On initial conditions and global existence for accelerating cosmologies from string theory

On initial conditions and global existence for accelerating cosmologies from string theory

Makoto Narita

Center for Relativity and Geometric Physics Studies, Department of Physics, National Central University, Jhongli 320, Taiwan

[email protected] 01/08/2005 at Taipei

• Motivations

• Definitions and assumptions

• Results

• Summary

1 Motivations

• Accelerating cosmologies from string/M-theory

Simple compactifications of higher-dimensional theories lead to lower-dimensional effective actions with exponential potentials.

These actions typically of the form

S = ^Z d⁴x√

−g

"

− ⁴R + 1

2(∇ψ)² + V₀e^aψ

#

. (1)

a is a coupling (positive) constant.

V₀ is set by the magnitude of fluxes of four-form field strenghs and/or the internal curvature.

Initial condition:

The field ψ has a large negative value with very large positive velocity.

Question:

Are there solutions satisfying the initial conditions to the Einstein- matter equations in generic?

2 Definitions and assumptions

• Reduced effective action in the Einstein frame:

S_IIA = ^Z d⁴x√

−g

"

−⁴R + 1

2(∇φ)² + 1

2 · 3!e^−2λφH² + 1

2 · 4!e^−2λφF²

#

, (2)

4R: Ricci scalar with respect to g, φ: dilaton, λ: coupling constant,

H = dB: anti-symmetric three-form field strength, F = dC: anti-symmetric four-form field strength.

In four dimensions, there is a duality between the three-form field strength and a one-form. Then, we define the pseudo-scalar axion field σ as follows:

H^µνρ = ²^µνρκe^2λφ∇_κσ. (3)

The field equation and the Bianchi identity

∇_µ^³e^−2λφF^µνρκ´ = 0, ∂_[αF_µνρκ] = 0, (4)

for the four-form field strength can be solved by

F^µνρκ = Q²^µνρκe^2λφ, (5) where Q is an arbitrary constant.

⇓

SIIA∗ =

Z

d⁴x√

−g

"

− ⁴R + 1 2

n(∇φ)² + e^2λφ(∇σ)² + Q²e^2λφo

#

. (6)

Hereafter, we assume Q 6= 0.

• Gowdy symmetry spacetimes: T² isometry group with spacelike orbits and the twists associated to the group vanish

• Spacetime topology: M₄ = T³ × R

• Metric in the areal time coordinate:

ds = −e^{2(η−U )}αdt² + e^{2(η−U )}dθ² + e^2U(dx + Ady)² + e^−2Ut²dy² (7)

∂/∂x, ∂/∂y: Killing vector fields generating the T² group action

• η, α, U, A, φ and σ are functions of t ∈ (0, ∞) and θ ∈ S¹.

• Initial singularity is at t = 0.

• Constraint equations:

˙η

t = ˙U² + αU⁰² + e^4U

4t²( ˙A² + αA⁰²)

+ 1 4

h˙φ² + αφ⁰²+ e^2λφ( ˙σ² + ασ⁰²) + αQ²e2λφ+2(η−U )i

, (8)

η⁰

t = 2 ˙UU⁰ + e^4U

2t²AA˙ ⁰− α⁰ 2tα + 1

2( ˙φφ⁰ + e^2λφ˙σσ⁰), (9)

˙α = −tα²Q²e2λφ+2(η−U ). (10)

• Evolution equations:

U − αU¨ ⁰⁰ = −U˙

t + ˙α ˙U

2α + α⁰U⁰

2 + e^4U

2t²( ˙A² − αA⁰²) + 1

4αQ²e2λφ+2(η−U ), (11)

A − αA¨ ⁰⁰ = A˙

t + ˙α ˙A

2α + α⁰A⁰

2 − 4( ˙A ˙U − αA⁰U⁰), (12)

φ − αφ¨ ⁰⁰ = − ˙φ

t + ˙α ˙φ

2α + α⁰φ⁰

2 + λe^2λφ( ˙σ² − ασ⁰²) − λαQ²e2λφ+2(η−U ), (13)

¨

σ − ασ⁰⁰ = − ˙σ

t + ˙α ˙σ

2α + α⁰σ⁰

2 − 2λ( ˙φ ˙σ − αφ⁰σ⁰). (14)

• ˙ and ⁰ denote derivatives with respect to t and θ, respectively.

