Chaire Galaxies et Cosmologie. Françoise Combes

Transcription

1 Chaire Galaxies et Cosmologie Inflation and new paradigms Françoise Combes

2 Overview Why inflation? Observational justification History of the model Principle of «slow roll» inflation Hybrid inflation, eternal inflation, Curvaton Inflation in string theory Observational constraints : Planck 2015 Critics and problems -Alternatives?

3 Why is inflation needed? Exponential expansion (~10 30 ) between s and s the problems of horizon, of homogeneity the problem of universe flatness problem p of monopoles (+strings, textures, etc..)

4 Problem of monopoles (and other textures) At very high energy in the very hot Big-Bang, all forces of physics are unified (GUT) T = GeV Then occur several breaks of symmetry in cascade, and in particular the electro-magnetic group At each symmetry break, magnetic monopoles are created (stable) The energy of a monopole ~ GeV, mass ~ 1.8x g With one per horizon, ~1 1.7x10 65 g/cm 3 (GUT epoch), then g/cm 3 today >> c =10-29 g/cm 3 Monopole, in Onion shape ~10 15 x m p

5 The horizon problem The horizon at the CMB epoch was < 2 Why T CMB is the same within 10-5 everywhere? (regions non causally linked) Homogeneity T =10-35 GUT 14 GeV, at t=10 s and today T CMB = ev (2.77K) Expansion of a= Horizon(GUT)= cm Homogeneity (t=0) ~10 2 cm!!

6 Inflation solves the horizon problem In absence of inflation, the horizon is cdt/a(t) l (t dec, 0) = 2ct dec /a dec, and the ratio a l(t 0,t dec ) = 3c/a dec t 2/3 dec t 1/3 0 R =2/3(t dec /t 0 ) 1/3 =0.02<< 02<< 1 R becomes , i.e. 2x10 6 more oewith inflation. y Universe horizon today During inflation, the horizon in comoving coordinates contracts! (system at rest % expansion) Time Elementary region causally linked t dec Inflation dilutes also the magnetic monopoles Guth (1997)

7 Rf Referential tilat rest The observer accompanies the inflation of space The horizon is constant Comoving referential The observer is at rest The horizon shrinks

8 The inflation solves the problem of flatness Whatever initial conditions The exponential expansion of a factor ~10 30, reduces the curvature term kc 2 /a 2 by a factor 10 60

9 Inflation, source of fluctuations at t < s today The scalar field which is the source of inflation is called inflaton The universe is empty at the start (just contains vacuum fluctuations) Corresponds to the de Sitter solution Aft th d i t db di ti th b tt After the era dominated by radiation, then by matter, the Universe becomes empty again (at 70%?) & starts another inflation

10 Fluctuations of quantum origin Quantum Mechanics (QM): virtual particles in the vacuum During inflation,, regions causally connected, are suddenly disconnected: particles cannot annihilate any more Wavelength th( (mode) Quantic ~a~exp(ht) Black hole Horizon =c/h ~cste Frozen waves > horizon Creation of gravit waves (tensor mode) Temperature ~1/H Kinney

11 Inflation between t~10-35 s GUT scale Until t=70/h~10-32 s H= km/s/mpc! a(t) e Ht = 2.5 x10 30 The temperature at the end of inflation is the same, because of the latent heat liberated in the phase transition T= GeV Otherwise T~1/a(t) Amplitude of perturb. ~same when they cross the horizon

12 Inflation: the only model for the anisotropies of the cosmic background Before Planck with WMAP + Ground experiments Eliminates the generation of fluctuations by cosmic defects Guth 2007

