Quantum noise in time-dependent media and cosmic expansion

Abstract

In spatially uniform, but time-dependent dielectric media with equal electric and magnetic response, classical electromagnetic waves propagate exactly like in empty, flat space with transformed time, called conformal time, and so do quantum fluctuations. In empty, flat space the renormalized vacuum energy is exactly zero, but not in time-dependent media, as we show in this paper. This is because renormalization is local and causal, and so cannot compensate fully for the transformation to conformal time. The expanding universe appears as such a medium to the electromagnetic field. We show that the vacuum energy during cosmic expansion effectively reduces the weights of radiation and matter by characteristic factors. This quantum buoyancy naturally resolves the Hubble tension, the discrepancy between the measured and the inferred Hubble constant, and it might resolve other cosmological tensions as well.

I. Introduction

Imagine a dielectric medium with time-dependent electric permittivity ε and magnetic permeability µ. Suppose the medium is made of a spatially uniform, infinitely extended block of material with

ε = µ = n(t). (1)

Note that n(t) describes the refractive index, as n is the square root of ε times µ. Consider, in this medium, the quantum fluctuations of the electromagnetic field (Fig. 1). What is their energy density? What is their gravitational force? Could they influence the expansion of the universe, and if so, how? These are the questions of this paper. As in analogues of gravity they aim at cosmological problems, but are grounded in condensed matter physics, and they do have answers.

Let us explain step by step. From the wave equation of the electromagnetic vector potential A in Coulomb gauge — c² times the curl of one over µ times the curl of A, plus the time derivative of ε times the time derivative of A, equal to zero — follows that A satisfies the free-space equation with the transformed time

τ = the integral of dt divided by n. (2)

So, like in transformation optics the medium of Eq. (1) performs a coordinate transformation of empty space for electromagnetic fields. Similar media have been proposed and experimentally demonstrated as temporal cloaking devices, or “history editors”, and time-dependent dielectrics have attracted considerable recent attention.

Artificial materials with the properties of Eq. (1) are still difficult to manufacture, but one of them occurs in nature: the “material” of space (Appendix A). Averaged over cosmological distances — more than 100 megaparsecs — space appears uniform and flat while expanding with scale factor a(t). Distances measured in wavelengths of electromagnetic radiation grow with the factor a (Fig. 1) as if the wavelength is reduced by the refractive index

n = a(t). (3)

Fig. 1: Visualization of quantum noise. Space-time diagram of Gaussian noise in the medium of the expanding universe (for t from 0 to the present time, with the actual cosmic parameters, Sec. IV C). Plot of 64 normalized modes summed up with Gaussian random complex amplitudes. One sees how the wavelength changes due to expansion or, equivalently, the time evolution of the medium.

The electric and the magnetic response of space are the same, and so, for the electromagnetic field, the scale factor acts like the refractive index of the spatially uniform, time-dependent medium of Eqs. (1) and (3). Moreover, this “dielectric medium” of space is presumed dispersionless — having dielectric properties independent of frequency or wavenumber — until wavelengths smaller or comparable with the Planck length

ℓP = the square root of ħG over c³ = 1.616255 × 10⁻³⁵ m. (4)

This is a consequence of the equivalence principle (Appendix A): space-time acts on everything equally, including every frequency component of the electromagnetic field, until classical general relativity reaches the limit of its validity, which is expected near ℓP. This fantastic range of frequencies carried without distortion by the medium of space is the reason why the quantum noise of the field might be modified, even as the expansion rate, the Hubble parameter

H = ȧ over a (5)

is as astronomically low as astronomers have measured: about 70 km/s/Mpc, about 2 × 10⁻¹⁸ Hz.

What is the quantum state of the electromagnetic field? Since the field propagates like in flat space-time with the transformed time τ of Eq. (2) — called conformal time — there is a natural ground state of the field called the conformal vacuum: the ground state of modes made of plane waves with positive frequencies with respect to τ. The universe is filled with the Cosmic Microwave Background (CMB) and the light of stars etc., which is negligible in energy compared with electromagnetic vacuum fluctuations with wavelengths approaching the Planck length. We may thus assume the field to be in the conformal-vacuum state.

