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暴涨宇宙学及其 检验. 郭宗宽(中科院理论物理研究所). 北京工业大学 应用数理学院 2012.10.18. 报告提纲. 暴涨宇宙学 宇宙微波背景辐射 微波背景对暴涨模型的检验 展望. 一 . 暴涨宇宙学. some problems in the hot Big Bang model : flatness problem , horizon problem , relic density problem. slow-roll inflation. criterions: cosmic acceleration e-folding number

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  • 暴涨宇宙学
  • 宇宙微波背景辐射
  • 微波背景对暴涨模型的检验
  • 展望
一. 暴涨宇宙学
  • some problems in the hot Big Bang model:
  • flatness problem, horizon problem, relic density problem



cosmic acceleration

e-folding number


successful exit


V (φ)





old inflation,large-field, small-field, hybrid, curvaton, k-inflation, G-inflation, trapped, warm, eternal, …

  • phenomenological models
  • fine-tuning problems
  • nature of inflaton field
  • to predict perturbations

potential, field, kinetic, coupling

Higgs field, D-brane, …

Single-field, minimally-coupled, canonical kinetic, slow-roll inflation generates almost scale-invariant, adiabatic, Gaussian perturbations.

large-scale structure, CMBR

1 power law inflaton coupled to the gauss bonnet gb term
(1) power-law inflaton coupled to the Gauss-Bonnet (GB) term
  • It is known that there are correction terms of higher orders in the curvature to the lowest effective supergravity action coming from superstrings. The simplest correction is the GB term.
  • Does the GB term drive acceleration of the Universe? If so, is it possible to generate nearly scale-invariant curvature perturbations? If not, when the GB term is sub-dominated, what is the influence on the power spectra? How strong WMAP data constrain the GB coupling?

our action:

Z.K. Guo, D.J. Schwarz, PRD 80 (2009) 063523


power-law inflation

an exponential potential and an exponential GB coupling

In the GB-dominated case, ultra-violet instabilities of either scalar or tensor perturbations show up on small scales.

In the potential-dominated case, the GB correction with a positive (or negative)coupling may lead to a reduction (or enhancement) of the tensor-to-scalar ratio.

constraints on the GB coupling

2 slow roll inflation with a gb correction
(2) Slow-roll inflation with a GB correction
  • Is it possible to generalize our previous work to the more general case of slow-roll inflation with an arbitrary potential and an arbitrary coupling?

introduce Hubble and GB flow parameters:

to first order in the slow-roll approximation

the scalar spectral index contains not only the Hubble but also GB flow parameters.

the degeneracy of standard consistency relation is broken.

Z.K. Guo, D.J. Schwarz,PRD 81 (2010) 123520


Consider a specific inflation model:

n = 2

Defining in the case, the spectral index and the tensor-to-scalar ratio can be written in terms of the function of N:

n = 4

The Gauss-Bonnet term may revive the quartic potential ruled out by recent cosmological data.

二. 宇宙微波背景辐射

(1) formation of the CMB

Shortly after recombination, the photon mean free path became larger than the Hubble length, and photons decoupled from matter in the universe.


(2) story of the CMB observation

  • George Gamow et al. estimated a temperature of 50K in 1946
  • the first discovery of CMB radiation in 1964-1965

the Nobel Prize in Physics 1978: A.A. Penzias and R.W. Wilson

  • COBE (Cosmic Background Explorer), launched on 18 Nov. 1989, 4 years

the Nobel Prize in Physics 2006: J.C. Mather and G.F. Smoot

  • WMAP (Wilkinson Microwave Anisotropy Probe), launched on 30 June 2001, 9 years
  • Planck, launched on 14 May 2009, 30 months (5 full sky)
  • Other experiments

ground based experiments:QUaD, BICEP, SPT, SPTpol from 2012, ACT, ACTPolfrom 2013

balloon borne experiments: BOOMRANG, MAXIMA


(3) CMB data analysis pipeline

time-ordered data

full sky map


parameter estimates

 time-ordered data

 the temperature anisotropies can be expanded in spherical harmonics


 for Gaussian random fluctuations, the statistical properties of the

temperature field are determined by the angular power spectrum

For a full sky, noiseless experiments,

 cosmological parameter estimation

likelihood function for a full sky:

the sky-cut, MCMC


(4) physics of CMB anisotropies

  • primary CMB anisotropies (at recombination)

Fourier space

spherical harmonics


the Einstein equations:

the linearized Einstein equations:


secondary CMB anisotropies (after recombination)


thermal Sunyaev-Zel’dovich effect

lensing effect

integrated Sachs-Wolf effect


三. 微波背景对暴涨模型的检验

  • primordial power spectrum of curvature perturbations: scale-invariant? slightly tilted power-law? running index? suppression at large scales? local features?

a critical test of inflation!

