f(r) theory, Cosmology and Beyond

Transcription

1 f(r) theory, Cosmology and Beyond Li-Fang Li The institute of theoretical physics, Chinese Academy of Sciences / 17

2 Outline 1 Brief review of f(r) theory 2 Higher dimensional f(r) theory: case study 3 f(r) theory and Numerical Relativity 2 / 17

3 Brief review of f(r) theory Brief review of f(r) theory Main Motivations of f(r) theory 1 Many quantum gravity theories can reduce to an effective action including high order correction terms compared with the standard Einstein-Hilbert action 2 Our universe is accelerately expanding. These high order correction terms can help us explain the dark components of our universe References: A. Felice, and S. Tsujikawa, Living Rev. Relativity 13, 3 (2010) T. Sotiriou and V. Faraoni, Rev. Mod. Phys. 82, 451 (2010) 3 / 17

4 Brief review of f(r) theory Attracting features of f(r) theory Explain inflation, consistent to CMB anisotropies observation and without the graceful exit problem J. Hwang and H. Noh, Phys. Lett. B 506, 13 (2001) Explain the current accelerate expansion of the Universe without dark energy S. Capozziello, Int. J. Mod. Phys. D 11, 483 (2002) Explain the dark matter S. Capozziello, V. Cardone and Troisi, J. Cosmol. Astropart. Phys. 2006(08), 001 (2006) Pass the local gravity constraints G. Olmo, Phys. Rev. Lett. 95, (2005) 4 / 17

5 Brief review of f(r) theory Possible f(r) gravity models: A. f (R) = αr n B. f (R) = R α/r n C. f (R) = R µr c (R/R c) 2n (R/R c) 2n +1 D. f (R) = R µr c [1 (1 + R 2 /R 2 c ) n ] E. f (R) = R µr c tanh(r/r c ) F. higher dimensional f(r) theory(extra dimension)... Which one can describe the true physics? How to settle down the parameters such as α, n and µ? 5 / 17

6 Higher dimensional f(r) theory: case study Five dimensional f(r) theory: case study We begin with the action in the five-dimensional spacetime S = 1 d 5 x gf (R) + S M (1) 2κ Taking the variation of Eq.(1), we get the corrected Einstein equation f (R)R ab 1 2 g abf (R) ( a b g ab c c )f (R) = κt ab where T ab = 2 g δs M δg ab. Contracting the above equation, we get the dynamical equation for the scalar field φ := f (R), a a φ = 1 4 [κt Rφ f (R(φ))] 6 / 17

7 Higher dimensional f(r) theory: case study Considering the universe is isotropic and homogeneous, the background 5d line element reads ds 2 = (dt) 2 + a 2 (t)(d 2 x + d 2 y + d 2 z) + λ(t) 2 d 2 x 5 The Ricci tensor and the dynamical equation of λ in 5d and 4d bear the following two relations R 4 ab = 1 2 λ 1 D a D b λ 1 4 λ 2 (D a λ)d b λ + h m a h n b R mn D a D b λ = 1 2 λ 1 (D a λ)d a λ 2R ab ξ a ξ b where D a denotes the covariant derivative on four dimensional spacetime and is defined as D a T c 1..c m b 1..b n = h d a h e 1 b 1..h cm f m d T f 1..f m e 1..e n. B. Huang, S. Li and Y. Ma, Phys. Rev. D 81, (2010) 7 / 17

8 Higher dimensional f(r) theory: case study The stress-energy tensor here is regarded as a perfect fluid in 5d spacetime with the expression T (5) ab = L 1 λ 1 2 [T (4) ab + Pλξa ξ b ] where ρ and P are the 4d energy density and the hydrostatic pressure. Here we use f (R) = αr m, and the evolution equation can be reexpressed as ä = 1 a 2 (ȧ a )2 + 1 ȧ λ 2 aλ m ( φ αm ) 1 1 φ m φ (ȧ a + λ λ ) κ 4 λ λ = 3 2 (ȧ a )2 3 ȧ λ 2 a λ m ( φ αm ) 1 1 φ m φ (3ȧ a λ λ ) κ 4 φ φ = 3 2 (ȧ a )2 + 3 ȧ λ 2 aλ m ( φ αm ) 1 1 φ m 1 2 φ (3ȧ a + λ λ ) κ 4 ρ φλ ρ φλ ρ φλ κp φλ where φ = f (R). Next, we will constrain our parameters m and α in our model. 8 / 17

9 Higher dimensional f(r) theory: case study First fix the initial values of a 0, ȧ 0, λ 0, λ 0, φ 0 and φ 0. a 0 and λ 0 has no physical meaning, we can set a 0 = 1 and λ 0 = 1. And considering H 0 = ȧ0 a 0 = 1, therefore ȧ 0 = 1. For any given a 0, ȧ 0, λ 0, and φ 0, we have the initial data for λ and φ: 1 A = (m 1)( αm m ) 1 m 1 B = 1 ( αm m ) 1 m 1 F = 2A + B q 0 + 2κρ 0 where q 0 is the current deceleration parameter and ρ 0 is the current density of our universe. 9 / 17

10 Higher dimensional f(r) theory: case study λ = 1 16 F ± F 2 16( 3 8 F 3 + κρ 0 + A 2 ) φ = 1 16 F F 2 16( 3 8 F 3 + κρ 0 + A 2 ) So given parameters m and α, the initial data are determined. In the following, we use SNIa data to constrain parameters m and α. 10 / 17

11 Higher dimensional f(r) theory: case study Constrain our results with the SNIa experiment. The luminosity distance can be expressed as t d L = a 0 (1 + z) 0 dt a(t). According to the astrophysical convention, we adopt the logarithmic measure of the luminosity distance instead of luminosity distance itself µ = 5 log 10 d L In the supernova observation, the χ 2 SN statistic is given by N χ 2 SN = (µ obs (z i ) µ th (z i )) 2 i=1 A. G.Riess, et al, Astrophys. J. 607, 665 (2004) σ 2 i 11 / 17

12 χ Higher dimensional f(r) theory: case study χ 0.5 given α 0.5 given m m α We can see: given any α we have best m; given m we have best α also. And all of these choices are consistent with SNIa data. 12 / 17

13 Higher dimensional f(r) theory: case study Several examples of evolution of deceleration parameter and effective gravitational constant deceleration parameter effective gravitational constant 13 / 17

14 Higher dimensional f(r) theory: case study Resulted viable parameters domain 0.95 "d:/m_plus.dat" u 1: α m 14 / 17

15 f(r) theory and Numerical Relativity Possible future study f(r) theory is very attractive from cosmological consideration f(r) theory seems to pass the local gravity check How about description of binary black holes? How about the related gravitational wave from BBH in f(r) theory? Numerical relativity study of f(r) theory 15 / 17

16 f(r) theory and Numerical Relativity Scalar description of f(r) theory for vacuum spacetime df (R) R ab 1 2 g abr = 8π φ T (φ) ab 3 φ + 2V (φ) φ dv dφ = 0 8πT (φ) ab = a b φ g ab ( φ V (φ)) V (φ) = R(φ)φ f (R(φ)) where φ = dr. So for given specific f (R) form, it is straight forward to simulate BBH in the framework of f(r) theory based on current numerical relativity code. M. Shibata, K. Nakao, and T. Nakamura, Phys. Rev. D 50, 7304 (1994) V. Paschalidis, S. Halataei, S. Shapiro and I. Sawicki, CQG 28, (2011) 16 / 17

17 f(r) theory and Numerical Relativity Thank you! 17 / 17