Brans-Dicke scalar field as a chameleon

Sudipta Das* and Narayan Banerjee+,‡

Relativity and Cosmology Research Centre, Department of Physics, Jadavpur University, Calcutta-700 032, India(Received 8 March 2008; published 6 August 2008)

In this paper it is shown that in Brans-Dicke theory, if one considers a nonminimal coupling between

the matter and the scalar field, it can give rise to a late time accelerated expansion for the Universe

preceded by a decelerated expansion for very high values of the Brans-Dicke parameter !.

DOI: 10.1103/PhysRevD.78.043512 PACS numbers: 98.80.Jk

I. INTRODUCTION

During the present decade, the speculation that theUniverse at present is undergoing an accelerated phase ofexpansion has turned into a certainty. The high precisionobservational data regarding the luminosity-redshift rela-tion of type Ia supernovae [1], the cosmic microwavebackground radiation (CMBR) probes [2] suggest thisacceleration very strongly. This is confirmed by the veryrecent WMAP data [3] as well. This observation leads to avigorous search for some form of matter, popularly calleddark energy, which can drive this acceleration as normalmatter cannot give rise to accelerated expansion due to itsattractive gravitational properties. A large number of pos-sible candidates for this dark energy has already appearedin the literature and their detailed behaviors are beingstudied extensively. For excellent reviews, see [4].

Although the expansion of the Universe is accelerated atpresent, it must have had a decelerated expansion in theearly phase of the evolution so as to accommodate fornucleosynthesis in the radiation dominated era. The earlymatter dominated era also must have seen a deceleratedphase for the formation of galaxies in the Universe. Thereare observational evidences too that beyond a certain valueof the redshift z, the Universe surely had a positive value

for the deceleration parameter (q ¼ � €a=a_a2=a2

> 0) [5]. It has

also been indicated that unless there is a signature flip froma positive to a negative value of q, the supernovae data arenot a definite indicator of an accelerated expansion con-sidering the error bars of the observation [6].

So, we are very much in need of some form of matter, thedark energy, which maintained a low profile in the earlypart of the history of the Universe but evolved to dominatethe dynamics of the Universe later in such a way that theUniverse smoothly transits from a decelerated to an accel-erated phase of expansion during the later part of the matterdominated regime. Apart from the cosmological constant�, which can indeed generate a sufficient negative pressureand hence drive this acceleration, the most talked about

amongst the dark energy models are perhaps the ‘‘quintes-sence models’’—a scalar field endowed with a potentialsuch that the potential term evolves to dominate over thekinetic term in the later stages of evolution generatingsufficient negative pressure which drives the acceleration.A large number of quintessence potentials have appearedin the literature (for an extensive review see [7]). However,most of the quintessence potentials do not have a soundbackground from field theory explaining their genesis.Hence it might appear more appealing to employ a scalarfield which is already there in the realm of the theory. Thisis where the nonminimally coupled scalar field models stepin as the driver of this alleged late time acceleration. Brans-Dicke (BD) theory [8] is arguably the most natural choiceas the scalar tensor generalization of general relativity(GR). BD theory or its modifications have already provedto be useful in providing clues to the solutions for some ofthe outstanding problems in cosmology (see [9,10]) andcould generate sufficient acceleration in the matter domi-nated era even without the help of the quintessence field[11]. Attempts have also been made to obtain a nondecel-erating expansion phase for the Universe at present byconsidering some interaction between the dark matterand the geometrical scalar field in generalized Brans-Dicke theory [12]. However, the form of interaction chosenwas ad hoc and did not follow from any action principle.A different approach is now being considered in general

relativity, where the quintessence scalar field is allowed tointeract nonminimally with the matter sector rather thanwith geometry and this interaction is introduced through aninterference term in the action. This type of scalar field isgiven the name ‘‘chameleon field’’ [13]. Many interestingpossibilities with this chameleon field have been recentlystudied [14]. It has also been shown recently [15] that thischameleon field can provide a very smooth transition froma decelerated to an accelerated phase of expansion of theUniverse. For similar work where the scalar field isstrongly coupled to matter, see also [16]. However, theproblem remains the same as that of the genesis of thescalar field.In Brans-Dicke theory or its modifications, there is an

interaction between the scalar field and geometry. Thechameleon field is also ‘‘nonminimally coupled,’’ but tothe normal matter sector rather than with geometry. It

*sudipta 123@yahoo.com+narayan@iiserkol.ac.in‡Present Address: IISER-Kolkata, Sector-III, Salt Lake,

Kolkata-700 109, India.

