Observationally constrainable free parameters of scalar field inflation models

Volume 260, number 1,2

PHYSICS LETTERS B

9 May 1991

Observationally constrainable free parameters of scalar field inflation models Bharat Ratra Theoretical Astrophysics and Theoretical Physics, California Institute of Technology, Pasadena, CA 91125, USA Received 25 January 1991; revised manuscript received 27 February 1991

We present an approximate estimate, in the slow "reheating" scenario, of the late time power spectrum of adiabatic energy density inhomogeneities generated during an early, scalar field dominated, de Sitter inflation epoch, when the scalar field potential, V(O) ~ Vo-2q~ (where Vo,2 and a are positive parameters). I f a is not too large, the power spectrum is fairly sensitive to the value of the Hubble parameter at "reheating", except in the a = 4 case (when it only depends logarithmically on the value of this parameter). When a is sufficiently large, the power spectrum is much more sensitive to the value of the Hubble parameter at "reheating", than it is to the value of the dimensionless coupling constant (2 multiplied by the appropriate power of Vo), which is, therefore, not unduly constrained by observational data; also, in this regime, the slow "reheating" prescription resembles the rapid "reheating" prescription. For a moderately large a (e.g., a = 8 or 10), the magnitude of the energy density perturbation is consistent with the late time large-scale observational limit, provided the energy scale of inflation is not too high (for a = 8 or 10 it needs to be ~ 2 × 10l ° o r 1011 GeV); the dimensionless coupling constant does not have to be small (in both cases it is > 10-2).

O u r analyses o f the generation and evolution o f spatial irregularities in simple, scalar field d o m i nated, r a p i d " r e h e a t i n g " [ 1 ], inflation models, leads to a late time, gauge-invariant, power spectrum o f large-scale adiabatic energy density inhomogeneities that [ 2 - 4 ], ( i ) for the case o f a fairly flat inflation epoch scalar field potential and de Sitter inflation, increases as the slope o f the inflation epoch potential is decreased, and that, ( i i ) for the general case (a not necessarily fiat scalar field potential a n d not necessarily de Sitter inflat i o n ) , d e p e n d s about as sensitively on the redshift o f "reheating" as it does on the slope o f the inflation epoch potential. Although both o f these results differ greatly from the conclusions o f the s t a n d a r d analysis [ 5-8 ] o f the quartic potential scalar field inflation model, they cannot be directly c o m p a r e d to the results o f the s t a n d a r d analysis since it makes use o f the slow "reheating" prescription (described below), which differs from the r a p i d " r e h e a t i n g " prescription we have used [ 1 ]. The purpose o f this p a p e r is to show that a class o f

slow "reheating", scalar field d o m i n a t e d , de Sitter inflation models has a late time energy density power spectrum that depends, in a part o f the model p a r a m eter space, about as sensitively on both the redshift o f " r e h e a t i n g " and on the slope o f the inflation epoch potential, to show that the quartic potential inflation m o d e l is nongeneric (i.e., in this case the d e p e n d e n c e o f the power spectrum on the redshift o f " r e h e a t i n g " is only l o g a r i t h m i c ) , and to show that in a different region o f the model p a r a m e t e r space the slow "reheating" prescription resembles the r a p i d "reheating" prescription and, when this happens, the d i m e n sionless coupling constant o f the model is not unduly constrained by observational d a t a [ 2 - 4 ]. The inflation m o d e l we study is described by the scalar field effective potential, V ( ~ ) = 2 M 4- ½ ~ M 4 - a ~ a ,

( 1)

where M is the energy scale o f inflation, )Tand c~ are constant positive dimensionless p a r a m e t e r s a n d the factors o f 2 are the consequence o f a partial a t t e m p t to be consistent with the conventions o f section 4 o f ref. [ 1 ]. P r i o r to outlining our analysis o f the evolution o f

0370-2693/91/$ 03.50 © 1991 - Elsevier Science Publishers B.V. ( North-Holland )

21

Volume 260, number 1,2

PHYSICS LETTERSB

energy density irregularities in this inflation modified hot big bang model we illustrate the reasons for some of the conclusions listed in the paragraph above the previous one. The dominant 2and M dependence of the late time baryon dominated epoch power spectrum of energy density perturbations is [see eq. (22) below] // . \2(o~--4)/(¢~--2) \mpl ] where rnp~ is the Planck mass and we only consider the range o~>2 (see below). I f a ¢ 4 (and is not too large), the power spectrum is sensitive to both 2 a n d M, while the ot = 4 case is nongeneric (since the M dependence drops out). If ~ is sufficiently large, the power spectrum is relatively insensitive to 2~ (which can, therefore, not be unduly constrained by observational data). We now turn to describing the derivation of the above formula. We assume, during inflation, that the parts of the spatially homogeneous scalar field (q)b) energy density satisfy

