f(R) and f(T) theories of modified gravity

We briefly review f(R) theories, both in the metric and Palatini formulations, their scalar-tensor representations and the chameleon mechanism that could explain the absence of perceptible consequences in the Solar System. We also review f(T) theories, a different approach to modified gravity consisting in a deformation of the teleparallel equivalent of General Relativity. We show some applications to cosmology and cosmic strings. As f(R)'s, f(T) theories are not exempted from additional degrees of freedom; we also discuss this still open issue.


I. INTRODUCTION
In the last five decades many theories of modified gravity have been proposed in connection with different physical purposes. In the 60's, Brans and Dicke coupled a scalar field φ to the metric g µν to get a variable effective gravitational constant. 1 In Brans-Dicke theory, the scalar field is a new degree of freedom of the gravitational field, which is not directly coupled to the matter, but it exerts influence by entering the dynamical equations that govern the spacetime geometry. The gravitational Brans-Dicke action contains a new constant ω that should be dictated by the experiment: where κ ≡ 8πG, and the signature + − −− was adopted. Also in the 60's, the Einstein-Hilbert Lagrangian was added with terms quadratic in the curvature to tackle the renormalization of the theory. 2 In 1971 Lovelock 3 considered terms of higher order in the curvature as well; but he was driven by another motivation. While this kind of Lagrangians leads, in general, to fourth order dynamical equations because they contain second order derivatives, Lovelock obtained the more general Lagrangian polynomic in the curvature and leading to conserved second order equations for the metric. Lovelock discovered that the bigger the spacetime dimension is, the bigger is the number of terms these Lagrangians can contain. For instance, the Einstein-Lanczos Lagrangian, is the Lovelock Lagrangian for dimensions 5 or 6; it is the Einstein-Hilbert Lagrangian with a cosmological constant plus a quadratic term. If the dimension is 4, then a) Electronic mail: ferraro@iafe.uba.ar; Member of Carrera del Investigador Científico (CONICET, Argentina) the added quadratic term becomes a topological invariant (Euler's characteristic); so it does not contribute to the variation of the action. Then we recover the Einstein equations as the sole conserved second order equations for the metric in four dimensions.
In 1970 Buchdahl proposed to replace the Einstein-Hilbert scalar Lagrangian with a function of the scalar curvature, and studied its cosmological consequences. 4 This type of modified gravity is nowadays called a f (R) theory. In 1983 Milgrom 5 thought that the galactic rotation curves were an evidence of the fail of Newtonian gravity to describe gravitation in the weak field regime (a g << a o ≈ 10 −10 ms −2 ). According to Milgrom, no dark matter was needed to explain the data but a theory of modified gravity. In the deep MOND (Modified Newtonian Dynamics) regime of Milgrom's theory, the acceleration of gravity goes to a g = √ a o a gNewton . In the last decade Bekenstein 6 developed a relativistic theory of gravity named TeVeS, because it combines the metric tensor, a vector field and a scalar field. TeVeS includes Milgrom's weakening of Newtonian gravity in the weak field regime, and has also consequences for lensing phenomena, cosmology, etc. String theory has been also a source of inspiration for theories of modified gravity. Just to mention a case, DGP gravity 7 describes the 4-dimensional universe as immersed in a 5-dimensional manifold. Thus a "normal" 5D gravity can cause large scale effects in 4D, as the accelerated expansion with no presence of dark energy. These 4D consequences are driven by a scalar field named galileon because of the symmetries it obeys. 8,9 II. f (R) THEORIES The simplest way of modifying Einstein's General Relativity is to replace the scalar Lagrangian R with a function f (R) of the scalar curvature: 4,10,11 By properly choosing the function f , one could generate "f (R)" theories departing from General Relativity both at small and large scales. So, deformations at large curvatures could be employed for smoothing singularities; while deformations at large scales could be useful to geometrically explain the accelerated expansion without resorting to dark energy. The weak field regime of the deformed theory also opens a way to explain phenomena otherwise attributed to dark matter.  are comprehensive reviews on f (R) theories.
As it happens in General Relativity, there are two ways of varying a metric theory of gravity. One can assume the Levi-Civita connection to write the Ricci tensor; so the metric is left as the sole dynamical variable. Alternatively, one could regard the affine connection Γ λ µν and the metric g µν as independent dynamical variables. The first way of variation is called metric formalism; the second one is the Palatini formalism. 19 In General Relativity there is no difference between the results of metric and Palatini formalisms, since the affine connection making stationary the Einstein-Hilbert action is precisely the Levi-Civita connection. Actually, the equivalence between both formalisms is valid for all the Lovelock's Lagrangians. 20 Instead, in f (R) theories both procedures should be separately studied because they yield different dynamics. Before choosing one of both formalisms, the variation of the f (R) Lagrangian density can be written as where the formula δ ln(det[g µν ]) = −g µν δg µν , valid for any matrix, was used to vary the determinant of the metric. Besides, δR µν can be expressed in terms of variations of the affine connection Γ λ µν (whatever Γ λ µν is; see for instance Ref. 21): Notice that the connection is not a tensor; but the difference δΓ λ µν between two different connections does transform as a tensor.

