Multifield Dark Energy (Part of the Master Thesis)

Transcription

1 Multifield Dark Energy (Part of the Master Thesis) Valeri Vardanyan Institut für Theoretische Physik Universität Heidelberg Bielefeld 2015 Valeri Vardanyan (ITP, Uni. Heidelberg) Multifield Dark Energy Bielefeld / 15

2 Outline 1 General Introduction 2 Multiscalar Brans-Dicke Theory 3 Perturbations 4 Non Cancellation of Polynomials Order Valeri Vardanyan (ITP, Uni. Heidelberg) Multifield Dark Energy Bielefeld / 15

3 General Introduction Modified Theories of Gravity Based on V.V., Luca Amendola, "How can we tell whether dark energy is composed by multiple fields?", gr-qc: Modifications of the standard theory of General Relativity to explain many fundamental phenomena in cosmology (the accelerated expansion of the Universe, Inflation, etc.) Brans-Dicke (strong motivations from higher dimensional theories, such as String Theory) Standard Brans-Dicke Action S BD = d 4 x g(φr ω BD φ g µν µ φ ν φ V (φ)) + d 4 x gl matter Valeri Vardanyan (ITP, Uni. Heidelberg) Multifield Dark Energy Bielefeld / 15

4 Multiscalar Brans-Dicke Theory Introduction One might think to generalize single-field theories to multi-field ones. (From compactifications of extra dimensions.) 2-Field Brans-Dicke Action S 2 field = d 4 x g[φ 1 R ω 1 φ 1 g µν µ φ 1 ν φ 1 + φ 2 R ω 2 φ 2 g µν µ φ 2 ν φ 2 -V(φ 1, φ 2 )] + d 4 x gl matter. N-Field Brans-Dicke Action S N field = d 4 x g[ N i=1 (φ ir ω i φ i g µν µ φ i ν φ i ) V (φ 1, φ 2,..., φ N )]+ d 4 x gl matter. Valeri Vardanyan (ITP, Uni. Heidelberg) Multifield Dark Energy Bielefeld / 15

5 Multiscalar Brans-Dicke Theory Introduction There exist a single-field potential V (φ) that reproduces any observed Hubble parameter. From Friedmann equations: ( ) dφ 2 H(z) 2 (1 + z) 2 = 2(1 + z)h(z) dh dz dz ρ m(z)(1 + w m (z)) (Find ρ m (z) from continuity eq.) V (z) = 3H(z) 2 (1 + z)h(z) dh(z) dz Linear Cosmological Perturbations!! + ρ m(z) (w m (z) 1) 2 Valeri Vardanyan (ITP, Uni. Heidelberg) Multifield Dark Energy Bielefeld / 15

6 Perturbations g µν = g (0) µν + δg µν and ds 2 = a 2 (η)[ (1 + 2Ψ)dη 2 + (1 + 2Φ)dx i dx i ] also φ(t, x) = φ(t) + φ(t, x) Interested in: η = Φ Ψ The effective gravitational constant Y which enters the modified Poisson equation: k 2 a 2 Ψ = 4πGY δρ Valeri Vardanyan (ITP, Uni. Heidelberg) Multifield Dark Energy Bielefeld / 15

7 Perturbations Continuation η and Y for Standard Brans-Dicke Theory ( ) ( ) η = h 1+k 2 h k 2 h 5, Y = h 1+k 2 h k 2 h 3 with h i some known time-dependent functions. η and Y for Multiscalar Brans-Dicke Theory P n η = h (1) (k) 2 P (2) n (k), Y = h 1 P (2) n (k) P n (3) (k) The exact forms of the coefficients are important!! Valeri Vardanyan (ITP, Uni. Heidelberg) Multifield Dark Energy Bielefeld / 15

8 Perturbations Continuation 2-Field Brans-Dicke Action S 2 BD = d 4 x g[φ 1 R ω 1 φ 1 g µν µ φ 1 ν φ 1 + φ 2 R ω 2 φ 2 g µν µ φ 2 ν φ 2 V (φ 1, φ 2 )] + d 4 x gl matter Equations of Motion (φ 1 + φ 2 )G µν + [ (φ 1 + φ 2 ) + 1 ω 1 2 φ 1 ( φ 1 ) ω 2 2 φ 2 ( φ 2 ) 2 ]g µν µ ν (φ 1 + φ 2 ) ω 1 φ 1 µ φ 1 ν φ 1 ω 2 φ 2 µ φ 2 ν φ 2 + V 2 g µν = 8πGT µν φ i R + 2ω i φ i ω i φ i α φ i α φ i φ i V,φi = 0, i = 1, 2 (3 + 2ω 1 ) φ 1 + (3 + 2ω 2 ) φ 2 + 2V (φ 1, φ 2 ) V,φ1 V,φ2 = 8πGT. Valeri Vardanyan (ITP, Uni. Heidelberg) Multifield Dark Energy Bielefeld / 15

