On the Viability of a Non-Analytical f(R)-Theory
aa r X i v : . [ g r- q c ] S e p ON THE VIABILITY OF A NON-ANALYTICAL f(R)-THEORY
Nakia Carlevaro a , Giovanni Montani a, b, d and Massimiliano Lattanzi a, c a Department of Physics - “Sapienza” University of Romec/o Dip. Fisica - “Sapienza” Universit`a di Roma, P.le A. Moro, 5 (00185), Roma (Italia) b ENEA – C.R. Frascati (Rome), UTFUS-MAG c ICRA – International Center for Relativistic Astrophysics d ICRANet – International Center for Relativistic Astrophysics Network [email protected] [email protected] [email protected]
Abstract:
In this paper, we show how a power-law correction to the Einstein-Hilbert action provides a viable modified theory of gravity, passing the Solar-System tests, when the exponent is between the values 2 and 3. Then, we imple-ment this paradigm on a cosmological setting outlining how the main phases ofthe Universe thermal history are properly reproduced.As a result, we find two distinct constraints on the characteristic length scaleof the model, i.e. , a lower bound from the Solar-System test and an upper oneby guaranteeing the matter dominated Universe evolution.
PACS : 95.30.Wi, 51.20.+d
From the very beginning, the possibility to riformulate General Relativity by using a genericfunction of the Ricci scalar (see, for example, [1] for a recent review and references therein)appeared as a natural issue offered by the fundamental principles established by Einstein.However, it is important to remark that any modification of the Einstein-Hilbert (EH) La-grangian is reflected onto a deformed gravitational-field dynamics at any length scale investi-gated or observed. Thus, the success of such f ( R ) gravity in the solution of a specific problemhas to match consistency with observation in different length scales [2, 3, 4]. A viable self-consistent model can be often obtained at the price to consider a generalized gravitationallagrangian containing a large number of free parameters. Nevertheless, the wide spectrum ofpossible choices for f ( R ) can appear as a weakness point in view of the predictivity of thetheory, because a significant degree of degeneracy is expected in the model.Here, we consider an opposite point of view, by studying the viability of a power-lawcorrection to the EH action having a single free parameter (a length scale) once the power-law exponent is fixed. We investigate the implementation of the Solar-System test to our1odel [5] and then we pursue a cosmological study of the resulting modified Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) dynamics. As expected, this scenario gives us a ratherstringent range of variation for the free length scale where searching for new gravitationalphysics. In this paper, we consider the following modified gravitational action in the so-called
Jordanframe S = − χ R d x √− g f ( R ) , f ( R ) = R + qR n , (1)where n is a non-integer dimensionless parameter and q < L ] n − (inthe equation above χ = 8 πG , using c = 1 and G being the Newton constant, moreover, thesignature is set as [ + , − , − , − ]). Such a form of f ( R ) gives the following constraints for n :if R >
0, all n -values are allowed; if R <
0, the condition n = ℓ/ (2 m + 1) must hold (where,here and in the following, m and ℓ denote positive integer). It is straightforward to verifythat S in eq.(1) is non-analytical in R = 0 for non-integer, rational n , i.e. , it does not admitTaylor expansion near R = 0.Let us now define the characteristic length scale of our model as L q ( n ) ≡ | q | / (2 n − , (2)while variations of the total action S tot = S + S M (where S M denotes the matter term) withrespect to the metric give, after manipulations and modulo surface terms: f ′ R µν − g µν f − ∇ µ ∇ ν f ′ + g µν f ′ = χ T µν , (3)where T µν is the Energy-Momentum Tensor (EMT). Here and in the following ( ... ) ′ indicatesthe derivative with respect to R , ≡ g ρσ ∇ ρ ∇ σ and ∇ µ or ( ... ) ; denotes the covariantderivative (Greek indices run form 0 to 3).We can gain further information on the value of n by analyzing the conditions that allow fora consistent weak-field stationary limit. Having in mind to investigate the weak field limit ofour theory to obtain predictions at Solar-System scales, we can decompose the correspondingmetric as g µν = η µν + h µν , where h µν is a small (for our case, static) perturbation of theMinkowskian metric η µν . In this limit, the vacuum Einstein equations read R µν − η µν R − nq ( R n − ) ; µ ; ν + nqη µν R n − = 0 , R = 3 nq R n − . (5)The structure of such field equations leads us to focus our attention on the restricted regionof the parameter space 2 < n <
3. This choice is enforced by the fulfillment of the conditionsby which all other terms are negligible with respect to the linear and the lowest-order non-Einsteinian ones.
