COSMOLOGICAL CONSTANT VIS-À-VIS DYNAMICAL VACUUM: BOLD CHALLENGING THE CDM111Invited review paper based on the invited plenary talk at the International Conference on New Physics at the Large Hadron Collider, NTU, Singapore, 2016. Some of the results and discussions presented here have been further updated and expanded with respect to the published version.

Joan Solà

Departament de Física Quàntica i Astrofísica ,

Av. Diagonal 647, E-08028 Barcelona, Catalonia, Spain


Institute of Cosmos Sciences (ICCUB)

Univ. de Barcelona, Av. Diagonal 647, E-08028 Barcelona


Abstract. Next year we will celebrate 100 years of the cosmological term, , in Einstein’s gravitational field equations, also 50 years since the cosmological constant problem was first formulated by Zeldovich, and almost about two decades of the observational evidence that a non-vanishing, positive, -term could be the simplest phenomenological explanation for the observed acceleration of the Universe. This mixed state of affairs already shows that we do no currently understand the theoretical nature of . In particular, we are still facing the crucial question whether is truly a fundamental constant or a mildly evolving dynamical variable. At this point the matter should be settled once more empirically and, amazingly enough, the wealth of observational data at our disposal can presently shed true light on it. In this short review I summarize the situation of some of these studies. It turns out that the const. hypothesis, despite being the simplest, may well not be the most favored one when we put it in hard-fought competition with specific dynamical models of the vacuum energy. Recently it has been shown that the overall fit to the cosmological observables SNIa+BAO++LSS+BBN+CMB do favor the class of “running” vacuum models (RVM’s) – in which is a function of the Hubble rate – against the “concordance” CDM model. The support is at an unprecedented level of and is backed up with Akaike and Bayesian criteria leading to compelling evidence in favor of the RVM option and other related dynamical vacuum models. I also address the implications of this framework on the possible time evolution of the fundamental constants of Nature.

Keywords: Cosmology; vacuum energy; fundamental constants
PACS numbers: 95.36.+x, 04.62.+v, 11.10.Hi

1 Introduction

On February 15th 1917 the famous seminal paper where Einstein introduced the cosmological term (actually denoted “” in it) along with the fundamental observation that such term does not destroy the general covariance of his original (1915) field equations, was issued 222“Wir können nämlich auf der linken Seite der Feldgleichungen [(13)] den mit einer vorläufig unbekannten universellen Konstante - multiplizierten Fundamentaltensor hinzufügen, ohne daß dadurch die allgemeine Kovarianz zerstört wird…” (A. Einstein [1]). Since then, and despite its lengthy and tortuous history, has traditionally been associated to the concept of vacuum energy density: , where is Newton’s constant. This association is somehow natural if we take into account that the vacuum energy is conceived as being uniformly distributed in all corners of space. In other words, it perfectly preserves the Cosmological Principle. Thus it cannot be related to any form of matter content of the Universe, which always tends to cluster through gravitational collapse. However, it is difficult to understand the origin of the vacuum energy (what is it after all?) unless one makes a connection with quantum physics, where the vacuum fluctuations of the fields are part of everyday’s life. Such connection was first pointed out by Zeldovich in 1967  [2].

Unfortunately, while the vacuum fluctuations in the presence of real particles (external lines of Feynman diagrams) are well understood in quantum field theory (QFT) – they are called “radiative corrections” to the classical processes —, the pure vacuum-to-vacuum diagrams, i.e. closed diagrams without external legs, are not so well understood and find themselves in a peculiar situation. They cannot be considered radiative “corrections” because they are not related to any classical or tree-level amplitude to be “corrected”! In other words, they are pure quantum effects with no classical counterpart. In both cases we are dealing with quantum effects, and as such they are usually UV-divergent and require regularization and renormalization procedures. In the case of the radiative corrections, the final result is always a small finite contribution to the classical quantity with which we started the zeroth order computation (e.g. a cross-section, decay rate, energy level etc). In contrast, if we attempt to renormalize a zero-point energy (ZPE) diagram with no external legs, i.e. a vacuum-to-vacuum diagram, we find that the renormalization procedure despite being perfectly possible as in the radiative calculation case, the resulting finite contribution is always much larger (by many orders of magnitude) to the measured value of [3, 4, 5, 6, 7]. And this is so no matter what is the sort of (boson or fermion) field we are considering. For example, take the standard model (SM) of strong and electroweak interactions. Let CeV be the (measured) Higgs mass and GeV the Fermi scale. The contribution to the ZPE from the Higgs field is of order GeV, and the ground state value of the (classical) Higgs potential reads GeV. In magnitude this is of order , where GeV is the vacuum expectation value of the scalar field. Equally significant is the ZPE from the top quark (with mass GeV), which is of order GeV and negative (because it is a fermion). After adding up all these effects a result of the same order ensues which is times bigger than what is needed. It is senseless to expect that these contributions may cancel, all the more so bearing in mind that they must be retuned order by order in perturbation theory!

