Historical research guide · 0684 · 2024

Quantum noise in time-dependent media and cosmic expansion

Ziv Landau · Ulf Leonhardt

In one minute

Ulf Leonhardt and Ziv Landau bring a condensed-matter tool to the loudest argument in cosmology. To light, they point out, expanding space behaves exactly like a block of glass whose refractive index changes with time — and in such a medium the energy of the quantum vacuum is not zero. It cannot be, because the subtraction that removes the infinite part of that energy has to be local and causal: it may know the medium only at the moment in question and how fast it is changing, never the whole history. What survives the subtraction is a real energy density that tracks the expansion, and it pushes back. Every ingredient of the universe then weighs less than it otherwise would, each by its own factor — the authors call this quantum buoyancy. Put in a cut-off length near twice the Planck length and the Hubble constant comes out about 8 percent above the value inferred from the microwave background, which is the discrepancy astronomers actually measure.

Connection

Chapter 2 argues that the vacuum is a real, structured medium, and chapter 5 that you can read gravitational physics off the behaviour of a medium; this paper does both at once, and at the scale of the whole universe. Chapter 3’s question — does zero-point energy have weight — is answered here quantitatively, with each cosmic fluid given its own buoyancy factor. And chapter 13 gets a rare thing: a theory in which a number from the smallest scale in physics, the Planck length, is fitted directly to an astronomical measurement, with no new fields and no modification of general relativity.

From the source

In a spatially uniform medium whose electric permittivity and magnetic permeability are equal and vary in time, classical electromagnetic waves propagate exactly as they do in empty flat space, with ordinary time replaced by conformal time — the integral of the time increment divided by the refractive index — and the quantum fluctuations of the field propagate the same way. The expanding universe is exactly such a medium for the electromagnetic field, with the refractive index equal to the scale factor.Section I, Equations (1) to (3); Appendix A
In empty flat space the renormalized vacuum energy is exactly zero, but in a time-dependent medium it is not. Renormalization is local and causal: the renormalizing Green function may depend only on the medium’s response and its first and second derivatives, taken at the earlier of the two times being correlated. It therefore cannot fully compensate for the transformation to conformal time, and a finite energy density survives that tracks the time evolution.Section I, the renormalization discussion; Section III; Conclusion, opening paragraphs
The surviving vacuum energy makes gravity partially repulsive, and each cosmic fluid is affected individually: the gravitational weight of a fluid with equation-of-state parameter w is its density divided by one plus three times one plus w times kappa. Radiation is divided by one plus four kappa, matter by one plus three kappa, and the cosmological constant is unaffected. The authors call this quantum buoyancy, and note that no other hypothetical vacuum energy would have this feature.Section IV B, Equations (76) to (80); stated in advance at Equation (12) of the introduction
The single free parameter kappa is eight over nine pi times the square of the ratio of the Planck length to the cut-off length. Taking the cut-off near twice the Planck length gives kappa of order ten to the minus two; solving the model numerically with the Planck satellite values for the matter and dark-energy fractions, kappa equal to 1.28 times ten to the minus two — a cut-off length of 4.69 Planck lengths — reproduces the measured ratio of 1.084 between the directly measured Hubble constant and the value inferred from the microwave background.Section I, Equation (13); Section IV C, Equations (87) to (90)
Of the order of one hundred theories proposed to explain the Hubble tension, the authors state that theirs is the only one requiring no new fields, no modification of general relativity and no new cosmological principle; what it does require is a coupling parameter set by an unknown dispersion length near the Planck scale, which they say must in practice be fitted to astronomical data.Section I, closing paragraphs; Conclusion, the paragraph on the coupling parameter
The hypothesis underneath the whole calculation — that the part of the vacuum energy which gravitates is the same part that exerts mechanical Casimir forces — has a laboratory test being prepared: an experiment to weigh the quantum fluctuations inside a material with time-dependent parameters. The authors add that although a medium with equal electric and magnetic response is hard to build, similar effects should appear in a material with a time-dependent electric and a constant magnetic response.Section I, the paragraph on weighing the fluctuations; Conclusion, the paragraph on tests other than in astronomy

The way in

https://doi.org/10.1103/PhysRevB.110.224202

Historical source-access record. This guide links to the original; it does not reproduce the paper text.

LICENCE. This sheet was promoted from abstract-only after checking the licence at the source: the arXiv posting 2501.02495, submitted 5 January 2025, carries an explicit Creative Commons Attribution 4.0 International licence, so the text is reproduced here under it. The version of record is Physical Review B 110, 224202 (2024), published by the American Physical Society under its own licence; the text below is the CC BY 4.0 arXiv:2501.02495v1 edition, not the APS typesetting. Ziv Landau is at the Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot, and at St John’s College, University of Cambridge; Ulf Leonhardt is at the Weizmann Institute. TEXT. Reproduced in full: the abstract, the whole of the introduction, section IV on cosmic expansion with its three parts, the conclusion and the acknowledgements. Sections II and III — the derivation of the Green functions, the fluctuation–dissipation theorem, the de Sitter case, the geometrization on the complex time plane and the renormalized energy density — and Appendix A and the reference list are omitted for length and are at the source; the omissions are marked where they fall. Running heads, page numbers and reference-number markers are dropped as page furniture. Displayed equations reached the library with Greek letters, integral signs and superscripts damaged in extraction, so they are given with their original numbers, restored where the extraction is unambiguous and stated as named results in plain words where it is not; inequalities are written in words. The six figures are simulation plots and geometric diagrams that cannot be reproduced as text, so the figures used in the reproduced sections are given as their captions, which carry the content. REGISTRY CORRECTION: the record reached the library with chapter 6, getting energy out of the vacuum, which this paper does not address; chapter 6 is dropped and chapters 2, 3, 5 and 13 are kept.

How to cite it

Ziv Landau, Ulf Leonhardt (2024) Quantum noise in time-dependent media and cosmic expansion doi:10.1103/PhysRevB.110.224202

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