Scalar perturbations in braneworld cosmology

Takashi HiramatsuResearch Center for the Early Universe (RESCEU),

School of Science, University of Tokyo

Collaboration with A.Cardoso, K.Koyama, S.S.Seahra

Institute of Cosmology and Gravitation, University of Portsmouth

arxiv/0705.1685 [astro-ph]submitted to JCAP

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Braneworld

• String theories imply 10/11 D space-time.

a simple description : Randall-Sundrum II model

• Cosmological perturbations in RSII model– Tensor perturbations

• During inflation

• After inflation

– Scalar perturbations• During inflation

brane

bulk 5D anti de Sitter space-time

Hiramatsu, Koyama, Taruya, PLB(2004)Ichiki, Nakamura, PRD(2004)Ichiki, Nakamura, astro-ph(2004)Kobayashi, Tanaka, JCAP(2004)Kobayashi, Tanaka, PRD(2005)Hiramatsu, Koyama, Taruya, PLB(2004)Hiramatsu, PRD(2006)Kobayashi, PRD(2006)Seahra, PRD(2006)

Langlois, Maartens, Wands, PLB(2000)Gorbunov, Rubakov, Sibiryakov, JHEP(2001)Kobayashi, Kudoh, Tanaka, PRD(2003)

Maartens, Wands, Bassett, Heard, PRD(2000)Calcagni, JCAP(2003)Liddle, Smith, PRD(2003)Tsujikawa, Liddle, JCAP(2004)Calcagni, JCAP(2004)Ramirez, Liddle, PRD(2004)Seery, Taylor, PRD(2005)Liddle, Taylor, PRD(2005)Koyama, Mizuno, Wands, JCAP(2005)Hiramatsu, Koyama, JCAP(2007)Koyama, Mennim, Rubakov, Wands, Hiramatsu, JCAP(2007)

• After inflation

Randall, Sundrum, PRL (1999)

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Scalar perturbations in RS model

• Observables– : curvature perturbations

– : density perturbations

CMB anisotropy, large-scale structure, etc.

• High-energy corrections caused by extra-dimensions– Friedmann equation slower expansion law

– Interaction with the bulk metric perturbations

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Evolution of primordial fluctuations

• High-energy effects appear above the critical wave number

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Metric perturbations (bulk)

• 5D metric perturbations (5D-longitudinal gauge)

• Master variable

No matterin the bulk

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Density perturbations (brane)

• Perturbation of energy-momentum tensor

• Gauge-invariant density/velocity perturbation

• Evolution equation

assumption

Correction toFiredmann eq.

Bulk metricperturbations

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Junction conditions (brane)

• Perturbed effective Einstein equation

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Growing / Decaying modes

• High-energy limit

Junctioncondition

Evolutionequation

3rd order equation for

dominant growing mode :

subdominant growing mode :

decaying mode :

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Curvature perturbations / consistency relations

• Two definitions for g.-i. curvature perturbations

• Consistency relations for the dominant growing mode on superhorizon scales– At high-energies,

– In 4D GR,

• On subhorizon scales,

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Numeric analysis

• Using two independent codes– Pseudo-spectral method (PS)

– Characteristic integration method (CI)

Hiramatsu, PRD(2006)

Seahra, PRD(2006)

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Initial conditions / critical wave number

• Initial condition– Only the dominant growing mode survived on the brane,

– and bulk metric perturbations become negligible

at very early time

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Typical waveforms (high-frequencies)

const

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Large scale fluctuations

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Enhancement factor (Comparing with GR)

• To separate out the two effects, …

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Enhancement factor (Comparing with GR)

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Enhancement factor (Comparing with GR)

~7.1

~3.0

~2.4

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Summary / Implications

• EOMs and junction condition for scalar perturbations in RD epoch in RSII model.

• Analytic approximations (3 modes).• Applied two numerical algorithms to solve EOMs.• All consistency relations are satisfied, and GR results

are recovered for low-frequency modes.• Above critical wave number, amplitude of is

enhanced, which leads to

宇宙初期における時空と物質の進化 , University of Tokyo, 28-30 May 2007 18 / 18

Summary / Implications

• Why is the amplitude enhanced ?

The self-gravity of supercritical perturbative modes will be greater than subcritical ones (, or ?)

• ImplicationsThe enhancement is important on comoving scales 10AU. too small to be relevant to CMB/galaxy observations through predictions of PBH abundance, we can derive new

limits on RS cosmology ?

Gravitational force ～

Cf. Sendouda et al., PRD (2003); PRD (2005); JCAP (2006) Guedens et al., PRD (2006a); Majumdar, PRL (2003)