PHYSICAL REVIEW D VOLUME 40, NUMBER 10 15 NOVEMBER 1989

Goldstone bosons to gravitons: A new gravity-wave production mechanism

R. Holman and Yun Wang Department of Physics, Carnegie Mellon Uniuersity, Pittsburgh, Pennsylvania 15213

(Received 5 December 1988; revised manuscript received 16 August 1989)

We propose a new mechanism for gravity-wave production: the conversion of Goldstone bosons to gravitons in the presence of an external gravitational field. We compute the gravity-wave lumi- nosity for the cases in which the external fields are due to various types of black holes: Schwarzschild, Reissner-Nordstrom, and Kerr. We also consider the case of the Schwarzschild constant-density star. The Goldstone-boson-induced gravity-wave luminosities are largely indepen- dent of the mass of the black hole. For the Goldstone bosons we have considered (the omion, re- cently proposed by Sikivie) these luminosities are small. However this is not due to effects inherent in the conversion process, rather this smallness comes about due to the incoming Goldstone-boson spectrum.

I. INTRODUCTION

The weak-field solutions of the Einstein equation obey a wave equation and can be interpreted as gravitational radiation. Because of the weakness of the gravitational interaction, only astrophysical sources are relevant in our quest of detecting such radiation. The detection of gravi- tational waves from astrophysical sources1 would open a new frontier for astronomy and would doubtless have great impact on other branches of physics. Therefore, we recognize the importance of understanding all possible sources of gravitational waves.

In this paper we point out the existence of a new mech- anism for the production of gravitational radiation: the conversion of Goldstone bosons of a spontaneously bro- ken, global, axial symmetry to gravitons in the presence of an external gravitational field. This process is the gravitational analog of the conversion of axionsQo pho- tons in an external magnetic field.

For the external gravitational field, several types of black holes (Schwarzschild, Reissner-Nordstrom, Kerr) are considered, because they not only emit no gravita- tional waves on their own, but also can be heavy enough to provide a strong background gravitational field. Furthermore, the black-hole metrics are asymptotically flat, thus enabling us to define the in and out states in the conversion process. The Schwarzschild constant-density star is also considered for its close relation to the Schwarzschild black hole.

The Goldstone bosons need not be specified in this work; they can be axions, om ion^,^ or something else. For concreteness, we will use the omion in most of our calculations.

In Sec. 11, we present the derivation of the stress-tensor source responsible for generating the gravitational waves; in Sec. 111, we study the spherically symmetric metrics and the corresponding stress tensors and transition am- plitudes. Section IV considers the same calculations for the Kerr metric while Sec. V is devoted to the computa- tion of the conversion cross sections and gravitational- wave luminosities. Section VI contains our conclusions.

11. STRESS-TENSOR SOURCE

For Goldstone-boson-graviton conversion to happen, we first require the existence of a theory with chiral fer- mions coupled to Higgs scalars in such a way that at least one of these fermions acquires a mass due to a nonzero Higgs vacuum expectation value. We further require that the Lagrangian of this theory exhibit a global chiral symmetry under which the Higgs scalar has a nonvanish- ing charge. Finally, the symmetry current J:' must be gravitationally a n o m a l o ~ s : ~

where D, is the gravitational covariant derivative, n = & T ~ [ Q ' ~ ' ] , Q ' ~ ' is the axial charge corresponding to J:' and the trace is taken over all spin-; fermions (taken to be left handed). The quantity *R,,,,,, is the dual cur- vature tensor:

1 1 -- aS *R,VP0 - 2 v: E,"USR pu . (2) g

We may use the chiral symmetry to eliminate the phase ~ ( x ) of the Higgs field @ ( x ) (the Goldstone mode) from the Yukawa couplings. Because of the anomalous nature of the chiral current, this will induce an interaction be- tween ~ ( x ) and the metric of the form

where ( @ ) =u. Note that for constant X, I y is a topolog- ical invariant (just as in the Yang-Mills case).

