SMALL COSMOLOGICAL CONSTANT FROM RUNNING GRAVITATIONAL COUPLING

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1 SMALL COSMOLOGICAL CONSTANT FROM RUNNING GRAVITATIONAL COUPLING arxiv: , arxiv: Andrei Frolov Department of Physics Simon Fraser University Jun-Qi Guo (SFU) Sternberg Astronomical Institute MSU, Moscow, Russia 1 June 2011 Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration SAI / 20

2 WHAT S THE MATTER WITH COSMOLOGY? Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration SAI / 20

3 DARK ENERGY IS TROUBLE! Value problem: expected ρ Λ m ρ Λ ρ M 0 observed Planck scale Coincidence problem: Ω Λ = 0.7 Ω M = 0.3 Ω R = ρ Λ a 0 ρ M a 3 ρ R a 4 Ω R Electro Weak scale Big Bang Nucleosynthesis ΩM Now ΩΛ UNKNOWN baryons dark matter dark energy log a Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration SAI / 20 10

4 GRAVITY IS AN EFFECTIVE FIELD THEORY! MAYBE... g S[λ] = λ 4 2n g (n) (λ) (n) +... d 4 x n=0 8πG = αm 2 µ d α d µ = β(α) d α β(α) = d µ µ GR = R 16πG m 2 2 R α = f (R) Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration SAI / 20

5 GRAVITY IS AN EFFECTIVE FIELD THEORY! MAYBE... g S[λ] = λ 4 2n g (n) (λ) (n) +... d 4 x n=0 8πG = αm 2 µ d α d µ = β(α) d α β(α) = d µ µ GR = R 16πG m 2 2 R α = f (R) Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration SAI / 20

6 GRAVITY IS AN EFFECTIVE FIELD THEORY! MAYBE... g S[λ] = λ 4 2n g (n) (λ) (n) +... d 4 x n=0 8πG = αm 2 µ d α d µ = β(α) d α β(α) = d µ µ GR = R 16πG m 2 2 R α = f (R) β(α s ) = 11 2 α 2 3 n s f 2π Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration SAI / 20

7 THIS HAS BEEN TRIED BEFORE... WHAT IF INSTEAD OF CURVATURE IN EINSTEIN-HILBERT ACTION WE HAD S = f (R) g 16πG + m d 4 x UV MODIFICATION: f (R) = R + R 2 M 2 Starobinsky (1980) IR MODIFICATION: f (R) = R µ4 R Capozziello et. al. [astro-ph/ ] Carroll et. al. [astro-ph/ ] FOR F(R) THEORY TO MAKE SENSE WE NEED: f > 0 otherwise gravity is a ghost f > 0 otherwise gravity is a tachyon Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration SAI / 20

8 THIS HAS BEEN TRIED BEFORE... WHAT IF INSTEAD OF CURVATURE IN EINSTEIN-HILBERT ACTION WE HAD S = f (R) g 16πG + m d 4 x UV MODIFICATION: f (R) = R + R 2 M 2 Starobinsky (1980) IR MODIFICATION: f (R) = R µ4 R Capozziello et. al. [astro-ph/ ] Carroll et. al. [astro-ph/ ] FOR F(R) THEORY TO MAKE SENSE WE NEED: f > 0 otherwise gravity is a ghost f > 0 otherwise gravity is a tachyon Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration SAI / 20

9 THIS HAS BEEN TRIED BEFORE... WHAT IF INSTEAD OF CURVATURE IN EINSTEIN-HILBERT ACTION WE HAD S = f (R) g 16πG + m d 4 x UV MODIFICATION: f (R) = R + R 2 M 2 Starobinsky (1980) IR MODIFICATION: f (R) = R µ4 R Capozziello et. al. [astro-ph/ ] Carroll et. al. [astro-ph/ ] FOR F(R) THEORY TO MAKE SENSE WE NEED: f > 0 otherwise gravity is a ghost f > 0 otherwise gravity is a tachyon Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration SAI / 20

10 THIS HAS BEEN TRIED BEFORE... WHAT IF INSTEAD OF CURVATURE IN EINSTEIN-HILBERT ACTION WE HAD S = f (R) g 16πG + m d 4 x UV MODIFICATION: f (R) = R + R 2 M 2 Starobinsky (1980) IR MODIFICATION: f (R) = R µ4 R Capozziello et. al. [astro-ph/ ] Carroll et. al. [astro-ph/ ] FOR F(R) THEORY TO MAKE SENSE WE NEED: f > 0 otherwise gravity is a ghost f > 0 otherwise gravity is a tachyon Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration SAI / 20

11 ... BUT NOT QUITE IN THIS WAY! Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration SAI / 20

