Dipole of the luminosity distance as a test for dark energy models
OO. SergijenkoAstronomical Observatory ofTaras Shevchenko National University of Kyiv
Dipole of the luminosity distance as a test for dark energy models
The dependence of Hubble parameter on redshift can be determined directly from the dipole of luminosity distance to Supernovae Ia. Weinvestigate the possibility of using the data on dipole of the luminosity distance obtained from the Supernovae Ia compilations SDSS,Union2.1, JLA and Pantheon to distinguish the dark energy models.Key words: dark energy, Supernovae Ia, cosmological parameters
Introduction.
According to the Planck data the dark energy in current epoch is close to the cosmological constant. This makes it muchharder to detect the temporal variation of the dark energy equation of state parameter and to distinguish the dark energymodels. The reliability of w(z) determination will be significantly higher if the dependence
H(z) is measured directly insteadof (or along with) the luminosity or angular diameter distances since taking numerical derivatives from the currentobservational data leads to the inaccurate results. Recent interest to testing the isotropy of the Supernovae Ia magnitude-redshift relation [1,2] brings attention to the dipole of luminosity distance as a possible direct measure of the Hubbleparameter [3]. In this paper we make an attempt to re-estimate the potential of the luminosity distance dipole to discriminatethe dark energy models using current data on Supernovae Ia.
Dipole of the luminosity distance.
According to [3,4] the first-order expansion of directional dependence of the luminosity distance reads:,where the monopole isand the dipole is .The variance of dipoleleads to the following estimate for precision of the
H(z) determination [3]: .
Model and data.
As the dark energy model we use the minimally coupled classical scalar field with the equation of state parameter obtainedfrom the condition [5]: .We investigate 2 cases: distinguishing between the best-fit quintessential and phantom models with the parametersobtained from the same dataset (as it was done in [5,6]) and distinguishing the mean model from the model with allparameters at 1σ or 2σ confidence limits (in the manner of [7]).The cosmological parameters and their confidence ranges are obtained by the Monte Carlo Markov chain (MCMC) [8]method implemented in the CosmoMC code (http://cosmologist.info/cosmomc). We assume the spatial flatness of theUniverse.In the first case we use the bets-fit parameters (Table 1) estimated in [5] from the following datasets: CMB temperature fluctuations and polarization angular power spectra from the 7-year WMAP data (WMAP7)[9-11]; Baryon acoustic oscillations from SDSS DR7 (BAO) [12]; Hubble constant measurements from HST (HST) [13]; Big Bang Nucleosynthesis prior on the baryon abundance (BBN) [14,15]; Supernovae Ia from SDSS compilation (SN SDSS) [16] (SALT2 [17] and MLCS2k2 [18] light curve fittings).In the second case we determine the mean values of cosmological parameters and their confidence ranges (Table 2) fromthe combined dataset including: CMB TT, TE, EE angular power spectra and lensing from Planck [19]; B-mode polarization for 2 frequency channels from BICEP2/Keck Array (BK) [20]; power spectrum of galaxies from WiggleZ Dark Energy Survey [21]; Supernovae Ia from JLA compilation [22];
Hubble constant determination [23]. Here we apply flat priors with ranges of values [-2,-0.33] for w and [-2,0] for c a2 , so the dark energy model involves bothquintessence and phantom subclasses.Table 1. The best-fit values of cosmological parameters for the models with quintessential (QSF) and phantom (PSF) scalarfields determined from 2 observational datasets: WMAP7+HST+BBN+BAO+SN SDSS SALT2 ( q , p ) andWMAP7+HST+BBN+BAO+SN SDSS MLCS2k2 ( q , p ) (from [5]).Table 2. The mean values, 1σ and 2σ confidence limits for cosmological parameters obtained from the observationaldataset Planck2015+WiggleZ+SN JLA+BK.For estimates based on the luminosity distance dipole we use the following Supernovae Ia compilations: SDSS [16]: 288 SNe (MLCS2k2, SALT2 light curve fitters): only statistical uncertainties; Union2.1 [24]: 580 SNe (SALT2): both statistical and systematic uncertainties; JLA [22]: 740 SNe (SALT2): both statistical and systematic uncertainties; Pantheon [25]: 1048 SNe (SALT2): both statistical and systematic uncertainties.We assume v =369.0 km/s (from the CMB dipole which is due to the same motion)[26].Fig. 1. Left: the theoretical relative differences ΔH model (z)/H (z)≡|H phant (z)-H quint (z)|/H quint2 (z) compared to ΔH(z)/H (z) fromSupernovae compilations. Right: the minimal number of Supernovae that is necessary for distinguishing the models in leftpanel if the uncertainties of Supernovae magnitudes are the same as in the compilation from legend. After a comma wequote the data (type of Supernovae light curve fitting) used to estimate the best-fit parameters for the pair of comparedmodels. Results and discussion.
