Chaotic inflation and cold baryosynthesis in a supersymmetric gut

Volume 220, number 3

PHYSICS LETTERS B

6 April 1989

CHAOTIC INFLATION AND COLD BARYOSYNTHESIS IN A SUPERSYMMETRIC GUT Larry C O N N O R S , Ashley J. DEANS and John S. H A G E L I N Department of Physics, Maharishi International University, Fairfield, 1A 52556, USA Received 7 November 1988

We consider chaotic inflation followed by Affieck-Dine cold baryosynthesis in a recently proposed flipped SU ( 5 ) X ( 1) GUT derived from the superstring. We show that the presence of a "quasi-flat" direction 2q~vin the scalar potential with 2 = [2cl2~ 10- ~ leads naturally to density perturbations Op/p~ 10-4-10 -5. We also show that the decay of an Affieck-Dine-like direction @Ao leads naturally to nB/s~ 10-6-I 0-~o taking account of the large increase in entropy A~ 108 produced by the subsequent decay of the "flaton" field @ responsible for GUT symmetry breaking.

Inflation provides a natural solution to the flatness, homogeneity, and horizon problems associated with the standard hot big bang model of cosmogenesis [ 1 ]. Inflationary models utilizing standard inflationary potentials [ 1 ] and chaotic inflation [2 ], in which scalar fields have as initial conditions large VEVs of O (Mp), both require a weakly-coupled scalar field in order to give density perturbations Op/p ~ 10-4-10 -5 consistent with constraints imposed by galaxy formation and the observed isotropy of the cosmic microwave background radiation [1 ]. A principal weakness of both types of models is that this weakly coupled scalar field is introduced in an ad hoc fashion that is often inconsistent with string theory. In this paper we identify a natural candidate for this scalar field within a recently proposed [ 3 ] supersymmetric flipped SU (5) X U ( 1 ) grand unified theory ( G U T ) derived from the superstring [ 4 ]. This SU (5) X U ( 1 ) G U T contains a natural and economical mechanism for splitting Higgs doublets and triplets (and hence no gauge hierarchy problem), a naturally long-lived proton, and a fully realistic lowenergy spectrum and phenomenology [3]. It is, in addition, the only viable G U T group obtainable from any known string formulation [4,5 ]. We show that, under plausible assumptions, adequate inflation followed by Affieck-Dine [6] cold baryosynthesis occurs naturally in this SU (5) × U ( 1 ) model, yielding O p / p ~ l O - 4 - 1 0 -5 and a baryon a s y m m e t r y nR/s ~ 10-6_ 10-1°. Our results extend to 368

a large class of intermediate scale models with Affleck-Dine directions and G U T symmetry breaking along a flat direction in the scalar potential. It has previously been shown [7] that adequate baryosynthesis can be achieved in the flipped SU (5) X U ( 1 ) model in a "weak reheating" scenario where the temperature following inflation is below that required to restore the G U T symmetry. (Strong reheating leads to problems in intermediate scale models with the overpopulation of certain metastable particles [8].) The weak reheating scenario, however, in addition to requiring an ad hoe inflaton field, also requires the presence of extra Higgs generations. Although extra Higgs have been shown to occur in a recent string-derived flipped SU (5) X U ( 1 ) model [4], in that model the Yukawa couplings needed to generate a net baryon asymmetry are missing. It therefore seems useful to explore alternative mechanisms for inflation and baryosynthesis. The minimal SU (5) X U ( 1 ) model contains three generations of matter fields with SU (5) X U ( 1 ) transformation properties [ 3 ] F,=(10,½),

i',=(5-~),

£i-(1,~)

(1)

the following Higgs representations: H = ( I O , ½),

IZI= (lO, - ½),

h=(5,-1),

fi=(5,1),

and four SU (5) X U ( 1 ) singlet fields (~rnm

(2) (

1, 0). The

0370-2693/89/$ 03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division)

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superpotential for this minimal model is W= 2;(F;~h + 2 ~F;~fi + 2 ~i',~ h + 24 HHh + 25 IZlIZIh + 2 ~'~F;I'qI0,,,+ 2 ~"hh~)m+ 2 ~"~p ¢,,,~9p.

