Origin of density fluctuations in extended inflation

Fermi National Accelerator Laboratory FERMILAB-Pub-90/II6-A (May 1990) OILIGIN OF DENSITY EXTENDED IN Edward W. Kolb, 1,2 David FLUCTUATIONS INFLATION S. Salopek, 1 and Michael S. Turner 1'_'3 ..2 Astrophysics *NASA/Fermilab Fermi National Accelerator Batavia, 2Department Enrico Illinois Institute, Chicago, Fermi t/- of Chicago -,j i ,-9 fT, <) C_ :) of Physics The Chicago, Astrophysics University 6O6ST-14ss Illinois Institute, Center Laboratory and The S Department Enrico //, 60510-0500 of Astronomy Fermi s University of Chicago aoasT-14ss IL Abstract c: r-3 £ We calculate the density fluctuations--both curvature and isocurvature--- t..J LL that arise inflation tions due based arise eral have to quantum upon arise fluctuations a nonscale-invariant spectrum, acceptable constant and to a very due to the inflaton theory, ations arise in these fields the massless particles associated inflation frame curvature too. The density The inflation potential perturbations fluctuations and closely are other resembles formula 0 eL _ c t/ L,,w and field conformal When slow-rollover viewed inflation for the amplitude of applies. .'MI, .q_ Operated by Universities Research Association Inc. under contract with the United E u !.L. _" of extended Einstein is emphasized. the usual fluctu- waves models V = _'' 7 L _ the' of the Brans-Dicke of the any massless isocurvature at more realistic importance that to tune of gravitational with excitations curvature extended then in gen- perturbations If there or au ilion, field, having >,. fluctua- an amplitude without Production attempts analyzed. in calculating an exponential an axion Several are also in this fra_ne, with e.g., Curvature and can have (_" of extended in the Brans-Dicke field are subdominaat. in the also discussed. value. model theory. interesting small fidds are in a simple the 3ordan-Brans-Dicke due to quantum is cosmologically coupling fluctuations States Department of Energy .J L 0 o I. INTRODUCTION Extended inflation flation. In old symmetry inflation breaking; turbations gauge associated while gravity the value In extended of some field that up in a false the but is hung not "gravitational state---a so that the vacuum bubble, cleation rate). At trapped in the false probability e(t) (per ,,_ the start gravitational constant during extended and Density during While the slowly, phase it is possible interesting, be very perturbations it seems few bubbles there that the uncertain: these density will be too many Because of the in slow-rollover arise as remnants large variation of the of the bubbles remains transition been perturbations that arise due efllcient. are nucleated elsewhere. to the bubbles _ are there will turns on nucleation the and by bubble on rapidly, with by the of time addressed size; if bubble back gravitational very that nu- exit" is accomplished to be consistent varying a true tr field as a power turns power exponentially, phase inflation--be nucleation the is circumvented have bubbles and the perturbations interesting a large so the only grows hung potential. of a "graceful old inflation If the bubble of cosmologically lack The P is the bubble unity Reheating gravitational of nucleating (here plagued inflation. reheating transition; The al- 4 it gets grow time) to order upon of its scalar e is small the scale factor during certainly inflation to a significantly. not time to be is determined the energy factor with based rapidly--as Hubble bubbles. that inflation, should--unlike per field so because vacuum weakly Jordan-Brans-Dicke. minimum scale increases vacuum constant: e(t) increases the been determines very per- inflation constant tr--does false it increases Universe of the and when have field--vary expands volume of extended of true field infiaton "tying" being that density to be a very again gravitational not global, that _ t', had 2 in- for GUT to achieve inflation field to the responsible for the once Brans-Dicke Universe Hubble r/H'(t) variation coUisions the but field theory inflaton the vacuum; nucleation decreases that It is crucial to a radiation-dominated g simplest exponentially---owing constant." by the the local, up, inflaton of the to as the field in order 3 of extended inflation--the vacuum _ field refer Higgs thereby the value field, the old 1 and slow-roUover it is possible breaking, where we shall the inflation Models on both inflation, level, it is crucial precipitates of time, small scalar inflation was in slow-rollover symmetry theories constant--which field In extended GUT variation inflaton phase transition. ternative ends the singlet. with cosmological While interesting of an acceptably coupled by is a very isotropy of the cosmic microwave background is just so, it is not possible acceptable. In any quantum we will focus in the various comparison, in slow-rollover lead dominant to the curvature the inflaton not quite and field, s Curvature scale-invariant fluctuations it is these have in the generalized constant). needs to be set small fluctuations Most to ensure that inflation. coupling For constant but and they arise value controls no dimensionless they to field that inflation, field whose importantly, due (Harrison-Zel'dovich) spectrum), (the rate arise in the inflaton arise in extended field that extended a very small a power-law value during nucleation and observationally fluctuations scale-invariant Brans-Dicke of the gravitational the bubble interesting density also necessitate value even on the perturbations (they to a very to be both perturbations: that s Unless fields in the theory inflation density perturbations, (CMBR). for relic bubbles case, fluctuations radiation for they are due to the parameter are of an acceptable--or interesting--size. In this paper we compute Einstein frame, extended the