f(R) Theories of Gravity Vin´ıcius Miranda Bragan¸ca and Ioav Waga Introduction The f(R) theories belong to the class of dark energy models which allows for a dynamic behavior of the variables ρEE and wEE . That result is obtained by a modification in the General Relativity action. The main idea consists in the replacement of The Einstein-Hilbert lagrangian £ = R with an arbitrary function of Ricci Scalar,£ = f (R) [?]. Z √ 1 4 S= d x −gf (R) + SM gµν , ψ , (1) 2κ h i 1 f ′(R)Rµν − gµν − ∇µ∇ν − gµν ∇θ ∇θ f ′(R) = κTµν , (2) 2 where ∇ν indicates covariant differentiation with Levi-Civita condf (R) nection and f ′(R) ≡ dR . The following conformal transformation can be applied, changing the lagrangian into a canonical form. √ g˜µν = χgµν ≡ gµν exp 2/ 6 φ, (8) Z p 1 µν 1 4 ˜ Sgrav = d x −˜ g R − g˜ ∂µ∂ν φ − V (φ) . (9) 2κ 2 That transformation defines the Einstein Frame. In contrast with (??), the lagragian (??) defines the Jordan Frame. In both frames we can calculate the trace of equation of motion, 2 dV 8πG 2χ T+ , ∇µ∇µχ = 3 3 dχ 8πG ˜ dV µ T+ . ∇ ∇µφ = 3 dφ Some examples 1. f (R) = R − α/Rn com n > 0 This model violates at least two constrains. The second derivative is lower than in some domain region and a matter dominated epoch does not exist [?]. (10) (11) Fig. 1: The evolution of densities for different components for the model f (R) = R − µ6/R2 com µ/H = 11.04 [?]. Necessary conditions for the viability of f(R) gravity Equivalence between f(R) gravity and Scalar- Tensor theories. Consider this example, f (R) = R + R2/6m2. We can define a new scalar field λ and we can drop the degree of equations of motion. To do that we must write a new equivalent lagrangian with a linear dependence on Ricci Scalar (using the new variable) [?]. " # Z 2 2 √ 1 R 1 2λ Sgrav = d4x −g R + R − 3m − = (3) 2 2 2κ 6m 6m Z √ 3 2 2 1 4 d x −g (1 + λ)R − m λ . (4) 2κ 2 The λ equation of motion is algebric, giving R = 3m2λ. The action (??) can become similiar to an action of Brans-Dicke Scalar-Tensor theory with ωB.D. = 0 if we define the scalar field χ ≡ (1 + λ) = f ′(R). This result can be generalized to arbitrary functions [?], Z √ ′ 1 4 d x −g f (Q)(R − Q) + f (Q) . (5) Sgrav = 2κ Assuming that f ”(R) > 0, we obtain Q=R as an equation of motion. By defining χ ≡ f ′(R) we complete the proof, Z √ 1 (6) Sgrav = d4x −g χR − χ2V (χ) 2κ where 1 V (χ) = 2 [Q(χ)χ − f (Q(χ))] 2χ (7) 1. To avoid the presence of t´aquions and the presence of instabilities in the shear (σ ≡ T<µν>), the following inequalities must hold [?]: f ′(R) > 0 f ′′(R) > 0. (12) 2. To allow the existence of a matter dominatted epoch ( Ωm = 1 , Ωr = 0 e ΩEE = 0 ) connected with an accelerator attractor, the following must hold [?]: dm(r) m(r ≈ −1) ≈ 0 (r ≈ −1) > −1, dr 0 ≤ m(r = −2) ≤ 1 > −1 (13) 2 2. f (R) = αRn com n 6= 1 e f (R) = R − µ41/R + µ−2 R 2 This model violates the necessary conditions for the existence of an matter dominated epoch followed by an accelerator attractor [?]. 2 − 1 com n, λ > 0 e R0 ≈ Λef f 3. f (R) = R − λR0 1 + R R2 0 4. f (R) = R−m2 2 n c1(R/m ) c2(R/m2 )n+1 com n > 0 e m2 ≡ κ2ρ(a = 0)/3 This models respect most of the constrains. Although the second derivative change it signs, the cosmology never reach that domain region [?] [?][?]. However both models do not obey the limit (??) and they can generate singularities in the Ricci Scalar [?][?] . (14) onde m ≡ Rf ”(R)/f ′(R) e r ≡ −Rf ′(R)/f (R). 3. Using the equivalence between f(R) gravity and Brans-Dicke theories we are able to show the necessary conditions that must hold in order to avoid singularities in the Ricci Scalar [?][?]: lim χ2V (χ) −→ ∞ (15) χ→χ˜ onde χ˜ ≡ limR→∞ χ(R). 