# Running vacuum in the Universe and the time variation of the fundamental constants of Nature

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## Abstract

We compute the time variation of the fundamental constants (such as the ratio of the proton mass to the electron mass, the strong coupling constant, the fine-structure constant and Newton’s constant) within the context of the so-called *running vacuum models* (RVMs) of the cosmic evolution. Recently, compelling evidence has been provided that these models are able to fit the main cosmological data (SNIa+BAO+H(z)+LSS+BBN+CMB) significantly better than the concordance \(\Lambda \)CDM model. Specifically, the vacuum parameters of the RVM (i.e. those responsible for the dynamics of the vacuum energy) prove to be nonzero at a confidence level \({\gtrsim } 3\sigma \). Here we use such remarkable status of the RVMs to make definite predictions on the cosmic time variation of the fundamental constants. It turns out that the predicted variations are close to the present observational limits. Furthermore, we find that the time evolution of the dark matter particle masses should be crucially involved in the total mass variation of our Universe. A positive measurement of this kind of effects could be interpreted as strong support to the “micro–macro connection” (viz. the dynamical feedback between the evolution of the cosmological parameters and the time variation of the fundamental constants of the microscopic world), previously proposed by two of us (HF and JS).

### Keywords

Dark Matter Particle Masse Drift Rate Grand Unify Theory Dark Matter Particle## 1 Introduction

The possibility that the so-called “constants” of Nature (such as the particle masses and the couplings associated to their interactions) are actually not constants, but time evolving quantities following the slow pace of the current cosmological evolution, has been investigated in the literature since long time ago [1]. The history of these investigations traces back to early ideas in the 1930s on the possibility of a time evolving gravitational constant *G* by Milne [2, 3] and the suggestion by Dirac of the large number hypothesis [4, 5], which led him also to propose in 1937 the time evolution of *G*. The same year Jordan speculated that the fine-structure constant \(\alpha _\mathrm{em}\) together with *G* could be both space and time dependent [6, 7] – see also [8, 9]. This, more field-theoretically oriented, point of view was retaken later on by Fierz [10] and, finally, in the 1960s, *G* was formally associated to the existence of a dynamical scalar field \(\phi \sim 1/G\) coupled to the curvature. Such was the famous framework originally proposed by Brans and Dicke (“BD” for short) [11, 12, 13, 14], which was the first historical attempt to extend General Relativity (GR) to accommodate variations in the Newtonian coupling *G*. To avoid conflict with the weak equivalence principle, the variability of \(\phi \) in the BD-theory was originally restricted to a possible time evolution only, i.e. \(\phi (t)\) with no spatial dependence [15]. A generalization of the BD approach in the 1970s [16, 17, 18] subsequently led to a wide variety of scalar–tensor theories, which are still profusely discussed in the current literature. Furthermore, the subject of the time variation of the fine-structure constant (cf. [19] for other early proposals) is particularly prolific. Numerous studies have been undertaken in our days from different perspectives, sometimes pointing to positive observational evidence [20, 21, 22, 23] but often disputed by alternative observations [24, 25] – see e.g. the reviews [26, 27, 28, 29, 30].

The possibility that the particle masses could also drift with the cosmic evolution has also become a favorite target of the modern astrophysical observations. Thus e.g. the dimensionless proton-to-electron mass ratio, \(\mu \equiv m_p/m_e\), has been carefully monitored using quasar absorption lines. Claims on significant time variation \(\dot{\mu }/\mu \) at \(\sim \)4\(\sigma \) c.l. are available in the literature [31], although still unconfirmed by other observations [32]. Future high precision quantum optic experiments in the lab are also planned to test the possible time variation of these observables, and they will most likely be competitive [1]. Clearly, the time and space variation of the fundamental constants is a very active field of theoretical and experimental research that may eventually provide interesting surprises in the near future [33].

From a more modern perspective the dark energy (DE) theories of the cosmological evolution also predict the cosmic time variation of the fundamental constants. These include the string-dilaton models, Kaluza–Klein theories, chameleon models etc., in which the underlying dynamical fields are either massless or nearly massless – cf. e.g. [34, 35, 36, 37, 38, 39] for some early attempts and [28, 29] for a review of more recent proposals and the observational situation. The possibility that dark matter theories could also impinge on the time variation of fundamental constants has also been put forward [40]. Many other proposals are available in the literature; see e.g. [41] for more contemporary developments.

