COSMIC INFLATION AND THE REHEATING OF THE UNIVERSE Francisco Torrentí - IFT/UAM Valencia Students Seminars - December 2014
Contents 1. The Friedmann equations 2. Inflation 2.1. The problems of hot Big Bang Theory 2.2. The inflationary idea 2.3. Slow-roll inflation 3. Reheating theory 3.1. Introduction to reheating 3.2. Perturbative reheating 3.3. Preheating 4. Reheating phenomenology 4.1. Primordial gravitational waves 4.2. Reheating the universe from the SM
1. THE FRIEDMANN EQUATIONS
1. The Friedmann equations G µ =8 GT µ GEOMETRY ENERGY CONTENT The Einstein field equations relate the geometry of a spacetime with its energy content. Application to the universe as a whole Friedmann equations Assumption (RHS): ENERGY CONTENT OF THE UNIVERSE Content of the universe is a PERFECT FLUID T µ = 0 B @ 0 0 0 0 0 g ij p 0 1 C A 4
1. The Friedmann equations Assumption (LHS): GEOMETRY OF THE UNIVERSE (Large scales) Homogeneous (Assumption: no preferred points in the universe) Isotropic (at one point) (confirmed by experiments) Homogeneous and isotropic AT ALL POINTS ds 2 = dt 2 FLRW metric: apple a 2 dr 2 (t) 1 Kr 2 + r2 d 2 K =1 K = 1 K =0 a(t): Scale factor 5
1. The Friedmann equations ȧ a The FRIEDMANN equations: 2 = 8 G 3 k a 2 ä a = 4 G 3 ( +3p) Content of the Universe modelized by:! i = p i / i i = a 3(1+! i)! M =0! R =1/3! = 1 Matter Radiation Cosm. Const. j = 8 G 3H 2 j j = M,R, Curvature k c = H 2 a 2 X i i = M + + R + c =1 Now 6
1. The Friedmann equations The Universe goes through different epochs. In each one, a specific kind of fluid dominated the expansion. i = a 3(1+! i) log[a(t)] a(t) =e Ht ΛD a(t) t 1/2 a(t) t 2/3 RD MD log[t]
2. INFLATION
2.1. The problems of hbb theory The three problems of hot Big Bang Theory: a(t) t p p = 1 2, 2 3 1. The horizon problem CMB is incredibly homogeneous. Not enough time for light to propagate and get thermal equilibrium. 1 2 d hor (t) = Z t t i dt 0 a(t 0 ) RD/MD and ti=0 Comoving horizon distance: comoving distance travelled by light since ti to tf dc hor(t dec ) (a dec H dec ) 1 At decoupling time d c hor(t 0 ) (H 0 ) 1 Now T (1) T (2) d c hor (t dec) d c hor (t 0) a 0H 0 a dec H dec 1 9
2.1. The problems of hbb 2. The flatness problem Current observations give c 0,m a(t) + 0,R a 2 (t) c 0. Unstable point in Friedmann equations! = 3k 8 G a(t) " c (t) " We need incredible fine-tuning! c (t pl ) 10 60 3. The primordial monopole problem Not observed cosmic relics predicted by GUT models 10
2.2. The inflationary idea An early phase of exponential expansion can solve the three problems at once a(t) =e Ht Alan Guth The idea (but model failed) Andrei Linde First successful inflationary model 11
2.2. The inflationary idea a(t) =e Ht Number of e-folds: N log a a i = Ht 1. Horizon problem: Due to the inflationary epoch, all points in the CMB were causally connected in the past. 2. Flatness problem: During inflation, Ωk=0 is an attractor point. c 0, a 2 (t) = 3. Primordial monopole problem: 3k 8 G a(t) " c(t) # Inflation washes out any cosmic relics. THE THREE PROBLEMS ARE SOLVED WITH N 60 12
2.3. Slow-roll inflation Definition of INFLATION: d 2 a dt 2 =0 HOW TO IMPLEMENT IT? Action of the inflaton: S = Z d 4 x p g 1 2 @ µ @ µ V ( ) = (t) Inflationary potential H(t) ȧ a E.o.m: Field and Friedmann equations: +3H(t) @V ( ) + @ H 2 = 1 3m 2 p =0 1 2 + V ( ) 2 Energetic content: = 1 2 + V ( ) 2 p = 1 2 V ( ) 2 13
2.3. Slow-roll inflation First requirement: V ( ) >> 1 2 2 =3 Potential energy dominates over kinetic 1 2 2 V ( ) << 1 First SLOW-ROLL parameter 1 d Fr. eqns: 2 a H 2 V ( ) a dt 2 +V ( ) INFLATION! 3m 2 p 3m 2 p a(t) a i e R t H( )dt0 Energy:! p = 1 2 2 V ( ) 1 2 2 + V ( ) 1+2 3 (Quasi) de Sitter 14
