Anisotropic asymptotic behavior in chaotic inflation

Volume 253, number 1,2

PHYSICS LETTERS B

3 January 1991

Anisotropic asymptotic behavior in chaotic inflation M. H e u s l e r Institute for Theoretical Physics, University of Zfirich, Schrnberggasse 9, CH-8001 Zurich, Switzerland Received 19 June 1990

We show that homogeneous cosmologicalmodels which are coupled to a scalar field can only approach isotropy at infinite times if the underlying Lie group is admitted by a Friedmann-Robertson-Walker model. This result generalises a theorem of Collins and Hawking for matter satisfying the positive pressure condition to scalar fields with an arbitrary positive, convex potential having a local minimum at ~o with V[~o] = 0.

In most investigations o f cosmology the universe is assumed to emerge from the Planck era in a highly symmetric F r i e d m a n n - R o b e r t s o n - W a l k e r ( F R W ) state. Since this assumption is based on technical reasons rather than on physical ones, it is important to investigate whether the present symmetric state of the universe can be obtained from more general initial conditions. Dropping the assumption o f isotropy, Collins and Hawking [ 1 ] have shown that this is not possible within the class of homogeneous but initially anisotropic cosmological models. Their theorem, which states that only a subclass o f measure zero in the space of all homogeneous models approach isotropy at infinite times, is essentially based on the dominant energy condition and the positive pressure criterion for the matter content of the Universe. On the other hand, as was pointed out by Gibbons and Hawking [2 ] and Hawking and Moss [3 ], the situation may change if the universe can undergo an inflationary stage. Due to the cosmic "no-hair" theorem of Wald [4] all initially expanding homogeneous models with a positive (and sufficiently large, in the Bianchi type IX case) cosmological constant asymptotically approach the isotropic de Sitter solution [ 5 ]. The success o f the inflationary scenarios [ 6 - 1 0 ] is based on a mechanism creating an effective cosmological constant. In the chaotic inflationary model [ I 1 ] the cosmological term decays in a neutral way and the de Sitter epoch is smoothly followed by the post-inflationary era.

It has been pointed out by several authors [ 12-15 ] that the initial anisotropies are almost damped out during the inflationary phase whenever it lasts sufficiently long. This is due to the fact that the scalar field acts like a cosmological term as long as its energy is dominated by the potential part. However, in all models with a convex, positive potential having a local minimum, there exist time intervals where the scalar field behaves like "ordinary" matter, i.e. like a perfect fluid with non-negative pressure [ 16 ]. Thus, it is not clear whether the homogeneous models asymptotically approach isotropy in a technical sense [ 17 ]. In order to decide this question, it turns out to be convenient to consider the effective equation of state for the scalar field with respect to a suitable defined time average. It is the aim o f this letter to extend the theorem of Collins and Hawking [ 1 ] to scalar fields. We shall show that it remains true for all convex and positive potentials having a local m i n i m u m with V[¢o] = 0 . We shall conclude that the only homogeneous models which eventually might approach isotropy are those Bianchi types, which admit a FRW model. Let us consider the evolution o f homogeneous cosmological models coupled to a scalar field and to "ordinary" matter, i.e. matter with stress-energy tensor T,~ satisfying the dominant and strong energy conditions as well as the positive pressure criterion. The e n e r g y - m o m e n t u m tensor of the scalar field is

S,~ = 0,,~ O ~ O - g , ~ L ,

0370-2693/91/$ 03.50 © 1991 - Elsevier Science Publishers B.V. ( North-Holland )

( 1) 33

Volume 253, number 1,2

PHYSICSLETTERSB

L = ½0u0 0 u 0 - V[q~],

(2)

where the potential V[ ~ ] is supposed to be positive and convex. Let Pu be the Ricci tensor of the threedimensional riemannian space (G, h), where G denotes the Lie group acting on the space-like surfaces S and hu:= -go are the components of the induced metric h on S. The Gauss, Roo and R u equations and the evolution equation for ~ are ½[ K 2 - t r ( K 2) ] = Too + E - ½e,