9 Perturbations Continuation 2-Field Perturbation Equations 2(φ 1 + φ 2 ) k2 Φ + 2 k2 (φ a 2 a φ 2 ) + 2(φ 1 + φ 2 ) k2 Ψ = 24πGc 2 a 2 s δρ 4 k2 Φ + 2 k2 Ψ [2 ω a 2 a 2 i k 2 φ i + M 2 a 2 i ]φ i = 0, i = 1, 2 2(φ 1 + φ 2 ) k2 a 2 Φ + k2 a 2 (φ 1 + φ 2 ) = 8πGδρ N-Field Perturbation Equations 2 k φ 2 i Φ + 2 k2 a 2 a 2 φi + 2 k φ 2 i Ψ = 24πGc 2 a 2 s δρ 4 k2 Φ + 2 k2 Ψ [2 ω a 2 a 2 i k 2 φ i + M 2 a 2 i ]φ i = 0, i = 1, 2,..., N 2 φ i k 2 a 2 Φ + k2 a 2 φi = 8πGδρ Valeri Vardanyan (ITP, Uni. Heidelberg) Multifield Dark Energy Bielefeld / 15

10 Perturbations Continuation η and Y 1+ N η = C i=1 C i k 2i 0 1+, Y = 1+ N C i=1 D i k 2i 0 Ni=1 C i k 2i 1+ N i=1 D i k 2i Coefficients where C 0 = 1, C 1+3cs 2 0 = 1+3c2 s Ni=1, aaaand... φ i Valeri Vardanyan (ITP, Uni. Heidelberg) Multifield Dark Energy Bielefeld / 15

11 Perturbations Continuation Coefficients C d = 1 [( N a 2d C i=1 φ i)2 d ω i1 ω i2 i 1 >i 2 >...>i d φ i1 φ i2... ω i d φ id j i 1,i 2,...,i d Mj 2 + ( ) 2 + 3c 2 s 2 d 1 N ω i1 ω i2 i=1 φ i1 φ i2... ω i d 1 φ id 1 j i,i 1,i 2,...,i d 1 Mj 2] C = ( N i=1 φ i) N (d = 1,..., N) j=1 M2 j ; i j i, j=1,...,d 1 i 1 >i 2 >...>i d 1 Valeri Vardanyan (ITP, Uni. Heidelberg) Multifield Dark Energy Bielefeld / 15

12 Perturbations Continuation Coefficients C d = 1 a 2d C [ ( 1 + 3c 2 s 2 ( 2 + 3cs 2 ) 2 d 1 N i=1 Coefficients ) ( N i=1 φ i)2 d i 1 >i 2 >...>i d ω i1 i j i, j=1,...,d 1 i 1 >i 2 >...>i d 1 D d = 1 a 2d D [( N i=1 φ i) 2 2 d ω i1 i 1 >i 2 >...>i d 3( N i=1 φ i)2 d 1 N i=1 i j i, j=1,...,d 1 i 1 >i 2 >...>i d 1 ω i2 φ i1 φ i2... ω i d φ id j i 1,i 2,...,i d Mj 2 + ω i1 ω i2 φ i1 φ i2... ω i d 1 φ id 1 j i,i 1,i 2,...,i d 1 Mj 2] ω i2 φ i1 φ i2... ω i d φ id j i 1,i 2,...,i d Mj 2 + ω i1 ω i2 φ i1 φ i2... ω i d 1 φ id 1 j i,i 1,i 2,...,i d 1 Mj 2] C = ( 1 + 3c 2 s ) ( N i=1 φ i) N j=1 M2 j ; D = ( N i=1 φ i) 2 N j=1 M2 j ; (Note: D d = C d ) Valeri Vardanyan (ITP, Uni. Heidelberg) Multifield Dark Energy Bielefeld / 15

13 Non-Triviality Effectively lower order polynomials only in one of the following cases: a) c 2 s = 1/3, b) at least one of the coupling constants is infinitely large, c) one or more M 2 i, d) one or more φ i 0, e) there are i and j such that ω i,j 0, f ) there are i and j such that ω i φ i = α ω j φ j and Mi 2 = αmj 2. (Note the difference between α = 1 and all the other α s.) Valeri Vardanyan (ITP, Uni. Heidelberg) Multifield Dark Energy Bielefeld / 15

14 Summary Multiscalar Brans-Dicke Theory. Generic fluid matter source with equation of state w and sound speed c s. This might turn out useful to compare real data to models in which dark matter includes hot or warm components. There are no non-trivial cases in which the higher-order k terms in the polynomials cancel out the k-structure of Y, η is a unique signature of the number of dark energy fields coupled to gravity. Valeri Vardanyan (ITP, Uni. Heidelberg) Multifield Dark Energy Bielefeld / 15

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