Cosmological dynamics and perturbations in Light mass Galileon models
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- Κανδάκη Τισιφόνη Ασπάσιος
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1 Cosmological dynamics and perturbations in Light mass Galileon models Md. Wali Hossain APCTP Pohang, Korea Collaborators: A. Ali, R. Gannouji and M. Sami 13 th December, 2017, CosPA 2017, YITP, Kyoto University, Japan
2 Plan Introduction to Light Mass Galileon Background Cosmology First and Second Order Perturbations 2/28
3 Cosmic History Figure: Cosmic history. Picture is taken from wfirst.gsfc.nasa.gov. 3/28
4 Cosmic acceleration = Equation of state = w = Pressure/Density < 1 3. Observationally = w inf 1 and also currently w DE,0 1. 4/28
5 Cosmic acceleration = Equation of state = w = Pressure/Density < 1 3. Observationally = w inf 1 and also currently w DE,0 1. Inflation: L = µ φ µ φ+v(φ). φ is a scalar field. V Φ Slow Roll Reheating Φ 4/28
6 Cosmic acceleration = Equation of state = w = Pressure/Density < 1 3. Observationally = w inf 1 and also currently w DE,0 1. Inflation: L = µ φ µ φ+v(φ). φ is a scalar field. V Φ Dark Energy: Simplest candidate can be Λ. Fine tuning problem = ρ Λ,obs ρ Λ,theo = Slow Roll Reheating Cosmic Coincidence = ρ Λ ρ m0. Φ Alternatives = Make DE dynamical = Modification of gravity. 4/28
7 Horndeski Lagrangian Most general scalar-tensor Lagrangian 5 L = c i L i i=1 L 2 = K(φ,X), L 3 = G 3 (φ,x) φ, [ L 4 = G 4 (φ,x)r +G 4X (φ,x) ( φ) 2 ( µ ν φ) 2], L 5 = G 5 (φ,x)g µν µ ν φ 1 [ 6 G 5X(φ,X) ( φ) 3 3 φ( µ ν φ) 2 +2( µ ν φ) 3], with K and G i (i = 3,4,5) are arbitrary functions of φ and X = 1 2 µφ µ φ and G ix = G i / X. Second order equation of motion. 5/28
8 Light Mass Galileon K(φ,X) = X V(φ), G 3 (φ,x) = 2X and G 4 = G 5 = 0. L 2 = 1 2 ( µφ) 2 V(φ), L 3 = ( µ φ) 2 φ, 6/28
9 Light Mass Galileon K(φ,X) = X V(φ), G 3 (φ,x) = 2X and G 4 = G 5 = 0. L 2 = 1 2 ( µφ) 2 V(φ), L 3 = ( µ φ) 2 φ, if V(φ) φ then φ-field preserves galilean shift symmetry in the flat space time = Galileon field. φ φ+b µ x µ +c µ φ µ φ+b µ There are five Lagrangians which preserve the shift symmetry and give second order equation of motion. 6/28
10 Galileon Lagrangian 5 L = c i L i i=1 L 1 = M 3 φ, L 2 = 1 2 ( µφ) 2, L 3 = 1 M 3( µφ) 2 φ, L 4 = 1 M 6φ ;µφ ;µ[ 2( φ) 2 2φ ;µν φ ;µν 1 2 Rφ ;µφ ;µ], L 5 = 1 M 9φ ;µφ ;µ[ ( φ) 3 3( φ)φ ;µν φ ;µν +2φ µν φ ;µρ φ ;ν ;ρ 6φ ;µφ ;µν φ ;ρ G νρ ]. Second order equation of motion. Can explain late time acceleration of the Universe. Local physics is restored through the Vainshtein mechanism. 7/28
11 The Model A. Ali, R. Gannouji, MWH and M. Sami, Phys. Lett. B 718, 5 (2012) MWH, Phys. Rev. D 96, no. 2, (2017) Action S = d 4 x [ M 2 pl g 2 R 1 2 ( φ)2( 1+ α ) ] M 3 φ V(φ) [ ] +S m ψ m ;e 2βφ/M pl g µν EH action is modified with Galileon Lagrangian L (2) and L (3) and with a potential = Only L (2) and L (3) can t give late time accelaration. Potential is added phenomenologically to get accelaration. Potential = breaks shift symmetry even in the flat background. Non-linear self interaction term of the galileon field plays the main role to preserve the local physics through Vinshtein mechanism. 8/28
