On tidal forces in f(R) theories of gravity
Alvaro de la Cruz-Dombriz, Peter K. S. Dunsby, Vinicius C. Busti, Sulona Kandhai
aa r X i v : . [ g r- q c ] D ec On tidal forces in f ( R ) theories of gravity ´Alvaro de la Cruz-Dombriz , , , ∗ , Peter K. S. Dunsby , , , † , Vinicius C. Busti , , ‡ , Sulona Kandhai , , § Astrophysics, Cosmology and Gravity Centre (ACGC),University of Cape Town, Rondebosch 7701, Cape Town, South Africa Department of Mathematics and Applied Mathematics,University of Cape Town, Rondebosch 7701, Cape Town, South Africa Instituto de Ciencias del Espacio (ICE/CSIC) and Institut d’Estudis Espacials de Catalunya (IEEC),Campus UAB, Facultat de Ci`encies, Torre C5-Par-2a, 08193 Bellaterra (Barcelona) Spain and South African Astronomical Observatory, Observatory 7925, Cape Town, South Africa. (Dated: June 7, 2018)Despite the extraordinary attention that modified gravity theories have attracted over the pastdecade, the geodesic deviation equation in this context has not received proper formulation thusfar. This equation provides an elegant way to investigate the timelike, null and spacelike structureof spacetime geometries. In this investigation we provide the full derivation of this equation insituations where General Relativity has been extended in Robertson-Walker background spacetimes.We find that for null geodesics the contribution arising from the geometrical new terms is in generalnon-zero. Finally we apply the results to a well known class of f ( R ) theories, compare the resultswith General Relativity predictions and obtain the equivalent area distance relation. PACS numbers: 04.50.Kd, 04.25.Nx, 95.36.+x
I. INTRODUCTION
The limitations faced by the cosmological concordancemodel or ΛCDM model have led cosmologists to proposea range of alternative theories. Modifications inside theframework of General Relativity (GR), with the pres-ence of a new component called dark energy have beenproposed [1], where a possible time evolution in its en-ergy density is encoded in the equation of state. Anotherpossibility consists of replacing the theory of gravity onlarge scales, where a different gravitational action mayexplain the current accelerated phase experienced by theUniverse. In this way, instead of a new fluid driving theacceleration, this effect results directly from the geomet-ric part of the gravitational field equations.There are several ways of modifying the gravitationalaction ( c.f. [2] for a thorough review), giving rise todifferent modified gravity theories. One of the simplestforms is to consider functions of the Ricci scalar R ,dubbed f ( R ) theories [3] and this class will be the focusof our investigations. These theories are constrained by anumber of requirements, which include: a ) the positivityof the effective gravitational constant [4]; b ) the existenceof a stable gravitational stage related to the presence ofa positive mass for the associated scalar mode [5] and,last but not least, c ) the recovery of the GR behavior onsmall scales and at early times in the history of the uni-verse in order to be consistent with Big Bang Nucleosyn- ∗ E-mail: alvaro.delacruzdombriz [at] uct.ac.za † E-mail: peter.dunsby [at] uct.ac.za ‡ E-mail: vinicius.busti [at] iag.usp.br § E-mail: kndsul001 [at] myuct.ac.za thesis and Cosmic Microwave Background (CMB) con-straints. There also exist several constraints for the valueof | d f / d R | R = R , where R holds either for the current orpast cosmological background curvature. The latter con-straint arises from the Integrated Sachs-Wolfe effect andcorrelations with foreground galaxies ( c.f. [6]) .With the aim of providing a satisfactory explana-tion for a range of cosmological and astrophysical phe-nomenon, modified gravity theories have been studiedfrom different points of view including the growth of den-sity [7] and gravitational waves [8] perturbations, deter-mining the existence of GR-predicted astrophysical ob-jects such as black holes [9] as well as research on theirstability [10].One important aspect which has not received a propertreatment so far relates to the timelike, null and spacelikestructure of spacetimes in the framework of fourth ordergravity theories in general and the example of f ( R ) the-ories in particular. An elegant way to study this canbe done through an analysis of the Geodesic DeviationEquation (GDE), also known as the
Jacobi equation . Thisequation encapsulates many results of standard cosmol-ogy [11] such as the observer area distance, first derivedby Mattig [12] for the dust case, the dynamics governedby the Raychaudhuri equation [13] and how perturba-tions affect the kinematics of null geodesics, leading togravitational-lensing effects [14].As a first application of the GDE in metric f ( R ) theo-ries, we restrict our attention to Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) spacetimes and derive the These constraints are obtained using several assumptions and aretherefore in general model dependent.
