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String Bounce R. Branden- berger Fluctuations in Non-Singular Bouncing Motivation Cosmologies from Type II Superstrings Alternatives Perturbations Superstring Bounce Robert Brandenberger S-Brane McGill University Fluctuations

slide-1
SLIDE 1

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Fluctuations in Non-Singular Bouncing Cosmologies from Type II Superstrings

Robert Brandenberger McGill University March 16, 2012 Work in collaboration with C. Kounnas, H. Partouche, S. Patil and N. Toumbas arXiv/1203xxxx

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slide-2
SLIDE 2

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Outline

1

Motivation

2

Alternatives

3

Cosmological Perturbations

4

Bouncing Type II Superstring Cosmology

5

Fluctuations in Bouncing Type II Superstring Cosmology

6

Conclusions

2 / 51

slide-3
SLIDE 3

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Plan

1

Motivation

2

Alternatives

3

Cosmological Perturbations

4

Bouncing Type II Superstring Cosmology

5

Fluctuations in Bouncing Type II Superstring Cosmology

6

Conclusions

3 / 51

slide-4
SLIDE 4

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Space-time sketch of inflationary cosmology

The Inflationary Universe Scenario is the current paradigm

  • f early universe cosmology.

Note: H = ˙

a a

curve labelled by k: wavelength of a fluctuation

4 / 51

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SLIDE 5

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Space-time sketch of inflationary cosmology

The Inflationary Universe Scenario is the current paradigm

  • f early universe cosmology.

Note: H = ˙

a a

curve labelled by k: wavelength of a fluctuation

4 / 51

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SLIDE 6

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Successes of Inflation

inflation renders the universe large, homogeneous and spatially flat classical matter redshifts → matter vacuum remains quantum vacuum fluctuations: seeds for the observed structure [Chibisov & Mukhanov, 1981] sub-Hubble → locally causal

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SLIDE 7

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Credit: NASA/WMAP Science Team

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SLIDE 8

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Credit: NASA/WMAP Science Team

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SLIDE 9

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Trans-Planckian Problem for Fluctuations

Success of inflation: At early times scales are inside the Hubble radius → causal generation mechanism is possible. Problem: If time period of inflation is more than 70H−1, then λp(t) < lpl at the beginning of inflation → new physics MUST enter into the calculation of the fluctuations.

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SLIDE 10

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Trans-Planckian Problem for Fluctuations

Success of inflation: At early times scales are inside the Hubble radius → causal generation mechanism is possible. Problem: If time period of inflation is more than 70H−1, then λp(t) < lpl at the beginning of inflation → new physics MUST enter into the calculation of the fluctuations.

8 / 51

slide-11
SLIDE 11

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Trans-Planckian Problem for Fluctuations

Success of inflation: At early times scales are inside the Hubble radius → causal generation mechanism is possible. Problem: If time period of inflation is more than 70H−1, then λp(t) < lpl at the beginning of inflation → new physics MUST enter into the calculation of the fluctuations.

8 / 51

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SLIDE 12

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Recent Reference: A. Linde, V. Mukhanov and A. Vikman, arXiv:0912.0944 It is not sufficient to show that the Hubble constant is smaller than the Planck scale. The frequencies involved in the analysis of the cosmological fluctuations are many orders of magnitude larger than the Planck mass. Thus, “the methods used in [1] are inapplicable for the description

  • f the .. process of generation of perturbations in this

scenario."

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SLIDE 13

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Applicability of GR

In all approaches to quantum gravity, the Einstein action is only the leading term in a low curvature expansion. Correction terms may become dominant at much lower energies than the Planck scale. Correction terms will dominate the dynamics at high curvatures. The energy scale of inflation models is typically η ∼ 1016GeV. → η too close to mpl to trust predictions made using GR.

10 / 51

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SLIDE 14

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Cosmological Constant Problem

Quantum vacuum energy does not gravitate. Why should the almost constant V(ϕ) gravitate? V0 Λobs ∼ 10120

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SLIDE 15

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Zones of Ignorance

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SLIDE 16

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Message

Current realizations of inflation have serious conceptual problems. This motivates the search for alternatives. Key principles of string theory lead to alternatives to cosmological inflation. Alternative A: String Gas Cosmology Alternative B: Matter Bounce. Topic of this lecture! Specific predictions allow the models to be distinguished from inflationary cosmology.