The above situation is of course (part of) the so-called “old cosmological constant problem” [8, 9] – see also [10, 11]. As mentioned, this problem was first formulated by Zeldovich about half a century ago [2] and is the main source of headache for every theoretical cosmologist confronting his/her theories with the measured value of . Furthermore, the purported discovery of the Higgs boson at the LHC has accentuated the CC problem greatly, certainly much more than is usually recognized. Owing to the necessary spontaneous symmetry breaking (SSB) of the electroweak (EW) theory, an induced contribution to is generated which is appallingly much larger (viz. ) than the tiny value GeV (“tiny” only within the particle physics standards, of course) extracted from observations. So the world-wide celebrated “success” of the Higgs finding in particle physics actually became a cosmological fiasco, since it instantly detonated the “modern CC problem”, i.e. the confirmed size of the EW vacuum, which should be deemed as literally “real” (in contrast to other alleged – ultralarge – contributions from QFT) or “unreal” as the Higgs boson itself! One cannot exist without the other.

In this work we will not further address the formal QFT issues mentioned above, as they go beyond the scope of the kind of presentation adopted here, which is essentially phenomenological. I refer the reader to review works, e.g.  [8, 9, 10, 11], for a more detailed exposition of the CC problem and the general dark energy (DE) problem [12]. Setting aside the “impossible” task of predicting the value itself – unless it is understood as a “primordial renormalization” [13] – I will focus here on a more phenomenological presentation, which may help to decide if the -term is a rigid cosmological constant or a mildly evolving dynamical variable. In the latter case there would be more room for considering as a more sophisticated QFT object within the kind of scenarios that I will describe here. Having this purpose in mind I will review several types of dynamical vacuum models (DVM’s) and shall put special emphasis on a subclass of them in which appears neither as a rigid constant nor as a scalar field (quintessence and the like) [9], but as a “running” quantity in QFT in curved spacetime. This is a natural option for an expanding Universe. As we will show, such kind of dynamical vacuum models are phenomenologically quite successful; in fact so successful that they are currently challenging the CDM [14, 15, 16, 17, 18, 19, 20, 21, 22].

The layout of the paper is as follows. In Sect. 2 we introduce the general dynamical vacuum models (DVM’s), which in some cases implies the dynamical evolution of the gravitational coupling and/or the anomalous conservation of matter. In Sect. 3 we present the running vacuum models (RVM’s) and other related models. These are more specific DVM’s and some of them have recently been shown to fir the overall cosmological data substantially better than the CDM. The results of the fitting procedure are presented and discussed in Sect. 4. The possible impact of these successful dynamical vacuum models for a possible explanation of the the cosmic time evolution of masses and couplings (in particular the fine structure constant) is addressed in Sect. 5. Finally, in Sect. 6 we deliver our conclusions.

2 Dynamical vacuum models (DVM’s)

The traditional hypothesis const. for the -term in Einstein’s field equations


is the simplest one compatible with the Bianchi identity satisfied by the Einstein tensor on the l.h.s. of Eq. (1), provided one assumes local covariant matter conservation, , and const. (i.e. rigid gravitational coupling). However, it is obvious that this scenario is just one particular set of assumptions among many other possibilities. If we define the vacuum energy density in the usual way, , and move it to the r.h.s. of Einstein’s equations, then the Bianchi identity leads in general to the following generalized conservation law for the product of times the full energy-momentum tensor on its r.h.s. (i.e. the energy-momentum tensor after we include in it the vacuum energy density):


Writing it out in the FLRW metric, we find:


where we have assumed the usual perfect fluid form for the matter energy-momentum tensor for all the fluid components , characterized in each case by the density and pressure in the proper frame. These components receive contributions from matter (both relativistic and non-relativistic) as well as from the vacuum energy density, whose pressure satisfies . Therefore, we can be more explicit and write the previous equation as follows:


Here is the Hubble rate, denotes the relativistic (or radiation) component () and the non-relativistic matter component (hence with ). Equation (4) is actually a first integral of the basic system of equations emerging from the explicit form of Einstein’s equations (1) in the Friedmann-Lemaître-Robertson-Walker (FLRW) metric. We will hereafter assume that the FLRW metric is characterized by spatially flat sections. This does not affect the structure of the conservation laws but can affect other equations, specially those related with the Hubble function and its time derivatives. In the flat FLRW case, Friedmann’s equation together with the accelerating equation read


They are equally valid if and/or are homogeneous and isotropic functions of the cosmic time or the scale factor, thereby evolving with the expansion. One can easily retrieve equation (4) from these two equations.

Remarkably, the local conservation law (4) mixes the matter-radiation energy density with the vacuum energy . In other words, the vacuum can decay into matter or the matter can decay into vacuum energy. This framework is perfectly compatible with the Cosmological Principle since the generalized conservation law was derived from the FLRW metric and hence respecting the postulate of homogeneity and isotropy.

Let us mention the following possibilities or cosmological vacuum types:

  • Standard type or CDM: const. and const.:

    If there are no other components in the cosmic fluid apart from relativistic and nonrelativistic matter, this is the standard case or “concordance” CDM cosmological model, implying the local covariant conservation law of matter-radiation:


    Except for the epoch near the equality of matter and radiation, these components are usually assumed to be conserved separately, hence




    Integration of these equations with respect to the scale factor variable immediately leads to


    where and are the respective current densities, i.e. the corresponding values at .

  • Type-A model: const and :

    Here Eq.(4) leads to an “anomalous” type of conservation law (this is what A stands for):


    in which there is a nonvanishing source causing the interaction of matter with the vacuum and therefore the non-conservation of matter. An exchange of energy between the matter and the vacuum takes place, in such a way that for the vacuum decays into matter, whereas for it is the other way around. Obviously we assume since we know that the standard conservation laws (10) and (11) are to a great extent correct.

