Quantum origin of dark energy and the Hubble tension

Local measurements of the Hubble parameter obtained from the distance ladder at low redshift are in tension with global values inferred from cosmological standard rulers. A key role in the tension is played by the assumptions on the cosmological history, in particular on the origin of dark energy. Here we consider a scenario where dark energy originates from the amplification of quantum fluctuations of a light field in inflation. We show that spatial correlations inherited from inflationary quantum fluctuations can reduce the Hubble tension down to one standard deviation, thus relieving the problem with respect to the standard cosmological model. Upcoming missions, like Euclid, will be able to test the predictions of models in this class.

for estimation of H 0 from BAO with a big bang nucleosynthesis prior on baryon density). It is also possible to study the variation in time of the Hubble parameter up to redshift z ∼ 2 using the so-called cosmic chronometers [4,5], which are galaxies with known time evolution.
Local measurements comprehend distances of SNeIa calibrated with Cepheids by the SH0ES collaboration [6] and time delays of multiple images of strongly lensed quasars studied by H0LiCOW [7]. Among other local probes which have been considered, one can mention calibrating SNeIa with the tip of the red giant branch (TRGB) (finding a lower H 0 compared to Cepheids calibration, see [8]) or with Mira variables [9], measures of distances from surface brightness fluctuations [10] and observations of galaxies hosting megamasers [11].
Several ideas have been proposed in order to reconcile these two values. A possible approach consists in introducing new physics such that the Hubble parameter inferred from CMB is larger. Such modifications can affect the cosmological evolution at late times or even the early Universe. As an example of late-time proposal, it has been shown that a phantom dark energy equation of state shifts the Hubble parameter predicted from CMB towards higher values [12]. Other studies consider possible late-time solutions coming from interacting dark energy [13], metastable dark energy [14] or decaying dark matter [15]. Indeed these scenarios give higher values of H 0 when fitting CMB data only, but in general the other independent H 0 measurements coming from BAO and cosmic chronometers leave little room for such late-times modifications, see [16,17]. As for early-Universe proposals, an example is early dark energy [18,19] where a scalar field dilutes faster than matter starting from an initial state frozen by Hubble friction. Apart from fine-tuning difficulties (e.g. in the required potential of the scalar field, in its initial conditions or in making early dark energy appear and dilute at the right times), early dark energy still has troubles related to the σ 8 tension between large-scale structures and CMB [20]. Further early-Universe proposals try to reduce the sound speed of the primordial photon-baryon plasma before recombination or attempt to make recombination occur earlier [21]. While all these ideas deal with changing the prediction of H 0 from CMB, other approaches focus instead on local measurements by studying how the SNeIa calibration could be affected by new physics, like chameleon dark energy [22] or late-time changes in the effective Newton constant [23].
As an alternative line of thought, in this paper we want to explore the scenario where latetime dark energy originates from quantum fluctuations in the early Universe and we start a study of the consequences expected on the spatial correlations of dark energy today. The analysis will allow us to make quite general predictions on the relief of the Hubble tension within this framework. In such a scenario the Hubble parameter is a stochastic variable and the stochasticity source comes not only from (dark) matter fluctuations due to classical inhomogeneities, but also from the intrinsic quantum nature of dark energy. At time t and for every comoving coordinate x the effect of quantum dark energy is such that the expansion rate at that point is an operator satisfying the first Friedmann equation where M P ≡ (8πG) −1/2 is the reduced Planck mass, ρ C (t) is the classical contribution to energy density,1 1 is the identity operator andρ Q (t, x) is the quantum energy density operator 1 , which can depend onĤ 2 (t, x).
A natural hypothesis for the form of the space dependence of dark-energy correlators is a power law behavior, which is in fact realised in the simple model that we discuss below.
In order to establish whether a dark energy of quantum origin could be enough to relieve the Hubble tension, we evaluate the conditional probability for local measurements (averaged over a spatial volume V 1 ) of the squared Hubble parameter to find a value above a threshold H 2 1 , given that global measurements (averaged over a larger volume V 2 containing V 1 ) find a value below H 2 2 . The square of the Hubble parameter is a natural fundamental variable to consider, as it appears in the Friedmann equation (1). We write the aforementioned conditional probability as where and similarly for V 2 . 2 The notation t 0 indicates cosmological time today. 1 In general, only a part of the full quantum energy density operator behaves as dark energy and other terms can act as additional matter contents. 2 More precisely, we are implicitly defining a classical stochastic variable H 2 (t 0 , x), whose statistical properties are the same as those of the quantum operatorĤ 2 (t 0 , x).
The values of H 1 and H 2 are chosen as the mean values of the results from SH0ES and Planck mentioned before: H 1 = 74.03 km/s Mpc −1 , H 2 = 67.4 km/s Mpc −1 . For the volumes V 1 and V 2 , we assume spheres centered around the observer with radii R 1 and R 2 that we specify below.