13 / 51

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SLIDE 17

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Message

Current realizations of inflation have serious conceptual problems. This motivates the search for alternatives. Key principles of string theory lead to alternatives to cosmological inflation. Alternative A: String Gas Cosmology Alternative B: Matter Bounce. Topic of this lecture! Specific predictions allow the models to be distinguished from inflationary cosmology.

13 / 51

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SLIDE 18

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Message

Current realizations of inflation have serious conceptual problems. This motivates the search for alternatives. Key principles of string theory lead to alternatives to cosmological inflation. Alternative A: String Gas Cosmology Alternative B: Matter Bounce. Topic of this lecture! Specific predictions allow the models to be distinguished from inflationary cosmology.

13 / 51

slide-19
SLIDE 19

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Message

Current realizations of inflation have serious conceptual problems. This motivates the search for alternatives. Key principles of string theory lead to alternatives to cosmological inflation. Alternative A: String Gas Cosmology Alternative B: Matter Bounce. Topic of this lecture! Specific predictions allow the models to be distinguished from inflationary cosmology.

13 / 51

slide-20
SLIDE 20

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Message

Current realizations of inflation have serious conceptual problems. This motivates the search for alternatives. Key principles of string theory lead to alternatives to cosmological inflation. Alternative A: String Gas Cosmology Alternative B: Matter Bounce. Topic of this lecture! Specific predictions allow the models to be distinguished from inflationary cosmology.

13 / 51

slide-21
SLIDE 21

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Message

Current realizations of inflation have serious conceptual problems. This motivates the search for alternatives. Key principles of string theory lead to alternatives to cosmological inflation. Alternative A: String Gas Cosmology Alternative B: Matter Bounce. Topic of this lecture! Specific predictions allow the models to be distinguished from inflationary cosmology.

13 / 51

slide-22
SLIDE 22

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Plan

1

Motivation

2

Alternatives

3

Cosmological Perturbations

4

Bouncing Type II Superstring Cosmology

5

Fluctuations in Bouncing Type II Superstring Cosmology

6

Conclusions

14 / 51

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SLIDE 23

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Space-Time Sketch of Inflationary Cosmology

N.B. Perturbations originate as quantum vacuum fluctuations.

15 / 51

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SLIDE 24

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Emergent Universe Scenario

Background dynamics for the scale factor a(t):

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SLIDE 25

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Space-Time Sketch of the Emergent Universe Scenario

  • A. Nayeri, R.B. and C. Vafa, Phys. Rev. Lett. 97:021302 (2006)

N.B. Perturbations originate as thermal fluctuations.

17 / 51

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SLIDE 26

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Addressing the Problems of Standard Big Bang Cosmology

No horizon problem as long as the emergent phase lasts long. No flatness problem: Jeans length large. Size and entropy problems NOT addressed. Note: non-singular. Note: no fluctuation problem for fluctuations.

18 / 51

slide-27
SLIDE 27

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Addressing the Problems of Standard Big Bang Cosmology

No horizon problem as long as the emergent phase lasts long. No flatness problem: Jeans length large. Size and entropy problems NOT addressed. Note: non-singular. Note: no fluctuation problem for fluctuations.

18 / 51

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SLIDE 28

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Space-Time Sketch in the Matter Bounce Scenario

  • F. Finelli and R.B., Phys. Rev. D65, 103522 (2002), D. Wands, Phys. Rev.

D60 (1999)

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slide-29
SLIDE 29

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Addressing the Problems of Standard Big Bang Cosmology

No horizon problem. Flatness problem mitigated. No size and entropy problems Instability towards anisotropies. Note: non-singular. Note: no fluctuation problem for fluctuations.

20 / 51

slide-30
SLIDE 30

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Addressing the Problems of Standard Big Bang Cosmology

No horizon problem. Flatness problem mitigated. No size and entropy problems Instability towards anisotropies. Note: non-singular. Note: no fluctuation problem for fluctuations.

20 / 51

slide-31
SLIDE 31

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Addressing the Problems of Standard Big Bang Cosmology

No horizon problem. Flatness problem mitigated. No size and entropy problems Instability towards anisotropies. Note: non-singular. Note: no fluctuation problem for fluctuations.

