Variable G correction for dark energy model inhigher dimensional cosmology

Shuvendu Chakraborty, Ujjal Debnath, and Mubasher Jamil

Abstract: In this work, we have considered N( = 4 + d)-dimensional Einstein field equations in which 4-dimensionalspace–time is described by a Friedmann–Robertson–Walker metric and that of the extra d-dimensions by an euclidean met-ric. We have calculated the corrections to statefinder parameters due to variable gravitational constant G in higher dimen-sional cosmology. We have considered two special cases: whether dark energy and dark matter interact or not. In a universewhere the gravitational constant is dynamic, the variable G-correction to statefinder parameters is inevitable. The statefinderparameters are also obtained for generalized Chaplygin gas in the effect of the variation of G correction.

PACS Nos: 95.36.+x, 98.80.–k

Résumé : Nous étudions ici les équations de champ d’Einstein en N = 4 + d dimensions, où un espace–temps à quatre di-mensions est décrit par une métrique de Friedmann–Robertson–Walker, alors que l’espace des d dimensions additionnellesest décrit par une métrique euclidienne. Nous calculons les corrections aux paramètres cosmologiques géométriques (state-finder) dues à la variation de la constante gravitationnelle G en cosmologie à plusieurs dimensions (correction G). Nousconsidérons deux cas spéciaux, selon que la matière noire et l’énergie sombre interagissent ou non. Dans un univers où laconstante de gravité est dynamique, les corrections G aux paramètres cosmologiques géométriques sont inévitables. Nousobtenons également les paramètres cosmologiques géométriques pour un gaz généralisé de Chaplygin dans l’effet de la va-riation de la correction G.

[Traduit par la Rédaction]

1. Introduction

Cosmological observations obtained by various cosmic ex-plorations of supernovae of type Ia [1], cosmic microwavebackground analysis of Wilkinson Microwave AnisotropyProbe data [2], extragalactic explorer Sloan Digital Sky Survey[3], and X-ray [4] convincingly indicate that the observableuniverse is experiencing an accelerated expansion. Althoughthe simplest and natural solution to explain this cosmic be-havior is the consideration of a cosmological constant [5], itleads to two relevant problems (namely “fine-tuning” and“coincidence”). Recently, the new dynamical nature of darkenergy is considered in the literature, at least at an effectivelevel, originating from various fields, including a canonicalscalar field (quintessence) [6], a phantom field that is a sca-lar field with a negative sign on the kinetic term [7], or thecombination of quintessence and phantom in a unifiedmodel named quintom [8].There are numerous indications that G can be varying and

that there is an upper limit to that variation, with respect totime or with the expansion of the universe [9]. In thisconnection the most significant evidence comes from the ob-servations of Hulse–Taylor binary pulsar [10, 11], helioseis-

mological data [12], type Ia supernova observations [1], andastereoseismological data coming from the pulsating whitedwarf star G117-B15A [13]; all of which combine to lead toj _G=Gj � 4:10� 10�11 a�1, for z ≲ 3.5, thereby suggesting amild variation on the cosmic level [14]. On a more theoreti-cal level, varying the gravitational constant has some benefitstoo, for instance it can help alleviate the dark matter problem[15], the cosmic coincidence problem [16], and the discrep-ancies in the value of the Hubble parameter [17]. In theliterature, a variable gravitational constant has been accom-modated in gravity theories including the Kaluza–Klein [18],Brans–Dicke framework [19], and scalar–tensor theories [20].From the perspective of new gravitational theories includ-

ing string theory and braneworld models, (see ref. 21 andreferences therein), there is a lot of speculation that therecould be extra spatial dimensions (six extra spatial dimen-sions in string theories) although there is no convincing evi-dence for their existence. The size of these dimensions(whether as small as the Planck scale or infinitely long likeusual dimensions) is still open to debate. In the literature,various theoretical models with extra dimensions have beenconstructed to account for dark energy [22]. Previously,some works on variable G correction have been investigated

Received 9 January 2010. Accepted 1 March 2012. Published at www.nrcresearchpress.com/cjp on 20 March 2012.

