PHYSICAL REVIEW D VOLUME 50, NUMBER 12 15 DECEMBER 1994
Coupling of pseudo Nambu-Goldstone bosons to other scalars and its role in double field inflation
Katherine Freese Physics Department, University of Michigan, Ann Arbor, Michigan 48109
(Received 8 August 1994) We find a coupling of pseudo Nambu-Goldstone bosons to other (ordinary) scalars, and consider
its importance in various contexts. Our original motivation was the model of double field idation. We also briefly consider the role of this coupling for the case of the QCD axion.
PACS number(s): 14.80.Mz, 98.80.Cq
The idationary scenario was proposed in 1981 by Guth [I] to explain several unresolved problems of the standard hot big band cosmology, namely, the horizon, flatness, and monopole problems. During the idation- ary epoch, the energy density of the Universe is domi- nated by vacuum energy, p N p,,, and the scale factor of the Universe expands superluminally, R > 0. In many models the scale factor grows as R(t) cx eHt, where the Hubble parameter H = R/R (8?r~p, , /3)~/~ during idation. If the interval of accelerated expansion satis- fies At 2 60H-l, a small casually connected region of the Universe grows sufficiently to explain the observed homogeneity and isotropy of the Universe, to dilute any overdensity of magnetic monopoles, and to flatten the spatial hypersurfaces, R 8?rGp/3H2 + 1.
In the original model of ida t ion [I], now referred to as old idat ion, the Universe supercools to a tempera- ture T
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the energy scale f , via a potential
where the scalar self-coupling X1 can be of order unity. Below this scale, the scalar part of the Lagrangian is then
Lscslar = lap@i12 + i(ap@2)2 - m:I@112 +im:@: + VR(@I, @2) , ( 5 )
with -m: < 0 since the symmetry is spontaneously bro- ken. VR is defined to be all contributions to the potential other than the mass terms. We can write
= (p + f)e"/f . ( 6 ) Below the scale f , we can neglect the superheavy radial mode p since it is so massive that it is frozen out, with mass ml = (X1/2)l/' f . The remaining light degree of freedom is the angular variable X, the Goldstone boson of the spontaneously broken U(l) symmetry [one can think of this as the angle around the bottom of the Mexican hat described by Eq. (4)].
Subsequently the symmetry is explicitly broken at the scale A. For example, U ( ~ ) P ~ is explicitly broken at the QCD scale N 1 GeV, when instantons become the dominant contribution to the path integral. Then, for the case of QCD, the Goldstone mode x becomes the axion. In general the potential for the PNGB below the scale A becomes
V(x) = A4[l f cos(Nx/f )I , (7) where N is the number of minima of the potential. We take N = 1.
Because of the U( l ) symmetry, the interaction poten- tial VI(al, a 2 ) must be independent of X. The poten- tial must always be invariant under x + x + c, so that the only coupling we can hope to get between the Gold- stone mode x and the field is through derivative cou- plings involving 8,~. At tree level, the kinetic term in the Lagrangian for the field is Ltree lap@112; note that this contains a term (apX)'. We will obtain one- loop corrections to this kinetic term, Ltree + L1 loop N [Z(@1,@2) + l]lap@112. Part of this one-loop correction is the interaction term we are looking for.
To obtain this we must calculate the effective action for the field dl as a function of a temporally and or spa- tially varying classical Goldstone background field. We follow Ref. , where an effective action expansion in per- turbation theory was found. One performs a derivative expansion of the functional determinant for the Gaussian fluctuations around the classical background. Reference  shows the leading order correction to be
where U = -m: + sgzi. Again, VR is defined to be all contributions to the potential other than the mass terms. Chan  also found higher order terms in the expansion in %; we will assume that this ratio is small so that we canJignore higher order derivatives and consider only the leading term above.
For example, we consider
(9) Thus, to one loop, our interaction term is
111. ROLE OF THE COUPLING IN DOUBLE FIELD INFLATION
In double field inflation we take the field to be the ida ton , a field that tunnels from false to true vacuum via nucleation of bubbles at a first order phase transition. Note that we use the word "inflaton" to refer to the field whose vacuum energy dominates the energy density of the universe; thus the inflaton is the field responsible for inflation. We take the potential V2(@2) to be an asym- metric potential with metastable minimum @- and abso- lute minimum @+ [see Eq. (12)]. The energy difference between the vacua is E . In the zero-temperature limit, the nucleation rate rN (per unit time per unit volume) for producing bubbles of true vacuum in the sea of false vacuum through quantum tunneling has the form 
where SE is the Euclidean action and A is a determinan- tal factor which is generally of order T,4 (where T, is the energy scale of the phase transition).
