Probing the physical and mathematical structure of f(R) gravity by PSR J0348+0432
aa r X i v : . [ g r- q c ] N ov Probing the physical and mathematical structure of f ( R ) gravity by PSR J Mariafelicia De Laurentis ∗ Tomsk State Pedagogical University, Tomsk, ul. Kievskaya 60, 634061 Russian Federation
Ivan De Martino
Departamento de Fisica Teorica, Universidad de Salamanca, 37008 Salamanca, Spain.INFN Sezione di Napoli, Compl. Univ. di Monte S. Angelo, Edificio G, Via Cinthia, I-80126, Napoli, Italy.
There are several approaches to extend General Relativity in order to explain the phenomenarelated to the Dark Matter and Dark Energy. These theories, generally called Extended Theories ofGravity, can be tested using observations coming from relativistic binary systems as PSR J . Using a class of analytical f ( R ) -theories, one can construct the first time derivative of orbitalperiod of the binary systems starting from a quadrupolar gravitational emission. Our aim is to setboundaries on the parameters of the theory in order to understand if they are ruled out, or not, bythe observations on PSR J . Finally, we have computed an upper limit on the gravitonmass showing that agree with constraint coming from other observations. PACS numbers:Keywords: Alternative gravity; spin polarization; gravitational radiation.
I. INTRODUCTION
Einstein’s theory of gravity, known as General Relativity (GR), is a well description of gravitational phenomenaat astrophysical and cosmological scales. The theoretical predictions are in good agreement with the measurementsin particular at the Solar System scale [1]. Even if, the local gravity is very well described by GR, there are severalobservations at astrophysical and cosmological scales for which we need to add extra ingredients in the amount ofmatter and energy densities of the Universe to fit the data. Observations based on Supernovae Type Ia (SNeIa),Cosmic Microwave Background (CMB) temperature anisotropies, Baryon Acoustic Oscillations (BAO) and otherobservables, pointed out that the Universe is in a period of accelerated expansion favoring the concordance Λ CDMmodel [2–5]. In this model, we need to input an extra component, known as Dark Energy (DE), that have the effectof a fluid with a negative pressure, to explain the acceleration of the Universe at cosmological scales [6]. Furthermore,we also need to input an extra contribution in the amount of matter, known as Dark Matter (DM) to explain theclustering of the Large Scale Structure. There are observational evidences point out that the baryonic matter is notenough to explain phenomena at galaxy and cluster scales, like rotation curves, gravitational lensing, cluster profilesand others [7]. However, we do not know the nature, at particle level, of this two dark ingredients, even if there aremany explanations in literature: quintessence, string theory, holographic principle are examples of DE models, whileWeakly Interacting Particles (WIMPS), Axions, or Massive Compact Halo Objects (MACHOs) was been proposedlike candidates for DM. As an alternative, instead to introduce two unknown components to explain the dynamicsand the evolution of Universe at all scales, it is possible change its the geometrical description. In this frameworkthe Extend Theories of Gravity (ETGs) are been developed [8–11]. By introducing the higher curvature terms inthe Lagrangian allow us to reconstruct the DE and DM phenomenology at all scales, from planetary dynamics, toflat rotation curves of spiral galaxies and the velocity dispersion of ellipticals, and from cluster of galaxies until thedynamics of the Universe as a whole [12, 13].However, to constraint and validate ETGs could be extremely important take also into account the gravitationalwaves (GWs) emission. In GR, the field equations linearized around a Minkowskian metric, show that small pertur-bations of the metric propagate following a wave equation [14, 15]. All astrophysical systems that emit GWs are verywell described in the framework of GR, where the gravitational interaction is mediate by a massless boson known asgraviton. But, the further degrees of freedom that come out considering ETGs lead to have also massive gravitationalmodes.In particular, from the