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

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…

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… : A.Starobinsky A. Guth

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

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

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

? curvature Mp4Mp ) « historical arguments » : flatness problem ln a ln  matter radiation dark energy today

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

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

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

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

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

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

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

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

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

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

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

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

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 

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 

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

2) Basic model : primordial cosmological fluctuations fluctuations today

2) Basic model : primordial cosmological fluctuations fluctuations at decoupling

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

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

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

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

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

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

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

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

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

3)Agreement between CMB maps and inflation

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

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

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

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)

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)

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)

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)

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 :

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)

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

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

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

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

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 !!

 future CMB experiments (B-polarization) : r ~ (factor 50 pour V)  future space-based GW interferometers : r ~ (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 GeV (extra-D ?) ordinary QFT (SUSY, PNGB…) 4)current constraints on inflation Energy scale of inflation still unknown !! Self-consistency relation still not checked !!

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

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????

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

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…

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)…

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

r Small field models  «m P CONCAVE, V’’>0 CONVEX, V’’<0 n large field models  ~m P V 3/2 /V’ ~ m p if V’ ~ V/ ,  V~(10 16 GeV) 4  ~m P

r CONCAVE, V’’>0 CONVEX, V’’<0  =6  =4  =2 monomial potentials V= (...)    =1 n

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

r 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

r n 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