13 Conservation of energy? Nothing is created from nothing! nihil fit ex nihilo Parmenides (greek philosopher, ~500 av J-C), Lucrece (Rome ~100 av. J-C), de Rerum Natura Scalar field (from GUT?) has the property to be in equilibrium in a false vacuum, where E 0 Then the field could slowly roll towards the true vacuum E=0 Negative pressure of false vacuum P=- c 2 Acts as a negative gravity The universe starts very small ~10-25 cm, Its size increases by 10 30, and its energy of (volume); but its gravitational energy is very negative, and compensates exactly. The energy of all the created matter is taken on the gravitational energy

14 on Inflatio GUT: Grand Unification of forces

15 Gaussian fluctuations, with a power spectrum The fluctuations are self-similar, similar with no characteristic size Except at horizon crossing Power law with flat slope ns = 1 + O(10-2 ) Compatible with Planck (2015) ns = Adiabatic fluctuations In the case of simple inflation, one predicts: --A flat universe, homogeneous and isotropic --Gaussian and adiabatic fluctuations, with a flat spectrum --Spectrum slope ns =

16 Scalar field? All gauge bosons correspond to «vectors» of spin 1 The graviton has spin 2 Has a tensor character As a scalar (spin 0) there is only the Higgs boson, could it be related to the inflaton?

17 Non-gaussianity Amplitude quantified by the coefficients f NL computed from the 3-point correlations Or also called «bi-spectrum» B(k1,k2,k3) =< (k1) (k2) (k3)> = f NL (2 3 3 (k1+k2+k3)b(k1,k2,k3) k2 k3) (k) is the Fourier transform of the fluctuation, and b(k1,k2,k3) defines the shape of the triangles Local shape of f NL Equilateral shape Komatsu 2008

18 Results from Planck The gaussianity is confirmed at the level of 0.03%, but finer constraints could reveal the cosmological model The gravitational lenses generate some non-gaussianity This must beco corrected edbefore emaking the test f local NL = , f equil NL = , f ortho NL = In the future, better constraints Flatness at 0.01%, and non-gaussianity at 0.005% (f NL ~5) Inflation confirmed, but ekpyrotic or cyclic scenario in difficulty

19 Some words of history Theory of Landau-Ginzburg, of symmetry breaking (1950) for superconductors: phase transition of 2nd order, order parameter Minimisation of Lagrangian: yields an order parameter cancelling for T= Tc Inflation proposed by Alexei Starobinski (1979/80, URSS) and Alan Guth (1980/81, USA). But the Guth mechanism requires a modification to get out of inflation New inflation, proposed by A. Linde, A. Albrecht, P. Steinhardt independantly in 1982 The first inflationary models: phase transition of 1st order, then of 2nd order Eventually without phase transition at all, as in the chaotic inflation, supposed deriving from chaotic initial conditions

20 Potential Depends on the ambiant temperature with respect to Tc Criticl temperature Tc~ GeV

21 Schema for an inflation with tunnelling effect The thermal fluctuations +tunnelling effect transition false true vacuum, terminating inflation in certain regions, or bubbles. All terminates when the expanding bubbles bbl touch each other, merge and heat the matter Schema for a chaotic inflation Without phase transition, if Ec << V Requires that the field is homogeneous on scales >> horizon!!

22 New potential In fact these models do not work, too inhomogeneous, not efficient for the re-heating The potential must have a smaller slope (slow roll) The inflaton decays into photons, and matter particles The quantum oscillations create initial perturbations Constant energy during inflation, efficient heating of matter by inflaton, or Higgs boson?

23 New and old inflation Smooth exit from inflation

24 Inflation scenario Quantum, L p = cm, M p = 10-5 g, = g/cm 3

25 Problems of models with phase transition The bubbles have a too fast expansion, Leaving the universe devoid of structures Fine tuning is necessary to avoid the collisions between bubbles (forming monopoles, domains, etc.) Fine tuning of the mechanism to exit the false vacuum, metastable Once the fine tuning is done, to agree with the CMB, the bubbles collide too frequently Use a field potential with a slow roll instead of Use a field potential with a slow roll instead of tunneling effect (solves the absence of monopoles, or cosmic defects) Use quantum fluctuations as initial conditions (avoids fine tuning)

