Scalar Tensor gravity with scalar matter direct coupling and its cosmological probe

Transcription

1 Scalar Tensor gravity with scalar atter direct coupling and its cosological probe Jik Su Ki Pyongyang Astronoical Observatory, Acadey of Sciences, Pyongyang, DPR Korea Chol Jun Ki, Sin Chol Hwang, Yong Hae Ko, Departent of Physics, Ki Il Sung University, Pyongyang, DPR Korea Abstract SNIa and CMB datasets have shown both of evolving Newton s constant and a signature of the coupling of scalar field to atter. These observations otivate the consideration of the scalar-atter coupling in Jordan frae in the fraework of scalar-tensor gravity. So far, ajority of the works on the coupling of scalar to atter has been perfored in Einstein frae in the fraework of inially coupled scalar fields. In this paper, we generalize the original scalar-tensor theories of gravity by introducing a direct coupling of scalar to atter in the Jordan frae. The cobined consideration of both evolving Newton s constant and scalar-atter coupling using the recent observation datasets, shows features different fro the previous works. The analysis shows a vivid signature of the scalaratter coupling. The variation rate of the Newton s constant is obtained rather greater than that deterined in the previous works. I. INTRODUCTION Cosological observation datasets are opening a wide possibility of test of the various cosological odels. Nesseris and Perivolaropoulos[] have shown that old dataset of SNIa yielded soe evidence of the scalar tensor property of gravitation. Making use of old dataset of SNIa[], they have found the Newton s gravitation constant to be evolved. On the other hand, Majerotto, Sapone and Aendola[3] and uo, Ohta and Tsujikawa [4] have found that cobined analysis of SNLS, CMB, and BAO datasets showed a signature of direct scalar atter coupling. The latter, however, had been based on the background of Einstein tensor gravity. The above both analyses are aking use of alost the sae observation datasets, but their results are quite contradictory, so we cannot be sure which of these odels should be accepted. As is well known, when the coupling of scalar to background space-tie vanishes the gravity returns to Einstein tensor gravity. Therefore, if we want to elucidate whether both of scalar background space tie and scalar atter couplings do exist or not, one should construct a ore inclusive odel than the previous ones[, 3, 4].

2 We construct a odel involving a scalar atter direct coupling in the scalar tensor gravity. eneralized theory of scalar-tensor gravity has already been studied in [5, 6, 7]. We investigate here the theory in Jordan frae as a physical frae rather than Einstein frae. To test our odel we use a new heterogeneous copilation Union of Type Ia supernovae[8], and the distance paraeter of baryon acoustic oscillation[9]. Cobining the above cosological observation datasets we obtain a definite constraint on the cosological paraeters, especially, a variation rate of gravitation constant and a paraeter characterizing the direct scalar atter coupling. In Sect.II we construct a general foralis for the scalar tensor gravity with direct coupling of scalar field to atter. And we apply the dynaical equations to the expanding FRW odel of the Universe. In Sect. III, aking use of cosological observation datasets of SNIa, and BAO, we find observational constraints on the coupling paraeters. Finally, in Sect. IV, we suarize results and discuss several iplications of the results. II. ENERAL FORMALISM FOR COUPLED SCALAR-TENSOR TEHORY OF RAVITY A. eneral equations The general foralis of the etric scalar-tensor theories of gravity is described in Refs. [5,6,7,,,,3]. We generalize it including the scalar-atter coupling in Lagrangian, thus aking the theory non-etric. We consider the action in Jordan frae S= d 4 x g[ ( F (φ ) R g μν μ φ ν φ ) V ( φ )+C (φ ) L ( ) (Ψ;g μν ) ] () where the action is characterized by three functions: scalar-curvature coupling function F, scalar-atter coupling function C, and self-interaction potential V. L describes the atter sector of the Lagrangian without scalar coupling, Ψ expresses generically the atter fields and F to ensure positivity of the energy of graviton. Superscript in the atter Lagrangian denotes that it does not include scalar field, but observations easure the coupled energy corresponding to C L ; g L, g. ; The kinetic ter of the scalar field in action() ay have a factor, but in the case of satisfying the observational constraint on the current coupling strength it can always be set equal to unity by a redefinition of the field [4]. In the action () we set 8 N where is Newton constant, so F, C and scalar field are diensionless, and in N this unit energy density and potential V have diension (length) -. The coupling function C akes the theory non-etric, and thereby presupposes the violation of WEP and the