In this paper, we would like to focus our attention on a specific class of DE models leading to a time variation of the fundamental constants [42]. These models are particularly interesting since it has recently been shown that they prove capable of fitting the main cosmological data in a fully competitive way with the concordance \(\Lambda \)CDM model [43]. These are the so-called running vacuum models (RVMs) of the cosmological evolution; see e.g. [44] for a recent summarized discussion and [45, 46, 47, 48] for a more extensive review and a comprehensive list of references. In these models there are no ultralight scalar fields and the time variation of the fundamental constants is effectively triggered via quantum effects induced by the cosmological renormalization group, whose flow is naturally set up by the expansion rate *H*, cf. [45, 46] and the references therein. The framework is characterized by a dynamical vacuum energy density, \(\rho _{\Lambda }\), which is a power series of *H* and its time derivatives. For the current Universe, it suffices to consider \(\rho _{\Lambda }=\rho _{\Lambda }(H,\dot{H})\) up to linear terms in \(\dot{H}\) and quadratic in *H* [47, 48, 49, 50, 51]. However, extensions with higher powers have also been considered for describing inflation; see [47, 52] and [53, 54, 55, 56, 57] for different kind of scenarios, including anomaly-induced inflation [58]. Because of the dynamical nature of the vacuum in these models, a natural feedback occurs between the cosmological parameters and the fundamental constants of the microscopic world, such as the particle masses and coupling constants. Remarkably, recent studies have shown that the class of RVMs can provide an excellent fit to the main cosmological data (SNIa+BAO+H(z)+LSS+BBN+CMB) which is highly competitive with that of the \(\Lambda \)CDM – see most particularly [59, 60, 61, 62], and previous studies such as [49, 50, 63, 64] and the references therein. Therefore, there is plenty of motivation for further investigating these running vacuum models. In this paper, building upon the aforementioned works which single out the especial status of the RVMs, we wish to estimate the possible variation of the fundamental constants triggered by the dynamical interplay between the evolution of the vacuum energy density \(\rho _{\Lambda }=\rho _{\Lambda }(H)\) and the concomitant anomalous conservation law of matter (which may lead to time-dependent particle masses) and/or the time evolution of the gravitational coupling *G*(*H*). Such feedback between the ultralarge scales of cosmology and the minute scales of the subatomic physics is what two of us have called elsewhere “the micro and macro connection” [42, 65]. Recently it has been shown that it could also lead to a possible explanation for the origin of the Higgs potential in the gravitational framework [66].

This paper is organized as follows. After a preliminary historical discussion in the introduction, in Sect. 2 we recall the possibility of the cosmic time variation of the cosmological parameters in the context of GR without committing to any particular model. In Sect. 3 we focus on the running vacuum models (RVMs) as a particularly appealing framework where to realize the time evolution of the fundamental constants. In Sect. 4 we consider in detail the specific prediction of the RVMs concerning the time variation of the particle masses and couplings. In Sect. 5 we briefly discuss alternative dynamical vacuum models. Finally, in Sect. 6 we present our conclusions.

## 2 Cosmological models with time evolving parameters

*t*and whether this feature might be related to the cosmic evolution of the fundamental constants of gravitation. If so the cosmic time evolution of \(\mathcal{P}\) should be typically proportional to the rate of change of the scale factor of the cosmic expansion, i.e. \(\dot{\mathcal{P}}/\mathcal{P}\propto \dot{a}/a\equiv H\) (the Hubble rate). From this point of view we should expect (at least in linear approximation) that the fractional cosmic drift rate of \(\mathcal{P}(t)\) near our time is proportional to \(H_0=1.0227\,h\times 10^{-10}\,\mathrm{year}^{-1}\,\), with \(h\simeq 0.67\), and hence very small in a natural way. Specifically \(\dot{\mathcal{P}}/\mathcal{P}\sim H_0\,\Delta \mathcal{P}/\mathcal{P}\lesssim (\Delta \mathcal{P}/\mathcal{P})\,10^{-10}\) year\(^{-1}\). From the existing bounds on the known particles, typically we expect that \(\Delta \mathcal{P}/\mathcal{P}\) should vary between a few parts per million (ppm) to at most one part in a hundred thousand over a cosmological span of time, depending on the specific parameter (such as couplings, masses etc: \(\mathcal{P}= G, m_i,\alpha _\mathrm{em}, \alpha _s, \Lambda _\mathrm{QCD}\ldots \)). However, the attributes of the dark matter (DM) particles (e.g. their masses and couplings) could vary faster. The possible cosmic time evolution of these parameters can equivalently be described as a redshift dependence:

*a*(

*t*) is the scale factor (normalized to \(a(t_0)=1\) at present) and \(\mathcal{P}'(z)=\mathrm{d}\mathcal{P}/\mathrm{d}z\). It will prove useful to present most of our results in terms of the cosmological redshift.

