameter which implements the anisotropy between the four dimensional spacetime and the extra dimension. We find that a range of value of the parameter can address the current dark energy density compared to the Planck energy density. Constraining the parameter and the other cosmological model parameters using the recent observational data consisting of the Hubble parameters, type Ia supernovae, and baryon acoustic oscillations, together with the Planck or WMAP 9-year data of the cosmic microwave background radiation, we find for Planck data and for WMAP 9-year data at 95% confidence level. We also obtained constraints on the rate of change of the effective Newtonian constant () at present and the variation of since the epoch of recombination to be consistent with observation.

The Effect of Anisotropic Extra Dimension in Cosmology

[17mm]

Seyen Kouwn,  Phillial Oh, and  Chan-Gyung Park

[3mm]

Institute of Convergence Fundamental Studies & School of Liberal Arts, Seoul National University of Science and Technology, Seoul 139-743, Korea

[2mm] Korea Astronomy and Space Science Institute, Daejeon 305-348, Republic of Korea

[2mm] Department of Physics, BK21 Physics Research Division,  Institute of Basic Science,

Sungkyunkwan University, Suwon 440-746, Korea

[2mm] Division of Science Education and Institute of Fusion Science, Chonbuk National University, Jeonju 561-756, Korea

[2mm] seyenkouwn@gmail.com, ploh@skku.edu, parkc@jbnu.ac.kr

1 Introduction

Nowadays, research on the higher dimensional gravity theories like Kaluza-Klein theory, string theory, and brane world scenario constitutes one of the mainstream of theoretical particle physics. In such theories, it is usually taken for granted that the higher dimensional spacetime is isotropic. Even though the isotropic spacetime appeals more aesthetical from the viewpoint of symmetry like Lorentz symmetry and general covariance, this has never been experimentally verified. Therefore, it is a fundamental question to ask whether higher dimensional spacetime has uniform physical properties in all directions [1, 2] and envisage the possibility that the extra dimensions might not share the same property with the four dimensional spacetime we are living in.

Recently, an attempt to construct a higher dimensional gravity theory in which the four dimensional spacetime and extra dimensions are not treated on an equal footing was made [3]. It is based on two compatible symmetries of foliation preserving diffeomorphism and anisotropic conformal transformation. The anisotropy is first implemented in the higher dimensional metric by keeping the general covariance only for the four dimensional spacetime. This was achieved by adopting foliation preserving diffeomorphism in which the foliation is adapted along the extra dimensions. Then, it was extended to conformal gravity with introduction of conformal scalar field. In order to realize the anisotropic conformal invariance a real parameter which measures the degree of anisotropy of conformal transformation between the spacetime and extra dimensional metrics was introduced. In the zero mode effective four dimensional action, it reduces to a scalar-tensor theory coupled with nonlinear sigma model described by extra dimensional metrics. There are no restrictions on the value of at the classical level. In this paper, we present a cosmological test of the scalar-tensor theory thus obtained in the case of five dimensional theory and check whether or not a specific value of is preferred.

In general, the conformal invariance constrains the theory in a very tight form in a conformal gravity [4], and contains at most one parameter, that is the potential coefficient The Brans-Dicke theory contains more parameters [5]: one is , which is the ratio between the nonminimally coupled term and kinetic energy term for Others are the potential and its respective coefficients, if introduced. It turns out that in the five dimensional anisotropic conformal gravity, the effective four dimensional scalar-tensor theory reduces to the Brans-Dicke theory with a potential, in which the parameter and the power of the potential, , are determined in term of the parameter . Therefore, from the view point of Brans-Dicke theory, is a hidden parameter and this is a consequence of anisotropic conformal invariance in higher dimensions.

In the gravitational theory with anisotropic conformal invariance, it is more convenient to work with a dimensionless scalar field in order to countercheck the arbitrary anisotropy factor . Recall that the kinetic coefficient of the Brans-Dicke theory can be allowed to be an arbitrary (positive definite) function of the scalar field, which results in a general class of scalar-tensor theories with a dimensionless scalar field and they can be tested with the solar system experiments[5]. In our case, the scalar field is also dimensionless. Nevertheless, is constrained to be a constant for the sake of the anisotropic conformal invariance, rendering the theory to be a Brans-Dicke type.