Note that in media where only ε varies and µ remains constant, classical electromagnetic waves get reflected and amplified and quantum particles are produced in a phenomenon called the dynamical Casimir effect. The effect is analogous to the cosmic particle production of non-conformally invariant fields such as the fluctuations of the gravitational field that, presumably, have been amplified from vacuum noise during cosmic inflation. This amplified noise then got modulated during the formation of the CMB and provided the seeds for structure formation in the universe. Analogues of cosmic particle production have been demonstrated with ultracold atoms. But as we have argued, as long as the electric and the magnetic response of a medium are the same — Eq. (1) — the electromagnetic field remains in the conformal vacuum.

In this paper, we calculate the vacuum energy density of the electromagnetic field in the time-dependent, spatially uniform medium of Eq. (1). We improve on a previous paper that relied on the analogy between expanding space and moving media. In such media, thermal radiation is produced at the cosmological horizon (where the expansion velocity reaches c) similar to the Bekenstein–Hawking radiation of black holes and their analogues. A more careful analysis, conducted here, reveals the limitations of this approach and opens the path to calculating the vacuum energy from first principles.

The classical energy density u of the electromagnetic field is given in terms of the electric field strength E, the dielectric displacement D, the magnetic field strength H and the magnetic induction B as (Appendix A) the sum of an electric part, E dotted into D over two, and a magnetic part, H dotted into B over two, (6) with the constitutive equations for the medium of Eqs. (1) and (3): D equals ε₀ a E and B equals µ₀ a H, in SI units where ε₀ denotes the electric permittivity and µ₀ the magnetic permeability of the vacuum, with ε₀ µ₀ equal to one over c². (7) As the fields are derivatives of the vector potential — E is minus the time derivative of A in Coulomb gauge, and B is the curl of A — we may regard the energy densities as derivatives of products of two vector potentials at different times and positions and then take the limit in which the second position and time approach the first. (8) We denote the vector potentials at the two space-time points by A1 and A2, and obtain from Eqs. (6) and (7) the electric density as ε₀ a over two times the mixed time derivative of A1 dotted into A2, and the magnetic density as ε₀ c² over two a times the curl of A1 dotted into the curl of A2. (9)

In spatially uniform media, the two polarizations of electromagnetic waves are separate and behave identically, so that the products of the vector potentials reduce to twice the product of one representative polarization component evaluated at each of the two points. For quantum fields, we need to replace that product by the expectation value of the symmetrized field operator products, such that these are Hermitian and their expectation values are real. We thus get for the energies the electric density equal to ħa over c times the mixed time derivative of a correlation function K, and the magnetic density equal to ħc over a times the mixed spatial gradient of K, (10) in terms of the correlation function (Fig. 2) K, defined as ε₀ c over two ħ times the expectation value of the symmetrized product of the two vector-potential operators. (11)

Note that such correlation functions have been measured inside materials. There, two ultrashort light pulses serve as probes of the quantum noise floor along their world lines. In appropriate nonlinear materials, the electromagnetic noise affects the polarizations of the probes. The polarization of each individual probe pulse becomes slightly noisy, but the polarization noise of the two pulses was found to be correlated, even outside the light cone.

In order to calculate the correlation function K we use a fundamental principle as our starting point, the fluctuation–dissipation theorem. The theorem relates the quantum correlation K to the classical Green function G of the electromagnetic field. This was first done in radar engineering where Rytov calculated the correlations of classical electromagnetic noise from the Green functions. Lifshitz and the Landau students Dzyaloshinskii and Pitaevskii extended this connection to quantum correlations and used it to calculate Casimir forces in dielectrics. Here we employ a general form of the fluctuation–dissipation theorem in terms of Hilbert transformations known in physics from the Kramers–Kronig relations. One finds (Sec. II) that the correlation function and energy densities of Eqs. (10) and (11) tend to infinity when the two times and positions approach each other according to Eq. (8). But the physical vacuum energies are surely not infinite.