frame inflation these perturbations where closely the role of the inflaton derived for curvature are directly particles, arise if there Finally, We production are other we analyze II. also massless The theory derives address fldds recent from the the and in the potential: at realistic FIELD the formulas in slow-roLlover of massless fluctuations as an models axlon field inflation Brans-Dicke that or an can ilion. of extended inflation. model a of extended inflation. FLUCTUATIONS eztended-inflation original such In this frame, 7 Moreover, isocurvature theory, to the with the Brans-Dicke production the transformation is constant. production the attempts BRANS-DICKE we consider inflation, and graviton a. Some For simplicity slow-roUover of gravitons, several constant with an exponential fluctuations applicable. the the gravitational resembles playing by a conformsl La-Steinhardt basics action 16_r + 1--_g _' _ + £;=,,tt_ , (2.1) where q_ = 2¢c_b2/_. This theory serves only as a toy model since the temperature 2 fluctuations that in the CMBR _ _ 20, s while model will serve field part fields: sits of the _m,tt,r quietly vacuum solar-system tests distribution of the theory the salient of bubble require features sizes requires w _ 500. 8 However, of the density fluctuations this that inflation. matter matter arise due to the well to illustrate arise in extended The that = (O_r)_/2 in the energy Lagrangian density false includes - V(cr)+.... vacuum, that the inflaton During and it contributes field extended affects the to the energy _r and inflation dynamics only density all the other inflaton through of the the Universe: PvAc = V(_ = 0) - M 4. In Eq. (2.1) coupling _ = -1/4w. be misleading: kinetic --7_/16_'GN derivatives Fluctuations _ in terms of a massless We warn The of the usual parts, we have expressed the reader term term, in • are is not tensor related rewriting for _b appears gravity of the metric that (from to those scalar field ¢ with curvature the action canonical, but canonical. 7_) may in terms because Moreover, be shifted of ¢ can of the absence by integrating to the _ kinetic fact will be of some The equation H 2 During evolve (2.2) utility of motion +3H_- extended term. in ¢ by: _ = v/_-_-/__¢. This by later. for • and the Friedmann are 8,_3.(, 0_ 3p); _72_ = 2w+ ----a2 = 3"-'_-+ inflation equation p -_ pvAc 6 _2 -- M 4 , p - (2.3) H_. -pvAc, and the scale factor a and as =(0 = =0(1+ m) _+_/' _ =o(B0=+_/' (form >>1), • (t) where = _0(1 B is defined B¢_/2 Eq. (2.4) + B$)' in terms = 1M,, pa_ implies h =__ -a = =_ _0B'$' that P = _(6w during (_ + 1/2)B l+Bt of w, M, (for B$ :>> 1), and w -t- 1/2 t of _ at the start of inflation + 5)(2w q- 3) 32_-_ _ " inflation, _ the value (2.4) the expansion by (2.5) rate (forBt >>1). is time dependent: (2.6) Since there the value is little of • variation in • during at the end of inflation the matter is approximately 2 t_M the time t, corresponds ill-defined time the _ field makes nucleation of Coleman-De to a temperature to the end of the order M. regimes, to its value today: rap: =_ t._ _.,,-#-_, of extended the transition Luccia equal 4 ,_.__c_v _=__p,___oB_t_ = _,, where or radiation-dominated bubbles, inflation. Around to the true vacuum and bubble The quantity collisions (2.7) this slightly through reheat p is a dimensionless the rapid the Universe constant of order m unity and for _o ::_ 1, ? --_ _/3/8_'. b. Production •The physical the Universe: that crossed = where a(to) taking the wavelength _ph_ of a linear oc _(t). outside the Consider horizon o/ fluctuations perturbation grows a fluctuation at time t during with of present extended the scale physical inflation; factor wavelength )_ is given by M .(t.) H__(t)' 2.75K act) reheat temperature = 1 is the scale a(t.)/a(t) _Mp¢ __ Cryt) --- (2.s) is assumed factor of today. to be Writing _'+1/2 it foUows M, aCto)/aCte _ -- _Mpc Mpc ) = M/2.75 K, and -_ _Mpcl0 as GeV -1 and that 10-_SPMP!(te/t)_-I/2; (2.9) It is interesting be reserved to exhibit the effective for the present the fluctuation of wavelength G GN t for w = 10 and value value of Newton's A went outside 10 _5 M = 1014 GeV, G/GN of the gravitational constant) the of the scale e(t) OC AMpc that 4/(_-1/2). is leaving the horizon of epoch wiU when (2.10) ; lt_410.21| _v "_Mpc" nucleation rate per Hubblevolume _(t) : r/m terms as a function G (GN horizon: XMpc "_ coupling In addition, since the bubble __ (t/t,)', we can express _(t)in at time t instead of t, (2.11) As bubble nucleation cross outside of scales the horizon: (A(ln proportional to (w - exponentially bound to w from on," say e increases From the relation A)) that varies Now "switches cross outside 1/2)/4. with bubble implies one that can 0.1 to I, a range of scales we see that the logarithmic the horizon This w, and above from easily as bubble nucleation the of bubble range appreciate why interval commences sizes there is expected is an upper nucleation. let'scompute the horizon-crossingamplitude of a fluctuationin the Brans- Dicke field(i.e., when it crossesoutside the horizon during extended inflation).We estimate itsamplitude by setting the fluctuationamplitude in the equivalent field_b equal to the value of H/2a" at the epoch of horizon crossing: _]_A _ ('_a+lJ/(_a-l/') "_ w-x12105°l('_-x12) P (_p/) I "mpc 'l('--al') (2.12) • (Since _b is only minimally coupled in the limit that _o >> 1, 5_b = H/2_r is only technicallycorrectin thislimit.)We see that the sizeof the fluctuationcan be large--just likethe value of m_,Jq_--and for the same reason: During extended inflation, can be very small compared exponential inflation, is independent to itspresent value. Moreover, we see that in the limit of i.e., _ _ of A--as 1, the spectrum one would decreases to zero as w _ expect. inflation that super-horizon-sized remains that constant