M4>> 1½ ( ~ b ) 2 - l ' ~ M * - " ( q ~ b ) " [

(2)

(where a dot denotes a time derivative); as a result the Hubble parameter of a spatially fiat FLRW model, during inflation, is time independent and given by

87l" M 4 3 rn ~,, '

(3)

and the FLRW scale factor obeys the de Sitter law,

a(t)=a~exp[H(t-ti) ] ,

(4)

where aj is the value of the scale factor at the initial reference time ti. We have shown, in section 3 of ref. [ 1 ], that the spatial momentum dependence and important free parameter dependence of the baryon dominated epoch, large-scale, gauge-invariant, adiabatic energy density power spectrum is adequately summarized by the standard expression [ 9 ]

e°(k, t,= × lim k << a H

22

k

4

(a(,,m,)) P(m" ) + Lt~] (O(k,t)~J*(k,t))+.~.]

, ..Jt=tc

(5)

9 May 1991

where ~ is the spatially inhomogeneous part of the inflation epoch scalar field, k is the magnitude of spatial coordinate momentum, complex conjugation is denoted by the star, the subscript G.I. on the scalar field perturbation two-point correlation function means that one is to remove from this quantity the contribution of the gauge-dependent time translation solution, and tc is the time at which the spatial momentum mode of interest crossed the Hubble radius during inflation [i.e., it is defined by the equation k=a(tc)H(tc), where a and H h a v e the form appropriate to the inflation epoch of the model ]. We emphasize that the variables within the square parentheses on the right hand side of eq. (5) are inflation epoch variables, while those outside the square parentheses are baryon dominated epoch variables. With our conventions, and ignoring the second term in eq. ( 1 ), the de Sitter spacetime scalar field perturbation two-point function results in IH 2 lim (O(k,t)O*(k,t))G.L= k3 ; k << a H 2

(6)

this should not be a very inaccurate approximation. From eqs. (1) and (4), the spatially homogeneous scalar field equation of motion is 4ib + 3H
(7)

To solve this equation in the slow "reheating" scenario, one makes the "slow-rolling" approximation, q6b << 3Hq~b ;

(8)

with this approximation the solution of eq. (7) is

¢2bb(t) = ((qbb * ) 2 - a c~(a-2)

+ 8 ~

"X- 1 / ~ - 2 )

2mp'M2-"(t*-t))

,

(9)

where t,>t and the constant of integration, q~b, - q~b(t,) [because of the approximation (8), one solution of eq. (7) has been discarded and we need to determine only one constant of integration ]. We note that for t, > t this solution implies ~b, > q~b(t), independent of the value of (finite) c~. In the slow "reheating" scenario, t, is the time at which the "slowrolling" approximation breaks down, i.e., it is defined by

Volume260, number 1,2

PHYSICSLETTERSB

~b(t,)=3H~b(t,) .

(10)

9 May 1991

~ b ( t ) = 24n M 4 (__96n

M '~ ,]1/(,~-2)

From eq. (9), we have ~b(/) Or(a-- 1 ) m~l,~ ]a-2 3H~b(t) = 96n M ~ [~b(t) ,

1 Jvnpl (t,--t)

(11)

(17)

and

--

\ 3 n ~ b J, < t. k ~b

.Jr=

t.

~-~(*b (t) x~o:-2 \

Cb, ]

(12) '

since ~b, > 4~b(t), if a > 2, W< 1 and the "slow-rolling" approximation eventually breaks down, while if a < 2, W> 1 and the "slow-rolling" approximation improves with time, so the slow "reheating" prescription cannot be used to fix the constant of integration in this case (in what follows we focus on the range a > 2 ). We note, from eq. ( 11 ), that the acceleration vanishes when a = 1 and that when a < 1 the force is such that the field is decelerated. Using eq. ( I 1 ) and the definition of eq. (10), we find ( 96n qbb* = \Or (~---2---1 )

M".~'/("-e)

mg,2]

(13)

Me 2] +...}, q~b.=M { 1+ 1ot [ln(97 n--~pi)--

(14)

i.e., in this limit q~b, is only very weakly dependent on From eqs. (9) and (13) we have

96n

ma.'~ l/(a-2)

× ( 1 + 2 v / ~ Ot--~mp~a--2 M2

q)b(t)=¢b.--Mln(l+2x/r~(t,--t))+..., ~bb(t) =

M a

( t , - t ) )X-l/(a-2) ,

4ib(t)--

M

'

