General Relativity and Gravitation, Vol. PS, No. 5, 1996

Particle Decay and Buli~ Dissipative Stress in the Early Universe

Diego Pavdn, 1,2 Jdr6me Gariel 1,3 and Gerard Le Denm~t I'4

Received July 13, 1995. Rev. version October 15, 1995

In this paper we show that the combined action of particle decay and a physically motivated dissipative bulk stress can lead to a period of generalized inflation and temperature increase. This effect may have occurred at the baryogenesis era.

1. I N T R O D U C T I O N

As has been recently shown [1], change in the number of particles by either direct production or decay of pre-existing particles has a non-negligible im- pact on the dynamics and thermal evolution of the early universe. This may be brought about by the decay of a nonrelativistic fluid into a rela- tivistic one (as for instance the electron-positron annihilation, Ref. 2, and the decay of heavy bosons into quarks and leptons, Ref. 3) or the creation of particles and/or strings out of the quantum vacuum [4]. These processes have been stuoied phenomenologically in a Robertson-Walker background

1 Laboratoire de Gravitation et Cosmologie Relativistes, Universitd Pierre et Marie Curie - CNRS/URA 769, Tour 22/12, 4~me dtage, B.C. 142, 4, Place Jussieu, 75252 Paris cedex 05, France

2 Permanent address~. Departamento de Fisica, Facultad de Ciencias, Edificio Co, Uni- versidad Autdnoma de Barcelona, 08193 Bellaterra, Spain. E-mail: diego@ulises.uab.ss

3 E-mail: gele~ccr.jussieu.fr 4 Permanent address: Universitd de Montpellier If, GRAAL-URA 1368, Place Eugene

Bataillon, 34095 Montpellier cedex 5, France

573 0001-7701/96/0500-0573509.50/0 {~) 1996 Plemtm Publishing Corporation

574 Pavdn, Gariel a n d Le D e n m a t

by re-interpreting the particle source as an effective bulk stress and ignor- ing the fact that, in general, a physical bulk stress may well enter the pic- ture too. A kinetic theoretical basis to this procedure has been provided recently [5]. Our target here is to incorporate consistently the physical bulk stress that naturally arises from the interaction between both fluids during the decay process. This will be implemented from the outset by explicitly separating the particle production from the physical bulk stress 7r [6]. This is crucial from the point of view of temperature evolution since the evolution of the latter predicted by viewing the particle production as a fictitious bulk stress sharply differs at early times from the one that follows from the physical bulk stress [7]. In this way we will obtain the time dependence of the scale factor, the comoving Hubble length, and the temperature of the relativistic fluid during the decay process. As we will see, the comb~_ned action of particle decay and a moderate lr can lead to a period of inflationary expansion with a brief increase in the radiation temperature.

There exists the widespread belief that the scalar field responsible for the ordinary inflation decayed into unstable heavy particles (among them supermassive bosons) and light particles (photons, quarks, etc). In this light our study may apply to the epoch of the decay of these heavy bosons, a process at the heart of the baryogenesis mechanism.

2. BASIC EQUATIONS AND NUMERICAL ANALYSIS

Let us consider a FRW universe with flat space sections filled with a nonrelativistic fluid consisting of massive particles, Pl ~- n l M , P1 "~ O, and a radiation-like fluid (photons, neutrinos and light particles in general) with p2 = (Tr2/30)g.T42, P2 = p2/3, where g. stands for the total number of relativistic degrees of freedom, which we will consider constant during the period of interest. The former fluid decays spontaneously into the latter with constant rate F1 = I"Vt/Nt, and so F2 = 1(I2/N1. Here Ni = n i R 3 (i = 1, 2), ni being the particle number density of the corresponding fluid, R the scale factor of the P~obertson-Walker metric and F1 and F2 negative and positive constant rates respectively, such that F1 -t- F2 > 0. From the above equations it follows that the number of radiation-like particles grows up with time according to

F2 eFlt), = + Yl0 (1 - (1)

where the subscript zero refers to the time at which the nonrelativistic fluid starts to decay.