  • Type-G model: and assuming self-conservation of matter.

    Since we assume the standard local covariant conservation laws in the separated forms (10) and (11), then Eq. (4) leads immediately to a dynamical interplay between the gravitational coupling and the vacuum energy density:


  • Type-AG model at fixed : and const. Obviously this situation corresponds to non self-conservation of matter, and the matter densities obey


    Here we admit the possibility that does not stay constant with the cosmic evolution, but we assume that the -term is a true cosmological constant. This situation is kind of complementary to the previous one since in the present instance there is a dynamical interplay between and the anomalous matter conservation at fixed vacuum density. A simpler and milder assumption would be to assume that radiation is conserved and that there is only exchange of energy between non-relativistic matter and vacuum. In this case it would simplify as follows:


    where here is given by Eq, (11). The simplified form (15) could be solved e.g. for , if would be given by some non-conservation ansatz, or vice versa.

  • General Type-AG model: , const. and anomalous matter conservation. Needless to say this is the most general situation and is represented by Eq. (4), in which one may have both and dynamical, together with an anomalous matter conservation law. We may rewrite (4) in the more expanded form


    Obviously, Eq. (14) is the particular case when .

  • Inflationary Universe: Another possibility with and is the case that there is no matter in the universe: . Then Eq. (4) obviously implies const. Of course one possibility is that both and stay constant, as in the usual inflationary scenarios. But we cannot exclude that both parameters can be time evolving while the product remains constant. Whether and are both constant or not, such situation could only be of interest in the early universe, when matter still did not exist and only the vacuum energy was present.

We will consider in the next sections some uses of the generalized cosmological scenarios indicated above. Although we will not address the type-AG models in this study, we will deal with some detail the type-A and type-G ones, as well as some more specific cases within the type-A models in which baryons and radiation are conserved but dark matter (DM) is not. The type-AG models are dealt with e.g. in  [23, 24, 25].

3 The running vacuum model (RVM) and related models

Among the general class of the DVM’s, a particular subclass is the so-called running vacuum model (RVM), which can be motivated in the context of QFT in curved space-time (cf.  [10, 11] and references therein). In the RVM the dynamical nature of the vacuum is governed by a renormalization group equation. The model can be extended to provide an effective description of the cosmic evolution starting from the early inflationary phase of the universe [10, 11, 13, 26, 27]. We will not consider here the applications to the early Universe since we want to test the model using the current data. We only need to know that well after inflation and up to our days, the RVM energy density can be written in the relatively simple form:


where we are omitting the higher order terms which could be important only for the (inflationary) physics of the early universe. The additive constant is fixed by the boundary condition , where and are the current values of these quantities, and is the normalized vacuum energy density with respect to the critical density now. The dimensionless coefficient encodes the dynamics of the vacuum and can be related with the -function of the running. Thus, we naturally expect . An estimate of in QFT indicates that it is of order at most [28], but here we will treat as a free parameter and hence we shall deal with the RVM on pure phenomenological grounds. It means we will fit to the observational data.

A more general form of (17) for the current universe is when we include both the and the term, both being dimensionally consistent. In this case the vacuum energy density reads


Here, apart from , there is also the second dimensionless coefficient, ; altogether they parametrize the dynamics of vacuum. These coefficients can be related with the -function of the renormalization group running of the vacuum energy and therefore we naturally expect and . In this generalized form of the RVM, the constant reads a bit more complicated: .

In all formulations of the RVM we understand that can be either constant or variable with the cosmic evolution, in accordance to the various generic scenarios described in the previous section. We will next consider the RVM and other related models under some of these scenarios.

3.1 Type-A RVM’s

Let us display the explicit solution of model (18) as a Type-A model, i.e when we have anomalous matter conservation at const. One starts from Eq. (12)


with given in (18). After a straightforward calculation, in which we make use of and its time derivative so as to compute from (18), it is easy to see that Eq. (19) can be rewritten as follows:


where we have defined


For small (the expected situation), we can write in good approximation and for these effective vacuum parameters.

The solution is more easily obtained in terms of the scale factor. Thus, trading the cosmic time for the scale factor , i.e. using , we can easily solve (20) separately for the matter and radiation parts (the same kind of assumption as in the standard model case) as a function of . It immediately leads to the desired anomalous conservation laws for matter and radiation:


The anomalies are of course related to the fact that in general we have and for . Since the matter and radiation densities are now known, the corresponding vacuum energy density follows from integrating (19) once more in terms of the scale factor variable, with the result


The normalized Hubble rate with respect to the current value, , is easily obtained from Friedmann’s equation, and we find:


The normalized cosmological parameters with respect to the critical density satisfy the usual cosmic sum rule


Needless to say the solution of the simpler model (17) ensues by setting in the previous expressions, which leads to , and as a result for that model the anomaly in the radiation conservation law (23) coincides with the one in the (nonrelativistic) matter conservation law (22). Finally we note that all of the above formulas immediately boil down to the CDM case when (i.e. ), as they should.