extended it is simple to show in amplitude. Once a fluctuation that present matter-dominated back inside value both epoch by a(tR) using the grow _'_ horizon. I0 (2.3), at the same (A > 13 Mpc), where For fluctuations and roughly their ratio from re-enter Eq. the the decrease tH is the that thus that q_a/q_ remains (2.3) when constant constant.) that horizon back Likewise, remain in their time cross epochs consta_ut. in _ also it follows that and to show matter-dominated fluctuations slightly it is simple rate, and of @ remains horizon, -s _Mpc, a Eq. radiation- For fluctuations a-l(t). the the super-horizon-sized in @ re-enters as as t_; in • grow the pzecisdy, decreases til today is given grows During inflation (More of the @ fluctuations of @-fieId fluctuations fluctuations plitude crossed _ in amplitude. foUow the amplitude "flat"--that gener_ relativity). c. Evolution extended Finally, becomes oo (in the limit of w --_o% the Brans-Dicke field• freezes out and the theory becomes During of fluctuations its am- during the amplitude the inside un- fluctuation the horizon during a(_s) the radiation-dominated (A < 13 Mpc), the decrease in amplitude is -_ 10 -s _Mp¢. Using these Brans-Dicke while facts field. Again, of present = _A T 1¢_-12. -v "kMpc less than correspond correspond _Mpc fluctuations be discussed. of the system tests is truly a toy Brans-Dicke that of the present ,I, particles; fluctuations fldd w the (2.14) fluctuations are of on the largest fleld--and fluctuations relativity. field massive, A _ 3000 in the gravitational any should temperature pulsar, which (2.15) horizon, while causing millisecond of general the effective one might and certainly or even needs massive resulting have and the fluctuations _ _> 3000 constant. the Mpc, they of constraints--remain effects, in the model to including CMBR, various affecting precision we are modiflcation--perhaps unstable---we in The consequences numerous affecting because Mpc, scales, fluctuations possibly However, most source expect amplitude. to try to analyze known and ("'+l)l('_-ll')" solar- considering making the wiU not consider them here. Since similar of M "Mpc 13 Mpc M = 1014 GeV, d. Curvature p/_, about 13 Mpc (A _ 13Mpc). :l density, model than about in the (A < 13Mpc), in the Brans-Dicke energy timing 2., Brans-Dicke contributing values fluctuations less than ) ("_+l)l(_'-ll')_ for _a = 10 and of the , to massless to spatial greater p (p-_pt x.2 J.v amplitude wavelength wavelength interesting e.g., la-11_ -" On scales further for the amplitude; 8q_ of present lo-s+s°l("-11') we see that T the present For fluctuations = _a-ll' interesting such we can compute for fluctuations T6":_ the epoch that of Newtonian fluctuations As we shall the as the Jordan fluctuatior_ production gravity in • give see this is essential]y of curvature conformaIfrarne, because is proportional rise to density correct. fluctuations the effective to Gp and Gp fluctuations While in the frame gravitationai oc of a it is tempting of Eq. (2.1), constant is varying and coupled, because the fluctuating such a procedure The surest way rescaled frame where HiIbert to the to is very analyze the form. This frame Einstein conformal fieldmthe field _mis not minimally suspect. curvature gravitational fluctuations part is known frame Brans-Dicke as the is to work of the action Einstein is accomplished takes conformal by the in a conformally the usual .frame. following Einstein- The rescaling conformal transfor- mation: _ = n-'(t)g_, where n' = _,/_, qJ02 = (2t# + 3)m_,t/16z'. In the Einstein lfi_'GA, + exp(where overline gravitational vacuum indicates so that $/q_o)g_0_,#O,,# term frame a false-vacuum t >> t,, when q_ _ m_t , the conformal become energy equivalent. monotonically (Since to 1.) During a(f) that and action is given by: the inflaton • grows only effect (. Note too extended f_ --, 1, so that with inflation value field is anchored the exp(-2_/_0)M factor (2.1z) ; GN = rni,_ is the present can be neglected; to contribute frames the - exp(-2@/@0)M'] We will assume its kinetic frame (2.16) + the Einstein constant. ) = )01n[_/-_,]; time, the it is simple the in the of the that inflaton at late Jordan In the (w/2 + 3/4) (1 +ct-) imally = eq_o/m2p,. Einstein coupled equation + 3H_t Assuming that frame scalar of motion - _ w/2 + 3/4 __ -(dV/d$)/3[I. justifies the slow-roll times, conformal factor to show that 9 decreases (for Ct >> 1); These the field (2.18) facts will be of use shortly. Brans-Dicke with with field qt takes a potential, on the Y(9) appearance of a min- = M 4 exp(-2qJ/@o). The for gt is familiar: V'gt + dV()) d--"'_- = 0. the _ field is homogeneous, d_/d[ is and Einstein (1+ c_) = (1+ Bty; 2B/C false a0(1 + C{)_'12+st4; = = where of the (It is simple approximation.) (2.19) its evolution to show That that is, when is just _//I_ extended that of a "slow .-_ w-_, inflation which roller:" for w >> 1 is viewed from the Einstein frame, the rescMed Brans-Dicke Because we can _ behaves compute by taking scale "HOR") the (6p) 7 HOR outside d_-_dE the amplitude coincide Remembering evaluate Eq. in--is (2.20) equal easily f[2 = 81rV/3m_ of order worked _ from are given • --we t and inflation is given s When (denoted by by are to be evaluated well after the curvature when extended fluctuations the scale inflation in both in the Jordan frame---which computed in the Einstein frame---where computed. dV(_)/d_ = -2V/_o, it is simple to ' 6 p-21(_,-x12) ._(,,,.,+,)/(,.,-,/2)_ w this is precisely ,/(,-1/2) _Mpc the same (2.21) • as the fluctuation (6P/P)HOR by R/2_. those and was the What assumed assumption would that have that amplitude been 6_ = H/2_? quantum the in the fact J if we had The fluctuations in • are in _b by Eq. (2.1): (2.22) that 6q' = _0(60/O)--which find that L-_---_ fluctuations outcome = this and 