DE for the spacelike, timelike and null geodesics. Al-though some attention has been paid to this equation inthe Palatini formalism [15] and in arbitrary curvature-matter coupling scenarios [16], the present investigationrepresents the first attempt to address this issue in themetric formalism.We also derive the area distance relation and studythe results for a specific parameterization of f ( R ) darkenergy theories - the so-called Hu-Sawicki (HS) models[17], which provide a viable cosmological evolution andhave been investigated in a range of astrophysical andcosmological situations.This paper is organized as follows. In Section II webriefly review the 1 + 3 decomposition of variables whichallows us to obtain the cosmological equations for f ( R )theories in the metric formalism, assuming a FLRW back-ground. Section III is devoted to studying the GDE in thecontext of f ( R ) theories for homogeneous and isotropicbackgrounds and a derivation of the observer area dis-tance formula is presented in Section IV. In order to char-acterize the background FLRW cosmology, a dynamicalsystem approach is described in Section V and this is usedin Section VI to illustrate the rich phenomenology of theevolution of the GDE for the HS class of f ( R ) models.This demonstrates how the standard GR geodesic devia-tion evolution is distorted when new non-constant termsare included in the gravitational action. Finally, in Sec-tion VII we present the conclusions and give an outlineof future work to be done. II. THE COSMOLOGICAL EQUATIONS FOR f ( R ) GRAVITYBasic Notation
Unless otherwise specified, natural units ( ~ = c = k B = 8 πG = 1) will be used throughout this paper.Latin indices run from 0 to 3, the symbol ∇ representsthe usual covariant derivative, we use the ( − , + , + , +)signature and the Riemann tensor is defined by R abcd = W abd,c − W abc,d + W ebd W ace − W f bc W adf , (1)where the W abd are the Christoffel symbols (i.e., sym-metric in the lower indices), defined by W abd = 12 g ae ( g be,d + g ed,b − g bd,e ) . (2)The Ricci tensor is obtained by contracting the first andthe third indices R ab = g cd R cadb . (3)Finally, the action for f ( R )-gravity can be written inthese units as A = Z d x √− g (cid:20) f ( R ) + L m (cid:21) , (4) where R is the Ricci scalar, f is a general differentiable(at least C ) function of the Ricci scalar and L m corre-sponds to the matter Lagrangian.In the metric formalism, the modified Einstein equa-tions (EFEs), obtained by varying this action with re-spect to the metric takes the form f ′ G ab = T mab + g ab ( f − Rf ′ ) + ∇ b ∇ a f ′ − g ab ∇ c ∇ c f ′ , (5)where f ≡ f ( R ), f ′ ≡ dfdR and T mab ≡ √− g δ ( √− g L m ) δg ab , oralternatively R ab = 1 f ′ (cid:20) T ab + 12 f g ab − g ab (cid:3) f ′ + ∇ a ∇ b f ′ (cid:21) , (6)whose trace is R = 1 f ′ [ T + 2 f − (cid:3) f ′ ] . (7)Defining the energy momentum tensor of the curvature“fluid” (denoted by super or subindex R ) as T Rab ≡ f ′ (cid:20)
12 ( f − Rf ′ ) g ab + ∇ b ∇ a f ′ − g ab ∇ c ∇ c f ′ (cid:21) , (8)the field equations (5) can be written in a more compactform G ab = ˜ T mab + T Rab , ≡ T ab , (9)where the effective energy momentum tensor of standardmatter is given by ˜ T mab ≡ T mab f ′ . (10)Assuming that the energy-momentum conservation ofstandard matter T mab ; b = 0 holds, this leads us to concludethat T ab is divergence-free, i.e., T ab ; b = 0, and therefore˜ T mab and T Rab are not individually conserved [18]:˜ T mab ; b = T mab ; b f ′ − f ′′ f ′ T mab R ; b , T Rab ; b = f ′′ f ′ T mab R ; b . (11) decomposition Before proceeding further let us introduce the 1 + 3decomposition. This decomposition will prove to be veryuseful in the following calculations. Let us consider thefour velocity u a ( u c u c = −