(3)

The 2 ~couplings in (3) generate masses for charge- -~ quarks, the 22 couplings for charge + -~ quarks, and the 23 couplings for charged leptons. The 24 and 25 couplings provide the natural doublet-triplet splitting which is a major virtue of this flipped SU(5) XU(1 ) model. The 26 coupling provides a seesaw mechanism guaranteeing light left-handed neutrinos, and 27 provides natural hfi mixing. (For more details, see ref. [ 3 ]. ) The GUT symmetry is broken to SU(3)cX SU (2) LX U ( 1 ) r by a large VEV for a combination • of H and IZI along a D- and F-flat direction ( v ~ ) = ( g h ) of the supersymmetric potential [ 3 ]. In addition, there are Affieck-Dine type flat directions OAD involving VEVs for scalar quarks and leptons. Choosing a basis in which 2~ is diagonal, one can easily identify the following D- and F-flat direction. ( 1 ) First generation: F']2 = a

,

F[4 =z) ,

£~ ,=ei~'x/lvlZ-b lal2 ,

(2) Second generation: 722=a,

~2=v,

( 3 ) Third generation: ~l _ _ e i ¢ / i v 1 2 + [ a 1 2 .

(4)

Here I~], is the combination of ~'s orthogonal to the combinations which couple via 2 3 to the second and third generation ~'s. This ensures F-flatness with respect to the 2 3 term in W (3) in addition to D-flatness for the extra U ( 1 ) generator. The flatness of these directions is broken by soft supersymmetry-breaking mass terms of order rh.~ 102-10 3 GeV, which also enable the h, fi, and 0,, to acquire small VEVs of order mw. Finally, we note that there are, in addition, "quasi-flat" directions which are D-flat and F-flat apart from the small electron Yukawa coupling 12~ 12~ 10- i t, where

6 April 1989

and ( h °) / (~o) ~ O ( 1 ). One such direction, which we denote by OoF, is obtained by replacing ~], in OnO (4) by Ec. This leads to a departure from F-flatness that is at most [2~ 12 In the following cosmological analysis we assume that all scalar field VEVs are initially O (rap). This differs from some previous work [2 ] in which the contribution from each field 0; to the potential V(0;) was taken to be ~ M g , which implies large VEVs>>Mp for certain fields. We consider this somewhat unnatural in that it requires as input conditions relatively constant VEVs over distances >> 1/ Mp to avoid large contributions >> M 4 to the potential energy from gradient terms ~. Another reason for taking 0; z¢, Mp is the probable occurrence of nonrenormalizable terms M y " ¢~,+4 in the potential generated by the exchange of massive string excitations. These nonrenormalizable operators have been analyzed in the context of our specific flipped SU (5) X U ( 1 ) 4D string model [ 4 ] and shown to arise through identifiable couplings to massive string modes. A third reason for taking ~; :~ Mp is that it could mean that fields corresponding to flat directions get permanently stuck at values >> Mp if soft supersymmetry breaking goes to zero for sufficiently large VEVs. Conversely, if the/~2~2 t e r m s persist for large field VEVs along these flat directions, there is a danger of too much rhZOZ-type inflation, leading to a smaller than desired value for Op/p [ 2 ]. Starting with initial field VEVs of O(Mp), a typical evolutionary scenario proceeds as follows. (I) For a random set of initial conditions ~;~O(Mp), those directions in field space corresponding to F and D non-flat directions evolve quickly in a 2~ 4 potential: "6j+ 3HOj=

OV H2 = 8np OOj~, 3M 2 ,

(5a)

where

~[

OW 2

2

(5b)

The directions in field space with the steepest slopes evolve first, moving in toward the origin or into nearby valleys if the potential V(~) contains flat di-

g2 m e ~ l ,

2~ = x/~ m w V . t < h ° > 2 / < f i ° > 2

~ Not all authors agree with this argument [2].