S. Chakraborty. Department of Mathematics, Seacom Engineering College, Howrah, 711 302, India.U. Debnath. Department of Mathematics, Bengal Engineering and Science University, Shibpur, Howrah-711 103, India.M. Jamil.* Center for Advanced Mathematics and Physics (CAMP), National University of Sciences and Technology (NUST), H-12,Islamabad, Pakistan.

Corresponding author: Shuvendu Chakraborty (e-mail: shuvendu.chakraborty@gmail.com).*Present addess: Eurasian International Center for Theoretical Physics, Eurasian National University, Astana 010008, Kazakhstan.

365

Can. J. Phys. 90: 365–371 (2012) doi:10.1139/P2012-027 Published by NRC Research Press

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[23] to find the statefinder parameters for several dark energymodels. The main motivation of this work is to investigatethe role of statefinder parameters [24] (which can be writtenin terms of some observable parameters) in higher dimen-sional cosmology assuming a varying gravitational constantG for interacting, noninteracting, and generalized Chaplygingas (GCG) models.

2. Basic equations and solutionsWe consider the homogeneous and anisotropic N-dimensional

space–time model described by the line element [25, 26]

ds2 ¼ ds2FRW þXdi¼1

b2ðtÞdx2i ð1Þ

where d is the number of extra dimensions (d = N – 4) andds2FRW represents the line element of the Friedmann–Robertson–Walker metric in four dimensions and is given by

ds2FRW ¼ �dt2 þ a2ðtÞ dr2

1� kr2þ r2 dq2 þ sin2qdf2

� �� �ð2Þ

where a(t) and b(t) are functions of t alone and represent thescale factors of 4-dimensional space–time and extra dimen-sions, respectively. Here k (= 0, ±1) is the curvature indexof the corresponding 3-space, so that the preceding model ofthe Universe is described as flat, closed, or open, respec-tively.Einstein’s field equations for the preceding nonvacuum

higher dimensional space–time symmetry are

3_a2 þ k

a2

� �¼

_D

D� d2

8

_b2

b2þ d

8

_b2

b2þ 8pGr ð3Þ

2_a

aþ _a2 þ k

a2¼ _a

a

_D

Dþ d2

8

_b2

b2� d

8

_b2

b2� 8pGp ð4Þ

and

_b

bþ 3

_a

a

_b

b¼ �

_D

D

_b

bþ

_b2

b2� 8pGp

2ð5Þ

where r and p are energy density and isotropic pressure ofthe fluid filling the universe, respectively. We choose, D2 =bd(t), so we have

_D

D¼ d

2

_b

b

and

_D

D¼ d

2

_b

bþ d2 � 2d

4

_b2

b2

Hence (1), (2), and (3) become

3_a2 þ k

a2

� �¼ d

2

_b

bþ d2 � 2d

4

_b2

b2� d2

8

_b2

b2þ d

8

_b2

b2þ 8pGr ð6Þ

2_a

aþ _a2 þ k

a2¼ d

2

_a

a

_b

bþ d2

8

_b2

b2� d

8

_b2

b2� 8pGp ð7Þ

and

_b

bþ 3

_a

a

_b

b¼ �

_d

2

_b2

b2þ

_b2

b2� 8pGp

2ð8Þ

Defining

H1 ¼ _a

aH2 ¼

_b

b

we have from (6) to (8)

