PHYSICAL REVIEW D VOLUME 3 1, NUMBER 4 15 FEBRUARY 1985

Evolution equation for the Higgs field in an expanding universe

Gordon Semenoff and Nathan Weiss Department of Physics, University of British Columbia, Vancouuer, British Columbia, Canada V6T 2A6

(Received 30 March 1984)

Using a new perturbative technique for evaluating real-time finite-temperature Green's functions in an expanding universe, we derive the evolution equation for the Higgs field during an inflationary phase. We also indicate how to evaluate the fluctuations of the field in this approach.

The new-inflationary-universe scenario' provides a reasonable explanation of several outstanding cosmologi- cal problems.2 Recently, attempts have been made to ex- plain, within this model, both the magnitude and spectra of primordial density fluctuations3 which could lead to the presently observed fluctuation spectra.4 Starting with a grand-unified-theory (GUT) model such as SU(5) with a Coleman-Weinberg symmetry-breaking scheme, one derives a finite-temperature effective potential v$~(+) for the Higgs field 4. In the very early universe, one supposes that T=D-' is large and that (4)=O. As the universe expands, it supercools to well below the critical tempera- ture for this GUT. This supercooling in the "false" vac- uum leads to a constant energy density and consequently to an exponential (de Sitter) expansion. It is believed that when 4, - ( 4 ) fluctuates from zero in some region of space, it "rolls" down the potential hill via the equation

where H is the Hubble "constant" (to a good approxima- tion /3+ ). Veff is the static effective potential defined in flat spacetime. Since this equation plays a key role in calculating the details of the inflationary model, we have examined its validity in the nonstatic situation of an ex- panding universe. We derive an expression for the correct one-loop evolution equation for I$ which in a certain "adi- abatic" limit reduces to the above equation.

The essential ingredient of our work is a new method which we have developed for perturbative calculation of real-time thermal Green's functions in a fully interacting field theory which is applicable to the nonequilibrium sit- uation of an expanding ~ n i v e r s e . ~ This method is modeled after the work of Niemi and semenoff6 who have developed a similar method in flat space-time. ~ r u m m o n d ' has developed a similar method based on an alternate choice of the initial density matrix. We shall give a brief outline of the technique in this paper. The de- tails are presented in the preceding paper. In order to simplify matters and to clarify the discussion, we restrict our attention to a one-component scalar 44 field theory. (This model of course has the flaw that the location of the minimum in Veff for the "Coleman-Weinberg" case falls outside the realm of validity of perturbation theory.) The scalar field propagates in a background Robertson-Walker (RW) space-time. We shall suppose that at some early

time to (earlier than the GUT transition) the system is in thermal equilibrium at a temperature T =p-' > T,",YT and then follow the evolution of the system via the quan- tum equations of motion.

Finite-temperature effective potentials in a cosmologi- cal context have been previously considered, notably by H U ' , ~ who uses a quasiadiabatic approach for the analysis in 44 theories. Although the location of the minimum of the effective potential yields the expectation value of 4, it is not obvious that Veff can be used away from its minimum to study the nonequilibrium behavior of the system (even in the one-loop approximation) by simply writing D24+dVeff/d4=0. In this work, we shall show that an equation of the form D2q5 +F($J) =0 can be de- rived directly from the field theory but that, in general, F(4) will be a nonlocal functional of 4. The form of F(4) is explicitly displayed for spatially flat Robertson- Walker spaces and, as expected, it reduces the form D 2$ +d veff/d4 = 0 in a certain "adiabatic" approxima- tion.

Consider a spatially flat Robertson-Walker background described by the line element"

Suppose the scalar field has an action

with

where 6 is a real parameter and R is the curvature scalar,

(an overdot denotes d/dt). In RW coordinates

Suppose that at some initial time to the system is in thermal equilibrium at a temperature P-'. In other words, let us prepare the system at time to such that it is

699 @ 1985 The American Physical Society

700 GORDON SEMENOFF AND NATHAN WEISS

described by the density matrix

p ( t o ) = ~ - 1 e x p [ - / 3 ~ ( t o ) ] , (6)

Z=Trexp[--/3H(to)] ,

where H ( t ) is the time-dependent canonical Hamiltonian for the action ( 5 ) (Ref. 11). Consider the real-time thermal Green's functions defined by

( T ( # J ( x I ) . . - ~ ( x , ) ) ) - @ H ( t o )

= ~ - ' ~ r [ e ( 4 I 1 4 , ] , (7)

where

and

This Green's function describes the expectation value of the operators in question when the system is allowed to evolve via H ( t ) from the thermal initial condition (6) to time t.

