Singularities in cosmologies with interacting fluids

Transcription

1 Georgia Kittou co-author Spiros Cotsakis University of the Aegean, Research Group GeoDySyC MG Stockholm Parallel Session GT2: Cosmological Singularities and Asymptotics 02/07/2012

2 Nature abhors vacuum Modelling the universe using cosmological fluids with mutual interaction and energy exchange Finite-time singularities may develop in cosmologies with interacting fluids Which are the asymptotic properties of the solutions in a neighbourhood of a finite-time singularity? abhors vacuum.png

3 1 The dominant balance at cosmological singularities The dynamical system Decompositions of the vector field 2 3

4 The dominant balance at cosmological singularities The dynamical system Decompositions of the vector field Roughly speaking about the method... Dynamical method to study approach to spacetime singularities thus revealing their nature We are interested in the local behaviour of solutions in a small neighbourhood of the finite-time singularity A non-linear vector field admits a weight homogeneous decomposition 1 of the form f = f (0) + f (1) f (k) such that the asymptotic system has an exact scale invariant solution 2 and the original vector field is splitted in parts collecting the most non-linear terms together the decomposition reveals important dominant features when approaching the singularity 1 f (t d 1 x 1,..., t d k x k ) = t w f (x 1,..., x k ) 2 x = Ξt p

5 The dominant balance at cosmological singularities The dynamical system Decompositions of the vector field FRW universe with k = 0 We consider two fluids with equations of state p 1 = (Γ 1)ρ 1 (1) p 2 = (γ 1)ρ 2 (2) Friedmann and the continuity equations are respectively 3H 2 = ρ 1 + ρ 2 (3) ρ 1 + 3HΓρ 1 = βhρ 1 + αhρ 2 (4) ρ 2 + 3Hγρ 2 = βhρ 1 αhρ 2 (5)

6 The dominant balance at cosmological singularities The dynamical system Decompositions of the vector field The vector field Using Friedman and the continuity equations we eliminate the energy densities and get a master equation for the Hubble expansion Ḧ + AHḢ + BH 3 = 0 (6) For x = H the above equation is equivalent with the dynamical system ẋ = y (7) ẏ = Axy Bx 3 where A = α + β + 3γ + 3Γ and B = 3 2 (αγ + βγ + 3Γγ) with α > 0 and β > 0

7 The corresponding vector field is f (x, y) = (y, Axy Bx 3 ) (8) Dominant terms of decompositions f (1) = (y, Axy Bx 3 ) f (2) = (y, Axy) f (3) = (y, Bx 3 ) Classification of solutions using parameter δ = B/A 2 Decaying cosmologies and the Borderline case Cyclic cosmologies

8 , the δ 0 attractor The second decomposition of the system has a unique balance given by B (2) 1 = [Ξ, p] = [(2/A, 2/A), ( 1, 2)], (9) where A 0 The candidate subdominant part f (2, sub) = (0, Bx 3 ) is asymptotically subdominant in the sense that f (2, sub) (Ξ, t p ) t p 1 (0, 8Bt3p A 3 t q 1 ) = ( 0, 8δ A only if δ 0 The associated K matrix has characteristic equation λ 2 λ 2 = 0 hence the K-exponents are given by spec(k (2) ) = ( 1, 2) ) 0 (10)

9 Dominant features of the solution The general asymptotic solution around the singularity is x = 2 A t 1 + c 21 t A 10 c2 21t (11) Dominating solution on approach to the singularity x = H 2 A t 1 or in terms of the scale factor a(t) t 2 A On approach to the singularity and A > 0 Big Bang singularity at early times (a 0, H ) On approach to the singularity and A < 0 Big Rip singularity at late times (a, H ) Barrow Clifton family of solutions a BC (t) t 2 A our solution at early times is asymptotic to

10 , δ The third decomposition of the system has two balances that give analogous results: B (3) 1 = [(± 2/ B, 2/ B), ( 1, 2)], B < 0 (12) The candidate subdominant part f (3, sub) = (0, Axy) satisfies only if A = 0, i.e δ f (3, sub) (Ξ t p ) t p 1 (0, Aθξ) = (0, 0) (13) δ singularity occurs only if one of the gammas is negative (eg Γ 0, γ < 0) The K-exponents are spec(k (3) ) = ( 1, 4)

