Modified Theories of Gravity in Cosmology

Modified Theories of Gravity in Cosmology Gonzalo J. Olmo University of Wisconsin-Milwaukee (USA) Gonzalo J. Olmo

About this talk... Motivation: General Relativity by itself seems unable to justify the late-time cosmic acceleration. Corrections to the Einstein-Hilbert lagrangian relevant at very low cosmic curvatures have been proposed as a mechanism for the late-time cosmic speed-up. Aims: Introduce some families of modified theories of gravity. Show how cosmic speed-up arises in f(r) models. Discuss the influence of the cosmic dynamics on local systems. Use elementary observational facts to constrain the form of the lagrangian f(r). Gonzalo J. Olmo St.Louis, November 18 th, 2006 - p. 2/8

Gravity lagrangians and fields Modified lagrangians are functionals of curvature invariants: Popular models f(r) Examples: R = g µν R µν P = R µν R µν Q = R α µβν Rµβν α f(r, P, Q) Not so popular f(r) = R+ R2 f(r) = R R2 M 2 0 R f(r,p,q) = R R 3 0 ar 2 +bp+cq Dynamical fields: Metric formalism: g µν (fourth-order e.o.m.) Palatini formalism: g µν,γ α βγ (second-order e.o.m.) By requiring second-order equations and covariance in metric formalism we are uniquely led to the EH lagrangian R 2Λ. Gonzalo J. Olmo St.Louis, November 18 th, 2006 - p. 3/8

Gravity lagrangians and fields Modified lagrangians are functionals of curvature invariants: Popular models f(r) Examples: R = g µν R µν P = R µν R µν Q = R α µβν Rµβν α f(r, P, Q) Not so popular f(r) = R+ R2 f(r) = R R2 M 2 0 R f(r,p,q) = R R 3 0 ar 2 +bp+cq Dynamical fields: Metric formalism: g µν (fourth-order e.o.m.) Palatini formalism: g µν,γ α βγ (second-order e.o.m.) By requiring second-order equations and covariance in metric formalism we are uniquely led to the EH lagrangian R 2Λ. There is no selection rule for the lagrangian neither in Palatini nor in fourth-order theories. Could cosmology be a guide? Gonzalo J. Olmo St.Louis, November 18 th, 2006 - p. 3/8

New dynamics at low R in f(r) models The e.o.m. for f(r) = R+εg(R) are : ( f = d f/dr, g = dg/dr) G µν = κ2 f T µν + ε f [ gµν (Rg (R) g(r)) ( µ ν g (R) g µν g (R)) ] trace R = κ 2 T ε[3 g (R)+Rg (R) 2g(R)] Gonzalo J. Olmo St.Louis, November 18 th, 2006 - p. 4/8

New dynamics at low R in f(r) models The e.o.m. for f(r) = R+εg(R) are : ( f = d f/dr, g = dg/dr) G µν = κ2 f T µν + ε f [ gµν (Rg (R) g(r)) ( µ ν g (R) g µν g (R)) ] trace R = κ 2 T ε[3 g (R)+Rg (R) 2g(R)] There exist two dynamical regimes: R κ 2 T GR dominates (deceleration). R R ε New dynamics dominates (speed-up). The speed-up is due to a radical change of dynamical regime Is the solar system affected by that change of regime? Gonzalo J. Olmo St.Louis, November 18 th, 2006 - p. 4/8

New dynamics at low R in f(r) models The e.o.m. for f(r) = R+εg(R) are : ( f = d f/dr, g = dg/dr) G µν = κ2 f T µν + ε f [ gµν (Rg (R) g(r)) ( µ ν g (R) g µν g (R)) ] trace R = κ 2 T ε[3 g (R)+Rg (R) 2g(R)] There exist two dynamical regimes: R κ 2 T GR dominates (deceleration). R R ε New dynamics dominates (speed-up). The speed-up is due to a radical change of dynamical regime Is the solar system affected by that change of regime? NO: because R SS R ε due to a cloud of dust and plasma. Gonzalo J. Olmo St.Louis, November 18 th, 2006 - p. 4/8

New dynamics at low R in f(r) models The e.o.m. for f(r) = R+εg(R) are : ( f = d f/dr, g = dg/dr) G µν = κ2 f T µν + ε f [ gµν (Rg (R) g(r)) ( µ ν g (R) g µν g (R)) ] trace R = κ 2 T ε[3 g (R)+Rg (R) 2g(R)] There exist two dynamical regimes: R κ 2 T GR dominates (deceleration). R R ε New dynamics dominates (speed-up). The speed-up is due to a radical change of dynamical regime Is the solar system affected by that change of regime? NO: because R SS R ε due to a cloud of dust and plasma. Who knows: Since R is a dynamical object... Is the local R sensitive to the asymptotic boundary values? Gonzalo J. Olmo St.Louis, November 18 th, 2006 - p. 4/8