The path-integral representation for these Green's func- tions* is given by (8):

( T ( 4 ( x l ) . . . ( x , ) ) ) = N J Dd(x, t )exp[ iSp14(xl , t~) . . . 4 ( x n , t n ) , t€P

periodic B. C.

where P denotes the path in Fig. 1. N is a normalization PI and P2 and that 4, is constant on the path P3 (Ref. factor and q5 must satisfy periodic boundary conditions. 14). The times t l , . . . , t, are to be chosen along part PI of P. The quadratic part of the action on P1 is now The action Sp is given by

Sp = S for PI

= -S for P2

and

I

The terms linear in 64 are treated as a perturbation. They lead, on P I , to a vertex of the form

1

J ~ ~ ~ ' = - ~ d ~ x a ~ ( t ) ( ~ ~ + 3 ~ ~ ~ + m ~ $ ,

The Green's functions can now be evaluated by expanding - - exp(iSp) as a power series in h. This is discussed in detail in the preceding paper.5

Suppose that V(d is chosen so that ( 4 ) = f f 0 in flat spacetime at T =O. We expect that for T larger than some critical temperature T,,, the symmetry is restored

The remaining interaction vertices on PI are given by and ( 4 ) =O. We would similarly expect that in an ex- panding universe when /3-' > T,, at to that ( $(to ) ) =O. However, as a ( t ) increases, the temperature decreases (or, at least the wavelength red-shifts) and ( 4 ( t ) ) # 0 for large t. Thus, in general, we expect +,(t)= ($ ( t ) ) to be a func- tion of time. The usual effective potential techniques need to be generalized to this situation. T- i

We evaluate $,(t) using the "tadpole method" of Wein- berg.I2 The field 4 in the action Sp of Eq. (9) is shifted so thatI3

If 4, is the expectation value of 4, we have

( 6 4 ( x , t ) ) = O . (13) FIG. 1. The time path over which fields are defined in

Let us suppose that #,(t) is the same on the paths evaluating real-time thermal Green's functions.

3 1 - EVOLUTION EQUATION FOR THE HIGGS FIELD IN AN . . . 70 1

two functions f jk '( t) , i = 1,2 which are linearly indepen- ( I ) Sp, =- J d (16) dent solutions to the homogeneous equation

J k

aO2+3~ao+-+m2+3hmc2( t )+g~( t ) One can similarly write down the Gaussian and interac- a 2 ( t )

tion terms on the paths P2 and P3. It is now necessary to find the propagator for the

(17)

Gaussian part of S subject to the periodic boundary con- and which satisfy the Wronskian condition ditions at the ends of the path. (The details of this calcu-

- i lation are found in Ref. 5.) The ropagator for a given f l ( t ) f 2 ( t ) - f 1 ( t ) f 2 ( t ) = ~ . (18) spatial mode is of the form G?"(t,tt) with t E P a and a ( t )

t l E P b . The form of G ' " , ~ ' is presented in terms of any In terms of these functions, we have, for example,

Note that G"'" and G " , ~ ' are, respectively, the time-ordered propagator and the Wightman function for the free field theory. The coefficients are determined by continuity and periodicity requirements. To exhibit the form of let us define

with

In terms of these functions

Finally we exhibit the form of G ' ~ , ~ ' ,

We shall not need the explicit form of the other propagators for the purposes of this calculation. Inspection of Eq. (9) shows that an evaluation of a Green's function in perturbation theory will require three types of

vertices which we call type 1, 2, and 3 corresponding to the path on which the field at the vertex finds itself. Type-1 ver- tices can be read off of Eqs. (15) and (16). A type-2 vertex is just minus the corresponding type-1 vertex and the type-3 vertex can be derived from Eq. (1 1) by shifting the field in SP3. The type-1 vertices are of the form

702 GORDON SEMENOFF AND NATHAN WEISS - 31

In the one-loop approximation, the equation ( 64( x, t ) ) = 0 can be written diagrammatically as

