Chaire Galaxies et Cosmologie
Inflation and new paradigms
Françoise Combes Overview
Whyyj inflation? Observational justification
History of the model
Principle of « slow roll » inflation
Hybrid inflation, eternal inflation, Curvaton
Inflation in string theory
Observat iona l constra ints : Pl anck 2015
Critics and problems -Alternatives? Why is inflation needed?
Exponential expansion (~1030) between 10-36s and 10-32s
the problems of horizon, of homogeneity
the problem of universe flatness
ppp(g)roblem of monopoles (+strings, textures, etc..) Problem of monopoles (and other textures)
• At very high energy in the very hot Big-Bang, all forces of physics are unified (GUT) T = 1014 GeV • Then occur several breaks of symmetry in cascade, and in particular the electro-magggpnetic group • At each symmetry break, magnetic monopoles are created (stable) • The energy of a monopole ~ 1016 GeV, mass ~ 1.8x101.8 x 10-8 g
• With one per horizon , ~17x101.7 x 1065 g/cm3 (GUT epoch) , -15 3 -29 3 then 10 g/cm today >> c =10 g/cm
Monopole, in Onion shape 15 ~10 x mp The horizon problem • The horizon at the CMB epoch was < 2° -5 • Why TCMB is the same within 10 everywhere? (regions non causally linked) Homogeneity
14 -35 TGUT=10= 10 GeV, at t=10 s and today -4 TCMB = 2..04 10 eV (2 .77K) Expansion of a= 4 1026
Horizon(GUT)=3 10-25cm Homogeneity (t=0) ~102cm!! Inflation solves the horizon problem In absence of inflation, the horizon is cdt/a(t) 2/3 1/3 l (tdec,0) = 2ctdec/adec, and the ratio a l(t0,tdec) = 3c/adectdec t0 1/3 R =2/3(= 2/3 (tdec/t0) =0. 02<< 1 4 R becomes 4 10 , i.e. Universe horizon 2x106 mooewre with inflatio n. today
During inflation, the horizon in comoving coordinates contracts! (system at rest
% expp)ansion) Time Elementary region causally linked tdec Inflation dilutes also the magnetic monopoles Guth (1997) RfReferen tiltial at rest The observer accompanies the inflation of space The horizon is constant
Comoving referential The obse rve r is at rest The horizon shrinks The inflation solves the problem of flatness
Whatever initial conditions
The exponential expansion of a factor ~1030, reduces the curvature term kc2/a2 by a factor 1060 Inflation, source of fluctuations at t < 10-32s
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
After the era ditddominated bditiby radiation, then by matter, the Universe becomes empty again (at 70%?) & starts another inflation Fluctuations of quantum origin Quantum Mechanics (QM): virtual particles in the vacuum During inflation,,g regions causall y connected , are suddenl y disconnected: particles cannot annihilate any more
WlthWavelength (d)(mode) Quantic ~a~exp(Ht)
Horizon =c/H ~ cste Frozen waves > horizon Creation of gravit waves (tensor mode) Black hole Temperature ~1/H
Kinney 2003 10 Inflation between t~10-35s GUT sca le Until t=70/H~10-32s H= 21053km/s/Mpc ! a(t) eHt = 2.5 x1030
The temperature at the end of inflation is the same, bfthbecause of the latent heat liberated in the phase transition T= 1014 GeV Otherwise T~1/a(t)
Amplitude of perturb. ~same whthhen they cross the horizon Inflation: the only model for the anisotropies of the cosmic background
Before Planck wihWMAPith WMAP + Ground experiments
Eliminates the generation of fluctuations by cosmic defects
Guth 2007 Conservation of energy?