20 / 51

slide-32
SLIDE 32

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Plan

1

Motivation

2

Alternatives

3

Cosmological Perturbations

4

Bouncing Type II Superstring Cosmology

5

Fluctuations in Bouncing Type II Superstring Cosmology

6

Conclusions

21 / 51

slide-33
SLIDE 33

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Theory of Cosmological Perturbations: Basics

Cosmological fluctuations connect early universe theories with observations Fluctuations of matter → large-scale structure Fluctuations of metric → CMB anisotropies N.B.: Matter and metric fluctuations are coupled Key facts:

  • 1. Fluctuations are small today on large scales

→ fluctuations were very small in the early universe → can use linear perturbation theory

  • 2. Sub-Hubble scales: matter fluctuations dominate

Super-Hubble scales: metric fluctuations dominate

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slide-34
SLIDE 34

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Theory of Cosmological Perturbations: Basics

Cosmological fluctuations connect early universe theories with observations Fluctuations of matter → large-scale structure Fluctuations of metric → CMB anisotropies N.B.: Matter and metric fluctuations are coupled Key facts:

  • 1. Fluctuations are small today on large scales

→ fluctuations were very small in the early universe → can use linear perturbation theory

  • 2. Sub-Hubble scales: matter fluctuations dominate

Super-Hubble scales: metric fluctuations dominate

22 / 51

slide-35
SLIDE 35

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Theory of Cosmological Perturbations: Basics

Cosmological fluctuations connect early universe theories with observations Fluctuations of matter → large-scale structure Fluctuations of metric → CMB anisotropies N.B.: Matter and metric fluctuations are coupled Key facts:

  • 1. Fluctuations are small today on large scales

→ fluctuations were very small in the early universe → can use linear perturbation theory

  • 2. Sub-Hubble scales: matter fluctuations dominate

Super-Hubble scales: metric fluctuations dominate

22 / 51

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SLIDE 36

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Quantum Theory of Linearized Fluctuations

  • V. Mukhanov, H. Feldman and R.B., Phys. Rep. 215:203 (1992)

Step 1: Metric including fluctuations ds2 = a2[(1 + 2Φ)dη2 − (1 − 2Φ)dx2] ϕ = ϕ0 + δϕ Note: Φ and δϕ related by Einstein constraint equations Step 2: Expand the action for matter and gravity to second

  • rder about the cosmological background:

S(2) = 1 2

  • d4x
  • (v′)2 − v,iv,i + z′′

z v2 v = a

  • δϕ + z

aΦ

  • z

= aϕ′ H

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SLIDE 37

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

where v ∼ aζ where ζ is the curvature fluctuation in co-moving coordinates.

24 / 51

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SLIDE 38

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Step 3: Resulting equation of motion (Fourier space) v′′

k + (k2 − z′′

z )vk = 0 Features:

  • scillations on sub-Hubble scales

squeezing on super-Hubble scales vk ∼ z Quantum vacuum initial conditions: vk(ηi) = ( √ 2k)−1

25 / 51

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SLIDE 39

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Application to Inflationary Cosmology

N.B. Perturbations originate as quantum vacuum fluctuations.

26 / 51

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SLIDE 40

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Origin of Scale-Invariance

Heuristic analysis [W. Press, 1980]: time-translation symmetry of de Sitter phase → scale-invariance of spectrum. Mathematical analysis [Mukhanov and Chibisov, 1982]: Pζ(k, t) ∝ Pv(k, t) ∼ k3 a(t) a(tH(k)) 2|vk(tH(k))|2 ∼ k3ηH(k)2|vk(tH(k))|2 ∼ k0 using a(η) ∼ η−1 in the de Sitter phase and ηH(k) ∼ k−1.

27 / 51

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SLIDE 41

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Application to the Matter Bounce Scenario

  • F. Finelli and R.B., Phys. Rev. D65, 103522 (2002), D. Wands, Phys. Rev.

D60 (1999)

28 / 51

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SLIDE 42

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Origin of Scale-Invariant Spectrum

The initial vacuum spectrum is blue: Pζ(k) = k3|ζ(k)|2 ∼ k2 The curvature fluctuations grow on super-Hubble scales in the contracting phase: vk(η) = c1η2 + c2η−1 , For modes which exit the Hubble radius in the matter phase the resulting spectrum is scale-invariant: Pζ(k, η) ∼ k3|vk(η)|2a−2(η) ∼ k3|vk(ηH(k))|2ηH(k) η 2 ∼ k3−1−2 ∼ const ,

29 / 51

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SLIDE 43

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Transfer of the Spectrum through the Bounce

In a nonsingular background the fluctuations can be tracked through the bounce explicitly (both numerically in an exact manner and analytically using matching conditions at times when the equation of state changes). Explicit computations have been performed in the case

  • f quintom matter (Y. Cai et al, 2008), mirage

cosmology (R.B. et al, 2007), Horava-Lifshitz bounce (X. Gang et al, 2009). Result: On length scales larger than the duration of the bounce the spectrum of v goes through the bounce unchanged.