Particle Decay and Bulk Dissipative Stress 575

Let us further assume that the fluids interact with one another and that their interaction is phenomenologically describable by a nonequilib- rium bulk stress

zc = - r (0 =--- 3H, H - R / R ) , (2)

which for expanding universes tends to decrease the overall pressure. Dis- sipative stresses like (2) are bound to arise whenever matter and radiation interact with each other. The corresponding phenomenological coefficient

(> 0) has been studied from the macroscopic [S] and kinetic [9] stand- points as well as from the standpoint of thermodynamic fluctuation theory [10]. This quantity can be approximated by ~ = olp27" where a denotes a positive constant parameter of order unity, and r the mean free interaction time between the matter and radiation particles. For simplicity we shall take T = f lH -1 , fl << 1, and so

~ = ~ p 2 H -1, (3)

where the constant ~ has been absorbed into a. This is a reasonable assumption since the mean interaction time between particles must be shorter than the expansion time of the fluid as a whole, for any hydrody- namic description to be valid.

Due both to the variation in the particle number density and the heat generated by the dissipative pressure, the radiation temperature evolves according to [7,11]

( O P2 as+ T2(Op2/OT2),

O 2 . (4)

In writing this we have implicitly assumed that the light particles are amenable to a fluid description right after their creation and that they become thermalized very quickly. With the help of the above, eq. (4) can be immediately integrated to give

T20 = \n~2o) \ ~ ) " (5)

The shortest way to obtain the time dependence of the scale factor is through the Friedmann equation H 2 = (8~rG/3)(pl + P2). Replacing Pl and P2 by their expressions above and inserting the integrated expressions

576 Pavdn , Gar ie l a n d Le D e n m a t

for the particle number densities one has

H2 = 8~rG n l o M e -Irll~ 3 -E

+ g. [1 + or(1 - e- IF. I , ) ] , /3_, (6)

where V/o -_-- n lo/n2o whereas F =- I~2[FI[ -1 stands for the number of radiation-like particles produced in a single decay, i.e. the decay multiplic- ity. In general this equation does not admit analytical solutions. However, by differentiating (6) with respect to t it can be readily shown that one will have/~(t = 0) > 0 (initial generalized inflation) if a > a*, where

36Ho 2 + - - Ho + - - Irll - (7) p20 / p20 ~ F2 ,

and/~( t = 0) < 0 otherwise. To ascertain the general behavior of the scale factor, the comoving

Hubble length 1 / (HR) , and the radiation temperature, we have solved eq. (6) numerically for different choices of the parameters and used the corresponding solutions in eq. (5). Figures 1-3 illustrate the evolution of R(t ) , ( H R ) -1 and X(t) =- T2(t)/T2o, respectively, for IF] I = 1, F = 3, 7/0 = 1, R0 = 1 (in this case c~* ~_ 0.024), and three different values of c~: c~ = 0 (no viscosity), 0.05 and 1/12, respectively. The pattern depends on whether c~ is higher or lower than c~*. In the first case, as Fig. 2 clearly shows, there is a generalized inflation period ( d ( H R ) - l / d t < O, i.e./~(t) > 0) along with an initial increase in the temperature. Notice that the expansion remains inflationary independent of the sign of :~(t). The larger c~, the higher the maximum of x(t) and the wider the time interval At* for which T2(t) > T20. For high enough c~ - - but still moderate; see the curve for c~ = 1/12 in Fig. 3 - - At* > IFll. In the second case (c~ < c~*) - - curves with c~ = 0 - - the expansion is non-inflationary from the start and the radiation temperature decreases monotonously.

From the acceleration equation

4~rG = 3 (p + 3Peff), (8)

one may realize that the dissipative bulk pressure alone, eqs. (2) and (3), cannot account for the inflationary period, since i~ /R is negative if one takes the effective pressure exclusively as Peff -- P1 + P2 + r . This period

P a r t i c l e D e c a y a n d B u l k D i s s i p a t i v e S t r e s s 5 7 7

7 0 , . . . . , , , , , , . , , . . . . , , , , . , , . , . , , . , . . . . . . , . . . . . . . , , , , , , , , , , , , ,

60

5 0

4 0

3 0

2 0

10

0 0 1 2 3 4 5 6

t

Figure 1. Time evolution of the scale factor for Iril = 1,1" -- 3,77o = 1,Ro = 1.

m +-4

2~ .,i.1.............,.........i......... .... ..,......,......

