Inflation from a ripple on a vanishing potential

Volume 159B, number 4,5,6

PHYSICS LETTERS

26 September 1985

INFLATION F R O M A RIPPLE ON A VANISHING POTENTIAL K. ENQVIST, D.V. NANOPOULOS fern, CH 1211 Geneva 23, Switzerland

and M. QUIROS t Instituto de Estructura de la Materia, Madrid, Spain Received 6 May 1985

We propose a very simple model of inflation having essentially one free parameter, the value of which is fixed by the amplitude and scale independence of energy density fluctuations. The model, based on the maximally symmetric supergravity with SU(n, 1) manifold, has an asymptotically flat inflaton potential. All inflationary conditions can be satisfied without any fine-tuning and all mass parameters can be O(Mvj ).

Cosmological inflation [1] is conceptually a very attractive idea; yet no particle physics model exists in which one can implement both inflation and reheating of the universe in a simple way. The shortcomings of the "new inflation" [2], based on the Coleman-Weinberg type flat potentials, led people to consider a supersymmetric model of inflation [3]. The simplest versions, based on the so-called minimal supergravity or trivial K~hler metric, were also shown to suffer from some diseases [4]. In addition, minimal supergravity has a gravitino of mass m3/2 = O(Mw), which in the absence of inflation is a cosmological embarrassment [5]. Inflation can dilute [6] any primordial gravitino number density, but unfortunately gravitinos will be regenerated after inflation by 2 ~ 2 scatterings. Recently, it has been shown [7] that if the regenerated gravitino density is to be suppressed below levels where the cosmic abundances of light elements will not be disrupted by gravitino decays, one finds a very low reheating temperature TRr~ < 10 8 GeV. Such a low reheating t Present address: Department of Theoretical Physics, University of Michigan, Ann Arbor, MI 48109, USA.

temperature is problematic for particle physics, especially for obtaining a correct baryon asymmetry while suppressing unwanted nucleon decays due to dimension-five baryon-number violating operators .1. This constraint is an obvious difficulty for models having an O(10 2) GeV gravitino, and although it is possible by clever choices to rotate away [9] the d = 5 ~ ~ 0 operators, the resulting models lack all aesthetic appeal. The decoupling of the gravitino mass from the scale of SUSY breaking, as felt by the light fields, can be achieved only in "non-minimal" versions of N = 1 supergravity with non-trivial K~hler metric. Here the efforts to write down an acceptable model of inflation and particle physics have focused on the no-scale models [10,11]. These have a maximally symmetric K~thler manifold with an SU(n, 1) global symmetry, where n is the number of chiral fields. Such models have flat tree level potentials and can lead to interesting low-energy particle physics as well as to a dynamical determination of all mass sales below the Planck mass. It can also give raise to a novel mechanism [11], whereby the QCD vacuum angle relaxes ,x For a recent review see ref. [8].

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dynamically to 0 = 0. In the no-scale models the source of the low-energy SUSY breaking is the gaugino masses, whereas the gravitino can be very heavy [10] or very light [11]. The various cosmological and particle physics constraints for a very low mass gravitino in no-scale models have recently been considered in ref. [11]. The SU(n, 1) structure of the K~ihler manifold has also recently been advocated [12] in the context of superstrings [13]. The compactification of the effective ten-dimensional theory to N = 1 supergravity in four dimensions dictates [14] the compact internal manifold to have SU(3) holonomy. Calabi-Yau manifolds have such a property, but otherwise little is known of them. Therefore it has been suggested [12] that a simple truncation of ten-dimensional supergravity preserving an SU(3) subgroup of the rotation group of the six-dimensional internal space may serve to mimic the properties of compactification on the C a l a b i - Y a u manifold. It is remarkable that the resulting effective theory can be shown to possess the SU(n, 1) Kahler manifold such as has been studied in refs. [10,11]. To obtain an acceptable inflationary scenario, the SU(n, 1) summery must however be broken; otherwise there would be no vacuum energy to drive inflation. A suitable generalization of the SU(n, 1) manifold, which can be shown to fulfill all requirements for inflation [15-18], is given by the K~ihler potential G

=. _

31n[z+z*-K(¢,q,*)

- aK(~,

ep*) + F +

F +

_ 1. i y . ]

written as V=[x3/(3x-a)]

X exp ( F + F + - a K )[ Fep - aK~,l 2/Kee,+ +xZexp(F+g+-aK)F,+F'+~D

D .