III. METRIC FORMALISM FOR f (R) THEORIES
If the Levi-Civita connection is assumed, then second derivatives of g µν are under variation in Eq. (5). As a consequence, fourth-order Euler-Lagrange equations should be expected as a result of the (double) integration by parts induced by the variation (5). It is, however, remarkable that g µν δR µν is, in this case, a four-divergence. This is because the Levi-Civita connection is metric, so the metric can enter the covariant derivative. This is the reason why General Relativity (f ′ (R) = 1) remains as a theory governed by second order dynamical equations. Contrarily, f (R) theories in the metric formalism are characterized by fourth-order dynamical equations: (6) Notice that f ′ (R) acts as renormalizing the gravitational constant κ; so, only functions with f ′ > 0 should be considered (besides, f ′′ > 0 to avoid instabilities [22][23][24]. Differing from General Relativity, these equations link the scalar curvature R and the trace T of the energymomentum tensor not algebraically but differentially. In fact, the trace of Eq. (6) is which displays the propagation of a new degree of freedom associated with f ′ (R) (this degree of freedom is absent in General Relativity since it is f ′ (R) = 1). A f (R) theory can be rephrased as a scalar-tensor theory governed by second order dynamical equations. [25][26][27] To show it, let us start from the following action containing a metric tensor g µν and a scalar field φ: The variation with respect to φ gives R = V ′ (φ), so linking the scalar field to the metric. This result also implies that the Lagrangian in (8) is nothing but the Legendre transform of the function V (φ); therefore, it depends just on R. So we can call it f (R): By anti-transforming, one also gets These results show that a f (R) theory in the metric formalism is dynamically equivalent to the action (8), where f (R) and V (φ) are related through the Legendre transformation (9). Notice that S grav [g, φ] in Eq. (8) resembles a Brans-Dicke theory with ω = 0 (absence of kinetic term). The action (8) is written in the so called Jordan frame representation of the theory. By transforming to the Einstein frame representation we will obtain second order dynamical equations. So let us define whereg µν is conformally related to g µν through the scalar field φ. Then, one applies the relation for the scalar curvatures of conformally related metrics, 28 and throws out a surface term to write the action in the form 91 As can be seen in Eq. (13), the action in the Einstein frame gets a canonical form: it describes a "gravitational field"g µν and a minimally coupled scalar fieldφ, governed by standard second order equations. So, we have success in reducing the order of the equations (of course, by widening the number of variables). An easy identification of the degrees of freedom is now possible: 2 degrees of freedom related to the tensorg µν plus 1 degree of freedom associated with the massive scalar fieldφ.

