Post on 11-Feb-2021
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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