30 / 51

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SLIDE 44

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Signature in the Bispectrum: formalism

< ζ(t, k1)ζ(t, k2)ζ(t, k3) > = i t

ti

dt′ < [ζ(t, k1)ζ(t, k2)ζ(t, k3), Lint(t′)] > , < ζ( k1)ζ( k2)ζ( k3) > = (2π)7δ(

  • ki)

P2

ζ

k3

i

×A( k1, k2, k3) , |B|NL( k1, k2, k3) = 10 3 A( k1, k2, k3)

  • i k3

i

.

31 / 51

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SLIDE 45

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Signature in the Bispectrum: Results

  • Y. Cai, W. Xue, R.B. and X. Zhang, JCAP 0905:011 (2009)

If we project the resulting shape function A onto some popular shape masks we get |B|local

NL

= −35 8 , for the local shape (k1 ≪ k2 = k3). This is negative and of

  • rder O(1).

For the equilateral form (k1 = k2 = k3) the result is |B|equil

NL

= −255 64 , For the folded form (k1 = 2k2 = 2k3) one obtains the value |B|folded

NL

= −9 4 .

32 / 51

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SLIDE 46

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Bispectrum of the Matter Bounce Scenario

  • Y. Cai, W. Xue, R.B. and X. Zhang, JCAP 0905:011 (2009)

0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 5 10 0.2 0.4 0.6 0.8 33 / 51

slide-47
SLIDE 47

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Plan

1

Motivation

2

Alternatives

3

Cosmological Perturbations

4

Bouncing Type II Superstring Cosmology

5

Fluctuations in Bouncing Type II Superstring Cosmology

6

Conclusions

34 / 51

slide-48
SLIDE 48

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Setup

C.Kounnas, H. Partouche and N. Toumbas, arXiv:1106.0946

Type II superstring theory compactified on M = S1(R0) × T 3 × F6 , Euclidean time radius R0 = β/(2π). Gravitomagnetic fluxes threading the Euclidean time cycle and cycles of the internal space. Leads to T-duality about the Euclidean time cycle (thermal duality) Z(β) = Z(β2

c/β) .

Large T 3 → effective field theory analysis under good control. Assumption: weak string coupling.

35 / 51

slide-49
SLIDE 49

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Setup

C.Kounnas, H. Partouche and N. Toumbas, arXiv:1106.0946

Type II superstring theory compactified on M = S1(R0) × T 3 × F6 , Euclidean time radius R0 = β/(2π). Gravitomagnetic fluxes threading the Euclidean time cycle and cycles of the internal space. Leads to T-duality about the Euclidean time cycle (thermal duality) Z(β) = Z(β2

c/β) .

Large T 3 → effective field theory analysis under good control. Assumption: weak string coupling.

35 / 51

slide-50
SLIDE 50

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Setup

C.Kounnas, H. Partouche and N. Toumbas, arXiv:1106.0946

Type II superstring theory compactified on M = S1(R0) × T 3 × F6 , Euclidean time radius R0 = β/(2π). Gravitomagnetic fluxes threading the Euclidean time cycle and cycles of the internal space. Leads to T-duality about the Euclidean time cycle (thermal duality) Z(β) = Z(β2

c/β) .

Large T 3 → effective field theory analysis under good control. Assumption: weak string coupling.

35 / 51

slide-51
SLIDE 51

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Setup

C.Kounnas, H. Partouche and N. Toumbas, arXiv:1106.0946

Type II superstring theory compactified on M = S1(R0) × T 3 × F6 , Euclidean time radius R0 = β/(2π). Gravitomagnetic fluxes threading the Euclidean time cycle and cycles of the internal space. Leads to T-duality about the Euclidean time cycle (thermal duality) Z(β) = Z(β2

c/β) .

Large T 3 → effective field theory analysis under good control. Assumption: weak string coupling.

35 / 51

slide-52
SLIDE 52

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Setup

C.Kounnas, H. Partouche and N. Toumbas, arXiv:1106.0946

Type II superstring theory compactified on M = S1(R0) × T 3 × F6 , Euclidean time radius R0 = β/(2π). Gravitomagnetic fluxes threading the Euclidean time cycle and cycles of the internal space. Leads to T-duality about the Euclidean time cycle (thermal duality) Z(β) = Z(β2

c/β) .