369

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rections. This initial phase of evolution in a generic 204 potential contributes ~ 7t(t~I/Mp) 2 e-foldings of inflation [ 2 ], where Cz characterizes the magnitude of the largest initial field VEVs #2. (II) The last of the F and D non-flat directions to evolve is the quasi-flat direction ~qF with its quartic self-coupling 2 = t2~ I-~~ 10- ~1. This inflationary epoch lasts until t l ) Q v ~ M p ( - = t l ) o ) a 3 at time to~ 10~'M~7 ~, where we have used the 2 ~ ~ chaotic inflationary formula [ 2 ]

CQv(t)=@~Fexp(-- ~

Mpt) ,

(6)

with 0~F ~ (few) M p . One can estimate the value of Op/p that results from this epoch as follows. During inflation, perturbations ~'gal that will later influence galaxy formation are inflated beyond the horizon H-~ since they grown proportionally to the universal scale factor: ln2~t =ln ~oo+k.

(7)

Here Ro is the value of R at the end o f (I~QF inflation, and k is determined by the size of 2 ga~~ I Mpc-~ exp ( 132 ) M~- 1 in our universe today. The behavior of H-1 as a function of R during @QF inflation [when the number of e-foldings before the end of inflation In Ro/R ~-7t(~Qr:--t~o)/Mp 2 2 2 for

OQF> ~o-----IMp] is

H-I=

6 April 1989

3 Mp ~ 1 f . Ro n@2~ -' 8n2@~F -- 4 8-2Mpk, tn-R-+-~-v,/

X

/

"

(8) Hence In H - 1 is a slowly varying function of R/Ro as can be seen in fig. 1. During the radiation- and matter-dominated periods that follow ~OF inflation (which will be discussed in later sections, but which are graphically summarized in fig. 1 ), the perturbations 2~) continue to grow proportionally to R, whereas H - 1 =at oc R a, where a = 2 (radiation) or 3 (matter). Therefore 2ga) eventually re-enters the horizon (at point F in fig. 1 ), giving [ 1 ] 0p

7[._3/2 H 2

_3/2 3 H 3

f~2N3/2

(9) where all quantities are to be evaluated when 2 ~ leaves the horizon (point A), the coefficientn -3/2 is relevant for radiation-dominated re-entry, and NAB~I~(d~QF/Mp) 2 is the number of e-foldings before the end of @oF inflation that 2galleaves the horizon. Finally, the dotted line between points C and D is a period of possible ~2 chaotic inflation (see section IV) during which In H - i is roughly constant and In R/Ro increases by NeD~--2n (tY~t/Mp)2 e-foldings if We can now calculate NAa, and hence Op/p (9), by starting from point B [whose coordinates (0, 14.4) are determined from eq. (8) ] and using the equations

InH-~=ln(at) , #2 In our cosmological analysis we will assume that the scalar fields 0r evolve according to their classical equations o f motion (5). Although we expect quantum fluctuations (~2) to play an important role during the initial phase (I) of chaotic inflation, our semi-classical approximation (5) becomes a good approximation by the onset of quasi-flat inflation (II) since H~R~/Mp<
370

12

A(In ff---o~)=f Hdt= 1-1n(

(10)

/I

for each period of radiation- and matter-dominated growth. The times associated with these periods are ta=to -~ 1 0 6 M F j , tea_to=fit -1~_ 1 0 1 6 M F l, tE~_F~l _ 1043Mb- 1, to (the transition from radiation to matter domination) -~ 1012 s ,~ 1055 M~-1 , and tn~-h ( = today) -~ 1.5× 101° y r ~ 10 61 M~-1 This results in a calculated value for the x-coordinate of point H (In R/Ro today) OfXH= XX= 75 + NeD, which combined with the present value of2gal ~ 1 Mpc yields k-~ 5 7 - NeD for the constant in eq. (7). Solv-

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6 April 1989

150 Matter-domlnated

era

Today

125 nated

,-4

era

I

' I~

ioo

decay

= r-I ,-. i,-4

75 2

'C A D

"~ ,-4

oscillations

50

2

2 '~AD i n f l a t i o n

25 oscillations

0 [

~

-60

-40

O~F i n f l a e l o n i

I

I

I

i

I

I

-20

0

20

40

60

80

100

I n ( R/R 0) Fig. 1. Evolution of the perturbation scale 28,, and the horizon size H - ' as functions of R/Ro, where Ro is the radius of the universe at the end o f ~ F inflation. Perturbations leave the horizon at point A and re-enter the horizon at point F.

ing (7) and ( 8 ) simultaneously for the x-coordinate o f A gives Xn ( = - NAB) --~ -- 48 + NCD. Inserting this value for NAB in (9) then gives