The basic idea of double field inflation is to have a time dependent nucleation rate of true vacuum bubbles: intially the rate is virtually zero, so that the universe re- mains in the false vacuum for a lone: time and sufficient inflation takes place. Then, after thishas taken place, the nucleation rate sharply becomes very large, so that bub- bles of true vacuum nucleate evervwhere at once and the phase transition is able to complete (unlike in old infla- tion). The Universe then has a chance to thermalize and return to radiation domination. In double field inflation, this sudden change in the nucleation rate is achieved by the coupling to a second scalar field, which is a rolling field. The purpose of this rolling field is to serve as a cat- alyst for the i d a t o n to go through the phase transition. While the rolling field is near the top of its potential, the nucleation rate is very small; once the rolling field nears the bottom, the nucleation rate becomes very large. Examples were given in , hereafter paper I. However, density fluctuations are produced by quantum fluctua- tions in the rolling field; the amplitude of fluctuations is too large (in excess of microwave background measure- ments) unless the rolling potential is very flat, namely the ratio of height to width must be less than N 10-lo 191. For this reason we have investigated using a PNGB as the rolling field. A PNGB can naturally provide the required flat potential. Unfortunately, as we show below, this requires a different unexplained small parameter.
Thus, using the notation above, we take a2 to be the
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ida ton , a tunneling field, and the PNGB x to be the rolling field. Then the overall Lagangian is given by Eq. (I) , where the interaction term is give by Eq. (10). For definiteness, we take the potential of the inflaton field to be
To leading order, the metastable minimum is given by 4- = -a and the absolute minimum by 4+ = +a.
For the interaction term we use Eq. (10) above. We take the spatial derivatives to be zero and focus on the time derivative x. We can take the amplitude of the rolling field to be 1@1 1' = f 2. Then the term becomes
Here we assume that the field 5 1 - f and ex- pand the denominator (later we will see that sufficient inflation requires f - mp1 so that this assumption is not unreasonable). Then the couplng part of the term is
The equation of motion for the rolling field is ji + 3Hx = -E . we will assume the slow roll limit in which the ji
d , ' term is negligible. Then, using V(X) from Eq. (7), from the equation of motion we find
where the f refer to the f in Eq. (7). We take Hz = 8.1rM4/3mgl, where M is the energy scale at which inflation takes place and mpl = lo1' GeV is the Planck mass. Thus the term (now written as a potential) is
For either sign, as the field x rolls down from the top to the bottom of the potential, it is during the first half of this rolling that sinZX/f and thus V' increase: dur- ing this period SE decreases and the inflaton tunneling rate increases with time, as desired. One can see this by looking at the total potential for the inflaton, namely, the sum of Eqs. (12) and (16a). During this first half of the rolling of X, the shape of the total potential for changes in the following way: the barrier height - Xza4 becomes smaller and the distance 2a between the minma becomes smaller. Thus the nucleation rate of true vac- uum bubbles grows in time. This is seen explicitly in the third constraint on the model discussed below. During the second half of the evolution of X, sinZX/f decreases and thus the nucleation rate decreases.
For the case of the plus signs, the field x rolls from 0 to
.lr f . We see that f ) and thus the nucleation rate peak at x = ( ~ 1 2 ) f , so that tunneling of the i d a t o n field happens by the time x rolls down to this value. The small angle approximation, sin(x/f) - X/ f , is good while x rolls to this value. For the case of the minus signs in Eqs. (7) and (15), we consider the field x rolling from ?rf to 0 (this time the potential is zero at x = 0 and reaches its maximum at x = ~ f ) . Here the nuculeation rate grows until x = (7~12) f , where it peaks. The tunneling of the inflaton field happens by the time x rolls to this value.
In paper I we discussed several constraints on the dou- ble field model. We apply these constraints here to the case where the rolling field is a PNGB with interaction term above, to find what the factor q must be. First, we want the field to dominate the dynamics of the Universe and be responsible for the inflationary epoch; hence we require V2(a2) > V2(x), i.e., E = M 4 2 A4.
Second, in order for the coupling of the x field to influence the inflaton and bring an end to inflation, we need the coupling term to be sufficiently large at the end of inflation, i.e., qX2@i - X2+$a2. We can see that this is the appropriate condition by writing V2(Qz) +VI = !jXz[@i -a&I2 - &(a2 -a) +const, where a2 eff = a2 - qX2/Xz. Since x - f towards the end of in- flation, this requirement becomes
Third, we need the nucleation rate to change from very small to very large. Bubbles will nucleate at a rate given by Eq. (11). In the thin wall limit, where the energy difference between vacua e
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fraction of the Universe [lo]. Here we have the num- ber of e-foldings NT = 3H25; 9, where the driving term F = dV/dx = A4+$ - 772xa2 - xpA4/f2, where /3 - 1. We find NT - ~ & l n ( f lX,-,). Requiring 60 e-foldings, ( f l m ~ ] ) ~ - 10-l1 can then only be obtained if xo/f = exp(-10"); the field must start ridiculously near the origin. The fraction of the Universe which will start out with that value is tiny.
Thus, to obtain a reasonable probability of sufficient inflation, we consider a second case, with f - mpl and A 5 10-3mpl for the width and height of the rolling PNGB field (101. Then, from Eq. (20), we can see that X2 is tiny. Thus, a n as yet unexplained small parameter is required in the inflaton potential, and one sees the resurgence of the problem we have been trying to avoid or explain. In addition, SE N 10 requires such a large value of E that Eq. (18) no longer applies as the barrier is now so small as to be almost irrelevant; the inflaton is practically a rolling field.
We also with to check the validity of the gradient ex- pansion. In Eq. (8) we have kept only the first order term and have neglected terms involving higher deriva- tives of x . This is valid as long as x