2.3. Slow-roll inflation Second requirement: We must ensure that ε<<1 is sustained for at least 60 e-folds or more. The field must not accelerate ( ""! ""! "") +3H(t) + @V ( ) @ =0 We need: << 3H, V 0 ( ) H << 1 Second SLOW-ROLL parameter SLOW-ROLL CONDITIONS IN TERMS OF POTENTIAL: V m2 p 2 V 0 ( V ) V 2 V m 2 p ( V V ) V 00 V, << 1 V, V << 1 15
2.3. Slow-roll inflation Working example: V ( )= 1 2 m2 2 END of inflation: = 2 m 2 p 2 = 2 m 2 p 2 = end = M P 2 p M P 3.5 the field starts to oscillate around the minimum of its potential 16
3. REHEATING (getting the bang from the Big Bang)
3.1. Introduction to reheating What is the origin of all matter and radiation present in our universe today? During inflation, the universe is empty and cold S 0 M 0 T 0 But now M 10 23 M S 10 89 (Inflation dilutes any relic species left from a hypohetical earlier period of the universe) (and T >> 0 in the early universe ) We need to reheat the universe after inflation 18
3.1. Introduction to reheating Inflation Reheating Hot Big Bang theory V ( ) (dominant energy) energy transfer to created particles (the universe gets hot) T r final reheating temperature the universe starts to get cold again.. 19
3.1. Introduction to reheating Model for reheating: L = 1 2 (@ µ ) 2 other V ( )+ 1 2 (@ + µ ) 2 interaction 1 2 m2 2 1 fields inflaton-fields 2 g2 2 2 Inflaton Scalar field Interaction term V ( )= 1 2 m2 2 (Parabolic potential) Inflation equation (neglecting interaction): (t) +3H(t) + m 2 =0 (+ Friedmann eqn.) (t) (t) sinmt a(t) t 2/3 Matter-dominated (t) = 0 t OSCILLATIONS AROUND THE MINIMUM OF THE POTENTIAL t 20
3.2. Perturbative reheating L = 1 2 (@ µ ) 2 V ( )+ 1 2 (@ µ ) 2 1 2 m2 2 Inflaton Scalar + (i µ 2 @ µ m ) h Fermion 1 g 2 g2 2 2 Interaction terms Inflaton couples (weakly) to other particles, and then it decays: (! )= g2 8 m (! ) = h2 m 8 = ( + i i ) + (! ) = h2 eff m 8 new friction term h 2 eff = X i h 2 i + g2 i m Inflaton effective e.o.m: Solution +3H(t) + + m 2 =0 (t) (t) sinmt (t) = 0 e 1 2 (3H+ )t d dt ( a3 )= a 3 d dt (n a3 )= n a 3 comoving inflaton energy density and particle number decays into particles 21
3.2. Perturbative reheating (1) << H = 2 3t total comoving energy and number of inflaton particles is conserved (2) H H 1 1 Inflaton decays suddenly into particles Age of the universe Inflaton lifetime It may take many many inflation oscillations to get to (2). (we need to wait until the Universe is old enough!) When (2) arrives, reheating is instantaneous. It realises all into, and in an exponential burst of energy. (3) Created particles interact among themselves THERMALIZATION to a reheating temperature T r 22
3.2. Perturbative reheating Estimation of the reheating temperature: t r 2 3 1 Reheating time For our chaotic model: T rh 0.1 p M p = h2 eff m 8 m 10 13 GeV h eff apple 10 3 T rh apple 10 11 GeV Reheating temperature Problems: 1) Low temperature. 2) In some models, always. H> BUT: Inflaton is not composed of individual inflaton quanta, it is a coherently oscillating field with large amplitude. We need new formalism! 23
3.3. Preheating New formalism: Particle production in the presence of strong background fields. L = 1 2 (@ µ ) 2 V ( )+ 1 2 (@ µ ) 2 1 2 m2 2 1 2 g2 2 2 Inflaton Scalar field Interaction term Minkowski spacetime (for the moment) (t) = sinmt r 2 + m 2 (t) m 2 (t)= 0 m 2 (t) =m 2 + g 2 2 sin 2 mt A free field with time-dependent mass! In momentum space f k : (field mode) Kofman, Linde & Starobinsky, Towards the theory of reheating after inflation (1997) d 2 f k dt 2 +!2 k(t)f k =0 Field-mode equation! 2 k(t) =k 2 + m 2 (t) Time-dependent frequency 24