6@= scale inflation, in amplitude _ \ WbpI / frame From extended fluctuations 1° in the Jordan computed ( --M in computing field to that 10s°/('_-1/_)4_" X in the quantum for (SP/P)HOR: "" Implicit after so that _ 9, cf. Eq. (2.12). is as per usual, for slow-roUover and unambiguously HOR Up to a factor from Moreover, is, the fluctuation is most that result side of Eq. (2.20) frames that that gravity (2.20) the Einstein we are interested and because dV--_/'; inflation. That field on that during the same! with role of the inflaton. the horizon the horizon frames potential, 3R s '_- on the right and ' is what inside off an exponetial developed of the fluctuation H' -- the Jordan are back amplitude the fluctuations of the machinery A crosses inttation like an inflaton the curvature the quantities crossed slo_-rollowr iqeld !P playLng just advantage a given where it resembles _+ 2--_" 8 follows from the definition of Thus, only in the limit w >> 1 is the result consistent with amplitude of H/27r canonical scalar kinetic field effectively field the result applies assuming and = H/27r coupled This massless • its limit 6_a = tt/27r The fluctuation scalar field is a minimally potential In the Jordan w --* oo limit, is the assuming = Ar/27r. frame because applies. in the 6_ coupled, conformal term, for w --_ oo: large it is still is an additional the curvature that quantum Einstein frame and thus in which coupled is very conformal _b the with flat it is frame the = H/27r only methods for two agree. there Note 6_ frame to a minimally kinetic Thus if w is not this case In the minimally estimating/_ Even only canonical massless. technically ¢ with applies term. with ¢ is only in the Einstein in the Jordan the to compute correction 6¢ in the to 6_b which arises Einstein from frame; in the interaction of that to scalar. power-law fluctuations The amplitude possible spectrum in _, given of these of curvature by Eq. (2.21), fluctuations is very fluctuations becomes flatter arise as w becomes for w = 10 and interesting: due M large. : 1014 GeV, 6p \/(P_ __ 4 X _,,.., ln-'_ _Mp© " o.21 (2.24) HOR The associated corresponding temperature to scales 6T) on large A ,,_ 100 Mpc to 1000 Mpc, 1 _'-- fluctuations angular scales, are given by 11 /_-2 1050/(W--1/"_[2"_--_ \T'rl, p| ]_) (,w-+-l,/(w--1/2) (-M p-2/(_-1/2, ×1o,/(._l/,)(aoh)_,/(._l/, {_o \ l° ) (where we use the fact that 34.4"(i20h)AMp¢) are certainly quadrupole anisotropy, can remedy on smaller not too too problem, scales. That large < while bubble bubbles scale ; A corresponds (2.25) to an angular For w = 10 and M = 1014 GeV, large 6T/T this many a comoving at recombination). fluctuations 0 ,,_ 1° to 180 °, to be consistent 3 x 10 -5. still requires Increasing predicting nucleation that with occur w must the limit w or decreasing M to the slightly of an interesting enough be less than of 6 = the temperature currents fluctuations rapidly size so that about there 20. s This size are fact together with the to imply that the fluctuations invariant. The scale may scales. desire fact scales that be of some (According M with will be both the amplitude importance to some, to be consistent motions, to associate of the density in that it boosts with the observed correlation of order of an interesting a scale-lnvariant the cluster-cluster a scale GUT magnitude the fluctuation large-scale lacks scale increases amplitude sufficient voids with on large power structure--large-scale and the large seems and not scale perturbations spectrum function, the on large streaming seen in the CfA red shift survey.) Again, certainly we remind requires significant that the reader modification. variation this simple realistic model fluctuations that the model However, since in the gravitational toy model with of extended are most In Section IV we will realistic models. III. w _< 20 would and analyze Finally, most FLUCTUATIONS resting in the inflaton tunnels nucleation that inflaton IN vacuum due inflation. However, suppressed 0_ M 2, is much effective previous section, fluctuations value M of the must density larger Planck the find them! ,,, the during small; less will certainly at more Quantum reason: inflation. than rapt vein, quietly inflation the arise from the in the perturbations before the fluctuations The mass G-'/' As we have to ensure that m_,_ cannot end of in the of the o"field, temperature, where ra;,'_ - in a similar 10 frame. role, of extended Gibbons-Hawking vf_(ra_/M) mass attempts in the (r field long for a very simple than Einstein a very passive end fluctuations fluctuations curvature _r Here we are interested be significantly are acceptably At expect of a more that in the is FIELDS field plays 0. inflation features in several which one might emphasize addressed field we will not Very roughly, m,/(lt/21r) is the _r = of bubbles. to quantum field are highly ///_.a-. state, mimic OTHER the inflaton to the true vacuum; arise m 2. = V"(0) Dicke false inflation and thermalization might extended of extended at least fluctuations a. Inflaton During inflation, we again a toy model of extended during unambiguously curvature is truly the key feature constant inflation. directly considered seen the be too Taw = < ms, in the Bransmuch less than rapt. than Gibbons-Hawking the Thus, highly suppressed. Jordan frame, address the mass of the inflaton temperature, Note that the fluctuations addressed the kinetic kinetic term, Any nearly of order H/2_" contributed these massless imprinted give it on rise to isocurvature isocurvature axion To analyze fluctuations fields are have their usual by factors tors of f/4 = exp(-2_/_0).) Since neous degrees the expansion Consider fo exp(-iS), where The axion before, or early getically effects become axion degree important, The misalignment and _- m_/(H/2_). will have fluctuations that the energy