II. THE MODEL
Let us consider a light spectator field during inflation that, for the sake of simplicity, will be taken as a scalar Φ. The key ingredient of the class of theories that we want to study is the assumption that some mechanism determines the effective mass M 2 (t) of the spectator field in a cosmological background in such a way that, by the end of inflation, correlators reach super-Planckian values and that these correlators are then translated into an effective dark energy at late times. A simple realization of such a mechanism is discussed in [24] (see also [25][26][27]) and consists in a non-minimally coupled massive scalar field with action where R is the Ricci curvature scalar of the metric g µν and g = det[g µν ]. It is clear that here the effective mass of the scalar field, M 2 = m 2 + ξR, receives contributions from the bare mass m and from a non-minimal coupling to the metric with coefficient ξ. As predicted in [24], in this model the field correlators of a light scalar grow during inflation and their enhancement is larger when the non-minimal coupling is negative and it dominates over the bare mass. Other mechanisms contributing to an effective mass are conceivable, like dynamical mass generation via interaction with another field acquiring a non-zero vacuum expectation value in a symmetry-breaking scenario. In the general case the effective mass is a field operator with a non-trivial derivative structure.
Notice that assuming a quadratic action for the spectator field at the classical level ensures a minimal amount of non-Gaussianity. However, at the level of quantum effective action, interactions with the metric perturbations (or in general with other fields) introduce additional non-Gaussianities in the evolution of the spectator scalar field, e.g. by generating effective self-interactions.
We consider a spatially flat FLRW Universe in D = 4 spacetime dimensions with scale factor a(t), i.e. ds 2 = −dt 2 + a 2 (t)d x 2 , where t is cosmological time and x are comoving coordinates.
The spectator scalar field is Φ(x) = Φ(t, x) and its action, computed by specializing (3) to the FLRW metric, is of the form where the dot (˙) denotes the derivative with respect to cosmological time d/dt and we intro- After introducing the canonical momentum Π(x) = a 3Φ (x), quantization of the model goes in the standard way (see e.g. [24]) by promoting field and momentum to operatorsΦ andΠ obeying canonical commutation relations.
The effect of the scalar field on cosmological evolution is determined by its energy-momentum /δg µν , which in its operator form reads, where G µν is the Einstein curvature tensor.
Friedmann equations tell us that quantum fluctuations of the spectator field enter the expansion history of the Universe through expectation values of energy density ρ Q ≡ ρ Q ≡ − T 0 0 and pressure p Q δ i j ≡ T i j , evaluated on a homogeneous and isotropic state that we take to be the vacuum |Ω . Neglecting spatial gradients of the field, the energy density operator from (5) simplifies tô In [24], the relevant expectation values for Friedmann equations are coincident correlators, The result is that quantum fluctuations of the spectator field behave at late times as an effective dark energy.
Here we want to investigate the consequences of the spatial correlations inherited by dark energy. Therefore we turn our attention to 4-point functions built from field and momentum operators which are relevant for density-density correlators.