PHYSICAL REVIEW D 78, 043512 (2008)

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deserves mention at this stage that there are attempts alsoto build up models where the dark energy and the darkmatter do not conserve themselves individually, but has aninteraction amongst them [17]. One important motivationof considering these interactions is of course to seek for asolution of the coincidence problem—why the dark energysector dominates over the dark matter sector now.

The motivation of the present work is to investigate theinteracting models in a more general framework. A Brans-Dicke framework is considered, where there is already anonminimal coupling between the scalar field and geome-try. The action is modified to include a nonminimal cou-pling of the scalar field with the matter sector as well. Thiswork is actually motivated by the recent work by Cliftonand Barrow [18] where they studied the behavior of anisotropic cosmological model in the early as well as in thelate time limits in this framework. However, the nonmini-mal coupling of a scalar field with both of geometry and thematter sector has been in use for quite a long time, courtesyof the dilaton gravity, the low energy limit of string theory.The present work uses the ansatz for a particular purpose,namely, to check if the required signature flip in the decel-eration parameter q can be obtained from this model. Theactual form of the coupling of the scalar field with mattercertainly has to be introduced by hand, but the model hasthe advantage of having the scalar field in the theory itself.

As already mentioned, although Brans-Dicke theoryproved useful for the solution for many a cosmologicalproblem, it has the serious drawback that the Brans-Dickeparameter ! has to have a small value of order unity. Thissquarely contradicts the local astronomical requirement ofa pretty high value of !. It has been shown that the presentmodel works even for very high values of ! (� 104) andthus can have good agreement with the observational limits[19]. Hence this kind of general interaction has features,which might solve the cosmological problems as well astake care of the observations on the solar systems, etc.

In the next section the model is described and it is shownthat this type of nonminimally coupled interacting modelscan provide a smooth transition from the decelerated to theaccelerated phase of expansion for a wide range of valuesof the BD parameter!. Section III gives two cases of exactsolutions and the last section discusses the results.

II. FIELD EQUATIONS AND RESULTS

The relevant action in BD theory is given by

A ¼Z ffiffiffiffiffiffiffi�g

pd4x

��R

16�Gþ!

��;��;� þ Lmfð�Þ

�; (1)

where R is the Ricci scalar, G is the Newtonian constant ofgravitation,� is the BD scalar field which is nonminimallycoupled to gravity, ! is the dimensionless BD parameter.The last term in the action indicates the interaction be-tween the matter Lagrangian Lm and some arbitrary func-tion fð�Þ of the BD scalar field. If fð�Þ ¼ constant ¼ 1,

one gets back the usual BD action. For a spatially flatFriedmann-Robertson-Walker (FRW) model of theUniverse, the line element is given by

ds2 ¼ dt2 � a2ðtÞ½dr2 þ r2d�2 þ r2sin2�d�2�; (2)

where aðtÞ is the scale factor of the Universe. Variation ofthe action (1) with respect to the metric components yieldsthe field equations as

3_a2

a2¼ �mf

�þ!

2

_�2

�2� 3

_a

a

_�

�; (3)

2€a

aþ _a2

a2¼ �!

