On the cosmological viability of the Hu-Sawicki type modified induced gravity
aa r X i v : . [ g r- q c ] A ug On the cosmological viability of the Hu-Sawicki typemodified induced gravity
Kourosh Nozari and Faeze Kiani Department of Physics, Faculty of Basic Sciences,University of Mazandaran,P. O. Box 47416-95447, Babolsar, IRAN
Abstract
It has been shown recently that the normal branch of a DGP braneworld scenario self-accelerates if the induced gravity on the brane is modified in the spirit of f ( R ) modifiedgravity. Within this viewpoint, we investigate cosmological viability of the Hu-Sawickitype modified induced gravity. Firstly, we present a dynamical system analysis of a gen-eral f ( R )-DGP model. We show that in the phase space of the model, there exist threestandard critical points; one of which is a de Sitter point corresponding to acceleratingphase of the universe expansion. The stability of this point depends on the effectiveequation of state parameter of the curvature fluid. If we consider the curvature fluid tobe a canonical scalar field in the equivalent scalar-tensor theory, the mentioned de Sitterphase is unstable, otherwise it is an attractor, stable phase. We show that the effectiveequation of state parameter of the model realizes an effective phantom-like behavior. Acosmographic analysis shows that this model, which admits a stable de Sitter phase in itsexpansion history, is a cosmologically viable scenario. PACS : 04.50.-h, 98.80.-k
Key Words : Braneworld Cosmology, Phantom Mimicry, Dynamical System, Cosmogra-phy [email protected] [email protected] Introduction
The late-time accelerating phase of the universe expansion which is supported by data relatedto the luminosity measurements of high red shift supernovae [1], measurements of degree-scaleanisotropies in the cosmic microwave background (CMB) [2] and large scale structure (LSS)[3], is one of the challenging problems in the modern cosmology. The rigorous treatment ofthis phenomenon can be provided essentially in the framework of general relativity. In theexpression of general relativity, late time acceleration can be explained either by an exotic fluidwith large negative pressure that is dubbed as dark energy in literature, or by modifying thegravity itself which is dubbed as dark geometry or dark gravity proposal. The first and simplestcandidate of dark energy is the cosmological constant, Λ [4]. But, there are theoretical problemsassociated with it, such as its unusual small numerical value (the fine tuning problem), nodynamical behavior and even its unknown origin [5]. These problems have forced cosmologiststo introduce alternatives in which dark energy evolves during the universe evolution. Scalar fieldmodels with their specific features provide an interesting alternative for cosmological constantand can reduce the fine tuning and coincidence problems. In this respect, several candidatemodels have been proposed: quintessence scalar fields [6], phantom fields [7] and Chaplygingas [8] are among these candidates. Nevertheless, we emphasize that the scalar field models ofdark energy are not free of shortcomings.As an alternative for dark energy, modification of gravity can be accounted for the latetime acceleration. Among the most popular modified gravity scenarios which may successfullydescribe the cosmic speed-up, is f ( R ) gravity [9,10]. Modified gravity also can be achievedby extra-dimensional theories in which the observable universe is a 4-dimensional brane em-bedded in a five-dimensional bulk. The Dvali-Gabadadze-Porrati (DGP) model is one of theextra-dimensional models that can describe late-time acceleration of the universe in its self-accelerating branch due to leakage of gravity to the extra dimension [11,12].Recent observations constrain the equation of state parameter of the dark energy to be w X ≈ − w X < − w eff < − f ( R ) gravity [10,15,16,17]. Firstly, we study the cosmological dynamicsof this model within a dynamical system approach. We show that there exists a standard deSitter point in the phase plane of the model. In this respect, this model has the potential to2xplain accelerated expansion of the universe. The stability of this