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probing i nflation with CMB anisotropies. Zong-Kuan Guo ( ITP, CAS). ICFPC 2012 ( Weihai ) August 12, 2012. content. inflation cosmic microwave background (CMB) CMB constraints on inflation outlook. 1. inflation. slow-roll inflation. criterions: cosmic acceleration

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probing i nflation with cmb anisotropies

probing inflation with CMB anisotropies

Zong-KuanGuo (ITP, CAS)

ICFPC 2012 (Weihai)

August 12, 2012

  • inflation
  • cosmic microwave background (CMB)
  • CMB constraints on inflation
  • outlook
1 inflation
1. inflation

slow-roll inflation


cosmic acceleration

e-folding number


successful exit


V ()




it solves some problems

  • flatness problem, horizon problem, relic density problem
  • phenomenological models
  • some fine-tuning problems
  • potential parameters, initial value of the field, kinetic, coupling
  • nature of the inflaton field
  • it predicts perturbations

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

large-scale structure, CMB

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/negative coupling may lead to a reduction/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:

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

n = 4

The GB term may revive the quartic potential model ruled out by recent cosmological data.


2. cosmic microwave background (CMB)

(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)CMB experiments

  • the first discovery of CMB radiation in 1964
  • COBE (Cosmic Background Explorer), launched on 18 Nov. 1989, 4 years
  • WMAP (Wilkinson Microwave Anisotropy Probe), launched on 30 June 2001, 9 years
  • Planck satellite, launched on 14 May 2009, 30 months
  • other experiments:

ground-based experiments(QUaD, BICEP,

ACT, ACTPolfrom 2013, SPT, SPTpol from 2012)

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

3 cmb constraints on inflation
3. CMB constraints on inflation
  • primordial curvature perturbations: exact 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?

a 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

4 outlook
4. outlook
  • theoretical prospects
  • observational prospects
  • new physics in the early Universe?
  • the primordial scalar perturbations?
  • entropy perturbations?
  • the primordial non-Gaussianity?
  • the primordial gravitational wave?