Fig. 2: Correlations. Density plot of the same noise as in Fig. 1. Time is measured in units of one over H₀ and space in units of c over H₀ (Sec. IV C). Correlations in the noise — curves of equal amplitudes — appear along the light cones of Eq. (23): wave noise is organized. In Eq. (11) we sample the quantum noise at two space-time points (dots) and calculate the correlation function.

In the Casimir effect, vacuum fluctuations in spatially varying media exert mechanical forces, and these forces are finite and typically quite small. In gravity, if we would take the Planck length of Eq. (4) as cut-off, the energy density would be in the order of the Planck mass mP divided by the cube of the Planck length, with mP the square root of ħc over G, about 20 micrograms. Concentrating this macroscopic mass into a Planck-scale volume would change everything we know in physics, and already therefore must be wrong, despite the empirical fact that the bare vacuum correlations of Eq. (11) do exist. How to reconcile these conflicting aspects of vacuum fluctuations? Not all of the vacuum energy can do mechanical work. One assumes that the part of the energy density that causes gravitational forces is the same as the one causing mechanical forces. An experiment is being prepared to test this hypothesis by weighing the quantum fluctuations inside a material with time-dependent parameters.

The procedure of extracting from the vacuum correlations the part that can do mechanical work is called renormalization. Lifshitz renormalization is based on the fluctuation–dissipation theorem: there one subtracts from the Green function G of the full problem the Green function G₀ of an infinitely extended, uniform block of material matching the local values of ε and µ, and then calculates the difference of the corresponding correlation functions K and K₀. In our case of spatially uniform, but time-dependent media, Lifshitz renormalization amounts to subtracting from G the Green function G₀ in a constant material with ε and µ set to their values at the time t₀ we are interested in. Empirical evidence from Casimir forces in fluids supports the idea that renormalization is local: each local region appears to have its own renormalizer G₀. Otherwise the theory would not converge, but the locally renormalized theory does and it agrees well with the experimental data.

Dzyaloshinskii and Pitaevskii suggested to use Lifshitz renormalization in spatially inhomogeneous dielectrics. They did not have the computational tools at the time to realise that this is wrong. One might approximate an inhomogeneous medium by a piecewise homogeneous one and make the approximation finer and finer, or build up the medium from the bottom as particles interacting with each other by retarded van der Waals forces; the continuum limit does not exist, and the force and energy densities do not converge, but approach infinity. It turns out that the renormalizing Green function G₀ should not only depend on the local values of the dielectric response functions, but also on their derivatives. Note that the renormalizer must not depend on all local derivatives, because then G₀ would be identical with G and nothing were left after renormalization.

In dispersive media, the minimal number of derivatives for renormalization to converge is two. In our case, we should make G₀ dependent on a, on its first time derivative and on its second. This is an absolutely critical assumption, and it was not made in the literature on renormalizing vacuum energies in cosmology. Another, more obvious assumption was not made either prior to our work: causality. The renormalizing Green function can only depend on the local dielectric functions and their derivatives at the earlier of the two times t₁ and t₂. In our case, if t₁ is the earlier one then G₀ should depend on the scale factor and its first two derivatives at t₁, and if t₂ is the earlier one then on their values at t₂. Without dispersion, up to fourth derivatives of the scale factor would be needed to get a converging result after renormalization. The resulting energy densities are in the order of ħc times the square of the space-time curvature. They could only play a role in inflation but not in standard cosmic expansion.

Cosmologists have ignored dispersion, but in condensed matter physics it is a natural feature of dielectric media. If space appears like a medium for electromagnetic fields, it ought to be dispersive, too. Dispersion combined with causality raises the leading order of the vacuum energy density to a level where it matters in cosmic expansion. This is because the “material” of space is almost dispersionless until the Planck scale. The renormalization nearly diverges with the inverse square of the cut-off length (Sec. III D) and produces a vacuum energy of the same order of magnitude as the measured cosmological constant. There is still some ambiguity on how the renormalizing Green function depends on the scale factor and its first two derivatives; we hope to have made the most natural choice (Sec. III B).