point depends completelyon the effective equation of state parameter of the curvature fluid. If we consider the curvaturefluid to be a canonical scalar field in the equivalent scalar-tensor theory, the mentioned de Sitterphase is unstable, otherwise it is an attractor, stable phase. Since the late-time acceleratingphase of the universe expansion is explained by a stable de Sitter phase, we can investigate thecosmological viability of such theoretical models based on the phantom-like behavior of this f ( R )-DGP gravity. To be more specific, in which follows we focus on the cosmological viabilityof the Hu-Sawicki type modified induced gravity and show that this model has capability torealize a stable, attractor de Sitter phase. We point out that the phantom mimicry discussed inthis study has a geometric origin. To be more realistic, we compare our results with observationvia a cosmographic approach. f ( R ) -DGP scenario In this section, possible modification of the induced gravity on the brane is investigated inthe spirit of f ( R ) theories [10,15,16,17]. It has been shown that 4 D f ( R ) theories in thepresent time can follow closely the expansion history of the ΛCDM universe [18]. Here westudy an extension of f ( R ) theories to a DGP braneworld setup. The motivation behind thisstudy is that modified induced gravity on the normal branch of a DGP scenario provides somenew interesting features, one of which is self-acceleration of the normal DGP branch in thissituation (see Refs. [10,16,17] for details). Similar to the normal branch of the standard DGPcosmology, the resulting generalized normal branch is also ghost-free and therefore the issueof ghost-instabilities is irrelevant in this case [17]. The action of this model can be written asfollows S = M Z d x √− g R + Z d x √− q (cid:16) M K + L (cid:17) , (1)where by definition L = m p f ( R ) + L m . (2)By calculating the bulk-brane Einstein’s equations and using a spatially flat FRW line element,the following modified Friedmann equation is obtained [15,16,17] H = 8 πG (cid:18) ρ ( m ) + ρ ( rad ) + ρ ( curv ) (cid:19) ± H ¯ r c (3)where ρ ( curv ) = m p (cid:18) h f ( R ) − Rf ′ ( R ) i − RHf ′′ ( R ) (cid:19) , (4)is energy density corresponding to the curvature part of the theory. This energy density can bedubbed as dark curvature energy density. ¯ r c is the re-scaled crossover distance that is definedas ¯ r c = r c f ′ ( R ) and a prime marks differentiation with respect to the Ricci scalar, R . We note3hat in this scenario there is an effective gravitational constant, which is re-scaled by f ′ ( R ) sothat G = G eff ≡ πm p f ′ ( R ) [15]. In order to study the phase space of this scenario, it is moresuitable to rewrite the normal branch of the Friedmann equation (3) in the following morephenomenological form E = Ω m (1 + z ) + Ω rad (1 + z ) + Ω curv (1 + z ) w curv ) − q Ω r c E , (5)where by definition Ω curv = 8 πG H ρ ( curv )0 , Ω r c = 14[ r c f ′ ( R )] H , and also w curv = − Rf ′′ ( R ) + ˙ R h ˙ Rf ′′′ ( R ) − Hf ′′ ( R ) i [ f ( R ) − Rf ′ ( R )] − H ˙ Rf ′′ ( R ) . (6)We note that w curv is not a constant and varies with redshift. f ( R ) -DGP model To investigate cosmological dynamics of this model within a dynamical system approach, weexpress the cosmological equations in the form of an autonomous, dynamical system. For thispurpose, we define the following normalized expansion variables p = √ Ω m a / E , r = √ Ω rad a E , s = √ Ω curv a w curv ) / E , u = √ Ω r c E . (7)In this way, equation (5) with minus sign (corresponding to the generalized normal DGP branch)and in a dimensionless form, is written as follows1 + 2 u = p + r + s . (8)This constraint means that the allowable phase space of this scenario in the p - r - s space isoutside of a sphere with radius 1, which is defined as p + r + s ≥ p ′ = p h p +(1+2 w curv ) r + s − i p + r + s +1) ,r ′ = r h p + s +(1+ w curv )( r − p − s − i p + r + s +1) ,s ′ = s h s + p +(1+3 w curv ) r − i p + r + s +1 . (9)4able 1: Eigenvalues and the stability properties of the critical points.points ( p, r, s ) character eigenvalues w curv < − w curv > − A (0 , ,
1) radiation h , (1 − w curv ) , i unstable unstable B (1 , ,