26 Inflation as an harmonic oscillator V( ) = m 2 /2 2 V ( ) = m 2 V ( ) = m 2 Eternal Inflation generated by the quantum fluctuations several expansions possible Chaotic inflation Linde 1986

27 The equations Einstein equation H 2 = 8 G/3 H 2 = ( a / a ) 2 = 8 G m 2 /6 2 Energy density V( ) = m 2 /2 2 X Klein-Gordon (Relativistic Shrödinger) with V = m 2 slow roll = 3H, k=m 2 Similar to the equation of an harmonic oscillator with friction (Hubble friction) x x x kx 0

28 Principle of the inflation Large values of field + large values of H large friction The field varies very slowly, therefore its value in energy is quasi constant a(t) e Ht In this case there is no false vacuum In this case, there is no false vacuum, nor phase transition

29 Conditions for inflation There must exist a slow roll, therefore a slow decrease of the potential, but how much? Requires H ~constant, or dh/dt << H 2 H 2 = V /(3 M p2 ) dh/dt /H 2 = M p2 /2 (V /V) 2 = = M p2 /2 (V /V) 2 << 1 = M p2 V /V << 1 N=ln(a f /a i ) V/V p Si V( ) = m 2 /2 2 V ( ) = m 2 V ( ) = m 2 = 2 (M p / ) 2 << 1 >> M p

30 Chaotic or eternal inflation It is difficult to stop inflation everywhere. One can stop it in a bubble in particular, produce a re-heating, and the creation of particles in a universe, but space continues its expansion elsewhere Each region of the Universe evolves independantly, according to the Initial values of quantum parameters The inflation self-maintains, iti in a chaotic way, and undefined dfi dtime There is no start, no end eternal inflation (Linde 1986) Inevitable! Guth 2007

31 Evolution of the inflaton field Because of quantum fluctuations, the probability to come back to large values of is not zero These regions produce a strong inflation and involve a large fraction of the volume Film A. Linde

32 Eternal in the future, but not in the past? The regions in inflation co-exist with the thermalised universes.. A particle or photon travel along a geodesic of the expanding universe and sees its frequency shifted to the red rouge In the past, this shift is towards the blue A particle computes the Hubble constant with comoving test particles 1 and 2 F( ) = 1/ At the origin, F= 0, and There exists a singularity, and one must rely on a new physics, or quantum conditions Borde, Guth, Vilenkin 2003

33 Fractal structure of eternal inflation Linde Vanchurin Vilenkin Winitzki simulations

34 f(r) = R +ar 2 Multiple models! V( ) ~(1+cos ) Max Camenzind

35 Hybrid inflation Two scalar fields and Slow roll inflation in the plane =0 The field transforms in a horse saddle shape Then sudden drop in the perpendicular plane Oscillations and reheating of the Universe U( ) inflaton

36 Can there exist strong non-gaussianities? V( ) The ordinary inflation does not produce non-gaussianities Strings can produce them, on the contrary Or the curvaton? V( ) Inflaton Curvaton Adiabatic perturbations Iso-curvature perturbations is determined by the quantum fluctuations, thus the perturbations amplitude is different in the various regions

37 The curvaton The curvaton is an extra scalar field, producing fluctuations of curvature, towards the end of inflation During inflation, the perturbations are adiabatic: photons and matter fluctuate together:, CDM, baryons same n/n After the inflaton, the curvaton is the main energy density Enqvist & Sloth, 2001, Lyth & Wands, 2001 It could even replace inflation Simple model of curvaton, field The principal contribution to these fluctuations is given by wavelengths exponentially increasing Mukhanov 1996, 2005

