Constraining Palatini cosmological models using GRB data
aa r X i v : . [ g r- q c ] M a r Constraining Palatini cosmological models using GRB data.
Michał Kamionka
Astronomical Institute, University of Wrocławul. Kopernika 11, 51-622 Wrocław, Poland.e-mail: [email protected]
Abstract.
New constraints on previously investigated Palatini cosmological models [1] have been obtained by adding GammaRay Burst (GRB) data [2].
Keywords: modified gravity, cosmological simulations, dark energy theory, cosmic singularity
PACS:
COSMOLOGY FROM THEGENERALIZED EINSTEIN EQUATIONS
Recently, we have investigated cosmological applica-tions and confronted them against astrophysical data thefollowing class of gravitational Lagrangians: L = √ g ( f ( R ) + F ( R ) L d ) + L mat ≡≡ √ g (cid:16) R + a R + b R + d + R + s L d (cid:17) + L mat (1) within the first-order Palatini formalism [1]. Here L d = − g mn ¶ m f¶ n f is a scalar (dilaton-like) field Lagrangiannon-minimally coupled to the curvature and L mat repre-sents perfect fluid Lagrangian for a dust (non-relativistic)matter. The numerical parameters a , b , d , s are to be de-termined by astrophysical data.Applying (Palatini) variational principle compiledwith flat FLRW metric one arrives to general Friedmannequation: H = ( f ′ + F ′ L d )[ f − f ′ R + ( F − F ′ R ) L d ] h f ′ − F ′ L d + [ f − f ′ R +( F ′ R − F ) L d ][ f ′′ +( F ′′ − F − ( F ′ ) ) L d ] f ′′ R − f ′ +[ F ′′ R + F ′ − F − ( F ′ ) R ] L d i (2) where H = ˙ aa denotes the Hubble parameter related to theFLWR cosmic scale factor. This reconstructs the L CDMmodel under the choice f = R − L , F =
0, which is thelimit a = d = − b = L . Setting further L = H = G ( a ) (3)(which is always the case for the Palatini formalism)leads to one-dimensional particle like Newton-type dy-namics which is fully described by the effective poten-tial V ( a ) = − a G ( a ) . This relevant property allows usto compare various cosmological models on the levelat the effective potential functions and the correspond-ing phase-space diagrams. Particularly, the dynamics of L CDM model is described by V L CDM = − ( L a + h a − ) where h is a density parameter for the dust mat-ter.As it was shown in [1] the equation (2) leads to twoclasses of cosmological models implemented by differ-ent solutions of generalized Einstein equations. Model I
Solving equations of motion by R = r = h a − , s = − d (4) one obtains generalized Friedmann equation under theform (cid:18) HH (cid:19) = + W , a ( + z ) − − dd W , b ( + z ) d h − W , a ( + z ) − ( − d )( − d ) d W , b ( + z ) d i × (5) × (cid:20) W , m ( + z ) + W , a W , m ( + z ) − − dd W , b W , m ( + z ) ( d + ) (cid:21) where W , m = h H , W , b = bh d , W , a = ah (6) are dimensionless (density like) parameters. Model II
Another cosmological model can be determined by R = (cid:20) h ( − d ) b (cid:21) + d a − + d , s = d (7) which leads to (cid:18) HH (cid:19) = + dd + W , a ( + z ) + d + + d − d W , m W − , b ( + z ) d + d (cid:20) + dd + d − + d W , a ( + z ) + d + − d − d W , m W − , b ( + z ) d + d (cid:21) (8) × (cid:20) + dd W , b ( + z ) + d + W , a W , b ( + z ) + d + − d − d W , m ( + z ) (cid:21) here now W , m = h H , W , b = H (cid:20) h ( − d ) b (cid:21) + d , W , a = a H W , b (9) Both models have W , m , W , a , W , b , d as free param-eters. By the normalization condition H ( ) = H , onlythree of them are independent ( H denotes the Hubbleconstant). FITTING PARAMETERS OF THEMODELS
In order to estimate the parameters of our models weuse a sample of N =
557 supernovae (SNIa) data [3],the observational H ( z ) data [4], the measurements of thebaryon acoustic oscillations (BAO) from the SDSS lumi-nous red galaxies [5], information from CMB [6] and, asan adition to [1], information coming from observationsof GRB [2].The entire likelihood function L TOT is characterizedby: L TOT = L SN L H z L BAO L CMB L GRB . (10)We have assumed flat prior probabilities forall model’s parameters. We also assumed that H = . [ kms − M pc − ] [8].The likelihood function is defined in the followingway: L SN (cid:181) exp " − (cid:229) i ( m theor i − m obs i ) s i , (11) where: s i is the total measurement error, m obsi = m i − M is the measured value ( m i –apparent magnitude, M –absolute magnitude of SNIa), m theori = D Li + M = d Li + M = − H +
25 and D Li = H d Li ,where d Li is the luminosity distance given by d Li = ( + z i ) c R z i dz ′ H ( z ′ ) (with the assumption k = H has beenused (which is obtained after analytical marginalizationof formula (11) over H ).For the H ( z ) data the likelihood function is given by: L H z (cid:181) exp " − (cid:229) i ( H ( z i ) − H i ) s i , (12) where H ( z i ) is the Hubble function, H i denotes observa-tional data.For BAO A parameter data the likelihood function ischaracterized by: L BAO (cid:181) exp (cid:20) − ( A theor − A obs ) s A (cid:21) , (13) where A theor = p W m , (cid:16) H ( z A ) H (cid:17) − h z A R z A H H ( z ) dz i and A obs = . ± .