A. Stochastic formalism
The stochastic formalism of Alexey Starobinsky makes use of the fact that on super-Hubble scales non-conformal quantum fields (Φ,Π) grow during inflation, such that their correlators can be accurately described by their classical stochastic equivalents (φ,π). These infrared fields obey, wheref φ (t, x) andf π (t, x) are stochastic sources, which arise because of coupling between the infrared and ultraviolet modes (see e.g. [24] for the precise definition of stochastic sources).

B. IR correlators
We can now proceed to extend the treatment of [24] to study spatial correlations of long modes. Let us recall that, in eqs. (30-32) of that paper, three 2-point functions were introduced because they appear in the expectation value of the energy-momentum tensor (conveniently rescaled with a power of a 3 (t)H(t) for each occurrence of a momentum op- Here we are interested in spatial fluctuations, therefore the relevant quantity is the density-density correlator at two different comoving coordinates , where the energy density operator isρ Q = −T 0 0 . Since eachT 0 0 is quadratic in field/momentum operators, we infer that ρ(t, x 1 )ρ(t, x 2 ) depends on 4-point functions of fields/momenta. Due to the homogeneity and isotropy of the FLRW background, comoving coordinates x 1 and x 2 only appear in correlators through their relative comoving distance r ≡ x 2 − x 1 .
The relevant correlators are six symmetric combinations (see [28] We can study the time evolution of 4-point functions and identify their stochastic noise sources induced by the stochastic forcesf φ (t, x) andf π (t, x) of (7-8), to obtain a system of six coupled differential equations. Here we give two equations, where we have traded the time t for the number of e-foldings N ≡ ln a(t), and the source in Eq. (10) is, The complete set of equations and sources is found in [28]. The stochastic sources can be computed in terms of the mode function ϕ(t, k) ofΦ. The source in (12) is thus, where j 0 (kr) ≡ sin(kr)/(kr) is the spherical Bessel function and µaH (µ 1) is an UV cut-off.

C. Time evolution of correlators
Analogously as it was done in [24], one can solve the equations of motion for the correlators (10-11) from an early inflation, through radiation and matter era. Even though the correct evolution of the correlators in matter era requires taking account of the dark energy backreaction, for simplicty here we model the late time universe by non-relativistic matter and cosmological constant. This introduces a few percent error [24], but has the advantage that one can solve the evolution of the relevant four-point functions exactly. The result is [28] ∆ (4) where N 0 = ln Ω M Ω R 8.1 e-foldings since matter-radiation equality, and s(r) contains spatial dependence, which is well approximated by, where a in is the scale factor at the beginning of inflation. Therefore the comoving length scale ruling spatial variations is the quantity (µa in H I ) −1 , i.e. the comoving Hubble horizon at the beginning of inflation (up to the µ −1 factor). Furthermore, Ω R , Ω M and Ω Λ 1 − Ω M are the energy density fractions in radiation and matter today, taken to be Ω R = 9.1 · 10 −5 and Ω M = 0.3.
The quantum backreaction of the scalar field contributes to energy density and pressure during matter domination as in [24], Matching the cosmological constant-like term to its value today Ω Λ determines the number of inflationary e-foldings N I in terms of the model parameters m and |ξ|, and ξ < 0 .
We also need m/H DE < 1 so that the scalar field stays light throughout the cosmological expansion.
The spatial dependence in (14) is the one inherited from inflation through the function s(r) in (16). Before discussing the Hubble tension problem, it is useful to work on the comoving length scale (µa in H I ) −1 associated to s(r) in order to relate its physical size today to the model parameters. Assuming an instantaneous reheating after inflation, we find the following ratio between the physical length scale of spatial variations today and the current Hubble horizon: Spatial variations are relevant for the Hubble tension problem discussed in section (III) only if they are significant at sub-Hubble scales. Therefore, we require that the ratio appearing in the left-hand side of (20) is at most of order one, which then limits the number of inflationary e-foldings. For H I 10 13 GeV, H 0 10 −33 eV (and µ not too small), this gives N I 60.
The condition on N I translates into a requirement on the parameters of the model, because N I is determined by |ξ| and m via the combination in (18).