0) matter h − , − w curv , i unstable unstable C (0 , ,
0) de Sitter h w curv − , w curv )2 , w curv )8 i stable unstableHere a prime marks differentiation with respect to the new time variable τ = ln a that a isscale factor of the universe. The critical points in the phase plane are obtained by solving theequations p ′ = 0, r ′ = 0 and s ′ = 0, that is, setting the autonomous system (9) to be vanishing.The results are shown in table 1. To investigate the stability of these points, we apply the linearapproximation analysis to achieve the Jacobian matrix. Note that the critical points and theirstability depend on the value of w curv . Here we investigate the stability of the standard criticalpoints in two different subspaces of the model parameter space where EoS of the curvature fluidhas either a phantom or a quintessence character. As we see in table 1, the radiation dominatedphase (point A ) and matter dominated phase (point B ) in this scenario, are unstable phases ofthe universe expansion independent on the value of w curv . Whereas, the accelerating phase ofthe universe expansion (point C ) is a stable phase if the curvature fluid is considered to be anon-canonical (phantom) scalar field ( w curv < −
1) in equivalent scalar-tensor theory; otherwiseit is an unstable phase. It is necessary to mention that whenever w curv = −
1, the variables s and u are not independent and the phase space is 2D (here the curvature fluid plays therole of a cosmological constant, the same as Λ DGP model. For more details see Ref. [19]).Figure 1 shows the 3D phase space trajectories of the model. In this figure, the point C as ade Sitter point is an attractor for w curv < −
1. Therefore, a model universe which is describedby modified induced gravity on the normal DGP branch, has a stable, positively acceleratedexpansion phase if the modified gravity indicates a phantom-like behavior. We note that points O and C ′ do not belong to physical phase space of our model universe.After exploration of the cosmological dynamics in a general f ( R )-DGP setup within a phasespace analysis, in the next section we study cosmological viability of an specific f ( R )-DGPmodel. 5 pr s OC’ C
Figure 1:
The 3D phase space of the autonomous system (9) for w curv < −
1. There are three criticalpoints: C is an attractor de Sitter point, O is a saddle point and C ′ is an attractor point. Now we focuss on the cosmological viability of the model by considering a Hu-Sawicki typemodified induced gravity on the DGP brane. It is shown in the Ref. [18] that the expansionhistory of the mentioned model in 4 dimensions is widely close to the ΛCDM model in thehigh-redshift regime. Now in a braneworld extension, we expect the Hu-Sawicki induced gravitymimics the ΛDGP model in the mentioned regime. In other words, in this regime curvatureterm in the Friedmann equation is close to the cosmological constant which is screened by theterm H ¯ r c . In fact, the dynamical screening effect is the main origin of the phantom-like behaviorof the curvature term in the normal branch of this DGP-inspired braneworld scenario [15]. TheHu-Sawicki model [18] is given by f ( R ) = R − m c ( Rm ) n c ( Rm ) n + 1 , (10)where m , c , c and n are free positive parameters that can be expressed as functions of densityparameters. Now we explore the dependence of these parameters on density parameters definedin our setup. Variation of the action (1) with respect to the metric yields the induced modifiedEinstein equations on the brane G αβ = 1 M S αβ − E αβ , (11)where E αβ (which we neglect it in our forthcoming arguments), is the projection of the bulkWeyl tensor on the brane E αβ = (5) C MRNS n M n R g Nα g Sβ (12)6nd S αβ as the quadratic energy-momentum correction into Einstein field equations is definedas follows S αβ = − τ αµ τ µβ + 112 τ τ αβ + 18 g αβ τ µν τ µν − g αβ τ . (13) τ αβ as the effective energy-momentum tensor localized on the brane is defined as [10] τ αβ = − m p f ′ ( R ) G αβ + m p h f ( R ) − Rf ′ ( R ) i g αβ + T αβ + m p h ∇ α ∇ β f ′ ( R ) − g αβ ✷ f ′ ( R ) i . (14)The trace of Eq. (11), which can be interpreted as the equation of motion for f ′ ( R ) , is obtainedas R = 524 