As an illustration, in Fig. 1 we display the evolution of the vacuum energy density for the simpler model (17) as a function of the redshift . Specifically, we plot the relative variation with respect to the current value of the vacuum energy density:


Here we have neglected the radiation component since we are considering low values of . For the plot in Fig. 1 we use , which is the typical value obtained in the numerical fits discussed in section 4, and is also the typical theoretical expectation for in QFT in curved spacetime [28]. The corresponding anomalous (nonrelativistic) matter conservation law is 333This equation was first proposed and tested in the literature in Ref. [29], and only later on it was also exploited by other authors – see e.g. [30].


By inserting and , from (27) and (28) respectively, in the generalized conservation law (12) for type-A models (in the matter-dominated epoch) we find that it is identically satisfied, as it should.

Figure 1: The evolution of the vacuum energy density within the RVM for We plot the relative variation of with respect to the current value, see Eq. (27).

3.2 RVMc: the running vacuum model with conserved baryon and radiation densities

In this section we still consider type-A RVM’s but now introduce a further specification about the conservation of matter which was not imposed thus far. Recall that can be split into the contribution from baryons and cold dark matter (DM), namely . In the following we assume that the dark matter density is the only one that carries the anomaly, whereas radiation and baryons are self-conserved, so that their energy densities evolve in the standard way (10-11), i.e. and . We call this version of the RVM, the RVMc. In this case we limit ourselves to consider the case when in (18). In other words, we focus now on the canonical RVM (17). The solution is not just (22-24) with since Eq. (19) cannot be written in the form (20) for the entire matter density owing to the conservation of the individual and components. Imposing this conservation condition in Eq. (19) we find that the interaction of matter with the vacuum becomes exclusively associated to the exchange of energy with the dark sector, and hence we can rewrite (19) as follows:


It is for this reason that the conservation law for the DM component is anomalous. The source function is a calculable expression from (17) and Friedmann’s equation. We find:


Finally, we can solve routinely for the energy densities of the DM and vacuum and we arrive at the following formulas:




As can be easily checked, for we recover the corresponding results for the CDM, as we should. The Hubble function can be immediately obtained from these formulas after inserting them in Friedmann’s equation, together with the conservation laws for baryons and radiation, and . We refrain from writing out these expressions.

3.3 Other DVM’s with conserved baryon and radiation densities

Let us now explore two additional type-A phenomenological DVM’s with conserved baryon and radiation densities, in which the source function in (29) is introduced purely ad hoc, i.e. without any special theoretical motivation. Recall that in the RVMc case discussed previously we could calculate because was known from Eq. (17). Now we leave this RVMc scenario for a while and introduce alternative forms for on merely phenomenological grounds. Two possible ansatzs discussed in the literature are the following:


Model was previously studied e.g. in [31], but as mentioned in [17, 18] we do not concur with their analysis, see also  [32]. Model was considered recently in [33]. It is closer to the RVM than , but not identical, compare equations (30) and (33).

In the above alternative models we have introduced new dimensionless coefficients and , which are in principle differen from for the RVM. Altogether these coefficients parametrize the evolution of the vacuum energy density and the strength of the dark-sector interaction for each model. For (hence ) the vacuum decays into dark matter (which is favorable from the point of view of the second law of thermodynamics) whereas for () is the other way around. We can easily account for models (33) and (34) by solving the differential equations (29) in each case.

As for model we find:


whereas for model :


In section 4 we will compare these alternative DVM’s with the RVM in their different ability to describe the cosmological data, and of course we will compare them all to the CDM to check which model performs better observationally.

3.4 Type-G RVM’s

Let us consider anew the more general sort of RVM indicated in (18). However, we shall deal with it here as a type-G model, meaning that is now a cosmological variable that satisfies the differential equation (13) with both conservation of nonrelativistic matter and radiation separately. Trading once more the cosmic time for the scale factor in that equation we can integrate it and determine as a function of . This can be done as follows. Combining Friedmann’s equation (5) and the acceleration equation (6), and using the matter conservation equations, we arrive at


where is the present value of , and is the normalized Hubble rate to its present value. The above equation links to . The final step is to insert (38) and (18) into Eq. (5), and integrating the resulting differential equation for :


The result after some calculations is:


where and have been defined previously in (21). In the radiation-dominated epoch, the leading behavior of Eq. (40) is , while in the matter-dominated epoch is . Furthermore, for , one can easily show that . This is the CDM form, as expected in that limit. From the above rather unwieldy expression for one can finally obtain the explicit form of by computing the derivative and inserting it in (38). The explicit scale factor dependence of the vacuum energy density, i.e. , ensues similarly upon inserting (40) into (18). We refrain from writing out these cumbersome expressions and we limit ourselves to quote some simplified forms. For instance, the form of when we can neglect the radiation contribution is simple enough to be quoted here:

CDM - 90.44/85 - -
XCDM - 74.91/84 13.23 11.03
A1 71.22/84 16.92 14.72
A2 70.27/84 17.87 15.67
G1 71.19/84 16.95 14.75
G2 70.68/84 17.46 15.26
Table 1: The best-fit values for the CDM, XCDM and the RVM’s, including their statistical significance (-test and Akaike and Bayesian information criteria, AIC and BIC, see Sect. 4.5). The large and positive values of AIC and BIC strongly favor the dynamical DE options (RVM’s and XCDM) against the CDM (see text). We use a total of data points from SNIa+BAO++LSS+CMB observables in our fit: namely points from the JLA sample of SNIa [4], from BAO [34, 35, 36, 37, 38, 39], from [40, 41, 42, 43, 44, 45, 46], from linear growth [37, 47, 48, 49, 50, 51, 52, 53, 54, 55], and from CMB [56]. For a summarized description of these data, see [16]. In the XCDM model the EoS parameter is left free, whereas for the RVM’s and CDM is fixed at . The specific RVM fitting parameter is , see Eq. (21) and the text. For G1 and A1 models, . The remaining parameters are the standard ones (). The quoted number of degrees of freedom () is equal to the number of data points minus the number of independent fitting parameters ( for the CDM, for the RVM’s and the XCDM). The normalization parameter M introduced in the SNIa sector of the analysis is also left free in the fit, but it is not listed in the table. For the CMB data we have used the marginalized mean values and standard deviation for the parameters of the compressed likelihood for Planck 2015 TT,TE,EE + lowP data from [56], which provide tighter constraints to the CMB distance priors than those presented in [57].

In the expression above, is the current critical density and is the current value of the gravitational coupling. Quite obviously, for we recover the CDM form: const. As for the gravitational coupling, it evolves logarithmically with the scale factor in the limit and hence changes very slowly444This is a welcome feature already expected in particular realizations of type-G models in QFT in curved spacetime [28].. As mentioned, the exact form of is a bit too bulky. It suffices to say here that it behaves as


where is a smooth function of the scale factor, which tends to one at present ( for ) irrespective of the values of the various parameters involved in it; and also in the remote past () for (i.e. ). As expected, for , and has a logarithmic evolution for . Notice that the limit is relevant for the BBN (Big Bang Nucleosynthesis) epoch and therefore should not depart too much from according to the usual bounds on BBN. This restriction is tacitly incorporated in our numerical analysis of the RVM models in the next section.

Figure 2: Likelihood contours in the plane for the values , , , (corresponding to 1, 2, 3, 4 and 5 c.l.) after marginalizing over the rest of the fitting parameters indicated in Table 1. We display the progression of the contour plots obtained for model A2 using the 90 data points on SNIa+BAO++LSS+BBN+CMB, as we evolve from the high precision CMB data from WMAP9 [5], Planck 2013 [6] and Planck 2015 [6], see  [16] for details. In the sequence, the prediction of the concordance model () appears increasingly more disfavored, at an exclusion c.l. that ranges from (for WMAP9), (for Planck 2013) and up to (for Planck 2015). Using the Fisher matrix and numerical integration, we find that of the area of the contour for Planck 2015 (and of the corresponding region) satisfies . The CDM becomes once more excluded at c.l. Subsequent marginalization over increases slightly the c.l. and renders the fitting values indicated in Table 1, which reach a statistical significance of for all the RVM’s. The corresponding AIC and BIC criteria (cf. Table 1) consistently imply a very strong support to the RVM s against the CDM.

4 Fitting the DVM’s to observations

In this section, we put the vacuum models discussed above to the test, see [14, 15, 16, 17, 18, 19, 20, 21, 22] for more details. We consider the general RVM model (18), which depends on the two vacuum parameters and , under the two modalities type-A and type-G. Let us start with type-A (cf. section 3.1) and then we will address the type-G case (cf. section 3.4). Subsequently we will focus on the RVMc in the canonical form (17), together with the and models (cf. sections 3.2 and 3.3). It proves useful to study them altogether in a comparative way.

4.1 Type-A and type-G

We confront now the various DVM’s to the main set of cosmological observations compiled to date, namely we fit the models to the following wealth of data: i) the data from distant type Ia supernovae (SNIa); ii) the data on baryonic acoustic oscillations (BAO’s); iii) the known values of the Hubble parameter at different redshift points, ; iv) the large scale structure (LSS) formation data encoded in the weighted linear growth rate ; v) the BBN bound on the Hubble rate (when applicable); and, finally, vi) the CMB distance priors from WMAP9 and Planck 2013 and 2015. In short, we use the essential observational data represented by the cosmological observables SNIa+BAO++LSS+BBN+CMB to fit the vacuum parameters 555The bounds on the vacuum parameters from the analysis of cosmic perturbations, including the power spectrum, are also compatible with the fitting results presented here, cf. [58]. See also [59] for a detailed analysis of the matter and vacuum perturbations in running vacuum models.. For the analysis we have defined a joint likelihood function from the product of the likelihoods for all the data sets discussed above. For Gaussian errors, the total to be minimized reads:


Each one of these terms is defined in the standard way, including the covariance matrices for each sector [60]. The fitting results are given in Table 1, see [16, 18] for more details. The contour plots for type-A and type-G are displayed in Figs. 2 and 3, respectively. In these figures the models referred to as A2 and G2 mean that we are fitting the two parameters and of (18). In the plots we indicate the result for , which we defined previously in (21). We can see in these figures the increasingly favorable evolution of the fits in support of these DVM’s when we successively use data from WMAP9 [5], Planck 2013 [6] and Planck 2015 [6]. This situation is not particular of A2 and G2, Table 1 clearly shows that if we fit only the parameter (i.e. taking , corresponding to models A1 and G1), we find always a similar quality fit [16]. In all these cases, remarkably enough, the fit is significantly better than the CDM one. We can see in Figs. 2 and 3 that we can indeed reach evidence that at least one of these vacuum parameters is nonvanishing and positive, i.e. evidence against the CDM. The fitting results indicated in Table 1 actually lead to evidence owing to the fact that in the final numerical results we have marginalized over as well. Therefore, the c.l. that we can read from the fitting tables is always slightly higher than the one that can be visually observed from the contour plots.

