Composite running vacuum in the Universe: implications on the cosmological tensions

As promised, I shall skip all technical QFT details here – see  [37, 38, 39, 40] or the reviews [13] in case you are interested. Here only phenomenological aspects will be touched upon. I will say though only one qualitative technicality, which is important enough: the VED that we measure today through the physical cosmological term, namely $\rho^{0}_{\rm vac}=\Lambda_{\rm obs}/(8\pi G_{N})$, is not just the bare term $\rho_{\Lambda}=\Lambda_{b}/(8\pi G_{b})$ that we may construct from the bare CC value $\Lambda_{b}$ (and the bare Newton’s coupling $G_{b}$) in the action, not even the one that we may construct from the renormalized value $\rho_{\Lambda}(M)$ (which depends on some appropriate renormalization scale $M$). To construct the physically renormalized VED we have to add the renormalized $\rho_{\Lambda}(M)$ and the renormalized zero-point energy (ZPE) in the corresponding renormalization scheme. We may write this very qualitatively as follows: ${\rm VED}=\rho_{\Lambda}+{\rm ZPE}$. Hence the physical cosmological term that we measure is $\Lambda_{\rm obs}=(8\pi G_{N}){\rm VED}\equiv(8\pi G_{N})\rho^{0}_{\rm vac}$. It is thanks to the presence of the ZPE in curved spacetime that the VED acquires a dependence on the expansion rate $H$ and therefore $\Lambda_{\rm obs}=\Lambda_{\rm obs}(H)$ becomes a dynamical quantity in an expanding universe. I repeat: there is no such thing as a rigid cosmological constant in the quantum field theory of an expanding universe! Previous approaches to the subject over the years failed blatantly in realizing this crucial fact.

Indeed, the renormalized result still requires a physical interpretation since it depends on the renormalization scale $M$. The vacuum effective action $W_{\rm eff}$ [27] is explicitly dependent on the renormalization scale $M$ despite the fact that the full effective action is not. An adequate choice of $M$ at the end of the renormalization program obtains if one picks it equal to the value of $H$ at each cosmic epoch under consideration. This corresponds to choose the renormalization group scale around the characteristic energy scale of FLRW spacetime at any given moment, and hence it should have physical significance. This is similar to the standard practice in ordinary gauge theories, where the choice of the renormalization scale is made near the typical energy of the process. As previously indicated, the setting $M=H$ has also been recently confirmed as being fully consistent within the lattice QG approach of [41]. It follows that the renormalized VED that we find is an evolving quantity with the cosmic expansion (see the above mentioned papers). The simplest situation which allows to illustrate this fact is by considering a quantized scalar field $\phi$ with mass $m_{\phi}$ endowed with a non-minimal coupling to gravity (i.e. having a term $\sim\xi\phi^{2}R$ in the action, $\xi$ being the non-minimal coupling)⁴⁴4One can generalize the QFT computations for an arbitrary number of quantized scalar fields and even an arbitrary number of quantized fermions fields, see [40], but for the sake of simplicity it will suffice to describe here the case with one single scalar field. The form (1) of the VED is the same in all cases. The differences are encoded in the structure of the effective parameter $\nu$. . Its quantum matter fluctuations in the expanding FLRW background produce an evolution of the VED with the cosmic expansion. The final result for the running VED at low energies between two expansion history times, say the current epoch (characterized by the value $H_{0}$ of the Hubble parameter) and some nearby epoch $H$ of the cosmic evolution can be rendered in the following very compact form (see the aforementioned papers):

where the leading form of $\nu$ is constant (with only a mild log evolution with $H$[38]) and reads