(2)

where D " refers to a correctly normalized D-term and x = exp (G0/3) (for a discussion on normalizations of fields and lagrangian terms in nonminimal supergravities, see ref. [19]). In the domain of positive kinetic energies where 3x - a > 0 and K**+ > 0 the potential (2) is positive semi-definite. The properly normalized kinetic energy terms are obtained after a point transformation ~ [ ( 3 X - a ) g q , ep.]l/2l,¢,=4, ° . . . . odP, fla ._+ x l / 2 ,

(3)

. a

Iq,=,l,o,x=xoY ,

valid at the minimum (q'0, Xo). The minimization of the potential (2) leads to F, = b ~ = 0, and at the metastable minimum with non-zero vacuum energy one also has the condition x = a/2.

(4)

At very high temperatures, all non-singlet fields y~ have been shown [19,20] to choose their symmetric minima so that for the purposes of inflation, we will consider the case y i = 0, which, together with (4), leads to V = ( a / 2 ) 2 exp ( F + F + - a K )IF~, _ aK~, 12K~,~,+,-1

(5)

~.v i J

(1)

= G o - a K + F + F +.

Here z is a gauge singlet field responsible for SUSY breaking, the yi are the matter fields, ~ is a gauge singlet inflation field and F = F(~, y~) is the superpotential. The function K can be assumed to be a function of q~@+, and a is the parameter responsible for SU(n, 1) breaking with a--+ 0 corresponding to the limit where one recovers the SU(n,1) structure of the K~ihler manifold. The choice a 4= 0 is thus dictated by the requirement of non-zero vacuum energy. The scalar potential corresponding to (1) can be 250

26 September 1985

where now F = F(¢). However, at the global minimum F~ = aK4,, x, and the gravitino mass, is no longer restricted by (4) but only by the positivity condition x > a / 3 . The potential (5) is attractive for inflationary purposes because of its simple structure and because of its positive definiteness. A model based on this generalization of the maximally symmetric K~ihler manifold is defined once F(@) and K(Iq, D are given. In previous attempts [15-19,21] rather complicated forms were used, leading to a potential suitable for inflation but bearing no relation to the SU(n, 1) manifold. Here we will consider the extremely simple possibility of potentials that are globally flat in the sense that V ~ 0 as

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I'/'l ~ ~ , but show local deviations from flatness. Surprisingly enough, although such potentials are very constrained, having only one free parameter, we are able to obtain an inflationable potential in a natural way for a large range of this one parameter. The aim of this paper is to show that, even in the simplest of the models described by (5), all inflation conditions are satisfied and scale-independent energy density fluctuation with the correct amplitude can be made consistent with all mass parameters being O(Mpl ). We will take the simplest imaginable ansatz for the function K: K = q,q, +,

(6)

so that the inflaton field has canonical kinetic terms up to an irrelevant numerical factor (a/2) 1/2. The superpotential can be written as r ( q s ) = F ( 0 ) + 4dp + ½(a - 42)d~2 + ~pdp3 + . . - .

