Coleman-Weinberg theory and the new inflationary universe scenario

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12 August 1982

COLEMAN-WEINBERG THEORY AND THE NEW INFLATIONARY UNIVERSE SCENARIO

A.D. LINDE Lebedev Physical Institute, Moscow 117924, USSR Received 5 May 1982

It is argued that a new inflationary universe scenario, which provides a possible solution of the horizon, flatness, homogeneity, isotropy and primordial monopole problems, can be naturally implemented in the context of grand unified theories of the type of the Coleman-Weinberg theory.

Several years ago the questions why there is no antimatter in the universe, why the universe is almost flat, homogeneous and isotropic etc. seemed almost metaphysical. Typical answers were that in the matter-antimatter symmetric, anisotropic and inhomogeneous universe there would be no observers who could ask such questions, see e.g. ref. [ 1 ]. However, after the discovery of a mechanism of quasi-isotropisation (i.e. local isotropisation) of the universe due to particle creation [2], and especially after the discovery of a possible mechanism of cosmological baryon production [3], it became clear that recent developments of elementary particle theory may provide us with physical answers to all these almost metaphysical questions. The situation became even more interesting after the important paper of Guth [4], who suggested a possibility to solve the horizon, flatness and primordial monopole problem [5] due to exponential expansion (inflation) of the universe is a supercooled symmetric state during symmetry breaking phase transitions [6, 7] in grand unified theories (GUTs). As it was shown in ref. [7], the stress tensor Tuv in a supercooled symmetric state ¢ = 0(¢ is the Higgs scalar field which breaks the symmetry) becomes equal to the vacuum stress tensor, Tuv = guvV(O), where V(~0) is the effective potential of the field ~ * a. .1 To be more precise, g/~uV(0) is equal to the canonical stress tensor at ~o= 0, and not to the improved stress tensor, which enters the Einstein equations. However, in our case the difference between these two tensors at ~0= 0 is negligibly small. 0 031-9163/82/0000-0000/$02.75

© 1982 North-Holland

This leads to the exponential expansion of the universe, a(t) ~ eHr. Here a is the scale factor, H is the Hubble constant at that time, H = [-~TrGV(0)] 1/2, where G is the gravitational constant. Guth has pointed out that the horizon, flatness and primordial monopole problems can be solved simultaneously if the exponential expansion proceeds during some period of time r >~ 65 H - 1, so that the universe grows at least 1028 times during this period [4]. An important feature of the inflationary universe scenario [4] was an implicit assumption, that the field ~0inside the bubbles of the new phase, formed during the phase transition, immediately grows up to its equilibrium value ~00, which corresponds to the minimum of V(~0), and thermalization occur~ only after the bubble wall collisions. Unfortunately , this assumption, which is actually true for many types of GUTs, makes the inflationary universe scenario in its form suggested in ref. [4] unacceptable. As was pointed out by Guth and other authors who studied this problem later, this scenario leads to an extremely large inhomogeneity and anisotropy of the universe after the phase transition [8]. In our paper [9] a new version of the inflationary universe scenario was suggested, which is free of the above-mentioned difficulty. This scenario provides a possible solution of the horizon flatness, homogeneity, isotropy and primordial monopole problems, and may help also to solve the baryon asymmetry problem [10, 11 ] and the galaxy formation problem [ 12,13 ]. The main idea of this scenario, emphasized recently in the 431

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papers of Hawking and Moss [14] and of Albrecht and Steinhardt [15], is that the field ~ v~ 0, which appears at the beginning of the phase transition in the theories of the type of the Coleman-Weinberg theory [ 16], grows up to ~o~ ~00 during some time r, which may be much greater than H - 1. In this case the universe remains exponentially expanding during some time ~ r after the beginning of the phase transition, which provides a solution of the above-mentioned problems if ~->~ 65 H

1

(1)

(this inequality may be even too strong, since the universe expands exponentially during some period before the phase transition as well). During the greatest part of this period the field ¢ remains much less than tp0 , so that V(¢) is almost equal to V(0), and energy density fluctuations, connected e.g. with the bubble walls, are negligible small as compared with V(tp). Only after the end of this period the inhomogeneities connected with the bubble walls may become large, but the typical distance between them becomes much greater than the size of the observable part of the universe, and the total energy of the walls remains much less than the total energy of the almost homogeneous field ¢, convergently oscillating near its equilibrium value tp ~ tp0. This is just the reason why the universe after the phase transition does not become greatly inhomogeneous and anisotropic, though small inhomogeneities, which are necessary for the galaxy formation, may be produced due to the growth of quantum fluctuations of metric [12] and of the scalar field ¢ [13] in the course of the phase transition. The main idea of the new inflationary universe scenario is rather simple. However, as we have mentioned in ref. [9], the concrete version of this scenario presented in ref. [9] (see also ref. [I 5] ) is oversimplified since it does not take into account the effects, connected with the rapid expansion and nonvanishing curvature of the universe, which become important at T <~H. Some of these effects have been investigated in the very interesting paper [14], where a possibility that the tunneling with symmetry breaking occurs simultaneously in the whole universe was suggested. Indeed, such a process may occur in a closed universe with the spatial volume V ~
12 August 1982