From Ix, we can construct a source term for gravitons in the presence of an external field in the ususal way. First expand the metric around a background metric g,,,:

with h-v16n/~, , . Next, insert this expansion in Eq. ( 3 ) and keep the linear term in the fluctuations h,,. This gives us the effective source for h,,:

3204 01989 The American Physical Society

40 GOLDSTONE BOSONS TO GRAVITONS: A NEW GRAVITY- . . . 3205

where the barred quantities are constructed from the background metric. The ~ ( x )-h,, effective interaction Lagrangian is (to linear order in h,,)

We may check that the stress tensor above is the correct one by noting that the terms with no ,y deriva- tives vanish by the Bianchi identity. This is as it should be, since, as mentioned above, for constant X, Ix is a to- pological invariant and so must have vanishing metric variation. This is also consistent with the decoupling theorems for Goldstone bosons. Note also that TAf is traceless: gApTAp=O as can be seen by use of the cyclic identity for the Riemann tensor.

In order to compute the conversion cross section, we must first choose the background metric and then specify the relevant quantum states. The S-matrix element Sf, for the conversion process is

where first-order perturbation theory has been used. The fields in Lint are in the interaction picture and

I h ; k , p ), lx;p ) denote the states (in this picture) corre- sponding to a graviton of momentum k and polarization p and a Goldstone boson with momentum p.

111. SPHERICAL METRICS

In this section we consider the case of the Schwarzschild black hole, Reissner-Nordstrijm black hole, and Schwarzchild constant-density star, all of which are spherically symmetric.

First we consider the Schwarzschild metric. Since this metric is a vacuum solution of the Einstein equations, R,?=O and *Kku6 has exactly the same symmetry prop- erties as R 'pub (Ref. 7). From this and the Bianchi identi- ty we have

In order to make use of the asymptotic flatness proper- ty of the Schwarzschild metric, we must write T" in terms of the quasi-Minkowskian coordinate^:^ xO=r, x ' = r sin0 cos4, x = r sin0 sin$, x = r COSQ:

Only the components relevant to our discussion are list- ed, and A and rs are defined by

where M , is the hole mass. The values of (r ,Q,4) are to be interpreted as functions of ( x l , x 2 , x , ) and we note that for the Schwarzschild metric DtD4x = atadx,D,D,x =a t a,x.

In order to simplify the calculation, we take the line of sight of the observer of the outgoing graviton to define the z axis, i.e., k = / k / ( 0 , 0 , 1 ) . We may then use the gauge invariance of the graviton Lagrangian to gauge away all the components of the graviton polarization tensor ~ j P p ) except E ( I ~ ' , E : ; ) , E ~ ' = -E\c;' (Ref. 8). Going back to Sfi, we have

where we have expanded h,, in box normalized (Min- kowskian) plane waves and have defined wk to be the graviton energy and V to be the box volume. Writing ~i:'* TAp in a helicity basis, with €2' E:?) i i&', we have

Using Eq. (10) and decomposing x into plane waves we have

where

Furthermore, the angles Qp-k and +,-, satisfy the rela- tions

3206 R. HOLMAN AND YUN WANG 9

The case of the Reissner-Nordstrom metric is more (16) complicated, since it is not a vacuum solution of the Ein-

stein equations. Using the cyclic and Bianchi identities we find

Note that A , a Y:'( @ppk,d,-k); i.e., the radiation is pure quadrupole. The radiation is also unpolarized, which is not totally unexpected since the bosons are sca- lars and the hole is spherically symmetric. Also note that the boundary condition r > r, has been applied, since the Schwarzschild black hole has a horizon at r = r,.

A Schwarzschild constant-density star has the same form of metric as the Schwarzschild metric outside the star. Inside the star we have

where rs is the Schwarzschild radius of the star, and R is the radius of the star.

I t turns out that T'P is zero inside the star, while it is exactly the same as in the Schwarzschild black-hole case outside the star. So in our context the Schwarzschild constant-density star behaves like a Schwarzschild black hole with horizon at R instead of r,. Thus the modification of the expression for A ,:

The relevant components are found to be

Note that these reduce to the Schwarzschild case for Q =o.

The Reissner-Nordstrom black hole has two horizons for Q < M H 7 the larger of which behaves much like a Schwarzschild horizon and has the value r- =+r,[l +( 1 - 4 ~ ~ / r i ) ~ ' ~ ] , and we have

wherez =Ip-klr, . To sum up our results in this section: Eq. (14) is the

general expression for conversion amplitude s?', valid for all three spherical metrics, with different A , for each case.

