Gauss-Bonnet Braneworld Cosmology with Modified Induced Gravity on the Brane

We analyze the background cosmology for an extension of the DGP gravity with Gauss-Bonnet term in the bulk and $f(R)$ gravity on the brane. We investigate implications of this setup on the late-time cosmic history. Within a dynamical system approach, we study cosmological dynamics of this setup focusing on the role played by curvature effects. Finally we constraint the parameters of the model by confrontation with recent observational data.


Introduction
One of the most significant astronomical observation in the last decade is the accelerated expansion of the universe [1][2][3][4][5][6][7][8][9][10][11][12][13][14]. One way to explain this accelerating phase of the universe expansion is invoking a dark energy component in the matter sector of the Einstein field equations [15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31]. However, it is possible also to modify the geometric part of the field equations to achieve this goal [32][33][34][35][36][37][38][39][40][41][42][43][44][45]. In the spirit of the second viewpoint, braneworld model proposed by Dvali, Gabadadze and Porrati (DGP) provides a natural explanation of late-time accelerated expansion in its self-accelerating branch of the solutions [46][47][48][49][50][51]. Unfortunately, the self-accelerating branch of this scenario suffers from ghost instabilities [52,53] and therefore it is desirable to invoke other possibilities in this braneworld setup. An amazing feature of the DGP setup is that the normal branch of this scenario, which is not self-accelerating, has the capability to realize phantom-like behavior without introducing any phantom field neither in the bulk nor on the brane [54][55][56][57][58][59][60]. By the phantom-like behavior one means an effective energy density which is positive, grows with time and its equation of state parameter (ω ef f = p ef f ρ ef f ) stays always less than −1. The phantom-like prescription breaks down if this effective energy density becomes negative. An interesting extension of the DGP setup is possible modification of the induced gravity 4 on the brane. This can be achieved by treating the induced gravity in the framework of f (R)-gravity [61][62][63][64][65][66]. This extension can be considered as a manifestation of the scalar-tensor gravity on the brane since f (R) gravity can be reconstructed as General Relativity plus a scalar field [32][33][34][35][36][37][38][39][40][41][42][43][44][45]. Some features of this extension such as self-acceleration in the normal branch of the scenario are studied recently [61][62][63][64][65][66][67][68]. Here we generalize this viewpoint to the case that the Gauss-Bonnet curvature effect is also taken into account. We consider a DGP-inspired braneworld model that the induced gravity on the brane is modified in the spirit of f (R)-gravity and the bulk action contains the Gauss-Bonnet term to incorporate higher order curvature effects. Our motivation is to study possible influences of the curvature corrections on the cosmological dynamics on the normal branch of the DGP setup. We analyze the background cosmology and possible realization of the phantom-like behavior in this setup. By introducing a curvature fluid that plays the role of dark energy component, we show that this model realizes phantom-like behavior on the normal branch of the scenario in some subspaces of the model parameter space, without appealing to phantom fields neither in the bulk nor on the brane and by respecting the null energy condition in the phantom-like phase of expansion. We show also that in this setup there is smooth crossing of the phantom divide line by the equation of state parameter and the universe transits smoothly from quintessence-like phase to a phantom-like phase. We present a detailed analysis of cosmological dynamics in this setup within a dynamical system approach in order to reveal some yet unknown features of this kinds of models in their phase space. Finally confrontation of our model with recent observational data leads us to some constraint on model parameters. The action of a GBIG (the Gauss-Bonnet term in the bulk and Induced Gravity on the brane) braneworld scenario can be written as follows [69][70][71][72][73][74][75][76][77][78][79][80] where by definition with and G (5) is the 5D Newton's constant in the bulk and G (4) is the corresponding 4D quantity on the brane. L GB is the Gauss-Bonnet term, and α is a constant with dimension of [length] 2 . q is the induced metric on the brane. We choose the coordinate of the extra dimension to be y, so that the brane is located at y = 0. The DGP crossover distance which is defined as has the dimension of [length] and will be appeared in our forthcoming equations. We note that this scenario is UV/IR complete in some sense, since it contains both the Gauss-Bonnet term as a string-inspired modification of the UV (ultra-violet) sector and the induced gravity as IR (infra-red) modification to the General Relativity. The cosmological dynamics of this GBIG scenario is described fully by the following Friedmann equation [81][82][83][84] where E 0 is referred hypothetically as the mass of the bulk black hole and the corresponding term is called the bulk radiation term. Note that when one adopts the positive sign, the above equation can be reduced to the generalized DGP model as α → 0 , but the branch with negative sign cannot be reduced to the generalized DGP model in this regime. Therefore, we just consider the plus sign of the above equation [81][82][83][84]. We note that depending on the choice of the bulk space, the brane FRW equations are different (see [85] for details). The bulk space in the present model is a 5-dimensional AdS black hole. In which follows, we assume that there is no cosmological constant on the brane and also in the bulk, i.e. Λ (4) = Λ (5) = 0. Also we ignore the bulk radiation term since its decay very fast in the early stages of the evolution (note however that this term is important when one treats cosmological perturbations on the brane). So, the Friedmann equation in this case reduces to the following form It has been shown that it is possible to realize the phantom-like behavior in this setup without introducing any phantom matter on the brane [86][87][88][89][90][91]. In which follows we generalize this setup to the case that induced gravity on the brane is modified in the spirit of f (R) gravity and we explore the cosmological dynamics of this extended braneworld scenario.