than that curvature occurs density of the inflaton, fluctuations, in much interesting appropriate and the example, but same way it we will treat the to work in the Jordan potential Einstein frame terms, kinetic of f/2 = exp(-_/_0) the fluctuations the but where in the and potential terms of G is only frame, the grav- terms we are interested variation field _ that after which carries _ obtains fo = (1_1) is the vacuum during at 6 = 0. Within coherent scalar of freedom on, during favored the of freedom, breaking, metry. field matter by fac- in do not involve of interest in so far as rate H. a complex symmetry case to the This kinetic (In rescaled it affects a redefined mr./(fI/2_) In the significantly it is most is varying. the gravitational we to carefully fluctuations. constant Lagrangian Were by using i.e., much smaller 12 As a simple are in the _ field in the in the theory all scales. fluctuations. inflation. these matter itational upon field fields field, i.e., m 2 << _2 will not contribute does in slow-roUover where massless by that field is subdominant, fluctuations instead scalar frame find that larger in the inflaton term for _ is canonical. we would b. Other of magnitude the fluctuations in the ¢ field in the Einstein which has a canonical orders and so fluctuations we have as in this frame is several inflation. and of//1 with production, to #12.is Fluctuations produced and correspond Later, region, the with portional Since 6 develops inflationary a vacuum expectation is 8. Suppose inflation. that around 6 will take minimum density _on 11 that value on some of axions perturbations: PQ sym- value is ener- instanton m,r4 and minimum arbitrary of the axion = occurs of 1 GeV, about (_) breaking no particular of depth sponta- breaks PQ symmetry in 6 will lead to fluctuations to isocurvature expectation a temperature a potential number and undergoes value 8 is massless, eventual the PQ charge value potential produced leads being in the number (5_/_o) 61 _ 0. to pro- of a_ons _- 2(_0/6_). Quantum fluctuations in where H is value horizon. During of the axion Hubble extended to _, cf. Eq. (2.9). rise to quantum parameter inflation Bringing perturbations _ give when the H o_ (w + 1/2)/t, this all together, is given fluctuations scale in 0: 50_ __ H/fa, _ crossed and we have outside previously we find that the spectrum the related t ofisocurvature by: ~1_ M2 \ na /_ 01 f, met '_ When the these of the same - is matter-dominated isocurvature amplitude. perturbations ( M _(_+1/2)/('-1/_) _---_p/j M _1-1 _ 102S/(_-1/2) Universe the horizon, t ' and perturbations In slow-ro]]over 12 is identical a given scale will give inflation • 1/(_-1/_) "_Mpc has crossed rise to density the spectrum back of fluctuations inside perturbations of isocurvature to Eq. (3.1) in the limit that w :_ 1. For M/f,, 10, and M -- 1014 GeV the amplitude (3.1) • axlon 01 "_ O(1), is 0.11 6_" ""3 x 10 -'_ which is definitely Any field can also "ilion" that (3.2) cosmologically interesting. could isocurvature develop do so in extended field, asymmetry. which tuations case is In general, was scale dence because the power law rather A second in a particular 14 In this fluctuations, inflation. ilion Hubble than model in extended parameter gravitational tic perturbations) derived long after extended results we derive g_--the ones result rise in isocurvature the spectrum inflation they is not is provided gives inflation constant baryon baryon-number of isocurvature will have during to the by the some inflation scale flucdepen- (inflation is perturbations wave perturbations it is most for slow-rollover example inflation exponential). c. Gra_iton To analyze interesting in slow-rollover of baryogenesis fluctuations in slow-rollover invariant; fluctuations appropriate inflation inflation for tensor Jordan fluctuations we are interested transverse, to work in the Einstein is are directly the (the and in _ in. 12 applicable. Einstein at late traceless framer As mentioned frames times coincide are identical tensor met- as the results previously, so that to those the in The dimensionless outside amplitude the horizon during of a gravitational extended inflation wave perturbation as it crosses is B h'A --_ _, (3.3) _Ttpl where £r is to be evaluated mode is outside horizon after amplitude the crossing; horizon extended of the at horizon its crossing amplitude inflation. during remains It is a simple tensor-metric extended perturbation inflation. constant matter hA at until to evaluate Once it re-enters Eq. (3.3) the for the post-extended-inflation horizon x7 10so/(,_-1/2) ,_ For w = 10 and M - p-2/(_-112) 1014 GeV we find (____pl) "_Mpe i 2/(ta-1/2) 2/(ta-l/2) ' (3.4) that (3.5) hA "_ 5 x 10 -0 1"qMpco.21* The gravitational to a quadrupole parameters limits wave mode anisotropy above just re-entering in the CMBR corresponds to the quadrupole is given wave mode to _fT/T the horizon of amplitude ,-, 3 x 10-S--very today, A -,, 3000 gT/T ,,_ hA, which close Mpc, leads for to the current upper just horizon crossing inside crossing the ratio the horizon of energy to that density of the total in the gray- energy density by AdpGw ' ; Pror / dA "_ __4 3:r ( _______1) for the mode that flA~_OMv¢ is just crossing ap .,la = PCmT For the parameters above, It is straightforward they the anisotropy. At post-extended-inflation Rational the extend from (3.6) inside the horizon today (A ,-_ 3000 Mpc) (M '_ I01141(_'-112) P -41(_'-I12)_ \rapt this is • (3.7) / NA~zoooMp_ " 10 -I°. to compute the spectrum $ ,-_ 10 -2s (metM) Mpc--the 13 of relic gravitational mode that re-entered waves the today; is horizon just after reheating--to today. and The fraction oc A2+4/('-I/2) fix ¢x A41(_'-112) it not to be greater GeV would is that oc 41nt/3(2_a variation and theories and and from that equations 'i'+3H,i,- a potential