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Problems with the concept of the inflationary universe
Ad~. Space 7?eB. Vol.3, No.10-12, pp.441-442, 1984 Printed in Great Britain. All rights reserved.
0273-1177/84 $0.00 + .50 Copyright © COSPAR
PROBLEMS WITH THE CONCEPT OF THE I N F L A T I O N A R Y U N I V E R S E G. B6rner and E. Seiler Max-Planck-Institut f~ir Physik und Astrophysik, Miinchen, F.R.G.
The picture of an exponentially increasing, "inflationary" phase of the early universe (Guth 1981; Linde 1982; Albrecht and Steinhardt 1982) may point the way to an understanding of our present universe without reference to extremely specific initial conditions. The model rests, however, on several assumptions which deserve critical examination.
I. THE INITIAL CONDITIONS OF THE INFLATIONARY UNIVERSE The exponential expansion occurs in a homogeneous and isotropic Friedmann-Robertson-Walker in most approaches so far (one exception is Barrow Turner). In the spirit of showing how inflation irons out specific initial conditions, however, more general pre-inflation environments should be investigated such as inhomogeneous or anisotropic cosmological models.
2. THE ENERGY MOMENTUM TENSOR The energy difference between the syn~etric (high temperature) and unsymmetric state of the universe is inserted into the Friedmann eq. (Guth 1981) as a term %gD~ - in analogy to a cosmological constant. It is, however, by no means clear that the energy-density of a thermal state must appear in this form. %6~o6~o , or a perfect fluid type Tp~ with Too = % seem equally possible. The requirement of covarlance cannot be used here - as is mostly done - to deriveKT,~ ~ % g ~ , because a thermal state defines its own preferred rest system. Rather < T > should be calculated in a l-loop approximation to check whether really % is a constant.
3. THE PHASE TRANSITION The phase structure of the early universe is determined from an effective potential Vef f (Guth 1981; Linde 1982). The quantum corrections in the presence of symmetry breaking are determined from a one-loop approximation scheme. In this approximation Vef f seems to have a temperature dependence as in fig.l: At T > T C there is a stable minimum at = 0 (symmetric phase); at T < T C this state becomes metastable, while a state at @ # 0 is stable. At T = T C a firstorder phase transition seems to take place.
/=To Fig. I
441
442
G. Borner and E. Seiler
It has been realized (Symanzik 1970) that the effective potential has to be convex, a property violated by the naive one-loop expansion schemes. If the exact effective potential is strictly convex, it cannot have the suggested double-well structure used in the inflationary schemes. Recently Bender and Cooper (1983) have again pointed out this fact, and also discussed another defect of this approach, namely that it cannot work if a secondary minimum is present in the classical potential (cf. O'Raifeartaigh and Parravicini 1976). The definition of Vef f is -Veff(~ c) = inf (W(J) - ~c J) J where W(J) is the finite temperature W(J) is convex (Symanzik 1970)
action functional
Z(J) = expW(J).
~2W = <~2> _ <~>2 < O. ~j2 Hence Vef f is also convex. A convex Vef f must have a shape as in fig.2,
Fig. 2
where the dotted line corresponds to a mixture of different phases (Gibb's construction). But it has been shown (Elitzur 1975; De Angelis et al 1978) that in Higgs-models W does not have a cusp at J = O (as e.g. for spin systmes, ferromagnets W has a cusp). Therefore V~ff > 0 and the dotted line disappears. The effective potential can therefore not be used to describe the phase structure of the thermal states in the early universe.
4. CONCLUSION The presently used scheme to obtain an inflationary model of the universe does not work.
REFERENCES Albrecht A., Steinhardt P., 1982, Phys. Rev. Lett. 48, 1220 Barrow J.D., Turner M.S., 1981, Nature 292, 35 Bender C.M., Cooper F., 1983, Los Alamos preprint De Angelis G.F., de Falco D., Guerra F., 1978, Phys. Rev. D|7, Elitzur S., 1975, Phys. Rev. DI2, 3978 Guth A.H., 1981, Phys. Rev. D.23, 347 Linde A.D., 1982, Phys. Lett. I08B, 389 O'Raifeartaigh L., Parravicini G., 1976, Nucl. Phys. BIll, 501 Symanzik K., 1970, Commun. Math. Phys. 16, 48
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0273-1177/84 $0.00 + .50 Copyright © COSPAR
PROBLEMS WITH THE CONCEPT OF THE I N F L A T I O N A R Y U N I V E R S E G. B6rner and E. Seiler Max-Planck-Institut f~ir Physik und Astrophysik, Miinchen, F.R.G.