3 Results

Theorem 1 Suppose that 0 < k < 1/2, α₀ > 0 and λκ > −1/2 hold, and ² is a positive constant such that max{0, −2λκ} < ² <

min{4k, 2 − 4k}. For any choice of the analytic data η₀(θ), α₀(θ), k(θ), U₀(θ), h(θ), A₀(θ), κ(θ), φ₀(θ) and ω(θ), subject to the follow- ing condition

η₀⁰ − 2kU₀⁰ − e^4U0(1 − 2k)h⁰A₀ − κφ⁰₀

2 + α₀⁰

2α₀ = 0, (15) the Gowdy symmetric IIA system has a solution of the form

η =



k² + κ² 4



ln t + η₀(θ) + t^²µ(t, θ), (16)

α = α₀(θ) + t^²β(t, θ), (17) U = k(θ) ln t + U₀(θ) + t^²V (t, θ), (18) A = h(θ) + t^2−4k(A₀(θ) + B(t, θ)) , (19) φ = κ(θ) ln t + φ₀(θ) + t^²Φ(t, θ), (20) σ = ω(θ) + t^²Σ(t, θ), (21) where µ, β, V , B, Φ and Σ tend to zero as t → 0. 2

Method: Fuchsian algorithm

Solutions given by Theorem 1 contain ones satisfying the initial conditions.

We do not have the maximum number of free functions in this case.

Theorem 2 Suppose that 0 < k < 1/2, α₀ > 0 and −1 < λκ < 0 hold, and ² is a positive constant less than min{4k, 2−4k, −2λκ, 2+

2λκ, 2K}. For any choice of the analytic data η₀(θ), α₀(θ), k(θ), U₀(θ), h(θ), A₀(θ), κ(θ), φ₀(θ), ω(θ) and σ₀(θ), subject to the fol- lowing condition

η₀⁰ − 2kU₀⁰ − e^4U0(1 − 2k)h⁰A₀ − κφ⁰₀

2 + e^2λφ0κω⁰σ₀ + α⁰₀

2α₀ = 0, (22)

the Gowdy symmetric IIA system has a solution of the form (16)-(20) and

σ = ω(θ) + t^−2λκ(σ₀(θ) + Σ(t, θ)) , (23) where µ, β, V , B, Φ and Σ tend to zero as t → 0. 2

We have the maximum number of free functions.

Solutions given by Theorem 2 do not satisfy the initial conditions.

Thus, our solution satisfying the initial conditions for acceler- ating cosmologies are not generic in the sense that there is not maximum number of arbitrary functions.

Theorem 3 (Global existence theorem) Let (M, g, φ, σ) be the max- imal Cauchy development of C^∞ initial data for the Gowdy sym- metric IIA system. Suppose that the timelike convergence con- dition, which is R_µνW^µW^ν ≥ 0 for any timelike vector W^µ, holds and there is a positive constant ¯λ such that |λ| ≤ ¯λ < 1/2. Then, M can be covered by compact Cauchy surfaces of constant areal time t with each value in the range (0, ∞).

Method: Light cone estimate

Note that solutions given by Theorem 1 and Theorem 2 satisfy the timelike convergence condition near the initial singularities.

4 Summary

• For Gowdy symmetric spacetimes with a exponential potential, a solution satisfying the initila conditions for accelerated expan- sion has been constracted.

• A global existence theorem for the spacetimes in the areal co- ordinate can be shown.