Modified GBIG Gravity
In this subsection we firstly formulate a GBIG scenario that induced gravity on the brane acquires a modification in the spirit of f (R) gravity. To obtain the generalized Friedmann equation of this model we proceed as follows: firstly, the Friedmann equation for pure DGP scenario is as follows [49][50][51]92,93] ǫ where ǫ = ±1 is corresponding to two possible embeddings of the brane in the bulk. Considering a Minkowski bulk with Λ 5 = 0 and by setting E 0 = 0 with a tensionless brane (λ = 0), for a flat brane (k = 0) we find The normal branch of the scenario is corresponding to the minus sign in the right hand side of this equation. The second term in the right is the source of the phantom-like behavior on the normal branch: the key feature of this phase is that the brane is extrinsically curved in such a way that shortcuts through the bulk allow gravity to screen the effects of the brane energy-momentum contents at Hubble parameters H ∼ r −1 c and this is not the case for the self-accelerating phase [54][55][56][57][58][59][60].
In the next step, we incorporate possible modification of the induced gravity by inclusion of a f (R) term on the brane. This extension can be considered as a manifestation of the scalartensor gravity on the brane. In this case we find the following generalized Friedmann equation (see for instance [61][62][63][64][65][66][67][68]92,93]) where a prime marks derivative with respect to R. In the third step, we need to the GBIG Friedmann equation in the absence of any modification of the induced gravity on the brane, that is, without f (R) term on the brane. This has been obtained in the previous subsection, equation (6). Now we have all prerequisite to obtain the Friedmann equation of our GBIGmodified gravity scenario. The comparison between previous equations gives this Friedmann equation of cosmological dynamics as follows 5 where we have defined hypothetically the following energy density corresponding to curvature effect Note that to obtain this relation, E 0 , Λ 5 , λ and k are set equal to zero. From now on we restrict our attention to the normal branch of the scenario, i.e. the minus sign in equation (12) because there is no ghost instabilities in this branch if only the DGP character of the model is considered.
Note however that although we refer to the normal (ghost-free) branch of the DGP model (in the sense that for f (R) = R the obtained solutions reduce to this branch) as an indication of the ghost-free property of the considered solutions, it is not a-priori guaranteed that on the obtained dS backgrounds which generalize the normal DGP branch, the ghost does not reappear. In fact, the ghost on the self-accelerated branch of the DGP model is entirely the problem of the de Sitter background. Within the crossover scale r c which is the horizon for the self-accelerated branch, the theory reduces to a scalar-tensor model, with the scalar sector (brane bending mode) described by a simple Galileon self-interaction [94] as which, in spite of the presence of higher derivatives, propagates a single healthy degree of freedom. The ghost on the self-accelerated branch arises merely due to the fact that π gets a nontrivial profile and the kinetic term for its perturbation flips the sign on the background. A similar argument can be applied in the present work to overcome the ghost instabilities in this extended braneworld setup.