12 ACCELERATED RG FLOWS OF NEWTON S CONSTANT β stable? f = α β α 2 > 0 R f = 1 2 β β α + β α 2 > 0 α de Sitter? 2f f R = α + β α 2 R = 0 Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration SAI / 20

13 ACCELERATED RG FLOWS OF NEWTON S CONSTANT β forbidden region (ghosts) α β = 0 α stable? f = α β α 2 > 0 R f = 1 2 β β α + β α 2 > 0 de Sitter? 2f f R = α + β α 2 R = 0 Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration SAI / 20

14 ACCELERATED RG FLOWS OF NEWTON S CONSTANT β forbidden region (ghosts) α β = 0 α + β = 0 α stable? f = α β α 2 > 0 R f = 1 2 β β α + β α 2 > 0 de Sitter? 2f f R = α + β α 2 R = 0 effective Λ Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration SAI / 20

15 ACCELERATED RG FLOWS OF NEWTON S CONSTANT β forbidden region (ghosts) α β = 0 α + β = 0 UV fixed point GR flow α stable? f = α β α 2 > 0 R f = 1 2 β β α + β α 2 > 0 de Sitter? 2f f R = α + β α 2 R = 0 β(α) = α(α 1) ds in IR effective Λ f (R) = R 2Λ Einstein s gravity Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration SAI / 20

16 ACCELERATED RG FLOWS OF NEWTON S CONSTANT β forbidden region (ghosts) α β = 0 α + β = 0 UV fixed point GR flow α stable? f = α β α 2 > 0 R f = 1 2 β β α + β α 2 > 0 de Sitter? 2f f R = α + β α 2 R = 0 β(α) = +α(α 1) ds in IR effective Λ f (R) = R + R 2 M 2 Starobinsky Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration SAI / 20

17 ACCELERATED RG FLOWS OF NEWTON S CONSTANT β forbidden region (ghosts) α β = 0 α + β = 0 UV fixed point GR flow α stable? f = α β α 2 > 0 R f = 1 2 β β α + β α 2 > 0 de Sitter? 2f f R = α + β α 2 R = 0 unstable ds ds in IR effective Λ β(α) = 2α(α 1) f (R) = R µ4 R Carroll et. al. Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration SAI / 20

18 ACCELERATED RG FLOWS OF NEWTON S CONSTANT β forbidden region (ghosts) α β = 0 α + β = 0 ds in IR UV fixed point GR flow ds in IR α stable? f = α β α 2 > 0 R f = 1 2 β β α + β α 2 > 0 de Sitter? 2f f R = α + β α 2 R = 0 β(α) = α 2 f (R) = R α α 0 ln R R 0 f (R) flow effective Λ something new? Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration SAI / 20

19 SO WHAT DOES THIS MEAN FOR HIERARCHY?.. Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration SAI / 20

20 f (R) GRAVITY IS A SCALAR-TENSOR THEORY Einstein equations turn into a fourth-order equation: f G µν f ;µν + f 1 2 (f f R) g µν = m 2 T µν A new scalar degree of freedom φ f 2 appears: f = 1 3 (2f f R) + m 2 Can rewrite fourth-order field equation as two second order ones! G µν = m 2 T µν + µν, φ = V (φ) µν = (1 + φ)g µν + φ ;µν φ 3P(φ) g µν P(φ) 1 6 (f f R), V (φ) 1 3 (2f f R), m 2 3 T 3 (ρ 3p) Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration SAI / 20

21 EXAMPLE I: (SAFE) UV MODIFICATION V/M 2 f (R) = R + R 2 φ = 2R M 2 M in vacuum U(φ) = V (φ) + (φ φ) φ V = 1 3 R 2 M 2 = M 2 12 φ2 massive scalar field! scalar degree of freedom φ is heavy and hard to excite Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration SAI / 20

22 EXAMPLE I: (SAFE) UV MODIFICATION in matterv/m 2 f (R) = R + R 2 φ = 2R M 2 M in vacuum U(φ) = V (φ) + (φ φ) φ V = 1 3 R 2 M 2 = M 2 12 φ2 massive scalar field! scalar degree of freedom φ is heavy and hard to excite Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration SAI / 20

23 EXAMPLE II: (FAILED) IR MODIFICATION V/µ 2 in vacuum φ f (R) = R µ4 R φ = µ4 R 2 V = 2 µ 4 3 R µ8 R 3 = 2 3 µ2 φ φ U(φ) = V (φ) + (φ φ) field φ is unstable! Dolgov & Kawasaki (astro-ph/ ) Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration SAI / 20

24 EXAMPLE II: (FAILED) IR MODIFICATION V/µ 2 in matter in vacuum φ f (R) = R µ4 R φ = µ4 R 2 V = 2 µ 4 3 R µ8 R 3 = 2 3 µ2 φ φ U(φ) = V (φ) + (φ φ) field φ is unstable! Dolgov & Kawasaki (astro-ph/ ) Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration SAI / 20