power spectrum of weak lensing one can estimate an upper limit of × − eV for thegraviton mass. Then, from clusters of galaxies, it is possible to obtain an upper limit of × − h eV, where h isthe Hubble constant in units of 100 km s − Mpc − . Finally, studying the emission of GWs from the binary systems,one can infer an upper limit of graviton mass of . × − eV [1]. ∗ [email protected] Since the mass of graviton have also effect of the waveform and on the polarization modes, the detection of GWswill point out that GR is validated or that it must be extended [8]. At moment, the most important tool, regardingto GWs, to constrain both theories of gravity is related to the gravitational emission from binary systems of WhiteDwarfs (WDs), Neutron Stars (NS) and Black Holes (BHs) [14]. Timing data analysis on the well-known binarypulsar B have confirmed that the energy loss by the system can be explained with the emission of GWs.Let’s point out that those type of systems, due to the high precision of the mass estimation, and also for the strongfield gravity regime in which they are (PSR B or PSR J are two examples of system with veryprecise measuraments), represent a very good laboratory to test theories of gravity using Post-Keplerian parameters[16–19]. Let’s start to consider a generic class of ETGs, called f ( R ) -theories, where we replace the Einstein-HilbertLagrangian with a more general function of the Ricci curvature. For an analytic f ( R ) , it is possible evaluate the firsttime derivatives of the orbital period for a binary system and, comparing the theoretical estimation with the observedone [20], we put bounds on theory parameters and then we compute an upper limit on the graviton mass.The outline of the paper is the following: in Sec. II we briefly introduce the theoretical framework in which wedescribe binary systems computing the energy lost through GWs emission. In Sec. III we summarize how to computethe first time derivative on the orbital period for a binary system in f ( R ) -gravity. Furthermore, we test how wellthe PSR J can constrain the first time derivative of the orbital period in f ( R ) -theories of gravity and,estimate the bounds on graviton mass. Finally, in Sec. IV we give our conclusions and remarks. II. THEORETICAL FRAMEWORK: f ( R ) -GRAVITY In ETGs, the field equations are computed, extending the Hilbert-Einstein Lagrangian with adding of higher-ordercurvature invariants and minimally or non-minimally coupled scalar fields. The simplest way is to consider a moregeneral function of the curvature f ( R ) [8]. In this case, the field equations have the following form (looking for majordetails at [8–11]) f ′ ( R ) R µν − f ( R )2 g µν − f ′ ( R ) ; µν + g µν (cid:3) g f ′ ( R ) = X T µν , (1) (cid:3) f ′ ( R ) + f ′ ( R ) R − f ( R ) = X T , (2)where T µν = − √− g δ ( √− g L m ) δg µν represents the energy momentum tensor of matter ( T is the trace), X = 16 πGc is thecoupling, f ′ ( R ) = df ( R ) dR , (cid:3) g = ; σ ; σ , and (cid:3) = ,σ,σ indicates the d’Alembert operator. The adopted signature is (+ , − , − , − ) , and we indicate the partial derivative with " , " and the covariant derivative with respect to the g µν isindicated by " ; " ; all Greek indices go from to and Latin indices go from to ; g is the metric determinant. In amore general approach, we don’t fix the form of the f ( R ) Lagrangian but we just assume that it is Taylor expandablein term of the Ricci scalar in order to compute the Minkowskian limit of the theory [20–22], f ( R ) = X n f n ( R ) n ! ( R − R ) n ≃ f + f ′ R + f ′′ R + ... . (3)In this contest, using theoretical approach given in [22], the total average flux of energy emitted in GWs has thefollowing form (cid:28) dEdt (cid:29)| {z } ( total ) = G * f ′ (cid:16) ... Q ij ... Q ij (cid:17)| {z } GR − f ′′ (cid:16) .... Q ij .... Q ij (cid:17)| {z } f ( R ) + . (4)The ratio of f ′ and f ′′ defines an effective mass m g = − f ′ f ′′ related to massive modes in GWs [23]. Until now, theonly modes of gravitational waves that have been investigated are the massless ones, this exclude the possibility to For convenience we will use f instead of f ( R ) . All considerations are developed here in metric formalism. From now on we