Λ (cosmological constant)
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| Λ (cosmological constant) | |
|---|---|
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| Name | Λ (cosmological constant) |
| Introduced | 1917 |
| Introduced by | Albert Einstein |
| Field | General relativity; Cosmology |
| Units | s^−2 (or m^−2 in geometrized units) |
Λ (cosmological constant) The cosmological constant Λ is a term in Albert Einstein's field equations of General relativity introduced to permit static Universe models; it is now interpreted as a form of vacuum energy driving accelerated expansion observed in the Hubble Ultra-Deep Field, Supernova Cosmology Project, and Sloan Digital Sky Survey. Modern cosmology treats Λ alongside Cold Dark Matter in the Lambda-CDM model, and it connects to quantum expectations from Quantum field theory, the Casimir effect, and issues raised by Paul Dirac and Wolfgang Pauli.
Definition and notation
Λ denotes a constant scalar added to the Einstein field equations R_{μν} − (1/2)g_{μν}R + Λ g_{μν} = (8πG/c^4) T_{μν}; this form was proposed by Albert Einstein and is conventionally written with the Greek capital lambda. In cosmological models such as the Friedmann–Lemaître–Robertson–Walker (FLRW) metric used by Georges Lemaître and Alexander Friedmann, Λ contributes an energy density ρ_Λ = Λ c^2/(8πG) and pressure p_Λ = −ρ_Λ c^2, entering the Friedmann equations employed by teams like the Planck Collaboration and WMAP. Units and sign conventions vary between texts by Stephen Hawking, Roger Penrose, and Misner, Thorne, and Wheeler.
Historical development
Einstein introduced Λ in 1917 while corresponding with Willem de Sitter to obtain a static Universe solution; subsequent debate involved Arthur Eddington, Georges Lemaître, and Alexander Friedmann who developed expanding models. After Edwin Hubble's redshift-distance relation and the observational work of Vesto Slipher and Milton Humason, Einstein reportedly called Λ his "biggest blunder", a phrase discussed in biographies by John Stachel and A. Douglas Stone. Interest revived with theoretical contributions by Yakov Zel'dovich linking Λ to vacuum fluctuations, and later by observational discoveries of late-time acceleration from teams led by Adam Riess, Saul Perlmutter, and Brian Schmidt recognized by the Nobel Prize in Physics.
Role in general relativity and cosmology
In General relativity Λ modifies global geometry and the dynamics of FLRW cosmologies studied by Max Born and Hermann Weyl; positive Λ yields de Sitter spacetime discovered by Willem de Sitter, while negative Λ yields anti-de Sitter spacetime central to the AdS/CFT correspondence developed by Juan Maldacena. Λ affects cosmic fate scenarios explored by Martin Rees, Stephen Hawking, and Andrei Linde and is a core parameter in the Lambda-CDM model constrained by datasets from Planck Collaboration, Baryon Oscillation Spectroscopic Survey, and Type Ia supernova surveys. In solutions such as Schwarzschild–de Sitter and Kerr–de Sitter, Λ influences black hole horizons analyzed by Roy Kerr and Subrahmanyan Chandrasekhar.
Observational evidence and constraints
Empirical support for a nonzero Λ arises from distance-redshift measurements by the Supernova Cosmology Project and the High-Z Supernova Search Team (work by Brian Schmidt, Saul Perlmutter, Adam Riess), anisotropies in the Cosmic microwave background measured by WMAP and the Planck Collaboration, and large-scale structure surveys like Sloan Digital Sky Survey and 2dF Galaxy Redshift Survey. Constraints on Λ and related parameters such as Ω_Λ, H_0, and w come from joint analyses by collaborations including Dark Energy Survey, eBOSS, and the Euclid mission, with tensions noted between measurements by Riess et al. and the Planck Collaboration prompting study by Vera Rubin Observatory teams and panels convened by NASA and European Space Agency.
Theoretical interpretations and implications
Λ is interpreted as vacuum energy density in quantum field contexts by Richard Feynman and Steven Weinberg, as a constant of nature akin to the Cosmological constant problem discussed by Jerome Friedman and Zel'dovich, and as a parameter in landscape scenarios of String theory by Leonard Susskind and Joseph Polchinski. Λ has implications for inflationary scenarios by Alan Guth and Andrei Linde, for anthropic reasoning invoked by Brandon Carter and John Barrow, and for semiclassical gravity and backreaction studies by Niels Bohr's intellectual descendants and researchers at Perimeter Institute and Institute for Advanced Study.
Cosmological constant problem and fine-tuning
The cosmological constant problem highlights a vast discrepancy between vacuum energy predictions from Quantum field theory (estimates by Steven Weinberg and calculations influenced by Julian Schwinger) and observed Λ inferred by Planck Collaboration and Type Ia supernova teams; this fine-tuning issue motivates proposals such as supersymmetry by Howard Georgi and Savas Dimopoulos, dynamical adjustment mechanisms by P. J. E. Peebles and Ratra, and anthropic explanations in the string landscape proposed by Leonard Susskind. The problem intersects with work on naturalness debated at conferences in CERN and Perimeter Institute and drives alternative model building by groups at Princeton University, Caltech, and Stanford University.
Alternatives and extensions
Alternatives to a pure Λ include dynamical dark energy models like quintessence developed by R. R. Caldwell and Paul Steinhardt, modified gravity theories such as f(R) gravity studied by T. P. Sotiriou and Valerio Faraoni, massive gravity proposals by Claudia de Rham, and emergent gravity concepts advanced by Erik Verlinde. Extensions incorporate coupled dark sector models explored by researchers at University of Cambridge, Harvard University, and Kavli Institute collaborations, while observational programs by Euclid mission, Nancy Grace Roman Space Telescope, and Vera C. Rubin Observatory aim to discriminate between Λ and these alternatives.