12 V = V 0 (cosh(αφ/m Pl ) 1) p V. Sahni and L. M. Wang, PRD 62, (2000). ( ) ) p αφ V = V 0 (cosh 1 M Pl 9/28
13 V = V 0 (cosh(αφ/m Pl ) 1) p V. Sahni and L. M. Wang, PRD 62, (2000). ( ) ) p αφ V = V 0 (cosh 1 M Pl V(φ) = V 0 2 eαpφ/m Pl, for V(φ) = V ( ) 2p 0 αφ, for 2 M Pl αφ M Pl 1, φ > 0 αφ M Pl 1 M. S. Turner, PRD 28, 1243 (1983). 9/28
14 V = V 0 (cosh(αφ/m Pl ) 1) p V. Sahni and L. M. Wang, PRD 62, (2000). ( ) ) p αφ V = V 0 (cosh 1 M Pl V(φ) = V 0 2 eαpφ/m Pl, for V(φ) = V ( ) 2p 0 αφ, for 2 M Pl αφ M Pl 1, φ > 0 αφ M Pl 1 M. S. Turner, PRD 28, 1243 (1983). ρ φ = ρ φ0 a 6p/(p+1) 9/28
15 V = V 0 (cosh(αφ/m Pl ) 1) p V. Sahni and L. M. Wang, PRD 62, (2000). ( ) ) p αφ V = V 0 (cosh 1 M Pl V(φ) = V 0 2 eαpφ/m Pl, for V(φ) = V ( ) 2p 0 αφ, for 2 M Pl αφ M Pl 1, φ > 0 αφ M Pl 1 M. S. Turner, PRD 28, 1243 (1983). ρ φ = ρ φ0 a 6p/(p+1) V. Sahni and L. M. Wang, PRD 62, (2000). 1 2 < w φ >= φ 2 V(φ) 1 φ 2 2 +V(φ) = p 1 p +1. 9/28
16 V = V 0 (cosh(αφ/m Pl ) 1) p α = 20 and p = V Φ Ρc0 1.0 wφ Φ Mpl Log 10 1 z 10/28
17 V = V 0 (cosh(αφ/m Pl ) 1) p α = 20 and p = V Φ Ρc0 1.0 wφ Φ Mpl Log 10 1 z α = 30 and p = 0.1. V Φ Ρc wφ Φ Mpl Log 10 1 z 10/28
18 V = V 0 e µ 1φ/M Pl +e µ 2φ/M Pl T. Barreiro, E. J. Copeland and N. J. Nunes, PRD 61, (2000) ) V = V 0 (e µ1φ/m Pl +e µ2φ/m Pl V Φ Φ M pl Figure: µ 1 = 20 and µ 2 = /28
19 V = V 0 e µ 1φ/M Pl +e µ 2φ/M Pl T. Barreiro, E. J. Copeland and N. J. Nunes, PRD 61, (2000) ) V = V 0 (e µ1φ/m Pl +e µ2φ/m Pl. V Φ Φ M pl Figure: µ 1 = 20 and µ 2 = 0.1. V Φ Φ M pl Figure: µ 1 = 20 and µ 2 = 1. 11/28
20 Cosmological Evolution 30 ρm 20 ρ ϕ ρr 0.5 weff wϕ Log 10 (ρ/ρc0) 10 0 w Log 10 (1+z) Log 10 (1+z) m r Φ Log 1 z Figure: µ 1 = 20, µ 2 = 0.1 and β = /28
21 Cosmological Perturbation We consider the following metric in the Newtonian gauge ds 2 = a(τ) 2[ (1+2Φ)dτ 2 +(1 2Ψ)d x 2], where τ is the conformal time, Φ and Ψ are the scalar perturbations of the metric. We expand the perturbations in series, Φ = Φ ! Φ ! Φ 3 +, Ψ = Ψ ! Ψ ! Ψ 3 +, 13/28
22 Linear Perturbation A. Ali, R. Gannouji, MWH and M. Sami, Phys. Lett. B 718, 5 (2012) MWH, Phys. Rev. D 96, no. 2, (2017) Equation for the linear density contrast in subhorizon (k 2 H 2 ) and quasistatic ( φ H φ k 2 φ ) approximations, δ 1 +(H+ β M Pl φ) δ 1 4πG eff a 2 ρ m δ 1 = 0 G eff G eff = G 1+ ( α M 3 a 2 M φ2 Pl +2β ) 2 4α M 3 a 2 ( φ+h φ ) 2 α2 M 6 a 4 M 2 Pl φ 4 14/28
23 Evolution of G eff Geff G Log 1 z Figure: Evolution of G eff /G with µ 1 = 20, µ 2 = 0.5 and β = /28