$\rho_{\rm vac}^{0}\equiv\rho_{\rm vac}(H_{0})$ being the current value of the VED (accessible from observations) and $H_{0}$ today’s value of the Hubble function. It is necessary to remark that $\nu$ is an effective parameter expected to be small due to its proportionality to $m_{\phi}^{2}/m_{\rm Pl}^{2}$. The higher powers of $H$, indicated in the above equations as ${\cal O}(H^{4})$, are not considered here since they can be significant only for the physics of the very early universe, namely during the inflationary time, and hence the above low-energy formula applies virtually to any (post-inflationary) epoch. The above dynamical expression for the VED turns out to adopt the canonical RVM form, see [11, 12, 13] and references therein. This formula proves rather successful from the phenomenological point of view. Fitting the overall cosmological data with it indeed provides an estimate for $\nu$ at the level of $\nu\sim 10^{-4}-10^{-3}$ [71, 72], see also [61, 62, 63, 65, 64, 66]. Such a phenomenological determination of $\nu$ lies in the ballpark of old theoretical expectations [73] and is compatible with the recent lattice QG approach of [41]. Furthermore, the order of magnitude of $\nu$ is reasonable if we take into account that the masses involved here pertain of course to the scale of a typical Grand Unified Theory (GUT) where, in addition, a large factor must be included to encompass the large multiplicity of heavy particles. It is also worth noticing that the obtained order of magnitude for $\nu$ is compatible with the Big Bang nucleosynthesis (BBN) bound[74].

An important quantity is the effective equation of state (EoS) for the running vacuum. Its study requires the computation of the quantum effects on the vacuum pressure, which produce a deviation from the classical result $P_{\rm vac}=-\rho_{\rm vac}$. This is remarkable as it enables the quantum vacuum to mimic quintessence and phantom DE. The leading expression of the EoS for the current universe is the following[38, 39]:

where again the ${\cal O}(H^{4})$ effects are omitted. For very low redshift $z$ and in terms of the current cosmological parameters $\Omega^{0}_{i}=\rho^{0}_{i}/\rho^{0}_{c}=8\pi G_{N}\rho^{0}_{i}/(3H_{0}^{2})$ the above expression reduces to[39]

This result is worth emphasizing since it quantifies the small departure of the vacuum EoS from -1. Since $\nu\sim 10^{-4}-10^{-3}$ is not that small this EoS correction could perhaps be measured in the late universe. For positive (resp. negative) sign of $\nu$, Eq. (4) predicts that the vacuum energy behaves as quintessence (resp. phantom DE) for small $z$. From this point of view, a departure of the effective EoS of the DE from $-1$ does not necessarily imply the existence of quintessence or phantom fields, as the (quantum) vacuum itself can be responsible for these effects! This is a remarkable qualitatively new feature of the quantum vacuum in the RVM formulation.

As noted before, the EoS expression(4) is valid only for small values of the redshift $z$. One can show that the deviation is even bigger in the remote past (although the exact formula is more complicated), adopting a kind of ‘chameleonic’ behavior by which the EoS of the quantum vacuum tracks the EoS of matter at high redshifts, see [39, 75] for more details. It is important to stress that these results have been obtained from first principles, namely from explicit QFT calculations in the FLRW background, hence they rely on the properties of the quantum vacuum. There is therefore no need to introduce ad hoc fields to make the DE dynamical: the quantum vacuum is already so!

Next we move to a composite RVM scenario, the $\Lambda$XCDM. It was proposed for the first time in [58] and further expanded in subsequent works [59, 60, 76]. Assume the existence of a DE component $X$ which exchanges energy with the running vacuum. One possibility for $X$ would be a scalar field $\chi$, but we are not going to make any specific assumption on the ultimate nature of $X$, e.g. it could be an effective representation of dynamical fields of various sorts, or even the behavior of higher order curvature terms in the effective action. Since we still have the RVM, this means that we have a running vacuum term, so the total DE density, $\rho_{\rm DE}$, is given by the sum $\rho_{\rm vac}+\rho_{X}$, and the overall DE conservation law reads

where $w_{X}$ is the EoS of $X$, which for simplicity we will assume constant. In the previous conservation law it was assumed that $P_{\rm vac}=-\rho_{\rm vac}$. Despite we now know that the running vacuum may depart slightly from that classical relation, we will still take it to hold as in [58] since this feature does not affect the main purposes of that study. The departure of the EoS from $-1$ will, however, be important for the considerations of the $w$XCDM model discussed in the next section, which are of different nature despite it sharing the $X$ component in both cases. For running $\rho_{\rm vac}$, Eq. (5) shows that the dynamics of the vacuum and the new component $X$ become entangled. One can derive the above conservation law starting from the total energy-momentum tensor of the mixed DE fluid, with $4$-velocity $U_{\mu}$:

Then one can use the FLRW metric

and straightforwardly compute $\bigtriangledown^{\mu}\,{T^{D}}_{\mu\nu}=0$, which yields (5). The corresponding Friedmann’s equation reads

and for the time derivative of $H$ one finds

In practice we can simply set $K=0$ (flat three-dimensional spacetime).