M τ . (15) τ , the trace of the effective energy-momentum tensor localized on the brane is expressed as τ = m p h f ( R ) − Rf ′ ( R ) − ✷ f ′ ( R ) i − ( ρ m + ρ rad ) , (16)To highlight the DGP character of this generalized setup, we express the results in terms of theDGP crossover scale defined as r c = m p M . So, the equation of motion for f ′ ( R ) is rewritten asfollows 56 r c h f ( R ) − Rf ′ ( R ) i + 9 (cid:16) ✷ f ′ ( R ) (cid:17) + 6 Rf ′ ( R ) ✷ f ′ ( R ) − f ( R ) ✷ f ′ ( R ) ! + 56 r c M h Rf ′ ( R ) − f ( R ) + 3 ✷ f ′ ( R ) i ( ρ m + ρ rad ) + 524 M ( ρ m + ρ rad ) − R = 0 . (17)In the next stage, we solve this equation for ✷ f ′ ( R ) to obtain ✷ f ′ ( R ) = − "(cid:16) Rf ′ ( R ) − f ( R ) (cid:17) + ρ m + ρ rad M r c ± r c s R , (18)Now we introduce an effective potential V eff which satisfies ✷ f ′ ( R ) = ∂V eff ∂f ′ ( R ) . This effectivepotential has an extremum at h Rf ′ ( R ) − f ( R ) i + 1 m p ( ρ m + ρ rad ) = ± r c s R . (19)In the high-curvature regime, where f ′ ( R ) ≃ f ( R ) R ≃ R ± r c s R = 1 m p ( ρ m + ρ rad ) . (20)7he negative and positive signs in this equation are corresponding to the DGP self-acceleratingand normal branches respectively. In which follows, we adopt the positive sign correspondingto the normal branch of the scenario. To investigate the expansion history of the universe inthis setup, we restrict ourselves to those values of the model parameters that yield expansionhistories which are observationally viable. We note that the Hu-Sawicki f ( R ) function intro-duced in Ref. [18], was interpreted as a cosmological constant in the high-curvature regime.The motivation for that interpretation was to obtain a ΛCDM behavior in the high curvature(in comparison with m ) regime. Here we show that in a braneworld extension, the Hu-Sawickiinduced gravity mimics the ΛDGP model in the mentioned regime. As we have pointed outpreviously, the phantom-like behavior can be realized from the dynamical screening of the branecosmological constant. In this respect, we apply the same strategy to our model, so that thesecond term in the Hu-Sawicki f ( R ) function (that is, the second term in the right hand side ofequation (10)) mimics the role of an effective cosmological constant on the DGP brane. Thenthis term will be screened by Hr c term in the late time (see the normal branch of Eq. (3)).In the case in which R ≫ m , one can approximate Eq. (10) as followslim m R → f ( R ) ≈ R − c c m + c c m (cid:16) Rm (cid:17) n . (21)During the late-time acceleration epoch, f ′ ( R ) ≃ R ≫ m and we can applythe above approximation. Also the curvature field is always near the minimum of the effectivepotential. So, based on Eq. (19), we have R + 1 r c s R = 1 m p ( ρ m + ρ rad ) + 2 c c m . (22)Since R in the f ( R ) function is induced Ricci scalar on the brane, we except crossover scaleto affect on the constant parameters c , c and m . In Ref. [18] the authors obtained 3 m ≡ R c = ρ m m p that ρ m is the present value of the matter density. But, in our setup the presentvalue of the matter density (see Eq. (20)) is given by R c + 0 . √ R c r c = 1 m p ( ρ m + ρ rad ) . (23)If we solve this equation for R c , we find3 m ≡ R c ≈ Ω r c + 3Ω m + 3Ω rad ± r Ω r c (cid:16) . r c + Ω m + Ω rad (cid:17) . (24)Therefore, the DGP character of this extended modified gravity scenario is addressed through m . As we have mentioned, at the curvatures high compared with m , the second term on theright hand side of equation (10) mimics the role of an effective cosmological constant on thebrane. In this respect, the second term in the right hand side of equation (21) also mimics the8ole of a cosmological constant on the brane in the high curvature regime. With this motivation,we find c c ≈ Λ Ω r c + 3Ω m + 3Ω rad ± r Ω r c (cid:16) . r c + Ω m + Ω rad (cid:17) . (25)There is also a relation for c c as follows c c = 1 − f ′ ( R ) n (cid:18) R m (cid:19) n +1 , (26)where R m in our setup can be calculated as follows: firstly, by using Eqs. (22) and (25), we find R + 1 r c s R = 1 m p ( ρ m a − + ρ rad a − ) + 12Ω Λ , (27)where ρ m can be omitted through Eq. (23) to obtain R + 1 r c s R = (cid:18) m + 1 . mr c − m p ρ rad (cid:19) a − + 1 m p ρ rad a − + 12Ω Λ . (28)Finally, if we solve this equation for √ R , we find the following relation for R m R m ≈ − . √ Ω r c m + (cid:20) (cid:16) . √ Ω r c m (cid:17) + 12Ω Λ m (cid:21) / ! , (29)where m is given by Eq. (24). Note that we have set H and a ( t ) equal to unity. Theserelations tell us that the free parameters of this model are n , Ω m , Ω rad , Ω r c and f ′ ( R ), whereasthe latter one is constrained by the Solar-System tests. In fact, experimental data show that f ′ ( R ) − < − , when f ′ ( R ) is parameterized to be exactly 1 in the far past. To analyzethe behavior of w curv , we specify the following ansatz for the scale factor a ( t ) = (cid:18) t + t − ν (cid:19) − ν , (30)where ν = 1 is a free parameter [20]. By noting that the Ricci scalar is R = 6( ¨ aa + ( ˙ aa ) ), onecan express the function f ( R ) of equation (10) in terms of the redshift z . Figure 2 shows thevariation of the effective equation of state parameter versus the redshift. As we see in thisfigure, in this class of models the curvature fluid has an effective phantom-like equation ofstate, w curv < −
1, at high redshifts and then approaches the phantom divide ( w curv = −
1) ata redshift that increases by decreasing n .The main point here is that a modified induced gravity of the Hu-Sawicki type in the DGPframework, gives an effective phantom-like equation of state parameter for all values of n , and9 K0.4 K0.3 K0.2 K0.1 0.0 0.1 0.2 w curv K1.0004K1.0003K1.0002K1.0001K1.0000 n=1n=2n=3n=4
Figure 2: w curv versus the redshift for a Hu-Sawicki type modified induced gravity with Ω m = 0 . rad = 0 .
3, Ω Λ = 0 .
9, Ω r c = 0 .
01 and f ′ ( R ) − ≈ − . As this figure shows, in this class of modelsthe curvature fluid has an effective phantom equation of state with w curv < − w curv = −
1) at a redshiftthat increases by decreasing the values of n . all of these models approach asymptotically to the de Sitter phase ( w curv = − f ( R ) model. Based on the analysis presented in the previous sectionwithin a phase space viewpoint and also the outcomes of this section, we can conclude that aHu-Sawicki type modified induced gravity on the normal branch of the DGP setup providesa cosmologically viable scenario. This is the case since it contains a radiation dominated erafollowed by a matter dominated era and finally a stable de Sitter phase in its expansion history.In the next section we compare our model with observational data via a cosmographic analysis.Our treatment here is mainly based on the Ref. [29,30]. While theoretical consistency of a physical theory is a primary condition for viability of thetheory, the observational consistency of the model is necessary too. For this goal, in whichfollows we discuss briefly observational status of our model via a cosmographic analysis. Beforethat, we note that the DGP model is a testable scenario with the same number of parametersas the standard ΛCDM model, and has been constrained from many observational data, suchas the SNe Ia data set [21], the baryon mass fraction in clusters of galaxies from the X-raygas observation [22], CMB data [23], the large scale structures [24] and the baryon acousticoscillation (BAO) peak [25], the observed Hubble parameter H ( z ) data [26], the gravitationallensing surveys [27]. The observational constraints on the DGP model with Gamma-ray bursts(GRBs) at high redshift also obtained recently from the Union2 Type Ia supernovae dataset [28]. In [28] the authors are shown that by combining the GRBs at high redshift withthe Union2 data set, the WMAP7 results, the BAO observation, the clusters baryon mass10raction, and the observed Hubble parameter data set and also in order to favor a flat universe,the best fit of the density parameter values of the DGP model are obtained as { Ω m , Ω r c } = { . +0 . − . , . +0 . − . } [28].Here to compare our f ( R )-DGP model with observational data we adopt the cosmographyapproach. Cosmography approach is a useful tool in order to constrain higher order gravity ob-servationally without need to solve field equations or addressing complicated problems relatedto the growth of perturbations [29,30]. In this case, one can define cosmographic parametersbased on the fifth order Taylor expansion of the scale factor. One can also relate the charac-teristic quantities defining the f ( R )-DGP model to the mentioned cosmographic parameters.Therefore, a measurement of the cosmographic parameters makes it possible to put constraintson f ( R ) and its first three derivatives. The likelihood function for the probe s is defined as[31] L s ( p ) ∝ exp ( − χ s ( p ) /
