REVIEWS OF MODERN PHYSICS, VOLUME 75, APRIL 2003 The cosmological constant and dark energy P. J. E. Peebles Joseph Henry Laboratories, Princeton University, Princeton, New Jersey 08544 Bharat Ratra Department of Physics, Kansas State University, Manhattan, Kansas 66506 (Published 22 April 2003) Physics welcomes the idea that space contains energy whose gravitational effect approximates that of Einstein’s cosmological constant, ⌳; today the concept is termed dark energy or quintessence. Physics also suggests that dark energy could be dynamical, allowing for the arguably appealing picture of an evolving dark-energy density approaching its natural value, zero, and small now because the expanding universe is old. This would alleviate the classical problem of the curious energy scale of a millielectron volt associated with a constant ⌳. Dark energy may have been detected by recent cosmological tests. These tests make a good scientific case for the context, in the relativistic Friedmann-Lemaıˆtre model, in which the gravitational inverse-square law is applied to the scales of cosmology. We have well-checked evidence that the mean mass density is not much more than one-quarter of the critical Einstein–de Sitter value. The case for detection of dark energy is not yet as convincing but still serious; we await more data, which may be derived from work in progress. Planned observations may detect the evolution of the dark-energy density; a positive result would be a considerable stimulus for attempts at understanding the microphysics of dark energy. This review presents the basic physics and astronomy of the subject, reviews the history of ideas, assesses the state of the observational evidence, and comments on recent developments in the search for a fundamental theory. CONTENTS 4. The redshift-angular-size and redshift- magnitude relations 587 5. Galaxy counts 588 I. Introduction 559 6. The gravitational lensing rate 588 A. The issues for observational cosmology 560 7. Dynamics and the mean mass density 589 B. The opportunity for physics 561 8. The baryon mass fraction in clusters of C. Some explanations 562 galaxies 590 II. Basic Concepts 563 9. The cluster mass function 590 A. The Friedmann-Lemaıˆtre model 563 10. Biasing and the development of nonlinear B. The cosmological constant 565 mass density fluctuations 590 C. Inflation and dark energy 566 11. The anisotropy of the cosmic microwave III. Historical Remarks 567 background radiation 591 A. Einstein’s thoughts 567 12. The mass autocorrelation function and B. The development of ideas 569 nonbaryonic matter 593 1. Early indications of ⌳ 569 13. The gravitational inverse-square law 593 2. The coincidences argument against ⌳ 570 C. The state of the cosmological tests 594 3. Vacuum energy and ⌳ 570 V. Concluding Remarks 595 Note added in proof 596 C. Inflation 572 Acknowledgments 597 1. The scenario 572 Appendix: Recent Dark-Energy Scalar Field Research 597 2. Inflation in a low-density universe 573 References 599 D. The cold-dark-matter model 574 E. Dark energy 576 1. The XCDM parametrization 576 2. Decay by emission of matter or radiation 577 I. INTRODUCTION 3. Cosmic field defects 578 4. Dark-energy scalar field 578 There is significant observational evidence for the de- IV. The Cosmological Tests 580 tection of Einstein’s cosmological constant, ⌳, or a com- A. The theories 580 ponent of the material content of the universe that var- 1. General relativity 580 ies only slowly with time and space and so acts like ⌳. 2. The cold-dark-matter model for structure ⌳ formation 582 We shall use the term dark energy for or a component B. The tests 584 that acts like it. Detection of dark energy would be a 1. Thermal cosmic microwave background new clue to an old puzzle: the gravitational effect of the radiation 585 zero-point energies of particles and fields. The total with 2. Light-element abundances 585 other energies, that are close to homogeneous and 3. Expansion times 586 nearly independent of time, acts as dark energy. What is 0034-6861/2003/75(2)/559(48)/$35.00 559 ©2003 The American Physical Society 560 P. J. E. Peebles and Bharat Ratra: The cosmological constant and dark energy puzzling is that the value of the dark-energy density has The reader is referred to Leibundgut’s (2001, Sec. 4) dis- to be tiny compared to what is suggested by dimensional cussion of astrophysical hazards. Astronomers have analysis; the startling new evidence is that it may be dif- checks for this and other issues of interpretation when ferent from the only other natural value, zero. considering the observations used in cosmological tests. The main question to consider now is whether to ac- But it takes nothing away from this careful and elegant cept the evidence for detection of dark energy. We out- work to note that the checks are seldom