"~-...

so when a becomes large, qbb(t ) evolves slowly and presumably can take a long enough time (depending on initial conditions) to ensure sufficient inflation before it gets to q~b, (which is almost 2~independent in this limit). When ot >> l, the slow "reheating" prescription is very similar to the rapid "reheating" prescription of refs. [2-4] - presumably, once q~b(t) reaches ~b,, it gets to the minimum of the complete potential in a time much shorter than that set by the Hubble parameter. Using eqs. ( 3 )- (6) and ( 16 ) we have at the present time,/now, t,ow)= ~ It

"

\mpl

l\ - (or- l )/(c~-2)

a - 1 real ( t , - t ) / (16) and

24nM4/m~,j

a [1 +2x/~(M2/mpl)(t,-t)]2

(15)

2X//~ M2 ( 96~ Ma~ ',(a-2) a - 1 mpl \ o ~ ( a - 1 ) m~,lZ]

X{t I + 2 x / ~ a - 2 M2

(18) (19)

+...

and

~k3p°(k, ~bb(t)

2w/-6-nM2/mp, 1 + 2 x / / ~ (M2/mp,)(t,-t)

(20)

"

In the limit when a >> 1 ( a ~ o o ) , this reduces to

(

we note that as 2is decreased, ~bb(t) increases - this is because the slow "reheating" prescription pushes q~b(t) further down the steep part of the potential as 2is decreased [eq. (2) provides a lower limit on ,~so ~b (t) cannot get arbitrarily large in this approximation]. In the limit when a > > l (c~--,oo), these expressions reduce to

16

(a- 1

)2 (a(oz-- 1 ).)2/~,~-2) \

\2(a--4)/(ot--2)

/

[ (~-2 In ( 4 \"12(c~--1)/(c~--2)/ _ a , )I ] _K ' ) × l+3a-1 ac anowH, ow " (21) 23

Volume260, number 1,2

PHYSICSLETTERSB

Here Hnow is the present value of the Hubble parameter and ao a . and anow are the values of the scale factor at time tc (when the scale leaves the Hubble radius during inflation), t. (when the "slow-rolling" approximation breaks down) and t .... Assuming that the present universe is spatially flat and dominated by pressureless matter, we have 16 ( a -

1)2

22k3P~(k' /now) = ~ /t

.

(a(a\ k

x2(Ot--4)/(~t-2)

X"~2/(a-2)'\mr,i M~)]

1 ) ) 2/('~-2)

96n 4

C/now~/now)

X{l+3~lln[(8~)l/4an°"vHn°Wk mm

,

( 1 +ZBR)1/4

(22) where ZBR is the redshift of equal radiation and baryon energy density. In the limit when c~ >> 1 ( a - , oo ) this equation reduces to

1 k3p~(k ' /now) = 7Z2

M

+ ....

1

2

(23)

which is ~Tindependent to leading order in 1/oz. Some preliminary conclusions that may be drawn from these expressions include: (i) the baryon dominated epoch power spectrum is approximately scale-invariant [ 10 ] as is expected of de Sitter inflation [ 5-8,11,12 ]; (ii) for a not too large, the power spectrum is roughly equally sensitive to the value of the dimensionless coupling constant, 2, and the value of the energy scale of inflation, M; (iii) the quartic potential inflation model, a = 4, is nongeneric - the power spectrum depends only logarithmically on M - conclusions drawn from an analysis of this model must be interpreted with due caution; (iv) for a not too large, decreasing ~Tdecreases the 24

power spectrum [5-8] - the slow "reheating" prescription has washed out the de Sitter inflation "divergence" [ 2-4 ]; (v) for ot < 4, decreasing M increases the power spectrum, while for a > 4, decreasing M decreases the power spectrum; and, (vi) for sufficiently large a, the power spectrum is almost ,~independent (~is, therefore, no unduly constrained by observational data), and as ot is increased [ which decreases ~b (t), eq. ( 19 ) ] the power spectrum increases - in this limit the slow "reheating" prescription resembles the rapid "reheating" prescription [ 1 ] and can no longer wash out the de Sitter "divergence" [ 2-4 ], although there is still an extra logarithmic dependence on scale [ 1 ]. The constraint (2) may be expressed as ~2/(ot-2) >> [R(ot,

M/rnpt) ] 2 / ( . - 2 ) ,

(24)

where we have defined

R( ot, M/mpl)

16 2[ M'~21" k ,4 -Ot - • -6757 ~kmpl) ~.a.owH.ow]

X{I+ 3 ln[(8/Q 1/4 an°w~/n°w

9 May 1991

- 12n°t(o~--l)2kmm] d

Ol--

(25)