Large T 3 → effective field theory analysis under good control. Assumption: weak string coupling.

35 / 51

slide-53
SLIDE 53

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Setup

C.Kounnas, H. Partouche and N. Toumbas, arXiv:1106.0946

Type II superstring theory compactified on M = S1(R0) × T 3 × F6 , Euclidean time radius R0 = β/(2π). Gravitomagnetic fluxes threading the Euclidean time cycle and cycles of the internal space. Leads to T-duality about the Euclidean time cycle (thermal duality) Z(β) = Z(β2

c/β) .

Large T 3 → effective field theory analysis under good control. Assumption: weak string coupling.

35 / 51

slide-54
SLIDE 54

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Thermal Phases

Low temperature phase: β ≫ βc Z V = n∗σrβ−3 + O(e−β/βc) , Small β phase: β ≪ βc Z V = n∗σrβ−6

c β3 + O(e−βc/β) .

Introducing the duality-invariant temperature T = Tce−|σ| with eσ = β βc . we obtain Z

V = n∗σrT 3 + O(e−T/Tc) .

36 / 51

slide-55
SLIDE 55

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Thermal Phases

Low temperature phase: β ≫ βc Z V = n∗σrβ−3 + O(e−β/βc) , Small β phase: β ≪ βc Z V = n∗σrβ−6

c β3 + O(e−βc/β) .

Introducing the duality-invariant temperature T = Tce−|σ| with eσ = β βc . we obtain Z

V = n∗σrT 3 + O(e−T/Tc) .

36 / 51

slide-56
SLIDE 56

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Thermal Phases

Low temperature phase: β ≫ βc Z V = n∗σrβ−3 + O(e−β/βc) , Small β phase: β ≪ βc Z V = n∗σrβ−6

c β3 + O(e−βc/β) .

Introducing the duality-invariant temperature T = Tce−|σ| with eσ = β βc . we obtain Z

V = n∗σrT 3 + O(e−T/Tc) .

36 / 51

slide-57
SLIDE 57

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Thermal Phases

Low temperature phase: β ≫ βc Z V = n∗σrβ−3 + O(e−β/βc) , Small β phase: β ≪ βc Z V = n∗σrβ−6

c β3 + O(e−βc/β) .

Introducing the duality-invariant temperature T = Tce−|σ| with eσ = β βc . we obtain Z

V = n∗σrT 3 + O(e−T/Tc) .

36 / 51

slide-58
SLIDE 58

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Effective Action

At the critical temperature: thermal winding states become massless. enhanced gauge symmetry at β = βc. Enhanced symmetry states enter the effective low energy action for the light degrees of freedom as an S-brane. S-brane: space-like topological defect: ρ = 0, p < 0. S-brane mediates violation of Null Energy Condition. S-brane allows for cosmological bounce.

37 / 51

slide-59
SLIDE 59

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Effective Action

At the critical temperature: thermal winding states become massless. enhanced gauge symmetry at β = βc. Enhanced symmetry states enter the effective low energy action for the light degrees of freedom as an S-brane. S-brane: space-like topological defect: ρ = 0, p < 0. S-brane mediates violation of Null Energy Condition. S-brane allows for cosmological bounce.

37 / 51

slide-60
SLIDE 60

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Effective Action

At the critical temperature: thermal winding states become massless. enhanced gauge symmetry at β = βc. Enhanced symmetry states enter the effective low energy action for the light degrees of freedom as an S-brane. S-brane: space-like topological defect: ρ = 0, p < 0. S-brane mediates violation of Null Energy Condition. S-brane allows for cosmological bounce.

37 / 51

slide-61
SLIDE 61

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Effective Action

At the critical temperature: thermal winding states become massless. enhanced gauge symmetry at β = βc. Enhanced symmetry states enter the effective low energy action for the light degrees of freedom as an S-brane. S-brane: space-like topological defect: ρ = 0, p < 0. S-brane mediates violation of Null Energy Condition. S-brane allows for cosmological bounce.

37 / 51

slide-62
SLIDE 62

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Effective Action

At the critical temperature: thermal winding states become massless. enhanced gauge symmetry at β = βc. Enhanced symmetry states enter the effective low energy action for the light degrees of freedom as an S-brane. S-brane: space-like topological defect: ρ = 0, p < 0. S-brane mediates violation of Null Energy Condition. S-brane allows for cosmological bounce.