2) (31)where χ s ( p ) = N s X n =1 h µ obs ( z i ) − µ th ( z i , p ) i σ µ i ( z i ) (32) µ obs ( z i ) are the observed distance modulus for the adopted standard candle (such as SNe Ia) atthe redshift z i with its error σ µ i . µ th ( z ) are the theoretical values of the distance modulus fromcosmological models which read as µ th ( z, p ) = 25 + 5 log D L ( z, p ) where D L = H d L is theluminosity distance. In the cosmography approach, one can obtain an analytical expression forluminosity distance versus the cosmographic parameters so that one require no priori model tosolve d L = (1 + z ) R z dz ′ E ( z ′ ) . By using the least squares fitting that means the getting of χ s min ,one can obtain the suitable cosmographic parameters. In the next step, one should relate the f ( R ) function and its first three derivatives to the cosmographic parameters to set constraintson the parameters of the f ( R ) function [29,30]. In this manner we constrain observationallythe parameters of a Hu-Sawicki type f ( R ) induced gravity on the normal DGP brane by thecosmography approach. Our strategy in this cosmographic approach is mainly based on therecent paper by Bouhmadi-L`opez et al. [30]. Firstly we relate the functions f ( R ), f ′ ( R ), f ′′ ( R ) and f ′′′ ( R ) to the parameters R , ˙ R , ¨ R , ( d Rdt ) and ˙ H which are expressed versusthe cosmographic parameters by using the Friedmann and Raychaudhuri equations at t = t .Now we have a system of two equations with four unknowns. To expand the f ( R ) function andits derivatives versus these cosmographic parameters, we need to two further equations to closethe system. In 4-dimensional f ( R ) gravity, the Newtonian gravitational constant G is replacedby an effective (time dependent) quantity as G eff = Gf ′ ( R ) . On the other hand, it is reasonableto assume that the present day value of G eff is the same as the Newtonian one G eff ( z = 0) = G or f ′ ( R ) ≃ f ( R ) = R . In order to resolve this problem, we canreplace the condition f ′ ( R ) = 1 with f ′ ( R ) = (1 + ǫ ) − . Another relation can also be obtainedby differentiating the Raychaudhuri equation [29,30]. We solve this system of four equations11or four unknowns to obtain the following relations f ( R )6 H = − A Ω m + B + C ( r c H ) − D + ǫ (2 − q ) , f ′′ ( R )(6 H ) − = − A Ω m + B + C ( r c H ) − D + ǫ B −C ′ D , f ′′′ ( R )(6 H ) − = − A Ω m + B + C ( r c H ) − ( j − q − D + ǫ B ′ −C ′ ( j − q − D , (33)where A i , B i , C i and D with i = 0 , , q , j , s and l (these functions aredefined in Ref. [30]). The new quantities C ′ , B ′ , and C ′ are defined as follows C ′ = j ( j − q −
1) + q ( q + q − − , (34) B ′ = 2 j (2 q + 6 q + j + 3) + 2 q (15 q + 42 q + 39) − l (1 + q ) − s ( q + j ) + 24 , (35) C ′ = j ( − j + [2 q + 8] q + s + 7 (cid:17) + s (1 − q ) − q (cid:16) q [ q + 8 q − − s − (cid:17) + 8 . (36)In the second step we have to determine the values of the cosmographic parameters that havethe best fit to the observational data (by the least squares fitting). Instead, here we usea minimal approach to parameterize the cosmographic parameters by the phenomenologicaldensity parameters. In other words, the cosmographic parameters will be calculated for a givenphenomenologically parameterized dark energy model. The best choice is the ΛCDM model. InRef. [30] the details of these calculations are done. They finally obtained the following results[30] q = − m , j = 1 , s = 1 −
92 Ω m , l = 1 + 3Ω m + 272 Ω m . (37)Now one can substitute these results into equations (33) and consider the observational conser-vative values Ω m = 0 .
266 and Ω r c = 10 − where Ω r c = (4 r c H ) − [2]. Finally, by consideringthe first order approximation in ǫ , one obtains the following results f ( R )6 H = 0 .