From the analysis of the weak-field limit in the Jordan frame, i.e. , eqs.(5), we learn thepossibility to find a post-Newtonian solution by solving eqs.(5) up to the next-to-leadingorder in h , i.e. , up to O ( h n − ), and neglecting the O ( h ) contribution only for the cases2 < n <
3. These considerations motivate the choice we claimed above concerning therestriction of the parameter n .The most general spherically-symmetric line element in the weak-field limit is ds = (1 + Φ) dt − (1 − Ψ) dr − r d Ω , (6)2here Φ and Ψ are the two generalized gravitational potentials and d Ω is the solid angleelement. Within this framework, the modified Einstein equations (5) rewrite R tt − R − nq ∇ R n − = 0 ,R rr + R − nq ( R n − ) ,r,r + nq ∇ R n − = 0 ,R θθ + r R − nqr ( R n − ) ,r + nqr ∇ R n − = 0 ,R + 3 nq ∇ R n − = 0 , h ∇ ≡ d dr + r ddr i , R = ∇ Φ + r ( r Ψ) ,r ,R tt = ∇ Φ ,R rr = − Φ ,r,r − r Ψ ,r ,R θθ = − Ψ − r Φ ,r − r Ψ ,r ,R φφ = sin θR θθ , where ( ... ) , denotes ordinary differentiation. The system above is solved by R = Ar n − , Φ = σ + δr + Φ n (cid:16) rL q (cid:17) n − n − , Ψ = δr + Ψ n (cid:16) rL q (cid:17) n − n − , A = h − nq (3 n − n − n − i − n , (8a)Φ n ≡ h − n (3 n − n − n − i − n ( n − n − n − , (8b)Ψ n ≡ h − n (3 n − n − n − i − n ( n − n − , (8c)where the integration constant δ has the dimensions of [ L ] and the dimensionless integrationconstant σ can be set equal to zero without loss of generality. The integration constant A hasthe dimensions of [ L ] (2 n − / (2 − n ) , and Φ n and Ψ n are dimensionless, accordingly. Moreover,one can check that Φ n and Ψ n are well-defined only in the case n = (2 m + 1) /ℓ while weget A > q <
0. In agreement to the geodesic motion as expanded in theweak field limit, the integration constant δ results equal to δ = − r S , where r S = 2 GM is the Schwarzschild radius of a central object of mass M .The most suitable arena to evaluate the reliability and the validity range of the weak-fieldsolution (8) is the Solar System [2, 4]. To this end, we can specify eqs.(8b)-(8c) for the typicallength scales involved in the problem and we split Φ and Ψ into two terms, the Newtonianpart and a modification, i.e. ,Φ ≡ Φ N + Φ M ≡ − r S ⊙ /r + Φ n ( r/L q ) n − / ( n − , (9a)Ψ ≡ Ψ N + Ψ M ≡ − r S ⊙ /r + Ψ n ( r/L q ) n − / ( n − , (9b)here, the integration constant δ of eqs.