Early dark energy from zero-point quantum fluctuations

We examine a cosmological model with a dark energy density of the form $\rho_{DE}(t)=\rho_X(t)+\rho_Z(t)$, where $\rho_X$ is the component that accelerates the Hubble expansion at late times and $\rho_Z(t)$ is an extra contribution proportional to $H^2(t)$. This form of $\rho_Z(t)$ follows from the recent proposal that the contribution of zero-point fluctuations of quantum fields to the total energy density should be computed by subtracting the Minkowski-space result from that computed in the FRW space-time. We discuss theoretical arguments that support this subtraction. By definition, this eliminates the quartic divergence in the vacuum energy density responsible for the cosmological constant problem. We show that the remaining quadratic divergence can be reabsorbed into a redefinition of Newton's constant only under the assumption that the energy-momentum tensor of vacuum fluctuations is conserved in isolation. However, in the presence of an ultra-light scalar field $X$ with $m_X<H_0$, as typical of some dark energy models, the gravity effective action depends both on the gravitational field and on the $X$ field. In this case general covariance only requires the conservation of the total energy-momentum tensor, including both the classical term $T^X_{\mu\nu}$ and the vacuum expectation value of T_{\mu\nu}. If there is an exchange of energy between these two terms, there are potentially observable consequences. We construct an explicit model with an interaction between $\rho_X$ and $\rho_Z$ and we show that the total dark energy density $\rho_{DE}(t)=\rho_X(t)+\rho_Z(t)$ always remains a finite fraction of the critical density at any time, providing a specific model of early dark energy. We discuss the implication of this result for the coincidence problem and we estimate the model parameters by means of a full likelihood analysis using current CMB, SNe Ia and BAO data.


Introduction
Understanding the origin of dark energy is one of the most important challenges facing cosmology and theoretical physics (see e.g. [1][2][3][4]). One aspect of the problem is to understand what is the role of zero-point vacuum fluctuations in cosmology. In a Friedmann-Robertson-Walker (FRW) metric with Hubble parameter H(t) the bare vacuum energy density takes the form where [ρ bare (Λ c )] Mink is the bare vacuum energy density in Minkowski space, whose leading divergence is O(Λ 4 c ), and we used for definiteness a momentum space cutoff Λ c . In the usual treatment this Λ 4 c divergence is reabsorbed into a renormalization of the cosmological constant, giving rise to the cosmological constant problem. The divergence ∝ H 2 Λ 2 c is instead absorbed into a renormalization of Newton's constant G [5,6].
In this paper, expanding on results presented in [7], we reexamine the role of vacuum energies in cosmol-ogy. First, we will propose theoretical arguments suggesting that the correct way of computing the physical vacuum energy is to subtract the bare vacuum energy density of Minkowski space, [ρ bare (Λ c )] Mink , from the FRW result given in eq. (1), before renormalizing the result. By definition this subtraction eliminates the troublesome Λ 4 c divergence and, therefore, the cosmological constant problem. Then we turn our attention to the left over term H 2 Λ 2 c which now becomes the leading term in the vacuum energy. It is usually believed that this quadratic divergence can be reabsorbed into a renormalization of G. We show that this is correct only under the assumption that vacuum expectation value (VEV) of the energy-momentum tensor is conserved in isolation (an assumption that was implicit in the literature). However, general covariance of General Relativity (GR) only implies the conservation of the total energy-momentum tensor T µν + 0|T µν |0 , including both the classical term T µν and the semiclassical term 0|T µν |0 . The separate conservation of 0|T µν |0 only takes place if we can define an effective action which depends only on the grav-itational field, by integrating out the matter degrees of freedom. This is possible only if the matter degrees of freedom are heavy with respect to the energy scale of the problem, and can then be integrated out. In a cosmological setting, this means that matter fields should satisfy m > H 0 . If, in contrast, there is an ultra-light scalar field with m < H 0 , as is typical for dark energy models such as quintessence, this field cannot be integrated out from the effective low energy action. We show that, as a result, in general ∇ µ T X µν = −∇ µ 0|T µν |0 0. In this case the effect of the quadratically divergent term in the vacuum fluctuations cannot simply be absorbed into a renormalization of Newton's constant G, and gives rise to interesting and potentially detectable cosmological effects. We construct a specific coupled early dark energy model and test it against current observations. We use natural units where = c = 1, G = M −2 Pl . If not specified otherwise, we work in a spatially flat FRW metric with signature (−+++), cosmic time t, scale factor a(t) and Hubble parameter H(t) = (da/dt)/a. Today, the Hubble parameter and the critical density take the values H 0 and ρ 0 = 3H 2 0 /(8πG), respectively.

