* a FermiiNaZionaI Accelerator Laboratory w FERnILAB-Pub-88/181-A November 1988 -_.-- COSMOLOGICAL COSMIC STRINGS Ruth Gregory N.A.S.A./ Fermilab AJtrophyJics Center, Fermi National Accelerator Laboratory P.O.Boz 500, Batavia, IL 60510, U.S.A. (USa-CB-184649) CGSIGLOGICAI CCSlltC N89- 16 5 25 SIBIIGS (Perri Bational dcce3erator lab.) 16 P CSCL 20c Unclas G3/77 0189648 ABSTRACT We investigate the effect of an infinite cosmic string on a cosmological back- ground. We find that the metric is approximately a scaled version of the empty space string metric, i.e. conical in nature. We use our results to place bounds on the amount of cylindrical gravitational radiation currently emitted by such a string. We then analyse explicitly the gravitational radiation equations and show that even initially large disturbances are rapidly damped as the expansion proceeds. The implications for the gravitational radiation background and the limitations of the quadrupole formula are discussed. 0 Oporrtod by Universities Research A8SOChtiOn Inc. under contract with the United States Department of Energy 1 Introduction. -_- -- Cosmic strings' have attracted a lot of attention recently as they give a compelling and plausible description of structure in the universe. They are topological defects which can be formed when the universe undergoes a suitable phase transition to a spontaneously broken symmetry state. The condition for string formation is that the vacuum manifold (the set of vacuum states of the theory) is not simply connected2. The strings responsible for galaxy formation are GUT strings, i.e. those formed during phase transitions at the GUT scale, 10I5GeV. Such strings are very thin (T, - 10'29cm) and extremely massive (p - lo2' g cm-l), but knowledge of their gravitational properties is not yet complete. The spacetime structure surrounding a local string in an otherwise empty spacetime has been analy~ed~-~,and the spacetime was found to be asymptotically conical. Whilst this is a useful test calculation, otherwise empty universes hardly correspond with our own, ' which currently on a large scale behaves like a matter dominated F'riedmann-Robertson- Walker (FRW) cosmology. Since we do on average expect at least one string per horizon volume', we must check that these properties are not modified in any essential way when we consider a more realistic background for the string. In particular, we expect the radius of the string to be fixed by the local microphysics, which will mean in general that the string is contracting with respect to the surrounding spacetime. This could give rise to interesting dynamical effects which could potentially be observable, such as cylindrical gravitational radiation, or particle radiation. As the most stringent constraints on the cosmic string scenario currently come from measurements of gravitational radiation7, this last feature could be of great relevance. In this paper we address the problem of a string in a cylindrically symmetric universe. There are two approaches one could take: one can either choose a particular model for the energy momentum tensor of the string and calculate the metric, or one can try the more complete treatment of regarding the string as a vortex solution to the equations of motion of some suitable spontaneously broken local gauge theory. The former has the advantage of simplicity, but neglects the more detailed structure of the string accounted for by the latter. 1 ORIGINAL PAGE IS OF POOR QUALITY We begin our investigation of the problem by using the most common model for a string: the wire-or delta function source. 'This gives an idea of the essential structure of the spacetime (which is again conical), however there are complications in using distributional sources in general relativity' and the validity of this approximation should be justified. Therefore we proceed to analyse the full coupled equations for a Nielsen-Olesen vortex in matter and radiation dominated F'riedmann universes which confirms the conical structure and also allows us to examine the question of cylindrical gravitational radiation from strings. 2 The Wire Approximation. We start by summarising the formalism for dealing with cylindrically symmetric sys- tems. We use the work of Thorne' as a basis. The metric of our universe may be written as ds' = e'('-*)(&' - &') - e2qdz2 - d2e-2qdfl2 9 (2.1) where A, \E and I' are functions of T and t only. We will assume that the energy momentum tensor splits into the form Tab + Sob, where Tab is the energy momentum tensor for the background matter fields (which are homogeneous and isotropic) and sab the string energy momentum tensor. With this splitting, the field equations take the form: A - d" = 8xGfi(T,O+ T: + S,O + S;) (2.24 d d -(d$) - -(A!&')= 4xG&(Tt - S," Si - Sz) (2.2b) dt dT + T,' + T: T,' + + Sr + (A" - d2)P= 8xGJ-g[df(T,0 + S,O) + d(Tl + Sl)] + Mf($+ !&") - 2dhk'4+ d'd"- &$ (2.2c) (d2- df2)f= 8aG&[A(Tt + S,")+ d'(T,' + S:)] + AA($ + qt2)- 2Mt\Ef@+ Ur- (2.2d) f' - r'' = - &' + ~TG~'(~-*)(T:+ Si). (2.2e) where fi= is the square root of the determinant of the metric, a dot denotes time derivative and a prime radial derivative. 2 i ORlGlNAL PAGE IS I3 OF POOR QUALITY In the otherwise empty space case the approximation of the string by a delta function or wire sourceteadily gave the essential asymptotic structure of the spacetime3. We adopt this approach here, using the wire model to investigate asymptotic structure. Therefore for the string source we set The boundary conditions are d'(0,t) = er('*'), I'(0,O) = 2\E(O,O) . These now become formal, since T = 0 is a (distributional) curvature singularity. We do not expect lim df(T, t) = d'(0, t) as prescribed by the boundary conditions, indeed it is this r-0 discontinuity which gives rise to the curvature singularity necessitated by the distributional string source. Examination of the field equations shows that the equation for d, (2.2a), is modified from the pure cosmology case The equation for \E, (2.2b), explicitly remains the same, as does the final equation, (2.2e). Although at first sight the first order equations for 1', (2.2c,d), appear modified, if we substitute for d'f we see that in fact the distributional term does not contribute. Notice also that all the equations are invariant under scaling of d by a constant. Integrating (2.4) over a disc of radius 6 gives A'((), t)- d'(6,t) = 8.rrGC1e2(r(0,t)-'(0*2)) 27r = 4Gpdf(0,t) + d'(6,t) = (1 - 4Gp)d'(O,t) . Thus we see that the effect of the string is to multiply the d in the original cosmology by a factor (1 - 4Gp) to give the required discontinuity in d' at the origin. The modified string-plus-cosmology metric is now 2 2 -29de2 ds2 = e2(r-')(dt2 - dr2)- e2*dz2 - (1 - 4Gp) d e 9 (2.6) 3 BR’lGtNAh PAGE IS OF. POOR QUALITY which is indeed a conical spacetime, with a deficit angle of 8xGp. This model is however static, in the sense that the effect of the string is a time inde- pendent one. The string, being infinitesimally thin, does not participate in the expansion of the universe. In a more realistic situation where the string has width and internal struc- ture, we might expect the dynamics of the surrounding spacetime to produce dynamical effects in the string. This is the problem we shall consider next. 3 Vortex solutions in Friedmann universes. We now try to take account of the fact that strings are in reality thick objects that will in general be affected by the dynamics of the surrounding spacetime. The string radius T, is generally seen as being fixed by the microphysics but a string of ked proper radius need not have fixed coordinate radius. In order to maintain a constant proper radius fixed by the microphysics, the string will have to resist the expansion (or otherwise) of the surrounding spacetime in which it sits. It is possible that this “variation” of the string radius with respect to the surrounding spacetime could give rise to interesting dynamical effects, for example cylindrical gravitational radiation or particle radiation, which may be observable in our universe. We will consider only local strings, since global strings have badly behaved asymptotic structures. In particular, we consider the case of a U(1) local string: the Nielsen-Olesen vortex. The Nielsen-Olesen string is a vortex solution to the lagrangian 1- x C[4,A,] = D,4tDp4 - -FPYfi””- -(q5t4 - #)’, 4 4 where D, = V, + ieA, is the usual gauge covariant derivative, and fiPYthe field strength associated with A,. However, we will express the field content in a slightly different manner, in which the physical degrees of freedom are more manifest. We define the (real) fields X,x and P, by 4(za) = qX(za)e’X(zP) (3.2~) 1 A,(za) = - [PP(za)- V,x(za)] . (3.2b) e 4 ORIGIEAL PAGE IS OF POOR QUALITY In terms of these new variables, the lagrangian becomes where Fpvis the field strength associated with P,,. We can see that a vacuum state is characterised by X = 1. However, this is not the only stable ground state which solves the equations of motion. Nielsen and Olesen'' showed that there exists a non-trivial stable ground state solution to the above equations of motion which has a vortex-like structure. The Nielsen-Olesen vortex solution corresponds to an infinite, straight static string aligned with the z-axis. In this case, we can choose a gauge in which in cylindrical polar coordinates. This string has winding number one.