LETTERE AL NUOVO CI~I]~NTO VOL. 44, ~-. 8 16 Dicembre 1985

Bulk Viscosity and the Inflationary Universe (*).

G. P_~PI~I and M. WEiss

D e p a r t m e n t o/ P h y s i c s a n d A s t r o n o m y , U n i v e r s i t y o] R e g i n a

l~egina, S a s k . S d S 0A2, C a n a d a

(r icevuto il 25 Se t t embre 1985)

PACS. 04.20. - General re la t iv i ty .

S u m m a r y . - Phase t rans i t ions f rom a universe w i th nonnegl igible bulk viscosi ty to an inf la t ionary one are possible only if v iscosi ty v i r tua l ly d i sappears in the process. This suggests the phase change involved is of the t ype no rma l fluid to superfluid. Some exac t solut ions of E ins te in equa t ions for a viscous fluid are also given. They show the subt le i n t e rp l ay of v iscos i ty and equa t ions of s ta te .

Though mos t cosmological models are based on an inviscid dus t or perfect - f lu id descr ip t ion of ma t t e r , t he i m p o r t a n c e of v iscosi ty in the t h eo ry of t he ear ly un iverse has a t t imes been emphas i zed (~). This f inds jus t i f icat ion in the work of Is rael and Vardalas (2) who have shown t h a t the bulk viscosi ty ~ of s imple gases does no t van ish a t t e m p e r a t u r e s i n t e rmed ia t e b e t w e e n the nonre la t iv i s t i c and ex t reme- re la t iv i s t i c l imits. Suppor t is also p rov ided by Weinbe rg ' s calculat ions regard ing fluid mix tu re s of ma te r i a l par t ic les w i th shor t mean free t ime, and r ad ia t ion q u a n t a such as pho tons , neu t r inos and g rav i tons (a).

The effects of bulk v iscos i ty on the ini t ial cosmological s ingular i ty h a v e been s tud ied (4) and exac t solut ions of E ins t e in ' s equa t ions have been found for classes of cosmological models t h a t incorpora te bulk (~) and even shear v iscosi ty (e). Pa r t i cu l a r ly

(*) "Work partially supported by thc Natural Sciences and Engineering Research Council of Canada. (1) See for instance: S. W. I{AWK1NG: Astrophys. J . , 145, 534 (1966); C. W. MISNER: .Astrophys. J . , 151, 431 (1968); S. ~VEINBERG" Gravitation and Cosmology (J. VViley and Sons, New York, N.Y., 1972), p. 469 ff. (~) \V. ISRAEL and J. N. VARDALAS: Left. Nuovo Cimento, 4, 887 (1970). (a) S. ~VEINBERG.* Aslrophys. J . , 168, 175 (1971). See also N. STRAUMANN: Helv. Phys. Acta, 49, 269 (1976). (4) •. HELLER and L. SUSZYCKI: .Acla Phys. Pol. B, 5, 345 (1974); 5L HELLER and Z. I~LIMEK: .dsirophys. Space Sci., 33, L37 (1975). (5) --~[. HELLER, Z. ]~LIMEK and L. SUSZYCKI: Aslrophys. Space Sci., 20, 205 (1973); M. SZYDI, OWSKI and M. ItELLER: Acla Phys. Pol. B, 14, 303 (1983); S. R. ROY and O. i ~ TIXVARI: Indian J. Pure Appl . Math., 14, 233 (1983); A. A. CO/~EY and ]3. O. J. TUPPER: Phys. Rev. D, 29, 2701 (1984); Astrophys. J . , 288, 418 (1985); N. O. SANTOS, R. S. DIAS and A. BANERJEE: J. Math. Phys. (N .Y . ) , 26, 878 (1985). (6) G. E. TALTBER: .Astrophys. Space Sei., 57, 163 (1978).

612

B'ULK VISCOSITY AND TILE INFLATIONARY UNIVERSE ~ 1 ~

interesting are the results of Neugebauer and Strobel (7), Treciokas and Ellis (s) and Nightingale (3) who have shown that for homogeneous, isotropic models and equa- tions of state of the type (lo)

(1) p = ~q, 0 < ~ < � 8 9

(2) p = ( 9 ) - - 1 ) e - - r 1 < ? < } ,

the cosmic scale factor /g increases with viscosity according to

(3) [ 2 [ [3~t] ]}2/3v

= [exp lvJ '

where r is constant. These results do not have a counterpart in classical cosmology where one expects the expansion to slow down because of viscosity. The question is readdres- sed here in an inflationary context. In the second part of the paper we present some new exact solutions which can be derived for particular choices of r

i) In order to determine the effect of viscosity on the onset of inflation, the requirements of homogeneity and isotropy are maintained, while (1) or (2) are replaced by the appropriate equation of state (11)

(4) ~o = - e .