assumephysical units G = c = 1 . observe further gravitational waveform coming out from massive terms, although, tests in this direction are alreadydone on stochastic background of GWs [24–27].From the equations of conservation we know that the gravitational radiation, predicted by GR, is proportional tothe third derivative of the quadrupole momentum, while the terms due to the monopole and the dipole momentumare zero. For a discussion on gravitational emission see [8, 28–30]. In f ( R ) -gravity we find a different situation. Thefact that we have extended the theory of gravity can be reflected by a mass of graviton not equal to zero. As wellknown, in ETG it is possible to obtain massive gravitons in a natural way [23]. The main feature is that higher-orderterms or induced scalar fields in the Lagrangian, give rise to massless, massive spin- gravitons and massive spin- gravitons. Such gravitational modes results in polarizations, according to the Riemann Theorem stating that in agiven n -dimensional space, n ( n − / degrees of freedom are possible. The fact that 6 polarization states emerge isin agreement with the possible allowed polarizations of spin-2 field [31]. In fact, the spin degenerations is • d = (2 s + 1) , m g = 0 = ⇒ s = 2 , d = 5 • d = 2 s, m g = 0 = ⇒ s = 1 , d = 2 • d = (2 s + 1) , m g = 0 = ⇒ s = 0 , d = 1 The massive spin- gravitational states, usually are ghost particles. The role of massive gravitons result relevant alsoin the case which we want define a cutoff mass at TeV scale. This limits allow us, both to circumvent the hierarchyproblem that the detection of the Higgs boson. For example, in such a case, the Standard Model of particles shouldbe confirmed without recurring to perturbative, renormalizable schemes involving new particles [32–34]. Furthermore,ETGs in post Newtonian regime gives rise at the Yukawa-like correction to the Newtonian potential. These correctionsare dependent by a characteristic length of self-gravitating structures that is connected to massive graviton modes.Upper limits on graviton mass come out when ones try to solve the connection between masses and sizes of self-gravitating structures without invoking huge amounts of DM. III. THE FIRST TIME DERIVATIVE OF THE ORBITAL PERIOD OF A BINARY SYSTEM: f ( R ) PARAMETERS AND BOUNDS ON GRAVITON MASS
Following the scheme that is generally used to compute GWs emission [14], and assuming a Keplerian motion ofthe stars in the binary system, we can define m p as the pulsar mass, m c as the companion mass, and µ = m c m p m c + m p as the reduced mass. The motion is reduced at the ( x − y ) -plane, so that averaging on the orbital period, P b , andusing the eqs. (4), we get the first time derivative of the orbital period [20] ˙ P b = − (cid:18) P b π (cid:19) − µG ( m c + m p ) c (1 − ǫ ) × " f ′ (cid:0) ǫ + 292 ǫ + 96 (cid:1) − f ′′ π T − ǫ ) ×× (cid:0) ǫ + 28016 ǫ + 82736 ǫ + 43520 ǫ + 3072 (cid:1)(cid:3) . (5)where ǫ is the eccentricity of the orbit, G is the Newtonian gravitational constant, and the quantities f ′ and f ′′ arethe ones that we need to constrain. Once we have a theoretical prediction of ˙ P b in f ( R ) -theories we can, accordingwith the prescription given in [35], compute an upper limit for the graviton mass. A. Application to PSR J0348+0432
The binary system PSR J , recently studied by Antoniadis et al. (2013) [19], gives a new possibility tounderstand which range of ETG’s parameters is allowed. It is a binary system composed by a pulsar spinning at ms with mass . ± . M ⊙ ( m p ), and a White Dwarf (WD) companion with mass . ± . M ⊙ ( m c ). Theorbital period of the system is P b = 0 . (days), and the eccentricity e = 2 . × − . For a correct estimation ofthe observed orbital decay, it was considered several kinematic effects that have to be subtracted by variation of theorbital period. The first one is the Shklovskii effect [36], that is an expression of the effect due to the proper motionof the the binary system. Its estimation for this system is ˙ P Shkb = P b µ dc = 0 . +0 . − . × − . (6) Figure 1: It is shown the result of our numerical analysis on the binary system PSR J . We use the following notation:the black line shows the behavior of the GR prediction for the first