Scalar perturbations in braneworld cosmology

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Scalar perturbations in braneworld cosmology

• Takashi Hiramatsu
• Research 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

Randall, Sundrum, PRL (1999)
bulk
5D anti de Sitter space-time
brane
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)
Langlois, Maartens, Wands, PLB(2000) Gorbunov,
Rubakov, Sibiryakov, JHEP(2001) Kobayashi, Kudoh,
Tanaka, PRD(2003)
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)

• After inflation

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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 matter in the bulk
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Density perturbations (brane)

• Perturbation of energy-momentum tensor
• Gauge-invariant density/velocity perturbation
• Evolution equation

assumption
Correction to Firedmann eq.
Bulk metric perturbations
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Junction conditions (brane)

• Perturbed effective Einstein equation

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Growing / Decaying modes

• High-energy limit

Evolution equation
3rd order equation for
Junction condition
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)
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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

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Summary / Implications

• Why is the amplitude enhanced ?
• The self-gravity of supercritical perturbative
  modes will be greater than subcritical ones (, or
  ?)
• Implications
• The 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)