Introduction Background model Construction of the basis Perturbation Theory Conclusions Perturbations on a cosmological model with non-null Weyl tensor Grasiele B. Santos 1 University of Rome "La Sapienza" and ICRANet Hot Topics in General Relativity and Gravitation Quy Nhon, Vietnam, August 2015. 1 In collaboration with E. Bittencourt and J. Salim, JCAP 06 (2015) 013.
Introduction Background model Construction of the basis Perturbation Theory Conclusions Outline Introduction Background model Construction of the basis Perturbation Theory Conclusions
Introduction Background model Construction of the basis Perturbation Theory Conclusions Outline Introduction Background model Construction of the basis Perturbation Theory Conclusions
Introduction Background model Construction of the basis Perturbation Theory Conclusions • We consider a class of Friedmann-type metrics with constant spatial curvature and with a stochastic magnetic field as matter content. • An anistropic pressure component sourced by this field is considered and it is found to be related to a non-null Weyl tensor. • We analyse the gravitational stability of this model under linear scalar perturbations using the covariant gauge-invariant approach in order to understand the role of the Weyl tensor in structure formation in this context.
Introduction Background model Construction of the basis Perturbation Theory Conclusions Outline Introduction Background model Construction of the basis Perturbation Theory Conclusions
Introduction Background model Construction of the basis Perturbation Theory Conclusions • Let’s consider ds 2 = dt 2 − a 2 ( t )[ d χ 2 + σ 2 ( χ ) d Ω 2 ] , (1) where t represents the cosmic time, a ( t ) is the scale factor and σ ( χ ) is an arbitrary function. • We then take as source the EM field with E i E i = 0 E i = 0 , B i = 0 , E i B j = 0 , (2) B i B j = − 1 3 B 2 h i j − π i j . (3) Therefore, T µν = ( ρ + p ) V µ V ν − pg µν + π µν , (4) 3 ρ and ρ = B 2 ( t ) with p = 1 . 2
Introduction Background model Construction of the basis Perturbation Theory Conclusions • Let’s consider ds 2 = dt 2 − a 2 ( t )[ d χ 2 + σ 2 ( χ ) d Ω 2 ] , (1) where t represents the cosmic time, a ( t ) is the scale factor and σ ( χ ) is an arbitrary function. • We then take as source the EM field with E i E i = 0 E i = 0 , B i = 0 , E i B j = 0 , (2) B i B j = − 1 3 B 2 h i j − π i j . (3) Therefore, T µν = ( ρ + p ) V µ V ν − pg µν + π µν , (4) 3 ρ and ρ = B 2 ( t ) with p = 1 . 2
Introduction Background model Construction of the basis Perturbation Theory Conclusions • Einstein equations admit a solution with constant spatial curvature and π µν only if 2 k π 22 = π 33 , π 11 = − 2 π 22 , π 11 = where a 2 σ 3 , (5) where k is an integration constant 2 . We can rewrite the metric as � � dr 2 ds 2 = dt 2 − a 2 ( t ) + r 2 d Ω 2 . (6) 1 − ǫ r 2 − 2 k r • FLRW is regained whenever 2 k ≪ r . From the evolution equation for the shear tensor and V µ = δ µ 0 we get 3 = − W µανβ V α V β = − 1 E µν . 2 π µν . (7) 2 E. Bittencourt, J. Salim and GBS, Gen. Rel. Grav. 46 (2014); Mc Manus and Coley, Class. Quant. Grav. (1994). 3 J. Mimoso and P. Crawford, Class. Quant. Grav. (1993).
Introduction Background model Construction of the basis Perturbation Theory Conclusions • Einstein equations admit a solution with constant spatial curvature and π µν only if 2 k π 22 = π 33 , π 11 = − 2 π 22 , π 11 = where a 2 σ 3 , (5) where k is an integration constant 2 . We can rewrite the metric as � � dr 2 ds 2 = dt 2 − a 2 ( t ) + r 2 d Ω 2 . (6) 1 − ǫ r 2 − 2 k r • FLRW is regained whenever 2 k ≪ r . From the evolution equation for the shear tensor and V µ = δ µ 0 we get 3 = − W µανβ V α V β = − 1 E µν . 2 π µν . (7) 2 E. Bittencourt, J. Salim and GBS, Gen. Rel. Grav. 46 (2014); Mc Manus and Coley, Class. Quant. Grav. (1994). 3 J. Mimoso and P. Crawford, Class. Quant. Grav. (1993).
Introduction Background model Construction of the basis Perturbation Theory Conclusions • The remaining equations are θ + θ 2 3 = − 1 ˙ 2 ( ρ + 3 p ) , (8a) ρ + ( ρ + p ) θ = 0 , ˙ (8b) E αµ ; α = 0 , (8c) E µν + 2 h ǫµ h νλ ˙ 3 θ E ǫλ = 0 . (8d) • The model can be extended to any equation of state (EOS) of the form p = ( γ − 1 ) ρ , which is also valid for a mixture of non-interacting fluids.