849 + 2 . ǫ , f ′′ ( R )(6 H ) − = 0 . − . ǫ , f ′′′ ( R )(6 H ) − = 1 . . ǫ (38)In the HS model, there are four parameters c , c , R c and n that can be constrained byobservational data via the values of the f ( R ) and its derivatives. So, we should create a systemof four equations in the four unknowns through equation (10) and its first three derivatives.By solving equation (10) and its first derivative for c and c , with f ( R ) = 0 .
849 + 2 . ǫ and f ′ ( R ) = (1 + ǫ ) − , one finds [29] e c ≡ c R − nc = n (1 + ǫ ) ǫ R − n (cid:20) − .
849 + 2 . ǫR (cid:21) (39)12 c ≡ c R − nc = n (1 + ǫ ) ǫ R − n (cid:20) − .
849 + 2 . ǫR − ǫn (1 + ǫ ) (cid:21) . (40)By substituting relations (39) and (40) in HS f ( R ) function and its derivatives, it is obviousthat parameter R c cancels out so that we have to work with two parameters e c and e c insteadof three parameters c , c and R c . In other words, R c cannot be obtained in this fashion. Bysetting the second derivative of the HS function equal to f ′′ ( R ) = 0 . − . ǫ , we get n = [(0 .
849 + 2 . ǫ ) /R ] + [(1 + ǫ ) /ǫ ](1 − [0 . − . ǫ ] /R ) − (1 − ǫ ) / (1 + ǫ )1 − (0 .
849 + 2 . ǫ ) /R . (41)In the last stage and in order to determine the value of ǫ , one can use the third derivative ofthe HS function and setting f ′′′ ( R ) = 1 . . ǫ to obtain the following constraint (see also[29])1 . . ǫ .
849 + 2 . ǫ = 1 + ǫǫ (0 . − . ǫ ) R (cid:20) R (0 . − . ǫ )0 .
849 + 2 . ǫ + ǫ (0 .
849 + 2 . ǫ )1 + ǫ (cid:16) − ǫ − (0 .
849 + 2 . ǫ ) /R (cid:17)(cid:21) (42)Using this constrain, the acceptable value of ǫ is ≃ .
03 (note that there are three othervalues of ǫ that are not acceptable since are very large). The value of R is determined by R = 6 H (1 + q ) with q = − Ω m . By equation (41), we get n ≃ e c ≃
10 and e c ≃ . R c here plays the role of a scaling parameter. We obtain c and c from equations (25) and (26) and then by using their relation with e c and e c , we find R c ≃ . Recently a mechanism to self-accelerate the normal branch of the DGP model, which is knownto be free from the ghost instabilities, has been reported [17]. This mechanism is based onthe modified induced gravity. In this paper, firstly we studied the cosmological dynamics ofthis model within a phase space approach. A de Sitter phase is the simplest cosmologicalsolution that exhibits acceleration. As we have shown in a dynamical system viewpoint, thisphase appears in our generalized setup. In fact, based on the dynamical system approach, weshowed that there exists a de Sitter fixed point in phase space of a general f ( R )-DGP model.In order to investigate the stability of this accelerating phase of expansion, we classified the f ( R ) functions in two different subspaces of the model parameter space. We have shown thatif the f ( R ) induced gravity plays effectively the role of a phantom scalar field in the equivalentscalar-tensor theory, it leads to a stable de Sitter solution and these models are cosmologicallyviable. Then, as an specific model, we studied the Hu-Sawicki type modified induced gravityin the DGP framework and we found that the equation of state parameter of the curvaturefluid has an effective phantom-like character. The origin of the phantom-like behavior in the13odel presented here can be due to the dynamical screening effect of the curvature term (whichplays effectively the role of a cosmological constant in high-redshift regime on the brane). Inother words, in this case the phantom-like behavior has a pure gravitational origin. We haveshown also that the Hu-Sawicki modified induced gravity mimics the ΛDGP model in thehigh-redshift regime. Since the Hu-Sawicki modified induced gravity contains an early timeradiation dominated era followed by a matter domination era and then a stable de Sitter phasein its expansion history, it is cosmologically a viable scenario. This result is independent onthe value of free parameter n of the Hu-Sawicki model. Finally we have tried to constrain ourmodel based on the observational data through a cosmographic procedure. In this manner weobtained reasonable values for parameters of the Hu-Sawicki induced gravity. Acknowledgement
We would like to thank an anonymous referee for insightful suggestions.
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