COSMOLOGICAL TESTS FOR PALATINI f(R) THEORY

JANILO SANTOS (UFRN) JAILSON S. ALCANIZ (ON) FÁBIO C. CARVALHO (INPE)

THE ACTION IN f(R) GRAVITY

VARYING THE ACTION WITH RESPECT TO THE METRIC WE OBTAIN THE FIELD EQUATIONS:

IN PALATINI FORMALISM THE METRIC AND THE CONECTIONS ARE TREATED AS INDEPENDENT VARIABLES WITH RESPECT TO WICH THE GENERALIZED ACTION IS VARIED. THE EQUATIONS OF MOTION ARE

WHERE

ARE THE LEVI-CIVITA CONNECTIONS

WE CONSIDER MATTER FIELDS SUCH THAT

IN THIS CASE THE SECOND EQUATION REDUCES TO

FROM THIS EQUATION WE OBTAIN THE INDEPENDENT CONNECTIONS AS

THE GENERALIZED FRIEDMANN EQUATION IN THIS FORMULATION IS GIVEN BY

CONSTRAINED BY(TRACE CONSTRAINT)

( WE ARE CONSIDERING ϸᵐ = 0 )

WE CONSIDER THE FUNCTION

WE TEST THIS MODEL USING THE HUBBLE PARAMETER DETERMINATIONS AT DIFFERENT REDSHIFTS OBTAINED FROM DIFFERENTIAL AGE TECHNIQUES: J. SIMON et al. [ PRD 71, 123001 (2005); R. JIMENEZ and A. LOEB, ApJ 573, 37 (2002)].

COSMOLOGICAL TESTS

EVALUATED AT Z = 0 THE TRACE CONSTRAINT GIVES

1) CONSTRAINTS FROM HUBBLE PARAMETER

USING H(Z) DETERMINATIONS ONLY H(Z) + BAO + CMB

FOR THE JOINT ANALYSIS ( H(Z) + BAO + CMB ) :

BEST FIT VALUES :

EFFECTIVE EQUATION OF STATE :

FOR MORE DETAILS SEE : “COSMOLOGICAL CONSTRAINTS FROM HUBBLE PARAMETER ON f (R ) COSMOLOGIES “F.C. CARVALHO, E.M. SANTOS, J.S. ALCANIZ, J. SANTOSJCAP 09 (2008) 008

2) CONSTRAINTS FROM SUPERNOVAE

WE ALSO TEST THIS MODEL WITH THE 307 SUPERNOVAE FROM THE “UNION SAMPLE” (M. KOWALSKI et al., arXiv: 0804.4142 [astro-ph] )

THE PREDICTED DISTANCE MODULUS FOR A SUPERNOVA AT REDSHIFT Z, IS

WHERE m AND M ARE THE APPARENT AND ABSOLUTE MAGNITUDES, AND

( LUMINOSITY DISTANCE )

CONFIDENCE INTERVALS AT 68.3%, 95.4%, 99.73%

FOR THE JOINT ANALYSIS ( SNe Ia + BAO + CMB ):

BEST FIT VALUES:

EFFECTIVE EQUATION OF STATE:

HUBBLE DIAGRAM FOR THE 307 SNe Ia FROM THE “UNION SAMPLE”

[ 6 ] M. AMARZGUIOUI et al., ( A & A 454, (2006) 707 )[ 7 ] S. FAY & R. TAVAKOL, ( PRD 75, (2007) 063509 )[ 10 ] F.C. CARVALHO et al., (JCAP 0809 (2008) 008 )[ 11 ] T. KOIVISTO, ( PRD 76 (2007) 043527 )[ This Letter ] J. SANTOS et al., ( PLB 669 (2008) 14 )

FOR MORE DETAILS SEE: “LATEST SUPERNOVAE CONSTRAINTS ON f (R) COSMOLOGIES”J.SANTOS, J.S. ALCANIZ, F.C. CARVALHO, N. PIRESPHYS. LETT. B 669 (2008) 14

CONCLUSIONS

I ) DIFFERENTIAL AGE METHOD DETERMINATIONS OF H(Z), WHEN COMBINED WITH BAO AND CMB, LEAD TO CONSTRAINTS ON f(R) COMPETITIVE WITH THOSE ACHIEVED WITH SNe Ia ;

II ) THE BEST FIT VALUE FOR THE DENSITY PARAMETER ( Ωо = 0.26 ) IS CONSISTENT WITH CURRENT ESTIMATES OF THECONTRIBUTION OF NON-RELATIVISTIC MATTER.

III ) THE UNIVERSE CORRESPONDING TO THE BEST FIT SOLUTION

SHOWS ALL THREE LAST PHASES OF THE COSMOLOGICALEVOLUTION: RADIATION ERA, MATTER AND A LATE TIME COSMIC ACCELERATION WITHAOUT NEED OF DARK ENERGY.

THANK YOU

F.C. CARVALHO, E.M. SANTOS, J.S. ALCANIZ, J. SANTOS (SUBMITTED)

GENERALISED FRIEDMANN EQUATIONS

THE EASIEST WAY TO SEE HOW f (R) – GRAVITYEXPLAIN THE LATE TIME ACCELERATED EXPANSIONIS TO DEFINE THE QUANTITIES:

EXAMPLES AND APLICATIONS

Problems: Solar System Tests, Stability

COSMOLOGICAL CONSTRAINTS

182 Sne Ia (Riess et al.) + SDSS galaxy survey + CMB shift parameter

( M. Fairbairn & S. Rydbeck, JCAP 12 (2007) 005 )

Luca Amendola et al. [PLB 660, 125 (2008); PRD 75, 083504 (2007)]examined the case

and find it cosmologically viable. However, if local gravityexperiments are included they find that

thus this model is very close to the LCDM model.

ENERGY CONDITIONS in f(R) – GRAVITY (for FLRW metric with a perfect fluid)

see J. Santos, J.S. Alcaniz, F.C. Carvalho and M.J. Reboucas, Phys. Rev. D, 083513 (2007)

W. Hu & I. Sawicki [PRD 76, 064004 (2007)] proposed the model

Can be consistent with both cosmological and local gravity

constraints for n ≥ 2

A. A. Starobinsky [JETP Lett. 86, 157 (2007)] proposed the model

SPIRES (search author): S. Capozziello, S.D. Odintsov, A. Troisi, S. Tsujikawa, J.D. Barrow.