Cosmological constant and dark energy

cosmological constant dark energy and cosmological constant order of magnitude
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Published Date:23-07-2017
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One Hundred Years of the Cosmological Constant The Big Bang: Fact or Fiction? Cormac O’Raifeartaigh FRAS WIT Maths/Physics Seminar Series June 2017 Overview I Introducing the cosmological constant Einstein’s 1917 static model of the universe Problems of interpretation II The fallow years Einstein in Berlin The expanding universe (1929) Einstein abandons the cosmological constant Many others abandon the term (age paradox resolved) III Resurrection A new age problem (1990s) The accelerating universe (1998) Dark energy and the cosmological constant The quantum energy of the vacuum The general theory of relativity The general principle of relativity (1907-) Relativity and accelerated motion? The principle of equivalence Equivalence of gravity and acceleration The principle of Mach Relativity of inertia Structure of space determined by matter A long road (1907-1915) Gravity = curvature of space-time Covariant field equations? The field equations of GR (1915) 10 non-linear differential equations that relate the geometry of space-time to the density and flow of mass-energy SR GR 4 4 2 𝜇 𝜈 2 𝜇 𝜈 = 𝑔 𝑑𝑠 = 𝑛 𝑑𝑥 𝑑𝑥 𝜇 ,𝜈=1 𝜇 ,𝜈=1 𝑛 : constants 𝑔 : variables 𝜇𝜈 𝜇𝜈 𝜇𝜈 𝜇𝜈 𝑑𝑥 𝑑𝑥 𝑑𝑠 I Einstein’s 1917 model of the cosmos A natural progression Ultimate test for new theory of gravitation Principle 1: stasis Assume static distribution of matter Principle 2: uniformity Assume no-zero, uniform distribution of matter Principle 3: Mach’s principle No such thing as empty space Boundary conditions at infinity? A spatially closed universe The Einstein World & the cosmological constant Assume stasis (no evidence to the contrary) Non-zero density of matter Introduce closed spatial curvature To conform with Mach’s principle Solves problem of 𝑔 Introduce new term in GFE Additional term needed in field equations Allowed by relativity Quantitative model of the universe Cosmic radius related to matter density Cosmic radius related to cosmological constant 𝜇𝜈Introduction of the cosmological constant 𝟏 From 3(a), in accordance with (1a) one calculates for the 𝑅 𝑹 − 𝒈 𝑹 = −𝜿 𝑻 𝟐 𝑥 =𝑥 =𝑥 =0 the values 1 2 3 2 − 0 0 0 2 𝑃 2 2 2 + + 2 1 2 3 2 2 2 = − 𝑐 0 − 0 0 2 2 2 𝑃 𝑟 1+ 2 2 2𝑃 0 0 − 0 2 𝑃 0 0 0 0 , 1 for 𝑅 − 𝑔 𝑅 , the values 2 1 0 0 0 2 𝑃 1 0 0 0 2 𝑃 1 0 0 0 2 𝑃 2 3𝑐 0 0 0 − , 2 𝑃 while for –𝜅𝑻 one obtains the values 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 − 𝑐 Thus from (1) the two contradictory equations are obtained Einstein 1933 1 =0 2 𝑃 (4) 2 3𝑐 2 λ term needed for (static) solution = 𝑐 2 𝑃 Interpretation? 𝜅𝜌 𝜅𝜌 𝜇𝜈 𝜇𝜈 𝑑𝑡 𝑑𝑠 𝑑𝑥 𝑑𝑥 𝑑𝑥 𝝁𝝂 𝝁𝝂 𝝁𝝂 𝜇𝜈 Einstein’s view of the cosmological constant Not an energy of space Incompatible with Mach’s Principle A necessary evil “An ugly modification of the GFE” (Einstein 1918) Einstein’s postcard to Hermann Weyl Dispensible? Purpose = non-trivial static solution “if the universe is not static, then away with the cosmic constant” A changing view New form of GFE (Einstein 1918) λ = constant of integration? Einstein’s view of the cosmological constant Introduced in analogy with Newtonian cosmology 2 𝛻 𝜙 =4𝜋G𝜌 (P1) Full section on Newtonian gravity (Einstein 1917) Indefinite potential at infinity? 2 𝛻 𝜙 −λ𝜙 = 4𝜋G𝜌 (P2) Modifying Newtonian gravity Extra term in Poisson’s equation A “foil” for relativistic models Introduce cosmic constant in similar manner Inexact analogy Modified GFE corresponds to P3, not P2 A significant error? 