1) and the projection operatordefined by h ab = g ab + u a u b , (12)which projects into the rest space orthogonal to u a andsatisfies h ab u b = 0 , h ca h b c = h ba , h aa = 3 . (13)2t follows that any spacetime 4-vector v a may be covari-antly split into a scalar V , which is the part of the vectorparallel to u a , and a 3-vector, V a , lying in the sheet or-thogonal to u a : v a = − u a V + V a , V = v b u b , V a = h b a v b . (14)The variation of the velocity with position and time isof interest here and therefore we consider its covariantderivative split into its irreducible parts: ∇ a u b = D a u b − u a ˙ u b , (15)then splitting the spatial change of the 4-velocity furtherinto its symmetric and anti-symmetric parts and the sym-metric part further into its trace and trace-free part: ∇ a u b = σ ab + ω ab + 13 Θ h ab − u a ˙ u b , (16)where σ ab , ω ab and Θ denote the shear tensor, vorticitytensor and expansion scalar respectively.Applying the previous decomposition to f ( R ) modi-fied gravity theories in the metric formalism, one gets forgeneral spacetimes: ∇ a ∇ b f ′ = − ˙ f ′ (cid:18) h ab Θ + σ ab + ω ab (cid:19) + u b u a ¨ f ′ + u a ˙ f ′ ˙ u b , (17)and consequently (cid:3) f ′ = − Θ ˙ f ′ − ¨ f ′ , (18)where terms involving the orthogonally projected deriva-tive have been dropped since our focus is on homogeneousand isotropic spacetimes. The Background FLRW equations
Considering a flat universe filled with standard matterwith energy density µ m and pressure p m with an FLRWmetric, the non-trivial field equations obtained from (4)lead to the following equations governing the expansionhistory of the model3 ˙ H + 3 H = − f ′ (cid:16) µ m + 3 p m + f − f ′ R + 3 Hf ′′ ˙ R + 3 f ′′′ ˙ R + 3 f ′′ ¨ R (cid:17) , (19)3 H = 1 f ′ (cid:18) µ m + Rf ′ − f − Hf ′′ ˙ R (cid:19) , (20)i.e., the Raychaudhuri and
Friedmann equations [13, 19].Here H is the Hubble parameter, which defines the scalefactor a ( t ) via the standard relation H = ˙ a/a and theRicci scalar is R = 6 ˙ H + 12 H . (21) The energy conservation equation for standard matter˙ µ m = − Hµ m (1 + w m ) (22)closes the system, where w m is its barotropic equation ofstate.Note that the Raychaudhuri equation can be obtainedfrom the Friedmann equation, the energy conservationequation and the definition of the Ricci scalar. Hence,any solution of the Friedmann equation automaticallysolves the Raychaudhuri equation.In a similar way, we can decompose the energy mo-mentum tensor of the curvature fluid to obtain the corre-sponding thermodynamical quantities (denoted in whatfollows by a R superscript or subscript). All these quanti-ties, unlike their matter counter-parts, vanish in standardGR, with a FLRW geometry µ R = T Rab u a u b = 1 f ′ (cid:20)
12 ( Rf ′ − f ) − Θ f ′′ ˙ R (cid:21) ,p R = 13 T Rab h ab = 1 f ′ h f − Rf ′ f ′′ (cid:18) ¨ R + 23 Θ ˙ R (cid:19) + f ′′′ ˙ R i , (23)where the anisotropic stress and energy flux (momentumdensity) vanish in these geometries. With the definitionof standard matter and the curvature fluid, one can definea total equation of state parameter ω total as follows: ω total ≡ p total µ total = p m /f ′ + p R µ m /f ′ + µ R , (24)where total density and pressure can be combined, sothat ˙ µ total + 3 H ( µ total + p total ) = 0 . (25)Let us stress that ω total does not represent the equation ofstate of any physical fluid or mixture thereof, but shouldinstead be regarded as a mathematical trick that allowsus to rewrite the EFEs and the conservation equation(25) in a more convenient way. III. GEODESIC DEVIATION EQUATION IN f ( R ) GRAVITY
