1. IFIC, 6 February 2007 Julien Lesgourgues (LAPTH, Annecy)

3. 1) « historical arguments » - flatness problem - horizon problem - monopoles & topological defects 2) basic model - slow rolling scalar field - primordial fluctuations 3) agreement between CMB maps and inflation - coherence - scale invariance - gaussianity - adiabaticity 4) current constraints on inflation, prospects…

4. 1) « historical arguments » - flatness problem - horizon problem - monopoles & topological defects 2) basic model - slow rolling scalar field - primordial fluctuations 3) agreement between CMB maps and inflation - coherence - scale invariance - gaussianity - adiabaticity 4) current constraints on inflation, prospects… 1979-1982: A.Starobinsky A. Guth

5. 1) « historical arguments » : flatness problem Definitions : -scale factor : a(t) ds 2 = dt 2 - a(t) 2 dx 2 c=1 -e-fold number : N = ln a e.g. “a stage lasts for  N=10 e-folds”  a(t) increases by factor e 10 =22000

6. Friedmann equation : or : matter (nr, r) spatial curvature a -2 a -3, a -4 1) « historical arguments » : flatness problem decelerated expansion H

7. ln a ln  matter radiation dark energy ? curvature today 1) « historical arguments » : flatness problem

8. ? curvature Mp4Mp4 10 62 1) « historical arguments » : flatness problem ln a ln  matter radiation dark energy today

9. ? TeV 4 10 32 1) « historical arguments » : flatness problem curvature ln a ln  matter radiation dark energy today

10. Inflation = stage of accelerated expansion Friedmann Energy cons. ä(t) > 0   + 3 p < 0    a n, -2 < n < 0 1) « historical arguments » : flatness problem

11. ln a ln  curvature matter radiation dark energy today inflation 1) « historical arguments » : flatness problem

12. What is the minimal duration of inflation ? 1) « historical arguments » : flatness problem

13. ln a ln  radiation a -4 today inflation  ~cst curvature a -2  N inflation =  N post-inflation

14. Minimal duration of inflation : 1) « historical arguments » : flatness problem  N inflation   N post-inflation transition infl. → rad.minimal  N inflation (10 16 GeV) 4 … (1 TeV) 4 ~67 … ~37

15. 1) « historical arguments » : horizon problem t x y

16. t x y last scattering surface (LSS) are all LSS points within causal contact ? photon decoupling

17. 1) « historical arguments » : horizon problem t x y Last scattering surface (LSS) ↓ initial singularity Hubble radius at decoupling: ~1° photon decoupling

18. 1) « historical arguments » : horizon problem t x y photon decoupling last scattering surface (LSS) x inflation

19. ? curvature 1) « historical arguments » : monopoles and other defects ln a ln  matter radiation dark energy today phase transition defects

20. ln a ln  curvature matter radiation dark energy inflation phase transition 1) « historical arguments » : monopoles and other defects today

21. 2) Basic model : a slow-rolling scalar field Inflation = stage of accelerated expansion Friedmann + e. c. : ä(t) > 0   + 3 p < 0 nearly homogeneous slow-rolling scalar fields :  = ½  ‘ 2 + V(  ) p = ½  ‘ 2 - V(  ) |dV/d  | < V/m P, |d 2 V/d  2 |< V/m P 2 V 

22. 2) Basic model : a slow-rolling scalar field Inflation = stage of accelerated expansion Friedmann + e. c. : ä(t) > 0   + 3 p < 0 nearly homogeneous slow-rolling scalar field :  = ½  ‘ 2 + V(  ) p = ½  ‘ 2 - V(  ) |dV/d  | < V/m P, |d 2 V/d  2 |< V/m P 2 V 

23. 2) Basic model : a slow-rolling scalar field Inflation = stage of accelerated expansion Friedmann + e. c. : ä(t) > 0   + 3 p < 0 nearly homogeneous slow-rolling scalar field :  = ½  ‘ 2 + V(  ) p = ½  ‘ 2 - V(  ) |dV/d  | < V/m P, |d 2 V/d  2 |< V/m P 2 V  end of inflation: field oscillates and decays in particles which finally thermalize

24. 2) Basic model : primordial cosmological fluctuations fluctuations today

25. 2) Basic model : primordial cosmological fluctuations fluctuations at decoupling

26. 2) Basic model : primordial cosmological fluctuations origin of fluctuations ?