In which follows, we assume that the energy density ρ on the brane is due to cold dark matter (CDM) with ρ m = ρ 0m (1 + z) 3 . We can rewrite the Friedmann equation in terms of observational parameters such as the redshift z and dimensionless energy densities Ω i as follows where , Ω curv ≡ κ 2 4 3H 2 0 ρ 0curv 5 We note that this equation can be derived using the generalized junction conditions on the brane straightforwardly, see [81][82][83][84].
p curv , the hypothetical pressure of the curvature effect, can be obtained by the following equation of continuityρ One can obtain a constraint on the cosmological parameters of the model at z = 0 as follows Note that we have used the normalization f ′ (R)| z=0 = 1 in this relation which is observationally a viable assumption.

Cosmological dynamics in the modified GBIG scenario
Now we study cosmological dynamics in this setup. To solve the Friedmann equation for the normal branch of this scenario, it is convenient (following the papers by Bohamdi-Lopez in Ref. [86][87][88][89][90]) to introduce the dimensionless variables as follows ρ ≡ 32 27 An effective crossover distance which is appeared on the right hand side of these relations can be defined as follows and by definition Ω r ≡ 1 . Then the Friedmann equation can be rewritten as The number of real roots of this equation is determined by the sign of the discriminant function N defined as where Q and S are defined as respectively. We can rewrite N as In which follows, we consider just the real and positive roots of the Friedmann equation (21).
Then, the number of real roots of the cubic Friedmann equation depends on the minimum energy density of the brane and the situation of ρ 1 relative to this minimum. Since in our setup, curvature effect plays the role of the dark energy component on the brane, we can consider two different regimes to determine the minimum value ofρ as follows: In this case, curvature fluid plays the role of quintessence component, then the minimum value happens asymptotically at z = −1 and we will obtainρ min = 0. In this situation we can define three possible regimes: a high energy regime withρ 1 <ρ ; a limiting regime withρ 1 =ρ and a low energy regime withρ 1 >ρ. In each of these cases, depending on the sign of N there are different solutions [86][87][88][89][90][91].

ω curv < −1
In this case, the curvature fluid plays the role of a phantom component (we will investigate its phantom-like behavior in the next section) and the minimum value ofρ happens at z = 0.18. So we find theρ min as follows where we have set Ω m = 0.26. We note that w curv is not constant and as we will show, it evolves from quintessence to phantom phase. We note also that the value of redshift thatρ min occurs (that is, z = 0.18), has no dependence on the values of w curv . Here we treat only the caseρ 1 <ρ min with details. When this condition is satisfied, the function N is positive and there is a unique solution for expansion of the brane described bȳ where η is defined as We note that this condition provides a constraint on the dimensionless parameters Ω i as follows  In this case there are three possible regimes as was mentioned above. A point that should be emphasized here is the fact that in the presence of modified induced gravity on the brane, the solution of the generalized Friedmann equation (12) is actually rather involved due to simultaneously presence of H,Ḣ andḦ. A through analysis of this problem is out of the scope of this study, but there are some attempts (such as cosmography) in this direction to construct an operational framework to treat this problem, see for instance [95]. Here we have tried to find a solution of equation (12) by using the discriminant function N , the result of which is given by (27). However, we note that a complete analysis is needed for instance in the framework of cosmography of brane f (R) gravity [96]. To investigate cosmology described by solution (27), we rewrite the original Friedmann equation in the following form in order to create a general relativistic description of our model that ρ curv is defined in (13). Comparing this relation with the Friedmann equation in GR we obtain an effective energy density which can be rewritten as follows The dependence of ρ ef f on the redshift depends itself on the regimes introduced above and the form of f (R) function. Figure 2 shows the variation of ρ ef f versus the redshift for with n = 0.25. This value of n lies well in the range of observationally acceptable values of n from Solar System tests 6 [32][33][34][35][36][37][38][39][40][41][42][43][44][45]. The effective energy density shows a phantom-like behavior i.e. it increases with cosmic time. This is a necessary condition to have phantom-like behavior but it is not sufficient: we should check status of the deceleration parameter and also equation of state parameter. In a general relativistic description of our model, one can rewrite the energy conservation equation as followsρ which leads to the following relation for 1 + ω ef f 1 + ω ef f = −ρ ef f 3Hρ ef f 6 We note however that the key issue with regards to passing Solar-System tests is not the value of n, but the value of f ′ (R) today. In fact experimental data tell us that f ′ (R) − 1 < 10 −6 , when f ′ (R) is parameterized to be exactly 1 in the far past. Figure 3 shows variation of the effective equation of state parameter versus the redshift for an specific f (R) model described as (34) with n = 0.25. The effective equation of state parameter transits to the phantom phase 1 + ω ef f < 0 but there is no smooth crossing of the phantom divide line in this setup. We note that adopting other general ansatz such as the Hu-Sawicki model [97]    In this relation H and f (R) are defined as (27) and (34). Figure 4 show variation of q versus the redshift for n = 0.25. The universe enters the accelerated phase of expansion at z ≃ 2.