24 Growing and decaying modes: Integral solutions MWH, Phys. Rev. D 96, no. 2, (2017) Using the transformation ã = a e βφ/m Pl Evolution equation of the density contrast can be written as δ 1 + H δ 1 4π G eff ã 2 ρ m δ 1 = 0 where G eff H = 1 dã ã dτ = H+ β dlnã φ = H M Pl dlna = G eff e (2β/M Pl)φ 16/28
25 Growing and decaying modes: Integral solutions Growing Mode D + (ã) = ã 7/4 m e 3βφm/4M Pl ( A(φm ) A(φ 0 ) ) 1/2 γ(ã) H 0 ã H Decaying Mode D (ã) = ãm 3/4 e7βφm/4m Pl [ ã m A(φ m ) ( ) 1/2 A(φm ) γ(ã) A(φ 0 ) ã ã dã ] ã m γ 2 (ã ) H 0 H 17/28
26 Growing and decaying modes: Integral solutions Where, A(φ) = dlnã dlna = 1+ β M Pl dφ dlna And γ 2 (ã) = e ã ãm dã g(ã ) Where g(ã ) satisfies dg(ã) dã 1 2 g2 (ã)+2i(ã) = 0 With I(ã) = A(ã)+ 1 d H ( ã H dã 1 d H ) d 2 H 4 H 2 dã 2 H dã 2 A(ã) = 4π G eff ρ m H 2 18/28
27 Density Contrast D+(z)/a β=0 β=0.1 β=0.2 ΛCDM Log 10 (1+z) D+(z)/a β= β= β= ΛCDM Log 10 (1+z) Figure: µ 1 = 20, µ 2 = 0.1 and β = 0, 0.1, /28
28 Density Contrast: Comparison δ1(z), D+(z) Log 10 (1+z) Figure: µ 1 = 20, µ 2 = 0.1 and β = /28
29 Power spectrum and fσ 8 (k) (h -3 Mpc 3 ) z=0 β=0.2 β=0.1 β=0 z=1 fσ8(z) β=0.2 β=0.1 β= k (hmpc -1 ) z Figure: µ 1 = 20, µ 2 = 0.1 and Ω b = 0.04, Ω m = 0.3 and n s = /28
30 Second Order Perturbation Equation for the second order density contrast, ( δ 2 + H+ β ) φ δ 2 4πG eff a 2 ρ m δ 2 =S δ M Pl Fourier transform of S δ can be written as S δ (a, k) = d 3 k 1 d 3 k 2 δ (3) ( k k 1 k 2 )K(a, k 1, k 2 )δ 1 (a, k 1 )δ 1 (a, k 2 ) 22/28
31 Second Order Perturbation Second Order Density Contrast ã δ 2 (ã, k) = D + (ã)δ 2 ( k) D + (ã) ã D + (ã )Ŝδ(ã, +D (ã) k) ã m ã 2 H 2 (ã )W r (ã ) dã D (ã )Ŝδ(ã, k) ã m ã 2 H 2 (ã )W r (ã ) dã W r (ã) = D + (ã) dd (ã) dã D (ã) dd +(ã) dã 23/28
32 Second Order Perturbation δ(ã, k) = δ 1 (ã, k)+ 1 2 δ 2(ã, k) = D + (ã)δ 1 ( k)+ d 3 k 1 d 3 k 2 δ (3) ( k k 1 k 2 ) F 2 (ã, k 1, k 2 )δ 1 (ã, k 1 )δ 1 (ã, k 2 ) Second Order Kernel F 2 (ã, k 1, k 2 ) = ã dã D(ã,ã )K(ã, k 1, k 2 ) ã m 2ã 2 H 2 (ã )W r (ã ) ( ) D(ã,ã ) = D2 + (ã ) D+(ã) 2 D (ã)d + (ã ) D + (ã)d (ã ) 24/28
33 Bispectrum δ(τ, k)δ(τ, k )δ(τ, k ) = δ (3) ( k + k + k )B(τ,k,k ) B(τ,k,k ) = 2F 2 ( k, k )P(k)P(k )+cyc Q = B(τ,k,k ) P(τ,k)P(τ,k )+P(τ,k )P(τ,k ) /28
34 Bispectrum (θ) (θ) θ/π θ/π Figure: µ 1 = 20, µ 2 = 0.1 and k = k = 0.01 hmpc 1 and 5k = k = 0.05 hmpc 1 for β = 0, 0.5 and ΛCDM. 26/28
35 Summary We have discussed the effect of the conformal coupling at the perturbation level in a tracker scalar field model with a cubic Galileon correction term. Integral solution of the growing and decaying modes are calculated in the subhorizon approximation. Effect of the conformal coupling constant on matter power spectrum and bispectrum has been observed. The power spectrum changes for different conformal constant but there is no significant change in the reduced bispectrum. Comparison with fσ 8 data shows that higher values of the conformal constant can be ruled out. 27/28
36 THANK YOU 28/28
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