Historical research guide · 0822 · 2013
Sequestering the Standard Model Vacuum Energy
Nemanja Kaloper · Antonio Padilla
In one minute
Quantum field theory says empty space carries an enormous energy density. General relativity says every form of energy curves spacetime. Put the two together and the universe should have curled up long ago. Nemanja Kaloper, at UC Davis, and Antonio Padilla, at Nottingham, propose a minimal change to general relativity that lets both statements stand. They make every mass scale in the matter sector depend on the total four-volume of spacetime, and add two global quantities varied like constants rather than like fields. What comes out is Einstein’s equation with one extra term: the whole-history average of the trace of the matter stress tensor, subtracted from the source. That subtraction removes the vacuum energy exactly — the classical piece and every quantum loop correction — so it never curves anything. Nothing new propagates, and local physics is untouched. The price is a prediction: for those averages to be finite the universe must be finite in time, so today’s steady dark energy is a passing phase and a collapse lies ahead.
Connection
Chapter 2 has to answer the obvious objection to a vacuum holding some 10¹¹³ joules per cubic metre — why is the sky not violently curved? Kaloper and Padilla give one clean answer: the energy is entirely real, it simply does not source curvature, and what is left over is small for a reason rather than by a tuning. Chapter 13 needs exactly that separation between how much energy the vacuum holds and how much of it gravitates.
From the source
A minimal reformulation of general relativity sequesters all of a matter sector’s vacuum energy from gravity. Every dimensional parameter in that sector is made a functional of the four-volume element of the universe, by coupling matter to a rescaled metric and treating the rescaling and the cosmological term as global auxiliary variables, with an external function placed outside the integral.Abstract; Eqs. (1) and (2)
The resulting field equations are Einstein’s with the four-volume historic average of the trace of the matter stress tensor subtracted from the source. Because that average of a constant is the constant itself, the hard cosmological constant — the classical piece plus vacuum loop corrections to any order — drops out of the equations completely and never contributes to curvature.Eq. (3), and Eq. (4) with the paragraph preceding it
The mechanism adds no new propagating degrees of freedom. All propagating fields obey standard second-order equations, the spectrum of fluctuations is that of conventional general relativity with minimally coupled matter, and the dependence of the matter scales on the four-volume is invisible to any non-gravitational local experiment by diffeomorphism invariance.Paragraph 4 of the Letter; closing discussion, penultimate paragraph
A residual cosmological constant survives — the historic average of the non-constant sources — and it is automatically small rather than tuned. Its size is set by the lifetime of the universe rather than by any ultraviolet cutoff, and two approximate symmetries, a scaling and a shift, make that smallness technically natural.Eq. (4) and the following two paragraphs; footnote 8
Vacuum energy released by early-universe phase transitions does not drop out of the field equations, but is automatically suppressed at late times: the leftover is smaller than today’s critical density by a power of the ratio of the transition’s curvature scale to the universe’s lifetime scale, which for the Standard Model is a very large ratio.Eq. (6) and the paragraph after it
Three observational consequences follow, and each is a test. The universe must be compact in spacetime — spatially closed, ending in a crunch, plausibly in around a hundred billion years — the vacuum curvature of a large old universe is small but not zero, and the measured dark-energy equation of state near minus one is a transient rather than a permanent state.Abstract; closing discussion, final two paragraphs
The way in
https://arxiv.org/abs/1309.6562How to cite it
Nemanja Kaloper, Antonio Padilla (2013) Sequestering the Standard Model Vacuum Energy doi:10.1103/PhysRevLett.112.091304