Introduction Background model Construction of the basis Perturbation Theory Conclusions Outline Introduction Background model Construction of the basis Perturbation Theory Conclusions
Introduction Background model Construction of the basis Perturbation Theory Conclusions • We take into account only the spatial scalar harmonic functions Q ( m ) ( x k ) and its derived vector and tensor quantities: Q i . Q ij . = Q , i , = Q , i || j = Q , i ; j . • These functions satisfy ∇ 2 Q ( m ) = m 2 Q ( m ) , (9) where m is a constant (the wave number) and ∇ 2 Q . = γ ij Q , i || j = γ ij Q , i ; j , (10) defines the 3-dimensional Laplace-Beltrami operator. • Then � R ( r ) Y n Q ( r , θ, φ ) = l ( θ, φ ) , l , n where Y n l ( θ, φ ) are the spherical harmonics.
Introduction Background model Construction of the basis Perturbation Theory Conclusions • We take into account only the spatial scalar harmonic functions Q ( m ) ( x k ) and its derived vector and tensor quantities: Q i . Q ij . = Q , i , = Q , i || j = Q , i ; j . • These functions satisfy ∇ 2 Q ( m ) = m 2 Q ( m ) , (9) where m is a constant (the wave number) and ∇ 2 Q . = γ ij Q , i || j = γ ij Q , i ; j , (10) defines the 3-dimensional Laplace-Beltrami operator. • Then � R ( r ) Y n Q ( r , θ, φ ) = l ( θ, φ ) , l , n where Y n l ( θ, φ ) are the spherical harmonics.
Introduction Background model Construction of the basis Perturbation Theory Conclusions • We take into account only the spatial scalar harmonic functions Q ( m ) ( x k ) and its derived vector and tensor quantities: Q i . Q ij . = Q , i , = Q , i || j = Q , i ; j . • These functions satisfy ∇ 2 Q ( m ) = m 2 Q ( m ) , (9) where m is a constant (the wave number) and ∇ 2 Q . = γ ij Q , i || j = γ ij Q , i ; j , (10) defines the 3-dimensional Laplace-Beltrami operator. • Then � R ( r ) Y n Q ( r , θ, φ ) = l ( θ, φ ) , l , n where Y n l ( θ, φ ) are the spherical harmonics.
Introduction Background model Construction of the basis Perturbation Theory Conclusions • We define the traceless operator Q ij = 1 m 2 Q ij − 1 ˆ 3 Q γ ij , (11) and its divergence can be computed yielding � 1 3 − ǫ � Q i − π ij Q j ˆ m 2 Q j . i || j = 2 (12) m 2 • In this model, we also need to consider the expansion of the terms π ij ˆ Q ij � ( m ) = a l ( m ) Q ( l ) , (13) l π ij Q j � ( m ) = b l ( m ) Q i ( l ) , (14) l and
Introduction Background model Construction of the basis Perturbation Theory Conclusions • We define the traceless operator Q ij = 1 m 2 Q ij − 1 ˆ 3 Q γ ij , (11) and its divergence can be computed yielding � 1 3 − ǫ � Q i − π ij Q j ˆ m 2 Q j . i || j = 2 (12) m 2 • In this model, we also need to consider the expansion of the terms π ij ˆ Q ij � ( m ) = a l ( m ) Q ( l ) , (13) l π ij Q j � ( m ) = b l ( m ) Q i ( l ) , (14) l and
Introduction Background model Construction of the basis Perturbation Theory Conclusions 1 Q ij ( l ) + γ ij 2 π k ( i ˆ � c l ( m ) ˆ � k Q j ) ( m ) = a l ( m ) Q ( l ) , (15) 3 l l where the coefficients a l ( m ) , b l ( m ) and c l ( m ) are constants for each of the modes m and l . • Assuming small deviations of the metric given in (6) wrt to FLRW, the quantities A ( m ) . B ( m ) . C ( m ) . � � � = a l ( m ) , = b l ( m ) , = c l ( m ) , l l l should be bounded. They are determined through the full solution for the basis and depend on k .
Introduction Background model Construction of the basis Perturbation Theory Conclusions 1 Q ij ( l ) + γ ij 2 π k ( i ˆ � c l ( m ) ˆ � k Q j ) ( m ) = a l ( m ) Q ( l ) , (15) 3 l l where the coefficients a l ( m ) , b l ( m ) and c l ( m ) are constants for each of the modes m and l . • Assuming small deviations of the metric given in (6) wrt to FLRW, the quantities A ( m ) . B ( m ) . C ( m ) . � � � = a l ( m ) , = b l ( m ) , = c l ( m ) , l l l should be bounded. They are determined through the full solution for the basis and depend on k .
Introduction Background model Construction of the basis Perturbation Theory Conclusions Outline Introduction Background model Construction of the basis Perturbation Theory Conclusions
Introduction Background model Construction of the basis Perturbation Theory Conclusions • According to the evolution equation for the shear tensor, we can define = E µν + 1 X µν . 2 π µν , (16) which is a good variable as it is null in the background. • Following Ellis & Bruni 4 , we also consider the fractional energy density gradient = a ( t ) h αν ρ ,ν χ α . ρ , (17) and the gradient of the expansion coefficient Z α . = a ( t ) h αν θ ,ν . (18) • To this set of variables we add: the acceleration a µ , σ µν and the divergence of the anisotropic pressure I µ ≡ h µǫ π ǫν ; ν . 4 G. F. R. Ellis and M. Bruni, Phys. Rev. D 40 , 1804 (1989).
Recommend
More recommend