With this renormalizer, we have obtained exceptionally simple results (Sec. IV B) as follows. The vacuum energy is generated by the cosmic expansion. In turn, the vacuum energy acts back on the expansion by its gravity — its weight. In cosmology, the matter and energy components in the universe appear as fluids with characteristic equations of state. For radiation (photons and neutrinos) the energy density is proportional to ħω over a³ with frequency ω falling with a, and so the density of radiation falls with a⁴. The density of matter (dark and baryonic) falls with a³, while the density of the cosmological constant remains constant. We will see (Sec. IV) that the vacuum energy makes gravity partially repulsive, which reduces the weights of the cosmic fluids as if they were becoming buoyant in the vacuum. Remarkably, each fluid turns out to have its own buoyancy. The cosmological constant is unaffected, while the weights of radiation and matter are reduced by characteristic factors:

the weight of radiation is its density divided by one plus four κ; the weight of matter is its density divided by one plus three κ; the weight of the cosmological constant is unchanged. (12)

Here the parameter κ quantifies the influence of the quantum vacuum. It is given by the cut-off length ℓ versus the Planck length of Eq. (4) as

κ = eight over nine π, times the square of ℓP over ℓ. (13)

We expect ℓ to be of the order of twice the Planck length or larger, as twice the Planck length plays the role of a fundamental length scale, for the following reasons. The Planck length of Eq. (4) is obtained from dimensional analysis and does not have a precise quantitative meaning. However, in Bekenstein’s formula for the entropy of the black hole, with Hawking’s prefactor, the horizon area divided by four times the square of the Planck length gives the entropy. In Jacobson’s thermodynamic derivation of Einstein’s field equations the entropy of causal horizons appears as the area divided by four times the square of the Planck length as well. In both cases, the horizon area is pixelled by elements of two Planck lengths by two Planck lengths to encode one bit of information each, suggesting that twice the Planck length is indeed a fundamental length.

For a cut-off of that size we obtain another remarkable result, assuming the same value for the matter density as obtained from measurements of the CMB fluctuations. Strictly speaking, the values of the densities should be modified due to the vacuum corrections of Eq. (12) in the dynamics of the CMB formation, but we shall argue (Sec. IV C) that these modifications are likely to be small (perhaps a few percent). What is not small is the correction to the Hubble constant H₀. The actual Hubble parameter at the present time was measured to deviate from H₀ by a factor of 1.084, and we match this value for a cut-off length of 4.7 Planck lengths. With our previous model we reproduced the measurement for a cut-off of one Planck length but only in perturbation theory.

The discrepancy between the directly measured and the CMB inferred value of the Hubble constant has been called the Hubble tension and it has become one of the most actively debated problems of contemporary astrophysics. While our latest result might still be a coincidence, and while there are in the order of 10² theories explaining the Hubble tension, our theory is the only one of them that does not require new fields, new modifications of general relativity or new cosmological principles. This does not mean of course that we do not make extrapolations and do not rely on hypotheses — but our assumptions are conservative, and we have clearly stated them here. While the Hubble tension and other cosmological tensions are some of the most urgent problems of astrophysics, their resolution might very well come from condensed matter physics.

(Sections II, Fluctuation and dissipation, and III, Renormalization — the Green functions, the fluctuation–dissipation theorem, the de Sitter case, the geometrization on the complex time plane and the renormalized energy density — are omitted for length; the complete text is at the source.)

IV. Cosmic expansion

A. Gravity

The only force acting over the vast distances of space is gravity. The content of the universe — radiation, matter and the quantum vacuum — generates a gravitational background field that, in turn, decelerates the cosmic expansion — if the force is attractive. This background field is weak and almost non-relativistic. We can use Newton’s law to work it out. Let us briefly review the known theory before we show how the quantum vacuum enters the picture.