2

_�2

�2�

€�

�� 2

_a

a

_�

�: (4)

Here �m is the energy density of dark matter and as theUniverse at present is dominated by matter, the fluid istaken in the form of pressureless dust, i.e., pm ¼ 0. Here, adot indicates differentiation with respect to the cosmic timet. Also, variation of action (1) with respect to the Brans-Dicke scalar field � yields the wave equation as

ð2!þ 3Þ�€�þ 3

_a

a_�

�¼ �mfþ �mf

0�; (5)

where a prime indicates differentiation with respect to �.From the two field equations and the wave equation onecan arrive at the matter conservation equation which comesout as

_�m þ 3_a

a�m ¼ � 3

2�m

_f

f(6)

and readily integrates to yield

�m ¼ �0

a3f3=2; (7)

where �0 is a constant of integration. It is evident fromEq. (7) that the usual matter conservation equation getsmodified here because the scalar field is now coupled toboth geometry and matter.Out of Eqs. (3)–(5) and (7), only three are independent

equations as the fourth one can be derived from the otherthree in view of the Bianchi identity. On the other hand, wehave four unknowns—a, �m, fð�Þ, and � to solve for. Inorder to close the system of equations, we make an ansatz

_�

�¼ � �

H; (8)

where � is an arbitrary positive constant. There is no apriori physical motivation for this choice; this is purelyphenomenological which leads to the desired behavior ofthe deceleration parameter q of attaining a negative valueat the present epoch from a positive value during a recentpast. The system of equations is closed now. Some char-acteristics of the model can now be discussed even withoutsolving the system. In the remaining part of the section, the

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(a) (b)

(c) (d)

(e) (f)

(g) (h)

FIG. 1. The q vs H plot for different values of ! and �.

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possibility of having a transition of the mode of the expan-sion from a decelerated to an accelerated one is studied.With the assumption (8), Eq. (4) easily yields an expressionfor the deceleration parameter q as

q ¼ H2 þ ð!2 þ 1Þ �2

H2 � 3�

2H2 þ �: (9)

In obtaining Eq. (9), the relation

_H ¼ �H2ðqþ 1Þ (10)

has been used.H gradually decreases with time from a verylarge value at the beginning of the evolution. Equation (9)indicates that q ¼ 1

2 at H ! 1, i.e., the model starts ex-

actly the same way as a matter dominated spatially flatmodel does.

Now the deceleration parameter q is plotted against H[Fig. 1(a)–1(h)] for different values of ! and �. Theyclearly show that the required signature flip in q can beobtained for any negative value of ! and also for smallpositive values of ! in some recent past (H > 1). Thenature of the behavior of q against H is hardly affectedby a small change in the value of !, which can only shiftthe epoch at which the acceleration sets in. This can againbe adjusted by choosing the value of � properly which is afree parameter. It must be mentioned that as 1

H is a measure

of the age of the Universe and H is a monotonicallydecreasing function of time t, ‘‘future’’ is given by H <1, ‘‘past’’ by H > 1 if H is scaled by the present value H0,i.e., H ¼ 1 at the present epoch.

Figures 1(e)–1(h) have the additional feature that q hastwo signature flips. For example, in Fig. 1(e) (i.e., for ! ¼� 3

2 ), the flips take place around H � 1:5 (i.e., past) and

H � 0:25 (future). So, in all these cases the Universereenters a decelerated phase of expansion again in nearfuture and thus a ‘‘phantom menace’’ is avoided, i.e., theUniverse does not show a singularity of infinite volume andinfinite rate of expansion in a ‘‘finite future.’’ The cases 1(a)–1(d), however, do not have this ‘‘double signatureflip’’, which can be seen from Eq. (9). If q is put equal tozero, the combinations of values of ! and � used in thesefigures do not yield two real positive roots for H.

III. TWO SPECIFIC EXAMPLES

Equation (8) along with Eq. (10) can be written as

d

dHðln�Þ ¼ �

H3ðqþ 1Þ : (11)

Replacing the expression for q from Eq. (9), the aboveequation takes the form

d

dHðln�Þ ¼ �ð2H2 þ �Þ

3H½H4 � 2�3 H2 þ 1

3 ð!2 þ 1Þ�2� : (12)

We try to solve Eq. (12) analytically for two special cases.Case I: 2!þ 3 ¼ 0

With this choice of !, Eq. (5) immediately gives

f ¼ �0

�; (13)

