5 D. f(r) gravity f(r) gravity is the first modified gravity model proposed as an alternative explanation for the accelerated expansion of the Universe [9]. We write the gravitational action as S = d 4 x g [R + f(r)]. (46) 6πG N GR with cosmological constant is clearly recovered through f(r) = 2Λ. Choosing f(r) = ar + b just corresponds to a rescaling of G N. Only if f is a non-linear function of R do we obtain an actual modification of GR. The field equations can be obtained through variation of the action Eq. (46), f f R G µν 2 g µν 2 f RR f R g µν µ ν f R = T µν. (47) Here, f R df/dr. Since R contains second derivative of the metric, these equations are fourth order in derivatives if f R is not simply a constant, i.e. if f is a non-linear function. Usually, higher order field equations are a bad sign, since they often entail instabilities. We will see however that there is a wide class of well-behaved f(r) models. Eq. (46) can be transformed into a somewhat more transparent form. We use make a trick, defining a new field φ through φ = + f R (R). (48) As long as we assume a non-trivial f(r) model with f RR d 2 f/dr 2 = 0, this relation is invertible to R(φ). We further define a potential Λ(φ) = [R(φ)φ f(r(φ))]. (49) 2 Then, Eq. (46) becomes S = 6πG N d 4 x g [φr 2Λ(φ)]. (50) This is just a scalar-tensor theory of the form Eq. (9), although a peculiar one with ω = 0, where f R plays the role of scalar field (this can already be noticed by comparing Eq. (47) with Eq. (20)). While in GR there is an algebraic relation, R = 8πG N T, (5) in f(r) this relation is replaced by a field equation for f R : 3f R R + f R R 2f = 8πG N T. (52)
6 This corresponds to a scalar-tensor theory with a potential V/ f R = R + f R R 2f. The curvature of the potential around the potential minimum is m 2 (f R ) = λ 2 C = + fr R 3 f RR 3 f RR, (53) where the approximate equality holds in the weak field limit, f R. f(r) gravity has attracted significant interest in the realm of cosmology. We can divide models into two classes: i) f is an increasing function of R; in this case the modifications become important at high spacetime curvature, e.g. in the early Universe. ii) f is a decreasing function of R; the modifications become important at low spacetime curvature, in particular in the late Universe. The first kind has attracted interest in the context of inflation, while the second kind has been proposed as an alternative to Dark Energy. We will discuss the second kind here. Since we have a free function to choose, f(r) gravity can in fact reproduce essentially any expansion history. One possible choice is which when R R c becomes f(r) 2Λ f(r) = 2Λ R/R c R/R c +, (54) R c + O((R c /R) 2 ), (55) R and thus effectively provides a cosmological constant, with corrections becoming important in regions of small spacetime curvature. On the other hand, as R 0, f(r) 0 and thus the model does not employ a true cosmological constant. The field then is R 2 0 f R = 2Λ R c R 2 = f R0 R 2 (56) where we have defined a new dimensionless parameter f R0 < 0, which gives the field value corresponding to the background curvature of the Universe today, R 0. Thus, R(f R ) f /2 R, and we see that Eq. (52) includes a run-away potential just like chameleon models. Matching Eq. (52) in the quasi-static case neglecting the potential ( 2 f R = 8πG N /3 ρ) immediately leads to the necessary requirement for the chameleon mechanism U(r ch ) = 2 3 f R. (57) Since f R = f R0 today, this implies a constraint of f R0 0 6 today from Solar System constraints, a significant constraint on f(r) models designed to produce an accelerated expansion. On the other hand, the Compton wavelength of such models (Eq. (53)) is still in the range of several Mpc, and thus can leave imprints in cosmological structure.