  • non-adiabaticity: matter isocurvature modes (axion-type, curvaton-type)? neutrino isocurvature modes?

a powerful probe of the physics of inflation!

  • non-Gaussianity: local form(multiple fields)? equilateral form(non-canonical kinetic)? orthogonal form(higher-derivative field)?

a powerful test of inflation!

  • primordial gravitational waves: the consistency relation?

smoking-gun evidence for inflation!


a single CDM isocurvature mode

 constraints on the power spectrum

for a pure power-law

for a running index

constraints on non-Gaussianity (95% CL)

 constraints on ns and r

1 cmb constraints on the energy scale of inflation
(1) CMB constraints on the energy scale of inflation
  • Determining the energy scale of inflation is crucial to understand the nature of inflation in the early Universe.

the inflationary potential can be expanded as

to leading order in the slow-roll approximation

Z.K. Guo, D.J. Schwarz, Y.Z. Zhang, PRD 83 (2011) 083522


We find upper limits on the potential energy, the first and second derivative of the potential, derived from the 7-year WMAP data with with Gaussian priors on the Hubble constantand the distance ratios from the BAO (at 95% CL):


Forecast constraints (68% and 95% CL) on the V0-V1 plane (left) and the V1-V2 plane (right) for the Planck experiment in the case of r = 0.1.

Using the Monte Carlo simulation approach, we have presented forecasts for improved constrains from Planck. Our results indicate that the degeneracies between the potential parameters are broken because of the improved constraint on the tensor-to-scalar ratio from Planck.

2 r econstruction of the primordial power spectrum
(2) Reconstruction of the primordial power spectrum

Relation between the inflation potential, the primordial power spectrum of curvature perturbations and the angular power spectrum of the CMB

It is logarithmically expanded


  • scale-invariant(As)
  • power-law (As, ns)
  • running spectral index (As, ns, as)

our method:


It is easy to detect deviations from a scale-invariant or a power-law spectrum because they are just straight lines in the lnk-ln P plane.

Negative values of the spectrum can be avoided by using ln P(k) instead of P(k) for the spline with steep slops.

The shape of the power spectrum reduces to the scale-invariant or power-law spectrum as a special case when N bin= 1, 2, respectively.

ZKG, D.J. Schwarz, Y.Z. Zhang, JCAP 08 (2011) 031






The Harrison-Zel’dovich spectrum is disfavored at 2s and the power-law spectrum is a good fit to the data.


(3) uncorrelated estimates from Planck simulated data

  • The spectrum parameters are correlated due to the geometrical project. With the localized principle component analysis we make uncorrelated estimates of the primordial power spectrum with five wavenumber bins.

ZKG, Y.Z. Zhang, JCAP 11 (2011) 032


(4) primordial power spectrum versus extension parameters


We find that a scale-invariant primordial spectrum is disfavored by the data at 95% CL even in the presence of massive neutrinos, however it can lie within the 95% confidence region if the effective number of relativistic species or the primordial helium abundance is allowed to vary freely.


ZKG, Y.Z. Zhang, PRD 85 (2012) 103519


(5) Lorentz invariance violation in the neutrino sector

  • Neutrino oscillations can be explained by small Lorentz invariance violation even without introducing neutrino mass.
  • The breaking of Lorentz symmetry may leave some imprints in astrophysical observations such as the CMB anisotropies.

the deformed dispersion relation for massive neutrinos can be generally parameterized by

we consider the n=2 case

ZKG, Q.G. Huang, R.G. Cai, Y.Z. Zhang, PRD 86 (2012) 065004


the FRW metric in the synchronous gauge

the Lagrangian for neutrino

the Boltzmann equation in the synchronous gauge


四. 展望

  • theoretical prospects
  • new physics?
  • observational prospects
  • the primordial scalar perturbations?
  • entropy perturbations?
  • non-Gaussianity?
  • the primordial gravitational wave?