Consider a sphere of radius a in the homogeneous and isotropic universe (Fig. 5). According to Gauss’ law the flux of the gravitational field across the surface of this sphere is given by 4πG times the integral of the density ρ (where G denotes Newton’s gravitational constant). As the universe is homogeneous and isotropic over cosmological scales the flux is isotropic and the density homogeneous. The force of the gravitational field gives the acceleration. We divide the flux by the volume, note that the surface of the unit sphere is 4π and the volume 4π over 3, and obtain for the gravitational acceleration:

the second time derivative of a, divided by a, equals minus four π over three, times G times ρ. (70)

Fig. 5: Gauss’ law. Top: Pick an arbitrary point in the homogeneous and isotropic universe. Take another point the distance a away. The gravitational force of the background on the second point, relative to the first one, tells how the universe is accelerating. Bottom: According to Gauss’ law the gravitational acceleration of the second point, with respect to the first one, depends only on the interior of the sphere around the first point. There gravity acts like the restoring force of an oscillator. As the density falls with a this is a nonlinear oscillator, and it may even reverse sign if the pressure is sufficiently negative, Eq. (71).

In this Newtonian picture, mass points the distance a away from the center of the sphere get decelerated by the total mass of the sphere; in the Einsteinian picture, the distance itself gets decelerated: the measure of length is modified for everything, including the electromagnetic field. Apart from this conceptual difference, relativity adds two quantitative features: all the energy density ϵ gravitates, not only the rest mass, and also the pressure p. Since the pressure acts in all three spatial dimensions, p receives a factor of three such that we get instead of the Newtonian Eq. (70) the Einsteinian

the second time derivative of a, divided by a, equals minus four πG over three c², times the quantity ϵ plus three p. (71)

The gravitational force is attractive if the pressure exceeds minus one third of the energy density, which is the case for matter and radiation, but according to Eq. (67) the anomaly generates a repulsive force and may cause the cosmic expansion to accelerate. Astronomical observations have indeed shown that the expansion is accelerating.

One can integrate the dynamic equation (71) combined with the thermodynamic relation (65). One obtains for the Hubble parameter (5):

H squared plus k over a squared equals eight πG over three c², times ϵ, with k constant. (72)

This is the other of Friedmann’s equations. The integration constant k quantifies the spatial curvature of the universe. CMB data have shown that k is approximately zero. (73) Space is flat to an excellent approximation. This concludes our miniature review of cosmic dynamics.

B. Quantum buoyancy

Consider now the influence of the vacuum energy on the cosmic dynamics. We have derived formula (69) for the total of the vacuum-energy densities. Let us write this formula for the corresponding mass densities in light of Friedmann’s equation (72):

the sum of the vacuum density and the cosmological-constant density equals a constant density ρ∞ plus two κ times three over eight πG times the time derivative of H, (74)

with κ given by Eqs. (4) and (13). In cosmology, time is typically measured in terms of the redshift, one over a minus one, that is, in terms of the scale factor a. Let us describe the evolution as a function of a. From Eq. (5) follows that the time derivative of H equals H times a times the derivative of H with respect to a, which is one half of a times the derivative of H squared with respect to a. (75)

Let ρM denote the total density of matter (baryonic and dark) and ρR the density of radiation (photons and neutrinos). The total density is the sum of the radiation, matter, vacuum and cosmological-constant densities. We obtain from Friedmann’s equation (72) with zero curvature k, and Eqs. (74) and (75):

ρ minus κ times a times the derivative of ρ with respect to a, equals ρR plus ρM plus ρ∞. (76)

This is an inhomogeneous linear differential equation for ρ. The homogeneous solution would be proportional to a power of the scale factor and hence grow indefinitely as a tends to infinity. This is impossible for a density. We thus put the homogeneous solution to zero and consider only the inhomogeneous one. As the differential equation (76) is linear, each fluid is influenced by the quantum vacuum individually. This is a highly nontrivial aspect of the renormalized vacuum energy specific to our result (64). No other hypothetical vacuum energy would have this feature.