Nothing is created from nothing! nihil fit ex nihilo Parmenides (greek philosopher, ~500 av J-C), Lucrece (R ome ~100 av. J -C), d e R erum N atura
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=-c2 AiiActs as a negative gravity
The universe starts very small ~10-25cm, Its size increases by 1030, and its energy of 1090 (volume); but its gravitational energy is very negative, and compensates exactly. The energy of all the created matter is taken on the gravitational energy GUT: Grand Unification of forces n oo Inflati Gaussian fluctuations, with a power spectrum
The fluctuations are self -similar, with no characteristic size Except at horizon crossing
Power law with flat slope ns = 1 + O(10-2) Compatible with Planck (2015) ns = 0.966 + 0.006
Adiabatic fluctuations
In the case of simple inflation, one predicts: --A flat universe,,g homogeneous and isotro pic --Gaussian and adiabatic fluctuations, with a flat spectrum --Spectrum slope ns = 1 + 0.1 Scalar field? All gauge bosons correspond to « vectors » of spin 1
The graviton has spin 2 HhHas a tensor character
As a scalar (spin 0) there is only the Higgs boson, could it be related to the inflaton? Non-gaussianity
Amplitude quantified by the coefficients fNL computed from the 3 -point correlations Or also called « bi-spectrum »
B(k1,k2,k3) =<(k1)(k2)(k3)> = 3 3 fNL(2 (k1+k2+k3)b(k1,k2 ,k3)
(k) is the Fourier transform of the fluctuation, and b(k1,k2,k3) defines the shape of the triangles Local shape of fNL
Equilateral shape
Komatsu 2008 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 Thissusbecoecedbeoe must be corrected before making gees the test
local f NL = 272.7 + 585.8, equil f NL = -42 +75, ortho f NL = -25 + 39
In the future, better constraints
Flatness at 0.01%, and non-ggyaussianity at 0.005% ( fNL~5) Inflation confirmed, but ekpyrotic or cyclic scenario in difficulty 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 Potential Depends on the ambiant temperature with respect to Tc Criticl temperature Tc~3 1014 GeV Schema for an inflation with tunnelli ng eff ect
The thermal fluctuations +tunnelling effect transition false true vacuum, terminating inflation in certain regions, or “bubbles”. All terminates when the expandi ng b ubbl es t ouch each oth er, merge and heat the matter Schema for a chaotic inflation Wihithout ph ase transi iition, if Ec << V Requires that the field is homogeneous on scales >> horizon!! 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 bifltby inflaton, or Hi ggs b oson ? New and old inflation
Smooth exit from inflation Inflation scenario
-33 -5 94 3 Quantum, Lp= 10 cm, Mp= 10 g, = 10 g/cm 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 tun ing of th e mech ani sm to exihflit the false vacuum, metastabl e
Once the fine tuningg,g is done, to agree with the CMB , the bubbles collide too frequently
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) Inflation as an harmonic oscillator
V() = m2/2 2 V’() = m2 V’’() = m2
Eternal Inflation
generated by the quantum fluctuations several expansions possible Chaotic inflation
Linde 1986 The equations Energy density V() = m2/2 2 Einstein equation H2 = 8G/3
H2 = ( a / a ) 2 = 8G m2/6 2 X
Klein-Gordon slow roll (Relativistic Shrödinger) with V’ = m2
= 3H, k=m2
Similar to the equation of an harmonic oscillator with friction (Hubble friction) xx kx 0 Principle of the inflation
Large values of field + large values of H large friction
The field varies veryyy, slowly, therefore its value in energy is quasi constant
a(t) eHt
In this case , there is no false vacuum , nor phase transition 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 << H2
2 2 2 2 2 H = V /(3 Mp ) –dH/dt /H = Mp /2 (V’/V) =
2 2 = Mp /2 (V’/V) << 1 2 = Mp V’’/V << 1 N=ln(af/ai) V/V’
Si V() = m2/2 2 V’() = m2 V’’() = m2
2 = 2 (Mp/) << 1 >> Mp
Chaotic or eternal inflation It i s diffi cult t o st op i nfl ati on everywh ere. O ne can st op it i n 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 i nfl ati on self -maitiintains, i n a ch aoti c way, and und dfidtiefined time There is no start, no end eternal inflation (Linde 1986)
Inevitable!
Guth 2007 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 Eternal in the future, but not in the past?
The regions in inflation co-exist with the thermalised universes..
A part ic le or ph oton travel al ong a geod esi c of th e expandi ng uni verse 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 comovinggp 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 Fractal structure of eternal inflation
Linde Vanchurin Vilenkin Winitzki
2000-07
simulations f(R) = R +aR2 Multiple models!
V() ~(1+cos)
Max Camenzind 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 Can there exist strong non-gaussianities ? The ordinary inflation does not produce V() non-gaussianities SiStrings can pro duce t hem, on t he 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 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 Dominant curvaton?