3 change with tie of the fundaental constant, such as, e.g. the fine-structure constant 5 of agnitude.7.8 EM EM EM for the red shift interval z [], and such observations, besides the otivation entioned in Sect. I, justify the consideration of the direct scalar-atter coupling. Many authors treat the scalar-atter coupling including it in Lagrangian L, ; g we represent it explicitly separating the factor explicitly separated the coupling[]. The variations of action () with respect to etric tensor dynaical equations for gravity and scalar field R g R F C T T but C to ephasize its role. Casas et al. have dv dc g and scalar field yield, () df R T, (3) d d d T,, (4),, g, F, ; g F gv where g T g g g g L, atter energy-oentu tensor is defined by other for using the trace of Eq. () and it reads, and T T. The equation of scalar field (3) can be expressed in dc d T dv df, df C F, 4V d d d d df where F 3. Taking covariant derivative of Eq. () and using Eq. (3) d we find energy-oentu conservation equation for the atter T T T, ; d d lnc, (6) d (5) F This for of the conservation equation is different fro the conventional expression on the conservation law for the atter. Expressing the energy-oentu tensor by T C T, (7) we can restore the conventional conservation equation T d lnc d dc ; T, T,, (8) d where T g g L g and T T. Eq. (8) resebled the equation 3

4 T ; C A T, in [3] where A C corresponds to our d ln C d. However, [3] considered this coupling in the Einstein frae and C A is constant, whereas, in our case, d ln C d is not constant and an arbitrary function of, so the coupling function C represents ost general coupling. Casas et al.[] have investigated the direct scalar-atter coupling in the Jordan frae in the fraework of Brans-Dicke theory of gravity, and their d ln C, coupling function is C, so,. d B. The coupled scalar-tensor theory in expanding Universe In this subsection, we apply the general equations to the expanding Universe and consider several probles of the theory in the presence of the direct scalar-atter coupling. We consider a flat Friedann-Robertson-Walker Universe whose etric in the Jordan frae is given by where t is cosic tie. To copare the outcoe of all the coputations with observations we perfor the coputation in the Jordan frae. In the following, atter will be described as a perfect fluid, so its energy-oentu tensor takes the for as T μν =( ρ+p )u μ u ν +g μν P () where u μ =dx μ /ds is the coponents of four velocity of the atter in Jordan frae. Taking a and a a a a 3 a (3) are given by 3F F ii the dynaical equations derived fro Eqs () and, () H C V 3HF H C P F HF, () dv dc 3dF 3H H H T d d d (9), (3) where subscript stands for the quantity corresponding to Lagrangian L in the action (). Fro Eq. (6) we get iediately P 3H. (4) In spite of the direct coupling of the scalar to the atter, the atter density which does not take into account the coupling is conserved. However, as entioned above, the observations easure the coupled atter energy corresponding to C L L, one can 4