Figure 3: As in Fig. 1, but for model G2. Again we see that the contours tend to migrate to the half plane as we evolve from WMAP9 [5] to Planck 2013 [6] and Planck 2015 [7] data. Using the same method as in Fig. 1, we find that of the area of the contour for Planck 2015 (and of the corresponding region) satisfies . The CDM becomes once more excluded at c.l.

4.2 Fiducial model

We should clarify an important aspect of the fitting results that we obtain for the dynamical vacuum models under study, namely the fact that all models are compared to the same fiducial model (including the CDM). This is important in order to fix the normalization factor of the power-spectrum. As a fiducial model we take the CDM model in which all parameters are taken from the Planck 2015 TT,TE,EE+lowP+lensing analysis  [7]. Let us explain in more detail the procedure by considering our treatment of the linear structure formation data. We have computed the density contrast for each vacuum model by adapting the cosmic perturbations formalism for type-A and type-G vacuum models. The matter perturbation, , obeys a generalized differential equation which depends on the RVM type. For type-A models such equation with respect to the cosmic time reads 666For a derivation and more details on this equation, see the comprehensive works [20], [19] and [15].


where . For const. we have and Eq. (44) reduces, of course, to the CDM form. For further convenience, let us write the last term of the CDM perturbation equation in terms of the Hubble function:


For type-G models the matter perturbation equation is more complicated [14, 15]:


This is a third order equation, but it is straightforward to show that the time derivative of the ordinary second order equation for the perturbations, i.e. the derivative of Eq. (45), coincides with (46) when const., as it should [15]. However for dynamical the perturbation equation that holds for type-G models is Eq. (46) – see also [22].

From the above perturbation equations we can derive the weighted linear growth for any of the type-A or type-G models, where is the growth factor and is the rms mass fluctuation amplitude on scales of Mpc at redshift . The latter is computed from


with a top-hat smoothing function (see e.g. [20] for a more elaborated discussion) and the transfer function, which we take from [61]. In addition, we have defined the fitting vectors for the vacuum models we are analyzing (including the CDM), and for the fiducial CDM model that we use in order to fix the normalization factor of the power-spectrum. From equation (47) we can now understand the point under discussion. The denominator of this equation is fully defined in terms of the aforementioned fiducial model in which all cosmological parameters are taken from the Planck 2015 TT,TE,EE+lowP+lensing analysis, [7]. These parameters are denoted by a subindex . As we can see, the calculation of and therefore of the the weighted linear growth , is computed for all models by taking the fiducial model as a reference. The comparison of the theoretical predictions and the data are indicated in Fig. 4. Since the CDM model is also compared to that fiducial model, all models (CDM, RVM’s and XCDM) are normalized in the same way. Clearly, this is an optimal strategy to compare the dynamical DE models to the CDM in the fairest possible way777This point was treated in a different way in a previous version of our analysis and this led to larger values of the AIC and BIC parameters for the dynamical models. However, after normalizing all models with respect to the same fiducial model the fitting results remain essentially unaltered and the AIC and AIC differences in favor of the dynamical DE option remain rather large, namely large enough to be ranked in the “strong” to “very strong” range of evidence in favor of dynamical DE. This is clear from Table 1 and also from Table 2 below. .

Figure 4: The data and the predicted curves by the RVM’s, XCDM and the CDM, using the best-fit values in Table 1. Shown are also the values of that we obtain for all the models. The theoretical prediction of all the RVM’s are visually indistinguishable and they have been plotted using the same (blue) dashed curve. The observational data points used (in green) and corresponding observational references are given in [16], see Table 4 of that reference.

4.3 Comparing with the XCDM model

In the XCDM model one replaces the -term with an unspecified dynamical entity , whose energy density at present coincides with the current value of the vacuum energy density, i.e. . It is not even a model, it is however the simplest possible parametrization for the dynamical DE, in which the equation of state (EoS) parameter is taken as constant, i.e. with conts. Such parametrization was first introduced long ago by Turner and White [62]. In a sense it mimics the behavior of a scalar field, quintessence () or phantom (), under the assumption that such field has an essentially constant EoS parameter near (but not exactly) . Since both matter and DE are self-conserved (i.e., they are not interacting), the energy densities as a function of the scale factor are simply given by and . The Hubble function is therefore given by

Figure 5: As in Fig. 1 and 2, but for model XCDM. The CDM is excluded at c.l.