Subtraction of the flat-space vacuum energy
In Minkowski space the divergence in the vacuum energy density is usually dealt with by normal ordering the Hamiltonian, which gives by definition a vanishing result for the physical vacuum energy density. However, it is useful to realize that the problem can be treated more generally in the context of renormalization theory, which rather allows us to fix the renormalized vacuum energy density to any observed value. In the standard language of renormalization, divergences in a generic N-point Green's function are cured by adding the corresponding counterterms to the Lagrangian density. The same procedure can be applied to vacuum energy, i.e. to the N = 0 Green's function: one simply adds a constant counterterm −ρ count (Λ c ) to the Lagrangian density. This corresponds to adding a term +ρ count (Λ c ) to the Hamiltonian density. Hence the renormalized, physical vacuum energy density is given by ρ ren = ρ bare (Λ c ) + ρ count (Λ c ). As always in renormalization theory, the counterterm ρ count is chosen so to cancel the divergences in ρ bare and leave us with the desired finite part that is fixed by comparison with the experiment.
Using the language of renormalization theory is useful in this context because it makes clear that the cosmological constant problem is not that quantum field theory (QFT) gives a wrong prediction for the cosmological constant (as it is sometimes incorrectly said). Strictly speaking QFT makes no prediction for the cosmological constant, just as it does not predict the electron mass nor the fine structure constant. Rather, it is a problem of naturalness, in the sense that the counterterm ρ count (Λ c ) must be fine-tuned to exceeding accuracy, in order to cancel the Λ 4 c divergence in ρ bare , leaving a physical vacuum energy density that, if one identifies Λ c with the Planck mass, is about O(10 120 ) times smaller than Λ 4 c . Posing the problem in terms of a cancellation between ρ bare (Λ c ) and ρ count (Λ c ) can also give a first hint for a possible solution. First of all, one should appreciate that neither the bare vacuum energy ρ bare (Λ c ) nor the counterterm ρ count (Λ c ) have a physical meaning and only their sum is an observable. Thus, this kind of cancellation is different from a fine-tuning between observable quantities. Indeed, the Casimir effect is a well-known example where a rather similar cancellation takes place. In that case the physical vacuum energy density of a quantum field in a finite volume is found by taking the difference between the bare vacuum energy density computed in this finite volume and the bare vacuum energy density in an infinite volume. Regularizing with a cutoff Λ c both terms diverge as Λ 4 c , but their difference is finite and depends only on the physical size of the system. This might suggest that, similarly, to obtain the physical effect of the vacuum energy density in cosmology, one should compute the vacuum energy density in a FRW space-time and subtract from it the value computed in a reference geometry, which could be naturally taken as Minkowski space, leading to a sort of "cosmological Casimir effect".
Before taking this analogy with the Casimir effect seriously, one must however face the obvious objection that in special relativity the zero of the energy can be chosen arbitrarily, and only energy differences with respect to the ground state are relevant. 1 In contrast, in GR we cannot chose the zero of the energy arbitrarily. One typically expects that "every form of energy gravitates", so the contribution of Minkowski space cannot just be dropped.
While it is certainly true that in GR the choice of the zero for the energy is not arbitrary, the point that we wish to make here is that what is the correct choice can be a non-trivial issue. As a first example, consider the definition of energy for asymptotically flat spacetimes. This is obtained from the Hamiltonian formulation of GR, which goes back to the classic paper by Arnowitt, Deser and Misner (ADM) [8,9]. To properly define the Hamiltonian of a given field configuration in GR one must work at first in a finite threedimensional volume V. Then the Hamiltonian takes the form H GR = H bulk + H boundary , where H bulk is given by an integral over the spatial volume V at fixed time, while H boundary is given by an integral over the twodimensional boundary ∂V. When one evaluates H bulk on any classical solution of the equations of motion one finds a vanishing result (since H bulk is proportional to the constraint equations of GR), so the whole contribution comes from the boundary term. On the other hand, H boundary diverges for any asymptotically flat metric g µν (including flat space-time), when the boundary is finally taken to infinity. The solution proposed by ADM is to subtract from this boundary term the same term computed in Minkowski space η µν . Accordingly, the energy E associated with a classical asymptotically flat metric g µν is obtained by defining This provides the standard definition of mass in GR, and reproduces the expected properties of asymptotically flat space-times. For instance, when applied to the Schwarzschild space-time, it correctly gives the mass that appears in the Schwarzschild metric. This underlines that our intuition that any form of energy gravitates according to GR is not entirely correct: eq. (2) tells us that the energy associated to Minkowski space does not gravitate. Similar subtractions also hold for non-asymptotically flat space-times, and can be performed either by subtracting the contribution of some reference space-time whose boundary has the same induced metric as the background under consideration [10][11][12][13], or even without introducing a reference background but just by adding some local counterterms to the boundary action, given by a coordinate-invariant functional of the intrinsic boundary geometry [14,15]. The latter prescription is particularly appealing for asymptotically AdS spacetimes. In fact, in the context of the AdS/CFT correspondence, this way of removing divergences in the gravitational action on the AdS side corresponds to the renormalization of the UV divergences in the conformal QFT that lives on the boundary [14][15][16][17][18].