The picture of an exponentially increasing, "inflationary" phase of the early universe (Guth 1981; Linde 1982; Albrecht and Steinhardt 1982) may point the way to an understanding of our present universe without reference to extremely specific initial conditions. The model rests, however, on several assumptions which deserve critical examination.
I. THE INITIAL CONDITIONS OF THE INFLATIONARY UNIVERSE The exponential expansion occurs in a homogeneous and isotropic Friedmann-Robertson-Walker in most approaches so far (one exception is Barrow Turner). In the spirit of showing how inflation irons out specific initial conditions, however, more general pre-inflation environments should be investigated such as inhomogeneous or anisotropic cosmological models.
2. THE ENERGY MOMENTUM TENSOR The energy difference between the syn~etric (high temperature) and unsymmetric state of the universe is inserted into the Friedmann eq. (Guth 1981) as a term %gD~ - in analogy to a cosmological constant. It is, however, by no means clear that the energy-density of a thermal state must appear in this form. %6~o6~o , or a perfect fluid type Tp~ with Too = % seem equally possible. The requirement of covarlance cannot be used here - as is mostly done - to derive
3. THE PHASE TRANSITION The phase structure of the early universe is determined from an effective potential Vef f (Guth 1981; Linde 1982). The quantum corrections in the presence of symmetry breaking are determined from a one-loop approximation scheme. In this approximation Vef f seems to have a temperature dependence as in fig.l: At T > T C there is a stable minimum at = 0 (symmetric phase); at T < T C this state becomes metastable, while a state at @ # 0 is stable. At T = T C a firstorder phase transition seems to take place.
/=To Fig. I
441
442
G. Borner and E. Seiler
It has been realized (Symanzik 1970) that the effective potential has to be convex, a property violated by the naive one-loop expansion schemes. If the exact effective potential is strictly convex, it cannot have the suggested double-well structure used in the inflationary schemes. Recently Bender and Cooper (1983) have again pointed out this fact, and also discussed another defect of this approach, namely that it cannot work if a secondary minimum is present in the classical potential (cf. O'Raifeartaigh and Parravicini 1976). The definition of Vef f is -Veff(~ c) = inf (W(J) - ~c J) J where W(J) is the finite temperature W(J) is convex (Symanzik 1970)
action functional
Z(J) = expW(J).
~2W = <~2> _ <~>2 < O. ~j2 Hence Vef f is also convex. A convex Vef f must have a shape as in fig.2,
Fig. 2
where the dotted line corresponds to a mixture of different phases (Gibb's construction). But it has been shown (Elitzur 1975; De Angelis et al 1978) that in Higgs-models W does not have a cusp at J = O (as e.g. for spin systmes, ferromagnets W has a cusp). Therefore V~ff > 0 and the dotted line disappears. The effective potential can therefore not be used to describe the phase structure of the thermal states in the early universe.
4. CONCLUSION The presently used scheme to obtain an inflationary model of the universe does not work.
REFERENCES Albrecht A., Steinhardt P., 1982, Phys. Rev. Lett. 48, 1220 Barrow J.D., Turner M.S., 1981, Nature 292, 35 Bender C.M., Cooper F., 1983, Los Alamos preprint De Angelis G.F., de Falco D., Guerra F., 1978, Phys. Rev. D|7, Elitzur S., 1975, Phys. Rev. DI2, 3978 Guth A.H., 1981, Phys. Rev. D.23, 347 Linde A.D., 1982, Phys. Lett. I08B, 389 O'Raifeartaigh L., Parravicini G., 1976, Nucl. Phys. BIll, 501 Symanzik K., 1970, Commun. Math. Phys. 16, 48
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