Casimir cosmology

Abstract

In 1998 astronomers discovered that the expansion of the universe is accelerating. Somehow, something must have made gravity repulsive on cosmological scales. This something was called dark energy; it is described by Einstein's cosmological constant; and it amounts to about 70% of the total mass of the universe. It has been conjectured that the cosmological constant is a form of vacuum energy, but its prediction from quantum field theory has failed by many orders of magnitude, until recently. Informed by empirical evidence on Casimir forces, Lifshitz theory has not only produced the correct order of magnitude, but is quantitatively consistent with the astronomical data. Moreover, the theory appears to resolve the tension between the measured and the predicted Hubble constant. There is therefore a good chance that Casimir physics explains dark energy. This article introduces cosmology for practitioners of vacuum forces as part of "The State of the Quantum Vacuum: Casimir Physics in the 2020s" edited by K. A. Milton. It may also be interesting for other physicists and engineers who wish to have a concise introduction to cosmology.

Ulf Leonhardt, Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 7610001, Israel. 9 February 2022.

1. Introduction

Observational cosmology has entered a golden age, the age of precision measurements. Gone are the days when "cosmologists were often in error but seldom in doubt" (Landau). Speculation and estimation have been replaced by precise data. What these data tell is astonishing: 95% of the current content of the universe is completely unknown. Dubbed "the dark sector" (for want of a more illuminating term) it is described in the cosmological standard model, giving it its name Λ Cold Dark Matter (ΛCDM) model. The true nature of the "dark sector" has been a mystery, or rather two mysteries, one less and the other more mysterious. For "dark matter" making up 25% of the total mass density of the universe there are many theories and several experimental programmes to detect its particles. Yet the lion share of the "dark sector" — 70% of all mass — has been an enigma. Called "dark energy", it is described by Einstein's cosmological constant Λ that causes a repulsive gravitational force. The constant Λ is commonly believed to be associated with the quantum vacuum, its energy contribution is called the "vacuum contribution", but the actual mechanism of how the quantum vacuum creates the cosmological constant has not been clear — by quite a bit. Estimations from quantum field theory are 120 orders of magnitude higher than the actual number. But these are based on a naive picture of vacuum fluctuations. Thanks to experimental and theoretical research on the Casimir effect we know a great deal more now about vacuum fluctuations. It is time for the practitioners of vacuum forces, the Casimir community, to take up the torch and shed light on "dark energy".

The first steps have been made and they seem encouraging. The 120 orders of magnitude in the mismatch between theory and measurement disappear if one accepts the idea that the quantum vacuum needs to be renormalized. Renormalization is normal in practical Casimir physics and its absence would contradict experimental facts. What remains after renormalization in cosmology is not exactly zero, as simple calculations would tell, and also not approximately zero, as experience would suggest — the Casimir force typically acts on the nanoscale, so how can it play any role on cosmological scales? The theoretical Λ is of exactly the right order of magnitude. Moreover, the theoretical prediction is consistent with the astronomical data up to the level of precision of that data. Furthermore, it fits and explains a tension in the astronomical data (and their interpretation) known as the Hubble tension that with each refinement in data and data analysis just gets sharper and has recently reached crisis level. The astrophysics community has discovered this and other inconsistencies between the ΛCDM model and the data, that urgently need explanations. The Hubble tension has been called the biggest crisis in contemporary astrophysics. But every crisis is an opportunity, and if the Casimir effect does indeed explain the cosmological constant, this is the golden opportunity for the Casimir community to make a decisive contribution to astrophysics.

How the Casimir effect may play a role in cosmology was explained elsewhere. Here I focus on the cosmology. This is primarily for bringing Casimir practitioners up to speed in a field that is normally alien to them, and secondly, for working out where exactly in the cosmic expansion the Casimir effect enters. There are excellent textbooks on cosmology, but it takes time and effort to extract and absorb the parts relevant to Casimir physics. Here is hopefully a primer that is sufficiently concise and still sufficiently correct. Following Einstein's well-known advice, I have tried my best to write it "as simple as possible, but not simpler".

