Volume 215, number 1 PHYSICS LETTERS B 8 December 1988

T H E R M A L INSTABILITY OF T H E DE SITTER VACUUM

Diego PAVON Department of Physics, Faculty of Sciences, Autonomous University of Barcelona, E-08193 Bellaterra (Barcelona), Spain

Received 10 June 1988

Assuming thermal equilibrium between the hot de Sitter vacuum and thermal radiation we obtain the second moments in the fluctuations of the energy around that equilibrium, lts stability imposes an upper limit on the temperature of this vacuum,

It has been known for some time now that the de Sitter space

1 1 "~ ~ Ar2 d r 2 d s 2 = - ( l - ] A r - ) d t 2 + 1 -

+r2(dO2+sin20 de 2) , ( 1 )

where A is a positive cosmological term arising from the vacuum energy density, A = 8~p .... enjoys some thermodynamic properties similar to those of black holes. In particular, this vacuum can be thought of as possessing a temperature [ 1 ] given by

T,.a,. = ( 2 x ) - ' (~A) '/2 (2)

Recently, it has been suggested that this vacuum may spontaneously decay into radiation. The obser- vational consequences of this suggestion have been explored by Freese et al. [ 2 ]. They found, amongst other things, that for times later than t~ 1 s the vac- uum energy density cannot dominate over the radia- tion energy density. Likewise, Gasperini [3,4] has pointed out that the de Sitter vacuum can reach a state of thermal equilibrium with radiation - say thermal radiation - at some temperature of the order of the Planck value T ~ Mp. By interpreting the cosmologi- cal term as a measure of the vacuum temperature and allowing A to depend on the cosmic time this author has provided a natural justification for the current extremely low value of that term (~< 10-~2°M~ ). As the universe experiences a fr iedmannian expansion the vacuum temperature goes down whence A de- creases too. This argument has the attractive feature

o f being free of whatsoever underlying fine-tuning assumption.

In view of the above it seems natural to ask whether the thermal equilibrium proposed by Gasperini be- tween the de Sitter vacuum and a bath o f thermal ra- diation is stable. The target o f our letter is precisely to answer this question. As we will show, by means o f an essentially classical approach, such an equilib- r ium becomes stable provided that the cosmological term fulfils the restriction

A < 9 9 3 / 2 a , (3)

where a = ~ lr 2gerr denotes the radiation constant and gcfr the effective number of relativistic degrees of freedom (we choose units so that c=G=h=kB= 1 ). Note that the instability we are suggesting is different in nature from that analysed by some authors [ 5 ].

It is interesting to see that despite the specific heat of the de Sitter vacuum Opvac/OTv,,. is positive, its heat capacity results negative. Effectively, the volume within the de Sitter horizon 47c(3/A)3/2 multiplied by the vacuum energy density yields Evac= (3/4A) ~/2. Hence, C,,c (=OE,,ac/OTvac) becomes

Cvac= -37~ /A . ( 4 )

This should not be surprising since, as pointed out by Landau and Lifshitz [ 6 ], systems bounded by long- range forces may exhibit negative heat capacity in spite of the fact that the specific heat of their constit- uents is positive. One illustrative example of this kind of systems can be found in ref. [ 7 ]. On the other hand, systems of negative heat capacity (e.g., normal starts,

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Volume 215, number l PHYSICS LETTERS B 8 December 1988

Schwarzschild black holes and the gas of massive modes of heterotic superstrings) are well known in the literature.

Before going any further, it is convenient to have in mind that any thermodynamic system, even at equilibrium - black holes included - experiences spontaneous fluctuations in its macroscopic thermo- dynamic quantities around their constant mean val- ues. In this connection it is permissible to consider the cosmological term as not strictly fixed but subject to small spontaneous fluctuations, 8,4, around its av- erage value ( A ) , which is supposed to remain con- stant. This is quite reasonable inasmuch as this term represents the vacuum polarization energy which we expect to fluctuate around some mean value.