37 / 51

slide-63
SLIDE 63

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Effective Action

At the critical temperature: thermal winding states become massless. enhanced gauge symmetry at β = βc. Enhanced symmetry states enter the effective low energy action for the light degrees of freedom as an S-brane. S-brane: space-like topological defect: ρ = 0, p < 0. S-brane mediates violation of Null Energy Condition. S-brane allows for cosmological bounce.

37 / 51

slide-64
SLIDE 64

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Effective Action

Low energy effective action S =

  • d4x
  • −˜

g

  • e−2φ ˜

R 2 + 2(∇φ)2 + P

  • + SB ,

Pressure: P = e−|σ| βc Z(|σ|) . S-brane action: SB = κ

  • d4x
  • ˜

he−2φδ(τ − τB) ,

38 / 51

slide-65
SLIDE 65

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Effective Action

Low energy effective action S =

  • d4x
  • −˜

g

  • e−2φ ˜

R 2 + 2(∇φ)2 + P

  • + SB ,

Pressure: P = e−|σ| βc Z(|σ|) . S-brane action: SB = κ

  • d4x
  • ˜

he−2φδ(τ − τB) ,

38 / 51

slide-66
SLIDE 66

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Effective Action

Low energy effective action S =

  • d4x
  • −˜

g

  • e−2φ ˜

R 2 + 2(∇φ)2 + P

  • + SB ,

Pressure: P = e−|σ| βc Z(|σ|) . S-brane action: SB = κ

  • d4x
  • ˜

he−2φδ(τ − τB) ,

38 / 51

slide-67
SLIDE 67

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Einstein Frame

Einstein frame low energy effective action: S = √−g R 2 − ∇µφ∇µφ

  • +

√−g Λ β4 κ

  • dτd3ξ

√ heφδ(τ) , Conformal rescaling of the metric: ˜ gµν = e2φgµν , Einstein frame temperature: β = eφβE ,

39 / 51

slide-68
SLIDE 68

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Background Cosmology

tc tS k

radiation domination matter domination

t x

matter domination

tS

radiation domination

H 1 40 / 51

slide-69
SLIDE 69

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Plan

1

Motivation

2

Alternatives

3

Cosmological Perturbations

4

Bouncing Type II Superstring Cosmology

5

Fluctuations in Bouncing Type II Superstring Cosmology

6

Conclusions

41 / 51

slide-70
SLIDE 70

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Matching Conditions

  • W. Israel, Nuovo Cim. (1966), J-C. Hwang and E. Vishniac, Ap. J. (1991), N.

Deruelle and V. Mukhanov, gr-qc/9503050, R. Durrer and F. Vernizzi, hep-ph/0203275

Matching two solutions of Einstein’s equations across a

  • brane. The following conditions must be satisfied:

Induced metric continous extrinsic curvature jumps by a value corresponding to the amplitude of the S-brane source.

42 / 51

slide-71
SLIDE 71

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Matching Conditions

  • W. Israel, Nuovo Cim. (1966), J-C. Hwang and E. Vishniac, Ap. J. (1991), N.

Deruelle and V. Mukhanov, gr-qc/9503050, R. Durrer and F. Vernizzi, hep-ph/0203275

Matching two solutions of Einstein’s equations across a

  • brane. The following conditions must be satisfied:

Induced metric continous extrinsic curvature jumps by a value corresponding to the amplitude of the S-brane source.

42 / 51

slide-72
SLIDE 72

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Matching Conditions

  • W. Israel, Nuovo Cim. (1966), J-C. Hwang and E. Vishniac, Ap. J. (1991), N.

Deruelle and V. Mukhanov, gr-qc/9503050, R. Durrer and F. Vernizzi, hep-ph/0203275

Matching two solutions of Einstein’s equations across a

  • brane. The following conditions must be satisfied:

Induced metric continous extrinsic curvature jumps by a value corresponding to the amplitude of the S-brane source.

42 / 51

slide-73
SLIDE 73

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Matching for Adiabatic Fluctuations

  • R. Durrer and F. Vernizzi, hep-ph/0203275

Start in longitudinal gauge. Matching surface: identified with a surface of ¯ η = const. ¯ η ≡ η + T , Metric in terms of the new time: ds2 = a2(¯ η)

  • d ¯

η2 1 + 2Φ − 2T ′ − 2TH

  • +

dxid ¯ ηT,i − dx2 1 − 2Φ − 2TH

  • .