Radiation and matter are characterized by a simple equation of state that relates the pressure to the energy density: the pressure of each fluid is a constant w times its energy density. (77) For radiation w is one third (Appendix A) while for matter w is zero (and for the cosmological constant w is minus one). From Friedmann’s thermodynamical relation (65) follows that the energy density of each fluid is proportional to a raised to the power minus three times one plus w. (78)

Let the effective weight of each individual density denote the contribution of that density to the total, including the vacuum contribution generated. We solve Eq. (76) for that weight with the fluid density as source,

the weight minus κ times a times the derivative of the weight with respect to a, equals the fluid density, (79)

set the homogeneous solution to zero, and obtain

the weight equals the fluid density divided by one plus three times one plus w, times κ. (80)

This is the effective gravitational density of the fluid. There the original density is reduced by the characteristic factor one plus three times one plus w times κ, as if the fluid were partially buoyant in the quantum vacuum.

C. Hubble tension

How would our result (80) appear in cosmological data? The most precise data in cosmology have been obtained from measurements of the fluctuations of the CMB. These fluctuations were modulated by sound waves in the early universe when the baryonic matter was sufficiently ionized to couple strongly to light. Sound waves are made by two counteracting forces: pressure and inertia. Light provided the pressure and baryonic matter, together with light, the inertia. The densities of light and matter producing those sound waves are not affected by the quantum vacuum, but the expansion should be, according to our theory. This has consequences on the scale of the correlation spectrum.

The measured quantities are temperature variations as a function of spherical angle, but they originate from waves at the time the CMB was released (called the time of last scattering) at which the universe was about 10⁻³ times smaller than it is today. In order to relate angles ϕ to wavelengths λ one needs to know the distance d (Fig. 6): the angle is approximately the wavelength divided by the distance. (81)

This distance is not simply given by c times the time of last scattering (13.8 billion years) because the universe has expanded and hence increased the distance, or equivalently, the speed of light has varied as c over n with the refractive index n given by the scale factor, Eq. (3). As light propagates with conformal time [Eq. (2)] like in empty Minkowski space, the distance traveled is c times the conformal time, which amounts to about 46.2 billion light years for the actual measured parameters of the universe. We denote the scale factor at the time of last scattering by a*, and have according to Eqs. (2), (3) and (5):

d equals c times the integral from a* to 1 of da over a squared H, (82)

where H is given by the Friedmann equation (72).

Fig. 6: Angular versus acoustical scale in the CMB. The CMB consists of thermal radiation with average temperature T₀ and variations δT. The variations were modulated by waves with wavelength λ when the CMB was released. They are observed as angular variations of temperature. The original temperature undulations depend on all cosmic parameters, except the cosmological constant Λ, because Λ did not play a significant role at the time of emission. But, in order to relate the angle ϕ to the wavelength λ one needs to know the distance d. As light travels in conformal time τ like in free space, d equals c τ, which does depend on the cosmological constant, see Eqs. (82) and (84) or (86). This is how Λ is inferred from CMB measurements.

In the standard model of cosmology, the Λ Cold Dark Matter (ΛCDM) model, we write the densities in the form of the Friedmann equation (72) with scaling law (78) as eight πG over three times the density, equal to H₀ squared times a dimensionless constant Ω, divided by a raised to the power three times one plus w, (83) with constant rate H₀, dimensionless constants Ω for each fluid and the w from the equation of state (77): one third for radiation, zero for matter and minus one for the cosmological constant. We thus have for the total Hubble parameter:

H squared equals H₀ squared times the sum of ΩR over a⁴, ΩM over a³, and ΩΛ. (84)

We require that the Ω quantify the relative weights of the cosmic constituents radiation, matter and the cosmological constant, which implies

ΩR plus ΩM plus ΩΛ equals one. (85)

From this and Eq. (84) follows that the constant H₀ is the Hubble parameter at the present time when a equals one, the Hubble constant — provided of course the ΛCDM model holds.

In our case, we obtain from Eq. (80) in the Friedmann equation (72) the modified Hubble parameter

H squared over H₀ squared equals ΩR over one plus four κ times a⁴, plus ΩM over one plus three κ times a³, plus Ω∞. (86)

Note that this effective modification of the ΛCDM model only appears in the cosmic dynamics; the densities themselves are not affected by the quantum vacuum. Now, during the time the CMB was formed, the cosmological constant is negligible compared with the other densities such that the dynamics is only governed by the radiation and matter terms. Furthermore, the radiation term is given by the Stefan–Boltzmann law. We may thus assume, to a good approximation, that the shape of the CMB correlation curve is not influenced by the quantum vacuum and use the same parameters H₀, ΩR and ΩM as obtained from CMB data.