We shall study the stability o f the thermal equilib- rium between the de Sitter vacuum and the bath o f thermal radiation by analysing the second moments in the fluctuations o f energy and temperature around the equilibrium Tv~c = Trad. The energy of the radia- tion and its corresponding heat capacity are

] -- tad, (5 )

and

~.3/2 qr'3 (6) C r a d = 4 a ' 4 ~ ( 3 / ( A ) J ~ a d ,

respectively. As is well known, the second moments in the fluc-

tuations o f the energy read [ 8 ]

(SEvac ~Evac ) = ( ~Erad ~ ) E r a d ) = C e f f T 2 , (7)

where T ( = T~d= Tvac) is the equilibrium tempera- ture and

67ra C~fr= 9 ~ 3 _ 2 a ( A ) (8)

denotes an effective heat capacity defined by Cg?~ = C ~ + C , ~ . The angular brackets mean statistical average.

In the first equality of eq. (7) we have made use of the conservation of energy 8 ( E , , ~ + E ~ d ) = 0 . In or- der that the second moments behave properly, C~f,. has to remain positive which automatically leads to relation (3). The procedure we have followed mim- ics the one which was utilised in the study of the fluc- tuations around the thermal equilibrium of a black hole immersed in a bath of thermal radiation [9 ].

Assuming the total number of relativistic degrees

of freedom at T ~ M p to be g~n-~ 150 one has a ~ 5 0 , hence from (3) it follows that the maximum value o f ( A ) compatible with stable equilibrium is close to 2.8M 2 and the corresponding value o f T v ~ 0.154

X Me. Starting from eq. (4) any other second moment is

easily derivable. For instance,

(ST~d 8T~.~) = ( 3 ~ / 2 a ) 2 C ~ T 6 ,

( S A S A ) = ( 2 / 3 ~ ) 2 C ~ ( A ) 4 .

(9)

(10)

The divergence in the second moments at ( A ) = 9n3/2a indicates that at this precise value of the cosmological term a phase transition from a sta- ble equilibrium situation to a non-stable one takes place. On physical grounds it is easy to see that for any value of ( A ) greater than 9n3/2a thermal equi- librium will no longer be stable. Effectively, let us as- sume that in that situation, a spontaneous statistical fluctuation makes the de Sitter vacuum gets hotter than the thermal bath. The former will immediately react trying to rise the temperature of the bath via the emission of more radiation. However, since its heat capacity, - 3 n / ( A ) , is negative and in absolute value greater than that of the bath, ] C,.acl > Cra~, the tem- perature difference between them will augment more and more. This, in its turn, will trigger a runaway process: as the vacuum temperature increases the cosmological term increases too, due to eq. ( 1 ). As a consequence the expansion of the universe ceases to be of the de Sitter type (R( t )ocexp[ (~ (A))~/2 t ] ) which assumes ( A ) constant, to evolve presumably into an era of faster expansion. It seems hard to rec- oncile such an era with current observational limits [ 10 ]. Accordingly, we may conclude that, in view of the thermodynamic properties of the de Sitter vac- uum and the suggestion made by Freese et al. as well as that by Gasperini, the temperature of the de Sitter hot vacuum, in thermal contact with black body ra- diation, has never been greater than 0.154Me.

It is a pleasure to thank Professor Davies for useful correspondence as well as Dr. Verdaguer and Profes- sor Kijowski for illuminating conversations. I am also obliged to the Institute o f Theoretical Physics of the Polish Academy of Sciences, where this work was partially done for warm hospitality and financial

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Volume 215, number 1 PHYSICS LETTERS B 8 December 1988

suppor t . P a r t i a l f i n a n c i a l s u p p o r t was also p r o v i d e d

b y t he C A I C Y T of the S p a n i s h G o v e r n m e n t .

References

[1] G. Gibbons and S.W. Hawking, Phys. Rev. D 15 (1977) 2752.

[2] K. Freese, F.C. Adams, J.A. Friemanand E. Mottola, Nucl. Phys. B 287 (1987) 797.

[3] M. Gasperini, Phys. Lett. B 194 (1987) 347. [4] M. Gasperini, Class. Quantum Gray. 5 (1988) 521, [5] P. Ginsparg and M.J. Perry, Nucl. Phys. B 222 (1983) 245. [6] L. Landau and E.M. Lifshitz, Statistical physics, 3rd Ed.

(Pergamon, New York, 1980 ) §2 l. [7] D. Pav6n and P.T. Landsberg, Gen. Rel. Grav. 20 (1988)

457. [ 8 ] See any standard text-book on statistical physics. [ 9 ] D. Pav6n and J.M. Rubi, Phys. kerr. A 99 ( 1983 ) 214.

[10] M.S. Turner, in: Cosmology and particle physics, eds. E. Alvarez el al. (World Scientific, Singapore, 1987 ).

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