The general GDE takes the form [20–22] δ η a δv = − R abcd V b V d η c , (26)where η a is the deviation vector, V a is the normalisedtangent vector field and v is an affine parameter. It isobvious that the contraction of the Riemann tensor withthe normalised tangent vector field V a and the deviationvector η a depend on the tensorial equations provided by3he gravitational theory under consideration. In order tomake explicit the f ( R ) dependence in the previous ex-pression, let us consider the usual Weyl tensor definition C abcd = −
12 ( g ac R bd − g ad R bc + g bd R ac − g bc R ad )+ R g ac g bd − g ad g bc ) + R abcd . (27)For homogeneous and isotropic spacetimes, the Weyl ten-sor is identically zero and therefore 27 when contractedwith V b η c V d yields R abcd V b η c V d = 12 (cid:0) η a V b V d R bd − V a V b η c R bc + ǫR ac η c ) − R η a ǫ , (28)with E = − V a u a , η a u a = η a V a = 0 and ǫ = V a V a . Theterms in (28) can be simplified as follows R abcd η c = 1 f ′ (cid:20) η a (cid:18) p m + f − (cid:3) f ′ (cid:19) + ( ∇ a ∇ c f ′ ) η c (cid:21) ,R bc V a V b η c = 1 f ′ (cid:2) ( ∇ b ∇ c f ′ ) V a V b η c (cid:3) ,R bd V b V d η a = 1 f ′ h ( µ m + p m ) E + ǫ (cid:18) p m + f − (cid:3) f ′ (cid:19) + V b V d ∇ b ∇ d f ′ i η a . (29)When assuming homogeneous and isotropic spacetimes(FLRW), i.e., ω ab = 0 = σ ab and using (17), we get V b V d ∇ b ∇ d f ′ = −
13 ˙ f ′ Θ (cid:0) ǫ + E (cid:1) + E ¨ f ′ , ( ∇ b ∇ c f ′ ) V a V b η c = 0 , ( ∇ a ∇ c f ′ ) η c = −
13 ˙ f ′ Θ η a . (30)Consequently (28) becomes R abcd V b V d η c = 12 f ′ " f + µ m − f ′ Θ3 − (cid:3) f ′ + p m η a ǫ + 12 f ′ (cid:20) µ m + p m −
13 ˙ f ′ Θ + ¨ f ′ (cid:21) η a E . (31)Using the fact that µ R + p R = 1 f ′ (cid:20) −
13 ˙ f ′ Θ + ¨ f ′ (cid:21) ,µ R + 3 p R = 1 f ′ h f + Θ ˙ f ′ + 3 ¨ f ′ i − R , (32)we obtain, after some manipulations, the final result forthe GDE in f ( R ) theories within the metric formalism: R abcd V b V d η c = 12 ( µ total + p total ) E η a + (cid:20) R µ total + 3 p total ) (cid:21) ǫ η a . (33) As expected from the homogeneous and isotropic ge-ometry, the GDE in these type of theories only resultin a change in the deviation vector component η a , i.e.,the force term is proportional to η a itself and, conse-quently, according to [23, 24] only the magnitude of η may change along the geodesic, whereas its spatial ori-entation remains fixed. Note also that the standard GRresult is recovered when f ( R ) = R . If anisotropic geome-tries are considered, a change also in the direction of thedeviation vector would result. This analysis will be leftto future work. IV. NULL GEODESICS IN f ( R ) THEORIES
Let us now restrict our investigation to null vectorfields, in this case V a = k a with k a k a = 0 and conse-quently ǫ = 0. Equation (33) then reduces to R abcd k b k d η c = 12 ( µ total + p total ) E η a , (34)which expresses the focusing of all families of past di-rected geodesics provided that( µ total + p total ) > p Λ = − ρ Λ does not affect the focusing of nullgeodesics [11]. Nevertheless, in the realm of modifiedgravity theories, Eq. (35) does not need to be satisfieda priori in order to guarantee the viability of a theory orclasses of models therein ( c.f. [25] and [26] for thoroughdiscussions on this issue). Past-directed null geodesics and area distance in f ( R ) theories Let us now consider V a = k a , k a k a = 0, k < η a = η e a , e a e a = 1, 0 = e a u a = e a k a , and using abasis which is both parallel propagated and aligned, i.e., δe a /δv = k b ∇ b e a = 0, one can rearrange (34) asd η d v = −
12 ( µ total + p total ) E η . (36)Provided that ( µ total + p total ) >