The conservation law (5) concerns the covariant conservation of the total composite DE density, namely the sum of the running VED – expressed by the RVM formula (1)– and the new component $X$. There is no exchange between DE and matter in this setup, hence matter is conserved in the usual way, $\rho_{m}\sim a^{-3}$. In these conditions the background cosmological equations of the $\Lambda$XCDM model can be fully solved analytically, see [58]⁵⁵5See also [59] for a variant $\Lambda$XCDM scenario, in which both matter and $X$ are conserved whereas the VED is running at the expense of another running quantity: the gravitational coupling $G$ Here, however, we will focus exclusively on the original scenario of [58] at fixed $G$. We shall not discuss these cumbersome equations here, but we would like at least to stand out the main reason for studying the $\Lambda$XCDM model in that reference, and hence the reason for the presence of the extra component $X$, which is to provide a possible explanation for the “cosmic coincidence problem” [25]. Recall that this problem is related to the behavior of the ratio $r(z)\equiv\rho_{\rm DE}(z)/\rho_{m}(z)$ (“cosmic coincidence ratio”) between the dark energy density and the matter energy density. In the standard $\Lambda$CDM model, that ratio unstoppably with the cosmic evolution because $\rho_{\rm DE}$ is just the strictly constant vacuum energy density $\rho^{0}_{\rm vac}$, whereas $\rho_{m}\to 0$ with the expansion, so there is no apparent reason why we should find ourselves precisely in an epoch where $r={\cal O}(1)$. This is the purported “cosmic coincidence” alluded to by the mentioned coincidence problem. If it is a problem at all [26], it has no known solution within the standard $\Lambda$XCDM model. However, it can be highly alleviated in the context of the $\Lambda$XCDM model. This is so thanks to the presence of the new DE component $X$, which can prevent the aforementioned cosmic coincidence ratio to increase forever. In fact, one can easily understand from Eq.(8) that for $\rho_{X}<0$ the function $r(z)$ will present a maximum at some redshift since we cannot have $H<0$. From explicit calculation using the analytical solution of the $\Lambda$XCDM model one finds  [58]:

where we have defined $\epsilon\equiv\nu(1+w_{X})$. One can show that this point cannot be in our past, so it must generally lie in the future (hence $-1<z_{\rm max}<0$). Beyond the maximum, there is a turning point $z_{s}<z_{\rm max}$ where there is a bounce, namely the universe stops and reverses its evolution. At the stopping point, $\Omega_{D}(z_{s})=-\Omega_{M}(z_{s})<0$. This is possible because we can have $\Omega_{X}<0$ in the $\Lambda$XCDM. Let us also notice that one can also achieve this situation with $\Omega_{X}>0$ and negative cosmological constant $\Omega_{\rm vac}<0$. The cosmic sum rule of the $\Lambda$XCDM allows this occurrence since today’s values of the cosmological parameters must satisfy

Mind, however, that in the $\Lambda$XCDM model the cosmological parameter associated to the physical cosmological term is not $\Omega^{0}_{\rm vac}$ but $\Omega_{\rm DE}^{0}=\Omega^{0}_{\rm vac}+\Omega_{X}^{0}$ since in this context the total VED is associated with the measurement of $\Omega_{\rm DE}$, not with $\Omega_{\rm vac}$. In fact, it is not possible to dissociate $\Omega_{\rm vac}$ from $\Omega_{X}$ without further information, as only their sum $\Omega_{\rm DE}$ can be determined at each point of the cosmic expansion.