(7)

26 September 1985

minimum at ¢o =

(10)

The potential will now be flat at large Re~, independently of 4, and globally fiat in the whole complex plane if 42 < 2a. Moreover, if 42 < a, the origin becomes a minimum along the imaginary direction and the path followed by the inflaton will be along the real direction. Let us now examine the conditions on inflation assuming for the moment that the initial distribution of the inflaton field is such that at the beginning of inflation q, = 0. We will return to this question at the end of the paper. A slow roll-over of the inflaton field can be achieved if [221 IV"(q ~) I~ < 9H2(~),

IV'(ep)/V('~)[Z v/6, (11a, b)

where the Hubble parameter is / 4 2 ( ~ ) = ~ [ V ( , ) = -'421 2 l"

(12)

Here F(0) = log m 0 and 4 4= 0 are related to the Hubble constant during the inflation and the particular form of the coefficient of the quadratic term is dictated by the condition of having a stationary point at the origin. Here we do not consider higher order than cubic terms. The curvature along the real and imaginary directions at the origin is given by

Eq. ( l l a ) is satisfied for 0 ~
m2=m~4(O -

During the slow roll-over, the semi-classical equation of motion of the inflaton

243),

m 2 = m ~ ( a a 2 - 6a42) - m~t.

(8)

If O =~ 0 the global minimum is not unique: there are two complex-conjugate global minima with

(13)

,~ + ( 3 H + F ) , ~ + V'(q0 = 0 ,

and the Hubble parameter (12) can be approximated by 3Hq,+V'(q,)=0,

v2 = ( 4 4 / 0 ) [ 1 + ( 2 a - 4 2 ) / 0 ] + [(2a -

42)/0]22/42,

(9a)

where ~ = 1/v~(u + iv). If O ~< 1/243, there are two real (v = 0) minima given by the solutions of 1 - (4/vt2-) u + ( 0 / 4 4 )

uz =

0.

(9b)

The most economical assumption is, however, that 0 = 0. In that case one finds a unique real

(14)

H2=~V(q~).

(lSa, b)

In that case, the number of e-foldings of the scale factor R(t) as q, rolls from q~a to q~b can be calculated exactly and is

N(a, b) 1

=f~rH(t)dt=-~ l°g

q~b[1 - ( 4 / 2 v ~ ) q~b] q~[1 (4/2vc2-)Oa] " (16)

A sufficient amount of inflation is obviously 251

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obtained for values of ~ that are less than O(1). The most stringent constraint on the parameters of inflationary potentials comes, however, from the magnitude of the energy density fluctuations, 8 p / p = 10 -4 on galactic scales. The amplitude of such perturbations, when they re-enter the Friedman horizon after the inflationary epoch, is given by

8p/p

= (V~-rr 3/2) - 1H2(~b,) 4 . - 1,

(17)

where ~ . = ~ . ( t . ) and t . is the time of the first horizon crossing during inflation. We estimate the number of e-foldings between ~ . and ~r, the end of inflation, to be N . = 43 + ~ log ( M / M o ) ,

(18)

where M~ is the solar mass, M is the mass contained in the comoving volume, and a reheating temperature T~H = 5 X 101~ GeV, consistent with our further results, has been used. For the scales M 1 = 1012M~ (galaxy mass), M2 = 101~M~ (supercluster mass) and M 3 = IO=M~ (present horizon mass), N , takes the value of 52, 55 and 60, respectively. Therefore the deviation from exact scale invariance is very small but will in general depend on f. We can obtain ~ . from the equations of motion (15a), whereas ~ , can be expressed as a function of q~r and N , by using (16). Then energy density perturbations can be written as

8p P

26 September 1985

H o -= H ( ¢ ) and Hr --- H(¢f) and thus the energy Vo1/4 and Vf1/4. Finally the total amount of e-foldings which the cosmic scale undergoes during the whole inflationary period can be read off from (16) as N t = (2~2)-11og ( dpf[1 -(~/2Vv2)q~,]/Ho }. (20) The deviation of given by

3p/o

from scale invariance is

A = ( 80/t,)2/( 80/O)1 = ( {1 - v~-~q~f[1 -(~/2V~-) q~f] exp [ - 2~ 2Nz] } { 1 - v~-~q~,[1 - ({/2V~-) q~]

x exp [-2

2Nl1} -1) 1/2

x exp [ - 2

(N2 - Ul)].