0, all families of past-directed (and future-directed) null geodesics experiencefocusing. For the pathological case, where the righthand side of (36) vanishes - in GR this scenario corre-sponds to a de Sitter universe - the solution of (36) be-comes η ( v ) = C v + C , equivalent to the case of flat(Minkowski) spacetime.4fter some manipulation that involves using expres-sions (19) and (20), as well as the fact thatd d v = (cid:18) d z d v (cid:19) (cid:20) d d z − d z d v d v d z dd z (cid:21) , dzd v = E H (1 + z ) , (37)equation (36) in redshift yieldsd η d z + (7 + 3 ω total )2(1 + z ) d η d z + 3(1 + ω total )2(1 + z ) η = 0 . (38)It follows that (38) depends only on ω total as a function ofredshift, i.e., as a function of the cosmological evolution.Equipped with the previous result, one can infer anexpression for the observer area distance r ( z ): r ( z ) := s (cid:12)(cid:12)(cid:12)(cid:12) d A ( z )dΩ (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) η ( z ′ ) | z d η ( z ′ ) / d ℓ | z ′ =0 (cid:12)(cid:12)(cid:12)(cid:12) , (39)where A is the area of the object and Ω the solid angle.We have used the fact that d / d ℓ = E − (1 + z ) − d / d v = H (1 + z ) d / d z and chosen the deviation to be zero at z = 0. Thus r is given by r ( z ) = (cid:12)(cid:12)(cid:12)(cid:12) η ( z ) H (0) d η ( z ′ ) / d z ′ | z ′ =0 (cid:12)(cid:12)(cid:12)(cid:12) . (40)Analytical expression for the observable area distancefor GR with no cosmological constant can be found in[11, 27], whereas for more general scenarios numericalintegration is usually required. V. DYNAMICAL SYSTEM FORMALISM
Finding solutions of the cosmological field equations(20) - (22) can in general become a cumbersome issue.We therefore employ a general dynamical systems strat-egy, following [29, 30], to significantly simplify the sys-tem of equations. For example, rewriting the Friedmannequation at (20) in the following way: H = µ m f ′ + 16 R − ff ′ − H f ′′ ˙ Rf ′ , (41)leads quite naturally to the definition of the following setof general dimensionless dynamical variables: x ≡ ˙ Rf ′′ f ′ H , y ≡ R H , χ ≡ f f ′ H , ˜Ω m ≡ µ m f ′ H , h ( z ) ≡ HH . (42)Substituting the modified field equations (20) - (22), fordust, into the redshift derivative of the above variables, leeds to the following set of five first order differentialequations(1 + z ) d h d z = h (2 − y ) , (43)(1 + z ) d x d z = x + x ( y + 1) − y + 4 χ − ˜Ω m , (44)(1 + z ) d y d z = y (2 y − xQ − , (45)(1 + z ) d χ d z = χ ( x + 2 y − − xyQ , (46)(1 + z ) d ˜Ω m d z = ˜Ω m ( x + 2 y − , (47)with the Friedmann constraint1 = y − χ − x + ˜Ω m . (48)where the term Q ≡ f ′ Rf ′′ specifies the theory under con-sideration. In order to close the system, Q must be ex-pressed in terms of the dynamical systems variables.To solve these equations requires fixing initial condi-tions for the normalised Hubble parameter h and the de-celeration parameter q , together with fixing the value ofΩ today. In this way we can compute the initial val-ues of { y, χ, ˜Ω m } directly using (42) and x through theconstraint (48). In general the background evolution willdiffer from ΛCDM, leading to a different predictions fromthe GDE.In terms of the DS variables introduced in (42), theGDE for models given by (50) can be rearranged as fol-lows: d η d z + 4 − y ( z )1 + z d η d z + 2 − y ( z )(1 + z ) η = 0 , (49)since ω total = (1 − y ( z )) / f ( R )theory under consideration as can be seen by a straight-forward calculation. VI. RESULTS FOR A CLASS OF f ( R ) THEORIES