III. RELIEVING THE HUBBLE TENSION
Note that the local expansion rate operatorĤ 2 (t, x) in (1) can be recast as, where we made use ofρ Q from (6) and neglected contributions from of {φ,π} andπ 2 , which is a reasonable assumption [28]. Next, we assume that H 0 H 2 = 67.4 km/s Mpc −1 , meaning that the statistical average value H 0 is estimated very well by the result of global measurements.
The translation of a probability statement forĤ 2 like that in (2) into an equivalent statement forφ 2 depends on whether these two quantities are related by a growing or decreasing function. A simple analysis of (21) shows that d( In what follows, we introduce some simplifications, whose accuracy is discussed in detail in Ref. [28]. Firstly, we shall assume that the fieldψ(t 0 , x) ≡φ 2 (t 0 , x) is a classical stochastic variable ψ(t 0 , x) which is Gaussian distributed. Secondly, we shall assume that the stochastic properties of ψ can be modeled by a Gaussian probability distribution, where the position-independent mean value is ψ 0 = ∆ φφ (t 0 ), while the covariances are As for the domain of definition, the probability distribution P [ψ] is assumed to be normalized in the global volume V 2 (which includes V 1 ).
Using the usual relation between conditional probabilities and joint probabilities, the problem of finding (2) amounts to study the joint probability distribution of the mean An important quantity is the correlation coefficient ρ defined as ρ = M 12 / √ M 11 M 22 . We can evaluate the covariance matrix M and the correlation coeffcient ρ by using for the volumes V 1 and V 2 two spheres of radii R 1 and R 2 centered around the observer. The conditional probability (2) can be reduced to, is Owen's T function and where I 11 = 2 0 dx 3 16 x 5 − 9 4 x 3 + 3x 2 s 1 (x) and For definiteness we choose the radius R 2 of the global volume such that µR 2 = H −1 0 4400 Mpc and the radius R 1 of the local volume such that µR 1 = 100 Mpc (we set µ = 1).
The conditional probability (24) depends on the two parameters of the model ξ and m.
Solid lines in Figure 1 show its behavior as a function of |ξ| for constant values of the ratio Notice that the condition (19) imposes to choose α > 1. Furthermore the condition of light field m/H DE < 1 means that, for a given constant α, only values of |ξ| with |ξ| < 1/(6α) have to be considered (this is automatically true for the values of α and the plot range chosen in Figure 1). The number of inflationary e-foldings throughout the range of parameters in Figure 1 is of order N I ∼ 60 and is therefore compatible with the approximate constraint N I 60. Figure 2 shows the corresponding Hubble tension zooming slightly on a more central region.
As can be seen from the plot, the model is able to reduce the tension down to almost 1σ which is a great improvement with respect to the > 4σ tension in ΛCDM. This has been possible thanks to spatial fluctuations of dark energy inherited from inflation.
In the limit of zero local volume one gets the largest conditional probability that can be reached, which can be obtained from (24) to be 1 2 − 1 π arctan 1 √ 2 30.4%, corresponding to a "tension" of only 1.03σ. This is in agreement with the solid lines in Figures 1-2, where the ratio R 1 /R 2 0.02 is small but finite. To mimic H0LiCOW measurements [7] we  Rubin Observatory. In [29] the authors test the same quantum dark energy model against cosmological data and compare its performance to ΛCDM, finding that the model is slightly favored, although not at a statistically significant level. We are currently investigating the form of luminosity distance correlators for type Ia supernovae induced by spatially correlated dark energy, which provide a test for deviations from ΛCDM. This is just an example of physical observables that could be used to test the features predicted by the model. It is also worth mentioning that, within a different approach using cosmological perturbations, dark energy with very large-scale inhomogeneities has been considered in [30] and their amplitude is constrained with CMB anisotropies data. and the source vector n(N, r) can be evaluated from the mode function. For 4 ξ < 0 and (m/H I ) 2 |ξ| 1, working at the lowest order in the non-minimal coupling ξ, we get (recalling that the scale factor is a(N ) = e N ) n(N, r) = H I 2π The differential problem (A3) is conveniently solved by diagonalizing the matrix A. Denoting by λ j the eigenvalues of A, by B the change-of-basis matrix (matrix whose columns are the eigenvectors of A) and by B −1 its inverse, the i-th component of the correlators vector ∆ (4) (N, r) is given by The integral can be evaluated by approximating the spherical Bessel function j 0 (x) = sin(x)/x appearing in (A5) as j 0 (x) θ(1 − x) where θ is the Heaviside step function.
Furthermore, in the limit of small X it can be shown that the sum over eigenvalues in (A6) is dominated by the smallest eigenvalue of A which is 6 − 2 With these simplifications one finds that, at the end of inflation (lasting N I e-foldings), the 4-pt correlators are where the function s(r) encoding spatial dependence was defined in (16).