A. The chameleon effect
Because of the observations in the Solar System, Brans-Dicke theory is constrained to values |ω| > 40000.
Despite that metric f (R) theories have ω = 0 in their equivalent representation (9), they are not ruled out. This is so because they contain a potential V (φ) (or U (φ)) that is absent in Brans-Dicke theory. This potential could be useful to hide the scalar degree of freedom within the Solar System. In other words, the observations in the Solar System could agree with metric f (R) theories, whenever the scalar degree of freedom does not appreciably distort the spherically symmetric outer static Schwarzschild solution for a typical stellar object. Even so, the scalar field could have physical effects at other scales, as to be the cause of the accelerated cosmological expansion, etc. This behavior, called the chameleon effect, has been proposed in Ref. 29 (cf. Refs. 16,[30][31][32][33][34][35], and is strongly dependent on the choice of the potential U . The idea can be exemplified by considering solutions to the dynamical equation (7) for φ = f ′ (R). Alternatively, this equation can be also obtained by adding the action (13) with an action for matter minimally coupled to the metric g µν = e (− √ 2κ/3φ)g µν , and varying with respect toφ. We are interested in static spherically symmetric solutions. In such case, the resulting equation reduces to where ρ ≡ T mat µ µ > 0. Here we are momentarily ignoring the back-reaction on the metric, by choosing a Minkowskian backgroundg µν = η µν .
We will divide the space in two regions of constant density: the region inner to a spherical star of radius R ⊙ and density ρ c , and the outer region filled by a medium of lower density ρ o . For a constant energy density ρ, we can define (in each region) the effective potential We have to choose a potential U allowing the chameleon effect to become apparent. In Figure 1, U has been chosen in such a way that U ef f has a minimum in both regions (we call themφ o andφ c ). We will search for a solution varying betweenφ c at the center of the star and φ o at infinity (grey strip in Figure 1). Outside the star, we expand U ef f at the minimumφ o : where m is the mass of the field in this approach. Thus, the outer solution is Inside the star, we assume that the exponential term U ef f dominates on U . Then if 8κ/3 |φ| << 1. So, the inner solution is Notice that the integration constant R s fulfills So, one can take Eq. (20) to be the inner solution for R s < r < R ⊙ , and extended it as the constantφ =φ c to r = 0 (in fact, U ′ ef f (φ c ) = 0). Thus, R s in Eq. (20) and C in Eq. (18) remain as two integration constants to be determined by the continuity of the solution (20) and (18) and its derivative at r = R ⊙ . By assuming that m R ⊙ << 1, one obtains the following two equations for R s and C: where Φ N = κ ρ c R 2 ⊙ /6 is the Newtonian potential on the surface of the star, and The chameleon effect happens when the potential U is such that ǫ << 1. In fact, in such case it is so the effect of the potentialφ around the star is negligible compared with Newtonian gravity. 92 Besides, then the inner solution differs fromφ c just in a thin-shell near the surface. The back-reaction on the metric has been considered in Ref. 32; it is proved that the PPN parameter characterizing the departure from the Schwarzschild metric is γ ≃ 1 + 2/3 ǫ. The Cassini tracking constrains ǫ to be ǫ 10 −5 in the Solar System. Since the Newtonian potential on the Sun surface is Φ Sun ∼ 10 −6 , one obtains The viable f (R) theories are those having a potential U accomplishing this relation. A typical f (R) used to model the accelerated expansion is: 32,34,36,37 because the R −n term dominates at low curvature. By replacing the potential U of Eq. (14) in Eq. (16), one gets where φ = exp[ 2κ/3φ]. If ρκ >> µ 2 , the minima of U ef f in each region are which are very near to zero as required by Eq. (27). This also means that there is no sensitive difference between metrics g µν andg µν (see Eq. (11)). In the Solar System, the density ρ o outside the Sun should be replaced by the mean density of the barionic matter in the galaxy (ρ galaxy κ is around 10 5 times the squared Hubble constant). Since ρ galaxy << ρ Sun , the Eq. (27) becomes in order that the model (28) be acceptable. The above used condition mR ⊙ << 1 can be accomplished too; in fact B

. Metric f (R) theories in cosmology
Within the framework of a FRW universe, it has been shown that the fieldφ is attracted to the minimum of the potential (29), and then adiabatically evolves following the Eq. (30) with ρ = ρ universe (t). Using the usual approximations, it is obtained that the net effect of the presence ofφ is the adding of a constant to the density of matter. 32,38 In such case, the cosmological effects resulting from the model (28) would be undistinguishable from a mere cosmological constant ("vanilla dark energy"). The growing of inhomogeneities are, however, a more promising arena to distinguish among f (R) theories and the ΛCDM model. [39][40][41] f (R) theories have also be applied to modify the high curvature regime. The simplest example is which produces inflation, withφ playing the role of inflaton. 42,43 Figure 2 shows that the potential U (φ) is nearly flat for large values ofφ, as required to get inflation.
Other cosmological effects, such as lensing due to overdensities of matter, have also been considered in the framework of metric f (R) theories. 44

IV. PALATINI FORMALISM FOR f (R) THEORIES
In Palatini formalism 19,45 the connection Γ λ µν and the metric g µν are regarded as dynamical variables to be independently varied. Thus, ∇ in Eq. (5) is just the covariant derivative for an arbitrary connection Γ λ µν . The variation of the action with respect to the connection involves the integration by parts of the first term in the Eq. (4); what results is not a dynamical equation but a constraint for the connection: 30,46,47 (this is the result when torsion is neglected 48,49 ). General Relativity is a special case, in the sense that the connection (34) is the Levi-Civita connection when f (R) = R; so, no difference exists between metric and Palatini formalisms in General Relativity. But, in a general case, the connection (34) is not metric for g µν , but for the conformal metricg µν = f ′ (R) g µν . On the other hand, the variation of the action with respect to the metric in Eq. (4) does yield dynamical equations: where R µν and R are built with the connection (34).
Here T µν is the usual energy-momentum tensor, whenever the action for matter does not contain covariant derivatives. As a remarkable difference compared with the metric formalism, the trace of Eq. (35) does not govern the propagation of a scalar degree of freedom but it is a mere algebraic relation between the curvature R and the matter distribution:

V. f (T ) THEORIES
General Relativity can be reformulated in a teleparallel framework by taking the field of orthonormal frames or tetrads as the dynamical variable instead of the metric tensor. 59 The tetrad is a basis {e a (x)}, a = 0, 1, 2, 3, of vectors in the spacetime. Each vector e a can be decomposed in a coordinate basis, so giving the components e µ a ; thus, the orthonormality condition reads: where η ab = diag(1, −1, −1, −1). This relation can be inverted with the help of the co-frame {e a }, defined as to obtain the metric starting from the tetrad: The Teleparallel Equivalent of General Relativity (TEGR) is a theory for the tetrad, whose dynamical equations are equivalent to Einstein equations whenever the tetrad is related to the metric through the Eq. (39). The TEGR Lagrangian does not contains second derivatives because it is quadratic in the tensor which is reminiscent of the electromagnetic field tensor (in fact, it is built of the set of four exact 2-forms T a ≡ de a ). The tensor (40) can be regarded as the torsion of the Weitzenböck connection, Weitzenböck spacetime has torsion but it is flat, because the Riemann tensor associated with the connection (41) is identically null. The connection (41) has the nice property that a vector is parallel-transported iff its projections on the tetrad remain constant; in fact, ∇ ν V µ = e µ a ∂ ν (e a λ V λ ). Moreover, Weitzenböck connection is metric compatible since ∇ ν e µ a ≡ 0. Weitzenböck connection could be compared with Levi-Civita connection by using Eq. (39). It results that they differ in a tensor named contorsion. The contorsion takes part in the TEGR Lagrangian, since the TEGR action is: 60,61 The equivalence between TEGR action and Einstein-Hilbert action comes from the fact that their Lagrangians differ in a four-divergence: where R[e a ] is the scalar curvature for the Levi-Civita connection, with the metric replaced with (39). In particular, the four-divergence encapsulates all the second derivatives contained in the Einstein-Hilbert Lagrangian.
In the same spirit than a f (R) theory, a f (T ) theory consists in a deformation of the TEGR Lagrangian: 62-66 where T ν λ is the energy-momentum tensor.

A. Cosmology
The first f (T ) model was proposed to avoid the Big-Bang singularity and obtain inflation without resorting to an inflaton. 62 But most of the cosmological applications concentrated in the late accelerated expansion of the universe. 64,65,67-71 A flat FRW universe is described by e a µ = diag[1, a(t), a(t), a(t)] in comoving coordinates, where H ≡ȧ/a is the Hubble parameter. Thus, the dynamical equations (46) become 12 where ρ and p are the energy density and pressure of the fluid of matter (the conservation lawρ = −3H(ρ + p) is guaranteed by Eq. (48)). In Ref. 62 a high curvature deformation, f (T ) = λ( 1 + 2T /λ− 1), was proposed to correct the evolution near the Big-Bang (more precisely, when |T | is of the order of λ). It was found that the Big-Bang is removed and replaced with an exponential expansion (H(t) goes to λ/12 when t → −∞) for any state equation p = w ρ with w > −1. As a consequence, the particle horizon diverges and the whole universe turns out to be causally connected.
Other no less important issues, such as the growth of fluctuations, 72-74 the observational constraints [75][76][77] or the variation of the universal constants, 78,79 have also been studied in f (T ) cosmology.