3H21 þ 3

k

a2� d

2_H2 þ � d

8� d2

8

� �H2

2 ¼ 8pGr ð9Þ

3H21 þ

k

a2þ 2 _H1 � d

2H1H2 þ d

8� d2

8

� �H2

2 ¼ �8pGp ð10Þ

_H2 þ 3H1H2 � d

2H2

2 ¼ � 8pGp

2ð11Þ

Now consider that the universe is filled with dark matter(with negligible pressure) and dark energy. Assume that p =urx, r = rm + rx, where rm and rx are the energy densitiesof dark matter and dark energy, respectively, and u is theequation of state parameter for dark energy. Note that u is adynamical time-dependent parameter and will be useful in la-ter calculations. Now eliminating _H1 and _H2 from (9), (10),and (11) we have

24k ¼ a2�� 24H2

1 � 12dH1H2 � ðd � 1ÞdH22

þ 16pGð4rm þ ð4� duÞrxÞ�

ð12Þ

This equation can be written as

ðd þ 3Þ2H2 þ 3H21 �

dðd � 1Þ2

H22 þ

12k

a2

¼ 32pGrm þ ð4� duÞ8pGrx ð13Þwhere H is the Hubble parameter defined by

H ¼ 1

d þ 3ð3H1 þ dH2Þ

This can be written as

Uþ 3

ðd þ 3Þ2 U1 � dðd � 1Þ2

U2 þ 12

ðd þ 3Þ2 Uk

¼ 4

d þ 3Um þ 4� du

d þ 3Ux ð14Þ

where

U1 ¼ H21

H2U2 ¼ H2

2

H2

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are dimensionless parameters and

Um ¼ 8pGrmðd þ 3ÞH2

Ux ¼ 8pGrxðd þ 3ÞH2

are fractional density parameters,

Uk ¼ k

a2H2

is another dimensionless parameter, represents the contribu-tion in the energy density from the spatial curvature, and Uis the total density parameter. Now solving (9), (10), and (11)we have the solutions of _H1 and _H2 as

_H1 ¼ 1

24

�� 48H2

1 þ dðd � 1ÞH22 �

24k

a2

þ 8pGð4rm � ½�4þ ðd þ 12Þu�rxÞ

ð15Þ

and

_H2 ¼ 2

d3H2

1 �1

8dðd þ 1ÞH2

2 þ3k

a2� 8pGðrm þ rxÞ

� �ð16Þ

The deceleration parameter

q ¼ �1�_H

H2

is given in terms of dimensionless parameters by

q ¼ �1� 3

d þ 3Uk þ d

8U2 þ 3

2Um

þ 12þ 12uþ du

8Ux ð17Þ

and the derivative of deceleration parameter is obtained as

_q ¼ 3

d þ 3HUk 2

ffiffiffiffiffiffiU1

p� 2ðqþ 1Þ

h iþ d

8_U2 þ 3

2_Um

þ 12þ 12uþ du

8_Ux þ d þ 12

8_uUx ð18Þ

Also from (14) we obtain the expression of the total den-sity parameter in the form

U ¼ 4

d þ 3

r

rcr� du

d þ 3Ux � 12

ðd þ 3Þ2 Uk

� 3

ðd þ 3Þ2 U1 þ dðd � 1Þ2

U2 ð19Þ

Now define the critical density

rcr ¼3H2

8pGðtÞwhich gives, after differentiation,

_rcr ¼ rcr 2_H

H�

_G

G

� �ð20Þ

which implies

_rcr ¼ �Hrcrð2ð1þ qÞ þDGÞ ð21Þwhere

DG � G0

G_G ¼ HG0

(prime denotes differentiation with respect to x ≡ ln a). Thebenefit of the previous rule _G ¼ HG0 relates the variations inG with respect to time _G and the expansion of the universe G′.Differentiating (19) we have