= 1 ? (attractor) fNL 1/
Planck f NL < 10
Byrnes (2014) Spatial distribution of curvaton
The curvaton becomes null on the « coast » Takes values between +H2/m On scales~ 0
0 The curvaton network and the non-gaussianity In s impl e i nfl ati on, one assumes a const ant amplit ud e f or th e -5 perturbations, H ~ 10 . But for the curvaton H can be very different, and thus introduce some non -gaussianity
Linde, Mukhanov, 2005
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 10500-101000 false vacua, metastable Each vacuum has different values of parameters () The anthropic ppprinciple ensures reasonable values of parameters
How to evaluate the probabilities? A theory is missing to measure these probabilities: a kind of renormalization 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 10100 to 101000 minima
Branes (linked by open strings) The inflaton field corresponds to the distance between branes of the Calabi-Yau space
This ki nd of mod el was hi stori call y th e fi rst proposed in the string theory Inflation in the string theory The problem of volume stabilisation:
One potential of the theory obtained by compactification in string theory V(X, 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 Stabilisation in volume KKLT Kachru, Kallosh, Linde, Trivedi 2003 Principal steps of the scenario 1) Start from a theory with an exponential potential 2) Make this pppgqotential 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
V V 0.5 s 1.2 100 150 200 250 300 350 400 1
-050.5 0.8
-1 0.6 0.4 -1.5 020.2
-2 100 150 200 250 300 350 400 s
minimum AdS Minimum dS metastable 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 1010120 yrs
Apparently, the stabilisation of vacuum can be done in 10100 -101000 different manners i.e. the potential energy V of the string theory can have 10100 -101000 minima, corresponding to possible universes Inflation in the theory
Inflation brane -anti-brane
Hybrid inflation D3/D7
Modular inflation
IfltiInflation DBI DBI(Di (Direct -Born-IfldInfeld) non-minimum kinetic terms Inflation Brane-Antibrane
When branes inflate, two can collide A problem for string inflation
In all versions of string theory inflation, the processus begins at V<<1 -16 4 Typically V=10 Mp 2/3 But a close and hot universe collapses in a time-scale t/tp = S (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>1012. Such a universe at Planck epoch consisted of 1012 horizons causally independent. Thus,toexplainwhytheuniverseisso huge and homogeneous, one must suppose that it was huge and homogeneous since the beginning… 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, whic h isaflat UiUniverse !
Its size increases as t1/2, while a relativistic particle travels ct
Therefore until the start of inflation the size of the Universe is smaller than the horizon
Linde 2006 Homogeneity
If the Universe initially had the Planck size (the smallest possible) ,then inacosmolillogical time t >> tp (in Planc k 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 Eternal inflation in a string landscape
The eternal inflation is a general property of all models based on ldlandscapes: the field s jump eternally from a miiinimum 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? Multiple Universes, with different values Initial conditions for inflation D3/D7 In the scenario D3/D7, the flat character of the inflaton does not depend on flux Eternal V inflation in a valley wi th different flux >> M p
Slow roll inflation >M> Mp s The fie ld ju mps in the hig h v alley bec ause o f qu antu m fluc tu atio ns, the n there is a tunnel effect due to flux change inside a bubble 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(r < 0.11 (cf Planck coll 2015) -12 4 V() in plateau favored, with VI = 10 Mp
4 4 76 4 i.e. VI ~MI << Mp (10 GeV ) What happens from tp to tI? Ijjas et al 2013 Planck coll, 2015 Constraints from CMB r 0.002 Dark 68% CL (at k=0.002/Mpc) Light 95% CL
R2 ns f(R) = R +aR 2 ns = 0.968+0.006 (dn/dk=0) V()~2, natural inflation ( 1+cos) r < 0.11 disfavored -- Inflation R2 OK (tensor/scalar ratio) Observationnal parameters of inflation
Slope of the spectrum for scalar perturbations P(k) kns-1 ns= 1- 6 +2 = 1-2/N
Tensor-scalar ratio r16r=16 = 12/N2 (V’/V)2
2 2 2 2 = Mp /2 (V ’ /V) = Mp (V’’ /V) N = 1/MP ∫(V/V’) d
N=ln(af/ai) 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 Vn
But f or p l ateau x very ddcuifficult Less inflation?
64 4 VI ~10 GeV
We measure today the last phase of inflation 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) Past cone of the observer O Remains the problem of initial singu larity What is the energy at the start? i ~ 10120 o e65 ~1030
Time Bubbles of false vacuum True vacuum Univers dS
Past con e ootef the 1(1 (unity observer O Rh) Temporal cut Turok 2002 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 Comparison: inflation/ cyclic
Radiation~1/R4 Matter~1/R3 Curvature~1/R2 inflation Vacuum=cste=
Inflationnary modldel cyclic 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
()(a)EiflExit false vacuum, end of accelileration (b)Scale-invariant Perturbations (c)Big Crunch, -∞, kinetic energy
a(t)
t Precise measure of Mt= mass of quark top Could the Higgs scalar field be the inflaton? Possible (Bezrukov & Shaposhnikov, 2008, Masina & Notari 2012)
Fine tuning of Mt to have Higgs ifliinflation
Hamada et al 2013, 2015 A needle in a hay stack?
Still many unknownss !
Future experiments 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 struct ures , with the right spectrum
-- very weak non-gaussianity fNL
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 fNL