5 define the coupled atter density C (8) yields H P, and corresponding conservation equation d ln C 3. (5) d This equation shows that the atter conservation is unaffected by scalar-curvature coupling function F and the direct coupling of scalar to the atter violates the atter conservation law. In the previous works[3, 4, 9,, 3, 5, 6, 7], the atter conservation equation in the presence of the scalar-atter coupling is custoary to be written in the for ρ +3 H ( ρ +P )=Γρ =δhρ. (6) Coparing with Eq. (5) one can find C d ln C. (7) C d Fro the relation Γ=δH, (6), we get iediately the following eaningful relation Ċ C =δ ȧ a C ~ aδ. (8) This siple relation iplies that in positive the coupling function C increased with expansion of the Universe and thereby particle ass increases, while in negative it reduces with tie. Let us define the energy density and pressure of the scalar field. It is straightforward to deduce the fro the definition of energy-oentu tensor for scalar field (4). V 3HF T, (9) P Tij 3 ij V F HF. () Multiplying the Eq. (3) by φ and taking into account of the definition (9) and () we get iediately the equation of energy conservation for the scalar field d lnc 3H P. () d The conservation equations for the atter and the scalar field (5) and () are derived in the Jordan frae, but, as entioned above, the conservation equations in the Jordan frae have the sae for as in the Einstein frae, and actually, alost all the previous studies of the scalar-atter coupling in Einstein frae are giving the sae equations as (5) and (). The difference is erely that the paraeter is expressed by d ln C dt, but this is an elucidation of the iplication of the paraeter. Both of the conservation equations (5) and () are giving the conservation of the cobined syste <atter + scalar field>. Bellow in Sect. III the coupling paraeter or will be deterined in the Jordan frae aking use of the recent SNIa and BAO datasets. Previous studies concerning the 5

6 direct scalar-atter coupling have been perfored in the Einstein frae[3, 4]. Recent observations show a signature of the scalar-tensor gravity[]. Therefore, to reexaine the scalar-atter coupling in the Jordan frae deserves further attention. III. OBSERVATIONAL CONSTRAINTS ON THE COUPLIN FUNCTIONS F z AND C z Recent observations of SNIa standard candles[, 8, 8, 9], CMB anisotropy and the baryon acoustic oscillations (BAO) in the Sloan Digital Sky Survey (SDSS) luinous galaxy saple []provide a new prospect to deterine the cosological paraeters. Nesseris and Perivolaropouls[], using the old dataset of SNIa, have shown an observational evidence for the evolving Newton s constant and thereby the scalar-tensor character of the gravitation. They have used the old dataset to deterine the paraeters of siple polynoial expressions for the functions H z and z of the for H 3 z H z a z a z a a, () z az. (3) In the deterination of evolving z, they have ade use of the fact that the peak luinosity of SNIa is proportional to the ass of nickel synthesized which is a fixed fraction of the Chadrasekhar ass M ch varying as. Therefore, the SNIa peak luinosity would vary like and the corresponding SNIa absolutely agnitude evolves like 5 M M log 4 where the subscript denotes the local values of M and. On the other hand, the Newton s constant enters the Friedann equation. In the scalar-tensor theories, it relates to the scalar-curvature coupling function F by a relation[3, 9] df d df d F F 4 z F F 3 (4). (5) The luinosity distance, therefore, involves the evolving gravitation constant z d L ( z )= (+z ) dz' ( (z' ) ) H ( z' ). (6) Then, the theoretical agnitude of observed SNIa in the context of scalar-tensor gravity is given by th z M 5log d z 5 z L log. (7) 4 6

7 Substituting the polynoial expression () and (3) into (6) and (7) and fitting (z ) and H (z ) to the old dataset of SNIa by -iniization, Nesseris and Perivolaropouls[] have deterined the paraeters a, a, and a in Eqs () and (3). In this paper, we follow the sae route as in [], but several odifications ust be appended in order to take into account of the scalar-atter coupling. In addition, at variance with (3), we take a new expression of Newton s constant as follows z bz az (8) which includes a linear ter. The inclusion of the linear ter in the expression (8) is justified by the fact that in the context of scalar-tensor theories of gravity the Newton s constant has the for[3, 9] z F (9) df d 3 df d. (3) F The result of the Doppler tracking of the Cassini spacecraft provides an observational constraint on the current value of the coupling constant [9] (3) This iplies that we can not yet entirely disregard the existence of the scalar partner in the gravitational interaction between two bodies, and that as we can see in Eq. (3), non- vanishing iplies the non-vanishing df d. Actually, fro the relation (8) z bz az z F z (3) we have F' z b, whereas the expression (3) yields F ' z. The expression (8) or (3) is iediately related to the rate of tie varying of the Newton s constant as follows Ġ/ z= = bh. (33) In point of fact, recent observations[8] show Ġ/ 9 3 yr (34) which we can copare with (33). Thus, only the linear ter in Eq. (8) allows to estiate the tie varying gravitational constant in the present epoch. To include the coupling function C in the Hubble paraeter H, we take an expression of H z different fro (). Fro Eqs () and (9) we have the Friedann equation as follows 3F H C. (35) On the other hand, fro Eq. (6) we have 7