The following observation is now in order: notice, from eq. (5), that for $\nu=0$ (which implies no running of the VED, i.e. $\dot{\rho}_{\rm vac}=0$ ) the energy density of the new component $X$ is conserved; so, if in addition $\Omega_{X}^{0}=0$ (vanishing current density of $X$), we must then have $\Omega_{X}(z)=0$ at all times. Thus, in the limit $\nu=\Omega_{X}^{0}=0$ we exactly recover the standard $\Lambda$CDM model. And it is precisely in this limit that the formula (10) for the redshift location of the maximum of the ratio $r(z)$ ceases to make sense, as it is evident by simple inspection of that formula. In contrast, when we make allowance for the parameter space of the $\Lambda$XCDM model, the maximum for $r(z)$ does indeed exist and moreover the ratio $r(z)$ stays bounded – typically $r<{\cal O}(10)$ – for the entire cosmic history. This is manifest in the numerical examples exhibited in Fig. 1, for various values of the parameters. Therefore, the $\Lambda$XCDM model can provide a nice solution to the cosmic coincidence problem without deviating exceedingly from the $\Lambda$CDM model. Furthermore, the detailed analysis of the cosmic perturbations in the $\Lambda$XCDM model indicate excellent compatibility with the present observational data on structure formation [60].

For the sake of a better contextualization of the PM option within the class of energy conditions, we display in Fig. 2 some of the most common possibilities for the EoS satisfied by the cosmic fluids. In it, we can see e.g. the locus of the phantom matter (PM) fluid. As can be seen, it is very different from the usual phantom DE, it lies actually in its antipodes. This must be stressed, these two phantom-like components are dramatically disparate. The existence of the aforementioned two different phases of the DE ($X$ and $Y$) in our universe separated by a transition redshift $z_{t}$, and in particular the possibility that $X$ embodies a PM phase, is not just a whimsical assumption since it is found in theoretical contexts such as the stringy RVM approach [31], which points to the existence of transitory domains or bubbles of PM when the universe asymptotes to a de Sitter epoch. The phantom-like component $X$ actually behaves as PM in the $w$XCDM . This entails negative energy density ($\Omega_{X}=\rho_{X}/\rho_{c}<0$) and hence positive pressure ($p_{X}>0$). The behavior of PM is, as indicated, far away from that of the usual phantom DE. For, despite they both satisfy $w\lesssim-1$, PM produces positive pressure at the expense of negative energy density, and hence it resembles usual matter (therein its name![58]). This is in stark contrast to phantom DE, which produces negative pressure with positive energy. ‘Conventional’ phantom DE has been amply considered in the literature to describe different features of the DE, including a possible explanation for the cosmological tensions on $H_{0}$ and structure formation in a variety of frameworks (see the Introduction). However, in our approach, conventional phantom DE is not used at all. As a matter of fact, phantom DE is not favored in our analysis of the cosmological data, only PM is singled out as the best option for a possible alleviation, or even full resolution, of the cosmological tensions.

While the PM component $X$ acts first during the cosmic expansion, restricted to a bubble of spacetime, the $Y$ component takes its turn in the most recent universe (below $z_{t}\sim 1.5$) up to our days, see Table 1. The $Y$ component is more conventional, we have $\Omega_{Y}>0$ and $p_{Y}<0$, and hence it could perfectly behave as quintessence or phantom DE. Nevertheless, it is the numerical fitting to the cosmological data which must decide phenomenologically which one of these two options is more favored. The result is that its EoS satisfies $w_{Y}\gtrsim-1$ (cf. Table 1 and Fig. 3), hence $Y$ is quintessence-like. The transition redshift $z_{t}$ separating the two components $X$ and $Y$ is also determined by the fitting procedure (cf. Table 1). Therefore, the characteristic free parameters of model $w$XCDM are just three, the transition point and the two EoS parameters of the DE components: $(z_{t},w_{X},w_{Y})$. Let us clarify that the density parameters for the DE components are not free since e.g. the value of $\Omega_{Y}^{0}\equiv\Omega_{Y}(z=0)$ depends on the fitting values of $H_{0},\omega_{b},\omega_{\rm dm}$ (cf. Table 1). In addition, the respective values of $\Omega_{X}$ and of $\Omega_{Y}$ immediately above and below $z_{t}$ are assumed to be equal in absolute value: $|\Omega_{X}(z)|=\Omega_{Y}(z)$ at $z=z_{t}$. This kind of assumption aims at reducing the number of parameters. We cannot exclude that by relaxing this assumption it may further improve the fit quality, but it would imply more modelling features which we wish to avoid at this point.