Einstein 's equations must also include viscosity in the source term. This is accomplished by writing the energy-momentum tensor for a viscous fluid in the form

where u u is the velocity vector,

~ a q ~ ~ 1

and

In (5) ~ and ~ are the coefficients of bulk and shear viscosity and

(6) P ~ = g ~ - uuu~

I t is convenient to consider the spherically symmetric metric (12)

(7) ds s = exp [2v] dt 2 -- exp [22] dr s -- lr2 dO ~ -- y~ sins 0 d9 s .

(7) G. ~EUGEBAUER a n d H . STROBEL: Vqissen. Ze i t s ch r . de r F r i e d r i e h - S e h i l l e r - U n i v e r s i t ~ t , J e n a J a h r g . 18 (1969). (s) R . TRECIOKAS a n d G. F. R . ELLIS: Commun. 1Vl'ath. Phys., 23, 1 (1971). (9) J. D. ~IGHTINGALE: .d8~'ophys. J., 18~, 105 (1973). (10) G. NEUG~BAUER a n d H . STROBEL, ref . ( ' ) , a s s u m e (1) as e q u a t i o n of s t a t e . The s a m e a s s u m p - t i o n is m a d e t h r o u g h o u t t h i s p a p e r . (11) A. H . GUTH: Phys. Rev. D , 23, 347 (1981). (~2) ]) . KRAMER, H . STEPHANI, ~VJ[. ~r a n d E. HERLT: Exact solutions o] Einstein ]ield

equaiions, V E B D e u t s c h e r V e r l a g de r W i s s e n s c h a f t e n (Ber l in , 1980), p . 166,

614 G. PAI'INI a n d )~. wi~iss

Here the u n k n o w n func t ions v, ~ a n d Y d e p e n d ex( ' lusively on r a n d t. I n the co -mov ing f r ame of t he fluid def ined b y u = exp [v]6~, E i n s t e i n ' s equa t i ons become

{ { y2} -- 1 . 2 exp [-- 2~] Y " - - Y ' ~ ' + 2Y J y (8) -y., = Y

(9) 2 exp [-- 2).]

y { - ~ - '+ ~-',~ = ~-r o,

{ (y2 } Y'2 / 2 e x p [ - - 2v] :~ + ]yp = 1 2 e x p [ - - 2 1 ] Y ' v ' + 2 Y ] Y

= 8x -- p -- exp [-- v] - - ~ - - - ~ ) , + - - 2 ~ ~ ,

f]~" 1~'~ ' ,~,~, + ~__~v'} + (11) e x p [ - - 2 ) , ] / - ~ - + v " ~ ' v'2 Y

+ e x p [ - - 2 v ] - - ~ - - ~ t ~ - ~ + i~ Y - - - -

where . = 5 / ~ t a n d ' = 5/~r. The canon ica l fo rm of t he R o b e r t s o n - W a l k e r me t r i c is o b t a i n e d f rom (7) b y s e t t i ng

(12) v = 0 , exp [2),] = R2(t)(1 -- kr2) -~ , y 2 = R ~ ( t ) r 2 .

T h e n eqs. (8)-(11) r educe to (13)

(13) ~ R ~ - 3 e ,

/~ R 4z (14) R 12,n~ ~. = -- ~ (e + 3p) ,

a n d no longer con t a i n ~ because of t he i so t ropy of t he met r ic . C o m b i n i n g (13), (14) a n d (4), one o b t a i n s

(15) ~ - /~.-- R 2 .