The Solar System in f(r) models Expanding about g µν = η µν + h µν h (2) 00 2G M r + Λ c 6 r2 h (2) i j δ i j [ 2γG M r Λ c 6 we find: r2] with M = Ê d 3 xρ sun, Λ c = R c f c f(r c ) G = κ2 8π f (R c ) γ = 3 e m cr 3+e m cr [ 1+ e m cr 3 ] f c where m 2 c f (R c ) R c f (R c ) 3 f (R c ) is a function of R c. { f = 1+εg (R) f = 0+εg (R) The change in the cosmic R c leads to dramatic changes: For R c R ε and ( f (R c ) 0, m c ) GR regime. As R c R ε then ( f (R c ) > 0, m c finite) New dynamics. Gonzalo J. Olmo St.Louis, November 18 th, 2006 - p. 5/8

The Solar System in f(r) models Expanding about g µν = η µν + h µν h (2) 00 2G M r + Λ c 6 r2 h (2) i j δ i j [ 2γG M r Λ c 6 we find: r2] with M = Ê d 3 xρ sun, Λ c = R c f c f(r c ) G = κ2 8π f (R c ) γ = 3 e m cr 3+e m cr [ 1+ e m cr 3 ] f c where m 2 c f (R c ) R c f (R c ) 3 f (R c ) is a function of R c. { f = 1+εg (R) f = 0+εg (R) The change in the cosmic R c leads to dramatic changes: For R c R ε and ( f (R c ) 0, m c ) GR regime. As R c R ε then ( f (R c ) > 0, m c finite) New dynamics. The local R also changes with the expansion R = Λ c + m2 cκ 2 4π f (R c ) Ê d 3 x ρ(t, x ) x x e m c x x Gonzalo J. Olmo St.Louis, November 18 th, 2006 - p. 5/8

The f(r) lagrangian according to experiments The cosmic expansion changes the effective mass [ ] m 2 c R c f (R c ) 3 R c f (R c ) 1 Gonzalo J. Olmo St.Louis, November 18 th, 2006 - p. 6/8

The f(r) lagrangian according to experiments The cosmic expansion changes the effective mass [ ] m 2 c R c f (R c ) 3 R c f (R c ) 1 The growth of f (R) increases the interaction range l c = m 1 c drives the cosmic speed-up. (In GR m 2 c =,l c = 0) Viable theories must lead to constant or decreasing l c. and If l 0 = bound to today s l c then lc 2 l0 2 is satisfied by [ ] f (R) R f (R) 1 1 l0 2R d ln f dr l2 0 1+l0 2R f(r) A+B(R+ l2 0 R2 2 ) Gonzalo J. Olmo St.Louis, November 18 th, 2006 - p. 6/8

The f(r) lagrangian according to experiments The cosmic expansion changes the effective mass [ ] m 2 c R c f (R c ) 3 R c f (R c ) 1 The growth of f (R) increases the interaction range l c = m 1 c drives the cosmic speed-up. (In GR m 2 c =,l c = 0) Viable theories must lead to constant or decreasing l c. and If l 0 = bound to today s l c then lc 2 l0 2 is satisfied by [ ] f (R) R f (R) 1 1 l0 2R d ln f dr l2 0 1+l0 2R f(r) A+B(R+ l2 0 R2 2 ) Since f > 0 and f > 0 it is also bounded from below: 2Λ f(r) R 2Λ+ l2 0 R2 2 See G.J.O. Phys.Rev.Lett. 95,261102 (2005), and G.J.O. Phys.Rev.D72,083505 (2005) Gonzalo J. Olmo St.Louis, November 18 th, 2006 - p. 6/8

Summary and conclusions f(r) gravities with nonlinear terms that grow at low curvatures lead to cosmic speed-up. Gonzalo J. Olmo St.Louis, November 18 th, 2006 - p. 7/8

Summary and conclusions f(r) gravities with nonlinear terms that grow at low curvatures lead to cosmic speed-up. The change in the late-time cosmic dynamics has dramatic effects in local systems via boundary conditions. The only f(r) lagrangians compatible with Solar System dynamics are bounded by: 2Λ f(r) R 2Λ+ l2 R 2 2 G.J.O. Phys.Rev.Lett. 95,261102 (2005), G.J.O. Phys.Rev.D72,083505 (2005) Moral The dynamics of local systems in modified theories of gravity might be very sensitive to the cosmic evolution via boundary conditions. In particular, such effects are likely to manifest in theories of the form f(r,p,q) [Work in progress] Gonzalo J. Olmo St.Louis, November 18 th, 2006 - p. 7/8

Thanks!!! Gonzalo J. Olmo St.Louis, November 18 th, 2006 - p. 8/8