017 for z A = . R shift parameter [7], which is related to the angular diameter distance( D A ( z ∗ ) ) to the last scattering surface: R = √ W m H c ( + z ∗ ) D A ( z ∗ ) . (14) The likelihood function has the following form: L CMB (cid:181) exp (cid:20) − ( R − R obs ) s A (cid:21) , (15) where R obs = .
725 and s − A = .
27 for z ∗ = . L GRB (cid:181) exp " − (cid:229) i (cid:20) m i − m th ( z i , W m , W L , ) s m i (cid:21) (16) The mode of joined posterior pdf as well as mean (to-gether with 68% credible interval) of marginalized pos-terior pdf were calculated, by means of Markov ChainsMonte Carlo analysis, using free accessible CosmoNestcode [9] which has been modified for our purpose. Theresults are presented on fig. 2,3.The numerical values of best fitted parameters fortwo our models as well as for L CDM are collected intable 1: the previous estimations without the GRB data(i.e. SNIa, H(z) and BAO and CMB) are shown in toppart of the table. The new estimations including the GRBdata occupy bottom part of the table.Quality of the estimation can be visualized on theHubble’s diagram (fig. 1). Both of our models are in goodagreement in the observational data.
34 36 38 40 42 44 46 48 0.01 0.1 1Union2 dataGRB datamodel Imodel II L CDM model
FIGURE 1.
Comparison of Hubble’s diagrams for models: I(blue), II (magenta) and L CDM (black).
CONCLUSIONS
In this paper we continued and completed analysis ofnew cosmological models which were previously de-scribed and investigated in our paper [1]. Adding GRBdata [2] allowed us to obtain better constraints of param-eter W a which wasn’t present previously.As it can be seen on the potential plots (fig. 4,5, bothmodels dynamically mimics L CDM model from the Big
50 0 50 −10 0 10 0 1 2 0 0.5 1 W m h Wb h −50 0 50−10010 d −50 0 50012 W a h W m h −50 0 5000.51 −10 0 10012 W b h −10 0 1000.51 d FIGURE 2.
Constraints of the parameters of model I . In 2Dplots solid lines are the 68% and 95% confidence intervalsfrom the marginalized probabilities. The colors describe themean likelihood of the sample. In 1D plots solid lines denotemarginalized probabilities of the sample, dotted lines are meanlikelihood. For numerical results see Table 1. −50 0 50 100 −2 0 2 0 0.5 1 0 0.5 1 W m h d −50 0 50 100−202 Wb h −50 0 50 10000.51 W a h W m h −50 0 50 10000.51 −2 0 200.51 d −2 0 200.51 W b h FIGURE 3.
Constraints of the parameters of model II . Themeaning of the colors and the lines this same as in the picture2. For numerical results see Table 1. Bang singularity until the present time. Discrepancieswill appear in the near future. Both of our models predictthe final finite size and finite time singularities (at a = .
673 for the model I, and at a = .
559 for the modelII). However, comparing with our previous simulations,adding new GRB data has changed properties of themodel II (Big Bounce is now replaced by Big Bang). - - - - @ a D V LCDM H a L V H a L FIGURE 4.
The diagram of the effective potential inparticle–like representation of cosmic dynamics for model Iversus L CDM model. Note that till the present epoch twopotential plots almost coincide. Particulary, one can observedecelerating BB era. Maximum of the potential function cor-responds to Einstein’s unstable static solution. Discrepanciesbecome important in the future time: e.g. discontinuities of thepotential functions (vertical, red line) denote that V → − ¥ , i.e.˙ a → ¥ for a → a final . It turns out to be finite–time (sudden)singularity. In any case the shadowed region below the graph isforbidden for the motion. - - - - @ a D V LCDM H a L V H a L FIGURE 5.
