Anthropic Bound on the Cosmological Constant
Steven Weinberg
Summary and citation · read the original at the source · none found
In one page
Steven Weinberg had spent years watching physicists fail to explain why the energy of empty space is so small. In this short 1987 letter he tries the other direction. Suppose the vacuum energy is not the same everywhere — suppose it takes different values in different regions or different big bangs. Then astronomers only ever exist where the value happened to permit them, and the question becomes: how large could it be and still allow galaxies to form? Weinberg answers it with ordinary gravitational physics. Vacuum energy pushes space apart; matter clumps by pulling together; once the push wins, clumping stops for good. So the vacuum energy must be no larger than the matter density at the moment galaxies first condensed — a few hundred times today’s cosmic mass density, rather than the hundred and twenty orders of magnitude larger that particle physics naively suggests. And the sharp part: if this selection is what holds the constant down, it should sit near the bound, not far below it, and therefore be measurable.
Why it matters hereChapter 2 keeps the vacuum-energy ledger, and this is the paper that put a real physical ceiling on one side of it — the first argument that told cosmologists roughly what number to look for; chapter 13 needs it because eleven years later the supernova teams found a vacuum energy of about twice the present mass density, inside the window Weinberg had drawn.
What it claims
01A large positive vacuum energy would have stopped galaxies from ever forming. Vacuum energy drives space apart at an accelerating rate, while matter clumps by gravitational condensation; once the vacuum term dominates, the growth of density perturbations freezes and no new structure forms. The requirement that gravitational condensation had already happened therefore places a real upper limit on the vacuum energy density in any region where astronomers exist.As restated by Weinberg in arXiv astro-ph/0005265, section 3, opening of the anthropic bound discussion, citing this paper
Published and peer-reviewed02The bound is quantitative, not a hand wave. Following the nonlinear growth of density perturbations with Peebles’s spherical infall model, Weinberg finds the vacuum energy density must be less than five hundred over seven hundred and twenty-nine times the mass density at recombination times the cube of the typical fractional density perturbation at that time. Everything in it is a measurable cosmological quantity.As restated in arXiv astro-ph/0005265, section 3, equation (7), attributed there to Phys. Rev. Lett. 59, 2607 (1987)
Published and peer-reviewed03In plain numbers the ceiling is a few hundred times the present cosmic mass density. Weinberg’s own paraphrase is that the vacuum energy should be no larger than the cosmic mass density at the earliest time of galaxy formation, which for a maximum galactic redshift of five is about two hundred times the present mass density; his fuller 1989 treatment gives four hundred and ten times for a flat universe. Set against a naive particle-physics estimate that misses by about a hundred and twenty orders of magnitude, that is an enormous improvement — and, as he says himself, still not good enough.arXiv astro-ph/0005265, section 3, paragraph following equation (7); Reviews of Modern Physics 61, 1 (1989), section V
Published and peer-reviewed04The argument only means anything if there are many big bangs. An anthropic explanation of the vacuum energy makes sense if and only if the observed big bang is one member of a large ensemble in which that energy takes different values — different expanding regions at different times and places in one spacetime, or different terms in the wave function of the universe. Weinberg is explicit that without such an ensemble the reasoning does not apply.arXiv astro-ph/0005265, section 3, opening paragraphs; Reviews of Modern Physics 61, 1 (1989), section V
What to watch05The prediction that made it testable: expect a value near the bound. If it is only this selection effect that keeps the vacuum energy within observational limits, then there is no reason for it to be very much smaller than the ceiling, so it should sit at roughly ten to a hundred times the present mass density — in Weinberg’s own closing phrase, rather large, and large enough to show up before long in astronomical observations. The supernova results of 1998 found a vacuum energy of about twice the present mass density.Reviews of Modern Physics 61, 1 (1989), section IX, Outlook, drawing on this paper
What to watch06What to watch: whether the observed value really sits where a selection argument would put it. Weinberg’s later refinement, with Martel and Shapiro, replaces the bare ceiling with a probability distribution — weight each value of the vacuum energy by the fraction of baryons that end up in galaxies there — so the prediction becomes a distribution to test rather than a single limit. The measurement that bears on it is the precision dark-energy programme now running: a vacuum energy that is constant and close to the anthropic peak supports the selection picture, while a value that drifts with time points somewhere else entirely.arXiv astro-ph/0005265, section 3, equations (8) and (9) and the Martel, Shapiro and Weinberg calculation described there
What to watch
The way in
https://doi.org/10.1103/PhysRevLett.59.2607LICENCE. Published as Physical Review Letters, volume 59, pages 2607 to 2610 (1987), under the APS default licence. Unpaywall, OpenAlex, Semantic Scholar and Crossref all record the article closed with no repository copy and no abstract released, so no text of it is reproduced here. SOURCE NOT REACHED DIRECTLY. The claims below were written on 2026-09-08 from Weinberg’s own later restatements of this result, read in full: his review The Cosmological Constant Problem, Reviews of Modern Physics 61, 1 to 23 (1989), section V on anthropic considerations, held on this site at /library/stm-5610822bc8; and his talk The Cosmological Constant Problems, given at Dark Matter 2000 and posted as arXiv astro-ph/0005265, whose section 3 states the bound as an inequality and cites Phys. Rev. Lett. 59, 2607 (1987) as its source. Each locator names which of those two restatements it comes from. REGISTRY NOTE: the record reached the library with chapters ch02, ch03 and ch13; chapter 3 is about inertia and gravity as zero-point-field effects, which this paper does not treat, so it is dropped here and the sheet carries chapters 2 and 13.
How to cite it
Steven Weinberg (1987) Anthropic Bound on the Cosmological Constant. doi:10.1103/PhysRevLett.59.2607
Where it sits in the curriculum