Slow roll inflation. 1 What is inflation? 2 Equations of motions for a homogeneous scalar field in an FRW metric

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1 Slow roll infltion Pscl udrevnge October 6, 00 Wht is infltion? Infltion is period of ccelerted expnsion of the universe. Historiclly, it ws invented to solve severl problems: Homogeneity: cusl ptch t the time of recombintion (i.e. t bout 0 5 yers fter the big bng) subtends n ngle of bout on the sky tody. How come the universe is so homogeneous nd isotropic? Relicts: Where re the relicts of phse trnsitions? Monopoles, domin wlls, strings, etc? (if you believe in GUTs)... Infltion s min selling point is the genertion of fluctutions (us!). Equtions of motions for homogeneous sclr field in n FRW metric We use units 8πG = Mp = = h = c. Tke the Einstein Hilbert ction plus sclr sclr field S = d 4 x ( g R + ) φ (φ), () where g = det g µν nd R = R µ µ(g µν ) the Ricci sclr. ry the ction with respect to the metric δs δg µν = 0 () R µν g µνr = T µν (φ), () where the lhs depends only on the metric nd the rhs only on the sclr field. Now let s plug in the homogeneous nd isotropic FRW metric ds = dx µ dx ν g µν (4) ( dr = dt (t) kr + ( r dθ + sin θdφ ) ),(5) where k determines the curvture > 0, closed universe k = < 0, open universe = 0, (sptilly) flt. (6) Let s focus on the flt cse (which is strongly preferred by observtions nywy). Letting Mple/Mthemtic/Mxim perform ll the tensor clculus we get H = ( ) φ + (φ) k, (7) Ḣ = φ + k, (8) where H ȧ nd the first eqution is the Friedmn eqution. The eqution of motion for the sclr field is φ + H φ + φ (φ) = 0, (9)

2 where the φ term is like friction term in clssicl mechnics. We hve three equtions, but only two of them re independent. If we wnt to solve them numericlly, use the Friedmn eqution for consistency of initil conditions s consistency check of the integrtion routine nd evolve the other dynmiclly. Another useful form of the FRW metric is ds = (τ) (dτ d x ), (0) written in conforml time dτ = dt, () sometimes lso clled η, but don t confuse this with the slow roll prmeter η. τ on the other hnd is lso used to denote the opticl depth to the surfce of lst scttering... A photon trvels coordinte distnce τ during conforml time τ. Its geodesic is determined by (ssuming purely rdil motion) ds = 0 dt = (t) dr dr = dt () r = dt = τ. () Accelerted Expnsion Now tht we hve ll bsic definitions nd the equtions of motion, we cn rigorously define wht we men by infltion/ ccelerted expnsion: ä = Ḣ + H = H ( + Ḣ ) = H }{{} H ( ɛ H ). (4) ɛ H The subscript H is for Hubble. ɛ is one of (mny) slow roll prmeters. There is n lterntive definition ɛ in terms of the potentil (see below) which cn be source of confusion. We shll drop the subscript H nd will refer to ll slow roll prmeters defined in terms of H unless noted otherwise. expnsion refers to ä Accelerted > 0 0 < ɛ H <. (5) A useful quntity is the number of efolds defined s dn = Hdt. (6) It is counted bckwrds in time from the end of infltion. In other words, N = 0 is t the end of infltion, nd N = 60 is before the end of infltion. 4 Infltion nd slow roll prmeters How do we get infltion from sclr field? This cn be seen independently of prticulr model in the following wy. Tke eqution (9) for very flt potentil. This should men tht we cn neglect the ccelertion φ. For if the field φ strts off with huge ccelertion φ, the friction term will tke cre of it. φ = H φ 0. (7) Then from the Friedmn eqution (7) we see tht H = const (8) ɛ H 0. (9) For the scle fctor this mens tht = 0 e H(t t0) which explins why infltion is sometimes lso referred to s exponentil expnsion. Most infltionry models function like outlined bove. Introduce the slow roll prmeter η H = φ H φ = Ḧ ḢH. (0) So the requirement tht we cn neglect the term φ compred to H φ is just the requirement tht η. (But note possible exceptions!)