Another important point to be mentioned is that in our four dimensional Brans-Dicke theory, the origin of the Brans-Dicke scalar can be identified with the conformal scalar that is necessarily introduced for the purpose of conformal invariance. It is well-known that in the isotropic case, the conformal or Weyl scalar field is a ghost field with a kinetic coefficient yielding a negative kinetic energy and they cannot become the Brans-Dicke scalar [4]. However, in the anisotropic case, the kinetic coefficient is determined as a specific function of and there exists a range of parameter where becomes positive. We will check that the actual cosmological test prefers the range of parameter with a positive value of

The paper is organized as follows: In Sec. 2, we give a formulation of the 5D gravity with anisotropic conformal invariance and perform dimensional reduction to obtain 4D Brans-Dicke theory. We perform cosmological analysis and give numerical results for evolution equations. In Sec. 3, comparisons with the recent cosmological data are made and the range of parameter is constrained. Sec. 4 contains conclusion and discussion.

2 Model

We start with a formulation of 5D anisotropic conformal gravity. The first part of this section is mostly redrawn from Ref. [3] to make the paper self-contained. Let us first consider the Arnowitt-Deser-Misner (ADM) decomposition of five dimensional metric:

(2.1)

Then, the five dimensional Einstein-Hilbert action with cosmological constant is expressed as

(2.2)

where is the five dimensional gravitational constant, is the spacetime curvature, is the cosmological constant, and is the extrinsic curvature tensor, . The above action (2.2) can be extended anisotropically by breaking the five dimensional general covariance down to its foliation preserving diffeomorphism symmetry given by

(2.3)
(2.4)
(2.5)
(2.6)

and non-uniform conformal transformations

(2.7)

where a Weyl scalar field to compensate the conformal transformation of the metric is introduced. In the above equation (2.7), a factor is introduced in the transformation of , which characterizes the anisotropy of spacetime and extra dimension 111 We assume that the the field is a dimensionless and is a scale related with Planck scale. We also consider only the case , because is not effected under the conformal transformation in (2.7). It can be actually shown that for , an anisotropic scale invariant gravity theory can be constructed without the need of the field . . The anisotropic Weyl action invariant under Eqs. (2.3)-(2.7) for an arbitrary can be written as

(2.8)

where are some constants, the potential , and are given by

(2.9)
(2.10)
(2.11)

A couple of comments are in order. The isotropic case with and leads to five dimensional Weyl gravity with a potential [4]. In the anisotropic case, the action (2.8) is, in general, plagued with perturbative ghost instability coming from breaking of the full general covariance of 5D. However, it can be shown that this problem can be cured by constraining the constants and , especially with [3].

Now we discuss 4-dimensional effective low energy action and let us consider only zero modes. We first go to a “comoving” frame with and impose -independence (cylindrical condition) for and . This enables to replace where is the size of the extra dimension and eliminates terms containing and The resulting action preserves the redundant conformal transformation

(2.12)

where in (2.7) is replaced with Using this, we further fix and find the resulting four dimensional action given by

(2.13)

where and are defined as

(2.14)

and is given by

(2.15)

Let us redefine the field as

Then, the action (2.13) becomes

(2.16)

where

We find the effective four dimensional action is given by Brans-Dicke theory with a potential. is positive for a range of . Note that is usually a ghost field in the isotropic case with . This status has been evaded for anistropic case, thus reproducing the Brans-Dicke theory in the effective four dimensional action. Moreover, becomes a big number for being close to , as is usually required to pass the solar system test. We will show that close to is indeed preferred in the cosmological test, which implies and . It corresponds to Kaluza-Klein reduction with , with being the five dimensional Newton’s constant.

From here on, we remove the tilde notation in Eq. (2.16). We include matter term to investigate the cosmology 222We assume that the matter term couples with only and breaks the anisotropic conformal invariance from the beginning.. The Einstein equations obtained from action (2.13) by varying with respect to the metric can be written in the following form:

(2.17)

and scalar field equation is given by

(2.18)

In this work we shall study the isotropic and homogeneous cosmology. Thus, we consider the space-time geometry is given by the Robertson-Walker metric:

(2.19)

In this metric, Einstein equations follow as

(2.20)
(2.21)

where is the Hubble parameter and and are the standard radiation and matter energy densities. The field equation for the scalar field can be rewritten as

(2.22)

where denotes the derivative of the potential with respect to .

As in [6], the effective Newtonian constant is

(2.23)

where is Newton’s constant measured in Cavendish-type and solar system experiments 333Note that when is fixed to be a constant , becomes the Newton’s coupling constant which corresponds to the vacuum solution of a de Sitter universe in the induced gravity context [7].. The present value of can be connected the Newton’s constant by the relation:

(2.24)

In order to compare with dark energy in Einstein gravity with a Newton’s constant , we identify dark energy density and pressure as follows [8, 9, 10]:

(2.25)
(2.26)