that its spatial volume at the beginning of the phase transition in the theories of the type of the C o l e m a n Weinberg theory is many orders greater than H 3, which makes the simultaneous tunneling in the whole universe practically impossible. Therefore it was not quite clear so far whether the new inflationary universe scenario can be actually realized, and whether it can be implemented in a context of more or less natural grand unified theories. In order to answer these questions let us recall that according to ref. [17] symmetry breaking in the theories of the type of the SU(5) Coleman-Weinberg theory, in which [m2(tp = 0)1 = [ V"(0) I ~ H 2, may occur only at T ~ H (this is not a general rule [13], but we shall consider here only those theories for which it is true). The time necessary for the phase transition to start due to high-temperature effects exceeds At T - 1 >~ H - 1 [9,15]. However during this time all temperature corrections to V(~0) and to the transition probability disappear completely. Thus for the investigation of symmetry breaking in such theories it is sufficient to consider V(~o) at vanishing temperature T = 0 *2 Since the main feature of our scenario is a slow growth of the field 9, let us consider first the growth of this field in the theory with m2(~) : V"(~0) < 0. Here 9 is the initial point from which the classical field begins to grow. The K l e i n - G o r d o n equation for the homogeneous field in the exponentially expanding universe is + 3H~ + m2~o = 0.

(2)

At Im21 >>H 2 this field grows with time as exp(mt). At Im 2 [ "~H 2 (this case will be most important for us) the field ¢ grows as exp(Im21t/3I-1), whereas the inhomogeneous fluctuations of this field grow even more slowly. Therefore symmetry breaking in the theory with Irn 2 1 4 H 2 proceeds during some time interval r >~ 3 H / l m 2 I [12,14]. According to (1), this is sufficient for the realization of the new inflationary uni-

,2 Note that from the point of view of a eomoving observer temperature falls down not to T = 0, but to the Hawking temperature in de Sitter space TH = H/2~r [ 18]. This does not alter our conclusions, since the contribution of TH to V(9) is automatically taken into account when computing V(~o)in the exponentially expanding universe [ 19,20]. For a more detailed discussion of this question see refs. [21,13],

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The simplest possibility is that the initial point from which the field ¢ begins to grow is ¢ = 0. In this case inequality (3) implies that

malization conditions (n.c.) on m 2, ~ and X. Here we shall discuss the n.c. for m 2 only, the other n.c. being irrelevant to the problem under consideration (for a discussion of n.c. for ~ and X see [13] ). Let us first recall that in Minkowski space (R = 0) the n.c. for m 2 in the Coleman-Weinberg theory is [16]

Im(0) [ <~ 0 . 2 H .

m2(0) = d2 g/d~ 2 I~,=o,n=o = 0 .

verse scenario if [tn(~0) I <~ 0 . 2 H ,

(3)

(3')

It is clear that the theories, in which inequality (3') holds, do actually exist. However this inequality is not quite innocent. For example, in the SU(5) theory w i t h M x ~ 5 × 1014 GeV, H ~ 1010 GeV [9], so that Ira(0) [ ~ 0.2 H ~ 2 × 109 GeV ~ M x. Therefore one may wonder whether it is possible to satisfy the hierarchy condition Im(0) l ~ M x in a natural way. The simplest strategy in considering this question would be to postpone answering it until the hierarchy problem in GUTs is solved. In any case, the possibility of an interplay between the hierarchy in GUTs and the horizon, flatness, homogeneity and isotropy problems in cosmology is very intriguing. There exists also another possibility to implement the new inflationary universe scenario, which seems to us more attractive. N a m e l y , let us consider the Coleman-Weinberg theory in the exponentially expanding Friedmann universe. The effective potential VCo) in this universe is equal to V(~0) in the Sitter space with the same curvature R. Therefore we will use the results of calculations of VC0) in de Sitter space which have been obtained by Shore for the U(1) C o l e m a n Weinberg model [20] (extension of these results for the SU (5) theory is trivial [ 13 ] ). Our interpretation of these results, however, will be slightly different from that of Shore [20]. After removing the divergences, an expression for V(~o) in de Sitter space with the curvature scalar R in the limit e2~o2 ~ R looks as follows [20] : v(~, R) = 111202 + (e2R~o2/647r2) ln(R//a2)