Although the shape of the CMB curve is hardly affected, the scaling is, because the modified Hubble parameter [Eq. (86)] alters the conformal distance [Eq. (82)] that relates, via Eq. (81), the angular scale to the acoustical scale (Fig. 6). Therefore, the value of Ω∞ in Eq. (86) will differ from ΩΛ in Eq. (84). The requirement that the distance d equal the ΛCDM distance (87) relates the two. For calculating the two distances we may put ΩR to zero, because, during most of the cosmic expansion since the release of the CMB, the contribution of radiation to gravity has been negligible. As the scale factor at last scattering is about 10⁻³ we may put it to zero as well. We obtain from Eq. (82) the ΛCDM distance as two c over H₀ times the square root of ΩM, times Gauss’ hypergeometric function with parameters one sixth, one half and seven sixths, evaluated at minus ΩΛ over ΩM. (88) For our case (86) we only need to replace ΩM by ΩM over one plus three κ and ΩΛ by Ω∞, which gives the distance as two c times the square root of one plus three κ, over H₀ times the square root of ΩM, times the same hypergeometric function evaluated at minus one plus three κ times Ω∞ over ΩM. (89)

We have argued (Sec. I) that a cut-off of about twice the Planck length or more is a good assumption, which implies [Eq. (13)] that κ is about 10⁻². For each κ we solve Eq. (87) numerically, using the values for the Ω from Planck satellite data (ΩM = 0.3153 and ΩΛ = 0.6847). For ΩR equal to zero we get from Eq. (86) the Hubble constant as the square root of Ω∞ plus ΩM over one plus three κ. (90) We obtain the measured value of 1.084 times H₀ for the parameter κ equal to 1.28 × 10⁻², that corresponds [Eq. (13)] to the cut-off length of 4.69 Planck lengths. In this case Ω∞ = 0.871, which significantly deviates from ΩΛ. With the expected order of magnitude for the cut-off length we have thus reproduced the deviation of the Hubble constant. Other, more subtle tensions might also be resolved with our theory, which requires the numerical solution of the relativistic Boltzmann equations for the CMB and is beyond the scope of this paper. Yet our simple, approximate result already indicates that one might actually resolve the most urgent problem of astrophysics, the cosmological tensions, with ideas from condensed-matter physics.

V. Conclusion

We have considered spatially uniform, time-dependent media with equal electric and magnetic response, Eq. (1). In such media, classical electromagnetic waves propagate like in empty, flat space with time transformed to the conformal time τ defined in Eq. (2). Quantum fluctuations propagate with conformal time, too, and are correlated like in empty space with transformed time τ, Eqs. (22) and (23). In empty, flat space the renormalized vacuum energy is exactly zero. Yet, as we have argued, in time-dependent media, the energy densities depend on the time evolution, because the renormalizing Green function must not remember nor anticipate the full evolution, the full conformal time τ, but may only depend on the present electromagnetic response and its first and second derivative. Renormalization is local — in our case, local in time.

Causality implies that the dielectric environment of the renormalizing Green function depends on the earlier of its two times. Causality combined with locality, Eqs. (41) and (43), creates a discontinuity in the third derivative of the parameter (46) that prevents the application of the standard fluctuation–dissipation theorem (Sec. II B). We can no longer directly infer the correlations from the fluctuations, but need to transform time on the complex plane (Sec. III C). The resulting energy density tracks the time evolution, Eq. (64), and becomes significant for nearly dispersionless media.

Such a dispersionless medium with equal electric and magnetic response is the “medium” of space in cosmology, because flat space expanding in time with scale factor a(t) is equivalent to the medium of Eq. (1) with refractive index n equal to a (Appendix A). According to the equivalence principle, this medium is dispersionless until the Planck scale of Eq. (4).