To illustrate the results in the previous sections, weconsider the following broken power-law form for f ( R ): f ( R ) = aR − m b (cid:0) Rm (cid:1) n c (cid:0) Rm (cid:1) n , (50)where the constants a, b and c are dimensionless modelparameters to be constrained by observations, and m is related to the square of the Hubble parameter. Inwhat follows we instead use the dimensionless parameter d ≡ m /H .This form of f ( R ), proposed by Hu & Sawicki, hasattracted much interest in the literature as a viable al-ternate for the gravitational interaction. Its popularity is5ue to its broken power-law nature. This enables the the-ory to assume the properties of standard GR in low cur-vature regimes, as well as mimic the observed late-timeaccelerated expansion behavior, accurately described byΛCDM, in the high curvature regimes. As can be seenby the form of (50), there is no explicit cosmological con-stant term, however, as R → ∞ , an effective cosmolog-ical constant appears, in the limiting case of b/c → f ( R ). When the initial value of the function (50) is cho-sen such that it lies comfortably on this plateau, an ap-propriately parameterized HS model mimics the behaviorof the ΛCDM model very well. Exponent n h q h ≡ H (today) /H and deceleration ( q ) parameters for models of the form (50)for different values of exponent n = 1, 1.1, 1.4, 1.8 and 2. H corresponds to the ΛCDM Hubble parameter value today.All the studied models provide h values as well as accelera-tion today ( q <
0) very close to ΛCDM counterparts. Ini-tial conditions for the cosmological evolution were imposedat z in = 10 (deep in the Hu-Sawicki plateau) matching the H and q values as given by ΛCDM. For illustrative purposesΩ m = 0 . By specifying the model parameters { a, b, c, d, n } andinitial values for the dimensionless Hubble parameter, h in , and deceleration parameter, q in at an initial redshift z in , we can fix the initial values of the dynamical vari-ables y, χ and ˜Ω m . The constraint equation (48) can beused to initialize x . In order for the HS model to mimicΛCDM as closely as possible, the values of h in and q in areset to their corresponding ΛCDM values, at the choseninitial redshift and study models of the type (50) withthe fixing of a = b = 1, c = 1 /
19 and d = 6 c (1 − Ω m )with Ω m = 0 . and varying theexponent n in the interval [1 , n = 1, 1.1, 1.4, 1.8 and 2. Fig. 1 depicts the evolutionof Hubble parameter and deceleration parameter of theaforementioned models. In Fig. 2 we have depicted theevolution of the deviation η as given by ΛCDM and sev-eral f ( R ) models of the type (50) and whose parameters The constraint c = 6 d (1 − Ω m ) was considered in order to guar-antee that lim R →∞ f ( R ) = R −
2Λ and therefore GR is recoveredat the early stages of the Universe. as well as cosmological evolutions are summarized in Ta-ble I. The right panel of Fig. 2 then showcases the areadistance evolution as well as its deviation from ΛCDMevolution.For all the studied models, the null geodesic devia-tion is very similar to the ΛCDM counterpart having as-sumed the same standard matter abundance today in allthe models. The relative deviation with respect to theΛCDM geodesic deviation remains almost indistinguish-able (order 10 − ) for very low redshifts and starts to devi-ate for redshifts z ≈ . f ( R )models, with the relative difference smaller for bigger val-ues of the exponent n . Thus n = 2 model provides theclosest geodesic deviation evolution to the concordancemodel. With respect to the area distance the evolutionsresemble with high accuracy that of ΛCDM, althoughthe latter evolution does not constitute a bound for thisquantity. Again n = 2 provides an area distance evolu-tion almost indistinguishable from ΛCDM in the stud-ied redshift range with a relative deviation smaller than10 − .As a next step, the equations for the area distance (39)and (40) can be used to constrain these models using sev-eral observational probes. For instance, the use of com-pact radio sources as cosmic rulers [31], the angular size-redshift relation derived from the Sunyaev-Zel’dovich ef-fect - X-ray technique [32]. By applying the relation be-tween luminosity distances and area distances it is alsopossible to extend our studies with type Ia supernovaedata [33] for both homogeneous and statistically homo-geneous cases [34]. VII. CONCLUSIONS
