Contents Previous


In several cosmological theories the observed big bang is just one member of an ensemble. The ensemble may consist of different expanding regions at different times and locations in the same spacetime, (7) or of different terms in the wave function of the universe. (8) If the vacuum energy density rhoV varies among the different members of this ensemble, then the value observed by any species of astronomers will be conditioned by the necessity that this value of rhoV should be suitable for the evolution of intelligent life.

It would be a disappointment if this were the solution of the cosmological constant problems, because we would like to be able to calculate all the constants of nature from first principles, but it may be a disappointment that we will have to live with. We have learned to live with similar disappointments in the past. For instance, Kepler tried to derive the relative distances of the planets from the sun by a geometrical construction involving Platonic solids nested within each other, and it was somewhat disappointing when Newton's theory of the solar system failed to constrain the radii of planetary orbits, but by now we have gotten used to the fact that these radii are what they are because of historical accidents. This is a pretty good analogy, because we do have an anthropic explanation why the planet on which we live is in the narrow range of distances from the sun at which the surface temperature allows the existence of liquid water: if the radius of our planet's orbit was not in this range, then we would not be here. This would not be a satisfying explanation if the earth were the only planet in the universe, for then the fact that it is just the right distance from the sun to allow water to be liquid on its surface would be quite amazing. But with nine planets in our solar system and vast numbers of planets in the rest of the universe, at different distances from their respective stars, this sort of anthropic explanation is just common sense. In the same way, an anthropic explanation of the value of rhoV makes sense if and only if there is a very large number of big bangs, with different values for rhoV.

The anthropic bound on a positive vacuum energy density is set by the requirement that rhoV should not be so large as to prevent the formation of galaxies. (9) Using the simple spherical infall model of Peebles (10) to follow the nonlinear growth of inhomogeneities in the matter density, one finds an upper bound

Equation 7 (7)

where rhoR is the mass density and deltaR is a typical fractional density perturbation, both taken at the time of recombination. This is roughly the same as requiring that rhoV should be no larger than the cosmic mass density at the earliest time of galaxy formation, which for a maximum galactic redshift of 5 would be about 200 times the present mass density. This is a big improvement over missing by 120 orders of magnitude, but not good enough.

However, we would not expect to live in a big bang in which galaxy formation is just barely possible. Much more reasonable is what Vilenkin calls a principle of mediocrity, (11) which suggests that we should expect to find ourselves in a big bang that is typical of those in which intelligent life is possible. To be specific, if Pa priori (rhoV) drhoV is the a priori probability of a particular big bang having vacuum energy density between rhoV and rhoV + drhoV, and N(rhoV) is the average number of scientific civilizations in big bangs with energy density rhoV, then the actual (unnormalized) probability of a scientific civilization observing an energy density between rhoV and rhoV + drhoV is

Equation 8 (8)

We don't know how to calculate N(rhoV), but it seems reasonable to take it as proportional to the number of baryons that wind up in galaxies, with an unknown proportionality factor that is independent of rhoV. There is a complication, that the total number of baryons in a big bang may be infinite, and may also depend on rhoV. In practice, we take N(rhoV) as the fraction of baryons that wind up in galaxies, which we can hope to calculate, and include the total baryon number as a factor in Pa priori(rhoV).

The one thing that offers some hope of actually calculating dP (rhoV) is that N(rhoV) is non-zero in only a narrow range of values of rhoV, values that are much smaller than the energy densities typical of elementary particle physics, so that Pa priori (rhoV) is likely to be constant within this range. (12) The value of this constant is fixed by the requirement that the total probability should be one, so

Equation 9 (9)

The fraction N(rhoV) of baryons in galaxies has been calculated by Martel, Shapiro and myself, (13) using the well-known spherical infall model of Gunn and Gott, (14) in which one starts with a fractional density perturbation that is positive within a sphere, and compensated by a negative fractional density perturbation in a surrounding spherical shell. The results are quite insensitive to the relative radii of the sphere and shell. Taking the shell thickness to equal the sphere's radius, the integrated probability distribution function for finding a vacuum energy less than or equal to rhoV is

Equation 10 (10)