(Sections 2 to 4 — Principles of cosmology, Cosmic dynamics and the Cosmic Microwave Background — are the primer itself: the Friedman-Lemaitre-Robertson-Walker metric, horizons and the Gibbons-Hawking temperature, the Friedman equations and the radiation, matter and vacuum eras, and the acoustic and angular scales of the microwave background. They are equation-dense and are omitted for length, as is the mathematical appendix on cosmic fluctuations; the complete text is at the source.)

5. Hubble tension

The Hubble constant H0 of Eq. (26) has been inferred from the Cosmic Microwave Background, but it can also be measured directly from Hubble's law (6) that relates redshift −dω/ω to distance dℓ. Frequencies ω and their shifts dω are measured precisely by spectroscopy. Astronomical distances are determined by distance ladders. The first ladder is constructed from geometrical measurements taking the orbit of Earth around the Sun as baseline. Parallaxes (angle deviations) of stars are observed when Earth moves and the distances to those stars determined by triangulation. This first ladder reaches to about 10Kpc. The second ladder makes use of bright, pulsating stars — Cepheids with a robust relation between brightness and pulsation rate. The apparent brightness depends on the actual brightness divided by the luminosity distance squared. The Cepheid relation is calibrated by geometrical measurements for the ones in sufficiently close proximity. Then, in turn, the established relation is used for inferring distances from the observed brightnesses, which constitutes the second ladder. This ladder reaches up to 40Mpc. The third ladder uses type Ia supernova explosions as "standard candles". These explosions are sufficiently bright to be measured up to redshifts z of about 3 and their actual brightness does not vary much (with deviations corrected from features of their spectra). Supernovae are rare events (1 per century in an average galaxy) but 19 of them have been calibrated with Cepheids. Having calibrated some type Ia supernovae as standard candles, many more similar supernovae serve to determine the distance as a function of redshift z (Fig. 1). Performing the limit z → 0 in the data gives H at the present time, with nearly 1% precision:

H0 = (73.04 ± 1.04) km/s per Mpc. (53)

The problem is that this value does not agree with the H0 in the cosmic parameters (26) inferred from the CMB. The uncertainty in the measured value (53) has been reduced to 5σ, which in high-energy physics would qualify as a discovery. In cosmology, there are still sceptics but the tension between H0 inferred from the early universe (from the CMB) and the Hubble constant measured by late-universe probes persists also in other observations based on completely different methods although with less precision yet. The overall evidence indicates that the Hubble tension is real.

There are about 10² theories explaining the Hubble tension, but all of them require untested modifications to the standard model of particle physics or to general relativity or to the cosmological principle, except one. This theory is based on the physics of the quantum vacuum. It does make extrapolations — it extrapolates quantum electrodynamics from dielectric media to curved space-time — and it contains assumptions on renormalization, but its principal steps are empirically tested or experimentally testable.

Here is the argument of how the Casimir effect explains the Hubble tension. We have seen in Sec. 4 that H0 is inferred from the relation between the angular and the acoustic scale of the CMB. This relation requires knowing the distance d to the surface of last scattering (Fig. 5) and d is given by Eq. (50). It is the only quantity that depends significantly on the cosmological constant Λ, because only during the free propagation of the CMB in the matter-vacuum era Λ plays a significant role. The shape of the correlation curve (Fig. 4) is formed in the radiation-matter era when the CMB is formed (Fig. 6). So the shape of the curve depends on Ω_R and Ω_M (and on the contributions from neutrinos and dark matter) but not on Ω_Λ H0², only the scale of the curve does.