_U ¼ 4

d þ 3

_r

rcrþ 4H½2ð1þ qÞ þDG�

d þ 3

r

rcr

� 24H

ðd þ 3Þ2 Uk

ffiffiffiffiffiffiU1

p� ðqþ 1Þ

h i� 3

ðd þ 3Þ2_U1

þ dðd � 1Þ2

_U2 � d

d þ 3u _Ux � d

d þ 3_uUx ð22Þ

where _U1 and _U2 are given by

_U1 ¼ H

�U1=2

1 Uk � 3U21 �

3d

2ðd þ 3ÞU3=21 U1=2

2 þ dðd � 1Þ8

U1=21 U2 � ðd þ 3ÞuU1=2

1 Ux � 9

d þ 3U1Uk

þ d

2U1U

1=22 þ dðd þ 7Þ

8ðd þ 3ÞU1U2 þ 4U1Um þ ð4þ 3uÞU1Ux

�ð23Þ

_U2 ¼ H

�12

dU3=2

1 � 3

d þ 3U3=2

1 U1=22 � d þ 1

2U1=2

1 U2 � 3d

2ðd þ 3ÞU3=21 U2 þ dðd þ 7Þ

8ðd þ 3ÞU1=21 U3=2

2

þ 12

dU1=2

1 Uk � 9

d þ 3U1=2

1 U1=22 Uk � 4ðd þ 3Þ

dU1=2

1 ðUm þUxÞ þU1=21 U1=2

2 ½4Um þ ð4þ 3uÞUx�

ð24Þ

The trajectories in the {r, s} plane corresponding to different cosmological models depict qualitatively different behaviour.The statefinder diagnostic along with future Supernova/Acceleration Probe Collaboration (SNAP) observations may perhaps beused to discriminate between different dark energy models. The preceding statefinder diagnostic pair for cosmology is con-structed from the scale factor a. The statefinder parameters are given by [24]

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r ¼ a���

aH2s ¼ r � 1

3ðq� 1=2ÞNow we obtain the expressions for r and s as follows:

r ¼ d2

32U2

2 �d

8

_U2

Hþ ½12þ ðd þ 12Þu�Ux � 3

8þ 3

4Um � 3

4ðd þ 3ÞUk þ d

16U2 � 1

8

_Ux

H

� �� 3

d þ 3Uk 3Um � 3þ 2U1=2

1

� �� 3d

4ðd þ 3ÞU2Uk þ 3d

8U2ð2Um � 1Þ þ 4U2

m � 3

2

_Um

H� 9

2Um þ 1

32½12þ ðd þ 12Þu�2U2

x

� �� ðd þ 12Þ8

_u

HUx þ 1 ð25Þ

s ¼ 8ðd þ 3Þ3 dðd þ 3ÞU2 � 24Uk þ ðd þ 3Þf�12þ 12Um þ ½12þ ðd þ 12Þu�Uxg

� d2

32U2

2 �d

8

_U2

H

�

þ ½12þ ðd þ 12Þu�Ux � 3

8þ 3

4Um � 3

4ðd þ 3ÞUk þ d

16U2 � 1

8

_Ux

H

� �� 3

d þ 3Uk 3Um � 3þ 2U1=2

1

� � 3d

4ðd þ 3ÞU2Uk

þ 3d

8U2ð2Um � 1Þ þ 4U2

m � 3

2

_Um

H� 9

2mþ 1

32½12þ ðd þ 12Þu�2U2

x

� �� ðd þ 12Þ8

_u

HUxÞ ð26Þ

These are the expressions for {r, s} parameters in terms of fractional densities of dark energy model in higher dimensionalcosmology for a closed (or open) universe where the derivative of the density parameters (i.e., _U1 and _U2) are given in (23)and (24). Now in the following subsections, we shall analyze the statefinder parameters for the noninteracting and interactingdark energy models.