27. 2) Basic model : primordial cosmological fluctuations decelerated expansion : - causal horizon = Hubble radius ( R H = c/H ) - R H (t) grows faster than a(t) causal acausal time MATTER DOMINATION  RADIATION DOMINATION RHRH primodial cosmological perturbations distance

28. RHRH  2) Basic model : primordial cosmological fluctuations phase transition no coherent fluctuations decelerated expansion : - causal horizon = Hubble radius ( R H = c/H ) - R H (t) grows faster than a(t) time MATTER DOMINATION RADIATION DOMINATION primodial cosmological perturbations

29. 2) Basic model : primordial cosmological fluctuations RHRH distance INFLATION accelerated expansion : - causal horizon » Hubble radius - R H (t) grows more slowly than a(t)  time MATTER DOMINATION RADIATION DOMINATION primodial cosmological perturbations

30. 2) Basic model : primordial cosmological fluctuations RHRH distance INFLATION 1 1 - quantum fluctuations of  and h  grow to macroscopic scales - normalization and evolution imposed by quantum mechanics  time MATTER DOMINATION RADIATION DOMINATION primodial cosmological perturbations

31. 2) Basic model : primordial cosmological fluctuations RHRH distance INFLATION 1 2 - Hubble crossing, Bogolioubov transformation - “squeezed state” → classical stochastic fluctuations 2  time MATTER DOMINATION RADIATION DOMINATION primodial cosmological perturbations

32. 2) Basic model : primordial cosmological fluctuations RHRH distance INFLATION 1 2 3 - perturbation amplitude frozen since - «primordial spectrum» of scalar and tensor perturbations 2 3  time MATTER DOMINATION RADIATION DOMINATION primodial cosmological perturbations

33. 2) Basic model : primordial cosmological fluctuations RHRH distance INFLATION 1 2 3 4 - insensitive to microscopical evolution (reheating, phase transition) - primordial spectrum mediated to , b,, CDM 4  time MATTER DOMINATION RADIATION DOMINATION primodial cosmological perturbations

34. 2) Basic model : primordial cosmological fluctuations RHRH distance INFLATION 1 2 3 4 5 - acoustic oscillations and decoupling - CMB anisotropies → primordial spectrum inherited from 3 5  time MATTER DOMINATION RADIATION DOMINATION primodial cosmological perturbations

35. 3)Agreement between CMB maps and inflation

36. inflation predict that perturbations are: 1.coherent 2.nearly gaussian 3.adiabatic* 4.nearly scale invariant* *for simplest inflationary models 3)Agreement between CMB maps and inflation

37. RHRH distance INFLATION decoupling time coherence of inflationary fluctuations : 3)Agreement between CMB maps and inflation primodial cosmological perturbations  time MATTER DOMINATION RADIATION DOMINATION

38. RHRH distance INFLATION absence of coherence in the case of topological defects : decoupling 3)Agreement between CMB maps and inflation primodial cosmological perturbations  time MATTER DOMINATION RADIATION DOMINATION

39. inflation predict that perturbations are: 1.coherent 2.nearly gaussian 3.adiabatic* 4.nearly scale invariant* *for simplest inflationary models 3)Agreement between CMB maps and inflation validated (existence of acoustic peaks)

40. inflation predict that perturbations are: 1.coherent 2.nearly gaussian 3.adiabatic* 4.nearly scale invariant* *for simplest inflationary models 3)Agreement between CMB maps and inflation validated (statistical analysis of CMB maps) validated (existence of acoustic peaks)

41. inflation predict that perturbations are: 1.coherent 2.nearly gaussian 3.adiabatic* 4.nearly scale invariant* *for simplest inflationary models validated (peak scale) 3)Agreement between CMB maps and inflation validated (statistical analysis of CMB maps) validated (existence of acoustic peaks)

42. inflation predict that perturbations are: 1.coherent 2.nearly gaussian 3.adiabatic* 4.nearly scale invariant* *for simplest inflationary models validated (peak scale) 3)Agreement between CMB maps and inflation validated (statistical analysis of CMB maps) validated (existence of acoustic peaks)

43. slow rolling scalar field : ASAS k amplitude  V 3/2 /V’ tilt (1-n S )  (V’/V) 2, V’’/V + hider order corrections (tilt running, …) ATAT k amplitude  V 1/2 tilt n T  (V’/V) 2 + higher order corrections (tilt running, …)  V  3)Agreement between CMB maps and inflation scale invariance :

44. inflation predict that perturbations are: 1.coherent 2.nearly gaussian 3.adiabatic* 4.nearly scale invariant* *for simplest inflationary models validated (peak scale) 3)Agreement between CMB maps and inflation validated (statistical analysis of CMB maps) validated (existence of acoustic peaks) validated (peak amplitudes)

45. single field slow-roll inflation : ASAS k amplitude  V 3/2 /V’ tilt (1-n S )  (V’/V) 2, V’’/V + next-order corrections (running of the tilt, …) ATAT k amplitude  V 1/2 tilt n T  (V’/V) 2 + next-order corrections (running of the tilt, …)  V  4)current constraints on inflation

46. ASAS k amplitude  V 3/2 /V’ tilt (1-n S )  (V’/V) 2, V’’/V + next-order corrections (running of the tilt, …) ATAT k amplitude  V 1/2 tilt n T  (V’/V) 2 + next-order corrections (running of the tilt, …) overall amplitude = 0.5x10 -5 m p 3 4)current constraints on inflation