These examples show that, already in classical GR, the energy that actually acts as a source of gravity can be obtained from a Hamiltonian only after performing an appropriate subtraction. It is quite natural to assume that the same should hold at the quantum level, so in particular for zero-point fluctuations of quantum fields in curved space. To understand what is the appropriate subtraction for the FRW metric, we consider the Friedmann equation that results from the Einstein equations sourced by T µ ν + 0|T µ ν |0 , where T µ ν = diag(−ρ, p, p, p) is the ordinary contribution of matter, radiation, etc, and 0|T µ ν |0 = diag(−ρ vac , p vac , p vac , p vac ) is the corresponding contribution of zero-point fluctuations: We then require that Minkowski space, H(t) = 0, should be a solution in the limit ρ → 0. This implies that all terms in [ρ vac ] FRW that do not vanish for H → 0 must be subtracted. In other words, we must subtract the vacuum energy computed in Minkowski space. 2 In contrast, all terms proportional to H 2 or H 4 are consistent with this requirement and thus generically allowed. This procedure eliminates the Λ 4 c term (as well as flatspace terms that appear for massive fields, such as m 2 Λ 2 c and m 4 ln Λ c , and are also much larger than the observed vacuum energy, for all known massive particles). Thus, this treatment of vacuum fluctuations solves the cosmological constant problem by definition, at least in its "old" form, i.e. it explains why the observed vacuum energy is many orders of magnitude smaller than M 4 Pl . We next turn to the quadratic divergence left over after the subtraction. We will see that it could teach us something about the more recent form of the cosmological constant problem, namely explaining why the observed dark energy (DE) density is just of the order of the critical density of the universe today, the coincidence problem.

Non-interacting zero-point fluctuations and renormalization of G
Let us now discuss the fate of the quadratic divergence ∝ Λ 2 c H 2 (t) in eq. (1). To understand what is the structure of the renormalized VEV of the energymomentum tensor, consider the semiclassical Einstein equations for the renormalized quantities, in which the vacuum expectation value of T µν is added as an additional source term. Together with the Bianchi identities the above equation implies This equation is therefore a consequence of the general covariance of the renormalized theory. 3 If T µν and 0|T µν |0 are separately conserved we further have The stronger condition ∇ µ 0|T µν |0 = 0 can indeed be derived by using the effective action for gravity, which is obtained by treating the metric g µν as a classical background and integrating out the matter degrees of freedom (see e.g. refs. [21][22][23]). The VEV of the renormalized energy-momentum tensor is then obtained by taking the functional derivative (2/ √ −g)δ/δg µν of the effective action. One can perform the calculation of the effective action using regularizations, such as dimensional regularization or point-splitting, that preserve general covariance explicitly. The effective action is therefore explicitly generally covariant, and the VEV of the energy-momentum tensor derived from it is automatically covariantly conserved. In this case, all divergences in 0|T µν |0 can be reabsorbed into generally covariant counterterms in the effective action. In particular, the divergence ∝ H 2 (t)Λ 2 c is cured by a counterterm proportional to the Einstein-Hilbert action. In fact, taking the variation of the Einstein-Hilbert term gives the Einstein tensor G µν , and G 00 , specialized to the FRW metric, is proportional to H 2 (t). This shows that a VEV 0|T 00 |0 ∝ H 2 (t) is obtained, in the effective action language, by an additional term proportional to d 4 x √ −g R, and therefore is reabsorbed into a renormalization of Newton's constant, as it is well known [5,6]. Such an effective action approach, however, assumes that we can integrate out all matter fields, i.e. that they are sufficiently massive with respect to the scale of interest. In our cosmological context this means that we are implicitly assuming that all fields have a mass m bigger than the Hubble parameter H(t) at the time of interest, which is the quantity that fixes the relevant scale. Equivalently, we are assuming that the wavelength 1/m is smaller than the horizon size H −1 (t). It is however interesting to consider the case in which in the spectrum there is a scalar particle X with a mass m X < H 0 . Such an ultra-light scalar field, with the mass protected against radiative correction by demanding that it is a pseudo Nambu-Goldstone boson, provides in fact a typical realization of quintessence [24]. In this case we cannot integrate out this field, and the low-energy effective action necessarily depends both on the metric and on this scalar field. Then the above derivation giving ∇ µ 0|T µν |0 = 0 no longer goes through. By taking the