IV. MERR METRIC

The case for the Kerr metric is more complicated than those considered in the preceding section, due to the fact that it is only axially symmetric. Choosing the symmetry axis as the z axis, we have

where a is the angular momentum of the black hole and

Since the Kerr metric is a vacuum solution of the Ein- stein solution, Eq. (8) holds. After much manipulation we have

40 - GOLDSTONE BOSONS TO GRAVITONS: A NEW GRAVITY- . . . 3207

where 6=sin28 and only terms no smaller than first order in a / rS are kept.

To get a qualitative idea of how the angular momen- tum of the black hole affects the produced gravity waves, we take the observation direction to be that of the angu- lar momentum, so that

Proceeding as in the previous section, we have

x ( E ~ ' A + +E?'A-) ,

where

where B r I p - ki. Note that this expression reduces to the Schwarzschild case when a =O.

From the above expression, we see that the lowest- order contribution from the angular momentum of the black hole leads to the polarization of the gravitational waves as observed in the direction of rotation of the black hole.

Because of the complexity of the Kerr metric, we have not obtained clean results for other directions of observa- tion.

where t is the cosmological time of omion emission. Since the omion is massless, the calculations are much simplified (compared to that of the massive Goldstone bo- sons).

For the Schwarzschild metric, we give the general for- mulas before specifying the Goldstone boson to be an omion. We have

V. CROSS SECTIONS AND LUMINOSITIES

sin'sp(sy'e "4.- € ' ~ l r - 2 i + ~ ) , Given the spectrum of the incoming Goldstone bosons, (29)

we are now in a position to calculate the cross sections of the Goldstone-boson-graviton conversion and the result- 2

d o - 9 ing luminosities of gravitational radiation. -- - s i n 4 8 , q

A candidate Goldstone boson is the omion4 recently d w d n [ 1 p-kI

proposed by Sikivie to explain a variety of astrophysical 2 data. If f ( up, Op, 4p)d 3P is the energy flux of Goldstone

bosons with momenta between p and p+dp, then for an 046( w -po ) , (30)

isotropic flux, f (w,, 8,, 4,) = f ( op ). In particular, for the omion,

where Ipl is the speed of the incoming bosons, w=/kl , v 2 1 f (w)=-- (28) and d f l is an element of solid angle about k. This may 4t2 o3 ' also be written as

3208 R. HOLMAN AND YUN WANG - 40

X l j l ( lp-k/ rs )126( P I - / P / o ) , (31) where

where / p / ; = ~ ~ - r n ~ ,

and we have summed over both graviton polarizations. The graviton luminosity per unit frequency interval and per steradian, dPg / d o d o , is given by

For an isotropic Goldstone-boson distribution [i.e., f ( up, @,,+,)= f ( w ~ ) ] , this becomes

where

with /p-k/,,= ( 2 0 p l o [ a ( o ) - X I ) 'I2. For the omion, a ( w ) = l , ( / p / o = w ) , and

where z = 2ors. This then yields

Since the conversion process is isotropic for the Schwarzschild black hole, we may compute the total luminosity in gravitational waves of all frequencies be- tween W - -2rr/t and o+ --u (these are the lower and upper bounds on the omion frequencies9):

with

z _ =2w-rs--5x

(41) U

z+ = ~ O + Y ~ = = ~ X 102~ [ l0Io GeV 1 12 1 ' where we have normalized the omion emission time t to the present epoch, and u to 101° GeV.

Note that I ( z - , z + )+O as M H - - t w , as expected, since, when the horizon of the black hole is infinitely large, no signal can escape outside the horizon to be ob- served, in our case, no gravitational radiation observed (L, =O).

Since z - -0,z + >> 1, we can use

and obtain

where we have normalized n to lo5 since in the omion models c o n ~ t r u c t e d ~ ~ it is constrained to be of this order.

Note that Eq. (34) holds for all three spherically symmetrical cases. For the Schwarzschild constant- density star with mass M and radius R ( rs=2GM), we have

Equation (34) gives the graviton luminosity density with

40 GOLDSTONE BOSONS TO GRAVITONS: A NEW GRAVITY-. . . 3209

where z 2 w R , and we have For a Reissner-Nordstrom black hole with charge Q - - 2 4 and larger event horizon r + , we have

L [ 1 [ I $ 1 I ( z - , z + 1 , (46) d o - 9 - ( 1 + c o s ~ ~ ) ~

w26( lp/ -w) ' where d o d n 256r5 (LJ;~, 1 1 -cosQD

L - - - And I ( z - , z + ) is given by Eq. (40), and we have 2 r 4Jl 'AJ-

r t

2 I

erg/sec . (48) where x = I - k 1 r + , and Eq. (34) gives the graviton lumi- nosity density with

[1- (z/zo )'I2 rx B ( ~ ) = 1 6 ~ ~ ~ ~ d z - j , (z)-21{~jl(z)-~[z2si(z)+sinz Q 2 +z cosz]]

0 z I r + +

where zo = 2wr + . For small Q2, the gravitational-wave luminosity from a

Reissner-Nordstrom black hole is of the same order of magnitude as that from a Schwarzschild black hole.

The Kerr black-hole case with small angular momen- tum gives the same order of magnitude for the gravitational-wave luminosity as in the Schwarzschild black-hole case, but the gravitational waves have polar- ization which is induced by the angular momentum of the black hole and is dependent on the direction of observa- tion.

In the above examples, the luminosities in gravitational waves have been rather small. Is this always the case? Suppose we had a true Goldstone boson with f (w)=aw, where a is a dimensionless constant. In the case of Schwarzschild black-hole background metric, the differ- ential power in gravitational waves is

And the total luminosity is

where

= + I ( Z L , Z + I= Jz d z z (z2-+)[ci(2z)-ln(2z)-y] I

with z - =2w-rs, and z + r 20+rs. Assuming that z + >>z-, we have

thus 2

101° GeV ~ ~ ( 2 . 4 ~ 1 0 ~ ~ ) [ $ I 2 [ ]

Note that this luminosity is comparable with the gravitational-wave luminosities from such sources as ca- taclysmic and x-ray binaries1 (these have Lg = 1028-32 erg/sec), and can be much larger if w+ is larger than 10" GeV, with reasonable values of u and a .

VI. CONCLUSIONS

We have exhibited a hitherto unknown mechanism for the production of gravitational waves. While the exam- ples discussed do not yield very large fluxes, this is not due to effects inherent in the conversion process, but de-

3210 R. HOLMAN AND YUN WANG 40

pends on the power spectrum of the incoming Goldstone boson. Other sources than the omion may give significantly larger fluxes.

I t is interesting to note that although we chose black holes as the background metrics because they are heavy (among other reasons) and would provide strong fields, the resulting gravitational-wave luminosities seem to be largely independent of the masses of the black holes (as clearly shown in the case of Schwarzschild black hole). This is hardly surprising since the black holes have event horizons which have sizes proportional to the black-hole masses, so only far fields contribute; this effect cancels out the strong-field effect (due to the large mass) and gives us a resulting gravitational-wave luminosity in- dependent of the black-hole mass.

We have seen that the gravitational-wave power spec- trum reflects that of the incoming Goldstone bosons and can be continuous (as seen in the example of omion). This is in contrast with most astrophysical gravitational- wave sources, their power spectra usually being discrete. This makes the detection of these waves rather difficult, since if resonant antennas are used, only the resonant fre- quency will be excited and most of the power will not be

sensed. Free-mass detectors may be more suitable for the purpose.

At completion of this work, the authors were made aware of a photon-graviton conversion process discussed by Raffelt and Stodolsky." Here we would like to note that this process was discussed in the context of QED, and that the transition rates there appear to be always negligibly small. Our work is exclusively devoted to the Goldstone-boson-graviton conversion in the background gravitational field, in which one is free to choose virtually any possible Goldstone boson to be converted; therefore, the cross sections and gravitational-wave luminosities need not be small and our results may possibly set certain restraints on undiscovered or unspecified Goldstone bo- sons in the Universe.

ACKNOWLEDGMENTS

R.H. would like to thank Professor P. Sikivie for numerous conversations about omions. We would also like to thank Dr. Clifford Will for very helpful sugges- tions on the manuscript. This work was supported in part by D O E Grant No. DE-AC02-76ER03066.

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