2.1 Noninteracting dark energy modelIn this subsection we study the noninteracting model where dark energy and dark matter do not interact with each other. We

assume that dark matter and dark energy are separately conserved. So the continuity equation for cold dark matter is _rm þ ðd þ3ÞHrm ¼ 0 and for dark energy is _rx þ ðd þ 3ÞHð1þ uÞrx ¼ 0. So solving (22) for two different cases we have the expres-sions for _Um and _Ux as

_Um ¼ 1

2ðd þ 3Þðd þ 3þ duÞ � 6 _U1 þ dðd � 1Þðd þ 3Þ2 _U2 þ 2 24H qþ 1�U1=2

1

� �Uk

�þ 4H ðd þ 3Þð2q� 1� d þDGÞ þ dð2qþDGþ 2Þuþ dðd þ 3Þu2

� �Um � d½4Hð2qþ 2þDGÞuþ ðd þ 3Þ _u�Uxg

�ð27Þ

_Ux ¼ 1

2ðd þ 3Þðd þ 3þ duÞ ½�6 _U1 þ dðd � 1Þðd þ 3Þ2 _U2

þ 2 24H qþ 1�U1=21

� �Uk � ðd þ 3Þf4ðd þ 3ÞHðuþ 1ÞUm þ ½�4Hð2qþ 2þDGÞ þ d _u�Uxg

� �� ð28Þ

In (25) and (26), we have calculated the general expres-sions for the statefinder parameters {r, s}. In this noninteract-ing dark energy model, the preceding parameters are also thesame where the _Um and _Ux are given by (27) and (28).

2.2 Interacting dark energy modelIn this subsection we study the interacting model where

dark energy and dark matter interact with each other. Thesemodels describe an energy flow between the components(i.e., they are not separately conserved). According to recentobservational data of supernovae and cosmic microwavebackground the present evolution of the Universe permitsan energy transfer decay rate proportional to the presentvalue of the Hubble parameter. Many authors have widelystudied this interacting model. Pavon and Zimdahl [27] statethat the unknown nature of dark energy and dark mattermake no contradiction about their mutual interaction. Zhangand Olivers et al. [28] showed that the theoretical interact-ing model is consistent with the type Ia supernova and cos-mic microwave background observational data.

Here we assume that dark energy and dark matter are inter-acting with each other, so the continuity equations of darkmatter and dark energy become

_rm þ ðd þ 3ÞHrm ¼ Q ð29Þand

_rx þ ðd þ 3ÞHð1þ uÞrx ¼ �Q ð30Þwhere Q is is the interacting term, which is a arbitrary func-tion. This interacting term determines the direction of the en-ergy flow between dark matter and dark energy. In generalthis term can be chosen as a function of different cosmologi-cal parameters like the Hubble parameter and dark energy ordark matter density. In this work we choose Q = (d + 3)dHrxwhere d is a coupling constant. Positive d represents the en-ergy transfer from dark energy to dark matter. If d = 0 themodel corresponds to the noninteracting case. Here negatived is not considered as it can violate the thermodynamicallaws of the universe. So _Um and _Ux are given by the equa-tions

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_Um ¼ 1

2ðd þ 3Þðd þ 3þ duÞ�� 6 _U1 þ dðd � 1Þðd þ 3Þ2 _U2

þ 2 24H qþ 1�U1=21

� �Uk þ 4H ðd þ 3Þð2q� 1� d þDGÞ þ dð2þDGþ 2qÞuþ dðd þ 3Þu2

� �Um

�þ f4H½�dð2qþ 2þDGÞuþ ðd þ 3Þdðd þ 3þ 2duÞ� � dðd þ 3Þ _ugUxÞ

�ð31Þ

_Ux ¼ 1

2ðd þ 3Þðd þ 3þ duÞ�� 6 _U1 þ dðd � 1Þðd þ 3Þ2 _U2

þ 2 24H qþ 1�U1=21

� �Uk � ðd þ 3Þf4Hðd þ 3Þð1þ uÞUm þ ½�4Hð2qþ 2þDGÞ þ 4ðd þ 3ÞHdþ d _u�Uxg

� �� ð32Þ

In (25) and (26), we have calculated the general expres-sions of the statefinder parameters {r, s}. In this interactingdark energy model, the preceding parameters are also thesame where _Um and _Ux are given by (31) and (32).