Continuity of the induced metric: [Φ + TH]|± = 0 ,

43 / 51

slide-74
SLIDE 74

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Matching for Adiabatic Fluctuations

  • R. Durrer and F. Vernizzi, hep-ph/0203275

Start in longitudinal gauge. Matching surface: identified with a surface of ¯ η = const. ¯ η ≡ η + T , Metric in terms of the new time: ds2 = a2(¯ η)

  • d ¯

η2 1 + 2Φ − 2T ′ − 2TH

  • +

dxid ¯ ηT,i − dx2 1 − 2Φ − 2TH

  • .

Continuity of the induced metric: [Φ + TH]|± = 0 ,

43 / 51

slide-75
SLIDE 75

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Matching for Adiabatic Fluctuations

  • R. Durrer and F. Vernizzi, hep-ph/0203275

Start in longitudinal gauge. Matching surface: identified with a surface of ¯ η = const. ¯ η ≡ η + T , Metric in terms of the new time: ds2 = a2(¯ η)

  • d ¯

η2 1 + 2Φ − 2T ′ − 2TH

  • +

dxid ¯ ηT,i − dx2 1 − 2Φ − 2TH

  • .

Continuity of the induced metric: [Φ + TH]|± = 0 ,

43 / 51

slide-76
SLIDE 76

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Matching for Adiabatic Fluctuations

  • R. Durrer and F. Vernizzi, hep-ph/0203275

Start in longitudinal gauge. Matching surface: identified with a surface of ¯ η = const. ¯ η ≡ η + T , Metric in terms of the new time: ds2 = a2(¯ η)

  • d ¯

η2 1 + 2Φ − 2T ′ − 2TH

  • +

dxid ¯ ηT,i − dx2 1 − 2Φ − 2TH

  • .

Continuity of the induced metric: [Φ + TH]|± = 0 ,

43 / 51

slide-77
SLIDE 77

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

For constant energy density hypersurfaces: Φ + TH = ζ . For adiabatic fluctuations, the constant temperature surface equals the constant energy density surface. Hence, for adiabatic fluctuations ζ is constant across the brane. Thus, the pre-brane-crossing scale-invariant spectrum transits to a scale-invariant post-brane-crossing spectrum.

44 / 51

slide-78
SLIDE 78

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

For constant energy density hypersurfaces: Φ + TH = ζ . For adiabatic fluctuations, the constant temperature surface equals the constant energy density surface. Hence, for adiabatic fluctuations ζ is constant across the brane. Thus, the pre-brane-crossing scale-invariant spectrum transits to a scale-invariant post-brane-crossing spectrum.

44 / 51

slide-79
SLIDE 79

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

For constant energy density hypersurfaces: Φ + TH = ζ . For adiabatic fluctuations, the constant temperature surface equals the constant energy density surface. Hence, for adiabatic fluctuations ζ is constant across the brane. Thus, the pre-brane-crossing scale-invariant spectrum transits to a scale-invariant post-brane-crossing spectrum.

44 / 51

slide-80
SLIDE 80

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

For constant energy density hypersurfaces: Φ + TH = ζ . For adiabatic fluctuations, the constant temperature surface equals the constant energy density surface. Hence, for adiabatic fluctuations ζ is constant across the brane. Thus, the pre-brane-crossing scale-invariant spectrum transits to a scale-invariant post-brane-crossing spectrum.

44 / 51

slide-81
SLIDE 81

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Matching in the General Case

Two mode functions of the “Bardeen variable": Φ(k, η) = A−(k) H a2 (η) + B−(k) . where A−(k) ∼ k−µ−1 , dominant B−(k) ∼ kµ−1 , with µ = 5 + 3w 2(1 + 3w) . For a matter-dominated phase µ = 5/2

45 / 51

slide-82
SLIDE 82

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Matching in the General Case

Two mode functions of the “Bardeen variable": Φ(k, η) = A−(k) H a2 (η) + B−(k) . where A−(k) ∼ k−µ−1 , dominant B−(k) ∼ kµ−1 , with µ = 5 + 3w 2(1 + 3w) . For a matter-dominated phase µ = 5/2

45 / 51

slide-83
SLIDE 83

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Matching in the General Case

Two mode functions of the “Bardeen variable": Φ(k, η) = A−(k) H a2 (η) + B−(k) . where A−(k) ∼ k−µ−1 , dominant B−(k) ∼ kµ−1 , with µ = 5 + 3w 2(1 + 3w) . For a matter-dominated phase µ = 5/2

45 / 51

slide-84
SLIDE 84

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Matching in the General Case II

Connection with ζ: Φ = 4πG k2 B(η)ζ′ , Hence, the A-mode in the contracting phase yields a scale-invariant power spectrum for ζ. In the expanding phase: Φ(k, η) = A+(k) H a2 (η) + B+(k) . B-mode is dominant. Adiabatic fluctuations: ζ conserved → scale-invariant spectrum for B after the bounce. Non-adiabatic fluctuations: B+ acquires the spectrum

  • f A− → spectrum of B after the bounce is not

scale-invariant.

