Historical research guide · 0690 · 2019
Lifshitz theory of the cosmological constant
Ulf Leonhardt
In one minute
Two things are measured and neither is explained by the other. Astronomers measure a cosmological constant that is accelerating the expansion of the universe. Optical physicists measure the Casimir and van der Waals forces, which are the quantum vacuum pushing on ordinary matter, and predict them to the percent level. Zel'dovich conjectured in 1968 that these are the same thing; nobody had made the connection quantitative. Ulf Leonhardt takes the tool that works in the laboratory — Lifshitz theory — and applies it to the expanding universe, using the fact that an expanding space looks to light exactly like a medium whose refractive index changes with time. The horizon around every observer radiates at a tiny temperature, and because gravity acts on every frequency up to the Planck scale, that tiny temperature turns out to be enough. The result is a cosmological constant that is not constant but responds to the expansion that creates it.
Connection
This is the strongest published bridge between the laboratory vacuum of chapter 2 and the cosmology of chapter 13: the same zero-point field that pushes two Casimir plates together is here doing the work usually attributed to dark energy, calculated with the theory that already matches the bench measurements. It also hands the site a live test — fit the measured expansion and you read off the one free parameter, which encodes the scale where the equivalence principle stops holding.
From the source
The two empirical facts can be treated with one theory: astrophysics gives evidence for the cosmological constant, atomic, molecular and optical physics has proven experimentally that the quantum vacuum exerts the van der Waals and Casimir forces on neutral matter, and this paper develops a version of Lifshitz theory that also accounts for the electromagnetic contribution to the cosmological constant — giving, if the other Standard Model fields behave similarly, a possible quantum-optical explanation for what has been called dark energy.Abstract; Section 1.1, Introduction
An expanding space is an optical medium: if three-dimensional space is flat and expands so that distances grow by a factor in time, then to electromagnetic waves that space and a uniform dielectric whose refractive index varies in time appear exactly the same — so calculations of the zero-point energy and pressure in such media give the electromagnetic contribution to the cosmological term.Section 1.1, Introduction, Eq. (1) and figure 1
Every observer sits inside a horizon that radiates. In coordinates affixed to a co-moving observer the expanding universe appears as an outward-moving fluid whose radial flow speed follows Hubble's law and reaches the speed of light at a cosmological horizon; because the quantum vacuum is universal it cannot adjust itself to all the conflicting horizons, so it bridges them by entangling the fields inside and outside in Einstein-Podolsky-Rosen states, and to the inside observer that pure state appears as a thermal state at the Gibbons-Hawking temperature, the reduced Planck constant times the Hubble constant divided by two pi — about 2 × 10⁻²⁹ K for the observed expansion rate.Section 1.4, Cosmological horizons, Eqs. (6)-(11) and figure 2
The equivalence principle is why so tiny a temperature can decide the fate of the universe: ordinary media respond only up to atomic frequencies, whereas space-time acts equally on the entire spectrum up to the Planck scale, and if the renormalized vacuum energy density diverges with the inverse square of a length set to the order of the Planck length, the first Friedmann equation is satisfied and the theory's one dimensionless constant lies in the order of unity.Section 1.5, Equivalence principle, Eqs. (12) and (13)
The result is a dynamical cosmological constant: the vacuum energy density depends on time derivatives of the refractive index up to fourth order, through the quantity given as the third time derivative of the reciprocal Hubble constant plus the Hubble constant times its second time derivative, and the trace anomaly — the missing recoil energy left by a renormalization that is not reciprocal — supplies an energy density accompanied by a pressure of equal magnitude and opposite sign, exactly the form of the cosmological term. Small perturbations of exponential de Sitter expansion then obey the equation of a damped harmonic oscillator, damped at half the Hubble rate.Section 1.6, Trace anomaly, Eqs. (17)-(20); Section 1.7, Results, Eqs. (21)-(23)
The theory is offered as testable rather than final: it does not predict a specific value of the cosmological constant because that value depends on the dynamics, it accounts for both the recent expansion and inflation without introducing any new field, and the single parameter — set by the characteristic length near the Planck scale where the equivalence principle ceases to hold and by the effective number of fields — may be inferred by fitting the measured expansion of the universe to the equation of motion. Leonhardt notes that the directly observed Hubble constant from galactic cepheids differs by 6 per cent from the value calculated using the constant from cosmic microwave background measurements, which suggests the cosmological constant then was different from the present one.Section 1.7, Results, closing paragraph; Section 4, Conclusion
The way in
https://doi.org/10.1016/j.aop.2019.167973How to cite it
Ulf Leonhardt (2019) Lifshitz theory of the cosmological constant doi:10.1016/j.aop.2019.167973