The diagram of the effective potential in particlelike representation of cosmic dynamics for the model II a = versus L CDM model. Maximum of the potential function cor-responds to unstable static solution. Again, until the presentepoch there is no striking differences between plots. One canobserve finite–size sudden singularity in the near future (ver-tical, red line). In any case the shadowed region below thepotential is forbidden for the motion.
ACKNOWLEDGMENTS
M.K. is supported by the Polish NCN grant PRE-LUDIUM 2012/05/N/ST9/03857.
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ABLE 1.
The values of estimated parameters (mean of the marginalized pos-terior probabilities and 68% credible intervals or sample square roots of variance,together with mode of the joined posterior probabilities, shown in brackets) fortwo investigated models. We compare estimations without GRB data (top part ofthe table) with the one employing GRB data (bottom part). models I - the parameters estimated without GRB data W , a W , b d W , m − . + . − . ( − . ) . + . − . ( . ) . + . − . ( . ) . ± . ( . ) models II - the parameters estimated without GRB data W , a W , c d W , b W , m − . + . − . ( − . ) . + . − . ( . ) . + . − . ( . ) . ± . ( . ) . + . − . ( . ) model L CDM - the parameter estimated without GRB data . + . − . ( . ) models I - the parameters estimated using GRB data W , a W , b d W , m . + . − . ( . ) . + . − . ( . ) . + . − . ( . ) . + . − . ( . ) models II - the parameters estimated using GRB data W , a W , c d W , b W , m . + . − . ( . ) . + . − . ( . ) . + . − . ( . ) . + . − . ( . ) . + . − . ( . ) model L CDM - the parameter estimated using GRB data . + . − . ( . ) indicators similar to Type Ia Supernovae?”,(2012), arXiv:1205.2954 .3. R. Amanullah et al. , “Spectra and Light Curves ofSix Type Ia Supernovae at 0.511 < z < 1.12 and theUnion2 Compilation”, Astrophys. J. , 712 (2010), arXiv:1004.1711 .4. J. Simon, L. Verde, R. Jimenez, “Constraints on the redshiftdependence of the dark energy potential”, Phys. Rev. D71 ,123001 (2005), astro-ph/0412269 .5. D. J. Eisenstein et al. , “Detection of the baryon acousticpeak in the large-scale correlation function of SDSSluminous red galaxies”, Astrophys. J. , 560-574 (2005), astro-ph/0501171 ;W. J. Percival, S. Cole, D. J. Eisenstein, R. C. Nichol,J. A. Peacock, A. C. Pope, A. S. Szalay, “Measuring theBaryon Acoustic Oscillation scale using the SDSS and2dFGRS”, Mon. Not. Roy. Astron. Soc. , 1053-1066(2007), arXiv:0705.3323 ;B. A. Reid et al. , “Baryon Acoustic Oscillations in theSloan Digital Sky Survey Data Release 7 Galaxy Sample”,Mon. Not. Roy. Astron. Soc. , 2148-2168 (2010), arXiv:0907.1660 .6. E. Komatsu et al. , “Seven-Year Wilkinson MicrowaveAnisotropy Probe (WMAP) Observations: CosmologicalInterpretation”, Astrophys. J. Suppl. , 18 (2011), arXiv:1001.4538 .7. J. R. Bond, G. Efstathiou, M. Tegmark, “Forecastingcosmic parameter errors from microwave background anisotropy experiments”, Mon. Not. Roy. Astron. Soc. ,L33-L41 (1997), astro-ph/9702100 .8. A. G. Riess et al. , “A Redetermination of the HubbleConstant with the Hubble Space Telescope from aDifferential Distance Ladder”, Astrophys. J. , 539(2009), arXiv:0905.0695 .9. P. Mukherjee, D. Parkinson, A. R. Liddle, “Anested sampling algorithm for cosmological modelselection”, Astrophys. J. , L51-L54 (2006), astro-ph/0508461 ;P. Mukherjee, D. Parkinson, P. S. Corasaniti, A. R. Liddle,M. Kunz, “Model selection as a science driver for darkenergy surveys”, Mon. Not. Roy. Astron. Soc. , 1725-1734 (2006), astro-ph/0512484 ;D. Parkinson, P. Mukherjee, A.R. Liddle, “A Bayesianmodel selection analysis of WMAP3”, Phys. Rev.
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