47. ASAS k amplitude  V 3/2 /V’ = 0.5x10 -5 m p 3 tilt (1-n S )  2.25 (V’/V) 2 - V’’/V = 0.5 m p -2 + next-order corrections (running of the tilt, …) ATAT k amplitude  V 1/2 tilt n T  (V’/V) 2 + next-order corrections (running of the tilt, …) overall slope 4)current constraints on inflation

48. ASAS k ATAT k amplitude  V 1/2 < (3.7x10 16 GeV) 2 tilt n T  (V’/V) 2 + next-order corrections (running of the tilt, …) amplitude  V 3/2 /V’ = 0.5x10 -5 m p 3 tilt (1-n S )  2.25 (V’/V) 2 - V’’/V = 0.5 m p -2 + next-order corrections (running of the tilt, …) absence of tensors 4)current constraints on inflation

49. ASAS k ATAT k amplitude  V 1/2 < (3.7x10 16 GeV) 2 tilt n T  (V’/V) 2 + next-order corrections (running of the tilt, …) amplitude  V 3/2 /V’ = 0.5x10 -5 m p 3 tilt (1-n S )  2.25 (V’/V) 2 - V’’/V = 0.5 m p -2 + next-order corrections (running of the tilt, …) absence of tensors 4) current constraints on inflation Energy scale of inflation still unknown !! Self-consistency relation still not checked !!

50.  future CMB experiments (B-polarization) : r ~ 10 -2 (factor 50 pour V)  future space-based GW interferometers : r ~ 10 -4 (BBO) (factor 5000 pour V) measure r, n t : inflationary energy scale + self-consistency r=-8n t measure r : inflationary energy scale no GW detected : inflation unconstrained new physics at 10 16 GeV (extra-D ?) ordinary QFT (SUSY, PNGB…) 4)current constraints on inflation Energy scale of inflation still unknown !! Self-consistency relation still not checked !!

51. ASAS k ATAT k amplitude  V 1/2 < (3.7x10 16 GeV) 2 tilt n T  (V’/V) 2 + next-order corrections (running of the tilt, …) amplitude  V 3/2 /V’ = 0.5x10 -5 m p 3 tilt (1-n S )  2.25 (V’/V) 2 - V’’/V = 0.5 m p -2 + next-order corrections (running of the tilt, …) ? 4)current constraints on inflation

52. ASAS k ATAT k amplitude  V 1/2 < (3.7x10 16 GeV) 2 tilt n T  (V’/V) 2 + next-order corrections (running of the tilt, …) amplitude  V 3/2 /V’ = 0.5x10 -5 m p 3 tilt (1-n S )  2.25 (V’/V) 2 - V’’/V = 0.5 m p -2 + next-order corrections (running of the tilt, …) ? 4)current constraints on inflation negative running, or no running????

53. 4)current constraints on inflation negative running, or no running???? no running (power law spectrum) negative running (convex spectrum) WMAP3+SDSS

54. 4)current constraints on inflation negative running, or no running????  Theoretical prejudice: Deep in the slow-roll limit, running ≈ 0 ( n s -1 ~ , n run ~  2 )  Do we expect to be deep in the slow-roll regime? Question of philosophy and aesthetics…

55. 4)current constraints on inflation negative running, or no running???? 1)Minimalistic aesthetics: simple potential (monomial, polynomial, simple function) slow-roll params  (  ) monotonically growing/decreasing 60 e-folds before the end, must be deep in slow-roll expect running ≈ 0 2)Modesty and pragmatism: V(  ) may have any shape (many scalars, landscape…) we can only reconstruct “observable region” (no assumptions on what’s before/after) possible large running (and beyond)…

56. 4)current constraints on inflation J.L. & W.Valkenburg, in preparation

57. 0 0.2 0.4 0.6 0.8 r 1 0.9 Small field models  «m P CONCAVE, V’’>0 CONVEX, V’’<0 n large field models  ~m P V 3/2 /V’ ~ 10 -5 m p if V’ ~ V/ ,  V~(10 16 GeV) 4  ~m P

58. 0 0.2 0.4 0.6 0.8 r 1 0.9 CONCAVE, V’’>0 CONVEX, V’’<0  =6  =4  =2 monomial potentials V= (...)    =1 n

59. 0 0.2 0.4 0.6 0.8 r 1 0.9 CONCAVE, V’’>0 CONVEX, V’’<0  =6  =4  =2 new inflation V=V 0 [1- (…)   +...]  =1 monomial potentials V= (...)   n

60. 0 0.2 0.4 0.6 0.8 r 1 0.9 CONCAVE, V’’>0 CONVEX, V’’<0  =6  =4  =2 monomial  =1 Loop correction monomial potentials V= (...)   Hybrid inflation  =1 new inflation V=V 0 [1- (…)   +...] n

61. 0 0.2 0.4 0.6 0.8 r n 1 0.9 CONCAVE, V’’>0 CONVEX, V’’<0  =4  =2 monomial  =1 Loop correction monomial potentials V= (...)   new inflation V=V 0 [1- (…)   +...]  =1  =6 WMAP-3 +SDSS