Thermodynamics or, equivalently, energy–momentum conservation adds another twist to the vacuum energy: a trace anomaly. The renormalized energy–momentum tensor of the electromagnetic fluctuations is no longer traceless, but contains an additional component of energy density and pressure with the pressure equal to minus the energy density. This is the equation of state of the cosmological constant, but note that this component is not necessarily constant. The total tracks the time derivative of the Hubble parameter, Eq. (69), while Einstein’s cosmological constant emerges as an integration constant.

Cosmic expansion, or equivalently, the time-dependence of the medium, Eq. (1), generates vacuum energies and pressures that then, due to their gravity, act back on the cosmic evolution. In the relativistic Gauss’ law of gravity, Eq. (71), the negative pressure causes a repulsive contribution that may overwhelm the gravitational attraction and accelerate the cosmic expansion, as has been observed. Although the vacuum responds as a whole to the various cosmic constituents, and it is their combined gravity that influences the evolution, we found a remarkable simplification: each constituent behaves as if its weight is reduced by a characteristic buoyancy factor, Eq. (80).

The quantum buoyancy of the vacuum modifies the cosmic evolution, Eq. (86), such that the Hubble constant — the Hubble parameter (5) at the present time — deviates from its projection without buoyancy by as much as 10%, in agreement with astronomical measurements. Our theory does not definitely resolve the Hubble tension, as it relies on an unknown coupling parameter that depends on the dispersion length ℓ, but for ℓ in the order of twice the Planck length we match the measurements. The physics on the Planck scale is unknown. However, it is reasonable to assume that a potential discreteness of space at twice the Planck length appears as dispersion for lengths exceeding that scale (in analogy to the dispersion of phonons in a crystal lattice). The dispersion of light propagation near the Planck scale does also resolve the trans-Planckian problem of Hawking radiation which has been the focus of many analogues of gravity in experiments.

We have exclusively focused on the electromagnetic field, as all the experimental evidence for vacuum forces comes from quantum electromagnetism. Tests of our specific theory — other than in astronomy — could perhaps be done with quantum fluctuations of the electromagnetic field as well. Although condition (1) of equal electric and magnetic response is difficult to achieve in the laboratory, one might probably observe similar effects from a time-dependent electric and constant magnetic response, in addition to the particle creation known and measured as the dynamical Casimir effect.

Which other fields could contribute to the vacuum energy in space and how would they affect it? Fermions polarize the vacuum due to virtual particle–antiparticle pairs, and their polarization energy might contribute to the cosmological constant. The fields of other elementary bosons — except the Higgs boson — are gauge fields, constructed from the same recipe as the electromagnetic field, but with mass (for the weak interaction) or nonlinearity (important for the strong interaction). In case they also contribute to the cosmological vacuum energy, one may trivially include them in our coupling parameter κ, simply by multiplying κ by the number of fields and enlarging the cut-off by the square root of that number. The κ parameter contains some information on the physics near the Planck scale, but not all, and in practice, κ should be fitted to astronomical data.

If our theory of quantum noise in time-dependent media, developed in condensed matter physics, is indeed applicable to cosmology, it would relate the physics near the smallest of scales with observations on the largest scale — the Planck scale written on the stars.

Acknowledgements

We are grateful for discussions with Ofer Aharony, Viktar Asadchy, Dror Berechya, Michael Berry, Kfir Blum, Nikolay Ebel, Eren Erkul, Mathias Fink, Uwe Fischer, Helmut Hörner, Jonathan Kogman, Amaury Micheli, Lukas Rachbauer, Scott Robertson, Stefan Rotter, William Simpson, Alexandre Tkatchenko, Grisha Volovik, Robert Wald, Eli Waxman, Chris Westbrook, and Anton Zeilinger. Our paper was supported by the Murray B. Koffler Professorial Chair of the Weizmann Institute of Science.

(Appendix A, a brief excursion into general relativity showing where the “medium of space” comes from, together with the reference list, is omitted for length; the complete text is at the source.)