variation of this action with respect to the metric we get the total energy-momentum tensor, including both 0|T µν |0 and the energy-momentum tensor T X µν of this scalar field, and general covariance now only implies ∇ µ ( 0|T µν |0 + T X µν ) = 0. Whether these two terms are separately conserved is now a dynamical question, and depends on whether there is an interaction among them. One can imagine mechanisms by which vacuum fluctuations can exchange energy with other forms of matter. Typical examples are the amplification of vacuum fluctuations [25,26], or the change in a large-scale scalar field due to the continuous horizon-crossing of small-scale quantum fluctuations of the same scalar field, which is also at the basis of stochastic inflation [27]. If ∇ µ 0|T µν |0 = −∇ µ T X µν 0, it is no longer possible to reabsorbe the effect of 0|T µν |0 into a renormalization of the Einstein-Hilbert term, nor of any other generally covariant local operator in the effective action. In fact, taking the functional derivative (2/ √ −g)δ/δg µν of a generally covariant term, we necessarily obtain a covariantly conserved tensor, so we can never obtain a quantity 0|T µν |0 that satisfies ∇ µ 0|T µν |0 0.

Cosmology with zero-point fluctuations
We now explore the cosmological consequences of the hypothesis that 0|T µν |0 and T µν are conserved in conjunction, as in eq. (5), but not separately. In this case, as shown above, the term Λ 2 c H 2 cannot be reabsorbed into a renormalization of G, and we rather expect that it will give a genuine contribution to the total energy density ρ Z (t) = O(H 2 (t)M 2 ), where M is the UV scale where new physics sets in (so that M could be typically given by the Planck mass M Pl , or by the string mass). Recalling that the critical density is ρ c (t) = 3H 2 (t)M 2 Pl /(8π) we see that, for M of order M Pl , ρ Z (t) is of order of the critical density ρ c (t) at any time t. Thus we write where ρ 0 is the present value of the critical density. The value of Ω Z is fixed by the renormalization procedure to the observed value, so we keep it as a free parameter. The same is true for the equation of state (EOS) parameter w Z defined by p Z = w Z ρ Z , that can in principle be a function of time (see [20] for an extended discussion of this point).
The late time acceleration of the Universe cannot be explained only by a DE density that scales like H 2 (t) [7,28,29], basically because observations tell us that the total DE density is at least approximately constant in the recent cosmological epoch. Therefore we assume ρ Z (t) only to provide a part of the total DE density that we write as ρ DE (t) = ρ X (t) + ρ Z (t). Here ρ X (t) is a second dynamical DE component that is the one dominating the energy budget at the current epoch. In this approach the physical origin of ρ X and ρ Z can a priori be completely different. For instance, ρ X could be due to a scalar field, as in quintessence models. In particular, we will take ρ X to be due to an ultra-light scalar field with m < H 0 as discussed in sect. 3. We define the EOS parameter of the X-component by w X = p X /ρ X . Since ρ X and ρ Z have different origins, w X and w Z can in principle be different and we will see that only for w X w Z a tracking mechanism for DE emerges.
Mechanisms such as the amplification of vacuum fluctuations can produce an interaction between ρ Z and scalar fields, but do not lead to interactions with photons nor massless fermions, that are not amplified in the FRW space-time because of conformal invariance. As we saw in sect. 3, massive particles with m > H 0 can be integrated out in the effective action and therefore cannot contribute to the violation of the condition ∇ µ 0|T µν |0 = 0. Thus, in our context it is natural to consider an interaction between ρ Z and ρ X , while nonrelativistic matter and radiation are conserved in isolation, so they scale in the standard way, ρ M ∼ a −3 and ρ R ∼ a −4 . In contrast, ρ Z and ρ X satisfy the coupled conservation equatioṅ In terms of the total DE density, ρ DE (t) = ρ X (t) + ρ Z (t), it readṡ We insert ρ Z (t) = Ω Z ρ 0 H 2 /H 2 0 on the r.h.s. and use the Friedmann equation, that now reads H 2 (t)/H 2 0 = Ω R a −4 + Ω M a −3 + ρ DE (t)/ρ 0 . With the definitions  and using d/dt = aH(d/da), we obtain a d da Note that the background evolution of the total DE density is fully determined by the two parameters w 0 and which can in principle be functions of time. We assume these functions to be constant for the scope of this work, i.e. we assume w Z and w X to be constant. A more general analysis will be presented in [20]. Then the evolution can be solved analytically, (11) where C is the integration constant that is fixed by the condition ρ DE (a = 1)/ρ 0 = Ω DE = 1 − (Ω R + Ω M ). Observe that the model reduces to wCDM when w X = w Z and the deviation from wCDM only depends on , so we are dealing with a one-parameter extension of wCDM, that we will call wZCDM.