0p ~ (48__NcD)3/2 . p - - - N/ 3n3

(11)

Thus with no ~2 inflation (NcD-~-0) we have Op/p = 6 × 10 -4, which is somewhat larger than the required Op/p~ 1 0 - 4 - 1 0 -5, but probably consistent within the uncertainties inherent in eq. (9) derived from de Sitter fluctuations [ 10]. However, we will see in section IV that some O2-type inflation is expected in this scenario, leading to a nonzero NCD and Op/pas small as (few) × 10 -s. ( I I I ) Once ~QF has fallen to ~ ¼Mp, ~4 chaotic inflation ends. The kinetic energy stored in the ~QF field causes it to oscillate within its ~4 potential, giving rise to a period o f radiation-dominated expansion. Dur-

ing this period, the energy density in the ~)4fields falls rapidly as 1/R 4, with p = 3M2H2/8n= 3M~,/32nt 2. By the time t > fit- ~, the only significant energy density left will be due to "flat" directions in the scalar potential ½fit a¢2 with ~ ~Me. (The time scale t ~ fit - l also corresponds to one period of oscillation within the fit2¢2 potential.) (IV) When H ~ fit, the "flat" directions O, OAD begin oscillating about their respective minima = MG, ~AD = 0 as a result of supersymmetry-breaking terms ~ fit 2~2 in the scalar potential. In addition, i f O or OAo are initially >~IMp, these oscillations will be preceded by a period of ~2-type chaotic inflation. Indeed, such ~2 inflation is to be expected, since according to our general assumptions we expect ( ~ , ~AD ) ~ M p - especially for ~AD, since the ~QF direction defined previously evolves naturally into the neighboring Affleck-Dine valley ~ n 9 (4), and by assumption • o v > (few) Me. This ~2-type inflation 371

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PHYSICS LETTERS B

ac

(a)

6 April 1989

u

ac

(b)

de

(c)

Fig. 2. (a) and (b) give a net contribution to the coefficient function Cax of the baryon-number-violating operator OBx in the presence of soft supersymmetry breaking. (c) can lead to an imaginary contribution in the presence of gaugino mass terms (denoted by × ). gives NCD = 2n ( ~ I M p ) 2 e-foldings o f inflation, with NCD < 48 needed to ensure that '~ga, is produced during O~v inflation (see fig. 1 ) #4. This NCD affects the estimate for Op/p ( 11 ), reducing Op/pfrom ~ 6 × 10 -4 to a value > (few) × 10 -5. The evolution of ~AD is also affected by baryonnumber-violating operators Oax generated by Feynman diagrams like those in fig. 2. In the limit o f exact supersymmetry, the two diagrams (figs. 2a, 2b) cancel, as explained in ref. [6]. However, in the presence of soft supersymmetry-breaking terms this cancellation is incomplete, leading to a small finite coefficient function Cax ~ rh2/M~ for the operator OBx = ( Q*d ~) (/~*fi~). In the presence of OBx, the (I)AD develops a net baryon number per particle [ 6 ] R~

Im CBx ( OBx ) 1 fit2[ ~AD I 2

Orb 2 = Im

<0~x >; '~

where the imaginary part results either from CP violation in the initial conditions < OBx >; or from an imaginary phase O ~ a/4~r in the coefficient function CBx generated by fig. 2c. These OAD oscillations ultimately decay when

/'AD=H= 4N/-~ r~aD IO~D I x ~M~r

(13a)

where 2

(13b)

[(:I:)dAD[ 2'

[(I)dAD[ is the amplitude o f (I)ADoscillations at the time o f decay, and r = r~ 2 1~12/fft2D I (I)ADI 2 is the ratio o f ~ , ~AD energies, which is expected to be O( 1 ) #5 F r o m (13) we can compute the energy density and ns/s of the OAt> decay products:

M ~ + I OAD I 2 rh21q~AD [2] ~< 1

p ~ D ~ ,2¢rtAD ~ 2 I,,,,d "a"AD12 (O~AD> M G ) ,

(12)

,,4/3 "'AD ~"P

1~4It is interesting to note that for 32~NcD<48, the perturbations generated at A (fig. 1 ) would temporarily re-enter the horizon between B and C, be re-inflated outside the horizon, and finally re-enter the horizon a second time at F. The initial OQV density perturbations would then be augmented by additional perturbations produced during rh ~(l)2-type inflation, but the small size of ~ makes these additional perturbations negligible.