8 3 3 a z. (36) Substituting Eq. (7) into Eq. () we get the following solution for the energy density of the scalar field 3 w 3 w z z z 3w Putting Eqs (36) and (37) in Eq. (35) and reebering C obtain final expression on the Hubble paraeter 3. (37) above Eq. (5), we H z E z, (38) H F z where 3F H. Thus, in the scalar-tensor gravity, the Hubble paraeter H z contains the coupling function F z. Therefore, to separate the function F z, we expressed the expression (39) as E z. We have in Eqs (38) and (39) three free paraeters w, (39), when we copare the theory with observations. Our odel thus has five free paraeters a b,, w, in all. In order to reduce the nuber of the paraeters to be deterined we, set the density paraeter of atter to equal.7 [, ]. As a SNIa dataset, we use the Union copilation of 37 SNIa [8]. The Original Union copilation consists of 44 SNIa and it reduces to 37 SNIa after selection cuts. This Union copilation includes the recent large saples of SNIa fro SNLS and ESSENCES Survey, the older datasets, as well as the recently extended dataset of distant supernovae observed with HST. To unify the various heterogeneous copilations, a single consistent and blind analysis procedure is used for all the various SNIa subsaples, and a new procedure is ipleented that consistently weights the heterogeneous datasets and reject outliers. The theoretical distance odulus is defined fro (7) th th 5 ( zi ) ( zi ) M 5log d L ( zi ) log F( zi ), (4) 4 where the luinosity distance d z ) is expressed, instead of (6), by d L z L ( i i / F( z) ( z i ) ( zi ) H dz. (4) E( z) Thus, the theoretical odel paraeters are deterined by iniizing the quantity SN N obs th [ ( z ) ( z ; a, b,, )] ( a, b,, ) c c C, (4) i i i tot sys ij i j ij 8

9 where σ tot represents an astrophysical dispersion obtained by adding in quadrature the dispersion due to lensing, σ lens, the uncertainty in the Milky Way dust extinction correction and a ter reflecting the uncertainty due to host galaxy peculiar velocities 3k/s. The dispersion ter σ sys contains an observed saple - dependent dispersion due to possible unaccounted for systeatic errors. The su in the denoinator represents the statistical uncertainty as obtained fro the light-curve fit with C ij representing the covariance atrix of fit paraeters: peak agnitudes, color and stretch corresponding correction paraeters. and c i = {,α, β } are the When we use only the SNIa dataset in the odel fitting, we find that Union data give a weak constraint on the coupling paraeter analysis adding BAO datasets. δ as in [9], so we perfor the cobined In this paper, we do not use CMB anisotropy data because the CMB anisotropy includes the inforation of epoch separated fro now by z 89, but the datasets of SNIa and BAO only range z, so our paraeterization (8) for varying gravitational constant does not suit for CMB. In the following paper we shall introduce a new paraeterization for CMB data. To avoid degeneracies intrinsic to the distance fitting ethods we can consider also the effect of the baryon acoustic peak of the large scale correlation function at h - Mpc separation detected by the SDSS tea[]. The position of the acoustic peak is related to the distance paraeter. / 3 / z / BAO z BAOF zbao F z A dz, (43) z BAO E zbao E z which takes value A and z. 35 [3]. The baryon acoustic peak is taken into account by adding BAO a, b,, w BAO A A, (44) A to the above SN and CMB, where A is the error of A. The We perfor a best fit analysis with iniization of total : SN BAO. (45) -function (45) has four independent paraeters a, b,, w. Through the iniization of the function (45) we obtained best fit values of the paraeters (Table). 9

