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Recent Developments in the Cosmic String Theory of Galaxy Formation 175 RECENT DEVELOPMENTS IN THE COSMIC STRING THEORY OF GALAXY FORMATION Neil Turok Department of Theoretical Physics Imperial College London SW7 2BZ, England ABSTRACT Recent progress in the cosmic string theory of galaxy formation is reviewed, and a remarkable agreement between the correlation function for cosmic string loops and that for ~bell clusters is reported. Lecture presented at the ESO Symposium on Cosmology and Particle Physics, Munich, 1986. 176 N. Turok In this talk I will review some of the recent developments in the cosmic string theory of galaxy 1 formation and report on an apparently remarkable agreement between the theory and the observed correlation function of 2 rich clusters of galaxies Of course I should begin by acknowledging our debt to the Ancient Greeks who were the first to suggest that galaxies lay on strings (see any picture of the constellations) although theirs had ends and branches! Over the past year there has been considerable progress in three main areas. First, many of the most interesting grand unified theories and in particular those based on superstring theories are now known to predict 3 4 cosmic strings ' • Second, as a result of numerical and analytical work, the evolution of a network of cosmic 5 6 16 strings is now significantly better understood ' ! Lastly, the numerical simulations have allowed a series of detailed calculations of the specific predictions of the string theory to be made, from which it has so far emerged very 7 8 9 successfully ' ' In spontaneously broken gauge theories, strings may form if the space of degenerate vacua v (the space of 0 1 minima of the higgs potential) is nonsimply connected • In the simplest case, the group U(l) is broken completely and is just a circle. Here we get the simplest, directional v0 strings (the higgs field can wind round the circle in either direction). Strings can also form if a simply connected group is broken to a subgroup with one or more disconnected components. For example, if the subgroup has one disconnected piece one gets 11 Z 11 strings, where two 2 Recent Developments in the Cosmic String Theory of Galaxy Formation 177 identical strings can annihilate. This case occurs in many 3 candidate grand unified theories • In the recently discovered superstring theories examples of both types occur. In the E 8x E 8 theory , most of the proposed symmetry breaking patterns involve breaking extra U(l)'s, giving directional strings. The S0(32) theory yields strings for slightly more subtle reasons. The universal covering group of S0(32) has centre z 2x z 2 which remains unbroken since all fields are in the adjoint representation, but is generally not part of a continuous symmetry in the unbroken subgroup. Hence we again get 4 nondirectional strings • In these theories, strings would form at a symmetry breaking phase transition in the very early universe. The 1 basic idea, due to Kibble , is that as the universe cools below some temperature T ~ m, the symmetry breaking scale, c the Higgs field ~ falls into the minimum of the potential 1 Vo. ~ is only correlated over some scale ~ ~T c < t and the universe is broken up into domains of size ~ ~. with ~ choosing a random position on v 0 in each. To minimise gradient terms in the energy ~ smooths itself out by choosing the shortest path along v 0 in going from one domain to the next. Edges where ~ winds right around Vo then become string segments. In these theories strings have no ends, so the segments are all part of closed loops or infinite lengths. One then has a tangled network of loops and lengths with a fixed number of segments of length ~ per volume ~3. Simulations of this process were first performed by Vachaspati and Vilenkin, who found that about 178 N. Throk 80% of the string is in_ the form of infinite lengths and the remainder in closed loops with a scale invariant size distribution i.e. a fixed number of loops of size ~r in a 3 volume r • Both infinite lengths and closed loops are in 11 the form of Brownian trajectories A crucial parameter in the cosmic string theory is the 2 ~ where µ is the mass per dimensionless number Gµ n(m/mp1 ) 1 6 unit length of the string. As we will see, m ~ 10 GeV is required for strings to form galaxies and clusters of galaxies. It has been argued that such heavy gauge strings are difficult to form after a period of inflation. Strings formed before inflation are obviously diluted to inconsequential levels if the inflation has 'solved' the flatness and horizon problems - there would be less than one string per present horizon volume. Gravitational radiation constraints force the reheating temperature after 17 inflation to be less than ~10 GeV. The reheating temperature usually calculated is much less than this, but is actually not the relevant one for forming strings. If 4 the energy density in the inflation field is M with ' 1 7 M<10 GeV, and the decay rate into radiation r then it is easy to see that in an expansion time or so after inflation 2 ends the universe reheats to a temperature TRH (rm M )~. p 1 2 The mass per unit length of the string is given by ~nm , 2 2 2 where the potential V(~) = A(~ - m ) whilst the temperature of the string-forming phase transition Tc~ A~,which can be much less than m. Requiring TRH > Tc 16 _2 11 and m ~ 10 GeV we find I'A >10 GeV which is easily 5 satisfied in several models for A~10 Recent Developments in the Cosmic String Theory of Galaxy Formation 179 This requires a small amount of "fine tuning" but to no greatewr an extent than is necessary in these models l 0 anyway Between the time when they are formed, at a l I 2 temperature T ~ (Gµ) mpl and the time when T - Gµmpl' the strings are heavily damped by collisions with relativistic l particles • This results in the string straightening out - the scale ~ on which the string is straightened out grows 5/4 1. as t and ~ ~ (G µ) T't at the time when damping ceases to be important. The subsequent evolution of this network is a complicated problem, and the early literature on the subject contained several errors. It is obviously crucial to know whether the string density comes to dominate over 2 the radiation, whose density Prad- 1/Gt • The earliest discussions claimed that the network would evolve by straightening out so that ~ ~ t, and the density in strings _2 p ~ µt , whatever the string-string interactions s l l 2 were ' This behaviour is desirable in the sense that the fractional density perturbation provided by the strings ps/ptot- Gµ is constant, so the strings neither come to dominate nor disappear as the universe expands. The perturbation they provide is also of the right order of 16 magnitude (for m ~ 10 GeV, a typical GUT scale) to seed the formation of galaxies and clusters of galaxies later on. However simulations by Albrecht and myself have shown that the evolution of the string network does depend crucially on how the strings interact. In particular if the strings simply pass through each other without 180 N. Turok interacting, the energy in infinite strings remains 3/2 constant, p a: t , and strings rapidly come to s 5 dominate • Luckily for the cosmic string theory it seems more likely that when two strings cross they almost always reconnect the other way. The first indications of this were in numerical simulations by Shellard of the crossing of "global" strings. He found that for all velocities of intersection less than ~ 0.9 the strings always reconnect the other way. Recently Copeland and myself have given an analytical treatment of the process which also applies to gauge strings and indicates similar behaviour. Crossing and reconnection of strings makes it possible for the infinite lengths of string to slowly chop themselves up into closed loops. Albrecht and myself also simulated the case where strings always reconnect the other way and found strong evidence that with ~ ~ t initially, as one would 1 expect for "grand unif iert" strings , and for directional strings the energy density in the string network does indeed fairly rapidly evolve towards the "scaling" 2 behaviour p ~ µt All our results are consistent with the following simple picture. The network of strings longer than the horizon straightens out on scales of the order of the horizon scale, with a fixed number of lengths crossing each horizon. The self-intersection of these long strings continually produces loops with radii r of order the horizon scale, and more or less str~ightened out on that scale i.e. with length R. ~ 2nr. These "parent" loops are Recent Developments in the Cosmic String Theory of Galaxy Formation 181 very likely to intersect themselves, breaking up into several smaller loops, but this process rapidly terminates, producing a clump of non-self-intersecting loops. These occupy a small and rapidly decreasing fraction of space and the probability of them reconnecting to the network of long strings or colliding with another loop rapidly becomes negligible. Whilst they are produced with considerable peculiar velocities v ~ 0.2 or so this rapidly redshifts away and the total distance loops of a given size move relative to a comoving frame is a small fraction of their mean separation. The loops just sit at almost fixed comoving location, oscillating in a periodic way and slowly radiating gravitational radiation at a calculable 14 1 5 rate ' , until they eventually disappear.
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