convincing, be- line the nature of the case in this section. After review- cause the astronomy is complicated and what can be ing the basic concepts of the relativistic world model in observed is sparse. What is more, we do not know ahead Sec. II, and in Sec. III reviewing the history of ideas, we of time that the physics well tested on scales ranging present in Sec. IV a more detailed assessment of the from the laboratory to the Solar System survives the enormous extrapolation to cosmology. cosmological tests and the evidence for detection of ⌳ or The situation is by no means hopeless. We now have its analog in dark energy. significant cross-checks from the consistency of results There is little new to report on the big issue for based on independent applications of the astronomy and physics—why the dark-energy density is so small—since 1 of the physics of the cosmological model. If the physics Weinberg’s (1989) review in this journal. But there have or astronomy was faulty we would not expect consis- been analyses of a simpler idea: can we imagine that the tency from independent lines of evidence—apart from present dark-energy density is evolving, perhaps ap- the occasional accident and the occasional tendency to proaching zero? Models are introduced in Secs. II.C and stop the analysis when it approaches the ‘‘right answer.’’ III.E, and recent work is summarized in more detail in We have to demand abundant evidence of consistency, the Appendix. Feasible advances in cosmological tests and that is starting to appear. could detect evolution of the dark-energy density, and The case for detection of ⌳ or dark energy com- perhaps its gravitational response to large-scale fluctua- mences with the Friedmann-Lemaıˆtre cosmological tions in the mass distribution. This would substantially model. In this model the expansion history of the uni- motivate the search for a more fundamental physics verse is determined by a set of dimensionless parameters model for dark energy. whose sum is normalized to unity, A. The issues for observational cosmology ⍀ ϩ⍀ ϩ⍀ ϩ⍀ ϭ M0 R0 ⌳0 K0 1. (1) We will make two points. First, cosmology has a sub- stantial observational and experimental basis, which supports many aspects of the standard model as almost ⍀ The first, M0 , is a measure of the present mean mass certainly being good approximations to reality. Second, density in nonrelativistic matter, mainly baryons and ⍀ ϳ ϫ Ϫ4 the empirical basis is not nearly as strong for cosmology nonbaryonic dark matter. The second, R0 1 10 ,is as it is for the standard model of particle physics: in a measure of the present mass in the relativistic 3-K cosmology it is not yet a matter of measuring the param- thermal cosmic microwave background radiation, which eters in a well-established theory. almost homogeneously fills space, and the accompanying To explain the second point we direct our attention to low-mass neutrinos. The third is a measure of ⌳ or the those more accustomed to experiments in the laboratory present value of the dark-energy equivalent. The fourth, ⍀ than to astronomy-related observations of astronomers’ K0 , is an effect of the curvature of space. We review Tantalus principle: one can look at distant objects but some details of these parameters in the next section, and never touch them. For example, the observations of su- of their measurements in Sec. IV. pernovae in distant galaxies offer evidence of dark en- The most direct evidence for detection of dark energy ergy, under the assumption that distant and nearby su- comes from observations of supernovae of a type whose pernovae are drawn from the same statistical sample intrinsic luminosities are close to uniform (after subtle (that is, that they are statistically similar enough for the astronomical corrections, a few details of which are dis- cussed in Sec. IV.B.4). The observed brightness as a purpose of this test). There is no direct way to check function of the wavelength shift of the radiation probes this, and it is easy to imagine differences between distant the geometry of spacetime, in what has come to be and nearby supernovae of the same nominal type. More called the redshift-magnitude relation.2 The measure- distant supernovae are seen in younger galaxies, because ments agree with the relativistic cosmological model of the travel time of light, and these younger galaxies ⍀ ϭ ⍀ with K0 0, meaning no space curvature, and ⌳0 tend to have more massive rapidly evolving stars with ϳ ⌳ ⍀ ϭ 0.7, meaning nonzero . A model with ⌳0 0 is two lower heavy-element abundances. How do we know that the properties of the supernovae are not also different? 2 ϭϪ The apparent magnitude is m 2.5 log10 f plus a constant, where f is the detected energy flux density in a chosen wave- 1Sahni and Starobinsky (2000); Carroll (2001); Weinberg length band.