Equation 11 (11)

with sigma the rms fractional density perturbation at recombination, and rhoR the average mass density at recombination. The probability of finding ourselves in a big bang with a vacuum energy density large enough to give a present value of OmegaV of 0.7 or less turns out to be 5% to 12%, depending on the assumptions used to estimate sigma. In other words, the vacuum energy in our big bang still seems a little low, but not implausibly so. These anthropic considerations can therefore provide a solution to both the old and the new cosmological constant problems, provided of course that the underlying assumptions are valid. Related anthropic calculations have been carried out by several other authors. (15)

I should add that when anthropic considerations were first applied to the cosmological constant, counts of galaxies as a function of redshift (16) indicated that OmegaLambda is 0.1+0.2-0.4, and this was recognized to be too small to be explained anthropically. The subsequent discovery in studies of type Ia supernova distances and redshifts that OmegaLambda is quite large does not of course prove that anthropic considerations are relevant, but it is encouraging.

Recently the assumptions underlying these calculations have been challenged by Garriga and Vilenkin. (17) They adopt a plausible model for generating an ensemble of big bangs with different values of rhoV, by supposing that there is a scalar field phi that initially can take values anywhere in a broad range in which the potential V(phi) is very flat. Specifically, in this range

Equation 12 (12)

It is also assumed that in this range V(phi) is much less than the initial value of the energy density rhoM of matter and radiation. For initial values of phi in this range, the vacuum energy density rhophi stays roughly constant while rhoM drops to a value of order rhophi. To see this, note that during this period the expansion rate behaved as H = eta / t, with eta = 2/3 or eta = 1/2 during times of matter or radiation dominance, respectively. If we tentatively assume that phi is roughly constant, then the field equation (1) gives

Equation 13 (13)

During the time that rhoM >> rhophi, the ratio of the kinetic to the potential terms in Eq. (3) for rhophi is

Equation 14 (14)

so rhophi is dominated by the potential term. The fractional change in rhophi until the time tc when rhoM becomes equal to rhophi is then

Equation 15 (15)

Following this period, rhophi becomes dominant, and the inequalities (12) ensure that the expansion becomes essentially exponential, just as in theories with the `tracker' solutions discussed in the previous section. Hence in this class of models, V(phi) plays the role of a constant vacuum energy, whose values are governed by the a priori probability distribution for the initial values of phi. In particular, if one assumes that all initial values of phi are equally probable, then the a priori distribution of the vacuum energy is

Equation 16 (16)

The point made by Garriga and Vilenkin was that, because V(phi) is so flat, the field phi can vary appreciably even when rhoV appeq V(phi) is restricted to the very narrow anthropically allowed range of values in which galaxy formation is possible. They concluded that it would also be possible for the a priori probability (16) to vary appreciably in this range, which if true would require modifications in the calculation of P (leq rhoV) described above. The potential they used as an example was


with V1 large, of order M4, A and B much smaller, and M a large mass, but not larger than the Planck mass. This yields an a priori probability distribution (16) that varies appreciably in the anthropically allowed range of phi.

It turns out (18) that the issue of whether the a priori probability (16) is flat in the anthropically allowed range of phi depends on the way we impose the slow roll conditions (12). There is a large class of potentials for which the probability is flat in this range. Suppose for instance that, unlike the example chosen by Garriga and Vilenkin, the potential is of the general form

Equation 17 (17)

where V1 is a large energy density, in the range mW4 to mPlanck4, lambda > 0 is a very small constant, and f(x) is a function involving no very small or very large parameters. Anthropically allowed values of phi / lambda must be near a zero of f(x), say a simple zero at x = a. Then V'(phi) appeq lambda V1 f'(a) approx lambda V1 and V''(phi) appeq lambda^2 V1 f''(a) approx lambda2 V1, so both inequalities (12) are satisfied if

Equation 18 (18)

Galaxy formation is only possible for |V(phi)| less than an upper bound Vmax, of the order of the mass density of the universe at the earliest time of galaxy formation, which is very much less than V1, so the anthropically allowed range of values of phi is

Equation 19 (19)

The fractional variation in the a priori probability density (16) as phi varies in the range (19) is then

Equation 20 (20)

justifying the assumptions made in the calculation of Eq. (10).