3. GCG

It is well known that Chaplygin gas provides a differentpath for the evolution of the universe; having behaviour atearly times as pressureless dust and at very late times as acosmological constant. An advantage of GCG is that it unifiesdark energy and dark matter into a single equation of state.This model can be obtained from a generalized version of theBorn–Infeld action. The equation of state for GCG is [29]

px ¼ � A

raxð33Þ

where 0 < a < 1 and A > 0 are constants. Inserting (33) of

the GCG into the noninteracting energy conservation equa-tion we have

rx ¼ Aþ B

a3bd� �aþ1

" #1=ðaþ1Þð34Þ

where B is an integrating constant.

u ¼ �A Aþ B

a3bd� �aþ1

" #�1

ð35Þ

Differentiating (35) we have

_u

H¼ �ðd þ 3ÞABð1þ aÞ

� 1

a3bd� �aþ1

Aþ B

a3bd� �aþ1

" #�2

ð36Þ

Now putting (36) into (25) and (26), we have

r ¼ d2

32U2

2 �d

8

_U2

H� 9

2Um þ 9

2U2

m � 3

2

_Um

Hþ 9

d þ 3Uk � 6

d þ 3U1=2

1 Uk � 3d

d þ 3U2Uk � 3d

8U2

� 36B2 � 3a6A2b2d a3bd� �2a

d � AB a3bd� �aþ1½dðd þ 18Þ þ ðd þ 3Þðd þ 12Þa�

8 A a3bd� �aþ1 þ B

h i2 Ux þ 1

32�12þ Aðd þ 12Þ

Aþ a3bd� ��a�1

B

" #2U2

x

þ 12� Aðd þ 12ÞAþ a3bd

� ��a�1B

" #3

4UmUx �

_Ux

8H

� �þ Bðd þ 12ÞUx

A a3bd� �aþ1 þ B

þ 12Um � dUx

" #d

16U2 � 3

4ðd þ 3ÞUk

� �þ 1 ð37Þ

s ¼ � 9

2þ 3dU2

8� 9Uk

d þ 3þ 9Um

2� 3dUx

8þ 3

8

Bðd þ 12ÞUx

A a3bd� �aþ1 þ B

" #�1d2

32U2

2 �d

8

_U2

H� 9

2Um þ 4U2

m � 3

2

_Um

Hþ 9

d þ 3Uk

�

� 6

d þ 3U1=2

1 Uk � 3d

8U2 �

36B2 � 3a6A2b2d a3bd� �2a

d � AB a3bd� �aþ1ðdðd þ 18Þ þ ðd þ 3Þðd þ 12ÞaÞ

8 A a3bd� �aþ1 þ B

h i2 Ux

þ 1

32�12þ Aðd þ 12Þ

Aþ a3bd� ��a�1

B

!2

U2x �

3d

d þ 3U2 þ 12� Aðd þ 12Þ

Aþ a3bd� ��a�1

B

!3

3UmUx �

_Ux

8H

� �

þ Bðd þ 12ÞUx

A a3bd� �aþ1 þ B

þ 12Um � dUx

!d

16U2 � 3

4ðd þ 3ÞUk

� ��ð38Þ

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These are the expressions for {r, s} parameters in terms of fractional densities for the noninteracting case of GCG model inhigher dimensional cosmology, where _Um and _Ux are given by (27) and (28).Again inserting equation of state (33) of the GCG into the interacting energy conservation equation (30) we have