In [31] it is argued that the PM-dominated era can be present in the early cosmic evolution near a de Sitter phase, but new episodes could also reflourish in the late universe. As shown in[32], the presence of chiral matter fields at the exit from inflation enables the cancellation of the gravitational anomalies and at large scales the universe can recover its normal FLRW background profile. Because the chiral matter fields will get more and more diluted with the cosmic expansion, this will entail an incomplete cancellation of the gravitational anomalies, and as a result these may eventually re-surface near the current quasi-de Sitter era. For this reason it is conceivable to admit the formation of phantom matter bubbles or domains tunneling into our universe at relatively close redshift ranges before the ultimate de Sitter phase takes over in the late universe. These bubbles of PM are characterized by positive pressure $p>0$ (see Fig. 2) and therefore could be the seed of structure formation at unusually high redshifts. If so, this could explain on fundamental grounds the overproduction of large scale structures at unexpected places and times deep in our past. This ideology provides an excellent framework for a fit to the overall cosmological data which proves to be (far) better than that of the standard $\Lambda$CDM model (cf. Table 1).

To constrain the $w$XCDM parameters, we make use of the following cosmological data sets (see [56] for a more detailed description and detailed references for all of them):

• •

  The full Planck 2018 CMB temperature, polarization and lensing likelihoods.

• •

  The SnIa contained in the Pantheon+ compilation, calibrated with the cosmic distance ladder measurements of the SH0ES Team.

• •

  $H(z)$ data from cosmic chronometers (CCH).

• •

  Transverse (aka angular or 2D) BAO data from Refs. [80]. These data deserve a particular consideration. In fact, transverse BAO data are claimed to be less subject to model-dependencies, since they are obtained without assuming any fiducial cosmology to convert angles and redshifts into distances to build the tracer map. These data points are extracted from the two-point angular correlation function or its Fourier transform. In anisotropic (or 3D) BAO analyses, instead, a fiducial $\Lambda$CDM cosmology is employed to construct the 3D map in redshift space, hence potentially introducing model-dependencies. Furthermore, some studies have suggested that uncertainties may be underestimated by roughly a factor of two in these analyses [81]. In contrast to the 3D BAO data, 2D BAO observations still leave room for low-redshift solutions to the Hubble tension while respecting the constancy of the absolute magnitude of SnIa, which is the kind of approach that we propose here.

• •

  The data on large-scale structure at $z\lesssim 1.5$. For these data we also wish to add some qualification. These are LSS observations on the weighted growth $f(z)\sigma_{8}(z)$, with $f(z)=-(1+z)d\ln\delta_{m}/dz$ the linear growth rate, $\delta_{m}=\delta\rho_{m}/\rho_{m}$ is the matter density contrast, and $\sigma_{8}(z)$ the rms mass fluctuations at a scale $R_{8}=8h^{-1}$ Mpc, see Table 3 of [72]. These data points, however, are taken using fiducial cosmology with $h\sim 0.67$, which translates into measurements at a characteristic length scale of $R\sim 12$ Mpc. At variance with this rather common practice, and as noted already in the Introduction, we adhere instead to the alternative procedure advocated in[42, 43, 44], and therefore we treat these LSS observations as data points on $f(z)\sigma_{12}(z)$, using a Fourier-transformed top-hat window function $W(kR_{12})$ in the computation of $\sigma_{12}(z)$. The advantage is that the scale $R_{12}=12$ Mpc is independent of the parameter $h$. Following this approach we find more consistent results for the growth of matter fluctuations.

Using the above data sources for our fit, we present our main numerical results in Table 1 and in the triangle plots of Figs. 4-3.