A very interesting feature of (11) is that ρ DE always scales as the dominant energy component, for 0. In fig. 1 we show the ratio ρ DE (a)/ρ tot (a), where ρ tot (a) = ρ DE (a) + ρ 0 Ω M a −3 + ρ 0 Ω R a −4 . Deep into the RD phase, as well as in the MD phase, ρ DE (a)/ρ tot (a) = O( ), while today it becomes O(1). Compared to ΛCDM, where the ratio ρ Λ /ρ tot is of order one today, but goes to zero as a 3 during MD and as a 4 during RD, the coincidence problem is sensibly alleviated.
We cannot claim that wZCDM entirely solves the coincidence problem; in this model, in fact, the transition between the regime where ρ DE /ρ tot = O( ) to the regime where ρ DE /ρ tot = O(1) takes place at the present epoch simply because we have fixed the integration constant C in (11) by the requirement that ρ DE (a = 1)/ρ 0 = Ω DE . Nevertheless the coincidence problem is certainly alleviated, compared to ΛCDM, where ρ DE is parametrically different from ρ tot , and the ratio ρ DE /ρ tot evolves from O(10 −120 ) at a Planck time to ∼ 0.7 today.
It is also interesting to note that our model provides a different theoretical justification for parameterizations of early DE models that have been proposed in the literature. To make the relation explicit, it is useful to define w DE = p DE /ρ DE and derive it from the total DE continuity equation,ρ DE + 3H(ρ DE + p DE ) = 0. Inserting the explicit solution for ρ DE (t) found in (11) we get This function is shown in fig. 2, setting w 0 = −1 for definiteness. For > 0 it evolves smoothly from a value w DE 1/3 during RD, to w DE 0 during MD and finally goes asymptotically to w DE w 0 . This EOS is quite similar to that obtained in a commonly used parameterization of early dark energy [30,31]. Note that, for < 0, w DE goes through infinity, as a consequence of the fact that ρ DE goes through zero withρ DE 0, see fig. 1, but the pressure w DE ρ DE stays finite, and the background evolution is regular.
Our model has some similarities, as well as important differences, with other DE models studied in the literature. In particular, in [32][33][34][35] a model was proposed where the DE density has the form ρ Λ (t) = n 0 + n 1 H 2 (t), inspired by the idea that the cosmological constant could run under renormalization group. In this case, however, the two components n 0 and n 1 H 2 (t) necessarily have the same EOS parameter. More closely related is the ΛXCDM model proposed in [36], in which DE has two components, an energy density ρ Λ = n 0 + n 1 H 2 (t) associated to a running of the cosmological constant interacting with an unspecified dynamical "cosmon" field, although in our case the interaction is rather between ρ X (that, for w X −1, plays basically the role of n 0 ) and ρ Z ∼ H 2 , which has a different EOS.
We have performed a detailed comparison of our model with current observations of CMB, SNe Ia and BAO using modified versions of CAMB and CosmoMC [37,38], treating perturbations in the DE by modeling it as a perfect fluid (without anisotropic stress) with the EOS parameter w DE given in (12) and a unit restframe sound speed. Full details will be reported in [20]. In fig. 3 we give a sample of our results. The plots on the left are the one-dimensional posterior probabilities marginalized over all parameters except w 0 or , respectively, while the plot on the right shows the 2dimensional posterior probability marginalized over all parameters except the pair ( , w 0 ). In particular, we find the marginalized limits −1.25 < w 0 < −0.908 and −0.0201 < < 0.0460 at 95% C.L., consistent with ΛCDM. The means of the marginalized posteriors are at w 0 = −1.07 and = 0.0104 and the standard deviations are σ w 0 = 0.0873 and σ = 0.0167, respectively. Thus, our wZCDM model is consistent with current data, and its deviations from wCDM, expressed by the parameter , are constrained at the level O(10 −2 ). Future data will be able to set more stringent limits or to detect a non-vanishing value of .