+ (3e4¢4/647r2) ln(R//l 2) + V(0, R ) ,

(4)

where/a/are some normalization constants, which are necessary for renormalization of mass m 2 = d 2 V/d~p2, of the parameter ~ = d 3 V/d~2dR and of the effective coupling constant X = 6 d4 V]d,p4 . At R ~ e2~02, V(~o) coincides with the usual Coleman-Weinberg effective potential in Minkowski space. To specify the values of ~. one should impose nor-

(5)

Calculation of the oneqoop effective potential V(¢) in the Coleman-Weinberg theory with the n.c. (5) shows that this theory is equivalent to the ordinary Higgs model with spontaneous symmetry breaking with some special relations between parameters of this model [22,7]. In particular, m2(0) (5) in the ColemanWeinberg theory is equal t o / l 2 - (3e4/167r2)~02, where/~ is the bare Higgs mass, 2//2 = d 2 V/d~p2 I~=~o [22,7]. Therefore at least at the one-loop level there exists a theory with m2(0) = V"(0) = 0 (the C o l e m a n Weinberg theory) and with symmetry breaking ¢ = ¢0 (since this theory is equivalent to the Higgs model). Unfortunately higher orders of perturbation theory for V(~) contain terms ~e4+n~p 4 lnntp, and the results of perturbative calculations of V@) become unreliable at small ~0. Therefore, strictly speaking, it is unknown whether there actually exists a theory, in which V"(0) = 0 and the absolute minimum of V(¢) occurs at ¢ = ¢0 4: 0. However such a possibility seems plausible, since according to the one-loop results, the value of m2(0) = V"(0) can be arbitrary varied if one varies the value of/l 2 in the corresponding Higgs model, and it is very likely that there exists such a value of/32, for which m2(0) = 0. Therefore it is commonly believed that at R = 0 the Coleman-Weinberg theory with m2(0) = 0 and with the dynamical symmetry breaking ~0= ¢0 does actually exist. Now let us note, that with an account taken of the gravitational effects the normalization condition (5) appears to be non-selfconsistent. Indeed, at ~ = 0 the curvature scalar R = 327rGV(0, R) :~ 0. The self-consistent generalization of the n.c. (5) in the C o l e m a n Weinberg theory is m2(0) = d2 V/dqo2 I~o=O,R=32,~GV(O,R) = 0 .

(5')

The one-loop effective potential V(¢) in such a theory has a minimum at some ¢ = ~o0 [20], and at the first glance no problems with higher orders of perturbation theory appear at small ¢, since the curvature scalar R serves here as an infrared cut off [there are 433

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no terms bin ~0in (4)]. However the situation is not so simple, since eq. (4) does not take into account the one-loop scalar field contribution to V(~0) [20]. Usually this is quite legitimate since in the ColemanWeinberg theory the scalar coupling constant is small, X ~ e 2 [16], but in de Sitter space the perturbation theory in X diverges at kR2/m 4 ~ M-I4/m 4 >~ 1 for the same reasons for which perturbation theory at finite temperature diverges at XT/m ~ 1 [7,13]. Therefore eq. (4) for V(~0) becomes unreliable at ~p4 ,~ R 2 ' and just as in the case R = 0 we cannot prove rigorously that the Coleman-Weinberg theory with m 2 (0) = 0 in de Sitter space does exist. However such a possibility seems plausible, since the same arguments, which we have used above in favour of existence of the Coleman-Weinberg theory in Minkowski space, can be applied in the case R v~ 0 as well. Therefore we shall suppose that a theory with dynamical symmetry breaking and with the normalization condition (5') does actually exist, and we shall call this theory hereafter the improved Coleman-Weinberg theory. In our opinion this theory is a most natural generalization of the usual Coleman-Weinberg theory [16] with an account taken of gravitational effects. Now let us study kinetics of the symmetry breaking phase transition in this theory. As we have argued above, it is sufficient to study symmetry breaking at T = 0 (TH = H/27r), when V"(0) = 0 according to (5'). There are several possible ways for the phase transition to occur in such a theory. (1) Bubbles of the field ~0are formed, which grow and gradually fill all the universe. Creation of such bubbles in the case R = 0 is described by the Fubini instantons ~0= (8/IXI) 1/2 p/(x 2 + t 2 + 0 2 ) ,