(3)

K - t r ( K 2) = (Too - ½T) + ½( E + 3 p ) ,

(4)

(Vr~Kj)'/~'~=(~

(5)

- ~ j,T ) + ~, j (, p - E ) - P J , i

~ ' - K b + Vo = 0 ,

(6)

where K u = - ½(gu)' is the extrinsic curvature of S with trace K,= K~ and E and p denote the energy density and pressure of the scalar field, respectively:

E=½((~)z+v[o],

p=½(~)z-v[0 ].

(7)

Using the usual decomposition of the structure constants of G [18], Cjk--~jkln ~_ u +~'tkaji, the Ricci curvature P'-= -P~ is found to be non-positive for all homogeneous models, except for Bianchi type IX, as was pointed out by Wald [4]. Before we discuss the behavior of the anisotropies, let us briefly recall some useful general asymptotic properties of the solutions of the field equations ( 3 ) (6).

Proposition 1. Let P ( t ) ~<0Vt and let Tu~ satisfy the strong and the dominant energy condition. Let V[~] >t Vo := V[0] >/0 for all q~. If there exists a time to with K(to)~< 0, i.e. if the universe is expanding at to, then ( i ) / ( ( t ) > / 0 , K(t)<~0

forall t>>.to,

(ii) P(t), Too(t), T(t), ~(t)-~0

a n d K ( t ) - - . - x / 3 V o as t - ~ . Proof (i) Adding eqs. (3) and (4) gives the following expression:

K=½[3 tr(KE)-K2]-½P +Too+ (Too-½T)+3(E+P), 34

(8)

3 January 1991

which is non-negative due to the Schwarz inequality, the assumptions on the curvature and the matter terms, and the fact that the combination E+p= (q~)2 is positive (although the pressure of the scalar field is indefinite). Eliminating the trace tr(K 2) from eqs. (3) and (4) yields K 2 - / ~ = - P + ( Too + ½T) + ~ ( E - p ) ,

(9)

which is again non-negative, due to the dominant energy condition and since E-p=2V>_.O. Using K = - (x/g)'/x/~ we obtain (x/~) "'>/0, from the inequality K 2- ]£ >t 0 and thus [ v/g (t) ]'>/ [x/~(to) ]] i.e., K(t) <~0for all t>_-toifK(to) ~<0. (ii) Since K(t) is a monotonically increasing negative function, we have K(t)-*0 and K(t)~Koo<~O as t--,ov. Since none of the terms on the RHS of eq. (8) are negative, we obtain P(t), Too(t), T(t) and 0(t)-*0 as t--,oo. From eq. (9) we can then see that K(t)-~-(3Vo) 1/2, since Vo is the only local minimum of V[ ~ ]. This proves Proposition 1. Before we shall investigate the case Vo=0, let us briefly consider the simpler situation where Vo> 0. In this case, the expansion rate - K ( t ) approaches the constant value ( 3 Vo) ~/2. Using the shear tensor,

aij ".=K o - ~guK ,

(10)

we immediately obtain from Proposition 1 and from eq. (8) t r ( a 2 ) ~ 0 . Since K-~ - (3Vo)1/2#0 we have tr(tTE)/K2--*0 and also au/K-+O as t-~oo. Thus, the anisotropy, defined as the shear measured with respect to the Hubble rate, vanishes asymptotically. Independent of the initial conditions, any homogeneous model with a convex scalar field potential which is bounded by a positive value V0 approaches the de Sitter solution with H = - ~K= ( ~ Vo)1/2. The lower bound Vo has the same effect as a cosmological constant which causes inflation and isotropisation due to the cosmic "no hair" theorem of Wald [4] for homogeneous cosmological models with a cosmological constant. Let us now consider positive, convex potentials which vanish at the local minimum, Vo=0. Most candidates for chaotic inflation, such as all potentials of the form V[ ~ ] ocO2n, n >/1 belong to this class. The fact that during the period of inflation the anisotropies are almost damped out, suggests the supposition that the "no-hair" theorem might be extended to this