25 CAN WE COME UP WITH SOMETHING BETTER?.. f (R) = R α α 0 ln R R 0 V Λ α 1 = α α 0 ln R 1 R 0 φ φ = ln R R α 0 α 0 = ln R R α0 1 R 4Λ = R 0 exp α 0 V (φ) = 4 3 Λe φ (φ 1) P(φ) = 2 3 Λe φ, V (φ) = 4 3 Λe φ φ 3 V (φ eq ) = = φ eq = W 4 Λ Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration SAI / 20

26 CAN WE COME UP WITH SOMETHING BETTER?.. f (R) = R α α 0 ln R R 0 V Λ α 1 = α α 0 ln R 1 R 0 φ φ = ln R R α 0 α 0 = ln R R α0 1 R 4Λ = R 0 exp α 0 V (φ) = 4 3 Λe φ (φ 1) P(φ) = 2 3 Λe φ, V (φ) = 4 3 Λe φ φ 3 V (φ eq ) = = φ eq = W 4 Λ Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration SAI / 20

27 CAN WE COME UP WITH SOMETHING BETTER?.. f (R) = R α α 0 ln R R 0 V Λ α 1 = α α 0 ln R 1 R 0 φ φ = ln R R α 0 α 0 = ln R R α0 1 R 4Λ = R 0 exp α 0 V (φ) = 4 3 Λe φ (φ 1) P(φ) = 2 3 Λe φ, V (φ) = 4 3 Λe φ φ 3 V (φ eq ) = = φ eq = W 4 Λ Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration SAI / 20

28 UNIVERSE AS A DYNAMICAL SYSTEM 2 scalar DoF, 4D phase space {φ,π,a,h} dynamical system equations φ = π, ȧ = a H Ḣ = R 6 2H 2 π+3hπ+v (φ) = m 2 subject to a constraint ρ 3p H 2 (φ+2)+hπ+p(φ) = m 2 3 ρ 3 P(φ) 1 6 (f f R) = 2 3 Λe φ V (φ) 1 3 (2f f R) = 4 3 Λe φ φ Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration SAI / 20

UNIVERSE AS A DYNAMICAL SYSTEM 2 scalar DoF, 4D phase space {φ,π,a,h} dynamical system equations φ = π, ȧ = a H Ḣ = R 6 2H 2 π+3hπ+v (φ) = m 2 subject to a constraint ρ 3p H 2 (φ+2)+hπ+p(φ) = m 2 3 ρ

29 UNIVERSE AS A DYNAMICAL SYSTEM 2 scalar DoF, 4D phase space {φ,π,a,h} dynamical system equations φ = π, ȧ = a H Ḣ = R 6 2H 2 π+3hπ+v (φ) = m 2 subject to a constraint ρ 3p H 2 (φ+2)+hπ+p(φ) = m 2 3 ρ 3 P(φ) 1 6 (f f R) = 2 3 Λe φ V (φ) 1 3 (2f f R) = 4 3 Λe φ φ vacuum evolution is constrained to a (non-trivial) 2D subspace Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration SAI / 20

30 UNIVERSE AS A DYNAMICAL SYSTEM 2 scalar DoF, 4D phase space {φ,π,a,h} dynamical system equations φ = π, ȧ = a H Ḣ = R 6 2H 2 π+3hπ+v (φ) = m 2 subject to a constraint ρ 3p H 2 (φ+2)+hπ+p(φ) = m 2 3 ρ 3 P(φ) 1 6 (f f R) = 2 3 Λe φ V (φ) 1 3 (2f f R) = 4 3 Λe φ φ vacuum evolution is constrained to a (non-trivial) 2D subspace Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration SAI / 20

31 UNIVERSE AS A DYNAMICAL SYSTEM 2 scalar DoF, 4D phase space φ {φ,π,a,h} dynamical system equations φ = π, ȧ = a H Ḣ = R 6 2H 2 π+3hπ+v (φ) = m 2 subject to a constraint ρ 3p H 2 (φ+2)+hπ+p(φ) = m 2 3 ρ 3 forbidden for vacuum solutions φ P(φ) 1 6 (f f R) = 2 3 Λe φ V (φ) 1 3 (2f f R) = 4 3 Λe φ φ Ω M,0 = Ω,0 = Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration SAI / 20