Now, the ΛCDM model assumes that the cosmological constant Λ is constant. Where is the evidence for this? The CMB data do not prove that Λ is constant, because Λ is contained in only one parameter, the scale of the correlation curve. As long as Λ is concerned, all the CMB data condense into one point. A single data point does not prove constancy. The Hubble diagram (Fig. 1) maps out the late stage of the cosmic expansion as a curve depending on the constancy of Λ. A constant Λ is consistent with the data, so Λ must not vary much, but the observed H0 is inconsistent. Is there good theoretical evidence? The most-common and most-convincing argument is this: the vacuum should be generally covariant — it should look the same in all reference frames. Consequently, the energy-momentum tensor of the vacuum must be a constant, the cosmological constant. This argument is wrong, because the premise is wrong: the vacuum is not generally covariant. For example, in the Unruh-Fulling-Davis effect the vacuum state of Minkowski space appears as thermal radiation to an accelerated observer. The vacuum is frame-dependent, as if it were a physical substance. The vacuum is an unusual "substance" though — it does not resist uniform motion, but it resists acceleration. In any case, the vacuum is not generally covariant, which invalidates the theoretical argument for the constancy of Λ. Can a constant Λ be calculated from theory? Zel'dovich suggested that Λ is given by the vacuum energy and calculated it for a simple model. However, this and other calculations disagree with the actual value of Λ by many orders of magnitude, depending on the cut-off of the theory. There is therefore no credible evidence for the absolute constancy of Λ.

Suppose now that Λ is given by the vacuum energy as obtained in Casimir physics. We can apply this theory, because curved space-time appears as a dielectric medium to electromagnetic fields. The many orders of magnitude disagreement with data disappear by renormalization. What remains is an energy density ε_vac of the correct order of magnitude. Why this is so is subtle and is discussed elsewhere. Here we describe how ε_vac influences the cosmic dynamics and resolves the Hubble tension using a thermodynamic argument.

In the standard Casimir effect, the vacuum exerts forces on the interfaces of dielectric media. Spatially varying media create spatially varying vacuum energies and stresses. In cosmology, space is homogeneous, but expands in time t. The varying scale factor a(t) creates a non-trivial vacuum energy similar to the dynamical Casimir effect briefly discussed in Sec. 2. Since ε_vac is created by variations in a it must depend on derivatives of a. But this implies that ε_vac is not like the other energy densities, ε_R and ε_M, that only depend on a. They follow the cosmic expansion adiabatically, whereas ε_vac is non-adiabatic, which violates the thermodynamic relation (14). This relation is necessary for the energy-momentum conservation and the validity of Einstein's field equations of gravity. For maintaining the energy balance we thus need to add an energy density ε_Λ to ε_vac. For the vacuum pressure p_vac we have

p_vac = ε_vac / 3 (54)

like in Eq. (19) for any other incoherent radiation pressure. The additional ε_Λ should just take care of the energy balance in Eq. (14), but not cause an imbalance itself. We thus require p_Λ = −ε_Λ [Eq. (21)] and obtain from Eq. (14):

∂t (ε_Λ + ε_vac) = −4 H ε_vac. (55)

The additional ε_Λ with pressure −ε_Λ appears like the trace anomaly of conformally invariant fields. It shows the characteristic feature, p_Λ = −ε_Λ, of the cosmological constant, but ε_Λ varies. However, the dynamical equation (55) allows for an integration constant we may put to

ε_∞ = lim (a → ∞) ε_Λ (56)

and interpret as the cosmological constant at infinity when space is expanding exponentially with constant Hubble parameter H. The vacuum energy ε_vac is generated by variations in the Gibbons-Hawking temperature of Eq. (13). In the era of exponential expansion, these variations vanish and hence ε_vac → 0, only ε_Λ remains with ε_Λ → ε_∞. The vacuum-dominated era is thus characterized by a constant cosmological constant. However, in the transition period between matter and vacuum domination — the period we live in — ε_vac varies and so does ε_Λ. The variation in the sum of the two energies affects the cosmic dynamics and hence the distance d required for relating the angular scale of the CMB to the acoustic scale (Fig. 5). The H0 inferred from that distance assuming constant ε_Λ is no longer the actual Hubble constant, because ε_Λ has varied.