3.3. Preheating It can be written as a Mathieu equation: d 2 f k dt 2 +(A k 2q cos(2mt))f k =0 A k = k2 + m 2 m 2 +2q 4m 2 q = g2 2 For certain values of (A k,q), the solution is unstable (exponential growth!): f k e µ kmt Re[µ k ]=0 Re[µ k ] > 0 n k (t) = 1 f 2! k 2 +! k k 2 f k 2 1 2 Particle number definition n k (t) e 2µ kmt Exponential growth of particle number!! (Just after inflation ends) instability chart for Mathieu equation 25
3.3. Preheating n k (t) e 2µ kmt ln n k (t) =2µ k mt This equations has two regimes: (for given mode k) q < 1 q >> 1 q = 0.1 q = 2000 Narrow resonance Continuous growth of particle number (t) Broad resonance Explosive particle creation when inflaton crosses the minimum of its potential. t 26
3.3. Preheating n j+1 Kofman, Linde & Starobinsky: analytical model k = f(n j k ) q>>1 PROPERTIES: 1. Step-like. Particles are produced only when inflation crosses zero. (when effective frequency changes rapidly) Adiabatic Non-adiabatic! k! 2 k >> 1 Non-adiabatic regime 2. Non-perturbative. n 1 k / e k2 g 3. Infrared effect. n j+1 k ' n j k ' 0 (k!1) 4. We can create particles with mass greater than the inflaton mass (forbidden in perturbative reheating). 5. Band structure. Only some momenta are excited (also valid for other potentials). 27
3.3. Preheating Including expansion The natural momentum redshifts: k a(t) 6. Stochastic preheating. The steps go down ~25% times) 7. Redshifts. For a given momentum, the parameter q redshifts, entering eventually to the narrow resonance regime (q<1) q = g2 2 (t) 4m 2 / 1 t 2 8. Behaviour Bosons: Fermions: n k (t) e 2µ kmt n k (t) < 1 (due to exclusion principle) Greene, Kofman (2000) 28
3.3. Preheating Analytical approach to reheating has limitations. It does not take into account: Backscattering of the created particles to the inflaton field and expansion rate of the universe. Spatial distributions of fields (structure formation). To go beyond, we need LATTICE TECHNIQUES. Solve the differential field equations numerically in a finite box. Quantum expectation values are spatial averages. (Example: LATTICEEASY program) 29
4. REHEATING PHENOMENOLOGY
4.1. Primordial gravitational waves Preheating in the early universe is a very violent phenomenon (huge masses colliding at nearly the speed of light) A source of primordial GRAVITATIONAL WAVES Can these GW be detected by any experiment in the future? 31
4.1. Primordial gravitational waves GW spectrum characterized by amplitude and frequency. Inflationary GW: Scale-invariant High-energy (p)reheating (p)reheating GW: Non scale-invariant Low-energy (p)reheating García-Bellido, Figueroa (2007) 32
4.2. Reheating the universe from the SM In order to do reheating phenomenology, we need to assume: 1. An inflationary model. 2. A set of couplings. ASSUMPTIONS: 1. The Standard Model is approximately correct up to inflationary energies 2. EW vacuum is stable up to inflationary energies 3. The Higgs does nos couple to the inflation (which is a beyond-the-sm particle) 33
4.2. Reheating the universe from the SM The Higgs starts to oscillate around the minimum of its potential a short time after inflation. It can reheat the universe V (h) (h) 4 h4 running coupling constant Well-known SM couplings to the other particles Fermions (quarks, leptons) Gauge bosons (W, Z, γ) Already studied Enqvist, Figueroa, Meriniemi (2012) (we re working on it!) 34
Summary Inflation provides a natural solution to the horizon and flatness problems of classical hot Big Bang theory (and also explains large-scale structure of the Universe). Reheating is a key process of the primordial universe. It requires non-ordinary techniques of QFT outside the equilibrium. GW emitted during reheating constitute a possible window to the primordial universe. Reheating takes place even in the Standard Model. 35
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