rx ¼A

dþ 1þ B

ðdþ 1Þ a3bd� �ðdþ1Þðaþ1Þ

" #1=ðaþ1Þð39Þ

and

u ¼ �AA

dþ 1þ B

ðdþ 1Þ a3bd� �ðdþ1Þðaþ1Þ

" #�1

ð40Þ

Differentiating (40) we have

_u

H¼ �ðd þ 3ÞABð1þ aÞ 1

a3bd� �ðdþ1Þðaþ1Þ

A

dþ 1þ B

ðdþ 1Þ a3bd� �ðdþ1Þðaþ1Þ

" #�2

ð41Þ

Now putting (41) into (25) and (26), we have

r ¼ d2

32U2

2 �d

8

_U2

H� 9

2Um þ 9

2U2

m � 3

2

_Um

Hþ 9

d þ 3Uk � 6

d þ 3U1=2

1 Uk � 3d

d þ 3U2Uk � 3d

8U2 � 9

d þ 3UkUm þ 3d

4U2Um

þ AB a3bd� �ð1þaÞð1þdÞðd þ 3Þðd þ 12Þð1þ aÞð1þ dÞ2

8 A a3bd� �ð1þaÞð1þdÞ þ B

h i2þ d

16U2 þ 3

4Um � 3

4ðd þ 3ÞUk � 3

8

� ��12þ Aðd þ 12Þð1þ dÞ

Aþ B a3bd� ��ð1þaÞð1þdÞ

!� �12þ Aðd þ 12Þð1þ dÞ

Aþ B a3bd� ��ð1þaÞð1þdÞ

!_Ux

8Hþ 1 ð42Þ

s ¼ 3dU2 � 72

d þ 3Uk þ 36Um � 36þ 3 �12þ Aðd þ 12Þð1þ dÞ

Aþ B a3bd� ��ð1þaÞð1þdÞ

" #Ux

( )�1

��d2

32U2

2 �d

8

_U2

H� 9

2Um þ 9

2U2

m � 3

2

_Um

Hþ 9

d þ 3Uk � 6

d þ 3U1=2

1 Uk � 3d

d þ 3U2Uk � 3d

8U2 � 9

d þ 3UkUm

þ 3d

4U2Um þ AB a3bd

� �ð1þaÞð1þdÞðd þ 3Þðd þ 12Þð1þ aÞð1þ dÞ2

8 A a3bd� �ð1þaÞð1þdÞ þ B

h i2þ d

16U2 þ 3

4Um � 3

4ðd þ 3ÞUk � 3

8

� ��12þ Aðd þ 12Þð1þ dÞ

Aþ B a3bd� ��ð1þaÞð1þdÞ

" #� �12þ Aðd þ 12Þð1þ dÞ

Aþ B a3bd� ��ð1þaÞð1þdÞ

!_Ux

8H

ð43Þ

These are the expressions for {r, s} parameters in terms offractional densities for the interacting case of the GCG modelin higher dimensional cosmology, where _Um and _Ux aregiven by (31) and (32).

4. ConclusionIn this work, we have considered N( = 4 + d)-dimensional

Einstein field equations in which 4-dimensional space–time isdescribed by a Friedmann–Robertson–Walker metric and thatof the extra d-dimensions by an euclidean metric. We havecalculated the corrections to statefinder parameters {r, s}and deceleration parameter q due to variable gravitationalconstant G in higher dimensional cosmology. These correc-tions are relevant because several astronomical observationsprovide constraints on the variability of G. We have first as-sumed that dark energy does not interact with dark matter.

Next we have considered that the dark energy and dark mat-ter are not separately conserved, that is, they interact witheach other with a particular interacting term in the form Q =(d + 3)dHrx where d is a coupling constant. In both cases,the statefinder parameters have been found in terms of the di-mensionless density parameters as well as EoS parameter uand the Hubble parameter. An important thing to note is thatthese are the G-corrected statefinder parameters and theyremain geometrical parameters as previously, because theparameter DG is a pure number and is independent of geometry.Finally, we have analyzed the preceding statefinder parame-ters in terms of some observable parameters for the nonin-teracting and interacting cases when the universe is filledwith GCG. These dynamical statefinder parameters maygenerate different stages of the anisotropic universe inhigher dimensional cosmology if the observable parametersare known for interacting and noninteracting models.

370 Can. J. Phys. Vol. 90, 2012

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