(21)

In fig. 1 we show q~m,~f, ~*, V~/4, Vo1/4, Nt and z~ for M 1 = 1012Mo and M 3 = 1022Mo as functions of ~. We see that N t > 60, which is the minimum amount of inflation needed to solve the cosmological problems of horizon and flatness, for the whole considered interval of ~. However, the deviation from scale invariance is z~ < 0(3), which is probably allowed, for ~ _<0.24 which is, as can be seen from fig. 1, the interesting range. On the other hand ~ . becomes << Mp~ for ( > 0.2. This can be understood analytically as follows: for ~2 > 1 / 2 N . , i.e., ~ >~0. 1, 4~. can be approximated by q~, = q~f[1 - (~/2V~-) q~f] exp ( - 2~2N.),

(22)

{1 - v~~q~, [1 -(~/2V'2)q~,] e x p ( - 2~2N.) } 1/2 (2rr)3/2f2q~, [1 - (f/2v/2-) ~e] X H , exp (2~2N,),

(19)

where H , = H(t,) is the Hubble parameter at. the time of the first horizon crossing. From now on, we will proceed in the following way: for a given value of f, we obtain the value of the inflation field at the end of inflation ~r, solving eq. (13), and at the time of horizon crossing, q~,, using (16) with N , = N3 = 60. The Hubble parameter at t , can be deduced from (19) and using the condition (17). From the very definition of H(~) and the values of H , , q~, and q,f, we can deduce 252

which approaches zero asymptotically as ~2 increases, The reheating temperature TRu is related to the available energy at the end of the inflation Vr1/4, in the case of good reheating, by Tp.R = (30/~r 2g.)l/4Ve1/4,

(23)

where g . is the effective number of relativistic degrees of freedom. Typically g . = O(10 z) and TRrI = (0.4-0.5)Vf 1/4. In this way one can obtain 1015 GeV < TRH _< 1016 GeV for ~ _<0.2. This range overlaps with the range of scale-invariant density fluctuations, and it is the interesting one.

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26 September 1985

while H o a n d V01/4 increase as 13

H 0 = 1.35 × 1013 G e V 4 -1,

12

1/'01/4= 7.49

11 10 9 8

7 6 5 4

3 2 I 0

0.1

0.2

0.3

Vd/4,

Fig. I. Graphs of 4m, 4f, 4*, I/fI/4,

Nt and A as

functions of ~ (for definitions, see text); 4m, 4f and 4 . are in units of Ml.1= 1.2 × 1019 GeV, and the total roll-over scale Nt is expressed in units of (Nt)min = 60.

F o r very small values of 4, H . a n d / - / f behave like H , = 2.4 × 1014 GeV, H r = 1.3 X 10 ~3 GeV, V~x/4 = 7.3 x 1015 GeV,

(24)

x 1015 GeV 4 -x/2.

(25)

F o r the sake of completeness we have tabulated, i n table 1, the n u m e r i c a l values of ~i = H0, ~ . , ~f, ~m, V 1 / 4 = ('rr2g*//30)l/4ZRH ' A a n d N t corres p o n d i n g to a typical set of values of the p a r a m e t e r 4It is a s t o n i s h i n g that we really can o b t a i n a successful description of inflation b y using a o n e - p a r a m e t e r m o d e l (with m 0 b e i n g only a n over-all scale factor which, as we will see, can be set e q u a l to M m ) a n d that the c o n d i t i o n o n 4, as d i c t a t e d b y A, is only the mild 4 --- 0.2. Therefore, having n o w established the feasibility of i n f l a t i o n a n d scale-independent energy d e n s i t y f l u c t u a t i o n s with the correct a m p l i t u d e for small 4, we n o w come to the value of the over-all mass scale mo. I n most inflationary models m 0 has to be a d j u s t e d to a very small value m 0 O ( 1 0 - 5 - 1 0 - 6 ) dictated b y 8 p / p = 10-4. However i n o u r model, using the expression (25) for s m a l l values of 4, we get m o --- 10-54 - 2 ,

(26)

so that m o gets small values only for values of 4 r u l e d out b y scale i n d e p e n d e n t energy density f l u c t u a t i o n s ; b u t m o - O(1) for 4 = 3 x 10 -3. I n c o n c l u s i o n , we have considered the simplest m o d e l of i n f l a t i o n based o n maximally symmetric supergravity. T h e model has two free parameters: a n d mo. All r e q u i r e m e n t s needed for a successful i n f l a t i o n a r y scenario are are satisfied without

Table 1 Numerical values of 4i = Ho, 4., 4f, 4m, (TRH)max A and Nt for a sample of values of ~.

