Inflation: a Status ReportInflation: a Status Report

Jérôme MartinJérôme Martin

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

Annecy, LAPTH, February 3, 2011

Outline

Introduction: definition of inflation

Perturbations of quantum-mechanical origin: the « cosmological Schwinger effect »

Constraints on slow-roll and k-inflation

An inflationary pipeline: testing inflationary models exactly (numerically)

Conclusions

Inflation is a phase of accelerated expansion taking place in the very early Universe.

This assumption allows us to solve several problems of the standard hot Big Bang model:

•Horizon problem

•Flatness problem

•Monopoles problem …

Defining inflation

The energy scale of inflation is poorly constrained

Accelerated expansion can be produced if the pressure of the dominating fluid is negative. A scalar field is a well-motivated candidate

Inflation

4

Inflation: basic mechanism

Slow-roll phase

Oscillatory phase

p=2

p=4

Slow-roll phase

Reheating phase

5

End of Inflation

The reheating phase depends on the coupling of the inflaton with the rest of the world

Γ is the inflaton decay rate

6

End of Inflation (II)

Slow-roll phase

p=4

After inflation, the radiation dominated era starts. The first temperature in the Universe is called the reheating temperature

Implementing Inflation

The common way to realize inflation is to assume that there is a scalar field (or several scalar fields) dominating in the early Universe

Implementing Inflation (II)

The common way to realize inflation is to assume that there is a scalar field (or several scalar fields) dominating in the early Universe.

There are plenty of different models

1- Single field inflation with standard kinetic term

Different models are characterized by

different potentials

Implementing Inflation (III)

The common way to realize inflation is to assume that there is a scalar field (or several scalar fields) dominating in the early Universe.

There are plenty of different models

1- Single field inflation with standard kinetic term

2- Single field with non-standard kinetic term (K-inflation)

Different models are characterized by

different potentials and different kinetic terms

Implementing Inflation (IV)

The common way to realize inflation is to assume that there is a scalar field (or several scalar fields) dominating in the early Universe.

There are plenty of different models

1- Single field inflation with standard kinetic term

2- Single field with non-standard kinetic term (K-inflation)

3- Multiple field inflation

Different models are characterized by

different potentials; the inflationary trajectory

can be complicated

Conditions for Inflation

Lorentz factor:

Slow-roll regime:

DBI regime:

During inflation, the Hubble radius is almost a constant

Conditions for inflation

Conditions for slow-roll inflation

Flat potential

Small sound velocity

- In order to have a more realistic description of the (early) universe (CMB, structure formation …) one must go beyond the cosmological principle.

- In the early universe, the deviations are small since T/T» 10-5. This allows us to use a linear theory

- The source of these fluctuations will be the unavoidable quantum fluctuations of the coupled gravitational field and matter.

- The main mechanism is a very conservative one: particles creation under the influence of an external classical field. Similar to the Schwinger effect.

small fluctuations of the geometry and matter on top of the FLRW Universe

Primordial fluctuations

The Schwinger Effect

Production of cosmological perturbations in the Early universe is very similar to pair creation in a static electric field E

The frequency is time-dependent: one has to deal with a parametric oscillator

One works in the Fourier space

J. Martin, Lect. Notes Phys. 738: 193-241, 2008, arXiv:0704.3540

The exact solution of the mode equation can be found but what are the initial conditions?

The WKB mode function is given by

wkb is valid

The initial conditions are chosen to be the adiabatic vacuum

The validity of the WKB approximation is necessary in order to choose well-defined initial conditions

particle creation

The Schwinger Effect (II)

Difficult to see in the laboratory:

With the previous Gaussian wave function, one can compute the number of pair created per spacetime volume. It is given

vacuum (WKB) initial state

particles creation

The “functional” integral can be done because it is still Gaussian

The Schwinger Effect (III)

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 Fourier amplitude of the fluctuations obey the equation of a parametric oscillator.

The shape of the effective potential depends on the shape of the inflaton potential through the sr Parameters

The initial conditions are natural in inflation because, initially, the modes are sub-Hubble. The initial state is chosen to be the Bunch-Davis vacuum

ììQ=! 2ìì ü 1

Inflation Radiation

These initial conditions are crucial in order to get a scale invariant power spectrum

Inflationary fluctuations

The ratio of dp to gw amplitudes is given by

Gravitational waves are subdominant

The spectral indices are given by

The running, i.e. the scale dependence of the spectral indices, of dp and gw are

Inflationary predictions: the two-point correlation function

- The amplitude is controlled by H (for the Schwinger effect, this was E)

- For the scalar modes, the amplitude also depends on 1

The power spectra are scale-invariant plus logarithmic corrections the amplitude of which depend on the sr parameters, ie on the microphysics of inflation

K-inflationary Perturbations

At the perturbed level, the Mukhanov-Sasaki variable obeys the following equation of motion

The “sound speed” is now time-dependent

- The usual calculation of the spectrum in terms of Bessel functions breaks down

- One has to worry about the initial conditions

- One needs to define a new hierarchy of slow-roll parameters

(DBI)

with

The ratio of dp to gw amplitudes is given by The spectral indices are given by

K-inflationary predictions

The amplitude and the spectral indices are modified by the « sonic flow » parameters

The « crossing point » is not the same for tensors and scalars

The spectral indices, runnings etc … can be determined at second order e.g. (agree with Kinney arXiv:0712.2043, disagree with Peiris, Baumann, Friedman & Cooray, arXiv:0706.1240, Chen, hep-th/0408084, Bean, Dunkley & Pierpaoli , astro-ph/0606685)

21

How can we test inflation?