(8b)-(8c) is δ = − r S ⊙ ≡ GM ⊙ ( M ⊙ being the Solarmass). While the weak-field approximation of the Schwarzschild metric is valid within therange r S ⊙ ≪ r < ∞ because it is asymptotically flat, the modification terms have the peculiarfeature to diverge for r → ∞ . It is therefore necessary to establish a validity range, i.e. , r Min ≪ r ≪ r Max , related to n and L q , where this solution is physically predictive [6].Since we aim to provide a physical picture at least of the planetary region of the SolarSystem, we are led to require that Φ M and Ψ M remain small perturbations with respect toΦ N and Ψ N , so that it is easy to recognize the absence of a minimal radius except for thecondition r ≫ r S ⊙ . The typical distance L ∗⊙ corresponds to the request | Φ N ( L ∗⊙ ) | ∼ | Φ M ( L ∗⊙ ) | , | Ψ N ( L ∗⊙ ) | ∼ | Ψ M ( L ∗⊙ ) | . (10)For r S ⊙ ≪ r ≪ L ∗⊙ , the system obeys thus Newtonian physics and experiences the post-Newtonian term as a correction. Another maximum distance L ∗∗⊙ can be defined, accordingto the request that the weak-field expansion (6) should hold, regardless to the ratios Φ M / Φ N and Ψ M / Ψ N . L ∗∗⊙ results to be defined by | Φ N ( L ∗∗⊙ ) | ≪ | Φ M ( L ∗∗⊙ ) | ∼ , | Ψ N ( L ∗∗⊙ ) | ≪ | Ψ M ( L ∗∗⊙ ) | ∼ . (11)3e remark that L ∗⊙ and L ∗∗⊙ are defined as functions of n and L q , i.e. , L ∗⊙ ∼ | r S ⊙ / Φ n | n − n − L n − n − q , L ∗∗⊙ ∼ L q (cid:14) | Φ n | n − n − , (12)and it is important to underline that, for the validity of our scheme, the condition L ∗⊙ ≫ r S ⊙ must hold, i.e. , L q ≫ r S ⊙ | Φ n | ( n − / (2 n − .Neglecting the lower-order effects concerning the eccentricity of the planetary orbit, wecan deal with the simple model of a planet moving on circular orbit around the Sun with anorbital period T given by T = 2 π ( r/a ) / ( a = ( d Φ /dr ) / T n = 2 πr / ( GM ⊙ ) / h n n − n − (cid:16) r n − n − (cid:17) / (cid:16) r S ⊙ L n − n − q (cid:17)i − / . (13)We now can compare the correction to the Keplerian period T K = 2 πr / ( GM ⊙ ) − / , withthe experimental data of the period T exp and its uncertainty δT exp . We then impose thecorrection to be smaller than the experimental uncertainty, i.e. , δT exp T exp > | T K − T n | T K ∼ | Φ n | n − n − (cid:16) r n − n − P (cid:17) / (cid:16) r S ⊙ L n − n − q (cid:17) , (14)where r P is the mean orbital distance of a given planet from the Sun.Let us now specify our analysis for the example of the Earth [2]. In this particular case, T exp ≃ . δT exp ≃ . · − days (with r P ≃ . × − pc). This way,for the Earth, we can get a lower bound L q > L Minq ⊕ for the characteristic length scale of ourmodel, as function of n , i.e. , L Minq ⊕ ( n ) = h . × | Φ n | r S ⊙ n − n − r n − n − P i n − n − , (15)where Φ n is defined in eq.(8b) and L Minq ⊕ ∼ × − pc, for a typical value n ≃ .
