The Theory of Inflationary The Theory of Inflationary PerturbationsPerturbations

Jérôme MartinJérôme Martin

Institut d’Astrophysique de Paris (IAP)Institut d’Astrophysique de Paris (IAP)

Indian Institute of Technology, Chennai 03/02/2012

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Introduction

A brief description of inflation

Cosmological perturbations during inflation

Constraining inflation with astrophysical data

Non-Gaussian aspects of the inflationary fluctuations

Conclusions

Outline

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The standard model of cosmology (the hot Big Bang phase) provides a convincing description of the Universe and of its history over a wide range of energy scales

The standard model of cosmology

Cosmology … in brief!

The model is based on two assumptions:

1- Cosmological principle: the Universe is homogeneous and isotropic. The evolution of the Universe is determined by a single function of time: the scale factor a(t)

2- Gravity is described by GR. The differential equation controlling the evolution of a(t) is given by the Einstein equations

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However, despite these impressive successes, the hot Big Bang phase has issues

These problems are related to the initial conditions needed at the onset of the hot Big Bang phase.

Inflation provides a solution

Inflation does not replace the hot Big Bang model but completes it.

It takes place before the hot Big Bang phase, at very high energy.

Inflation

The standard model of cosmology (II)

Inflation is a phase of accelerated expansion taking place in the very early Universe. One can show that such of phase can solve the standard problems of the hot big bang phase

In GR, one has acceleration if the pressure of the dominating fluid is negative

At very high energies, the relevant description of matter is field theory. The simplest model compatible with the cosmological principle is a scalar field:

In order to have inflation the potential must be flat

One of the main issues is to indentify(theoretically and experimentally) the correct potential of the inflaton field: who is the inflaton?

Defining inflation

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Large-field model

Small-field model

Hybrid model

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Inflation: basic mechanism

Slow-roll phase

Oscillatory phase

p=2

p=4

Slow-roll phase

Reheating phase

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COBE (1992)

WMAP (2003)

Planck (2013)

But the Universe is not homogeneous and isotropic as revealed e.g. by the CMBR anisotropies

In the early Universe, the amplitude of the fluctuations is small, δ T/T ~ 10-5. A linear theory is therefore possible

The initial fluctuations are amplified by gravitational instability … but what is the source of these fluctuations?

Inflation, combined with QM, provides an answer …

CMB anisotropies

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Fourier transform on the sphere

CMB anisotropies

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Inflationary fluctuations

The amplitude of the linear fluctuations is characterized by the Mukhanov-Sasaki variable; it is similar to a test scalar field in curved space-time

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The multipole moments are the consequences ot two different physical processes:

1- The statistical properties of the primordial fluctuations on large scales (produced during inflation)

2- The evolution of the perturbations when they re-enter the Hubble radius (a priori, well-known physics)

The astrophysical data are compatiblewith a scale invariant power spectrum

Inflationary fluctuations

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Inflationary fluctuations

The perturbed (i.e. linear) Einstein equations lead an equation for the amplitude of the fluctuations

The equation of motion is the equation of a parametric oscillator, i.e. an harmonic oscillator with a time-dependent frequency

The time-dependent frequency depends on the scale factor and its derivative (and on the comoving wavenumber); the background expansionis encoded into the amplitude of the fluctuations

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Quantization

Amplification mechanism: gravitational instability

Source of these fluctuations?

In inflation, these are the unavoidable vacuum quantum fluctuations of the inflaton field and of the gravitational field

In the Schroedinger picture, the wavefunction of the system is a Gaussian

Solution of the « quantum linear Einstein equations »

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Inflationary fluctuations

r

During inflation, the vacuum evolves into a strongly (two modes) squeezed statecorresponding to creation of pair of particles with opposite momenta

r is the squeezing parameter

The two point correlation function of the Mukhanov-Sasaki variable is given by

This leads to

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Inflationary predictions: the two-point correlation function

Scale invariant amplitude

Controlled by the strengthof the gravitational field, H

Scale depend logarithmic corrections, the amplitude of which is controlled by the microphysics of inflation

Slow-roll parameters

Schwinger effect Inflationary cosmological perturbations

- Scalar field

- Classical electric field

- Amplitude of the effect controlled by E

- Perturbed metric

- Background gravitational field: scale factor

- Amplitude controlled by the Hubble parameter H

Inflationary fluctuations vs Schwinger effect

The basic mechanism is in fact a well-known one: particles creations under the influence of a classical source, i.e. a quantum field interacting with a classical field

It is similar to the Schwinger effect

Towards an inflationary pipeline

Data:

Hot Big Bang:

Posterior distributions

What is the best model of Inflation?

NG on the celestial sphere

Model of inflation (or of the early Universe)

Inflation & Observations

In principle, measuring the fine structure of the power spectrum allows us

- to rule out models of inflation - to discriminate among models - to constrain parameters describing a model

The data are so accurate (WMAP, SPT, ACT etc …) that one can already constrain many inflationary models.

One can also constrain the pre/re-heating phase

Large field models are now under pressure:

WMAP7 and large field models

Large field models are now under pressure:

WMAP7 and large field models

Mean likelihood

Marginalized posteriors (p2 [0.2,5])

J. Martin & C. Ringeval, JCAP 08, 009 (2006)astro-ph/0605367

Fourier transform on the sphere

Non-Gaussianity

“ Non-Gaussianity”. One can also use the higher order correlation

functions to constrain further models of inflation

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Why NGs in the Starobinsky Model?

NG is negligible if

- Einstein Gravity

- Single scalar field

- Canonical kinetic term

- in Bunch-Davies state

- always slow-roll

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Example: the Starobinsky Model

Single field inflationary model with a « feature » in the potential

Representative of models where inflation never stops but where the slow-roll approximation is temporarily violated.

Nice playground since everythingcan be done analytically, ie the background evolution and the perturbations!

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Calculations of NG

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Results

Slow-roll parameters:

Energy scale:

Gravity waves

Tendency for red tilt (3 sigmas)

No prior independent evidence for a running

No entropy mode

No cosmic string

No non-Gaussianities

m^2 φ2 under pressure, λ φ4 ruled out, small field doing pretty well

The observational situation: recap

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ConclusionsConclusions

Inflation is a very consistent scenario, based on conservative physics and compatible with all the data

The continuous flow of high accuracy cosmological data allows us to probe the details of inflation, i.e. to learn about the microphysics of inflation.

The scenario is quite remarkable since it combines general relativity and quantum mechanics. Moreover, this now a driven data field.

This shows that cosmology is an interesting playground to understand the foundations of quantum mechanics

In particular, the decoherence of the perturbations (environment, pointer basis etc …) has recently been a subject of many discussions

The next step is to use the soon to come Planck data to constrain the inflationary scenario in detail and to answer the question “what is the best model of inflation?”.

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