Another important issue to be investigated in this setup is the big-rip singularity. To avoid super-acceleration on the brane, it is necessary to show that Hubble rate decreases as the brane expands and there is no big-rip singularity in the future. Figure 5 shows variation ofḢ versus z. We see that in this modelḢ < 0 always and therefore, there is no super-acceleration and future big-rip singularity in this setup. All the previous considerations show that this model accounts for realization of the phantom-like behavior without introducing a phantom field neither on the brane nor in the bulk. Nevertheless, we have to check the status of the null energy condition in this setup. Figure 6 shows the variation of (ρ + p) tot versus the redshift. We see that this condition is fulfilled at least in some subspaces of the phantom-like region of the model parameter space.

A dynamical system viewpoint
Up to this point, we have shown that there are effective quantities that create an effective phantom-like behavior on the brane. In this respect, one can define a potential related to the effective phantom scalar field φ ef f as follows [98] V ef f (z) We note that in principle these equations can lead to V ef f = V ef f (φ ef f ), but in practice the inversion cannot be performed analytically. Now we define the following normalized expansion variables [99][100][101][102] x With these definitions, the Friedmann equation (14) takes the following form This constrain means that the phase space can be defined by the relation x 2 + y 2 − 2υ ≥ 1, since f ′ (R) − 1 < 10 −6 by solar system constraints and υ is a positive quantity. Introducing the new time variable τ = ln a, and eliminating χ and E, we obtain the following autonomous system Here primes denote differentiation with respect to τ , and q = −ä ȧ a 2 stands for the deceleration parameter To study cosmological evolution in the dynamical system approach, it is necessary to find fixed (or critical) points of the model that are denoted by (x * , y * , υ * ). These points are achieved by fulfillment of the following condition where where in this table x B and y B are as follows y B = x 2 B (w curv − 1) + (1 + w curv ) and x B = −17 w curv + 4 + 6 w 2 curv + 297 w 2 curv − 120 w curv − 180 w 3 curv + 16 + 36 w 4 curv 2 (w curv − 1) Table 2: Eigenvalues and the stability regions of the critical points. ς(w curv ), ξ(w curv ) and ζ(w curv ) are complicated functions of w curv . The critical curve F for which w curv = −1 is a stable de Sitter phase.
A part of dynamical system analysis of this model is summarized in table 1. The critical points A and B demonstrate the early-time, matter dominated epoch which lead to a positive deceleration parameter. Points C and D which are phases with vanishing matter character, that is Ω m = 0, can explain the positively accelerated phase of the universe expansion for ω curv ≥ −6.88. Critical curve E also demonstrates a positively accelerated phase for all values of the equation of state parameter of the curvature fluid. Critical curve F which exists only for the case ω curv = −1, is a de Sitter phase in this model. Existence of a stable de Sitter point and an unstable matter dominated phase (in addition to radiation dominated era) in the universe expansion history is required for cosmological viability of any cosmological model. In order to investigate the stability of these points, one can obtain the eigenvalues of these points separately. Based on table 2, in order for point A to be an unstable point, it is necessary to have ω curv < 1 4 . Therefore, the point A as a saddle point agrees with what we have shown in figure 7. Now the stability of the positively accelerated phases of the model depends on whether the curvature fluid plays the role of a quintessence scalar field or not. Points C, D and E of table 2 are stable phases of this model if ω curv > −1. Whereas if the curvature fluid plays the role of a cosmological constant, the point F will be a stable de Sitter phase. We note that generally, if a nonlinear system has a critical curve, the Jacobian matrix of the linearized system at a critical point on the curve (line in our 2-dimensional subspace) has a zero eigenvalue with an associated eigenvector tangent to the critical curve at the chosen point. The stability of an specific critical point on the curve can be determined by the nonzero eigenvalues, because near this critical point there is essentially no dynamics along the critical curve [103]. We have plotted the phase space of this model in subspace x − y with ω curv = −1. As we see, point A is a saddle point and curve F is a stable de Sitter curve. This model is cosmologically viable since there are an unstable matter dominated point and a stable de Sitter phase.