of the dimensions, There field to acquire varying, rn, and M both provide in so doing of motion V_'_ a term for a(t) - ,-- with general The rub varies as G -1 = the most were some 1014 stringent mechanism form A(_ including to - rn_,l) 2, superstrings a field like the Brans-Dicke are a variety of reasons and extended imagine that (potential) for wanting inflation the Brans-Dicke term field provides field does in the Lagrangian, (4.1) "anchor" a mass it would < (16a'/w)l/_Ma/mp_ of such require inflation. which theories, a mass, model, to >> 1 theory w -,, 10 and today, there In many "dilaton." for _ would now discuss, The addition the dilaton higher with of extended inflation, in the form of an additional a potential provided as the theory Brans-Dicke to be consistent be circumvented. involve of the model that Aw ._ 1-_ ( - rn_")_" £ ---* E As we shall of @, e.g., (3.8) INFLATION constant extended For the sake of a simple a mass, constant that the yet another. Such could is known expecting acquire to G. If, after A as inflation. EXTENDED and viable too rapidly time in slow-roUover Brans-Dicke a very elegant with < A _< 13 Mpc, tests + 3), is changing difficulty arises, OF 500, varies the horizon for 12x~3OOOMpc. In the limit predicted gravitational the above other Mpc solar-system about today re-entering _< ,_ _ 3000 Mpc, the effective limits the that that is just contributed above MODELS than for a_ < 500, prevent as that fact provide solar-system by the result OTHER for the density mode 10 -2s (raptM) spectrum IV. Mpc--the 13 Mpc by normalized this is the same Were of critical Gx' can A ,,_ 3000 @, thereby for the ... 10 0 GeV <I>;in the 2to,3(P-3p)+ field, affect rn_ extended of the theory conformal frame _.,,,+-----_m_,t(m_,-_,); 14 gravitational = Am_, t. inflation (for M = 1014 GeV). density Jordan the Brans-Dicke not adversely to the Lagrangian and preventing would they modify become: During extended with the these equations. o" field of extended must inflation in the If the potential inflation, then also of motion extended massless g$ that field appLicable here. There is one introduced new the minimum before 10 eV). Universe the then the Universe (A similar problem The cure and decay. thorough acquires a mass is given in Ref. inflation, is not of the and implementation of Eqs. (4.2) These reheating coherent to become when the of 3 K at the to arrange of a model of extended how extended inflation density inflation fits into and mass when is term the value oscillating behave just of the Universe (at a temperature come to the dominate temperature the is 7'. >> 10 eV, age of 10 Gyr (T./IOeV) inflation a field the oscillations with the Polonyi The Brans-Dicke for (effectively) ff will be left dominated oscillations therefore, fluctuations with the mass (4.3) temperature) not end if-field matter if these is simple: Like an field must of such where a realistic < in the conditions curvature the m® terms << H/2_'); behaves need that new associated inflation in slow-rollover disease m+ for the to dominate Universe the Gibbons-Hawking wrinkle (4.3) provided met, that calculation a temperature difficult of both << 1. In addition, g' still the so after specific, was encountered discussion Steinhardt density for this dread This than and will come reaches field extended is supposed are subdominant. worrisome of its potential. energy sides sides m_pt inflation conditions our previous To be more the associated +)' 1 equivalently less to rn_t , and the Universe of about (and In general, matter with on the right-hand extended $ are and potential equal Like nonrelativistic long that energy right-hand to interfere "( Brans-Dicke much for the ff field: of ff is precisely about (mass implying and the vacuum the w rn+mpi 16¢ M' during << B/2_" inflation scalar =https:///27r, • controls terms if these for a (_) the (4.2) that met Likewise, guarantee during are - and for _ is not << I, conditions + -p, vacuum ) _, (16_r/_)X/2M2/mpt. equations false the additional _ 1 H M 4, p = this requires ( 2 2 w re+rapt 16¢ M' of these p _ being be subdominant; Both 3+ + 6 + large the -x/2. field. be unstable mass. A more Brans-Dicke particle TM) physics field model 19. and Accetta in an attempt z have to construct proposed another a realistic 15 model, model dubbed of extended hyperextended inflation. They start with the action S where (the [ # is a dimensionless GUT scale?), is to include the model be simplified regime Since (a), today theory, Today then, the inflation regime (a). of the gravitational gravitational the value of some significance. this is the There period of 6: during difference necessarily assumed possible relevant case the inflation however; be less that spirit (b), the that would the Planck the dominates. where Those gravitational theory resembles we are not in regime by regime (c), a fact in deduce Provided squared, (b) = GN. = M2/#c_--and was the gravitational be G = M -a > m_ mass In G = M -a. of the be described model can dominates. also has a time-varying constant--G_v in which analysis that that M < will be proposed we will assume inflation. First, case. of inflation action variation must in their complicated the term = ra_,(M/raptVr_). is less than In the third is the time constant = Mmpl/v_ of • today are several in which _ is quite terms relativity theory theirs; _ and of _.) of general we can be certain case we can read off the gravitational value the In regime field (b) M a <_ • _< M 2/_/, where where (c), the theory M _ mpl, effective vr'Bmvt, (b), address that the value present that of extended than However, dominates; mass The key modification frame g. the Planck Brans-Dicke in powers one of the three version + £,..,,,, ; (4.4) notation _ and (c) • _> M2/#, and in regime Supposing since which feature the to the Einstein where is a rescaled we