1. Inflation: a Status Report Jérôme Martin Institut d’Astrophysique de Paris (IAP) Annecy, LAPTH, February 3, 2011

2. 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

3. 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. 4 Inflation: basic mechanism Slow-roll phase Oscillatory phase p=2 p=4 Slow-roll phase Reheating phase

5. 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. 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

7. 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

8. 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

9. 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

10. 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

11. 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

12. - 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

13. 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

14. 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)

15. 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)

16. 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

17.  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 InflationRadiation These initial conditions are crucial in order to get a scale invariant power spectrum Inflationary fluctuations

18. 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

19. 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

20. 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. 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

22. 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

23. WMAP5 and K-inflation - Four parameters instead of two - The relevant parameters are because Jeffrey’s prior 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

24. 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] 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

25. 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)

26.  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 (p 2 [0.2,5]) J. Martin & C. Ringeval, JCAP 08, 009 (2006) astro-ph/0605367

27. The first calculation of the inflationary evidence J. Martin, C. Ringeval & R. Trotta, arXiv:1009.4157

28.  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

29. 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.

30. Waiting for Planck! Thank you! Galaxy foreground The CMB is just behind! First Planck data

31. The CMB can (also) constrain the reheating temperature! Radiation-dominated era Matter–dominated era

32. Large field inflation Constraining the reheating

33. The first CMB constraints on reheating!  Rescaled reheating parameter constrained - LF: - SF:  Reheating temperature (but with extra assumptions) w reh =0 _ w reh =-0.1 _ w reh =-0.2 w reh =-0.3 _ Mean likelihood Marginalized posterior pdf J. Martin & C. Ringeval, Phys. Rev. D82: 023511 (2010), arXiv:1004.5525

34. 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

35. Superimposed oscillations WMAP and super-imposed oscillations J. Martin & C. Ringeval, PRD 69 083515 (2004), astro-ph/0310382

36. 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]