[we have evaluated the right hand side of eq. (2) at t,]. If we require that the energy density perturbation, on the scale of the present Hubble radius, satisfy ~-2(k3p~)(k=anowHnow, tnow)<10 -8 (where the factor of n - 2 is a consequence of our normalization of the Fourier transform), then with the standard values, //now = 100h km s - l M p c - ~ and ZUR= 4 × 104h 2, and approximating (where we have taken h=0.5),

F

/

\1/2

ln|(~_n)l/4 [ mm ~ L \nnow. ]

1

7

( 1 +ZBR)i/4 J -~68.6,

(26)

by 70, we need, from eq. (22), )~2/(.-2) < [ Q ( a ,

Mlmpl) ]2/(.-2)

where we have defined

,

(27)

Volume 260, number 1,2

PHYSICS LETTERSB

Q(a, M/mp~) - -

4.3X,0-"[ 421 "~3 a ( c t - z-rr ) f5.5X

3 °-2 (~)]-3 -21 1 -oL2-~-421In

1+

-

10 - 6 mpl

x ~ - - 4--~T L c~--~Tr M

--1 t~--4

[

3 a - 24z~ In ( ~ P l ) ] × 1+ 211 a--z~T

}

.

(28)

We emphasize that eq. (27) is only an order of magnitude estimate - a more accurate method for comparing to observational data is described in ref. [ 2 ]. To illustrate the numerics we consider a few specific values o f a . We first consider the constraint (24); we find /

~ar\ 3

R ( 3 , M/FHpI ) ' ~ ' 8 9 ~ P t )

,

(29)

t' ~ , r \ 4

R(4, M / m p l ) - - - 53/~-~pj),

(30)

/mar\ 8

R(8, M/mp,)~-~-~PL)'

(31) / m a r x 1o

R( 10, M / m p l ) ~ 6 . 4 X

10-2~-~p1)

.

(32)

9 May 1991

ing" problem - choosing M = 10 -Smpl we find Q(4, 10-5) -~2.3x 10-12 and R(4, 10-5) _~5.3 X 10 -19, SO eq. (24) is satisfied. We note that (~)2Q(4, 10 -5) --- 1.3x l0 -1°, which agrees with the result of section 4 of ref. [ 1 ] (the extra numerical factor is a consequence of the different conventions used in refs. [ 8,1 ] ). When a = 3, if we take M = 0.1 mpl, we have Q(3, 0.1)-~2.8x10 -7 but R(3, 0 . 1 ) ~ 8 . 9 X 1 0 -2, which will not work. If, however, for a = 8, we take M = 2 X 10-9mpl, we find Q(8, 2 x 10 -9) --~1.1 X 10 -2 and R(8, 2X 10 -9) "-2.6× 10 -7°, and if for a = 10, we take M=10-Smpj we find Q(10, 10-8)-~ 1.3x 1 0 - 2 and R ( 10, 10 - 8) _~6.4 X 10- 82, both of which satisfy eq. (24). For large enough a eq. (23) should be used; if we take our value M = 10-8mpl for the a = 10 model, we find the maximum a is ~ 740 which is much larger than 10 - so even with a = 10 we are still some distance away from the region of the de Sitter "divergence". To summarize, if one wants the dimensionless coupling constant )~to be not very much smaller than unity, one needs to have the inflation energy scale lower than where it is conventionally placed and one also needs a larger value of a than is usually considered. One might wish to mimic the converse of the standard procedure for deriving the quartic potential inflation model [ 5-8 ]. If we define

~o =(4") TM M,

From eq. (28) we find

(37)

k,t/

Q(3, M/mpl) _~2.6X10_ 6 M mpl

[

and consider the effective potential l+~ln

(33) ~

'

V(@) =2M4{ 1 - ( ~ o ) a [ 1 - a l n ( ~ o ) ] } ,

(38)

Q(4, M/mpt) --3

"--1.3X 10-'2 D + T~T41 ln(~--~el)]

,

(34)

Q(8, M/mpl) ~- 1.7X 10 -38

X

,8 In 1 -~- 1267

'

(35)

Q( 10, M/mpl) ",. 8.3 X 10 -52 × ( ~ p l ) - 6 [ l + ~ 3 ln(~--~el)] -9 .

the standard assumption is that the logarithmic term does not change much on top of the hill and so this reduces to the effective potential of eq. (1) during inflation. Eq. (38) has a minimum at ~o; expanding about it, ~=~o+ ~,

(39)

it is straightforward to show that the mass, cubic and quartic couplings of the 7J field are given by (36)