66. Weremark that L Minq ⊕ , by virtue of eq.(8b), is defined only for n = (2 m + 1) /ℓ .Our analysis clarifies how the predictions of the corresponding equations for the weak-fieldlimit appear viable in view of the constraints arising from the Solar-System physics. Indeed,the lower bound for L q does not represent a serious shortcoming of the model, as we aregoing to discuss in Sec.6, where a plot of L Minq ⊕ ( n ) and of L ∗⊙ ( L q ) and L ∗∗⊙ ( L q ) will be alsoaddressed. In order to study how our f(R) model affects the cosmological evolution, we start from themodified gravitational action (1) and we assume the standard Robertson-Walker (RW) lineelement in the synchronous reference system, i.e. , ds = dt − a ( t ) (cid:2) dr / (1 − Kr ) + r d Ω (cid:3) , (16)where a ( t ) is the scale factor and K the spatial curvature constant. Using such expression, the00-component of eq.(3) results, for symmetry using the Bianchi identity, the only independentone and it writes as f ′ R − f + 3( ˙ a/a ) f ′′ ˙ R = χ T . (17)4here the dot indicates the time derivative. We assume as matter source a perfect-fluid EMT, i.e. , T µν = ( p + ρ ) u µ u ν − pg µν , in a comoving reference system (thus T = ρ ), where p is thethermostatic pressure, ρ the energy density and u µ denotes the 4-velocity. The 0-componentof the conservation law, i.e. , T νµ ; ν = 0 with ν = 0, assuming the equation of state (EoS) p = wρ , gives the following expression for the energy density: ρ = ρ [ a/a ] − w ) .Using now f = R + qR n with q <
0, we are able to explicitly write eq.(17):2 ˜ χ a − w + 6 n n q a − n ¨ a ( − K − ˙ a − a ¨ a ) n − ++ a (cid:2) − K − a + 6 n q a − n ) ( − K − ˙ a − a ¨ a ) n (cid:3) + (18)+ 6 n ( n − n q ˙ a a − n ) ( − K − ˙ a − a ¨ a ) n − (cid:2) − a − K ˙ a + a ˙ a ¨ a + a ... a (cid:3) = 0 , where ˜ χ = χρ a w ) . Let us now assume a power-law a = a [ t/t ] x for the scale factor and,for the sake of simplicity, we set ¯ a = a t − x (clearly, [¯ a ] = [ L − x ]). Here and in the following,we use the subscript ( ... ) to denote quantities measured today. In this case, eq.(18) can berecast in the form − a K t x − a x t x − + q ¯ a t x (cid:16) C t − − a − K t − x (cid:17) n + 2 ˜ χ ¯ a − w t x (1 − w ) == nqx ¯ a − n t x (cid:16) C ¯ a t − − K t − x (cid:17) n ( C K t + xC t x )( K t + C t x ) , (19)where C = 6 x (1 − x ), C = ( x (2 n − − C = x ¯ a ( x + 2 n − x − C = x ¯ a (2 x − Here, we assume the radiation-dominated Universe EoS w = 1 / ρ ∼ a − ). In the following,we will discuss the three distinct regimes, in the asymptotic limit as t →
0, for x < x > x = 1, separately.In the case x <
1, all terms containing explicitly the curvature K of eq.(19) results to benegligible for t → x n/ < n < q ¯ a C n [ 1 − ( C /C ) nx ¯ a ] t x − n = 0 , (20)and x = 1 / x = [2 + 3 n − n ± (4 + 8 n + n − n + 4 n ) / ] / n are the solutions. Suchsecond expression results to be negative or imaginary for 2 < n < x <
1, in the asymptotic limit for t →
0, is the well-knownradiation dominated behavior a ∼ t / . In the other two cases, i.e. , for x >
1, it is easy torecognize that no asymptotic solutions are allowed. Therefore, the approach to the initialsingularity is not characterized by power-law inflation behavior when spatial curvature isnon-vanishing.Let us now assume a vanishing spatial curvature in eq.(19). In can be show how, for K = 0, the radiation-dominated solution with w = 1 / x = 1 / exact solution(non-asymptotic and allowed for all n -values) giving ρ = 3 / (4 χt ), matching the standardFLRW case. In the case x >
1, the leading-order terms of eq.(19) read, for t → K = 0, q ¯ a C n [ 1 − ( C /C ) nx ¯ a ] t x − n + 2 ˜ χ = 0 . (21)Three distinct regimes have to be now separately discussed. For x > n/
2, the leading order ofthe equation above does not admit solutions since it writes simply 2 ˜ χ = 0 and, for x < n/ x <
1. Instead, for x = n/
2, anddefining H = ( n/ /t , one gets ρ = ˜ ρ ( n ) q χt , ˜ ρ ( n ) = n (1 − n ) ( n − n n ( n (4 + (6 − n ) n ) − n/ − n , (22)where we have introduced the dimensionless parameter q = H n − q . We remark that theconstraint n = (2 m +1) /ℓ (which is in agreement with respect to the one obtained from Solar-System test) must hold in order to have ρ > q < q <