Volume 253, number 1,2

PHYSICS LETTERS B

class of potentials. As is easily seen from the preceding proposition, the shear tensor still tends to zero, but since now the expansion rate also vanishes, it is not obvious whether ao/K--.O still holds. As a matter of fact, we shall show that this is no longer the case. As is well known, almost all homogeneous models (without scalar fields) do not approach isotropy. To be precise we quote the following theorem of Collins and Hawking [ 1 ]: " I f the dominant energy condition and positive pressure criterion are satisfied, the universe can approach isotropy only if it is one of the types I, V, VIIo and VIIh." Subsequently, we shall give an extension of this theorem to the scalar field case. At first glance one might think that this is not possible, since scalar fields do no longer satisfy the positive pressure criterion. However, defining

y(t) : = 2 [ 1 - V ( t ) / E ( t ) ] ,

(11)

the scalar field satisfies the time-dependent effective equation of state

p(t)= [y(t)-l]E(t).

(12)

Since the ratio V / E is oscillating between 0 ( ~ = 0 ) and 1 (0 = 0 ), y (t) covers the entire interval between 2 (stiff matter, p = E) and 0 (cosmological constant, p = - E ) . In order to discuss the asymptotic behavior of the solutions, we have to investigate the effective equation of state with respect to a suitably defined time average. We shall first give a necessary condition for isotropisation from the field equations ( 3 ) (5). In a second step we shall use this condition to derive a virial theorem for the damped oscillator equation (6) and show that the latter cannot be satisfied by a convex potential. We shall then conclude that the quoted theorem also holds for scalar fields. Following Collins and Hawking [ 1 ] we say that an expanding homogeneous cosmological model approaches isotropy, if (i) a~j/K~O

as t--.oo,

(ii) a o :=gijg-t/3--.a~j

where a,7 are finite constants and det(a a) = 1. Let us also recall the scaling property of Po,

Pj(gm,) =P~(am~)g-I/3,

ous model does not contain a FRW model: (P~+~¢SjP)(amn)#O,

(13)

and a theorem of Ellis and MacCallum [ 18 ] which states that Pj(a,,,) is not isotropic if the homogene-

Via.,. I < o o .

(14)

Proposition 2. Any homogeneous model which is coupled to a scalar field and which is not among the Bianchi types admitting a FRW model can only approach isotropy as t--,~ if (i) E / K 2 o ~ , E g l / 3 - , ~ , (ii) ( V / E ) >1~ , where ( ) denotes the time average with respect to the time coordinate

s(t) :=ln[x/~(t) ] .

(15)

Proof. (i) Let us first show that the models under consideration can only approach isotropy if the function K2g ~/3__,oo as t ~ 00. We shall construct a contradiction from the assumption that K2g 1/3 is bounded. Using the time coordinate s, the traceless part of eq. (5) can be written as ( a~gl/6 ), s + 2 (o-~gl/6) = (Kg 1/6) - ' (P~+ ~j:P) (amn) •

(16)

Since (Kg 1/6) is bounded we have (O'j-gt/6)= (Kg '/6) (a~/K)--*0 if the model approaches isotropy as s~oo. On the other hand, the RHS of eq. (16) tends to a non-vanishing constant since (Kg t/6) - ~ is monotonic and bounded from below by a positive value and since (PJ + ]8jP) approaches a nonvanishing constant as a o ~ a a . Thus [~tri (7'l/6J.s ~ also tends to ~'j~, a non-vanishing constant, which contradicts (o'~gl/6) --~0. Using again the scaling property of the three-dimensional Ricci scalar, the Gauss equation (3) yields for a~/ K--,O: E 1 ' ~P(amn) K-5--* ~ + g2gt/3

as t - - , ~ ,

3 January 1991

ast~oo.

