Inverse problem: Reconstruction of modified gravity action in Palatini formalism by Supernova Type Ia data

We introduce in $f(R)$ gravity--Palatini formalism the method of inverse problem to extract the action from the expansion history of the universe. First, we use an ansatz for the scale factor and apply the inverse method to derive an appropriate action for the gravity. In the second step we use the Supernova Type Ia data set from the Union sample and obtain a smoothed function for the Hubble parameter up to the redshift~1.7. We apply the smoothed Hubble parameter in the inverse approach and reconstruct the corresponding action in $f(R)$ gravity. In the next step we investigate the viability of reconstruction method, doing a Monte-Carlo simulation we generate synthetic SNIa data with the quality of union sample and show that roughly more than 1500 SNIa data is essential to reconstruct correct action. Finally with the enough SNIa data, we propose two diagnosis in order to distinguish between the $\Lambda$CDM model and an alternative theory for the acceleration of the universe.Comment: 8 pages, 8 figures, accepted in Phys. Rev.

Peculiar velocity measurement in a clumpy universe

In this work we address the issue of peculiar velocity measurement in a perturbed Friedmann universe using the deviations from measured luminosity distances of standard candles from background FRW universe. We want to show and quantify the statement that in intermediate redshifts ($0.5< z < 2$), deviations from the background FRW model are not uniquely governed by peculiar velocities. Luminosity distances are modified by gravitational lensing. We also want to indicate the importance of relativistic calculations for peculiar velocity measurement at all redshifts. For this task we discuss the relativistic correction on luminosity distance and redshift measurement and show the contribution of each of the corrections as lensing term, peculiar velocity of the source and Sachs-Wolfe effect. Then we use the SNe Ia sample of Union 2, to investigate the relativistic effects we consider. We show that, using the conventional peculiar velocity method, that ignores the lensing effect, will result in an overestimate of the measured peculiar velocities at intermediate redshifts. Here we quantify this effect. We show that at low redshifts the lensing effect is negligible compare to the effect of peculiar velocity. From the observational point of view, we show that the uncertainties on luminosity of the present SNe Ia data prevent us from precise measuring the peculiar velocities even at low redshifts ($z<0.2$).Comment: 15 pages, 5 figures, Int. J. Mod. Phys. D 27, 1850019 (2018