Radiation epoch
After the end of inflation, the correlators will evolve in radiation era starting from the initial conditions (A7) inherited from inflation. Similarly to the case of 2-pt functions studied in [24], we can safely neglect stochastic sources after inflation because their contribution will be irrelevant compared to that of initial conditions. This is true due to the enhancement factor e 16|ξ|N I /ξ 2 , which is present in (A7) but not in the stochastic sources of radiation epoch.
In radiation epoch ( = 2) the Ricci scalar is null and we can also neglect (m/H(t)) 2 (due to the assumption of very light scalar field, this term only becomes relevant at recent cosmological times, when matter and cosmological constant dominate the evolution of the Universe). This leads to a simplified evolution during radiation era, of the form where A is the constant matrix If N = 0 is the beginning of radiation epoch, then the system is easily solved as ∆ (4) is the initial condition taken from the end of inflation (A7).
The exponential of the matrix −A is computed by diagonalizing A. At leading order in |ξ| we find that, by the end of radiation epoch lasting N R ≈ 50 e-foldings, Since N R 1, we can say that, up to contributions suppressed by powers of e −N R , at the end of radiation epoch This is the initial condition that will be used in the final stage of cosmological evolution after matter-radiation equality. We remark that the spatial dependence is still that imprinted by inflation. This is a consequence of the smallness of stochastic sources after inflation, so that the post-inflationary evolution does not affect the form of s(r).

Matter + cosmological constant epoch
After matter-radiation equality, non-relativistic matter becomes the dominant component of the Universe leading its expansion. We know from eq. (90) of [24] that in matter period, the 2-pt functions of the scalar field evolve in such a way to develop a dark energy component (more precisely they have a matter-like contribution and a CC-like contribution).
We are interested here in 4-pt functions, that should be evolved numerically in a background containing the backreaction of the scalar field. However we can obtain a simplified analytical treatment by approximating the background evolution as an exact matter+CC Universe.
We set N = 0 at matter-radiation equality and the current (today) time corresponds to N 0 = ln Ω M Ω R where H DE was already defined right after (18) while B 1 , B 2 and B 3 are the following constant matrices The exact solution of (A14) is where T exp denotes the time-ordered exponential, which is there because B(x) matrices at different x's do not commute. However we checked numerically that neglecting the time ordering gives accurate results at a few percent level, with the advantage of displaying simpler formulas. 5 Then the integrals can be solved analytically and the matrix exponential can be also evaluated analytically with a symbolic computation software.