wiU not Brans-Dicke the theory expanded the first term and between involving regimes dominates; the _ crucial constant. terms (a) • _< M a, where term constant different field. difference transformation by considering are: second three is then • scale less than the Brans-Dicke a slightly of _ which + 16f M is an energy in the coupling adopted the conformal of the regimes terms have ta is a function because constant, order (We In general, 16 M2 .... and _ is, as before, higher curvature. the le - - _ the than = m_,t. scenarios for us--the analysis is just final ra_, I. last of the that value for hyperextended 60 or so e-folds--occurred previous Sections of a Brans-Dicke of the q_ during In deriving In particular, the if this 16 theory. phase various is not applies (b), formulas the case, that during since There denoted the regime effective is one by crucial q_, for fluctuations then we must the will we modify some of the previous (#/ _,mpl / results. To begin, Eq. (2.7) = ,oB't:= QM 2. whereq = _/mp_ which defines te becomes, (4.5) t. = iv1- islessthan1.Now Eq. (2.9) relating t,/ttoM/mr_, _Mp¢,and O) becomes _re_ 102s/(,,_1/2) t which is the amplitude same -\pmel/ as Eq. (2.9) of the curvature except for the fluctuations -;- .oR _Mpc of q. Equation (2.21) for the becomes _ __ factor , ¥ 10so/(_,-x/2)47r_2__ q-(2t_+l)/(,_-1/2)p-2/(_,-1/2) (4.7) The amplitude of the fluctuations The amplitude of graviton is increased perturbations vature perturbations are largest possible of _e is M2/_, turbations similar is at least factor (_P) P- Unless/9 value a factor of M/m_,t HOB increased by a factor is increased by a factor by the same factor, while isocur- of (m_,J_e)('+1/2)/2('-t/2). the increase of (v/-_M/mp_) of (m_,J_e)(_+1/2)/('_-1/2). in the amplitude 2(_'+t)/(_'-_n), Since of curvature which nearly the per- cancels a in Eq. (4.7): >10so/(,__l/2)4_r#__p_2/(,__l/2)_O_+1)/(,,_l/2)i << 1, it is now difficult to achieve curvature I'Mpc 2/(w--1/2). perturbations (4.8) of an acceptable magnitude. Now consider S = / By means regime (c). d'tz_ the action + of the following g,,, ._, _,,,, In this regime conformal is effectively + E.m_,, ; given by (4.9) transformation (4.10) f_2= M2m_,/_¢2, 17 the theory can written = as /IzV_ 16-_G + z where V(_) : (M*l_2m_v,)exp(-2_l_o)V(_), _G definition the of _o is _ ,_ << 6, in which case for _, cf. = 3m_t/161P. the kinetic Eq. (4.4), involving easy to analyze in the _ { 3/4 _. Einstein to that evolution of the conformal frame: and we have aries rescaled from scale frame that kinetic term the factor conformal a and _ are 2° (4.12) it follows that (4.13) ,I_ _ a 2 cct. In this regime there its final value, _ = MmPt/v_. inflationary is no inflation! regime new assumed from the original which the _ cc { 1/2 a _ t 1/2, during The compared exp(_12_o), theory, term for • that arises is negligible transformation In the Jordan In rescaling : TM However, Because (b), • does of the large this version evolve, density as it must perturbations of hyperextended inflation to reach that arise seems doomed to us occurred during to failure. There regime stein is one (c), which conformal transformation • o=- last requires frame above _ possibility, that that is not the _ _ 6. In this a simple is valid with inflation one. If the following relevant case, the transformation we treat • as slowly Ein- varying, the change: (4.14) 6fl_)" i+ to the In this case 3 a = _% Early on, when wM2/2fl a oc t 2"_-1, Using m = _ + s/_' _ _< wM=/2_, superluminal to _ = wM2/2fl. w M 2 inflation expansion During occurs ceases. (4.1s) as m > 1; as • increases The epoch of inflation to the value of lasts from _ -- M2/fl inflation, @ oc a 1/(2"-1). the usual formula (_P) -P- _81r,/-m102sl(m_1) V 6 HOR '_ = _ 1/= (4.16) for (6P/P)HOR, we find that (\71_pl/M _('m-l)/(m-1)_ 18 "_Mpc 1/(rn.-1) • (4.17) Sincem varies during inflation, the period of inflation, During we cannot that immediately is, while evaluate M2/_3 this expression. _ • _ wM2/2_, dln.=(2, -l)dln = Integrating this expression, a(t) inflation during achieve the Further, we find that is (w/2) 60 or so e-folds end interesting of inflation): the total _ exp(w/4 - of inflation it is straightforward cosmologically the (4.1s) 1/2); m _- N/2 went the ,_ 30. w must value outside of e-folds we immediately necessary, to compute scales number Returning in order of about around time (N to Eq. see that factor be in excess of m the horizon in the scale the we find, 240. that = 60 or so e-folds (25) to the before provided that w _> 240, (4.19) \ _n p-----il l HOR Since m __ N/2 (gg/P)aoa _ >> 1, this expression 300(M/mpt)_. can be attMned AMP2c/ (N-2)" is very Fluctuations for M ,,_ 10 is GeV fluctuations inflation in the passive role until thereby reheating the or so, provided inflaton it makes the dominant field. with bubbles). addressed inflation an exponential as in slow-rollover are typically any with slow-roUover parameter nonscale small are In the Unlike slow-rollover to ensure 19 important--it that from naturally Einstein they Brans-Dicke those and conformal associated unambiguously frame field playing of curvature inflation, extended the role of fluctuations these is fluctuations is not necessary have a very end of inflation, in the (aside most to quantum field plays at the fluctuations the calculation even more value inflaton with the Brans-Dicke potential; inflation. amplitude arise due vacuum fluctuations frame: inflation _Mpc: w _> 240. the true quantum fluctuations invariant--and to a very inflation curvature conformal and interesting fluctuations to the It is the curvature Einstein resembles the inflaton, the same These in the curvature its transition field that give rise to the dominant that of N REMARKS In extended Universe. independent of a cosmologically V. CONCLUDING In slow-roUover nearly an