0. The function ˜ ρ results to increase as n goes from 2 to 3 and, in particular, one canget 216 < ˜ ρ <
21 024. Finally, for x = 1 and K = 0, eq.(19) reads [1 − n (2 n − t − n = 0,giving n = [1 ± √ /
2. As the previous case, the regime x = 1 does not admit solutions inthe region 2 < n < Let us now study the matter-dominated Universe EoS w = 0 ( ρ ∼ a − ). As previously done,we analyze the three distinct regimes for x < x > x = 1, and, in the limit for t → ∞ , it is easy to recognize that there are no power-law solutions in all these cases for K = 0. Setting K = 0, the x > x < − x ¯ a ] t x − + 2 ˜ χ ¯ a t x = ¯ a qC n [ − C ¯ a nx /C ] t x − n . (23)Since 4 x − > x − n , the term on the right hand side can be neglected in the limit of large t and the equation above admits three distinct situations: x < / x > / x = 2 /
3. Bothcases with x = 2 / x = 2 / t → ∞ . In fact, eq.(23) reduces to 8¯ a = 6 ˜ χ and the FLWR matter-dominatedpower-law solution a = ¯ a t / is reached setting ρ = 4 / (3 χt ).In conclusion, we can infer that, for f ( R ) = R + qR n , the standard matter-dominatedFLRW behavior of the scale factor a ∼ t / is the only asymptotic (as t → ∞ ) power-lawsolution. As shown above, the matter dominated solution a ∼ t / is obtained for K = 0 andasimptotically as t → ∞ . In order to neglect all the K -terms in our f ( R ) model, we startdirectly from the expression of the Ricci scalar [7]. Using a power-law scale factor, we getthe t -range (if x = 1 / x < t ≪ (cid:12)(cid:12) [ x (2 x − / [ K/ ¯ a ] (cid:12)(cid:12) / (2 − x ) . (24)For the matter-dominated era and using standard cosmological parameters [8], one can getthe upper limit K/ ¯ a . .
006 ( H ) / , to estimate the value of K/ ¯ a . Thus, setting x = 2 / t ≪ /H , independently of the form of f ( R ).At the same time, if we set x = 2 /
3, the asymptotic solution ρ = 4 / (3 χt ) is reachedneglecting the right hand side ( ≪
1) of eq.(23), i.e , if t is constrained by the following lowerlimit (we remind that q = H n − q ) t ≫ µ ( n, q ) /H , µ ( n, q ) = (cid:12)(cid:12) q (cid:2) − (4 / n + 2 (2 n +1) − n n (2 n − / (cid:3) (cid:12)(cid:12) / n − . (25)6et us now recall that the matter-dominated era began, assuming H − ≃ . × s, at t Eq ≃ . × − /H . In this sense, we can safely assume µ ( n, q ) . × − , which impliesan upper limit for | q | , i.e. , | q | | q | Max , where | q | Max ( n ) = (cid:2) . × − (cid:3) − n ) (cid:12)(cid:12) − (4 / n + 2 (2 n +1) − n n (2 n − / (cid:12)(cid:12) − . (26)It is easy to check that the function | q | Max ( n ) is decreasing as n goes from 2 to 3, inparticular, one gets: 10 − . | q | Max . − . After discussing the power-law evolution of the Universe proper of the radiation- and matter-dominated eras, we now analyze the inflationary behavior characterizing the very early dy-namics (for an interesting approach to the inflationary scenario within the modified gravityscheme, see [9, 10]). In this respect, we hypothesize an exponential behavior for the scalefactor of the Universe a = a e s ( t − t ) = ¯ a e st , where s > a = a e − st . In the following,we concentrate the attention on the solution for vanishing spatial curvature K = 0 and, inthis case, eq.(18) rewrites as¯ a e st (cid:2) q ( − n s n (1 − n/ − s (cid:3) + 2 ˜ χ (¯ a e st ) − w = 0 . (27)Let us now assume w = − i.e. , ρ = ρ I = const. ) during inflation. Using the definition q = H n − q , the equation above reduces to (cid:2) ( − n n q (1 − n/ (cid:3) s n − s + κ = 0 , (28)where κ = 2 χρ I H − and s is a dimensionless parameter defined as s = s/H . Since H de-notes the Hubble parameter measured today and estimating H I = p χρ I / i.e. , accordinglyto its Friedmannian value) during inflation [7], one can obtain κ ∼ H I /H ∼ . For suchvalues, it is easy to realize that considering the case n = 2 ℓ/ (2 m + 1), the equation abovedoes not admit real solution, thus we now discuss, consistently with the previous analyses,only n = (2 m + 1) / (2 ℓ + 1).In order to integrate eq.(28), we focus on a particular value of the power-law f ( R ) exponent, e.g , n = 29 / ∼ .