372

3 ~,/3 k~// (1 + r ) -

1020 GeV 4 ,

()

R p AD m AD

nB

s

AD

(NAD),/4(pdD)3/4 25R

#5 However, see the later discussion.

l,,

(14a) (14b)

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PHYSICS LETTERS B

where NAp = 63 × ~ is the number of particle degrees of freedom at TOAD~ (pdD)l/4~ 105 GeV, and we have interpreted the oscillating (PAD field as a coherent state of particles of mass r~, number density pdAD rh-~, and baryon number + 1. Although some authors [11 ] contend that finite temperature effects place an upper bound on the reheating temperature of T~D 1 ) or soon becomes (if r < 1 ) dominated by (P oscillations, which ultimately decay when H=Fa,=efft3/M 2. At this time, the energy density and temperature due to (P decays are [ 7 ]

p

3 ~2,,~,6 ~ar2 tr¢t~lw 8n M4 '

(15a)

p$

(15b)

c

P$-

(30

)1/4

where NMev = ~ is the number of degrees of freedom at T ~ ~ 1 MeV. In order to preserve the successes of standard big bang nucleosynthesis (SBBN) calculations, it is necessary that T ~ > 1 MeV, which places a non-trivial constraint on the coefficient e = 9,~,4 2 . 3 o r 7 / 2 0 4 8 n 5 in F . [ 7 ]. The (P decays lead to an increase in entropy by a factor

Nl/4

/ . d ' )3/4 MeV\P q:, A___ •ATl/4g~d ~3/41D d /Dd''t3 ' , AD k / J A D / kl~-AD/l'-tla ]

(16)

where the ratio of radii

f RAD'~3

p~,

p,~,

- (p.)~D - rp~D"

(17)

Thus

A~_r( NMevpdAD) 1/4 NADp~, ~ lOSr( 1 + r ) - 1 / 1 2 ( rnAD~ 5/6 ~-e-e-eVJ '

(18)

where we have used T ~ ~ 1 MeV to calculate p~ in eq. ( 15 ). Hence (P decays dilute (nB/S)A D by a large factor d ~ 10 8, providing a natural solution to the

6 April 1989

general problem rlB/S~ 1 associated with AffieckDine cold baryosynthesis in the framework of chaotic inflation. Combining eqs. ( 14b ), ( 18 ), and (15b), our final expression for nB/s is particularly simple:

nB s -

1(/'/B) T

R T(~ r ff/AD

R_..~I (p(~

~1/4

AD-- r r h A D k N M e v /

10_6 R r

(19)

where we have used T ~ ~ 1 MeV and F~/AD~ 1 TeV. Note that nB/s~ 10 -9 requires R/r~ 10 -3. As we have seen, R reflects the magnitude of CP violation in the coefficient function CBx and/or initial conditions ( OBx ) 1 ( 12 ), and could easily be 10- l_ 1 0 - 3. Note also that r=ff/~[ OI2/ff/2D ] OAD[2~ (1-100) is consistent with our general assumptions (P/~O(Mp) and with previous arguments [8] and estimates [3] which favor a heavy Jfio> 1 TeV from SBBN constraints and dynamical calculations. Thus a value ofn,/s~ 10-~-10 - 1o seems highly plausible. In this paper we have considered chaotic inflation and Affieck-Dine cold baryosynthesis in the context of a recently proposed flipped SU (5) X U ( 1 ) G U T derived from the superstring. We have shown that the presence of a quasi-fiat direction 20~v with 2 = 12e [2~ 10- Jl leads naturally to density perturbations Op/p= 10-4-10 -5, without resorting to any additional ad hoc inflation field. We have also shown that the decay of the Affleck-Dine like direction (PAD followed by (P decay leads naturally to nB/S~ 10 -610-m, taking into account the large entropy factor A ~ 108 produced by (P decays. The conditions under which these results apply are (1) (P~F ~> (few) Mp is needed to provide adequate (P~F-type chaotic inflation, giving a characteristic Op/p~ 6 X 10 -4. [This condition can be relaxed to (PQF> Mp if there is subsequent (PZ-type inflation produced by the (P and/or (PAD fields (see condition 2 below). ] (2) (Pz or (I)/AD~Mp is useful (though perhaps not essential) to give (P2-type inflation (Nco) and thereby decrease Op/pfrom its above value to >~ (few) X 10 -5. Given these plausible conditions, our results for Op/ p and na/s can be considered generic features of our flipped SU (5) × U ( 1 ) GUT, and may extend to other theories with Affleck-Dine directions, a small elec373