A more sophisticated approximation to the behavior of a scalar field playing the role of dynamical DE is provided by the CPL prametrization [63], given by


However, it involves two parameters . Therefore, in order to better compare with the one-parameter family of dynamical vacuum models under consideration it is better at this point to stick to the XCDM parametrization, which has also one parameter only. Recall that the RVM’s have the parameter (models G1,A1) or the two parameters (models G2,A2), but in the last case we fix a convenient relation between and , so any of these models has actually one-parameter in the DE sector [16]. For this reason the XCDM, with also one single parameter, is more appropriate for comparison with the RVM’s. The XCDM serves as a baseline dynamical DE model to compare any more sophisticated model for the dynamical DE. If a given model claims to be sensitive to dynamical DE effects (as it is the case with the RVM’s and other DVM’s considered here), it is convenient to see if part or all of these effects can be detected through the XCDM parametrization. Let us note that we should not necessarily expect that the XCDM is sensitive to the DE effects in a way comparable to the DVM’s. The latter, for example, are in some cases vacuum models that interact with matter or have a variable . Nothing of this applies for the XCDM, so the mimicking of the vacuum dynamics need not be perfect through the XCDM. Still, since the vacuum dynamics is mild enough, we expect that a significant part of the vacuum dynamics should be captured by the XCDM parametrization, either in the form of effective quintessence behavior () or effective phantom behavior (). The result can be clearly appraised in Tables 1 and 2 (where the best fit parameters for the XCDM are also quoted), as well as in the contour plots of Fig. 5. We see that the effective quintessence option is definitely projected at exactly level: . Thus, a large part of the dynamical vacuum effect suggested by the RVM’s (at c.l.) is also disclosed by the XCDM parametrization.

Recently, in Ref.[64] further evidence on the time-evolving nature of the dark energy has been provided by fitting the same cosmological data as in the current work in terms of specific scalar field models. As a representative model in that work it was used the original Peebles & Ratra potential, . Remarkably enough, unambiguous signs of dynamical DE at c.l. have also been found, thus reconfirming through a nontrivial scalar field approach the strong hints of dynamical DE found with the dynamical vacuum models and the XCDM parametrization.

4.4 RVMc, and with conserved baryons and radiation

In Fig. 6, based on Table 2, we continue our numerical analysis by considering once more the fitting results to the the cosmological observables SNIa+BAO++LSS+CMB, but we now address the triad of models RVMc and and . Recall that for these particular models we assume that baryons and radiation are conserved and the interaction of the vacuum energy is only with dark matter (cf. sections 3.2 and 3.3). Thus, no BBN bound is necessary in this case since the radiation content at the BBN time is identical to that of the CDM. Also for simplicity we set in Eq. (18), so that the vacuum energy density takes here the simpler form (17). The details on the SNIa+BAO++LSS+CMB data points used are given in Ref. [17, 18]. Here we limit ourselves to show the final fitting results, see Table 2. We emphasize the use of updated observational inputs concerning BAO and linear growth data, particularly from [37] – rather than the older data from [65]. The contour plots in Fig. 6 correspond (from left to right) to the main dynamical vacuum model RVMc and the alternative models and discussed in section 3.3. It is interesting to see that the DVM’a are in all cases favored with respect to the CDM since the region is clearly projected in all the contour plots (in contrast to the CDM case ). But even more interesting is to realize that the more recent LSS data (especially in connection to the observable for linear structure formation [37]) do greatly enhance the quality fit of the DVM’s versus the CDM as compared to the older LSS data. Once more the confidence level by which the vacuum parameters are fitted to be non null (and hence departing from the CDM) is at the level for models RVMc and . The most conspicuous one is RVMc, in which is nonvanishing at c.l. (cf. Table 2).

4.5 Akaike and Bayessian criteria

The statistical results, in particular the relative quality of the fits from the various models, can be reassessed in terms of the time-honored Akaike and Bayessian information criteria, AIC and BIC [66, 67].

CDM - -1 90.44/84 - -
XCDM - 74.91/83 13.23 11.04
RVMc -1 70.32/83 17.82 15.63
-1 71.19/83 16.95 14.76
-1 83.08/83 5.06 2.87
Table 2: Best-fit values for the CDM and the three dynamical vacuum models (DVM’s) with conservation of baryons and radiation, including their statistical significance (-test and Akaike and Bayesian information criteria, AIC and BIC). Once more the AIC and BIC increments clearly favor the dynamical DE options. The RVMc and are particularly favored ( c.l.). Our fit is based on the same SNIa+BAO++LSS+CMB data set as in Tables 1 and 2. The specific fitting parameter for each DVM is (RVMc), (), (). The result for the XCDM is of course the same as in Tables 1 and 2 and is included here to ease the comparison with the DVM fitting results.

These information criteria are extremely useful for comparing different models in competition. The reason is that the models having more parameters have a larger capability to adjust a given set of data, but of course they should be penalized accordingly. In other words, the minimum value of should be appropriately corrected so as to take into account this feature. This is achieved through the AIC and BIC estimators. They are defined as follow [66, 67, 68]:




where is the number of independent fitting parameters and the number of data points. The larger are the differences AIC (BIC) with respect to the model that carries smaller value of AIC (BIC) – the DVM’s here – the higher is the evidence against the model with larger value of AIC (BIC) – the CDM. For AIC and/or BIC in the range we can speak of “strong evidence” against the CDM, and hence in favor of the DVM’s. Above 10, we are entitled to claim “very strong evidence” [66, 67, 68] in favor of the DVM’s. Specifically, Tables 1 and 2 render AIC and BIC virtually in all cases for the type-A, type-G, RVMc and . The results are outstanding since for all these models both AIC and BIC peak very strongly in the same direction. Thus, these DVM’s are definitely more favored than the CDM, and the most conspicuous one is the RVMc.