23. Using eq.(15), for this value of n one obtains that it can be safelyconsidered L Minq ⊕ ∼ . × − pc and, having in mind that L q = | q | / (2 n − /H with H − ≃ . × pc, we get | q | > | q | Min ∼ . × − .Let us now fix the parameter q to a reasonable value like q ∗ ∼ − | q | Min (such as-sumption will be physically motivated in the next Section). In this case, the solution ofeq.(28) is s ∼ . × . This analysis demonstrates that an exponential early expansionof the Universe is still associated to a vacuum constant energy, even for the modified Fried-mann dynamics. However, we see that the rate of expansion is significantly lower than theFriedmann-like one of about a factor in s of 10 . Although our estimation relies on theFriedmannian relation between H I and ρ I (the latter is taken of the order of the Grand Uni-fication energy-scale), nevertheless the values of s remains many order of magnitude belowthe standard value ∼ even if we change H I for several order of magnitude. Despitethis difference, it is still possible to arrange the cosmological parameter in order to have asatisfactory inflationary scenario, as far as we require a longer duration of the de Sitter phase. As already discussed in Sec.2, the parameter q has dimension [ L ] n − . We have thereforedefined a characteristic length scale of the model as L q ( n ) = | q | / (2 n − . Assuming f ( R )7orrections to be smaller than the experimental uncertainty of the orbital period of the Eartharound the Sun, the lower bound (15) for L q ( n ) was found. In order to identify the allowedscales for our model and in view of the upper constraint on the parameter q = H n − q derived in the cosmological framework, we can now define the upper limit for L q ( n ) as L Maxq ( n ) = [ | q | Max ] / (2 n − /H , (29)which, considering eq.(26), yields to the constraints 65 .
59 pc < L
Maxq < .
37 pc, for2 < n <
3. Assuming H − ≃ . × pc, the two bounds for the characteristic lengthscales here discussed, i.e , eq.(15) and eq.(29), are plotted in Fig.1(A). At the same time - - - n L q a ll o w e d @ p c D H A L L q Å Min L qMax L q A ll o w e dd i s t a n ce s @ p c D H B L L Ÿ * L Ÿ ** n = (cid:144) ~ Figure 1:
Panel A: L Minq ⊕ of eq.(15) and L Maxq of eq.(29). The gray zone represents theallowed characteristic-length scales of the model. We stress that L Minq ⊙ is definedonly if n = (2 m + 1) /ℓ , as represented by the dotted line. Panel B: L ∗⊙ and L ∗∗⊙ of eq.(12). The gray zone represents here the allowed distances for the model.two other typical lengths have been outlined in eq.(12) for the Solar System. L ∗⊙ representsthe minimum distance to have post-Newtonian and Newtonian terms of the same order.While L ∗∗⊙ was defined according to the request that the weak-field expansion holds. Settingnow n = 23 / ≃ .
55, one can show from eq.(15) and eq.(29) that the allowed scales are0 . . L q . .
72 pc. In this range, L ∗⊙ and L ∗∗⊙ can be plotted as in Fig.1(B).Summarizing, our analysis states a precise range of validity for the power-law f ( R ) modelwe consider. Indeed, for a generic value of n ( i.e. , not close to 2 or 3) the fundamental lengthof the model is constrained to range from the super Solar-System scale up to a sub-galacticone. Therefore, in agreement to eq.(9a), we have to search significant modification for theNewton law in gravitational system lying in this interval of length scales, like for instance,stellar clusters. Acknowledgment : NC gratefully acknowledges the CPT - Universit´e de la Mediterran´ee Aix-Marseille 2 and the financial support from “Sapienza” University of Rome.
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