2.0

1.5

1.0

0.5 ~ a :O .05

a : 1 / 1 2

0.0 , , , . , , , , , l , , , , , , , , , l . , , , , , , , + 0 1 2 3 4 5 6

t

Figure 2. Time evolution of the cornoving Hubble length for the same set of parameter v~lues as in Fig. 1. The decrease of II(HR) means inflationary expansion.

578 P a v S n , Gariel and Le D e n m a t

A

1 . 2 y , , , , i w , , , , 0 , 1 0 . 0 , , . , , , , , , , , l l l , , , o , , , , , , o , , , , , . . , ' w , ' ' ' ' '

1.0

0.8

0.6

0.4

=0 0.2

O. 0 l i i l l , * ' l l l ' , * l j i l | l i i | l l t l ' l l ' ~ I l l l .

1 2 3 4 5

t

Figure 3. T i m e evolut ion of x(t) = T2(t)/T2o for the same set of pa r ame te r values as in Fig. 1.

must be explained as the effect of the combined action of the bulk pressure and the pressure associated with particle production [1]; i.e. the latter - - which has to be included in the effective pressure - - is so high at the beginning of the decay that , when added to 7r, it overpowers the energy density p and the hydrostatic pressures taken together. Alternatively this can be viewed in the following way. At the beginning of the decay/~2 is very high, and so the radiation temperature increases very quickly and the Hubble factor augments so much tha t for some t ime H 2 > I/:/I . Later on, because of the sharp decrease in the number of massive particles,/~2 becomes much lower and the expansion ceases to be inflationary. The bulk stress 7r adds positively, of course, to the rate of expansion.

Due to the limitations of this two-fluid model, chiefly because of the assumption ~ -- constant, its validity range is restricted to a t ime interval not much greater than, say, five times ]Fll- Once this t ime has elapsed and the massive fluid essentially decayed into the light one, the bulk stress should accordingly be set to zero, and so R ~ t 1/2 and T c( R -1. However, in this "canonical" evolution the temperature is bound to drop below the rest mass of one or more of the light particles - - which initially behaved like radiation because of the high ambient temperature - - and a fresh

Particle Decay and Bulk Dissipative Stress 579

dissipative bulk stress will set in as long as these particles interact with the massless ones, e.g. photons and perhaps neutrinos.

3. CONCLUDING REMARKS

We have explored the evolution of the scale factor, the comoving Hub- ble length, and radiation temperature induced by the process of decay of massive particles into light ones, with the inclusion of the dissipative stress arising from the interaction between both fluids. It is generally believed that particle decay is unable by itself to drive inflation, and that the bulk pressure can't either, for it should have to take unrealistically high values. However, as it has turned out, the combination of particle decay and a moderate dissipative bulk stress may render the initial phase of the decay inflationary with a brief increase in the radiation temperature. Obviously for higher multiplicities the inflation, will be stronger. Since the quantity ln[(HR)f/(HR)i] is lower than 70 in all the cases considered here, this inflation cannot solve the horizon, flatness and smoothness problems of the standard big-bang scenario [12]. These are probably solved by some version of the ordinary scalar field-driven inflation theory at the GUTS era [3]. The former, however, must have occurred some time after the latter and prior to the nucleosynthesis age - - perhaps at the baryogenesis pe- riod - - and may have helped to dilute undesired relics left over by the phase transitions at GUTS scales [3,13,14].

A more realistic treatment would have dealt with a non-constant since this quantity should slowly evolve from a small positive initial value at the beginning of the decay down to disappear altogether at the end of it [8-10]. However, the calculation would have been terribly messy and the results should have not differed substantially from those reported here.

In using eq. (2) for the bulk stress we have limited ourselves to the sta- t ionary Eckart 's theory of irreversible processes [15] which, though widely employed, is now known to be of limited scope. In reality we should have resorted to a more general transport equation, such as the one proposed by nonstationary theories (see e.g. Ref. 16) but these are much more involved. We will deal with these elsewhere.

ACKNOWLED GEMENTS

We are indebt@d to Winfried Zimdahl for conversations and the anony- mous referee for helpful comments. We thank Tahar Melliti and Vicen~ Mendez for their kind computational assistance. One of us (DP) is deeply grateful to the members of the Laboratoire de Gravitation and Cosmologie Relativistes for warm hospitality and to the CNRS for financial support.

580 P a v 6 n , Gar ie l a n d Le D e n m a t

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