acceptably to tune small amplitude. In principle, be important; example, Just however, w--be field fluctuations tuned canonical kinetic computed in the While in the terms mode] theory, analyzed a couple viable model of extended inflation; of extended work by the NASA axlon as they frame, of attempts to illustrate This the can also the Brans-Dicke or the can any arise ilion. Because inflation, ordinary fluctuations in any Such isocurvature are in siow-rollover these density will serve in the bubbles a parameter--in fluctuations spectrum. models occurs e.g., magnitude these which arise from but matter are most they fields have appropriately frame. only and that isocurvature in the Jordan Jordan that so. a nonscale-invariant we have key feature to require inflation, are of a similar have perturbations seems to be just present typically density that as in slow-roUover massless simplest the inflation toy model in extended at a realistic the general the reason is significant analyzed in part through NAGW-1340 model, features to expect inflation there that that variation here was supported grant fluctuations is some one can for the hope that expect in a this is true is that in the gravitational the constant, for w _ 20. by the DOE (at (at Chicago and at Fermilab) and Fermilab). References 1. A. Guth, Phys. 2. A. D. Linde, Phys. 3. D. Rev. La and Rev. Phys. Left. 48, Phys. Bertschinger, B 108, 1220 Rev. D 39, Penn. preprint Left. 2854 (1990); 389 (1982); A. Albrecht and P. J. Steinhardt, (1982). Left. Phys. Phys. 4. P. Jordan, Left. P. J. Steinhardt, J. Steiahardt, 90/44-A D 23, 347 (1981). Phys. B 220, B 231, Rev. Lett. 62, 375 (1989); 231 (1989); 376 D. La, (1989); P. J. Steinhardt R. Holman, E. W. Kolb, and La and P. and E. P. J. Steinhardt, F. S. Accetta (1989); D. and J. J. Trester, F. S. Accetta, and Y. Wang, Univ. Fermilab of preprint (1990). Zeit. Phys. 157, 112 (1959); 2O C. Brans and R. H. Dicke, Phys. Rev. 24,924(1961). 5. E. J. Weinberg, Phys. Rev. D 40, 3950 (1989). 6. J. M. Bardeen, P. J. Steinhardt, A. H. Guth and S.-Y. Phys. B 117, Left. 7. Albeit Phys. Rev. Bardeen, Pi, Phys. context, potential D 32, Phys. Rev. 175 (1982); in a different an exponential and M. S. Turner, Rev. curvature been (1985), D 40, 1753 9. R. Holman Phys. tuations the to the have Their scalings fact that a canonical tor, same. kinetic over Jordan dard energy the frame, result traces applicability where they the stein conformal factor, 11. The but in other large-angle the to the Universe further frame). large angles was matter discussion is given on the term of this to a point J. M. of the curvature fluc- _ H2/¢ by a factor of principle: for the at hand be even the more and with the horizon amplitude see J. M. Bardeen with to the The in the stan- contezt theory is only not fac- quibble time. rescaled of (Ein- a numerical significant. arise due to the __ ($P/g)HOa. sky re-entered derived difference does as applied with in the traces remaining We would in although paper) The is changing applicable ', cf. Eq. (13) descrepancy fluctuations was inflation of O(30), of this inflation thus/s 21 of extended (¢ is as in our constant by: 6TIT point and (it is (0_,¢')2/32r). fluctuations dominated with S. Matarrese, R. Bond, model of _ ¢' = v/]'_¢ it could temperature amplitude ours A factor For the model models (1982). inflation and J. (6P/P)HOa than slow-rollover and B 115,295 L219 (1979). taking gravitational action, Lett. the amplitude of this formula use from the Einstein-Hilbert and field A. A. Starobinskii, for power-law in this simple and are the scalar (1983); 37 (1990). scale is smaller _-_/3p(2"-1)/(_'-1/2), them J. 234, answer the Phys. S. Salopek, a estimated frame (1982); D 28,679 by F. Lucchin, by D. B 237, horizon in the Jordan paper. other Left. Bertschinger on the present by working their and analyzed Rev. (1989). et al., Astrophys. 10. La, Steinhardt, 1110 fluctuations and 8. R. D. Reasenberg et al., 49, S. W. Hawking, have 1316 Left. Phys. The after Sachs-Wolfe scales that decoupling (_P/P)HOR effect, correspond when _ H2/57r_. et al. s or E. W. Kolb the For and M. S. Turner, The Early pp. 283-291, Universe and M. S. Turner, Left. 375 (1985), B 158, 13. J. Preskill, 15. M. Wise, P. Sikivie, 14. A. Cohen and City, CA, 1990), Nucl. Phys. M. T. Ressell and B244, M. S. Turner, Left. Phys. Left. Phys. B 120, 127 (1983); B 125, Rev. ibid, L. Abbott 137 (1983). 251 (1987). Phys. waves Universe A. D. Linde, and W. Fisctder, Left. B 216, Phys. Left. 445 B. Allen, Phys. of gravitational The Early Left. B 199, 541 (1984); M. S. Turner, of the production Phys. and A. Veryaskin, M. Pollock, (1985); therein. and D. Kaplan, M. Sazkin, and D 32, 3178 M. Dine Phys. A. Cohen, 16. V. A. Rubakov, Rev. F. Wilczek, ibid, 133 (1983); M. S. Turner, Wise, Phys. and references and D. Kaplan, R. Fabbri Redwood 383-390. 12. D. Seckel and (Addison-Wesley, B 115, (1983); Phys. D 37, see, (1982); and 2078 For a textbook inflation (Addison-Wesley, 189 L. Abbott Rev. D, (1990). during 20 (1989). (1988); discussion E .W. Kolb Redwood M. City, and CA, 1990), of the gravi- However, they pp. 288-291. 17. La, Steinhardt, tational carried stant and wave mode out the that 18. G. D. Cougtdan, 19. J. A. Harvey, 20. It is simple to show that Jordan we have the Left. frame, where • _ [ s/4 for the 22 result amplitude today. the about gravitational con- their calculation is a factor of V/_/3p 59 (1983). M. S. Turner, 7_. quibble their and term for _ is dominated involving horizon same B 131, the the 10. Numerically, Phys. E. W. Kolb, the kinetic transformation et al., estimated re-entering in the and in Footnote ours. s also is just calculation is time-varying, as mentioned smaller than when Bertschinger work in preparation gravitational by the contribution action (1990). _'_7_/M2('_-1) from the conformal