�0 being a constant of integration. Equation (12) then leadsto the solution for � as

� ¼ AH4ðH2 � �

2Þ2ðH2 � �

6Þ4; (14)

A being a constant of integration. Using this expression for�, one can obtain the solutions for a, �m, and H as

a ¼ a0ðH2 � �

6Þ2=3½ðH2 � �

2Þð3H4 � 3�H2 þ 3�2

4 Þ�1=3 ; (15)

�m ¼ 3A2

�0

H6ðH2 � �2Þ6

ðH2 � �6Þ8

; (16)

and �H � ffiffiffi

�2

pH þ ffiffiffi

�2

p ��Hþ ffiffiffi

�6

pH� ffiffiffi

�6

p �2=

ffiffi3

p¼ exp

� ffiffiffiffi�

2

rðt0 � tÞ

�: (17)

It deserves mention that ! ¼ � 32 is a special case because

in the conformally transformed version of the BD theory,ð2!þ 3Þ ¼ 0 indicates that the kinetic part of the energycontribution from the scalar field sector is exactly zero[20].As shown in Eq. (13), this particular choice of ! gives

fð�Þ � 1� . The converse is also true. If one starts by

assuming fð�Þ � 1� , ! can take only one value, i.e., � 3

2

and one arrives at the same results.As already mentioned this choice of ! ¼ � 3

2 provides

the important feature of ‘‘future’’ deceleration and thusdoes not suffer from the problem of ‘‘big rip.’’Case II: 2!þ 3 ¼ �8In this case, Eq. (12) yields the solution for � as

� ¼ �0

�1� 7�

6H2

�2=7

; (18)

�0 being a constant of integration. Using this expressionfor �, from Eq. (8) one can obtain the solutions for theHubble parameter H and the scale factor a as

H ¼ffiffiffiffiffiffi7�

6

scoth

�3

2

ffiffiffiffiffiffi7�

6

st

�; (19)

a ¼ a0

�sinh

�3

2

ffiffiffiffiffiffi7�

6

st

��2=3

: (20)

The interesting feature of Eq. (20) is that for small t, a�t2=3 which is the same as that for a dust dominated era. On

the other hand, for high values of t, a� eð3=2Þðffiffiffiffiffiffiffiffi7�=6

pÞt and

thus gives an accelerated expansion for the Universe.

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From the field equations, the solutions for f and �m alsocome out as

fð�Þ ¼ 196�20

a60�2

1

�2

�1

40ð��0Þ�ð7=2Þ � 24þ 33ð��0

Þ7=2�2

(21)

and

�mðtÞ ¼ a60�3�3

0

2744�20

ðsech12=7XÞ½40coth2X � 24cosech2X

þ 33sech2Xcosech2X�; (22)

where X ¼ 32

ffiffiffiffiffi7�6

qt.

Although here the equation system has been completelysolved for only small negative values of !, this modelworks even for high values of ! (� 105) as shown inFig. 1(a)–1(c). This is consistent with the limits imposedby solar system experiments which predict the value of !to be of the order of tens of thousands (! � 40 000) [19].

It deserves mention that in Fig. 1(a) or 1(b), wherevalues of ! chosen are very high [! ¼ 106 in Fig. 1(a)and ! ¼ 104 in Fig. 1(b)], the corresponding values of �required are very low [� ¼ 10�3 in Fig. 1(a) and � ¼ 0:02in Fig. 1(b)] in order to adjust the time of signature flip inthe observationally consistent region.

IV. DISCUSSION

Thus we see that for a spatially flat FRW universe (k ¼0), we can construct a presently accelerating model withthe history of a deceleration in the past in Brans-Dicketheory by considering a coupling between the matterLagrangian and the geometric scalar field. The salientfeature of this model is that no dark energy sector isrequired here to drive this alleged acceleration. Also itdeserves mention that the nature of the q vs H plot is notcrucially sensitive to the value of ! chosen; only the timewhen the signature flip in q occurs shifts a little but that toocan be adjusted by properly choosing the value of �, whichis a parameter of the model.