Eq. (34) illustrates the quartic potential "fine-tun-

m. = / ~ O L { ! ~ l/amPl ~ ~" ~[8~ ~4 J M " '

(40) 25

Volume 260, number 1,2

PHYSICS LETTERS B

)1"3'/'=~ / - ~ "-~O~2(2Ot-3) (1~'~3/c~ t - / 4 " ~mPI 'M "1 '

(41)

2 4 ~ , = ~ 4 0 ~ 2 ( 3 a 2 - 1 2 a + 11 ) (l~')4/a

(42)

,

where H is the Hubble parameter during inflation. Using the quartic potential values, a = 4 , ,~--2.3× 10 -12 and M = 10-Srnpl, we have m e = 120H, /l 3v,= 3 X 10-4H and 24~/t ~' 4 X 10-12; as mentioned in ref. [ 1 ], it seems that these values might be too small for the model to be able to "reheat" (we note that in this case one is allowed to adjust the energy scale of inflation to increase m ~,and 23 wwithout significantly changing ;~ 24~, is, however, more directly constrained by observational data). If we consider the c~=8 model with 2---0.011 and M = 2 × 10-9mpl , w e find m ~ 7 x l 0 S H , J . 3 ~ , ~ 3 X 1 0 9 H and J.4~--~15, while for the a = 1 0 case, with 2"-~0.013 and M~--10-Smpl, we have m ~ = 2 X l 0 S H , 23e---2× 109H and J . 4 ~ 8 0 , which are probably more than sufficient to ensure that the model "reheats". We emphasize that the numerical values we have used or derived above are only order of magnitude estimates. Furthermore, they are only meant to be illustrative - we have made no attempt to systematically study the parameter space of the model. We close with a few general comments about this class of models. The observational data and the requirement that ~be not too small seems to force one to pick a nonrenormalizable effective potential with a lower than usually considered energy scale for inflation. This is not disturbing since there is no reason to expect that an effective model will be renormalizable (even if it is experimentally acceptable, e.g., Fermi's weak interaction model or Einstein gravity) nor does a M/mpl much smaller than unity mean that the model is ruled out (the standard electroweak model comes to mind). However, i f M / m p l really turns out to be as small as we seem to find, then the GUT baryosynthesis mechanism probably cannot be used this also does not seem to be a serious problem since there is, at present, no indication that baryosynthesis must work this way - it would, of course, be nice if

26

9 May 1991

one could find an alternate, low-energy, baryosynthesis mechanism, It would also be useful to attempt to find a microphysical model that results in the effective potential of eq. (38) and to decide whether this effective potential is "natural" [ 13,14 ] ~ Finally, we emphasize that these results must be considered to be preliminary until the "reheating" prescription used here is actually shown to be an effective approximation of the microphysics of "reheating". I am indebted to J. Preskill for valuable advice. This work was supported in part by the NSF, grant AST8451725, by the DOE, contract DE-AC03-81ER40050, and by the California Institute of Technology. #~ This was emphasized by J. Preskill.

References [ I ] B . Ratra, Caltech preprint GRP-242/CALT-68-1701 (1990). [2] B. Ratra, Princeton preprint PUPT-1102B (1989). [3]B. Ratra, Caltech preprint GRP-217/CALT-68-1594 (1989). [4]B. Ratra, Caltech preprint GRP-229/CALT-68-1666 (1990). [ 5 ] S.W. Hawking, Phys. Lett. B 115 (1982) 295. [6] A.A. Starobinsky, Phys. Lett. B 117 (1982) 175. [7] A.H. Guth and S.-Y. Pi, Phys. Rev. Lett. 49 (1982) 1110. [8] J.M. Bardeen, P.J. Steinhardt and M.S. Turner, Phys. Rev. D28 (1983) 679. [9]J. Bardeen, Cosmological perturbations from quantum fluctuations to large scale structure, University of Washington preprint 40423-01 C8. [ 10] E.R. Harrison, Phys. Rev. D 1 (1970) 2726; P.J.E. Peebles and J.T. Yu, Astropyys. J. 162 (1970) 815; Ya.B. Zel'dovich, Mon. Not. R. Astron. Soc. 160 (1972) IP. [ 11 ] W. Fischler, B. Ratra and L. Susskind, Nucl. Phys. B 259 (1985) 730. [ 121B. Ratra, Phys. Rev. D 40 (1989) 3939. [ 13] L. Susskind, Phys. Rev. D 20 (1979) 2619. [ 14 ] G. 't Hooft, in: Recent developments in gauge theories, eds. G. 't Hooft et al. (Plenum, New York, 1980).