One must sum over vertices of type i = 1,2,3. Of course the. full tadpole will vanish if and only if the irreducible tadpole vanishes, i.e.,

Using Eq. (25), this leads, for the one vertex, to the equa- tion

d ; , ( t )+3Hd, ( t )+ [ (m2+C~ ) 4 , ( t ) ~ - h $ , ~ ( t ) I

Using Eqs. (19) and (23),

~ L " l ) ( t , t ) = ~ : l , ~ ) ( t , t )

so that Eq. (28) becomes

$ , + + ~ d , +[(m2+CR ) 4 , + ~ 4 , ~ 1

Finally, we derive the equation for 4, on the path P3. Assuming (as discussed previously) that 4, is independent of T on the imaginary time path P3, we find

One can now use the definition of p from Eq. (21) and change variables of integration to k / a z to obtain

with ~ 2 = z 2 + m 2 + ~ ~ ( t o ) + 3 h r ) c 2 ( t o ) . This is immedi- ately seen to be derivative of the usual (unrenormalized) effective potential with respect to 4 at to. 4, = O is always a solution to Eq. (32). Other solutions may exist when m and { are sufficiently small and T < some critical tem- perature T,. (Let us defer the question of renormalization for the moment.) We thus choose 4 at to to be at the minimum of the static effective potential at the initial temperature T.

Equations (30) and (32) yield a complete description of the semiclassical evolution of 4,. In the inflationary universe scenarios, will be chosen such that /3-' > T, so

that dc(to)=O. $, is also a solution to Eq. (32) so that it seems that 4, = O for all times. Suppose, however, that at some time t l there is a fluctuation in $, so that 4,( t l )=$o#O. Equation (30) will then describe the semi- classical time evolution of $, with spatial variation not in- cluded. We are presently investigating the time evolution of 4 via this new equation.

Having derived this improved equation for the evolu- tion of 4 in an expanding universe, it is useful to under- stand under what circumstances Eq. (30) reduces to the equation used in the literature:

with Veff(4,) the usual static effective potential for the field theory in question.

To answer this question we approximate f jk ' ( t ) in Eq. (30). To this end, let us return to Eq. (171, the defining equation for f. In the case &( t ) <<p2(t) , we may solve Eq. (17) in a "WKB" approximation to obtain

1 f j:)),(2)(t)~ , , exp [ ~ i $itp(t ')dt '] . (34)

[2a 3 ( t )p ( t ) l

This is usually called the "adiabatic" approximation.'5 To obtain the result (33), we require one further assumption which is that

This approximation may be difficult to satisfy in practice. It leads, however, to the result

The equation of motion for 4, [Eq. (30)] then becomes

At zero temperature (D+ oo 1, coth[Dp(to )/2]+ 1 and, after changing variables to k /a ( t ) , the final term in Eq. (37) is simply d/d+, of the one-loop correction to the zero-temperature (unrenormalized) effective potential. This is the commonly used equation in studying the GUT phase transition in the new inflationary scenarios. (Of course the fermion and gauge field corrections must be in- cluded and m 2 and CR must be chosen small enough so that the model is the Coleman-Weinberg model or at least nearly the Coleman-Weinberg model.) In the adiabatic approximation, the expression (37) is renormalized by re- normalizing m 2,h, and g in the usual manner. Renormal- ization in the general case [Eq. (30)] is more complicated and will be discussed in a future publication.

It is important to note that Eq. (37) describes a system that a t to was in thermal equilibrium. If we choose the Coleman-Weinberg model m '=o, then for k 2 >>CR + 3 ~ 4 2 , we have

EVOLUTION EQUATION FOR THE HIGGS FIELD IN AN . . . 703

B c L ( ~ o ) coth-

2 ~ c o t h - Bk = f i l p ( t ) ,

2 a ( t o )

where p=Ba( t ) / a ( t o ). Thus for sufficiently small wave- lengths, the system behaves as if it remained near thermal equilibrium with (temperature at t ) x a ( t ) = const, as ex- pected for massless particles. In the general case, howev- er, Eq. (37) allows only for a red-shift of all wavelengths. In general, the system will not be in thermal equilibrium at a later time t unless the expansion is sufficiently slow to allow the interactions to re-equilibrate the system.