The matter conservation equation obviously gets modi-fied in this framework due to the coupling between matterand the scalar field, i.e., matter is no longer conserved byitself. The right-hand side of Eq. (6) indicates that a trans-fer of energy between the matter and the scalar field takesplace due to the coupling factor fð�Þ. One may have anidea about the direction and amount of this energy transferif fð�Þ is exactly known. In the two specific examplesdiscussed in the present work, the energy in fact flows fromthe dark matter to the scalar field sector. In case I, with2!þ 3 ¼ 0, fð�Þ is given by Eq. (13) which yields (with

Eq. (8))_ff ¼ � _�

� ¼ þ �H . So the right-hand side of Eq. (6)

is negative and hence �m decreases more rapidly than whatis expected for a self-conserved matter sector. As we are

working in units where the present value of H is equal to 1and � is less than 1 (� ¼ 0:75 as used in Fig. 1(e)], thepresent transfer rate is obviously less than the Hubble rateof expansion. In case II, where 2!þ 3 ¼ �8, one can use

Eqs. (6) and (21) to find_ff , and if �< 1, the transfer rate is

of the same order of magnitude as the Hubble expansion

rate. In this case also, the present_ff is positive and hence the

energy flows from the dark matter sector to the scalar fieldsector. If fð�Þ ¼ constant, this interaction vanishes and thematter sector conserves itself as usual.As the nonminimally coupled scalar field theories allow

for a variation of the strength of gravitational interaction, itis worthwhile to comment on this aspect as well. As 1

�

behaves as the effective Newtonian gravitational constant

G, one has_GG ¼ � _�

� ¼ þ �H . As the present value of H ¼

1, and � � 1 in all the examples discussed,_GG at present is

less than the Hubble rate of expansion. As already men-tioned, the signature flip in q in Figs. 1(a)–1(h) can still beobtained with other choices of the pair of ! and �, the

value of_GG can be further lowered. In the early stages, when

H had a very high value,_GG was in fact negligible. In the far

future when H ! 0,_GG may have high values, but at that

epoch the hierarchy between gravitational and electroweakcouplings will hardly matter.As mentioned earlier, this particular model works for a

wide range of values of ! and even for high values of !(� 104). So this model is capable of solving two majorproblems at one go—the first one is to obtain the smoothtransition from a decelerated to an accelerated phase ofexpansion in the recent past without any dark energy sectorand the second one is to solve the nagging problem ofdiscrepancy in the values of ! as suggested by localexperiments and that required in the cosmological context.It had been shown before that the Brans-Dicke scalar fieldinteracting with dark matter can indeed generate an accel-eration [12] where ! is not severely restricted to lowvalues, but the parameter ! had to be taken as a functionof the scalar field �. The Brans-Dicke scalar field interact-ing nonminimally with the dark energy sector also has apossibility of having an arbitrary value of! [21]. But againthat required a dark energy sector as the driver of theacceleration.It deserves mention at this stage that the belief that BD

theory goes over to GR in the high ! limit suffered a jolt[22]. But in the weak field regime, relevant for the obser-vations on the solar system, a high value of ! is stillwarranted [19]. So the present work, and the work byClifton and Barrow [18] indeed opens up the possibilityof seeking solutions to cosmological problems in BDtheory. It should be noted that in view of the couplingbetween � and 8m as fð�ÞLm in the action, it is requiredthat the said weak field approximation of the field equa-tions be revisited. In the presence of a potential V ¼ Vð�Þ

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where � is the BD field, such investigations have alreadybeen there [23], where the results depend on derivatives ofthe potential. For the present work, the details of thecalculations will be different as there is no potentialVð�Þ as such. Investigations in this direction to find theactual order of magnitude of ! which passes the astro-nomical fitness test are in progress. However, it appearsthat although the expression for the post-Newtonian cor-rections for various modifications of BD theory will havedifferent features, all will require a large value of ! for thelocal astronomical tests [23,24].