Before comparing with observational data, we first perform a numerical analysis of the background evolutions equations. Basically, the evolutions are determined by the parameters , , and the present matter density . The Brans-Dike parameter and the current value of scalar field are determined by the relations (2.15) and (2.24), respectively. For initial conditions of the scalar field , the initial velocity is assumed to be zero, and the initial value is determined by the shooting method. In Fig. 1 we plot the evolution of the . At early times the scalar field dynamcis is frozen during the radiation-dominated epoch and begins to grow at the end of radiation-dominated epoch to realize the Newtonian gravitational constant at present time. In the first of Fig. 2 we display the time evolution of the energy density for dark energy defined (2.25). It has the same scaling behavior with the radiation energy density at early time and eventually it remains almost constant near the present time. In the second of Fig. 2 we plot the time evolution of the equation of state parameter for dark energy . We again find that its value is close to in the early radiation dominant epoch and approche until recently. It is worth mentioning that this behavior is similar in some respects to one in [11] although their origins are different, where the scalar field stays in a minimum of potential until radiation epoch and is shifted from the minimum during the transition from radiation to matter which leads to a suitable amount of dark energy explaining the present accelerated expansion.

Figure 1: The typical evolution of the Brans-Dicke field from radiation domination epoch to the present time as function of . Here, the cosmological parameters are , , , , and .
Figure 2: Evolution of energy density (left) and equation of state parameter (right) for different values of . Here, the cosmological parameters are , , , and .

3 Observational Constraints

In this section we constrain our model with the latest cosmological data described in [12], and investigate whether or not it can be distinguished from the -CDM model. For this purpose, we use the recent observational data such as type Ia supernovae (SN), baryon acoustic oscillation (BAO) imprinted in large-scale structure of galaxies, cosmic microwave background radiation (CMB), and Hubble parameters []. For numerical analysis, it is convenient to rewrite equations (2.20)-(2.21) in terms of as follows:

(3.27)

where a prime indicates a derivative with respect to . The equation of motion for the scalar field is rewritten as

(3.28)

where we have introduced dimensionless quantities,

(3.29)

Here, is the present value of the Hubble parameter, usually expressed as , and are the radiation and matter density parameters at the present epoch, respectively. The radiation density includes the contribution of relativistic neutrinos as well as that of photons, with the collective density parameter

(3.30)

where is the effective number of neutrino species, and is the photon density parameter with for the present CMB temperature  (WMAP9) and for  (PLANCK). Notice that the background dynamics is completely determined by a set of parameters . We need the baryon density parameter () to confront our model with the BAO and CMB data, and finally our model has six free parameters . It should be emphasized that the Hubble constant () is no longer a free parameter because it is derived from the integration of field equations for a given set of parameters chosen. The free parameters take the following priors: , , , , and . We apply the Markov chain Monte Carlo (MCMC) method to obtain the likelihood distributions for the model parameters [13]. The method propagates the parameter vector in random directions to explore the parameter space that is favored by the observational data, by making decisions for accepting or rejecting a randomly chosen parameter vector (or chain element) via the probability function , where denotes the data, and is the sum of individual chi-squares for , SN, BAO, and CMB data (defined below). We consider that the convergence of the MCMC chain is achieved if the means estimated from the first (after burning process) and the last 10% of the chain are approximately equal to each other.

3.1 Hubble Parameters

We use 29 data points of the Hubble parameters in a redshift range of , which include 23 data points obtained from the differential age approach [14] and 6 derived from the BAO measurements [15]. The chi-square is defined as

(3.31)

where and are theory-predicted and observed values of the Hubble parameter at redshift , respectively, and indicates the measurement uncertainty of the observed data point.

3.2 Type Ia Supernovae

In our analysis, the Union 2.1 compilation of 580 SNe in a redshift range of is used to constrain the energy content of the late-time Universe [16]. We use the chi-square that has been marginalized over the zero-point uncertainty due to the absolute magnitude and Hubble constant [17]:

(3.32)

where

(3.33)

where and denote the observed distance modulus and its measurement uncertainty of SN at redshift . The theoretical prediction of the distance modulus is defined as

(3.34)

where is the comoving distance at redshift ,

(3.35)

with the speed of light and for , respectively.

3.3 Baryon Acoustic Oscillations

As the BAO parameter, we use six numbers of extracted from the Six-Degree-Field Galaxy Survey [18], the Sloan Digital Sky Survey Data Release 7 and 9 [19], and the WiggleZ Dark Energy Survey [20]. These BAO data points were used in the WMAP 9-year analysis [21]. Here is the effective distance measure related to the BAO scale [22],

(3.36)

and is the comoving sound horizon size at the drag epoch. We use a fitting formula for the redshift of drag epoch () [23]:

(3.37)

where

(3.38)

Since the sound speed of baryon fluid coupled with photons () is given as

(3.39)

the comoving sound horizon size before the last scattering becomes

(3.40)