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PHYSICS LETTERS B

t r o n Y u k a w a c o u p l i n g 12e I 2 ~ 1 0 - ' ', a n d G U T symm e t r y b r e a k i n g a l o n g a fiat d i r e c t i o n in t h e scalar potential. We f i n d it i n t e r e s t i n g a n d e x c i t i n g that this single S U ( 5 ) X U ( 1 ) G U T d e r i v e d f r o m the s u p e r s t r i n g offers n a t u r a l s o l u t i o n s to so m a n y o u t s t a n d i n g p r o b l e m s in g r a n d u n i f i c a t i o n a n d c o s m o l o g y . T h i s enc o u r a g e s us to t h i n k that as physics b e c o m e s m o r e p r o f o u n d l y b a s e d o n the u n i f i e d field, s o l u t i o n s to o t h e r o u t s t a n d i n g p r o b l e m s , such as the c o s m o l o g i cal constant, will be f o r t h c o m i n g . We w o u l d like to t h a n k K e i t h O l i v e for his c o m m e n t s o n the m a n u s c r i p t .

References [ 1 ] M.S. Turner, in: Cosmology and particle physics, Proc. XVII GIFT Intern. Seminar on Theoretical physics, eds. E. Alvarez et al. (World Scientific, Singapore, 1987), and references therein; E.W. Kolb, in: From the Planck scale to the weak scale: toward a theory of the universe, Proc. Theoretical Advanced Study Institute (University of California, Santa Cruz, CA,

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1986), ed. H.E. Haber (World Scientific, Singapore, 1987), and references therein. [2] A.D. Linde, Phys. Lett. B 129 (1983) 177; B 162 (1985) 281; B 202 (1988) 194; Rep. Prog. Phys. 47 (1984) 925; Prog. Theor. Phys. Suppl. 85 ( 1985); Mod. Phys. Lett. A 1 (1986) 81. [3] I. Antoniadis, J. Ellis, J.S. Hagelin and D.V. Nanopoulos, Phys. Lett. B 194 (1987) 231; J. Ellis, J.S. Hagelin, S. Kelley and D.V. Nanopoulos, Nucl. Phys. B 311 (1988) 1. [4] I. Antoniadis, J. Ellis, J.S. Hagelin and D.V. Nanopoulos, Phys. Lett. B 205 (1988) 459; B 208 (1988) 209. [5 ] B. Campbell, J. Ellis, J.S. Hagelin, D.V. Nanopoulos and R. Ticciati, Phys. Lett. B 198 (1987) 200; H. Dreiner, J.L. Lopez, D.V. Nanopoulos, and D.B. Reiss, Phys. Lett. B 216 (1989) 283. [6] I. Affieck and M. Dine, Nucl. Phys. B 249 (1985) 361. [7] B.A. Campbell, J. Ellis, J.S. Hagelin, D.V. Nanopoulos and K. Olive, Phys. Lett. B 200 (1988) 483; J. Ellis, J.S. Hagelin, D.V. Nanopoulos and K.A. Olive, Phys. Lett. B 207 (1988) 451. [8] J. Ellis, J.S. Hagelin, S. Kelley, D.V. Nanopoulos and K.A. Olive, Phys. Lett. B 209 (1988) 283. [9] F. Graziani and K. Olive, Phys. Lett. B 216 (1989) 31. [ 10 ] S. Hawking, Phys. Lett. B 115 ( 1982 ) 295; A.A. Starobinskii, Phys. Lett. B 117 (1982) 175; A. Guth and S.-Y. Pi, Phys. Rev. Lett. 49 (1982) 1110; J.M. Bardeen, P.J. Steinhardt and M.S. Turner, Phys. Rev. D28 (1983) 679. [ 11 ] A.D. Linde, Phys. Lett. B 160 (1985) 243.