3 In terms of the potentil, ɛ nd η re defined s ɛ ( ) φ, () η φ. () They equl the Hubble slow roll prmeters only if they (nd the higher order ones which I did not bother defining here) re smll: ɛ ɛ H, () η ɛ H + η H, (4) There is higher order slow roll prmeters, defined either s higher derivtives of the potentil or Hubble. Note tht for successful infltion, the only criterion is 0 < ɛ <. The mgnitude of η does not mtter directly. Indirectly, lrge vlues of η re likely to mke ɛ grow s well dɛ dn = ɛ (η ɛ), (5) but there re exceptions. Observble quntities like the sclr nd tensor power spectrum (see upcoming tlk in two weeks) re commonly expressed in terms of Hubble or potentil slow roll prmeters. 5 Slow roll ttrctor for m φ Genericlly, the models of sclr field infltion possess ttrctor solutions. We shll now exmine these in the cse of m φ infltion. We strt off with the Lgrngin L φ = φ m φ, (6) to find the equtions of motion φ + H φ + mφ = 0, (7) H = 6 ( φ + m φ ), (8) Ḣ = φ. (9) Plugging (8) into (7) we obtin or φ + ( φ + m φ ) / φ + m φ = 0, (0) () ( ) / d φ φ dφ = + m φ φ + m φ, () φ where we used φ = φ d dφ φ. We shll explore its phse digrm, see Figure. In the limit φ m φ, () becomes d φ dφ = φ, () φ = φ 0 e φ + φ, (4) where we picked the right semi-plne φ > 0. In other words, the field is rolling exponentilly fst towrds the ttrctor φ = φ. Towrds the end of infltion, the friction term in the eom for φ (9) becomes subdominnt nd we re left with hrmonic oscilltor. φ + m φ = 0. (5) 6 Power lw infltion Power lw infltion is very useful model to benchmrk pproximtion schemes for the computtion of sclr power spectr s its spectrum is exctly solvble (see tlk in two weeks). The potentil is given by = M 4 e ± q p (φ φ0). (6) The scle fctor in this model behves s = t p. (7) Let s construct the potentil from the knowledge of the scle fctor, thereby proving tht this is the so-

4 Φ' Φ Φ 0. () (b) Figure : () Shpe of the potentil (b)phse spce for m φ infltion. Notice the existence of the ttrctor solution nd the oscilltions round the origin. lution. = t p, (8) H = p H 0 t, (9) Ḣ = φ p t H 0 = φ (40) φ ph0 =, (4) t φ = ph 0 ln t, (4) t 0 q ph φ t = t 0 e 0, (4) = H φ = p t H 0 = ph ( 0 t ph 0 ) ph 0 t (44) = M 4 e q H 0 p φ. (45) It goes to show tht knowledge of Hubble s function of time is sufficient to reconstruct the potentil (modulo some integrtion constnts). 7 Monomils Let s turn our ttention to monomil potentils = λφ n. (46) In this cse, it is best to work with the number of efolds N s time vrible. Nottion is φ N φ = H φ. Assume slow roll: φ = φ H φ = nλφn H (47) φ = n φdφ = ndn (48) φ φ = nn. (49) Now does this inflte? Yes: ɛ = Ḣ H = φ H = φ = n 4 N. (50) 8 Generl slow roll For generic potentil, let us ssume tht the friction term in (9) will dominte over φ nd the φ term in the Friedmn eqution (7) becomes smll compred to s well. Then we hve H, (5) H φ + φ 0 φ φ H φ,(5) 4

5 from which we cn get the time dependence of φ(t). Then we use H = ln t = φ φ ln φ H φ ln (5) H φ φ ln (54) φ ln φ = ( φ ln ), (55) = 0 e R dφ ( φ ln ) (56) where we plug in φ(t) to obtin the time dependence of. If we re only interested in H(t), simply use H(t) (φ(t)). (57) 9 Clsses of infltionry models old infltion: Infltion proceeds vi tunneling out of flse vcuum new infltion: Colemn-Weinberg potentils (φ 4 ln φ φ 0 4 φ4 + 4 φ4 0), fine-tuned initil condition: inflton field hs to sit ner mximum chotic infltion: chotic becuse of rbitrry initil conditions k-infltion: non-cnonicl kinetic terms multifield infltion curvton scenrio f(r) theories (conformlly equivlent to sclr fields)... 5