In the context of the present accelerated expansion of theUniverse, a nonlinear contribution from the Ricci scalar Rin the action has attracted a lot of interest [25]. This form ofaction, very widely dubbed as fðRÞ gravity, has beenshown to be formally equivalent to a Brans-Dicke actionendowed with an additional potential V which is a functionof the BD scalar field for a particular value of the BDparameter !, namely ! ¼ �3=2 [26]. This is particularly

true for the Palatini kind of variation for the fðRÞ gravityaction. The present work does not contain a V ¼ Vð�Þ, butthe term fð�ÞLm in the action serves as an effectivepotential and may serve a similar purpose.The particular interaction chosen in this work is con-

trived, but it at least serves as a toy model, where the veryexistence of the particular scalar field is not questioned, itis there in the theory. A form of fð�Þ for which the darkmatter sector redshifts close to a�3, and the accelerationtakes place for quite a high value of !, could indeed be avery interesting possibility. Furthermore, the model has alot of features and has the promise to reproduce other formsof modifications of gravity as special cases.

ACKNOWLEDGMENTS

One of the authors (S. D.) wishes to thank CSIR forfinancial support.

[1] A. G. Riess et al., Astron. J. 116, 1009 (1998); S.

Perlmutter et al., Bull. Am. Astron. Soc. 29, 1351

(1997); Astrophys. J. 517, 565 (1999); J. L. Tonry et al.,

Astrophys. J. 594, 1 (2003); arXiv:astro-ph/0305008.[2] A. Melchiorri et al., Astrophys. J. Lett. 536, L63 (2000);

A. E. Lange et al., Phys. Rev. D 63, 042001 (2001); A. H.Jaffe et al., Phys. Rev. Lett. 86, 3475 (2001); C. B.

Netterfield et al., Astrophys. J. 571, 604 (2002); N.W.

Halverson et al., Astrophys. J. 568, 38 (2002).[3] S. Bridle, O. Lahav, J. P. Ostriker, and P. J. Steinhardt,

Science 299, 1532 (2003); C. Bennett et al., Astrophys. J.

Suppl. Ser. 148, 1 (2003); G. Hinshaw et al., Astrophys. J.Suppl. Ser. 148, 135 (2003); A. Kogut et al., Astrophys. J.

Suppl. Ser. 148, 161 (2003); D. N. Spergel et al.,

Astrophys. J. Suppl. Ser. 148, 175 (2003).[4] V. Sahni and A.A. Starobinsky, Int. J. Mod. Phys. D 9, 373

(2000); T. Padmanabhan, Phys. Rep. 380, 235 (2003); E. J.Copeland, M. Sami, and S. Tsujikawa, Int. J. Mod. Phys.

D 15, 1753 (2006).[5] A. G. Riess, Astrophys. J. 560, 49 (2001); arXiv:astro-ph/

0104455.[6] T. Padmanabhan and T. Roychoudhury, Mon. Not. R.

Astron. Soc. 344, 823 (2003); arXiv:astro-ph/0212573;T. Roychoudhury and T. Padmanabhan, Astron.

Astrophys. 429, 807 (2005).[7] V. Sahni, Lect. Notes Phys. 653, 141 (2004).[8] C. Brans and R.H. Dicke, Phys. Rev. 124, 925 (1961).[9] D. La and P. J. Steinhardt, Phys. Rev. Lett. 62, 376 (1989);

C. Mathiazhagan and V. B. Johri, Classical Quantum

Gravity 1, L29 (1984).[10] N. Banerjee and D. Pavon, Classical Quantum Gravity 18,

593 (2001); S. Sen and T. R. Seshadri, Int. J. Mod. Phys. D

12, 445 (2003); A.A. Sen and S. Sen, Mod. Phys. Lett. A16, 1303 (2001); S. Sen and A.A. Sen, Phys. Rev. D 63,

124006 (2001); O. Bertolami and P. J. Martins, Phys. Rev.

D 61, 064007 (2000); E. Elizalde, S. Nojiri, and S.