7 E. Braneworld models Braneworld models are a class of higher-dimensional models of gravity. We know from lab tests and particle experiments that matter fields and the three forces apart from gravity follow a fourdimensional spacetime. There are thus two options for introducing extra dimensions: first, one can compactify them to a scale R TeV, thus hiding them from current measurements. This is the standard approach adopted in string theory (which needs at least 0 spacetime dimensions to work). Models with warped extra dimensions (such as the Randall-Sundrum model) can also be included in this class, as they do not modify the large-distance behavior of gravity. Another option is to think of our four spacetime dimensions as a sub-manifold (brane) embedded in a higher-dimensional spacetime. If we confine all matter and interactions to this brane, apart from gravity, we can reproduce the observable Universe. The key point is to allow gravity to propagate into the higher dimension(s) (bulk), allowing for interesting signatures in the gravity sector. One motivation for this behavior comes, again, from string theory: fermions and vector bosons are modeled as open strings, which can be confined to a brane by attaching their endpoints to it (Dirichlet boundary condition, hence D-branes). On the other hand, gravitons are modeled as closed strings which cannot be attached to branes, and hence propagate through the entire bulk. Here, I will only consider this second class, since the compact extra dimensions essentially do not have any large-scale observable signatures.. The Dvali-Gabadadze-Porrati (DGP) model The Dvali-Gabadadze-Porrati model [20] consists of a spatially three-dimensional brane in a 4+ dimensional Minkowski bulk. Matter and all interactions except gravity are confined to the brane. The gravitational action consists of the five-dimensional Einstein-Hilbert action plus a term which leads to the 4D gravity limit on small scales: S grav = 6π G (5) d 5 X g (5) R (5) + S boundary 6π G (4) d 4 x g R (4) (4) + L m. (58) Here, X, g (5) stand for the bulk coordinates and metric, while x, g (4) are the induced coordinates and metric on the brane, and R (5), R (4) denote the corresponding Ricci scalars. In the following, we will drop the (4) notation where no confusion can arise. The boundary term is added to the action in order to ensure that variation with respect to g (5) leads to the correct five-dimensional Einstein equations (e.g., [2]).
8 The two gravitational constants, or Planck masses M Pl (5), M Pl (4) appearing in Eq. (58) can be related via a length scale, the crossover scale: r c G (5) 2 G (4) = 2 M 2 Pl (4) M 3 Pl (5). (59) On scales above r c, gravity becomes five-dimensional, with forces falling off as /r 3. Below r c, gravity is four-dimensional, but not Einsteinian gravity, a point to which we return below. This can be summarized as a modified graviton propagator given by k 2 + k/r c. (60) Since all matter is thought as confined to the brane, the five-dimensional metric has to obey nontrivial junction conditions over the brane [22]. Assuming an empty Minkowski bulk, a spatially flat brane, and a homogeneous and isotropic matter distribution on the brane, the junction conditions lead to the following analogue of the Friedmann equation for the scale factor of the induced metric on the brane: H 2 ± H r c = 8π G 3 ρ, H ȧ a. (6) The junction conditions leave two possible branches of the theory, determined by the sign on the left hand side of Eq. (6). The branch with the sign asymptotes to a late-time de Sitter-Universe, H = /r c = const., and is correspondingly called the self-accelerating branch. On the other hand, the + sign leads to the so-called normal branch, which does not lead to any acceleration. Even in the normal branch, the expansion history is different from ΛCDM, and corresponds to an effective dark energy with w eff /2 in the matter-dominated era at high redshifts. 2. Cosmological perturbations and brane bending mode The propagation of light and particles on the DGP brane is completely determined by the perturbed 4D Friedmann-Robertson-Walker metric. However, in order to determine the evolution of the metric potentials on the brane, it is necessary to solve the full 5D Einstein equations [9, 23]. An additional scalar degree of freedom associated with local displacements of the brane appears, the so-called brane-bending mode ϕ which couples to matter. In our convention, ϕ is dimensionless, instead of being scaled to the Planck mass M Pl (4). In the decoupling limit of DGP [24], when setting gravitational interactions to 0, the selfinteractions of the ϕ field remain constant. Hence, while perturbations of the metric higher than