Confrontation with Recent Observational Data
In this section we use the combined data from Planck + WMAP + high L+ lensing + BAO [104] to confront our model with recent observation. In this way we obtain some constraints on the model parameters; especially the Gauss-Bonnet curvature contribution. For this purpose, we consider the relation between Ω curv and Ω m in the background of the mentioned observational data. We suppose that Ω curv plays the role of dark energy in this setup. Figure 8 shows the result of our numerical study. In this model with Ω rc ∼ 10 −4 (see [105] for instance), Ω α is constraint as follows ω curv = −0.5 : 0.008 < Ω α < 0.011 ω curv = −0.92 : 0.01 < Ω α < 0.055 On the other hand, if we consider the Ω ef f defined as Ω ef f = κ 2 4 3H 2 0 ρ ef f as our main parameter, the result will be as shown in figure 9. In this case we have the following constraint on Ω α Ω curv = 0.7 : 0.0043 < Ω α < 0.036 Ω curv = 0.5 : 0.0048 < Ω α < 0.0081

Summary and Conclusion
In this paper, we have constructed a DGP-inspired braneworld scenario where induced gravity on the brane is modified in the spirit of f (R)-gravity and higher order curvature effects are taken Figure 8: Ω curv versus Ω m in the background of Planck + WMAP + high L+ lensing + BAO joint data. Figure 9: Ω ef f versus Ω m in the background of Planck + WMAP + high L+ lensing + BAO joint data.
into account by incorporation of the Gauss-Bonnet term in the bulk action. It is well-known that the normal branch of the DGP braneworld scenario, which is not self-accelerating, has the potential to realize phantom-like behavior without introducing any phantom fields neither on the brane nor in the bulk. Our motivation here to study this extension of the DGP setup is to explore possible influences of the curvature corrections, especially the modified induced gravity, on the cosmological dynamics of the normal branch of the DGP setup. In this regard, cosmological dynamics of this scenario as an alternative for dark energy proposal is studied and the effects of the curvature corrections on the phantom-like dynamics of the model are investigated. The complete analysis of the generalized Friedmann equation needs a cosmographic viewpoint to f (R) gravity, but here we have tried to find an special solution of this generalized equation via the discriminant function method. In our framework, effective energy density attributed to the curvature plays the role of effective dark energy density. In other words, we defined a curvature fluid with varying equation of state parameter that incorporates in the definition of effective dark energy density. The equation of state parameter of this curvature fluid is evolving and the effective dark energy equation of state parameter has transition from quintessence to the phantom phase in a non-smooth manner. We have considered a cosmologically viable (Hu-Sawicki) ansatz for f (R) gravity on the brane to have more practical results. We have shown that this model mimics the phantom-like behavior on the normal branch of the scenario in some subspaces of the model parameter space without introduction of any phantom matter neither in the bulk nor on the brane. In the same time, the null energy condition is respected in the phantom-like phase of the model parameter space. There is no super-acceleration or big rip singularity in this setup. Incorporation of the curvature effects both in the bulk (via the Gauss-Bonnet term) and on the brane (via modified induced gravity) results in the facility that curvature fluid plays the role of dark energy component. On the other hand, this extension allows the model to mimic the phantom-like prescription in relatively wider range of redshifts in comparison to the case that induced gravity is not modified. This effective phantom-like behavior permits us to study cosmological dynamics of this setup from a dynamical system viewpoint. This analysis has been performed with details and its consequences are explained. The detailed dynamical system analysis of this setup is more involved relative to the case that there are no curvature effects. We have shown that with suitable condition on equation of state parameter of curvature fluid, there is an unstable matter era and a stable de Sitter phase in this scenario leading to the conclusion that this model is cosmologically viable. We have constraint our model based on the recent observational data from joint Planck + WMAP + high L+ lensing + BAO data sets. In this way some constraints on Gauss-Bonnet coupling contribution are presented. We note that no Rip singularity is present in this model since the Gauss-Bonnet contribution to this model is essentially a stringy, quantum gravity effects that prevents the Rip singularity (see for instance [106] for details).