38 Dominant curvaton? = 1? (attractor) f NL 1/ Planck f NL < 10 Byrnes (2014)

39 Spatial distribution of curvaton The curvaton becomes null on the «coast» Takes values between +H 2 /m On scales~ 0 0

40 The curvaton network and the non-gaussianity In simple inflation, one assumes a constant t amplitude for the perturbations, H ~ But for the curvaton H can be very different, and thus introduce some non-gaussianity Linde, Mukhanov, 2005

41 The huge number of «landscapes» in string theory After having demonstrated that inflation in all models is eternal, one realizes that the number of possible universes is enormous in the frame of string theory, given the extra dimensions (Susskind 2003, Bousso & Polchinski 2004) A large number of fields, with an enormous number of minima or false vacua The possibilities are estimated at false vacua, metastable Each vacuum has different values of parameters ( ) The anthropic principle p ensures reasonable values of parameters How to evaluate the probabilities? A theory is missing to measure these probabilities: a kind of renormalization

42 Two types of inflationary models Closed strings The simplest models: the inflaton is the module of the string Use scalar fields already present in the models of compactification, with their large number of minima A landscape with to minima Branes (linked by open strings) The inflaton field corresponds to the distance between branes of the Calabi-Yau space This kind of model was historically the first proposed in the string theory

43 Inflation in the string theory The problem of volume stabilisation: One potential of the theory obtained by compactification in string theory V(X,Y, ) Y ) ~ e ( 2X - 6Y) V( ) X and Y are the normalised canonical fields corresponding to the dilaton and to the volume of compactified space; is the field driving the inflation The potential is very steep with respect to X and Y, these fields evolve rapidly, and the potential energy V disappears. These fields must be stabilised Stabilisation of volume: construction KKLT Kachru, Kallosh, Linde, Trivedi 2003

44 Stabilisation in volume Principal steps of the scenario KKLT Kachru, Kallosh, Linde, Trivedi ) Start from a theory with an exponential potential 2) Make this potential drop with strong quantum effects (non perturbative) 3) Redress the minimum until the state of positive vacuum energy by energy addition from an anti-d3 brane of Calabi-Yau space 0.5 V s minimum AdS V s Minimum ds metastable

45 Too numerous results It is possible to stabilise the model in its own dimensions, and obtain a universe in acceleration. At the end, our region of universe decays and becomes 10-dimensions, but this occurs only in yrs Apparently, the stabilisation of vacuum can be done in different manners i.e. the potential energy V of the string theory can have minima, corresponding to possible universes

46 Inflation in the theory Inflation brane-anti-brane Hybrid inflation D3/D7 Modular inflation Iflti DBI(Di tb Ifld) Inflation DBI (Direct-Born-Infeld) non-minimum kinetic terms

47 Inflation Brane-Antibrane When branes inflate, two can collide

48 A problem for string inflation In all versions of string theory inflation, the processus begins at V<<1 Typically V=10-16 M 4 p But a close and hot universe collapses in a time-scale t/t p = S 2/3 (S entropy). To survive until the start of inflation at t=1/h =V -1/2 one must have S > V -3/4 The initial entropy (the number of particles) must be S> Such a universe at Planck epoch consisted of horizons causally independent. Thus,toexplainwhytheuniverseisso huge and homogeneous, one must suppose that it was huge and homogeneous since the beginning

49 One possible solution Difficult to start inflation: How to create a flat universe? Take a box (a fraction of the flat universe) and glue opposite faces to each other torus, which h isaflat Universe! Its size increases as t 1/2, while a relativistic particle travels ct Therefore until the start of inflation the size of the Universe is smaller than the horizon Linde 2006

50 Homogeneity If the Universe initially had the Planck size (the smallest possible), then in acosmological l time t >> t p (in Planck units) the particles have the time to run across the torus several times and to appear in all these regions with equal probability, This makes the universe homogeneous and keeps it homogeneous until the start of inflation

51 Eternal inflation in a string landscape The eternal inflation is a general property of all models based on landscapes: the fields jump eternally from a minimum i to another, and the universe continues its exponential expansion However, an epoch occurs when the fields stop their jumping as in classical inflation, and begin to slowly roll down like in the chaotic inflation: the last step of inflation is always of this type How to create the initial conditions of this slow-roll inflation after the tunnelling effect?