2 2 𝛻 𝜙 + 𝑐 λ=4𝜋G𝜌 (P3) Implications for interpretation Schrödinger and the cosmological constant Schrödinger, 1918 Cosmic constant term not necessary for cosmic model Negative pressure term in energy-momentum tensor Einstein’s reaction New formulation equivalent to original (Questionable: physics not the same) Erwin Schrödinger 1887-1961 Schrödinger, 1918 Could pressure term be time-dependent ? Einstein’s reaction −𝑝 0 0 0 0 −𝑝 0 0 If not constant, time dependence unknown 𝑇 = 𝜇𝜈 0 0 −𝑝 0 “I have no wish to enter this thicket of hypotheses” 0 0 0 𝜌 −𝑝 Measuring the cosmic constant Calculate orbits of astronomical bodies Newtonian calculation Compare with astronomical observation Difference = measure of cosmological constant Possible in principle (Einstein 1921a) Globular clusters Specific example (Einstein 1921b) Null result Future observations? More accurate data needed The stability of the Einstein World How does cosmic constant term work? Assume uniform distribution of matter Perturbation What happens if the density of matter varies slightly? Failed to investigate No mention of issue in 1917 paper No mention of issue until 1927, 1930 Lemaître (1927) Cosmos expanding from Einstein World Eddington (1930) Einstein World unstable II Non-static cosmologies Alexander Friedman (1922) Consider time-varying solutions for the cosmos Alexander Friedman Expanding or contracting universe (1888 -1925) Retain cosmic constant for generality Evolving universe Time-varying radius and density of matter Positive or negative spatial curvature Depends on matter Ω =d/d c Reception (1927) Rejected by Einstein Considered hypothetical (unrealistic) Ignored by community All possible universes Lemaître’s universe (1927) Georges Lemaître (1927) Allow time-varying solutions (expansion) Retain cosmic constant Georges Lemaître Compare astronomical observation (1894-1966) Redshifts of the nebulae (Slipher) Extra-galactic nature of the nebulae (Hubble) Expansion from static Einstein World Instability (implicit) Reception Ignored by community Rejected by Einstein “Vôtre physique est abominable” The watershed: Hubble’s law Hubble’s law (1929) A linear redshift/distance relation for the spiral nebulae Edwin Hubble (1889-1953) -1 -1 Linear relation: h = 500 kms Mpc (Cepheid I stars) Evidence of cosmic expansion? RAS meeting (1930): Eddington, de Sitter Friedman-Lemaître models circulated Time-varying radius and density of matter Einstein apprised Cambridge visit (June 1930) Sojourn at Caltech (Spring 1931) Expanding models of the cosmos (1930 -)  Eddington (1930, 31) On the instability of the Einstein universe Expansion caused by condensation?  Tolman (1930, 31) On the behaviour of non-static models Expansion caused by annihilation of matter ?  de Sitter (1930, 31) Further remarks on the expanding universe Expanding universes of every flavour  Einstein abandons λ (1931, 32) Friedman-Einstein model k =1, λ = 0 Einstein-de Sitter model k = 0, λ = 0 Problem: Age paradox Einstein’s steady–state model (1931): λ = energy of the vacuum? The Friedman-Einstein model (1931) Cosmic constant abandoned Unsatisfactory (unstable solution) Unnecessary (non-static universe) Calculations of cosmic radius and density 8 -26 3 10 Einstein: P 10 lyr, ρ 10 g/cm , t 10 yr 9 -28 3 9 We get: P 10 lyr, ρ 10 g/cm , t 10 yr Explanation for age paradox? Assumption of homogeneity at early epochs Not a cyclic model “Model fails at P = 0 ” Contrary to what is usually stated The age paradox (1930-1950) Rewind Hubble graph (Lemaître 1931) U smaller in the past Extremely dense, extremely hot (‘big bang’) Fr Georges Lemaître Expanding and cooling ever since 9 But time of expansion = 1/H = 10 years Universe younger than the stars? Lemaître’s universe (1931-33) Expansion from radioactive decay Retain cosmic constant Stagnant epoch Circumvents age problem Accelerated epoch λ = Energy of vacuum Lemaître’s universe 2 2 p = - ρ c , ρ = λc /8πG 0 0 Abandoning the cosmic constant λ used to address age problem Eddington, Lemaitre, Tolman Resolution of the age problem Recalibration of distance Walther Baade Allan Sandage Cepheid II stars; stellar intensities 10 New age 10 years Cosmic constant abandoned Unnecessary term Neglected for many years Redundant 1950s-1990s)

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