46 / 51

slide-85
SLIDE 85

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Matching in the General Case II

Connection with ζ: Φ = 4πG k2 B(η)ζ′ , Hence, the A-mode in the contracting phase yields a scale-invariant power spectrum for ζ. In the expanding phase: Φ(k, η) = A+(k) H a2 (η) + B+(k) . B-mode is dominant. Adiabatic fluctuations: ζ conserved → scale-invariant spectrum for B after the bounce. Non-adiabatic fluctuations: B+ acquires the spectrum

  • f A− → spectrum of B after the bounce is not

scale-invariant.

46 / 51

slide-86
SLIDE 86

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Matching in the General Case II

Connection with ζ: Φ = 4πG k2 B(η)ζ′ , Hence, the A-mode in the contracting phase yields a scale-invariant power spectrum for ζ. In the expanding phase: Φ(k, η) = A+(k) H a2 (η) + B+(k) . B-mode is dominant. Adiabatic fluctuations: ζ conserved → scale-invariant spectrum for B after the bounce. Non-adiabatic fluctuations: B+ acquires the spectrum

  • f A− → spectrum of B after the bounce is not

scale-invariant.

46 / 51

slide-87
SLIDE 87

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Matching in the General Case II

Connection with ζ: Φ = 4πG k2 B(η)ζ′ , Hence, the A-mode in the contracting phase yields a scale-invariant power spectrum for ζ. In the expanding phase: Φ(k, η) = A+(k) H a2 (η) + B+(k) . B-mode is dominant. Adiabatic fluctuations: ζ conserved → scale-invariant spectrum for B after the bounce. Non-adiabatic fluctuations: B+ acquires the spectrum

  • f A− → spectrum of B after the bounce is not

scale-invariant.

46 / 51

slide-88
SLIDE 88

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Matching in the General Case II

Connection with ζ: Φ = 4πG k2 B(η)ζ′ , Hence, the A-mode in the contracting phase yields a scale-invariant power spectrum for ζ. In the expanding phase: Φ(k, η) = A+(k) H a2 (η) + B+(k) . B-mode is dominant. Adiabatic fluctuations: ζ conserved → scale-invariant spectrum for B after the bounce. Non-adiabatic fluctuations: B+ acquires the spectrum

  • f A− → spectrum of B after the bounce is not

scale-invariant.

46 / 51

slide-89
SLIDE 89

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Power Spectrum

For adiabatic fluctuations, the power spectrum is: Pv(k, t) ≃ H(−˜ teq) mpl 2 .

47 / 51

slide-90
SLIDE 90

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Model B: Emergent Brane Phase

tc k

radiation domination

t x

matter domination

tS

radiation domination

H 1 48 / 51

slide-91
SLIDE 91

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Power Spectrum in Model B

Fluctuations exit the Hubble radius in the radiation phase. No growth of v on super-Hubble scales during the radiation phase. Pre-bounce spectrum is vacuum. Post-bounce spectrum is vacuum.

49 / 51

slide-92
SLIDE 92

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Plan

1

Motivation

2

Alternatives

3

Cosmological Perturbations

4

Bouncing Type II Superstring Cosmology

5

Fluctuations in Bouncing Type II Superstring Cosmology

6

Conclusions

50 / 51

slide-93
SLIDE 93

String Bounce

  • R. Branden-

berger Motivation Alternatives Perturbations Superstring Bounce S-Brane Fluctuations Conclusions

Conclusions

Bouncing cosmology based on Type II superstring theory in a background of gravitomagnetic fluxes. Crucial point 1: Temperature duality. Crucial point 2: S-brane emerges in the low energy effective action at the critical temperature. This yields a stringy realization of the matter bounce paradigm. Vacuum fluctuations exiting Hubble radius in a matter-dominated phase of contraction lead to a scale-invariant spectrum of adiabatic perturbations. Adiabatic curvature fluctuations on IR scales pass through the S-brane without change in the spectrum.

51 / 51