Finally, let us substitute z = k / a ( t ) in Eq. (37). The last term becomes

xco th Iii 2 [ q + A 2 ( t , - , ) a ( t o ) ] ' I 2 ] (38')

with A2=m2+<R +3h4,'. If a 2 ( t ) >>a2(to), the expres- sion (38) is seen to resemble closely the zero-temperature one-loop correction to dVe f f /d+.

Having evaluated $,( t ) , it is straigHtforward to estimate the fluctuations in 4 by computing the correlation func- tion ( S + ( x ) S b ( y ) ) in the background 4, field at equal time t. This is given by

In the adiabatic approximations (34) and (351,

For D-+ CO, the magnitude of the fluctuation on the scale ( k /a 1- ' is given by

If (and only if) m +<R + 3h4c2 << H ~ , the magnitude of these fluctuations on the scale of the horizon size ( k /a - H ) is given by

H A4 I horizon - - 2T

which is the result usually quoted in the literature. We have derived the time evolution equations for

I$,( t )= ( + ( t ) ) in an expanding universe in the one-loop approximation and identified the approximations which are needed.so that this equation reduces the often-quoted equation 4 + 3H4 + Vkff( I$ ) = 0. These are the adiabatic assumptions (34) and (35). For the Coleman-Weinberg model in de Sitter space these conditions will, at best, ap- ply only to wavelengths within the horizon whereas Eq. (37) involves an integral over all k . We have also shown that the same assumptions are required if the fluctuations on the scale of a horizon size are to be approximately H / 2 ~ .

We wish to thank Marc Sher, Michael Turner, Alan Guth, and Bill Unruh for helpful discussions. This work was supported in part by the Natural Science and En- gineering Research Council of Canada.

'A. D. Linde, Phys. Lett. 108B, 389 (1982); A. Albrecht and P. J. Steinhardt, Phys. Rev. Lett. 48, 1220 (1982).

2A. Guth, Phys. Rev. D 23, 347 (1981). 3J. Bardeen, P. Steinhardt, and M. Turner, Phys. Rev. D 28, 679

(1983); A. Guth and S. Y. Pi, Phys. Rev. Lett. 49, 1110 (1982).

4 ~ . R. Harrison, Phys. Rev. D 1 , 2726 (1970); Y. A. B. Zeldo- vich, Mon. Not. R. Astron. Soc. 160, 1P (1972).

5Gordon Semenoff and Nathan Weiss, preceding paper, Phys. Rev. D 31, 690 (1985).

6A. J. Niemi and G. W. Semenoff, Ann. Phys. (N.Y.) 152, 105 (1984); Nucl. Phys. B230 [FSIO], 181 (1984).

'1. T. Drummond, Nucl. Phys. B190 [FS3], 93 (1981). 8B. L. Hu, Phys. Lett. 123B, 189 (1983). 9B. L. Hu, Phys. Lett. 108B, 19 (1982). l0See, for example, S. Weinberg, Gravitation and Cosmology

(Wiley, New York, 1972). "This definition of the density matrix requires some justifica-

tion. One knows, for example, that if the energy-momentum tensor T p v is renormalized so that the vacuum state of H ( t l 1 has finite energy density, the consequence is that the vacuum state of H ( t 2 ) has infinite energy density. This question of which "vacuum" is relevant in any given circumstance is a

difficult and unresolved problem. The choice of initial condi- tion in our work, is quite general since 0 and to are both left as parameters. Many of the vacua discussed in the literature are special cases of these states. It turns out, however, that the Green's functions at nonzero space-time separation change smoothly as the initial time to is changed. Thus, al- though we have not resolved the issue of vacuum choice, we feel that our initial condition can give a reasonable description of any physics which is insensitive to problems with TNV.

12S. Weinberg, Phys. Rev. D 7, 2887 (1973). 13Since the system is expected to undergo a phase transition, it

may be advisable to have 4,=4,(x, t ) . It is technically too difficult to carry out our calculation for this case. We shall thus assume that 4,( t ) adequately describes the spatially aver- aged expectation value of 4 over the extent of a bubble. The simplest generalization of Eq. (30) would be to allow 4 to be a function of x and t, replace 6 by 04, without modifying the one-loot, term.

14This assumption is not essential and can be relaxed in a straightforward manner.

15N. D. Birrell and P.C.W. Davies, Quantum Fields in Curved Space (Cambridge University Press, Cambridge, 1982).