Strong-field tests of f(R)-gravity in binary pulsars

We develop the parameterized post-Keplerian approach for class of analytic f(R)-gravity models. Using the double binary pulsar system PSR J0737-3039 data we obtain restrictions on the parameters of this class of f(R)-models and show that f(R)-gravity is not ruled out by the observations in strong field regime. The additional and more strong corresponding restriction is extracted from solar system data.


Introduction
General relativity (GR) is a very beautiful theory which allows to go beyond the Newtonian picture of the world and explains many unaccounted phenomena. However our understanding of fundamental laws still has several shortcomings. The accelerated expansion of the Universe (i.e., dark energy) has been found from cosmological observations recently 1. Moreover already in 1930s the problem of galactic rotation curves arose 2 . One way to unriddle these puzzles is to add yet unknown particles and look for them on LHC 3 and in cosmic rays. Another way is to expand GR by including additional corrections in terms of the Ricci scalar in the Lagrangian. This method underlies l(R)-gravity 4•5 .
2 l(R)-gravity f(R)-gravity is actually a family of theories, each of them is defined by a different function of the Ricci scalar. In the simplest case the function equals to the scalar; that is GR. We can explain dark matter, dark energy and inflation 4 by different models of I (R)-gravity. The action of I (R)-gravity has the following form 6•7: (1) where K-= l67rG/c4 is the coupling coefficient, g is the determinant of the metric tensor, L m is the standard matter Lagrangian, I (R) is an analytical function of the general form. This function can be expanded in a series in terms of the Ricci scalar 6: where l o= canst, lb = dl (R) I , The flat Minkowskian background is recovered for R = Ro c:= 0. GR is recovered in the limit Jo = 0, !6 = 4/3, f t = 0 7 . Hereafter we assume Jo = 0, !6 = 4/3 whereas f t is a fr ee parameter. Our purpose is to restrict the possible value of this free parameter f t .
However any theory of gravity should be verifiable. Naturally, there are many other ways for testing theories of gravity but in this work we applied only PPN and PPK formalisms to J(R)-gravity. Each parameter is responsible for its effect . However, the considered f(R) gravity model is the conservative theory and, in this case, only two parameters ('Y, (3 ) are not equal to zero (see table 1) 10.
Drewing an analogy between the scalar-tensor gravity and the higher order theories of gravity, Capozziello and Troisi 6 developed the PPN formalism for f(R)-gravity. The similarity between the non-minimally coupled scalar models (Lagrangian of Brans-Dicke type 11•12 ) and the models of gravity with higher order curvature corrections have been discussed since 1983 13 . Basing on this similarity Capozziello and Troisi 6 obtained the Eddington's parameters for . f(R)-gravity in analytical form: where f (R) is an arbitrary function of R. Using the expansion (2), we carried out the Eddington's parameters for the considered model of f(R)-gravity: Using the fact that f (R)-gravity recovers GR at f6 = 4/3 7 and the observational values of parameters 'Y PPN and (3 PPN 10 (see. table 1), we can impose restrictions on the value of f t by solving the system of equations (5):   we considered only those parameters that have the most accurate measurements, so we didn't take into account the last three of them.
It should be noted that different theories of gravity can give different predictions for PPK parameters. We should compare predictions of the theory and the values of these parameters obtained from observations. Thus we have powerful instrument for testing extended gravity models in the strong field limit 9•1 6.  x ( 89le8 + 28016e6 + 82736e 4 + 43520e 2 + 3072) ) .

(7)
These parameters depend only on the orbit eccentricity, projection of the semi-major axis of the pulsar orbit, orbital period, masses of the pulsar and its companion and also the parameter fff of the f(R) gravity model. All of them, except the parameter and masses of the model,  can be obtained from observations. In our work we used the data for binary pulsar J0737-3039 which was presented in the article by Kramer and his colleagues17. It is the only known double binary pulsar. It is the smallest period that the known systems of this type may have. The extraordinary closeness of system components, small orbital period and also the fact that we see almost edge-on system allow to investigate the manifestation of relativistic effects with the highest precision. Also it is possible to measure semi-major axis of the orbit for each of components of the system J0737-3039 and hence their ratio is equals: a 2 = m 2 = R , a 1 m 1 i.e. the ratio of the masses can be measured directly!  fig. 1, fig. 2, fig. 3). That is the limitation that we receive for this parameter from the binary pulsar data: -0.05772 :::: Jg :::: 0.  In this work we impose restrictions on the considered model of f (R)-gravity from the observations in the strong and weak field limits. For our aims we used the data of double bynary pulsar system and accurate measurements of the PPN parameters in the Solar System, respectively. We show that the observational data of double pulsar system give the following limit on a value of parameter f g: -0.05772 :::; !3 :::; 0. (10) This parameter characterizes the contribution of the quadratic curvature correction in the action of f(R)-gravity. It is important to note that the obtained restriction on the possible values of f3 is small but at the same time it can not be considered negligible even within the measurement accuracy. This result allows the realization of GR as well as its extensions, including quadratic curvature corrections.
At the same time it is possible to receive the limitations on the value Jg from Eddington parameters measurements in the Solar system. The parameter I PPN gives a better limit than the parameter (3 PPN: / PPN : -0.0055 :::; f3 :::; 0, (3 PPN : -7 :::; f3 :::; 0, Thus, more strict limitation on the model parameters follows from the experiments in the solar system than from the data of bynary pulsar systems. On the one hand, it can be connected with the fact that measurement accuracy in the Solar system is much better than in the systems with the pulsar. On the other hand, in a system with a compact object gravity is much stronger (2GM/(c 2 R) P SR � 0.2), than in the solar system (2GM/(c 2 R)suN � 10-6), therefore, the contribution of corrections type R 2 should be more prominent.
Since J(R)-gravity is one of the ways to describe dark energy and dark matter, then obtaining the experimental constraints on the parameters of such models is an important step in solving these fundamental problems.