B. Cosmic strings
Static circular 80 or spherically 81,82 symmetric solutions are also analyzed in the f (T ) literature. In particular, it has been shown that the Schwarzschild geometry remains as a solution of f (T ) theories. 83 The issue of removing singularities in stationary configurations was studied in a slightly different framework of modified TEGR, by using a Lagrangian density inspired in Born-Infeld electrodynamics: 84 (g µν is that of Eq. (39)). TEGR is recovered in the limit λ → ∞ if T r(F ) = S · T. A possible choice, but not the only one, is F µν = S µλρ T λρ ν ; then (50) This expression shows that the theory (49) modifies General Relativity at high curvatures and differs from a mere f (T ) theory. These features make it potentially able of avoiding the singular Schwarzschild solution or any other solution having S · T = 0. 63,83 The Lagrangian (49) was used in Ref. 84 to heal the singular behavior of a cosmic string: In General Relativity it is Y = 1. In particular, if the dimension is reduced to D = 2+1 (z is removed), the Y = 1 case is a solution of Einstein equations for T 00 = µ δ(x, y) and T 0i = (J/2) ǫ ij ∂ j δ(x, y), where µ ≡ (1 − M )/4. So the solution (51) looks like the geometry associated with a particle of mass µ and spin J (a cosmon). 85 However, no gravitational field surrounds the cosmon since the metric is manifestly flat (in terms of the Levi-Civita curvature). Instead, the presence of a cosmon only produces topological effects: the deficit angle 8πµ (conical singularity), and the existence of closed timelike curves (CTC) of constant (t, ρ, z) when ρ < ρ o ≡ 4J/M : When the geometry (51) is treated within the modified gravity framework ruled by the Lagrangian (49), then Y becomes a J-depending function of ρ. Y (ρ) goes to 1 for ρ >> 4J/M (GR limit) but diverges for ρ → 4J/M . 84,86 Besides, the solution J = 0 coincides with the respective GR solution. While J in General Relativity has no local effects (J could be locally absorbed through the coordinate change t ′ = t + 4J θ), now the integration constant J is a physically relevant degree of freedom that fixes the scale ruling the GR limit. The curved geometry that replaces the GR cosmic string has remarkable features: i) the Levi-Civita curvature is well behaved at ρ o = 4J/M (R, R µν R µν and R λµνρ R λµνρ vanish at ρ o ), ii) an infinite proper time is required to reach ρ o , and iii) no CTC's are left. So, the theory (49) successfully smoothes the GR cosmic string.
C. Degrees of freedom in f (T ) theories f (T ) gravity is structurally simpler than metric f (R) theories, because it always produces second order dynamical equations. However, this nice feature does not prevent f (T ) gravity from displaying additional degrees of freedom. The circular symmetric solution of the previous section, obtained in the context of an extension of f (T ) gravity, exhibits a local degree of freedom associated with the integration constant J that is only globally apparent in the corresponding GR solution. Since f (T ) is a theory not for the metric but for the tetrad, it contains more degrees of freedom than GR. In fact, many different tetrads lead to the same metric, since the relation (39) is invariant under local Lorentz transformation of the tetrad field. In order that a theory for tetrads has the same degrees of freedom than a theory for the metric, its action should be invariant under local Lorentz transformation of tetrads in the tangent space. TEGR is a particular case accomplishing this condition: although S · T does vary under local Lorentz transformation of the tetrad field, the variation is located in the divergence term of Eq. (44); therefore the dynamics does not vary.
But in a f (T ) theory, the variation affects the dynamics because the divergence term remains encapsulated in the function f . Only a global Lorentz invariance survives in such case. Because of this reason, a f (T ) theory globally determines the field of tetrads; it provides the spacetime with a global frame that fixes its metric and endows it with a parallelization. A local Lorentz transformation would destroy the parallelization (consider, for instance, a Cartesian grid in Minkowski spacetime). As was proven in Ref. 87, the local Lorentz invariance cannot be restored by adding the action with a spin connection. The issue of counting the number of degrees of freedom in a f (T ) theory could be tackled by reformulating the f (T ) action in a Brans-Dicke-like form. 88 Nevertheless, the counting the first and second class constraints in the Hamiltonian formulation shows that f (T ) theories in four dimensions have five degrees of freedom. 89 Summarizing, in passing from TEGR to f (T ) gravity, we are replacing a local symmetry with a global one; in return, we would be converting global degrees of freedom (like the topological J in the cosmic string) into local degrees of freedom. These new local degrees of freedom could be essential to heal singularities. As a last remark, it should be realized that, even if the geometry is highly symmetric, it could be very hard to exploit the symmetry to anticipate aspects of the tetrad field parallelizing such a geometry. This causes that the naive diagonal choice we used in Eq. (47) does not work for open and closed FRW universes. 90 VI. CONCLUSIONS f (R) and f (T ) theories are alternative ways to modify General Relativity. Like metric f (R) theories, f (T ) gravity contains additional degrees of freedom. However, these additional degrees of freedom do not appear as a consequence of the higher order of the dynamical equations, since f (T ) gravity always leads to second order equations. They appear because f (T ) gravity provides the spacetime not only with a metric but with a global parallelization. An extension of f (T ) gravity -the one governed by the determinantal Lagrangian of Eq. (49)shows that the extra degrees of freedom can play a fundamental role in smoothing singularities. In the case of the cosmic string, a global (topological) property of the GR solution, encoded in the constant J, becomes a local degree of freedom entering the metric tensor. This generates a family of geometries parametrized by J, which includes a GR solution as a particular case (the J = 0 case).

ACKNOWLEDGMENTS
The author wishes to thank the organizers of I Cosmo-Sul. This work was supported by Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET) and Universidad de Buenos Aires.