52 Multiple Universes, with different values

53 Initial conditions for inflation D3/D7 In the scenario D3/D7, the flat character of the inflaton does not depend on flux V Eternal inflation in a valley with different flux >> M p Slow roll inflation >M p The field jumps in the high valley because of quantum u fluctuations, u then there is a tunnel effect due to flux change inside a bubble s

54 Critiques of inflation The initial conditions must be fine tuned (Steinhardt 2011) The probability of inflations incompatible with observations is very large, the scenario without inflation is even more probable (Penrose) Eternal inflation, which never stops (Linde) Produces an infinity of universes no predictions Weak value of r<011(cf 0.11 Planck coll 2015) V( ) in plateau favored, with V I = M 4 p i.e. V I ~M I4 << M p 4 (10 76 GeV 4 ) What happens from t p to t I? Ijjas et al 2013

55 Planck coll, 2015 r (at k=0.002/mpc) Constraints from CMB Dark 68% CL Light 95% CL R 2 ns f(r) = R +ar 2 ns = (dn/dk=0) r < 0.11 (tensor/scalar ratio) V( )~ 2, natural inflation (1+cos ) disfavored -- Inflation R 2 OK

56 Observationnal parameters of inflation Slope of the spectrum for scalar perturbations P(k) k ns-1 ns= = 1-2/N Tensor-scalarscalar ratio r=16 = 12/N 2 (V /V) 2 = M p2 (V /V) 2 /2 = M p2 (V /V) N = 1/M P2 (V/V ) d N=ln(a f /a i ) number of multiplications by e at the end of inflation N V/V, very easy to have a strong inflation (large N) with potentials in power-law V n But for plateaux very difficult

57 Less inflation? V I ~10 64 GeV 4 We measure today the last phase of inflation

58 Alternatives: cyclic models In our past, we will never see but only one inflation. The concept of eternal inflation is philosophical (the causality principle is not in question) Remains the problem of initial singularity What is the energy at the start? i ~ o e 65 ~10 30 Past cone of the observer O Time Bubbles of false vacuum True vacuum Univers ds Past cone of the observer O Temporal cut 1( (unity R h ) Turok 2002

59 Alternatives: cyclic models Ekpyrotic model, from colliding branes Cyclic, with rebound (Turok & Steinhardt 2005) The cyclic model however needs Gravity propagates outside the 3-brane

60 Comparison: inflation/ cyclic Radiation~1/R 4 Matter~1/R 3 Curvature~1/R 2 inflation Vacuum=cste= Inflationnary model dl cyclic

61 Test of gravitationnal waves The inflation predicts primordial gravitationnal waves Their measure could give access to the potential V( ) and eliminate the other cyclic models ()Eifl (a)exit false vacuum, end of acceleration (b)scale-invariant Perturbations (c)big Crunch, -, kinetic energy a(t) t

62 Precise measure of M t = mass of quark top Could the Higgs scalar field be the inflaton? Possible (Bezrukov & Shaposhnikov, 2008, Masina & Notari 2012) Fine tuning of M t to have Higgs inflation Hamada et al 2013, 2015

63 A needle in a hay stack? Still many unknownss! Future experiments

64 Conclusion The inflationary model is still favored by the observations -- solves the problems of flatness, horizon, homogeneity -- absence of magnetic monopoles, textures -- produces initial quantum fluctuations, able to develop the structures, res with the right spectrum -- very weak non-gaussianity f NL Problems: eternal inflation, multiple universes Fine tuning, non-predictibility Observations: inflations with only one field are favored shape: better a plateau, than a power-law, at least in the last phase Very soon: future observations of r=tensor/scalar, and f NL