I t follows f rom (15) t h a t un ive r ses for wh ich ~ oc o, or ~ = cons t > 0, or ~t = ~1 w i t h ~1= cons t > 0, choices usua l ly m a d e in t he l i t e r a t u r e (9), are no t c o m p a t i b l e w i t h an e x p o n e n t i a l expans ion . The i n f l a t i ona ry e x p a n s i o n is in fac t poss ible on ly if, a t t he same t ime, ~ decreases exponen t i a l l y . This can be easi ly a sce r t a ined f rom (15) b y sub- s t i t u t ion . P h a s e t r a n s i t i o n s f rom a un ive r se w i t h nonnegl ig ib le b u l k v i scos i ty to an

(la) S. IVEINBERG: Gravi tat ion a n d Cosmology (J. Wiley and Sons, New York, N. Y., 1972), p. 469 ff"

B U L K VISCOSITY A ~ D TI-I~ I ~ F L A T I O ~ A R Y UNIV~RS]~ 615

inflationary one can, therefore, occur only if in the process viscosity virtually disappears. This and the fact that the Higgs fields in metastable false vacua of inflationary cosine- logics correspond to superfluids (14) satisfying the special equation of state (4), strongly suggest the phase change involved corresponds to a normal fluid-to-superfluid transition.

ii) The argument advanced makes the viscous fluid model an obvious candidate for the pre-inflationary universe. Some new exact solutions are presented here. They correspond to different functional forms of ~ and the equation of state (1).

a) If ~t = ~1 > 0, a solution of (13) and (14) for a dust-filled Universe is

k )~ 9~ 1 (16) R(t) = 24zt~-- 1 t , ~ = t- ~

with ~1> 1/24zl, if k = 1 and 0 < ~1< 1/24z for k = - 1. The expansion (15) is (9 = 3/t .

A solution of (13) and (14) corresponding to a dust-filled uuiverse and the rather idealized choice r = const is given in ref. (5) for the very special case k = 0. A more general dust model can be obtained from (2) and (3) when ~, = 1. The one represented by (16) shows a marked decrease in the growth of/~ when ~ itself decreases, in qualitative agreement with the arguments of ref. (9).

Other conformally flat universes for which the conformal factor depends exclusively on t can be obtained by s e t t i n g

exp [2v] = exp [24] = y2/r2 = f2(t)

in eqs. (8)-(1). The following solutions have been found.

b) Viscous fluid with p - = ~ and $ constant. One obtains from (11) and (13)

2 4 ~ ]2__ el / l -ant2 = 0 (17) ] 3~ + ~ '

where c 1 is a constant. This equation can be integrated immediately when a choice for ~ is made. For the particular case c1= 0, eq. (17) integrates to

/ = 0 2 - (8~$/(~ + 1 ) ) t ' o~ # - 1 ,

where c 2 is constant. For a radiation-filled universe (~ = �89 one has in particular

27 0 ~ ~ zr~ 2 , 0 = 18zr~, ' l

(1~) G. PAPINI a n d M. WEISS: Phys. Le l t . .4 , 89, 329 (198~); Left. Nuovo Cimento 44, 83, (1985). (1~) See G. F . R . ELLIS: Relativistic cosmology, in Prec. S . I .F . , Course X L V I I ( A c a d e m i c P ress , N e w Y o r k , N . Y . , 1971).

616 G. pAPINI and M. w e i s s

and

a t = 0 , o ~ , = % ~ = 0 .

I n this mode l the bulk viscosi ty affects t he l ine e lement and O.

c) p = ~ , ~ = ~1/], ~1= const > 0. F r o m eqs. (8)-(11) one obta ins

(18)

] = {c~exp [ - - 1 § 3~ t ] Cl(~ 21(-1-3a) 4~1 y ~ ~- 8z~r

('+3=)! 1 2 4 ~ exp -- 8~1 * ( l e a ) 2 1 + ~ J

( ' + +

where c2 is a constant . The expansion is g iven by

o - ~ T ~ exp - 4 ~ 1 * c2 exp -- 4 ~ 1 1 + ~ J ~1 ~ J

and a~ = 0, ~o~-- @v = 0. The met r ic becomes flat for t -+ c~, while Q vanishes in the same l imit , for any posi t ive va lue of ~. For ~ < -- 1/3, ] --> 0 exponent ia l ly wi th t, while for -- 1/3 < ~ < 0 ] increases exponent ia l ly .

Though indica t ive of the va r i e ty of universes, one can genera te even wi thou t abandoning the const ra int of a conformal ly fiat line e lement , the solutions g iven depend on the par t i cu la r choices of ~(t) and of the equa t ion of s tate . Fo r lack of more detai led models of the evo lu t ionary stages before inflation, these choices are not, at present , v e r y s t r ingent . This suggests the type of research still needed (16) to place the theory of the viscous phase on a f i rmer basis.

(ls) See for instance B. L. HU: Phys. Lett. A, 90, 375 (1982).