In this paper we have presented a complete analysisof the geodesic deviation equation in the metric formal-ism of f ( R ) theories. We used a 1 + 3 decompositionwhich enabled us to simplify the intermediate calcula-tions and determine that the new geometrical contribu-tions contribute to the deviation for both null and time-like geodesics. Equation (33) encapsulates the generalresult for isotropic and homogeneous geometries.We proved that the extra terms introduced by thesetheories, as well as the standard matter content impacton the evolution of the geodesic deviation, as is clearlyrepresented in the aforementioned equation. The well-known fact that modified gravity theories do not need toaccomplish the standard energy conditions, which stan-dard fluids do [26], may lead the geodesic deviation equa-tion to exhibit a model-dependent behavior that mayserve to constrain the viability of classes of models insuch theories.We have illustrated our results for a class of fourth or-6 .91.02.05.010.0 1 2 5 H ( z ) / H z Λ CDMModel n=1Model n=1.1Model n=1.4Model n=1.8Model n=2 -4 -3 -2 -1 z -0.6-0.4-0.2 0 0.2 0.4 0 1 2 3 4 5 6 q ( z ) z Λ CDMModel n=1Model n=1.1Model n=1.4Model n=1.8Model n=2
Figure 1:
Evolution of Hubble parameter h = H ( z ) /H (left panel) and deceleration parameter q (right panel) as a function of redshiftfor several exponents n . H holds for the ΛCDM Hubble parameter value today. For all the models, initial conditions were fixed to matchΛCDM values of H and q at initial redshift z in = 10, i.e., deep in the plateau for this class of f ( R ) theories. All the models provide valuesof h ≈ q < m = 0 . η ( z ) z Λ CDMModel n=1Model n=1.1Model n=1.4Model n=1.8Model n=2 -4 -3 -2 -1 z r ( z ) / c z Λ CDMModel n=1Model n=1.1Model n=1.4Model n=1.8Model n=2 -6 -5 -4 -3 -2 -1 z Figure 2:
Geodesic deviation η (left panel) and area distance (right panel) as functions of redshift. Initial conditions η ( z = 0) = 1 and η ′ ( z = 0) = 0 were imposed in equation (38). The depicted redshift interval was [0 , r ( z ) evolution. The inner panels represent relative deviation with respect to ΛCDM (Ω m = 0 . η and r respectively. der gravity theories, the so-called Hu-Sawicki f ( R ) mod-els, which can be considered as a natural extension to theEinstein-Hilbert Lagrangian, able to recover the GeneralRelativity predictions at high curvatures and to providelate-time acceleration, while also satisfying weak fieldconstraints. First we solved the background equationsfor different values of the exponents n after having fixedthe remaining parameters, where the initial conditionswere imposed in the matter dominated epoch, with Hub-ble and deceleration parameters matching their ΛCDMcounterparts. Let us remind that the initial conditionsare fixed well deep in the f ( R ) Hu-Sawicki model plateauwhich appears for large curvatures. Therefore for suchinitial redshift the models effectively behave as ΛCDMonce the f ( R ) model parameters are chosen adequately. We then used the cosmological background to study theevolution of the deviation for null geodesics as well aspresent numerical results for the area distance formula.For all the cases considered the results are similar toΛCDM, which means that they remain phenomenolog-ically viable and can be tested with observational data.The analysis performed in this communication is eas-ily extensible to other f ( R ) models and modified gravitytheories. Work in this direction is in progress in order toapply our results to the most competitive fourth-ordergravity as well as scenarios combining gravity theoriesbeyond General Relativity in non-FLRW spacetimes. Acknowledgments:
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