4. Using again the equivalence between f(R) gravity and Brans-Dicke theories, the compatibility between f(R) and local test of gravity is conditioned to the following necessary constrain [?]: √ onde exp −2φ 6 V (φ) ≈ φ−n, exp(−φ) (16) Fig. 2: Potential V (φ) for the model (4). In that case φ ≡ f ′(R) − 1 [?]. ≡ f ′(R). m(r=-2) Is it possible to exist viable f(R) models ? Numerical results of the model (??) 1 1 0.8 0.8 0.6 It is not easy to transform some of the constrains into useful information about the shape of the function. For example, although the standard way to avoid contradiction with the second constrain is to made a function that resemble General Relativity for high values of the Ricci scalar, it is not clear that other kinds of functions will necessary fail the test. The third and fourth model showed above are the successfully f(R) that have appeared in literature. They can be written as f (R) = R + h(R) The main idea behind them is the following limits. In the Jordan frame it is easy to prove that χ˜ = 1. Now the limit (??) is satisfied, as we can see in the following picture lim (h(R)) = 0 (17) Fig. 4: The potencial V (χ) against χ − 1 for the model (??). (18) The expression (??) converges to general relativity for high values of R so, intuitively, it is a good candadite to respect the critical points constrains. In fact for α > 1 the model obey the inequalities (??) and (??) for all values of R0. R→0 and lim (h(R)) = const R→∞ Despite of his success in many constrains, they suffer from singularities in the Ricci scalar. 0.6 0.4 0.4 0.2 0.2 0 0 2 1.8 1.6 1.4 1.2 Ro 1 0.8 0.6 0.4 0.2 0 0 0.6 0.8 0.2 0.4 1 1.6 1.8 1.2 1.4 2 α To see the existence of matter dominated epoch consider the following particular initial condition. log[m(r=-1)] -13.85 -13.85 -13.9 Fig. 5: need a caption -13.9 -13.95 -13.95 -14 -14 -14.05 -14.1 -14.05 -14.15 -14.1 Referˆ encias -14.15 Fig. 3: The evolution of densities for different components for the model f (R) = R − µ6/R2 com µ/H = 11.04 [?]. To fix this problem it is necessary to modify the limit (??). One way to do that is to introduce a factor proportional to the square of the Ricci Scalar in the ultraviolet limit [?]. However a delicate fine tuning appear as shown in [?].The main idea of our work is to propose another way to avoid singularities problem. Instead of considering the limit ??, we will only assume that. h(R) = const (19) lim R R→∞ For example, we consider the following model: R . f (R) = R − αR0 log 1 + R0 Universidade Federal do Rio de Janeiro 2 1.8 1.6 1.4 1.2 Ro [1] T. P. Sotiriou e V. Faraoni, arXiv:0805.1726v2[gr-qc]. 1 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 [2] A. Hindawi et al, arXiv:9509142v3 [hep-th] . 2 α [3] T. Faulkner et al, arXiv:0612569v1 [astro-ph]. [4] Alexei A. Starobinsky, arXiv:0706.2041v2 [astro-ph] . [5] L. Amendola et al, arXiv:0612180v2 [gr-qc]. dm/dr(r=-1) [6] A. V. Frolov, arXiv:0803.2500 [astro-ph]. -0.075 -0.075 -0.08 -0.085 -0.08 -0.09 -0.085 -0.095 -0.09 -0.1 [7] T. Kobayashi and K. Maeda, arXiv:0807.2503 [astro-ph]. [8] S. A. Appleby and R. A. Battye, arXiv:0803.1081v1 [astro-ph]. -0.095 -0.105 -0.1 [9] P. Brax et al, arXiv:0806.3415v2 [astro-ph]. -0.105 2 1.8 1.6 1.4 1.2 Ro [10] W. Hu and I. Sawicki , arXiv:0705.1158 [astro-ph]. 1 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 (20) Grupo de Astrof´ısica, Relatividade e Cosmologia (ARCOS) 1 1.2 α 1.4 1.6 1.8 2 [11] L. Amendola e S.Tsujikawa, arXiv:0705.0396v2 [astro-ph]. [12] A. Dev et al, arXiv:0807.3445v2 [hep-th]. [13] T. Kobayashi and K. Maeda, arXiv:0810.5664v1 [astro-ph]. [email protected] http://omnis.if.ufrj.br/˜ arcos