(17)

Since P(a°d,) is finite for lam, l < o o and since K2g ~/ 3__,oo as t --, oo, we immediately obtain P ( gin. ) / K2--.0, E/K2--.] and Eg'/3--.oo as t--.oo. (ii) Let us now consider the conservation law

F,=K(O)2=2K(E - V),

(18) 35

Volume 253, number 1,2

PHYSICS LETTERS B

which derives either from the Bianchi identity or from eq. (6) after a multiplication with ~. Together with the definition ( 11 ) for 7(t) we obtain t

E(t)=E(to) exp(f7(t')K(t')dt').

(19)

t0

(~,} -'= lim -1 f ;,(s') ds'~ z . S

0"+

0'+ ~

V~=0,

(23)

where a dash denotes differentiation with respect to s. Multiplying this equation with 0 yields ( 0 0 ' ) ' - ( 0 ' ) 2+ ~00'+K2-[£ ~-~1Vo0= 0 .

Substituting ( 15 ) and ( 19 ) into Eg ~/3__.09 yields the condition exp[]s-fT(s)ds]--.09 as s--.09, or equivalently

s ~

3 January 1991

(20)

,/

Using 7= 2 ( 1 - V/E) we finally obtain the condition (V/E) >t~, which must be satisfied whenever the cosmological model asymptotically approaches isotropy. This completes the proof of the Proposition. In the case of "ordinary" matter which satisfies the positive pressure criterion 7(s) >t 1Vs, the inequality ( 20 ) is violated. For scalar fields, however, ~,(s) covers the whole interval between zero and two and one has to compute its time average from the damped oscillator equation (6), where the main difficulty is due to the fact that the damping function -K(t) is not explicitly known. Using the necessary condition E~ K 2 ~ } of the preceding proposition and the general properties of K(t), we shall now derive a virial theorem for the 0-field which contradicts the condition (V/E) >t 2, if the potential is assumed to be convex.

(24)

Using E = ½K2(•')2-¥ V and E/K2-,c2¢{O, ~ } , we can see that I0'1 is bounded. Since 0 is also bounded [E(to) < 09], the first term in (24) does not contribute in the time average. Since k is smaller or equal than K z, we have 0 ~<(K 2- / £ ) / K 2 ~<1 and, since 0--' 0, the third term in (24) tends to zero as s--*~. Thus, the time average of this term also vanishes. Solving E = ½ K 2 ( 0 ' ) 2 + Vfor (0') 2 we now obtain from eq. (24)

The factor ElK 2 can be eliminated since S

lim 1

s~

s

ff(s')g(s') d s ' = c 2 lim -1 ~f(s') ds'

S ,/

s~

S

holds for any pair of bounded functionsf(s) and g(s) with limg(s) =c2¢ {0, 09}. Any convex function V[0] satisfies the inequality V[Oz]-V[01]>_. V'[0~] X ( 02 -- ~1 ), which reduces to 0 V' [ 0 ] >t V[ 0 ] if Vhas a local minimum at 02 with V[02] =0. Since this inequality holds for all 0, it is also satisfied for the time average, and we obtain [since E ( t ) >I 0]

Proposition 3. Let V[0] be a non-negative, convex function

with

local

minimum

V[0]=0.

Let

K(t) <~O<~if(t)<~K2(t) for all t and fK(t) d t = - ~ . Then every solution of

~'-K(t)~+ Vo=O with finite initial energy and isfies the inequality

( V / E ) <~~ ,

and thus ( V / E ) (21)

E/K2~c2(~{O,09} sat(22)

where ( ) denotes the average with respect to the time coordinate s, defined as ds := -K(t) dr.

Proof As is seen from the assumptions on K(t), s is a well-defined time coordinate with s--,09 as t--.09 for which the oscillator equation (2 1 ) reads 36

<~~, which proves Proposition 3.

