The Theory of Inflationary Perturbations

Transcription

1 The Theory of Inflationary Perturbations Jérôme Martin Institut d Astrophysique de Paris (IAP) Indian Institute of Technology, Chennai 03/02/2012 1

2 Introduction Outline A brief description of inflation Cosmological perturbations during inflation Constraining inflation with astrophysical data Non-Gaussian aspects of the inflationary fluctuations Conclusions 2

3 The standard model of cosmology Cosmology in brief! 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 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 3

4 The standard model of cosmology (II) 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 4

5 Defining inflation 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 Hybrid model Large-field model One of the main issues is to indentify (theoretically and experimentally) the correct potential of the inflaton field: who is the inflaton? Small-field model 5

6 Inflation: basic mechanism Oscillatory phase p=4 p=2 Slow-roll phase Reheating phase Slow-roll phase 6

7 CMB anisotropies 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 ~ A linear theory is therefore possible COBE (1992) WMAP (2003) The initial fluctuations are amplified by gravitational instability but what is the source of these fluctuations? Inflation, combined with QM, provides an answer Planck (2013) 7

8 CMB anisotropies Fourier transform on the sphere 8

9 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 9

10 Inflationary fluctuations 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 compatible with a scale invariant power spectrum 10

11 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 expansion 11 is encoded into the amplitude of the fluctuations

12 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» 12

13 Inflationary fluctuations During inflation, the vacuum evolves into a strongly (two modes) squeezed state corresponding to creation of pair of particles with opposite momenta r is the squeezing parameter r 13

14 Inflationary predictions: the two-point correlation function The two point correlation function of the Mukhanov-Sasaki variable is given by This leads to Scale invariant amplitude Slow-roll parameters Controlled by the strength of the gravitational field, H Scale depend logarithmic corrections, the amplitude of which is controlled by the microphysics of inflation 14

15 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 Schwinger effect - Scalar field - Classical electric field - Amplitude of the effect controlled by E Inflationary cosmological perturbations - Perturbed metric - Background gravitational field: scale factor - Amplitude controlled by the Hubble parameter H

16 Towards an inflationary pipeline Model of inflation (or of the early Universe) Hot Big Bang: NG on the celestial sphere Data: What is the best model of Inflation? Posterior distributions

17 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

18 WMAP7 and large field models Large field models are now under pressure:

19 WMAP7 and large field models Mean likelihood Marginalized posteriors (p2 [0.2,5]) J. Martin & C. Ringeval, JCAP 08, 009 (2006) astro-ph/ Large field models are now under pressure:

20 Non-Gaussianity Fourier transform on the sphere Non-Gaussianity. One can also use the higher order correlation functions to constrain further models of inflation

21 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 21

22 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 everything can be done analytically, ie the background evolution and the perturbations! 22

23 Calculations of NG 23

24 Results 24

25 The observational situation: recap 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

26 Conclusions Conclusions 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?. 26