0.35 0.30 0.25 0.20 0.15 0.10 0.05

4i/Mm

4*/M m

4f/Mpl

4m/Mvl

(~r2g,/30)a/aTRH (GeV)

A

Nt

2.8 x 1.3 x 2.8 × 3.6 x 2.2 x 8 × 2.7 x

1.7 x 8.8 x 2.8 x 5.7 x 5.9 x 0.44 2.74

0.66 0.79 0.97 1.25 1.70 2.63 5.46

0.81 0.94 1.13 1.41 1.86 2.79 5.62

2.2 x 1013 1.3 x 1014 5.6 x 1014 1.8 x 1015 3.9 X 1015 6.2 X 1015 7.2 × 1015

11.5 6.1 3.5 2.2 1.6 1.3 1.14

95 109 133 183 281 588 2.5 x 103

10-tl 10 9 10-8 10-7 10 -6

10-6 10-5

10-7 10-6 10-4 10-3 10-2

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a n y fine-tuning of parameters. Energy density fluctuations are scale-independent for ~ < 0.24. T h e correct amplitude 8p/p = 10 -4 determines the H u b b l e constant at the end of the inflationary period H r (and hence the reheating temperature for the case of g o o d reheating as in the two-comp o n e n t inflationary model proposed in ref. [16]) a n d at the origin H 0. The further requirement of large reheating temperature (TRH > 1015 G e V ) needed to generate, after the inflation, the cosmological b a r y o n asymmetry by out-of-equilibrium decays of h e a v y (MIj > 1016 GeV) colour triplets - leads to the mild constraint ~ < 0.2. I n this p a p e r we have not addressed the question o f the initial pre-inflationary conditions. T w o different mechanisms have been so far p r o p o s e d : the high-temperature phase transition [23] and the chaotic inflation [24] scenario. The feasibility o f a high-temperature phase transition d e p e n d s on the value of the typical relaxation time ( A t - l / m o T ) as c o m p a r e d to the age of the (radiation-dominated) universe (t u - 1 / T 2). In fact, as has been argued b y Linde [24], if At > t U for T >_, T¢, then any imtial distribution of the inflaton field m a y remain effectively frozen and the theory of temperature phase transitions does not apply. I n that case one should consider some chaotic initial distribution for ~ and assume that in s o m e d o m a i n of the universe the field ¢ was initially near the origin 0 < ¢ _< H 0. O n the other hand, if At < t U then any initial distribution will have the time to relax dynamically to the origin at Tc and temperature corrections are relevant. In our model, small values of ~, i.e., large values of m 0, seem to favour this possibility. N o t only that, recently Mazenko et al. [25] have argued that large field fluctuations would f o r m d o m a i n s where the inflaton lies at the global m i n i m u m ¢ = ¢0 and no slow roll-over would take place. However, as was pointed out by Albrecht and Brandenberger [26], the domain formation rate will d e p e n d on the strength of the inflaton interaction rate as c o m p a r e d to the rate of the H u b b l e expansion which will red-shift large fluctuations. According to Albrecht and Brandenberger the correct initial state will be obtained for small e n o u g h coupling of the inflaton, which entails that the position of the global m i n i m u m should be larger than O(Mpl ). Also our simple 254

26 September 1985

m o d e l with small ~ will favour the formation of a correct initial state: in particular for ~ = 3 × 10 -3 we get if0 = 102 Mp~. The problem of initial conditions in inflationary models based on maximally symmetric supergravity is presently under investigation [27]. O n e of us (K.E.) gratefully thanks the A c a d e m y of F i n l a n d for financial support.

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