Let us briefly summarize why the $w$XCDM can deal effectively with the cosmological tensions. Let us consider $H_{0}$ first. Recall that $\theta_{*}$ (the angular size of the sound horizon) is given by $\theta_{*}=r_{d}/D_{A}(z_{*})$, where $r_{d}$ is the sound horizon at decoupling and $D_{A}(z_{*})$ is the angular diameter distance at this point: $D(z_{*})=c(1+z)^{-1}\int_{0}^{z_{*}}\frac{dz^{\prime}}{H(z^{\prime})}$. As indicated, we shall not modify the physics prior to decoupling (i.e. for $z>z_{*}$) and therefore we aim essentially at changes occurring after recombination. However, these changes cannot modify the extremely well known value of $\theta_{*}$. This riddle can be solved by implementing internal changes in $D_{A}(z_{*})$ that compensate each other but that leave a measurable imprint on the late universe (e.g. producing a change of the current value of the Hubble parameter $H_{0}$). In fact, this can be accomplished as follows. Notice that the integral defining $D_{A}$ can be split at the transition point $z_{t}$ which separates the two regimes (PM for $z>z_{t}$ and quintessence for $z<z_{t}$):

where $H_{\rm Q}$ and $H_{\rm PM}$ are the Hubble functions within the quintessence (Q) and phantom matter (PM) regimes depicted in Fig. 2. Using this composite RVM model we can easily explain within our PM scenario (based on the $w$XCDM model) why $H_{0}$ is found larger than in the $\Lambda$CDM. It is important to realize that because the energy of the $X$ entity is negative (and non-negligible at high redshift, in absolute value), this enforces $H_{\rm PM}$ to be smaller than in the standard regime and hence the second integral in (12) is larger than the usual contribution in the $\Lambda$CDM. This enforces a smaller value of the first integral on the r.h.s. of the above expression and hence a larger value of the expansion rate $H$ for $z<z_{t}$, This is possible in the quintessence stage characterized by $H_{\rm Q}$ provided we have a larger value of $H_{0}$ (to be fixed by our fit) . In this way the two changes can compensate each other and $D_{A}(z_{*})$ (and $\theta_{*}$) remains the same without modifying the physic of the early universe, i.e. the value of $r_{d}$. Quantitatively, the parameter $H_{0}$ emerging from our fit (cf. Table 1) is in full agreement with that of SH0ES ($H_{0}=73.04\pm 1.04$ km/s/Mpc)[82] to within $\sim 0.25\sigma$. The Hubble tension is therefore completely washed out (see also Fig. 4). At the same time the rate of LSS formation becomes suppressed below $z_{t}\sim 1.5$ during the quintessence-like regime, which is in accordance with the observations. This can be easily understood by throwing a glance to the equation for the matter density contrast $\delta_{m}$ in the $w$XCDM model. Before the transition at $z_{t}$ (i.e. at $z>z_{t}$), the matter perturbations equation reads

where the primes denote derivatives with respect to the scale factor. Since PM has negative energy density ($\Omega_{X}<0$) and positive pressure (due to $w_{X}<-1<0$) it induces a decrease of the friction term and an increase of the Poisson term in Eq. (13). Both effects are cooperative and cause an enhancement of the structure formation processes in the PM bubbles. Notice also that this would not occur for ordinary phantom DE or quintessence (cf. Fig. 2), for which the friction term gets enhanced and the Poisson term suppressed, i.e. just the opposite behavior of PM.

Focusing now on the $w$XCDM parameters, in Fig. 3 we show the constraints obtained in the EoS plane $w_{Y}$-$w_{X}$, which involves two of the characteristic new parameters of the $w$XCDM model (the third one being $z_{t}$, shared with $\Lambda_{s}$CDM). The central value of $w_{X}=-1.16$ falls in the phantom-like region (in fact, PM region since $\Omega_{X}<0$) but is compatible with $-1$ at $\sim 1\sigma$ CL. There is, in contrast, a non-negligible ($\sim 3.3\sigma$) preference for a quintessence-like evolution of the DE for the low redshift range nearer to our time ($z<z_{t}$): $w_{Y}=-0.90\pm 0.03$. This situation is beautifully consistent with the recent DESI results suggesting dynamical DE [57]. Besides, we can see in Table 1 and in Fig. 4 that the amplitude of the power spectrum at linear scales that is preferred by the data is optimal for the $w$XCDM , $\sigma_{12}\sim 0.77$. This can again be understood by looking at the equation for the density contrast, whose form in the range $z<z_{t}$ is identical to Eq. (13), but with the replacements $\Omega_{X}\to\Omega_{Y}$ and $w_{X}\to w_{Y}$.