On the problem of inflation in non-linear multidimensional cosmological models. Odessa National University

Transcription

1 On the problem of inflation in non-linear multiimensional cosmological moels Alexaner Zhuk, Tamerlan Saiov Oessa National niversity

2 -imensional factorizable geometry: g g ( ( x n i L Pl e β i ( x g ( i ( y Internal space scale factor: a i ( x L Pl e i β ( x Manifol: M M L M M { n External (our space-time Internal spaces (Ricci-flat orbifols with fixe points branes in fixe points niversal Extra imension moels: the Stanar Moel fiels are not localize on branes but can move in the bulk

3 imensional action: S x g { R[ g] Λ } Sm κ M S b bulk matter branes Monopole form fiels: S m n x g M i! i ( ( i F { S b fixe points x ( g ( x τ ( k fixe points Freun-Rubin ansatz Inuce metric tensions n i f a i i i const

4 imensional reuction (Einstein frame: n /( ( i β i ~ ( μν e gμν i g ; β i ( β i i x β ( x ( n Internal space stabilization position (present ay value S κ M ~ ~ { ( ( μν R g ] g ϕ ϕ ( ϕ } ( g [ ~ μ ν ϕ ( i β

5 Effective potential: ( ϕ e ~ f /( ϕ /( ϕ /( ϕ [ Λ e λe ] ~ f / ( λ κ τ κ f a k m ( k Zero global minimum conition: Λ f / λ Internal space stabilization qϕ Large ϕ limit: Λ e, q < Power law inflation: q 8πG a( t a ρ ( t t / q

6 The form of ( ϕ in the case, Λ f λ / 5 Internal space compactification - - ϕ Power-law inflation (internal space ecompactification Slow-roll parameter in local imum: η η < ϕ Topological inflation: ϕ ϕ min ϕcr.65 Inflation in local imum is absent in our case: ϕ ( / ln.5 <.65

7 Generalization: aitional scalar fiel ( ϕ, e ( ~ f /( ϕ /( ϕ /( ϕ [ Λ e λe ] Origin: nonlinear gravitational moels S x g f ( R κ g ab M /( [ f / R ] gab { e ( /( n x g i! M i ( x g ( x ( k τ ( ( i F μ ν M κ [ ~ ~ ( ( ] ~ ( μν { ~ ( μν g R g g ϕ ϕ g ( ϕ, } μ ν Internal space scale factor Nonlinearity scalar fiel

8 Quaratic moel: f ( R R ξ R Λ ( e B ξ A ( e Λ, A, B A Non-negative minimum conition of ( ϕ, : ( ϕ, : ξ, Λ > Zero global minimum conition of ( f / λ ( [ ] exp( A ( 8 / ( A B A A B B ξλ A B ( ϕ, Global minimum of ( ϕ, : Sale point of ( ϕ, : ( ϕ, ( ϕ ϕ, ϕ ( / ln.5 <.65

9 Contour plot of the ective potential ( ϕ, for parameters, ξ Λ. Colour lines escribe trajectories for scalar fiels starting from ifferent regions of the ective potential..5 esitter stage.5 Power-law stage.5 Φ.5 Friemann M stage

10 Φ t t - - Internal space scale factor an nonlinear scalar fiel as functions of time (in Planck units

11 Log a 8 H t 6 8 t Number of e-folings. We choose the following bounary conition (in Planck units: a( t N 9 Hubble parameter

12 Parameter of acceleration esitter stage a&& ( s / s, a t q Ht H a, a e s Power-law inflation ( s q ( ω EoS parameter q q t t -.5 { -.5 Friemann M stage s / ( q.5 Power-law inflation: exp 8ξ ( /( ϕ exp( (A B s >

13 Quartic moel: f ( R R γ R Λ B / A / ( e (γ ( e Λ Asymptotes: B ( e ( 8 (γ / ( γ Λ / e ({ B A/ ( 8 / ( ( if γ, Λ > if γ >, < 8 8 We shall consier the case < 8,, - critical value Non-negative local minimum: (, λ, f, γ, Zero local minimum: ( f / λ Λ >

14 ( - extremum conition: Λ M M M k R ( ( (, m, ( / / / / k M ω ω [ ], (6 ( ( 7 ( 7( k k ω > Λ k γ (6 ( ( 7 k k > ( ( ( R R Minimum conition: minimum imum [ ] [ ] ( ( min ln ln R A R A γ γ min min (, (

15 ( ϕ, - extremum conition (with respect to : ϕ ϕ Solutions: χ α λ / f ( ± χ ( ± α ± α ± α min α β f f min β,, χ λχ ; χ λχ α α β min β β α / f min ; e e α bϕ bϕ χ χ α < α < α α α α < α < α α α > α no extrema one extremum (point of inflection on the line min two extrema (one minimum an one sale on the line min three extrema (minimum an sale on the line min, inflection on the line four extrema (minimum an sale on the line min, imum an sale on the line

16 Four extrema case ( α > α with zero local minimum: ( ϕ, min ~ α > α k ( < k < k / min < / β /, α / k γλ >.5.5 Χ Χ Χ Contour plot of in the case Here, α /.57, α. 59 Χ Φ, β /, k., Λ an α /.67 > α...

17 Inflation? Slow-roll parameters in extrema: ε Η k Η Φ k Graphs of η ϕ an η as functions of k ( k ~, k for local imum χ an ( parameters β /, Λ.,,,. Η Φ k η Graph of as fuction of k for sale point an parameters χ ( β /, Λ.,,,.

18 Contour plot of the ective potential ( ϕ,, β /, α /, k., Λ.. for parameters Colour lines escribe trajectories for scalar fiels starting from ifferent regions of the ective potential. esitter stage Φ Friemann M (ust stage

19 Φ. 6 8 t 6 8 t.5 Internal space scale factor an nonlinear scalar fiel as functions of time (in Planck units

20 Log a H t 6 8 t Number of e-folings Hubble parameter N

21 Parameter of acceleration a&& H a q esitter stage q t { Friemann M (ust stage s / ( q.5 a t / q t

22 Topological (eternal inflation Φ Φmin Critical value Δ cr k ϕ ϕ χ Plots of min (for the profile ( as a function of k ( k ~, k for parameters β /, Λ.,,, (from left to right } 5 6 Φ Comparison of the potential ( ϕ χ, ( with a ouble well potential for parameters β /, Λ.,. In this particular case:. 6 > Δ The ratio of the characteristic thickness of omain wall to the horizon scale: r w H / / sale χ (. > r w H cr min.8 cr thick enough!

23 e-fols reuction: Gravexciton ecay: ψ ΔN γ 5 m ln Γ m Pl ψ Barenboim&Vives arxiv:86.89 Γ m ψ m / Pl Günther,Starobinsky,Zhuk PR69( If m ψ TeV ΔN 6 τ ψ / Γ m / m t ( life-time Pl Pl Pl 8 t presenthorizon scale Log(/(a H inflation matter raiation matter ark energy N e ΔN Log(a

24 Conclusions:. For all consiere moels, there are rangers of parameters where the internal space is stably compactifie.. At the same time nonlinear multiimensional moel can provie inflation of the external (our space. For consiere moels imal value of e-folings is N. This value is not sufficient to explain the horizon an flatness problem but enough for CMB. However, the spectral inex is less than. For example, in the case of R moel n s η χ.6<. (. The number of e-folings of the orer of is big enough to encourage the following investigation of the nonlinear multiimensional moels to fin theories where this number will approach to 5-6. R. Nonlinear moel can provie conitions for topological (eternal inflation.