10 paraeters Union( N 37 ) + BAO a b δ SN w.3 Table. Best fit values of paraeters in Jordan frae. According to forula(33), the best fit value of paraeter b akes it possible to deterine the rate of tie-variation of the gravitation constant. It gives Ġ/ z= =. 86 yr, (46) where we have taken H 7k / s Mpc 7.5 yr. Figure. The -distribution of estiated paraeters near the bets fit ones. Contours display, and 3 confidence level. The dashed lines ean the best fit value of paraeters

11 In ΛCDM odel, we can set w φ = and recalculate the best-fit values of a, b, δ (Table. ). It gives Ġ/ z= =.43 yr, (47) where we have taken H =7k/s Mpc= 7.5 yr. This value is siilar to, but a bit greater still than the value in [8]. paraeters Union( N 37 ) + BAO a.8 b. SN δ..4 Table. Best fit values of paraeters with w φ = Figure. The -distribution of estiated paraeters near the best-fit ones with Contours display, and 3 confidence level. The dashed lines ean the best fit value of paraeters w. IV. DISCUSSION AND CONCLUSION In contrast with [3, 4, 8], in this paper we constructed the generalized odel of the scalar-tensor gravity that has the direct coupling between scalar and dark atter and inspected the coupling, using the cosological dataset of SN and BAO. The odel we suggested can include the general relativity and it can exaine the scalar-atter coupling if any. In this paper we applied a new paraeterization for the function z including a z F linear ter in contrast with []. This akes it possible to estiate the tie-varying rate of

12 gravitational constant according to (33): Ġ/ z= =.86 yr. This constraint is weaker than the estiation in [3]. On the other hand, as regards the scalar-atter coupling, Table. shows a positive coupling paraeter.. uo et al. has given rather negative value. 3 [4]. The contrary results between ours and [4] see to ste fro the difference of the odels and the utilized observation datasets. Also, we recalculated and gained the tie-varying rate of gravitational constant with w φ = : Ġ/ z= =.43 yr, which is ore closer to the range Ġ/ 9 3 yr in [8], but it is a bit greater. In a future publication we wish to include the CMB anisotropy data and the cobined analysis of SNIa, CMB and BAO datasets is believed to give ore conclusive results. References [] Nesseris, S., Perivolaropoulos, L., Phys. Rev. D73, 35( 6) [] Riess, A.. et al., Astrophys. J. 67, 665( 4) [3] Majerotto, E., Sapone, D., Aendola, L., arxiv:astro-ph/4543 [4] uo, Z., Ohta, N., Tsujikawa, S., Phys. Rev, D76, 358( 7) [5] Bergan, P.. 968, Int. J. Theor. Phys., 5(968) [6] Nordvedt, K., Astrophys. J. 6, 59( 97) [7] Wagoner, R. Phys. Rev, D, 39(97) [8] Kowalski, M. et al., Astrophys. J. 686, 749(8) [9] Aendola, L., Capos,. C., Rosenfeld, R., Phys. Rev. D75, 8356( 7) [] Tiothy Clifton et al., Phys. Revt. 53() [] Webb, J. K. et al., Phys. Rev. Lett. 87, 93( ) [] Casas, J. A., arcia-bellido, J., Quiros, M., Classical Quantu ravity, 9, 37( 99) [3] Aendola, L., Phys. Rev. D6, 435( ) [4] Esposito-Farese., Polarsky, D., Phys. Rev. D63, 6354( ) [5] Daour T., Nordvedt, K., Phys. Rev. D48, 3436 ( 993) [6] Huey,., Wandelt, B.D., Phys. Rev. D74, 359( 6) [7] Aendola, L., Quartin, M., Tsujikawa, S., Waga, I., Phys. Rev. D74, 355( 6) [8] Riess, A.. et al., astro-ph/657( 6)

13 [9] Astier, P. et al., Astron. Astrophys. 447, 3(6) [] Eisenstein, D. J. et al., Astrophys. J. 633, 56(5) [] Benett, C.L. et al.,astrophys. J. Suppl., 8, (3) [] Hinshaw,. et al., Astrophys. J. Suppl., 8, 9(3) [3] Willias, J.., Turyshev, S.., Boggs, D. H., Phys. Rev. Lett. 93, 6(4) 3