derivative of the orbital period for a binary system; the redline represents the observed orbital period variation ˙ P b Obs and its errors, in particular the dashed lines show the experimentalerror on the observation; the blue line shows the f ( R ) -theory prediction as computed in eq. (5). Another effect is due to the difference of Galactic accelerations between the binary system and the Solar System ˙ P Accb = P b a c c = 0 . +0 . − . × − . (7)The last term is due to a possible variation of the gravitational constant ˙ G : ˙ P ˙ G b = − P b ˙ GG = (0 . ± . × − . (8)Finally, the observed value of the first time derivative of the orbital period is ˙ P b = ( − . ± . × − . Allthose terms are quoted in [19].As pointed out in [20], the ETGs are not ruled out when the orbital parameters of the binary systems are very wellestimated, and the range of the f ( R ) parameters is . ≤ | f ′′ | ≤ .In Fig. 1, we report the result of our numerical analysis on the binary system PSR J . We use thefollowing notation: the black line shows the behavior of the GR prediction for the first derivative of the orbital periodfor the binary system; the red line represents the observed orbital period variation ˙ P b Obs and its errors, in particularthe dashed lines show the experimental error on the observation; the blue line shows the f ( R ) -theory prediction ascomputed in eq. (5). Moreover, it is possible to see from Fig. 1 that the GR value of ˙ P GR is recovered for f ′′ = 0 (diamond black). The value of the second derivative of the gravitational Lagrangian for which ETGs are able toexplain the observed period variation is f ′′ = 4 . ± . (blue triangle). This value is comparable with the rangegiven in [20]. Using the range of the f ( R ) parameters allowed, we are interested to compute an upper limit for thegraviton mass to understand if this agree with other estimation. According with the prescription given in [35] we getthe following upper limit m g < . × − eV /c , (9)that is comparable with the constrain coming out from B , and is also comparable with the one obtainedanalyzing the Brans-Dicke theory [19]. IV. DISCUSSION
Several shortcomings coming out from astrophysical and cosmological observations, and theoretical problem relatedto have a full quantum description of space-time, suggest to extend the GR to overcome them. Approaches basedon extension or corrections of GR are in agreement with several observables at astrophysical and cosmological scales,explaining the dynamical effect related to the DM, and also the observed acceleration of the Universe.In the post-Minkowskian limit of ETGs provide an accurate description of the problem of gravitational radiation.In this approach, we found extra polarizations modes of the GWs signal with respect to the plus (+) and cross ( × )polarizations predicted by GR [8]. Indeed, the theories allows massive and ghosts modes that in principle couldbe detected using ground-based interferometric detectors, like VIRGO and LIGO [37, 38], and the future spaceinterferometric detector, LISA [39] including the additional polarization modes. Furthermore, results coming outfrom the timing array analysis on binary pulsars systems like PSR B and from other binary systems, canprovide a very useful tool to investigate the the viability of ETGs, like the f ( R ) -gravity theories. For that purpose, itis possible to develop a quadrupolar formalism of the gravitaional radiation using analytic f ( R ) -models, and showingthat expertimental bounds on the first derivative of the orbital period of the binary system, that are provided by PSR J , do not ruled out the f ( R ) -gravity [20, 21]. Furthermore, the upper limit on graviton mass, that wehave computed basing our calculation on the experimental constraint of the orbital parameters, and on the theoreticalprediction of the f ( R ) model, is comparable with the other constrains that come from clusters, galaxies etc [19]. Thisresult, together with other results from observations in the Solar System and Cosmology, indicates that the study andthe possible detection of massive modes of GWs could be the real discriminating between the theory of GR and itsextensions. Acknowledgments
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