We may adjust the integration constant ε_∞ such that we fit the angular scale of the CMB and then see what the dynamics of Eq. (55) does. For this we need to determine ε_vac from Casimir physics assuming a cut-off close to the scale where space-time presumably ceases to act like a medium, the Planck scale characterized in terms of the Planck length

ℓ_P = the square root of ħG/c³, about 10⁻³⁵ m. (57)

It turns out that, for a cut-off at exactly the Planck length and for the most naive model, the theoretical variation of the Hubble constant agrees with the latest data (8.4%, Eq. 53) with 1% precision, the very precision of the data.

6. Outlook

Casimir physics has a chance of resolving the biggest crisis in contemporary astrophysics. The first results seem promising, but much more work is needed to be certain that they are real and not coincidences. First, the theory developed so far is incomplete: while the theory takes the Gibbons-Hawking effect of cosmological horizons (Sec. 2) as its starting point, it ignores the Hawking partners created on the other side of the horizon. As the cosmological horizon is not an event horizon (Fig. 2) the partners eventually arrive and play a role of equal importance than the Hawking particles created this side of the horizon. Second, quantum electromagnetism and quantum electromagnetism alone seems to fit the bill. Where are the other fields of the standard model of particle physics, why are they silent here? Third, further and more consistent analysis of the astronomical data is needed. A great deal of modern cosmology is statistics: exploring variations in the parameters and their influences on several, disconnected data sets from vastly different cosmological eras (CMB, Baryon Acoustic Oscillations, supernovae, gravitational lensing etc). Fourth, there are other tensions in the cosmological data than the Hubble tension, for example inconsistencies in the standard fit to the CMB correlations (Fig. 4) for low angular Fourier components l or inconsistencies in the value of the matter density and the amplitude of cosmic structure. These tensions are more subtle and less statistically significant (about 3σ instead of 5σ for the Hubble tension) but they might be real, too. If Casimir physics resolves the Hubble tension it might also resolve the other tensions. Fifth, experimental tests are needed. These should be tests of the principle mechanism and could be performed in laboratory analogues of gravity. For example, the anomaly giving rise to the cosmological constant can be tested with ultracold atoms. Or the connection between the Gibbons-Hawking effect and the dynamical Casimir effect can be explored in the laboratory. There are probably more (and better) tests possible. They could give much needed empirical grounding to the lofty extrapolations of the theory, which brings us finally to: sixth, the foundations of the theory need to be better understood. While the Lifshitz theory of the cosmological constant is based on well-understood and well-tested physics, it still makes extrapolations and assumptions. In particular, the renormalization method of the theory needs a better foundation. The problem is related to the renormalization of the Casimir effect in inhomogeneous media. In cosmology, space represents a uniform medium inhomogeneous in time, on Earth graded-index materials are constant in time but inhomogeneous in space. While the Casimir effect in inhomogeneous media has been full of surprises the theory has remained underdeveloped since the foundation of the field by Lifshitz, Dzyaloshinskii and Pitaevskii. There was simply no need, no urgent problem to be solved in inhomogeneous media — and the theory is difficult and observable effects, if any, are likely to be obscure. So is it surprising that inhomogeneous media have been a backwater? Now there is an urgent problem worth the salt of the Casimir community. Let's get it solved.

Acknowledgements

I am most grateful to Dror Berechya for his helpful comments and the many discussions we had during our struggle to understand modern cosmology, and to David Bermudez and Nikolay Ebel for discussions and comments on analogues of gravity and beyond. The paper has been supported by the Israel Science Foundation and the Murray B. Koffler Professorial Chair.

(Sections 2 to 4 and the appendix on cosmic fluctuations are omitted for length; the complete text, with all figures, equations and references, is at the source. The companion paper carrying the calculation of the vacuum energy itself is on this site at /library/stm-748797c2ce.)