The BAO measurements provide the following distance ratios [21]

(3.41)
(3.42)
(3.43)

together with the inverse of the covariance matrix between measurement uncertainties

(3.44)

The chi-square is given as

(3.45)

where

(3.46)

3.4 Cosmic Microwave Background Radiation

We use the CMB distance priors based on WMAP 9-year data [21] and Planck data [24] to constrain our model. The first distance measure is the acoustic scale defined as

(3.47)

The decoupling epoch can be calculated from the fitting function [25]:

(3.48)

where

(3.49)

The second distance measure is the shift parameter which is given by

(3.50)

Recently, Shafer & Huterer [26] derived the distance priors for the WMAP and Planck data as an efficient summary of CMB information. Hereafter, we use these priors to constrain our model parameters.

The CMB distance prior has been widely used to constrain the dark energy property. However, it is worth mentioning that using the CMB distance prior to constrain the model parameters has some limitation in that the estimate of the distance prior itself is model-dependent. As mentioned in [27], the distance prior can be safely applied to constrain the dark energy model only when the model considered is based on the standard Robertson-Walker universe with the conventional radiation, matter, and neutrinos and on nearly power-law primordial power spectrum of curvature perturbations with negligible tensor modes. Since our model is based on the Robertson-Walker space-time geometry and the consequent background evolution equations include the matter and radiation together with the effective dark energy component, it is justified to use the WMAP and Planck distance priors to constrain our dark energy model at least in the background level.

3.4.1 WMAP 9-year data

According to WMAP 9-year observations (WMAP9) [21], the mean values for the three parameters are [26]

(3.51)

The inverse of the covariance matrix between the parameter uncertainties is

(3.52)

The chi-square is given as

(3.53)

where

(3.54)

3.4.2 Planck data

According to Planck observations (PLANCK) [24], the mean values for the distance priors are given as [26]

(3.55)

Their inverse covariance matrix is

(3.56)

The chi-square becomes

(3.57)

where

(3.58)

3.5 Results

We explore the allowed ranges of our dark energy model parameters using the recent observational data by applying the MCMC parameter estimation method. In the calculation, we use , , , , and as free parameters. The results are shown in Table 1 for a summary of parameter constraints with mean and confidence limits, and Fig. 3 for marginalized likelihood distributions of parameters that are common to our model and CDM model. The results for the other parameters of our model are presented in Fig. 4. We can see that the result obtained with Planck data gives tighter constraints on model parameters. The best-fit locations in the parameter space are

(3.59)

with a minimum chi-square of for the +SN+BAO+WMAP9, and

(3.60)

with for +SN+BAO+PLANCK.

To assess the goodness-of-fit of our model, in Table 1 we present the parameter constraints for the model and list the value of the minimum reduced chi-square () for each case. The minimum reduced chi-square is defined as , where is the number of degrees of freedom and and are the numbers of data points and free model parameters, respectively. In our analysis, , and for our model and for the model. Although the simple model gives the slightly better fit to the observational data with the smaller values of and , we judge that our model fits the data reasonably well in the sense that the reduced chi-square is very close to unity. We note that for our model to be compatible with observations the parameter must be lager than and the parameter should be close to , which give the following relation via (2.14) and (3).

(3.61)

The above relation can be satisfied, for example, with and , in which case .

5D Brans-Dicke Model Model
+ SN + BAO + SN + BAO + SN + BAO + SN + BAO
+ WMAP9 + PLANCK +WMAP9 +PLANCK





- -

- -

- -

- -

589.886 599.747 584.344 590.502
0.96073 0.97679 0.94861 0.95861
- -

- -

- -
Table 1: Summary of parameter constraints and derived parameters. The confidence levels are 68% unless otherwise stated.
Figure 3: Marginalized likelihood distributions of parameters that are common to our model and the CDM model for different combinations of the data sets. Here WMAP9 and PLANCK refer to + SN+BAO+WMAP9 and + SN+BAO+PLANCK, respectively. The contours indicate 68% and 95% confidence limits.
Figure 4: Marginalized likelihood distributions of our model parameters with 68% and 95% confidence limits, obtained by the joint parameter estimation with + SN+BAO+PLANCK (blue) and +SN+BAO+WMAP9 (red) data sets, respectively.

3.6 Local Constraints

The general relativity in weak-field conditions are confirmed by Solar-System experiments at the 0.04% level [28]. Thus, we should verify that the Brans-Dicke models are presently close enough to Einstein’s theory. In the first post-Newtonian approximation of general relativity, the deviations from general relativity can be parametrized by two real numbers, and , denoted by Eddington [28]. In the present models, they take the form

(3.62)

The Solar System experiments implies the following bounds

(3.63)
(3.64)