We conclude that the wealth of cosmological data at our disposal currently suggests that the hypothesis const. despite being the simplest may well not be the most favored one. The absence of vacuum dynamics is excluded at c.l. as compared to the best DVM’s considered here. The strength of this statement is riveted with the firm verdict of Akaike and Bayesian criteria. Overall we have collected a fairly strong statistical support of the conclusion that the SNIa+BAO++LSS+CMB cosmological data do favor a mild dynamical vacuum evolution 888See also the upcoming work [18] for a subsequent reanalysis of the DVM’s taking into account the last updated results from Ref.[37]. Conclusions are essentially unchanged..

Figure 6: Likelihood contours in the plane for the DVM’s with conserved baryons and radiation, after marginalizing over the rest of the fitting parameters indicated in Table 2. The CDM () appears disfavored once more at c.l. for the RVMc (denoted in the figure as RVM) and , and at c.l. for the . As in Figs. 2 and 3, subsequent marginalization over increases slightly the c.l. and renders fitting values with a statistical significance of for the RVMc and .

5 RVM’s and the cosmic time evolution of masses and couplings

The framework outlined in the previous sections suggest that owing to a small interaction with vacuum the matter density of the Universe might not be conserved during the cosmic expansion and in such case it would obey an anomalous conservation law. However, if there is a feedback of matter with the cosmic vacuum it would open a new window into the domain of the time variation of the fundamental constants of Nature999For a summarized introduction, see e.g. [69]. The idea traces back to early proposals in the thirties on the possibility of a time evolving gravitational constant by Milne [70] and the suggestion by Dirac of the large number hypothesis [71], including the ideas by Jordan that the fine structure constant together with could be both space and time dependent [72, 73].. Such window would permit a limited, but nonvanishing, exchange of information between the two widely different worlds of Quantum Physics and General Relativity. In point of fact it could be considered as a kind of living fossil (i.e. what is left at present) of the glorious unification of QP and GR in the remote past, and therefore the late testimony of such epoch in the very early Universe. We have elsewhere called that subtle interplay:“the micro and macro connection” [24, 23]. If by some chance the physical laws still use a narrow passage of this sort to communicate the two ostensibly divorced worlds of QP and GR, there might be still hope for a real understanding of the grand picture of the Cosmos!

A large number of studies have been undertaken in our days from different perspectives on the possible variation of the fundamental “constants”, sometimes pointing to positive observational evidence [74] but often disputed by alternative observations [75] – see e.g. the reviews [76, 77, 78, 79, 80]. Of essential importance is to count on a consitent theoretical framework. The DVM’s, and in particular the subclass of RVM’s might offer new clues for a possible explanation of the potential reality of these subtle and often evanescent effects, which are being constantly scrutinized by means of highly sophisticated lab experiments and accurate astrophysical observations [81, 82].

In the simplest RVM case (17) the anomalous matter conservation law takes on the form given in Eq. (28), which is concomitant with a mild vacuum evolution of the sort (27). However, here we wish to reinterpret the anomalous matter conservation law in a different way [23, 24], see also [83, 84]. Rather than assuming that the presence of a nonvanishing is related to an anomalous conservation of the number density of particles, we may conjecture that it is associated to the nonconservation of the particle masses themselves. In other words, we suppose that while there is a normal dilution of the particle number density with the expansion for all the particle species, i.e. , the corresponding mass values are not preserved throughout the cosmic expansion:


being the current values () of the particle masses, and the various are the different anomaly indices for the non-conservation of each species. This was the point of view adopted in [23]. Notice that in Eq. (28), i.e. the mass density for the ith-species of particles, is indeed equal to , with given by (52) and . In the remaining of this section it will be necessary to distinguish among the different values of for each particle species, but for simplicity we shall differentiate only between the two large families of baryons and DM particles to which we will attribute the generic indices and respectively 101010The anomaly mass indices here should not be confused with the vacuum parameters introduced for the different DVM’s in the previous sections. However, once a DVM is selected, e.g. the canonical RVM defined by (17), we have a collection of anomaly indices for the different particle species which are related to the RVM vacuum parameter through (63) below. ( denoting here the generic particles contributing to the DM). Formulated in this fashion, such scenario is in principle closer to the general RVM one discussed in section 3.1 rather than to the more specific RVMc framework discussed in section 3.2, in which baryons and radiation were assumed to be conserved. However, as we will discuss below, the observational measurements actually suggest that and, therefore, in practice we will effectively stay quite close to the RVMc, namely to that very successful dynamical vacuum model capable of surpassing the CDM ability to describe the cosmological observations at c.l. (cf. sect. 4). This may be viewed as an additional motivation for such a particular realization of the RVM since it is naturally compatible both with the cosmological observations as well as with the current bounds on the possible variation of the fundamental constants.

The above interpretation leads to a peculiar “Weltanschauung” of the physical world, in which basic quantities of the standard model (SM) of strong and electroweak interactions, such as the quark masses, the proton mass and the quantum chromodynamics (QCD) scale parameter, , might not be conserved in the course of the cosmological evolution, see [23, 24]. Let us take, for example, the proton mass, which is given as follows: , where are the quark masses and the last term represents the electromagnetic (em) contribution. Obviously the leading term is the first one, which is due to the strong binding energy of QCD. Thus, the nucleon mass can be expressed to within very good approximation as , in which is a non-perturbative coefficient. The masses of the light quarks , and also contribute to the the proton mass, although with less than 10 and can therefore be neglected for this purpose. It follows that cosmic time variations of the proton mass are essentially equivalent to cosmic time variations of the QCD scale parameter: