Entropy generation in the early universe by dissipative processes near the Higgs phase transition

Volume 95B, number 1

PHYSICS LETTERS

8 September 1980

ENTROPY GENERATION IN THE EARLY UNIVERSE BY DISSIPATIVE PROCESSES NEAR THE HIGGS PHASE TRANSITION L.Z. FANG The Institute for Theoretical Physics, State University o f New York at Stony Brook, Stony Brook, N Y 11 794, USA and Astrophysics Research Division, University o f Science and Technology o f China a t Hefei, Anhwei, People's Republic o f China 1

Received 4 June 1980

We suggest that the anomalous dissipation near the Higgs phase transition in the early Universe might be one of the nonnegligible mechanisms to produce the cosmic entropy. Such processes may well generate a value of S comparable with the presently observed result. In a simplified a-model the qualitative relations between the specific entropy and the mass and self-interaction of the vacuum field have been obtained.

The problem of cosmic entropy generation has an outstanding status in the fast developing field of particle astrophysics. This is because an effective mechanism to produce the presently observed specific entropy of 108 +-1 can be provided by thermodynamically nonequilibrium processes in the early Universe dominated by the grand unified theories of strong, weak and electromagnetic interactions (GUTs) [ 1 ]. There are mainly two kinds of models provided by the GUTs with spontaneous symmetry breaking. In the first, the relevant irreversible processes are the CPand B-violating decays o f super-heavy (1015 GeV) bosons (either Higgs or gauge bosons) and their antiparticles, when the temperature o f the Universe has decreased to lower than the rest masses of those particles [ 2 - 5 ] . In the second kind, the crucial process is the irreversible transition from the initial symmetry restoring phase to the symmetry breaking phase as the temperature and/or density decreases through the phase transition point [6,7]. While the decay explanation is in the fashion, it strongly depends on the masses of the Higgs and the gauge particles so that the predicted value of S may be either smaller or larger than 108. Therefore, it is important, at least for the proof of the non-baryon1 Permanent address. 1 54

conserving model, to examine various processes which possibly have a non-negligible effect on the evolution of the cosmic entropy. The purpose of this paper is to discuss the effect of Higgs phase transitions upon the value of S. While the phase transition model has the advantage of predicting cosmic entropy generated at the same time as inhomogeneity [ 8 - 1 0 ] , the specific entropy generated is unfortunately often smaller than the desired value [11]. We point out that the production of cosmic entropy could be closely related to one of the characteristic phenomena in the physics of phase transitions - an anomalous increase in the energy dissipation and in the magnitude of density fluctuations. Such processes may well generate a value of S comparable with that presently observed. Since our main aim is to demonstrate the basic mechanism, we will consider the simplified a-model with lagrangian £ = ~-(O#q~)2 + ~/a2~ 2 -- l~.q54 + ~ ( i 3 / r , , - g ~ ) ~ ,

(1)

where ~ is the fermion field interacting with the scalar field qS. If X ~ g2 ~ 1, a second-order phase transition with symmetry breaking should occur at the fermion density "c ~ },/5~0*/g) 3 ,

from the ordered state <~) ~/I/X/X (n > nc) to the

(2)

Volume 95B, number 1

PHYSICS LETTERS

disordered state ¢5 ~ 0 (n < n c ) or vise versa. The difference in energy of tile q5 field between the ordered and the disordered state is about tt4/4?t which corresponds to the A term in the Universe before the phase transition. If this energy goes completely into photons during the transition epoch, the specific entropy should increase from zero (in the zero-temperature initial fermion phase) to S 0 ~ (30rr 2) 1/4(g4/X)3/4 < 1 .

(3)

Since S O cannot exceed l, these processes cannot explain the generation of the observed cosmic entropy. Similar conclusions have been obtained for the phase transition in the Weinberg Salam theory [i 1 ]. It is known that tile spontaneously symmetry breaking vacuum o f the tliggs field can be considered as the condensation o f a boson system. One o f the characteristic properties of a second-order phase transition is that tire susceptibility increases anomalously as the phase transition point is approached. According to the Landau theory of phase transitions, tile anomalous behavior of the susceptibility X near the critical density n c can be described by [12] X ~ h e / I n - ncl •

(4)

This occurs due to the increase of the correlation length or mean free time scale at the critical point. What is the influence of these anomalous dissipative processes upon the expanding Universe? Consider a Universe described by the k = 0 R o b e r t s o n - W a l k e r metric and containing an imperfect fluid. The Einstein equations are

k2 = ~ a o R 2 ,

(51

d(pR 3 ) / d t = - 3R 2/}(p _ 3 f / } / R ) ,

(6)

where R is the cosmological scale factor, and p, p and f denote the pressure, the mass density and the coefficient o f bulk viscosity, respectively [13]. The A term in eqs, (5) and (6) has been neglected, since it is always smaller than the corresponding ternrs o f p and p under the condition (3). Due to X ~ g 2 the fennion fluid is not in the extreme relativistic or the extreme nonrelativistic limit during the phase transition. The coefficient f is then on the order o f the energy density times the mean free time, ~ ~ p r.

8 September 1980

Because R/[~ ~ t is on the order of the age of the Universe, when the mean free time is larger than the age of the Universe, tile first term o11 the right side of eq. (6) can be neglected as compared with the second. Then we have d(pR 3)/d t = q f R R 2 = 247rGfpR 3 .

(8)

Strictly speaking, the hydrodynanfical approximation should not be applicable when the mean free time is larger than the time scale o f the fluid movement. But in the absence of any better approximation we will still use the hydrodynamical approach to discuss the qualitative picture. The criterion for Landau's phase transition theory to be applicable is [12] l>ln

ncl/n c>>(x/g) 2 ~ x .

(9)

Hence we may, for simplicity but with no loss of generality, take the mean flee time to be

r/t~,~O,

In nJ/n~ >X,

~ ( r 0 / t c ) ? , 1,

In

(10)

ncllnc
and f = pc r, where Pc and t c are, respectively, the density and the age of the Universe at the phase transition epoch and r 0 is the time scale of collision between an average pair of elementary particles, outside the range of the phase transition. With these considerations, we can obtain the solution for the cosmic scale factor during the phase transition : R ~explSTrGf(t exp[~'rtc2(t

to)] tc) ] ,

(1 I)

where we have used the relation t c ~ (GPc)l/2, since, when n > nc, the fermion fluid is in the extreme relati.vistic limit. Because the particle conservation equation is unaffected by dissipation, we have n~R-3

~exp[

24~Gf(t

tc) l .

(12)

The rate of increase o f the entropy per particle by the dissipative processes is described by [13] = (qf/nkT)(R/R) 2 .

(13)

The total entropy generation during the phase transition epoch is, thus, no less than

(7) 155

Volume 95 B, number 1

S ~ S1/2(rO/tc)3/2?t -3/4 ,

PHYSICS LETTERS (14)

and the cosmic specific entropy may be completely generated by the anomalous dissipation of the cosmological phase transition only, if we have

(tc/rO)3/2~ 3/8 ~ 10 - 8 .

(15)

The Higgs mass/.t 2 can be determined by the phase transition time tc,

la ~ G - l / 4 tcl/2~.3/8 .

(16)

For the very early Universe, say t c ~< 10 -36 s ro/t may be on the order of unity [4]. Eqs. (15) and (16) can, thus, be satisfied by the Higgs mechanism. On the other hand, eqs. (15) and (16) may also be regarded as constraints on the mass and self-interaction of the Higgs particle. It has been emphasized [1] that in the non-baryonconserving process model the entropy can be generated only if the masses of the h e a w particles are larger than some critical mass; this condition is also the condition of the conservation o f previous baryon asymmetries. However, in the phase transition model the anomalous dissipation can occur in ranges of the GUT parameters different from the non-baryon-conserving model. Then, there exists wide parameter ranges in which both the generation of the final cosmic entropy and the erasure of previous baryon asymmetries may occur during the GUT epoch. In the model developed above the increase of the cosmic scale factor R goes like t ~ before and after the phase transition while it is exponential during the phase transition. This "explosive" solution of R may be a general effect of the anomalous dissipation on the evolution of the Universe, since it depends only on the time scales and not on the details of the phase transitions * 1 In this case, the entropy is produced not from the dissipation o f the movement belonging to the degree of freedom of matter itself but from that of the universe's own expansion. If the universe were regarded as an isolated thermodynamical system the entropy could be generated only by irreversible processes within the cosmic matter. However, if gravitation is taken #1 Exponential solutions of R have also been obtained recently by Sato (preprints, 1980), using the model of first-order phase transition to produce the domain structure of the cosmological baryon number. 156

8 September 1980

into account the Universe must not be regarded as a thermodynamical system in equilibrium or with a small deviation from equilibrium. The expanding Universe is not in the state of maximum entropy whether its initial state is hot and chaotic or zerotemperature and homogeneous. Therefore cosmic entropy can be generated only if there exists an effective mechanism to dissipate the ordered gravitational move ment of the Universe as a whole. In this sense, the dissipative mechanism of a phase transition is like that of particle production near singularities [15]. Indeed, the energy density of the fermion is constant during the phase transition in spite of the "explosive" evolution o f R . The constancy maintained by the anomalous dissipation is, in fact, the equivalent in hydrodynamical language of particle production. Similarly, the problem of the negative value of the effective pressure exists in both cases [16] The author wishes to thank Drs. B. Carr and B.L. Hu for stimulating discussions. He would also like to thank Professor M. Rees for hospitality at the Institute of Astronomy, Cambridge and Professor C.N. Yang for hospitality at the Institute for Theoretical Physics, State University of New York at Stony Brook.

References [1 ] B.J. Carr and M.S. Turner, Enrico Fermi Institute preprint (1980) No. 80-09. [2] M. Yoshimura, Phys. Rev. Lett. 41 (1978) 281. [3] S. Weinberg, Phys. Rev. Lett. 42 (1979) 850. [4] D.V. Nanapoulos and S. Weinberg, Phys. Rev. 20D (1979) 2484. [5] S. Barr, G. Segr~ and H.A. Weldon, Phys. Rev. 20D (1979) 2494. [6] A.D. Linde, Rep. Prog. Phys. 42 (1979) 25. [7] G. Lasher, Phys. Rev. Lett. 42 (1979) 1646. [8] L.Z: Fang, Kexue Yongbao 22 (1977) 258. [9] L.Z. Fang and G.Z. Xie, Kexue Tongbao 24 (1979) 548. [10] R.W. Brown and F.W. Stecker, Phys. Rev. Lett. 43 (1979) 315. [11 ] S.A. Bludman and M.A. Ruderman, Phys. Rev. Lett. 38 (1977) 255. [12] A.Z. Patashinskii, Fluctuation theory of phase transitions (Pergamon, 1979). [13] S. Weinberg, Astrophys. J. 168 (1971) 175. [14] J. Ellis, M.K. Gaillard and D.V. Nanopoulos, preprint (1979). [ 15 ] B.L. Hu, Proc. First Theoretical particle physics conf. (1980) (Academy of Science Press, Peking, China). [t61 B.L. ttu and L. Parker, Phys. Rev. 17D (1978) 933.