Odintsov, Phys. Rev. D 70, 043539 (2004); V.K. Onemli

and R. P. Woodard, Classical Quantum Gravity 19, 4607(2002); arXiv:gr-qc/0204065; V. K. Onemli and R. P.

Woodard, Phys. Rev. D 70, 107301 (2004); arXiv:gr-qc/

0406098; T. Brunier, V. K. Onemli, and R. P. Woodard,

Classical Quantum Gravity 22, 59 (2005); arXiv:gr-qc/0408080; E. Elizalde, S. Nojiri, S. Odintsov, D. Saez-

Gomez, and V. Faraoni, Phys. Rev. D 77, 106005 (2008).[11] N. Banerjee and D. Pavon, Phys. Rev. D 63, 043504

(2001).[12] S. Das and N. Banerjee, Gen. Relativ. Gravit. 38, 785

(2006).[13] J. Khoury and A. Weltman, Phys. Rev. Lett. 93, 171104

(2004); Phys. Rev. D 69, 044026 (2004); David F. Mota

and John D. Barrow, Mon. Not. R. Astron. Soc. 349, 291(2004); arXiv:astro-ph/0309273; Phys. Lett. B 581, 141(2004); arXiv:astro-ph/0306047.

[14] S. Das, P. S. Corasaniti, and J. Khoury, Phys. Rev. D 73,083509 (2006); arXiv:astro-ph/0510628.

[15] N. Banerjee, S. Das, and K. Ganguly, arXiv:gr-qc/

0801.1204.[16] D. A. Easson, J. Cosmol. Astropart. Phys. 02 (2007) 004;

arXiv:astro-ph/0608034; D. F. Mota and D. J. Shaw, Phys.

Rev. D 75, 063501 (2007); arXiv:hep-ph/0608078; Phys.Rev. Lett. 97, 151102 (2006); arXiv:hep-ph/0606204.

[17] W. Zimdahl and D. Pavon, Gen. Relativ. Gravit. 36, 1483(2004); W. Zimdahl, D. Pavon, L. P. Chimento, and A. S.

Jakubi, arXiv:astro-ph/0404122.[18] T. Clifton and J. D. Barrow, Phys. Rev. D 73, 104022

(2006).[19] B. Bertotti, L. Iess, and P. Tortora, Nature (London) 425,

374 (2003).

SUDIPTA DAS AND NARAYAN BANERJEE PHYSICAL REVIEW D 78, 043512 (2008)

043512-6

[20] R. H. Dicke, Phys. Rev. 125, 2163 (1962).[21] N. Banerjee and S. Das, Mod. Phys. Lett. A 21, 2663

(2006).[22] N. Banerjee and S. Sen, Phys. Rev. D 56, 1334 (1997).[23] G. J. Olmo, Phys. Rev. D 72, 083505 (2005).[24] T. Clifton, D. Mota, and J. D. Barrow, Mon. Not. R.

Astron. Soc. 358, 601 (2005); arXiv:gr-qc/0406001.[25] S. Capozziello, S. Carloni, and A. Troisi, Recent Res. Dev.

Astron. Astrophys. 1, 625 (2003); S. Capozziello, V. F.Cardone, S. Carloni, and A. Troisi, Int. J. Mod. Phys. D

12, 1969 (2003); arXiv:astro-ph/0307018; S.M. Carroll,V. Duvvuri, M. Trodden, and M. S. Turner, Phys. Rev. D70, 043528 (2004); arXiv:astro-ph/0306438; S. Das, N.Benerjee, and N. Dadhich, Classical Quantum Gravity 23,4159 (2006); arXiv:astro-ph/0505096.

[26] D. N. Vollick, Classical Quantum Gravity 21, 3813 (2004);G. J. Olmo, Phys. Rev. Lett. 95, 261102 (2005); S. Fay, R.Tavakol, and S. Tsujikawa, Phys. Rev. D 75, 063509(2007); B. Li, D. F. Mota, and D. J. Shaw, arXiv:gr-qc/0801.0603.

BRANS-DICKE SCALAR FIELD AS A CHAMELEON PHYSICAL REVIEW D 78, 043512 (2008)

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