Testing f (R) theories using the first time derivative of the orbital period of the binary pulsars

Transcription

1 Testing f (R) theories using the first time derivative of the orbital period of the binary pulsars Mariafelicia De Laurentis in collaboration with Ivan De Martino TEONGRAV- Meeting 4-5 February 2014, Roma

2 Outilnes f(r)-gravity background Radiated Energy Application to pulsar binary systems Comparing theory prediction with data Discussion and remarks

3 f(r)-gravity background Starting from the following field equations in f (R) gravity The Minkowskian limit can be calculated for a class of analytic f (R )-Lagrangian At the first order, in terms of the perturbations, the field equations become Here, the Ricci tensor and scalar read Capozziello S., De Laurentis M., 2011, Phys. Rep., 509, 167 Capozziello S., De Laurentis M., Faraoni V., 2009, Open Astron. J., 2, 1874

4 f(r)-gravity background By choosing a suitable transformation and a gauge condition One obtains that field equations and the trace equation solving this system of equations, we find wavelike solutions with massless and massive contributions A similar situation emerges also in the Newtonian limit: the Newton potential is recovered only as the weak field limit of GR. In general, Yukawa-like corrections, and then characteristic interaction lengths, are present The effective mass is and then f 0 has to be negative in order to have physically defined states. C. Bogdanos, S. Capozziello, M. De Laurentis, S. Nesseris, Astrop. Phys. 34, 236 (2010) S. Capozziello, C. Corda M. De Laurentis, Phys. Lett. B , (2008)

5 f(r)-gravity background Assuming that the source is localized in a finite region, as a consequence, outside this region the energy momentum tensor of gravitational field in f (R) gravity the energy momentum tensor consists of a sum of a GR contribution plus a term coming from f (R) gravity: which in terms of the perturbation h is the energy momentum tensor assumes the following form: De Laurentis M., Capozziello S., 2011, Astropart. Phys., 35, 257

6 Radiated Energy In order to calculate the radiated energy of a GW source suppose that h μν can be represented by a discrete spectral representation. The instantaneous flux of energy is given by Defining the following momenta of the mass energy distribution: and analysing the radiation in terms of multipoles, found the total average flux of energy due to the tensor wave Precisely, for f 0 = 0 and f 0 = 4/ 3

7 Application to pulsar binary systems Our goal is to use a sample of binary pulsar systems to fix bounds on f (R) parameters. We assume that the motion is Keplerian and the orbit is in the (x, y) plane the quadrupole matrix is whit the time derivatives of the quadrupole: where

8 Application to pulsar binary systems we can perform the time average of the radiated power by writing and finally, we get the first time derivative of the orbital period: we will go on to constrain the f (R) theories estimating f 0 from the comparison between the theoretical predictions of dt b and the observed one.

9 Comparing theory prediction with data

10 Comparing theory prediction with data We make an estimation of the second derivative f 0 imposing the strong hypothesis that the difference between the observed binary period variation and the one obtained by the relativistic theory of gravitation is fully justified by imposing that We also have subtracted the external contributions to the orbital decay as galactic or Shklovskii acceleration when those values are available in the literature Filnally we get an estimation of f 0 and its upper and lower limits corresponding, respectively, to δ. So we get where M. De Laurentis, I. De Martino, MNRAS 431, (2013), 741D.

11 Comparing theory prediction with data

12 Results for J C

13 Results for J

14 Numerical Results

15 PSR J It is a binary system composed by a pulsar spinning at 39 ms with mass 2.01 ± 0.04 M, and a White Dwarf companion with mass ± M. The orbital period of the system is P b =0.102 (days), and the eccentricity e= the observed value of the first time derivative of the orbital period is M. De Laurentis, I. De Martino, arxiv: , 2013

16 Discussion and remarks We have seen that where the GR theory is not enough to explain the gap between the data and the theoretical estimation of the orbital decay, there is the possibility of extending the GR theory with a generic f (R) theory to cover the gap Here, we simply verify that this possibility exists, but there is need to compute the post-keplerian parameters in the f (R)-theory to estimate correctly the masses of the binary systems to constrain correctly the analytic parameters of the f(r)-gravity We have seen that the first five systems have masses determined in a manner quite reliable, while for the remaining sample, masses are estimated by requiring that the mass of the pulsar is 1.4M and assuming for the orbital inclination one of the usual statistical values (i=60 or i=90 ), and from here then comes the estimate of the mass of the companion star. So a primary cause of major discrepancies, not only for the f(r)-gravity, but also for the GR theory, between the variation of the observed orbital period and the predicted effect of emission of GWs could be a mistake in the estimation of the masses of the system

17 Conclusions Here, we gave a preliminary result about the energy loss from binary systems, and we showed that, when the nature of the binary systems can exclude energy losses due to trade or loss of matter, we can explain the gap between the first time derivative of the observed orbital period and the theoretical one predicted by GR, using an analytical f (R)-theory of gravity In conclusion, to improve the estimation of the f (R) coefficients, we need to consider the hydrodynamic effects due to the transfer of the matter in the binary system, in order to analyse different systems from double NS; and to improve the estimations of the mass of the stars in the binary systems without prior on pulsar mass and orbital inclination.