Historical research guide · 0658 · 1989
The cosmological constant problem
Steven Weinberg
In one minute
Steven Weinberg wrote this review in 1989, and it is the paper that turned the cosmological constant into a problem every physicist knows by name. His starting point is an identification: the constant Einstein put into his field equations and the zero-point energy quantum field theory puts into empty space are the same quantity, so only their sum is observable. Then he does the sum. Add up the zero-point energies of the known fields to the Planck scale and you get about two times ten to the seventy-first in particle-physics units; the sky says the total is smaller than about ten to the minus forty-seventh. The two contributions must therefore cancel to better than a hundred and eighteen decimal places. Weinberg is careful that this does not depend on trusting physics all the way up: the strong interaction alone still demands forty-one places. He then reviews five ways out — supersymmetry and strings, anthropic selection, adjustment mechanisms, changing gravity, quantum cosmology — and finds none of them finished.
Connection
This is the canonical statement of the number chapter 2 leans on — the roughly hundred-and-twenty-order gap between the vacuum energy the laboratory implies and the vacuum energy the sky measures. Chapter 13 needs that ledger kept honestly, and Weinberg keeps it, right down to his closing prediction that if anthropic selection is what holds the constant down then it should be large enough to show up in astronomical observations before long — which is what the supernova teams found nine years later.
From the source
The Einstein cosmological constant and the quantum vacuum energy are one quantity, not two. Because the vacuum expectation value of the energy-momentum tensor must be proportional to the metric, adding it to the field equations has exactly the same effect as shifting the cosmological constant, so what any experiment can bound is the effective total: the constant divided by eight pi times Newton’s constant, plus the vacuum energy density. Weinberg works with that sum from this point on.Section III, The Problem; Equations (3.1) to (3.3)
Summing the zero-point energies of the normal modes of a field up to a wave-number cutoff at the Planck scale gives a vacuum energy density of about two times ten to the seventy-first in units of giga-electronvolts to the fourth power. Cosmological observation bounds the effective total below about ten to the minus forty-seventh in the same units, equivalent to about ten to the minus twenty-ninth grams per cubic centimetre. The two terms must therefore cancel to better than 118 decimal places.Section III; Equations (3.4), (3.5) and (3.6)
The problem does not depend on trusting quantum field theory up to the Planck scale. Counting only the zero-point energies of quantum chromodynamics gives a vacuum energy of order the QCD scale to the fourth over sixteen pi squared, about ten to the minus sixth in giga-electronvolts to the fourth power, which still requires the cosmological term to cancel it to about 41 decimal places. Zeldovich’s 1967 attempt, keeping only the gravitational self-energy of the vacuum fluctuations with a one giga-electronvolt cutoff, still overshoots the observational bound by some nine orders of magnitude.Section III, paragraphs following Equation (3.6); Equation (3.7)
The zero-point energies are real and measured. Weinberg notes that particle theorists were slow to worry about this despite the demonstration in the Casimir effect of the reality of zero-point energies, and gives the number in a footnote: Casimir showed in 1948 that quantum fluctuations between two flat conducting plates a distance d apart produce a force per unit area of h-bar c pi squared over 240 d to the fourth, which is 1.30 times ten to the minus eighteen dyne centimetres squared divided by d to the fourth, and Sparnaay measured a force per area of one to four times ten to the minus eighteen in those units for separations between two and ten micrometres.Section III, footnote on the Casimir effect
Five approaches to a solution are surveyed and none is finished: supersymmetry, supergravity and superstrings; anthropic considerations; adjustment mechanisms driven by a scalar field; changing gravity so that the determinant of the metric is not dynamical; and quantum cosmology with wormholes. In the Outlook, Weinberg rates quantum cosmology the most promising of the five at the time of writing and supersymmetry and adjustment mechanisms the least, and observes that any solution is likely to have a much wider impact on other areas of physics or astronomy — a light scalar showing up as a fifth force, or wormholes forcing all the constants of nature to their outer bounds.Sections IV to VIII; Section IX, Outlook
Weinberg’s anthropic bound, and the prediction he draws from it. Gravitational condensation had already begun by a redshift of about four, when the energy density exceeded the present mass density by a factor of roughly 125, so a vacuum energy density no larger than about a hundred times the present mass density would not have prevented galaxies from forming; his quantitative 1987 analysis gives 410 times for a flat universe. If it is only the anthropic constraint that keeps the effective constant within empirical limits, then the vacuum energy should sit at ten to a hundred times the present mass density, because there is no anthropic reason for it to be any smaller — in his own closing words, rather large, large enough to show up before long in astronomical observations.Section V, Anthropic Considerations; Section IX, Outlook
The way in
https://doi.org/10.1103/RevModPhys.61.1How to cite it
Steven Weinberg (1989) The cosmological constant problem doi:10.1103/RevModPhys.61.1