Historical research guide · 0659 · 2017
How the huge energy of quantum vacuum gravitates to drive the slow accelerating expansion of the Universe
Qingdi Wang · Zhen Zhu · William G. Unruh
In one minute
Qingdi Wang, Zhen Zhu and William Unruh take the enormous vacuum energy that quantum field theory predicts, decline to renormalise it away, and insist — as general relativity’s equivalence principle demands — that it gravitates. Their key observation is that this energy density is not a constant. The vacuum is an eigenstate of the total energy, so the total is fixed, but it is not an eigenstate of the energy density at a point: at every point the density swings as violently as its own huge magnitude, and neighbouring points swing out of step. So spacetime does not inflate uniformly. At each point it oscillates between expanding and contracting, and those Planck-scale oscillations very nearly cancel. Very nearly: a weak parametric resonance leaves expansion a little ahead of contraction on every cycle, and that tiny surplus accumulates across cosmological distances into the slow acceleration astronomers actually see. Their expansion rate falls toward zero as the cutoff rises — the exact opposite of the old, unbounded prediction.
Connection
Chapter 2 says the vacuum is a real, structured medium rather than an averaged-out constant, and this paper is the strongest published demonstration of how much that distinction changes: keep the fluctuations and the same huge energy density drives a slow expansion instead of a catastrophe. It gives chapter 3 a concrete account of vacuum energy obeying the equivalence principle, chapter 5 a picture of spacetime as a churning medium — Wheeler’s foam — and chapter 13 a way of reading dark energy and zero-point energy as one quantity seen at two scales.
From the source
The vacuum is an eigenstate of the total Hamiltonian but not of the local energy density operator, so the total vacuum energy is fixed while its density fluctuates at every point. Direct calculation gives a mean squared deviation equal to two thirds of the square of the mean, with the mean itself growing as the fourth power of the high-energy cutoff — the density fluctuates as violently as its own magnitude.Section III, Eqs. 19 to 21
The vacuum is also extremely inhomogeneous. The expectation of the squared difference between the energy density at two points climbs to the order of the density itself once their separation grows by only about one over the cutoff, so the homogeneous, isotropic assumption behind the standard step from vacuum energy density to Hubble rate does not hold, and a new method of relating the two is required.Section III, Eq. 22 with Figure 1
Allow the scale factor to depend on position as well as time and the spacetime sourced by the vacuum oscillates at each point between expansion and contraction, with neighbouring points out of phase. The violent gravitational effect is then confined to Planck scales and largely cancels at macroscopic ones — a picture the authors trace back to Wheeler’s spacetime foam.Introduction; Sections IV and V
Those oscillations are not perfectly balanced. A weak parametric resonance leaves expansion slightly ahead of contraction on each cycle, and the surplus accumulates into an observable global expansion rate equal to a constant times the cutoff times the exponential of minus a second constant times the square root of Newton’s constant times the cutoff. That rate falls toward zero as the cutoff is taken to infinity, where the older treatment gave a rate growing without bound.Section V D, Eq. 75; Section VI
On this account the observed acceleration needs neither a cosmological constant fine tuned to an accuracy of 10⁻¹²⁰ nor a dark energy with peculiar negative pressure. Matching the measured rate requires a cutoff of the order of the Planck energy or higher — about a thousand Planck energies if only two scalar fields contribute, and less as more fundamental fields are included.Section VI, Meaning of our results
The actual value of the zero-point energy density matters even where gravity is absent. The Casimir stress on a mirror is never constant: it fluctuates with a magnitude set by the vacuum energy density itself, and the measurable fluctuation of the time-averaged stress grows as the eighth power of one over the detector’s resolving time. What to watch: the faster the force detector, the more of that fluctuation it registers, which is a direct handle on the magnitude of the zero-point field rather than on the small mean force alone.Section IX C 1, Eqs. 202 to 207, citing Barton
The way in
https://arxiv.org/abs/1703.00543How to cite it
Qingdi Wang, Zhen Zhu, William G. Unruh (2017) How the huge energy of quantum vacuum gravitates to drive the slow accelerating expansion of the Universe doi:10.1103/PhysRevD.95.103504