(6)

with the action S = 8n2[31XI for arbitrary p, so that the bubble formation probability p ~ exp (-87r2/ 3 IX I). In de Sitter space these instantons remain unaltered for O ~ T~ 1 = 2n/H. In the SU(5) ColemanWeinberg theory of the effective coupling constant X at ~0~ H i s of the order of 0.1 [13]. This means that p ~ e - 200, and bubbles with p ~ 27r/H, ~o~ HX/~/~x/~ practically cannot be produced. Thus, the only bubbles which may appear during the phase transition are bubbles with ~0~ HV~/zrVt-~, m (~o) ~ V~X~o "~HV~[ rr. As it will be shown in ref. [23], the probability of formation of such bubbles in the exponentially expand434

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ing universe in the context of the improved ColemanWeinberg theory is not exponentially suppressed, and therefore just the bubbles with Im(~0)I ~ H are produced during the phase transition. According to (3), this is sufficient for the realization of the new inflationary universe scenario. (2) The long-wave vacuum fluctuations of the field ~0, which are responsible for the above-mentioned infrared problem at m4(¢) ~< M/4, can initiate the phase transition without the usual bubble formation [13]. This may lead to the desirable inflation of the universe, at least for sufficiently small X. A subsequent growth of these fluctuations may give rise to galaxy formation [ 12,13 ]. However a thorough investigation is needed in order to understand whether or not these fluctuations may destroy the large-scale homogeneity of the universe after the phase transition. (3) Due to nonperturbative effects [e.g. due to the SU(5) or gravitational instantons] there may appear a a condensates of the type of (~- ~) ~ A 3 , (GuvGuv) A 4 and (~02 ) ~ A 2. These condensates may lead to the phase transition without tunneling. For example, the term ~(~$)~0, which may appear in V(~0) if superheaw fermions exist, leads to the growth of the field ~0 [24]. Since the expected value of A is very small (A "
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which of these possibilities (or some other ones [13] ) could be actually realized in realistic GUTs. A discussion of this question and of some other problems connected with the new inflationary universe scenario will be contained in a series of subsequent publications [11,13,231. In conclusion I would like to express my deep gratitude to G.V. Chibisov, P .C .W. Davies, A.D. Dolgov, V 2 . Frolov, A.S. Goncharov, S.W. Hawking, R.E. Kallosh, D.A. Kirzhnits, V.F. Mukhanov, V.A. Rubakov, A.A. Starobinsky, A.V. Veryashkin and Ya.B. Zeldovich for many enlightening discussions.

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[7] D.A. Kirzhnits and A.D. Linde, Ann. Phys. (NY) 101 (1976) 195; A.D. Linde, Rep. Prog. Phys. 42 (1979) 389. [8] S.W. Hawking, l.G. Moss and J.M. Steward, DAMPT preprint (1981) ; A.H. Guth and E. Weinberg, MIT preprint (1982). [9] A.D. Linde, Phys. Lett. 108B (1982) 389; A.D. Linde, in: Quantum gravity (Plenum Press, New York, 1982). [10] P. Steinhardt, M. Turner and F. Wilczek, in preparation (1982). [ 11 ] A.D. Dolgov and A.D. Linde, Baryon asymmetry in the inflationary universe, ITEP preprint (1982). [12] G.V. Chibisov and V.F. Mukhanov, Pisma Zh. Eksp. Teor. Fiz. 33 (1981) 549. G.V. Chibisov and V.F. Mukhanov, Lebedev, Phys. Inst. preprint No 198 (1981), to be published in Zh. Eksp. Teor. Fiz. [ 13] A.D. Linde, Coleman-Weinberg theory and the new inflationary universe scenario, Lebedev, Phys. Inst. preprint (1982) (extended version). [14] S.W. Hawking and I.G. Moss, Phys. Lett. ll0B (1982) 35. [ 15 ] A. Albrecht and P.J. Steinhardt, Pennsylvania University preprint UPR-0185 T (1982). [16] S. Coleman and E. Weinberg, Phys. Rev. D7 (1973) 1888. [17] G.P. Cook and K.T. Mahanthappa, Phys. Rev. D23 (1981) 1321; M. Sher, Phys. Rev. D24 (1981) 1847; A. Billoire and K. Tamvakis, Nucl. Phys. B200 [FS4] (1982) 329; K. Tamvakis and C.E. Vayonakis, Phys. Lett. 109B (1982) 283; [18] G.W. Gibbons and S.W. Hawking, Phys. Rev. D15 (1977) 2738. [19] G.W. Gibbons, J. Phys. A l l (1978) 1341. [20] G.M. Shore, Ann. Phys. (NY) 128 (1980) 376. [21] N.D. Birrell and P.C.W. Davies, Quantum fields in curved space (Cambridge U.P. 1982). [22] A.D. Linde, JETP Lett. 23 (1976) 73. [23] A.S. Goncharov and A.D. Linde, Bubble formation in the inflationary universe, Lebedev, Phys. Inst. preprint (1982). [24] E. Witten, Nucl. Phys. B177 (1981) 477.

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