32 UNIVERSE AS A DYNAMICAL SYSTEM 2 scalar DoF, 4D phase space φ {φ,π,a,h} dynamical system equations φ = π, ȧ = a H now Ḣ = R 6 2H 2 π+3hπ+v (φ) = m 2 subject to a constraint ρ 3p H 2 (φ+2)+hπ+p(φ) = m 2 3 ρ 3 forbidden for vacuum solutions φ P(φ) 1 6 (f f R) = 2 3 Λe φ V (φ) 1 3 (2f f R) = 4 3 Λe φ φ Ω M,0 = Ω,0 = Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration SAI / 20

33 UNIVERSE AS A DYNAMICAL SYSTEM 2 scalar DoF, 4D phase space φ {φ,π,a,h} dynamical system equations φ = π, ȧ = a H Ḣ = R 6 2H 2 π+3hπ+v (φ) = m 2 subject to a constraint ρ 3p H 2 (φ+2)+hπ+p(φ) = m 2 P(φ) 1 6 (f f R) = 2 3 Λe φ V (φ) 1 3 (2f f R) = 4 3 Λe φ φ 3 ρ 3 now forbidden for vacuum solutions ds Ω M,0 = Ω,0 = tracker φ Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration SAI / 20

34 UNIVERSE AS A DYNAMICAL SYSTEM 2 scalar DoF, 4D phase space φ {φ,π,a,h} dynamical system equations φ = π, ȧ = a H Ḣ = R 6 2H 2 now future ds φ π+3hπ+v (φ) = m 2 ρ 3p 3 forbidden for vacuum solutions subject to a constraint H 2 (φ+2)+hπ+p(φ) = m 2 ρ 3 tracker P(φ) 1 6 (f f R) = 2 3 Λe φ V (φ) 1 3 (2f f R) = 4 3 Λe φ φ Ω M,0 = Ω,0 = Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration SAI / 20

35 UNIVERSE AS A DYNAMICAL SYSTEM 2 scalar DoF, 4D phase space φ {φ,π,a,h} dynamical system equations φ = π, ȧ = a H Ḣ = R 6 2H 2 now future ds φ π+3hπ+v (φ) = m 2 ρ 3p 3 forbidden for vacuum solutions subject to a constraint H 2 (φ+2)+hπ+p(φ) = m 2 ρ 3 past tracker P(φ) 1 6 (f f R) = 2 3 Λe φ V (φ) 1 3 (2f f R) = 4 3 Λe φ φ Ω M,0 = Ω,0 = Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration SAI / 20

36 UNIVERSE AS A DYNAMICAL SYSTEM 2 scalar DoF, 4D phase space φ {φ,π,a,h} dynamical system equations φ = π, ȧ = a H Ḣ = R 6 2H 2 now future ds φ π+3hπ+v (φ) = m 2 ρ 3p 3 forbidden for vacuum solutions subject to a constraint H 2 (φ+2)+hπ+p(φ) = m 2 ρ 3 past tracker P(φ) 1 6 (f f R) = 2 3 Λe φ V (φ) 1 3 (2f f R) = 4 3 Λe φ φ Ω M,0 = Ω,0 = Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration SAI / 20

37 COSMOLOGICAL EVOLUTION: TRACKER SOLUTION present m 2 ρ 3 = ΩM,0 + Ω R,0 H 2 a 3 a, 4 0 φ trac W 3H 2 0 4Λ Ω M,0 a 3 Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration SAI / 20

38 COSMOLOGICAL EVOLUTION: DISTANCE MODULUS µ = for z [0, 2] µ = 5 log 10 d L, f (R) d L,ΛCDM, 1 d L = 1 d a a a 2 H a Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration SAI / 20

39 COSMOLOGICAL EVOLUTION: EFFECTIVE Ω & w present Ω M Ω R Ω Ω = ρ 3m 2 H 2, ρ = 3m 2 H 2 ρ p = m 2 (H 2 R/3) p Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration SAI / 20

40 COSMOLOGICAL EVOLUTION: EFFECTIVE Ω & w w 1/3 w equilibrium approximation w = p ρ, ρ = 3m 2 H 2 ρ p = m 2 (H 2 R/3) p Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration SAI / 20

41 INVOKE CHAMELEON TO SATISFY LOCAL TESTS 1 ρ R/κ 2 (n=4, f R0 =0.1) ρ (g cm -3 ) R/κ 2 (g cm -3 ) f R0 = f R0 =0.01 f R0 =0.05 f R0 =0.1 ρ n= r/r Hu & Sawicki ( ) r/r Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration SAI / 20

42 N-BODY SIMULATIONS WITH F(R) DARK ENERGY Oyaizu, Lima & Hu ( ) Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration SAI / 20

43 LONGITUDINAL GRAVITY WAVES FROM COLLAPSE! Harada, Chiba, Nakao, & Nakamura (gr-qc/ ) t/m Andrei Frolov (SFU) Modified Gravity and Cosmic Acceleration SAI / 20

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