9 linear order can be neglected for cosmological studies, it is crucial to consider the self-interactions in the brane-bending mode. The field equation reads ϕ + r2 c [(ϕ)2 ( µ ν ϕ)( µ ν ϕ)] = 8πG N δρ, (62) where δρ = ρ m ρ m is the matter density perturbation, and the function β(a) is given by: β(a) = ± 2H(a) r c + Ḣ(a) 3H 2. (63) (a) Note that β is always negative in the self-accelerating branch, signaling that gravity is weakened by a repulsive brane-bending mode. This is unexpected for scalar field, and in fact it has been shown that perturbations around the de Sitter limit of the self-accelerating branch are ghosts (i.e. the kinetic term has the wrong sign). The normal branch does not suffer from this issue. The brane-bending mode influences the dynamics of particles and light in the same way as in standard scalar-tensor theories. In particular, g 00 = g GR 00 + 2 φ g ij = g GR ij 2 φδ ij. (64) Thus, we again see that dynamics of non-relativistic objects couple to φ, whereas light does not. 3. Weak brane regime If the gradient of the ϕ field is small, i.e. for small gravitational accelerations, Eq. (62) can be linearized, yielding a standard Poisson equation: 2 ϕ L = 8πG N δρ ϕ L = 2 U. (65) In this regime, also called weak-brane phase, ϕ = ϕ L becomes proportional to the Newtonian potential U, corresponding to a constant rescaling of the gravitational constant through Eq. (64): G N G eff = + G N. (66) Substituting the linear solution into Eq. (62), we obtain a rough estimate for the overdensity δ δρ/ ρ m at which the non-linear interactions become important: r2 c ( 2 ϕ L ) 2 / 2 ϕ L = 8πG r2 c 9β 2 δρ = 3H2 0 r2 c 9β 2 Ω m δ. (67) For a self-accelerating DGP model that leads to cosmic acceleration today, r c H0, β(a = ), and the prefactor in Eq. (67) is of order unity today. Hence, the self-coupling of the brane bending mode ϕ becomes important as soon as the matter density field becomes non-linear, δ.
20 4. Vainshtein effect In general, Eq. (62) is difficult to solve in full generality due to the non-linearity in the derivative terms. However, an instructive test case, a spherically symmetric matter distribution, can be solved analytically. In the spherically symmetric case, Eq. (62) becomes [23]: d 2 dr 2 + 2 d (ϕ + r dr Ξ) = 8πG N δρ, (68) r Ξ 2rc 2 dr dϕ 2 r dr. 0 For simplicity, we assume a spherical mass M of radius R with uniform density, and set a =. Then, we can integrate Eq. (68) once and obtain the gravitational acceleration in DGP: g = g N + dϕ 2 dr = g N[ + Δ(r)], (69) where g N is the Newtonian acceleration of the spherical mass, and: Δ(r) = 2 r 3 /r 3 + r r 3, r R R 3 /r 3 + r R 3, r < R. Here, r denotes a characteristic scale of the solution, the Vainshtein radius: (70) r 3 = 8 r2 c r s 9β 2, (7) and r s = 2GM is the gravitational radius of the mass. For very large distances, r r, Δ(r) approaches the constant /(), which exactly matches the linear solution, Eq. (66). Note also that by substituting δρ M/r 3 in Eq. (67), we see that the non-linearity criterion is directly proportional to (r /r) 3. In the opposite limit, r r, ϕ becomes suppressed with respect to the linear solution, and Δ(r) approaches the small constant 2/()(R/r ) 3/2 inside the mass (assuming R r ). This Vainshtein effect [25, 26] amounts to restoring GR in deep potential wells, while far away from the mass, gravity is in the scalar-tensor regime. Interestingly, outside the mass but well within the Vainshtein radius, Δ(r) scales as (r/r ) 3/2, and thus g N Δ r /2. This departure from a pure /r 2 force leads to anomalous precession of objects such as the moon. Those constraints could become competitive with future measurements. Another exact solution to Eq. (62) exists: for a simple plane wave density perturbation, δρ(x) = A exp(ik x). In this case, the two non-linear terms exactly cancel, and we are left with the linear solution, Eq. (65). The plane wave and spherically symmetric solutions can be considered as two limiting cases for understanding the ϕ field behavior in the cosmological context.