If the potential is a homogeneous function of degree 2n (n>~l), then eq. (25) reads ( V / E } = ( 1 + n) - l ~<½< ] or equivalently 2n

(7)=~>11

ifn>_-l,

(26)

which may be considered as the effective time-averaged equation of state for the scalar field. It obviously contradicts the necessary condition (20) for isotropisation. For any strictly convex potential eq. (26) is valid for the first non-vanishing term in the

Volume 253, number 1,2

PHYSICS LETTERS B

Taylor expansion o f V[~] at ~ = 0 . This m e a n s that at late times the averaged equation o f state satisfies the positive pressure criterion and ( V / E ) is thus strictly smaller than -]. Let us finally establish the a s s u m p t i o n s on which the preceding virial argument is based: The function K ( t ) satisfies K ( t ) ~<0 a n d K ( t ) >/0 for all t by Proposition 1. The i m p o r t a n t relation K ( t ) ~t ~ o f Proposition 2 and the result o f the preceding virial argum e n t yields the following extension o f the t h e o r e m o f Collins and Hawking [ 1 ] to scalar fields:

Theorem. A n y homogeneous cosmological m o d e l coupled to a scalar field with a convex, positive potential having a local m i n i m u m at ~o with V[ ~o ] = 0 can only a p p r o a c h isotropy if it is a m o n g the Bianchi types I, V, VII or IX. This d e m o n s t r a t e s that cosmological " n o - h a i r " theorems cannot exist for scalar fields (at least not for a rather general class o f p o t e n t i a l s ) . We suggest that it might be possible to show that the generic solutions o f type VIIh a n d IX do also not a p p r o a c h isotropy at infinite times, as it is the case for " o r d i n a r y " m a t t e r [ l ]. If this turns out to be correct, then

3 January 1991

only a subclass o f vanishing measure in the space o f all homogeneous initial conditions can a p p r o a c h isotropy. This statement would then also generalise the second, stronger theorem o f Collins and Hawking [ 1 ] to scalar fields. The a u t h o r is very grateful to Professor N. Straum a n n and Dr. C. Kiefer for valuable discussions. This research was supported by the Swiss N a t i o n a l Science F o u n d a t i o n .

References [ 1] C.B. Collins and S.W. Hawking, Astrophys. J. 180 ( 1973 ) 317. [ 2 ] G.W. Gibbons and S.W. Hawking, Phys. Rev. D 15 ( 1977 ) 2738. [3] S.W. Hawking and I.G. Moss, Phys. Lett. B 110 (1982) 35. [ 4 ] R.M. Wald, Phys. Rev. D 28 ( 1983 ) 2118. [ 5 ] J.B. Barrow and G. Goetz, Phys. Len. B 231 ( 1989 ) 228. [6] A. Guth, Phys. Rev. D 23 ( 1981 ) 389. [7] A.D. Linde, Rep. Prog. Phys. 47 (1984) 925. [8] A.D. Linde, Phys. LeU. B 108 (1982) 389. [9] D. La and P.J. Steinhardt, Phys. Rev. Lett. 62 (1989) 376. [ 10] D. La and P.J. Steinhardt, Phys. Lett. B 220 (1989) 375. [ 11 ] A.D. Linde, Phys. Lett. B 129 (1983) 177. [ 12] T. Futamase, Phys. Rev. D 29 (1984) 2783. [13] O. Gron, Phys. Rev. D 32 (1985) 2522. [14] G. Steigman and M.S. Turner, Phys. Lett. B 128 (1983) 295. [ 15 ] I. Moss and V. Sahni, Phys. Lett. B 178 ( 1986 ) 159. [ 16] M.S. Turner and L.M. Widrow, Phys. Rev. Len. 57 (1986) 2237. [ 17] T. Piran and R. Williams, Phys. Lett. B 163 (1985) 331. [ 18 ] G.F.R. Ellis and M.A.H. MacCallum, Commun. Math. Phys. 12 (1969) 108.

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