11 Dominant features of the solution The general solution is given by 2 x = ± B t 1 + c 41 t 3 B 2 12 c2 41 B t (14) Dominating solution while approaching the singularity a(t) t ± 2 B, B < 0 (15) For 2 B irrational number and tɛc The origin t = 0 is a logarithmic branch point For tɛr The scale factor is an increasing/decreasing function of time with Big Bang/Big Rip singularity located as t 0

12 Decaying cosmologies, case δ < 1 8 For the range of values 0 < δ 1 8 then the two fluids decay into each other in proportion to their energy densities We are interested in the second balance of the first decomposition f (1) The corresponding balance is given by the expression spec(k (1) 2 ) = ( 1, 1) The general solution reads B (1) 2 = [(3/A, 3/A), ( 1, 2)] (16) x = 3 A t 1 + c 11 + A 3 c2 11t +... (17)

13 The dominant features of the solution On approach to the singularity the dominant part of the solution is given by a(t) t 3 A (18) The 1-parameter solution has again a dual behaviour depending on the value of A A family of solutions for δ = 1/9 is found by Luis Chimento and reads [ a a 0 ] A 3 = 1 + C5 t + C 6 t 2 (19)

14 The borderline case δ = 1/8 δ = 1 8 is the limit for real power-law solutions For the first decomposition of the system f (1) there are two balances which give identical results for δ = 1/8; B (1) 1 = B (1) 2 = [(4/A, 4/A), ( 1, 2)] (20) spec(k (1) 1 ) = spec(k(1) 2 ) = ( 1, 0) The solution of the system is particular and given by x H = 4 A t 1 (21) A more general solution for δ = 1 8 is found by Luis Chimento H 2 = a A 2 (c3 + c 4 ln a) whereas an identical to ours is found by Barrow-Clifton

15 Cyclic universes, case δ > 1/8 The first decomposition of the system f (1) reveals a very complicated singularity for the specific value of δ For both balances and δ = 1 2 the second eigenvalue of the Kovalevskaya matrix is complex with positive real part so the solution has no Puiseux series representation The solution for both balances is given by x = 1 ± i 3 t 1 (22) A In terms of the scale factor a t 1±i 3 A

16 Analysis of the behaviour of solution For tɛr a(t) t 1 A [ cos ( ) ( )] 3 3 ± i sin A A Setting t = exp τ then ( a A (τ) exp (τ) cos( 3τ) ± i sin( ) 3τ) (23) For tɛc The scale factor is a multifunction with a logarithmic branch point placed at the origin Oscillatory behaviour Big Bang singularity at τ

17 Anti-decaying fluids, case δ Anti-decaying fluids are met in the decomposition f (3) and B > 0 For the specific singular value of δ, ρ 2 decays and ρ 1 instead of loosing energy it gains energy The singularity δ occurs for α > 0 and β < 0 The two balances are given by B (3) 1 = B (3) 2 = [(±i 2/B, i 2/B), ( 1, 2)] (24) The components of the vector Ξ are complex The decomposition is asymptotically acceptable only if A = 0, i.e. δ

18 Oscillatory behaviour of solution The general solution is given by x = i 2/B t 1 + c 41 t 3 i B 12 2/Bc 2 41 t (25) For the dominating part of the solution the scale factor is given by a(t) t i 2 B For tɛr For t = exp (τ) then the scale factor is given by 2 2 a(τ) cos( τ) i sin( τ) (26) B B For tɛc The origin represents a logarithmic branch point

19 Complete universes, case δ < 1/8 Another sort of solution is found when proceeding with the first decomposition of the system, for the case δ = 1/9 The K-exponents are all negative therefore the solution is driven away from the singularity The general solution is x = 6 A t 1 + c 11 t 2 + c 21 t (27)

20 Discussion We have examined the behaviour of the solution near a finite-time singularity when considering interacting fluids with energy exchange We found different types of solution depending on the value of the δ parameter The solution corresponding to the case δ 0 is an attractor of the Barrow-Clifton solutions at early times There is an interesting behaviour regarding the Phantom singularities case where the origin is a logarithmic branch point In decaying cosmologies the scale factor collapses to zero size or decays into big rip singularity when approaching t 0 The scale factor also behaves as an oscillator in early times There is also a family of solutions that drives the scale factor away from the finite-time singularity

21 Future work The addition of a curvature term differentiate the form of the asymptotic solutions? Is the flat case scenario an attractor to the curved solutions when approaching a singularity?

22 Thank you for your attention

23 J.D. Barrow and T. Clifton, Phys. Rev. D73 (2006) L.P. Chimento, J.Math. Phys. 38(1997) 2565