1- Using the slow-roll approximation for the power spectrum

Simple and model independent

Usually quite accurate

Important to understand the model

Not exact

Prior choices not very appropriate

Not well-suited for reheating

breaks down if we go beyond slow-roll

Pros Cons

2- Model by model exactly (ie numerically)

Pros All the sr Cons!

Perfect to compute the Bayesian evidence

Cons Obviously, it requires to specify models so maybe it is not generic enough?

We should do both (important: there is also the reconstruction program!). The two approaches are complementary!

Two strategies to constrain inflation

The slow-roll pipeline

Slow-roll power spectrum

Data

Hot Big Bang:

Slow-roll parameters:

Energy scale:

Gravity waves

J. Martin & C. Ringeval, JCAP 0608, 009 (2006), astro-ph/0605367

WMAP5 and K-inflation

- Four parameters instead of two

- The relevant parameters are because

Jeffrey’s prior

Uniform prior in [-0.3,0.3] Uniform prior in [-0.3,0.3]

Jeffrey’s prior

Mean likelihood

Marginalised posterior probability distribution

- The main constraints are

2D Marginalised posterior probability distribution

L. Lorenz, J. Martin & C. Ringeval, Phys. Rev D78, 083543 (2008), arXiv:0807.2414

Including non-Gaussianity: DBI

-Including non-Gaussianity means a prior on 2

Uniform prior:

2 2 [1,467]

Uniform prior in [-0.3,0.3] Uniform prior in [-0.3,0.3]

Jeffrey’s prior

Mean likelihood

Marginalised posterior probability distribution

-This breaks the degeneracies between 1 and

2D Marginalised posterior probability distribution

2D Mean likelihood

L. Lorenz, J. Martin & C. Ringeval, Phys. Rev D78, 083543 (2008), arXiv:0807.2414

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)

This approach allows us to constrain directly the parameters of the inflaton potential

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

The first calculation of the inflationary evidence

J. Martin, C. Ringeval & R. Trotta, arXiv:1009.4157

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 2 under pressure, 4 ruled out, small field doing pretty well

The observational situation: recap

Conclusions

Inflation is a very consistent paradigm, based on conservative physics and compatible with all known astrophysical observations.

The continuous flow of high accuracy cosmological data allows us to probe the details of inflation ie to learn about the microphysics of inflation. I have presented the first calculation of the evidence for some inflationary models= first steps towards a complete inflationary pipeline.

For a given model, one can also put constraints on the reheating temperature. First constraints in the case of large and small field models are available.

On the theoretical side, the case of multiple fields inflation is very important.It must be included in the inflationary pipeline … more complicated.

On the observational side, polarization, Non-Gaussianities, entropy modes and direct detection of gravity waves have an important role to play.

Waiting for Planck!

Thank you!

Galaxy foreground

The CMB is just behind!

First Planck data

The CMB can (also) constrain the reheating temperature!

Radiation-dominated era Matter–dominated era

Large field inflation

Constraining the reheating

The first CMB constraints on reheating!

Rescaled reheating parameter constrained

- LF:

- SF:

Reheating temperature (but with extra assumptions)

wreh=0_

wreh=-0.1_

wreh=-0.2wreh=-0.3_

Mean likelihood

Marginalized posterior pdf

J. Martin & C. Ringeval, Phys. Rev. D82: 023511 (2010), arXiv:1004.5525

Testing the initial conditions?

J. Martin & R. H. Brandenberger, PRD 68 063513 (2003), hep-th/0305161

Is the Bunch-Davies state justified?

Below the Planck length, we expect corrections from string theory

Inflation has maybe the potential to keep an inprint from this regime: window of opportunity.

If physics in non-adiabatic beyond the Planck, then one expects corrections.

Any new physics will generate the other WKB branch and, therefore superimposed oscillations the shape of which will be model dependent. In the minimal approach the amplitude is proportional to

Superimposed oscillations

WMAP and super-imposed oscillations

J. Martin & C. Ringeval, PRD 69 083515 (2004), astro-ph/0310382

WMAP and super-imposed oscillations

Power-spectrum of super-imposed oscillations

Usual SR power spectrum

Results

Logarithmic oscillations

From the Baeysian point of view (ie taking into account volume effects in the parameter space), the no-oscillation solution remains favored

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

Marginalized probalities Mean likelihood

2 [0,2]

|x |2 [0,0.45]

flat 1

Log(1/ )2 [1,2.6]