I should emphasize that no fine-tuning is needed in potentials of type (16). It is only necessary that V1 be sufficiently large, lambda be sufficiently small, and f(x) have a simple zero somewhere, with derivatives of order unity at this zero. These properties are not upset if for instance we add a large constant to the potential. But why should each appearance of the field phi be accompanied with a tiny factor lambda? As we have been using it, derivatives of the field phi appear in the Lagrangian density in the form - 1/2 ðµphi ðµphi, as shown by the coefficient unity of the second derivative in the field equation (1). In general, we might expect the Lagrangian density for phi to take the form

Equation 21 (21)

where f(x) is a function of the sort we have been considering, involving no large or small parameters, M is a mass perhaps of order (8 pi G)-1/2, and V1 is a large constant, of order M4. With an arbitrary field-renormalization constant Z in the Lagrangian, the field phi is not canonically normalized, and does not obey Eq. (1). We may define a canonically normalized field as phi' ident sqrtZ phi; writing the Lagrangian in terms of phi', and dropping the prime, we get a potential of the form (16), with lambda = 1/M sqrtZ. Thus we can understand a very small lambda if we can explain why the field renormalization constant Z is very large. Perhaps this has something to do with the running of Z as the length scale at which it is measured grows to astronomical dimensions.

There is a problem with this sort of implementation of the anthropic principle, that may prevent its application to anything other than the cosmological constant. When quantized, a scalar field with a very flat potential leads to very light bosons, that might be expected to have been already observed. If we want to explain the masses and charges of elementary particles anthropically, by supposing that these masses and charges arise from expectation values of a scalar field in a flat potential with random initial values, then the scalar field would have to couple to these elementary particles, and would therefore be created in their collisions and decays. This problem does not arise for a scalar field that couples only to itself and gravitation (and perhaps also to a hidden sector of other fields that couple only to other fields in the hidden sector and to gravitation). It is true that such a scalar would couple to observed particles through multi-graviton exchange, and with a cutoff at the Planck mass the Yukawa couplings of dimensionality four that are generated in this way would in general not be suppressed by factors of G. But in our case the non-derivative interactions of the scalars with gravitation are suppressed by a factor V'(phi) propto lambda, which according to Eq. (18) is much less than sqrt[8 pi G], yielding Yukawa couplings that are very much less than unity. Thus it may be that anthropic considerations are relevant for the cosmological constant, but for nothing else.

7 A. Vilenkin: Phys. Rev. D 27, 2848 (1983);
A. D. Linde: Phys. Lett. B175, 395 (1986). Back.

8 E. Baum: Phys. Lett. B133, 185 (1984);
S. W. Hawking: in Shelter Island II - Proceedings of the 1983 Shelter Island Conference on Quantum Field Theory and the Fundamental Problems of Physics, ed. by R. Jackiw et al. (MIT Press, Cambridge, 1985);
Phys. Lett. B134, 403 (1984);
S. Coleman: Nucl. Phys. B 307, 867 (1988). Back.

9 S. Weinberg: Phys. Rev. Lett. 59, 2607 (1987). Back.

10 P. J. E. Peebles: Astrophys. J. 147, 859 (1967). Back.

11 A. Vilenkin: Phys. Rev. Lett. 74, 846 (1995);
in Cosmological Constant and the Evolution of the Universe, ed. by K. Sato et al. (Universal Academy Press, Tokyo, 1996). Back.

12 S. Weinberg: in Critical Dialogs in Cosmology, ed. by N. Turok (World Scientific, Singapore, 1997). Back.

13 H. Martel, P. Shapiro, and S. Weinberg: Astrophys. J. 492, 29 (1998). Back.

14 J. Gunn and J. Gott: Astrophys. J. 176, 1 (1972). Back.

15 G. Efstathiou: Mon. Not. Roy. Astron. Soc. 274, L73 (1995);
M. Tegmark and M. J. Rees: Astrophys. J. 499, 526 (1998);
J. Garriga, M. Livio, and A. Vilenkin: astro-ph/9906210;
S. Bludman: astro-ph/0002204. Back.

16 E. D. Loh: Phys. Rev. Lett. 57, 2865 (1986). Back.

17 J. Garriga and A. Vilenkin: astro-ph/9908115. Back.

18 S. Weinberg: astro-ph/0002387. Back.

Contents Previous