In the left panels of Fig. 1-4 we present the calculated quantities ΔH model (z)/H (z) and compare them with the correspondingquantities ΔH(z)/H (z) obtained from the luminosity distance dipole using the data from Supernovae compilations in 16redshift bins with the width 0.1 (0< z <1.6). In the right panels we estimate the number of Supernovae that is needed todistinguish between the models from the left panels.From Fig. 1-4 it is clear that distinguishing between the best-fit models and between the mean model and the model with allparameters at 1σ limits is not realistic at all. The number of Supernovae necessary to distinguish the model with meanparameters from the model with all parameters at the limits of their 2σ confidence ranges is minimal in the first redshift bin(0< z <0.1). For SN SDSS with MLCS2k2 fitting it is 1998 or 4063, for SN SDSS with SALT2 fitting 2735 or 5563, for SNUnion2.1 4466 or 9083, for SN JLA 5411 or 11006, for SN Pantheon C11 3074 or 6252 and for SN Pantheon G10 3040 or6183 for the upper or lower limits correspondingly. For higher redshift bins the needed numbers of Supernovae are larger atleast by one order of magnitude.ig. 2. Left: the theoretical relative differences ΔH model (z)/H (z)≡|H (z)-H mean (z)|/H mean2 (z) (for upper and lower limits)compared to ΔH(z)/H (z) from Supernovae compilations. Right: the minimal number of Supernovae that is necessary fordistinguishing the models in left panel if the uncertainties of Supernovae magnitudes are the same as in the compilationfrom legend.Fig. 3. Left: the theoretical relative differences ΔH model (z)/H (z)≡|H phant (z)-H quint (z)|/H quint2 (z) compared to ΔH(z)/H (z) fromSupernovae compilations. Right: the minimal number of Supernovae that is necessary for distinguishing the models in leftpanel if the uncertainties of Supernovae magnitudes are the same as in the compilation from legend. After a comma wequote the data (type of Supernovae light curve fitting) used to estimate the best-fit parameters for the pair of comparedmodels.Fig. 4. Left: the theoretical relative differences ΔH model (z)/H (z)≡|H (z)-H mean (z)|/H mean2 (z) (for upper and lower limits)compared to ΔH(z)/H (z) from Supernovae compilations. Right: the minimal number of Supernovae that is necessary fordistinguishing the models in left panel if the uncertainties of Supernovae magnitudes are the same as in the compilationfrom legend. Conclusion.
We have found that despite the major increase in number of Supernovae in available compilations over the last 12 yearsthe current prospects of using the dipole of luminosity distance for distinguishing the dark energy models are not bright. Thisis partly due to the fact that the uncertainties in determination of the cosmological parameters from other data are nowmuch smaller and the tests for dark energy equation of state parameter should be more precise. Another reason is that nowtaken into account systematic errors result in the larger total ones. So, to make the luminosity distance dipole useful as thecosmological test it is necessary not only to increase largely the number of Supernovae (especially the low-redshift ones) ina dataset, but also to reduce the uncertainties of distance moduli by improving the light curve fitting and to control betterthe systematics. cknowledgements
This work has been supported in part by the Department of target training of Taras Shevchenko National University of Kyivunder National Academy of Sciences of Ukraine (project 6Ф). Author acknowledges the usage of CosmoMC package.
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