The above micro–macro-connection scenario [42, 65] can be implemented from Einstein’s field equations in the presence of a time evolving cosmological constant, \(\Lambda (t)\). In fact, without violating the cosmological principle in the context of the Friedmann–Lemaître–Robertson–Walker (FLRW) Universe, nothing prevents \(\Lambda (t)\) and/or \(G=G(t)\) from being functions of the cosmic time.^{1} The field equations can be written \( G_{\mu \nu } = 8 \pi G \tilde{T}_{\mu \nu }\), where \(G_{\mu \nu } = R_{\mu \nu } - 1/2 g_{\mu \nu } R\) and \(\tilde{T}_{\mu \nu } = T_{\mu \nu } + g_{\mu \nu } \rho _{\Lambda }\) stand, respectively, for the Einstein tensor and the full energy-momentum tensor, in which \(T_{\mu \nu }\) is the ordinary part (involving only the matter fields) and \(g_{\mu \nu } \rho _{\Lambda }\) carries the time evolving vacuum energy density \(\rho _{\Lambda }(t) = \Lambda (t)/ 8 \pi G(t)\).

*G*and \(\Lambda \) constants. The corresponding generalization of the Friedmann equation reads

*G*. For \(G=\hbox {const.}\) and \(\rho _{\Lambda }=\hbox {const.}\) (i.e. within the context of the \(\Lambda \)CDM) Eq. (3) integrates trivially and renders the canonical conservation law of non-relativistic matter:

*G*, we can see that this scenario may well lead us to a concrete realization of the time variation of the fundamental constants that can be consistent with GR.

## 3 Running vacuum in the expanding Universe

### 3.1 Running *G* and running \(\rho _{\Lambda }\)

*G*is also a slowly varying cosmic variable. More specifically, we shall suppose that it varies logarithmically with the Hubble function as follows:

*G*is given in our case by (7). Notice that for \(\nu _m=0\) (matter conservation) and \(\nu _G=0\) (\(G=\)const.) the Hubble function (9) boils down to the \(\Lambda \)CDM, as should be expected.

^{2}. Therefore we concentrate here on the simplest form of the RVM density, which we can rewrite after an appropriate redefinition of \(a_0\) and \(a_2\) as follows:

^{3}. In particular, for \(\nu =0\) we recover the \(\Lambda \)CDM case. Notice that \(M_P^2\,H^2\) is of order \(\rho _{\Lambda }^0\) and hence for \(\nu \ne 0\) the term \(\sim \nu \,M_P^2\,H^2\) represents a small correction (for \(|\nu |\ll 1\)) to the constant value \(\rho _{\Lambda }^0\) and endows \(\rho _{\Lambda }\) of a mild dynamical behavior which can be favorable to observations. Indeed, by fitting the parameter \(\nu \) to the overall cosmological data one finds an improved fit as compared to the \(\Lambda \)CDM, provided \(|\nu |\sim 10^{-3}\)—for details, see [49, 50, 59, 60].

The connection between (11) and (8) can now be elucidated as follows [47, 49]. To start with, take \(G=\)const. as this simplifies the structure of the generalized conservation law (3). Inserting (5) in that law we can solve for \(\rho _{\Lambda }(z)\) and we find Eq. (8) with \(\nu _G=0\). Knowing the matter and vacuum densities, Friedmann’s equation immediately provides *H* as a function of the redshift and we arrive at Eq. (9) (with \(G=G_0\)). Combining this expression for *H*(*z*) with that of \(\rho _{\Lambda }(z)\) we find (11).

*G*(

*H*):

*H*when \(\nu _m\) and \(\nu _G\) are both non-vanishing. From (8) and (9) we find

*H*in all possible cases within this framework, i.e. even when matter is non-conserved and at the same time there is a running of the gravitational coupling. In other words, the running of \(\rho _{\Lambda }\) as a function of

*H*in (11) is controlled in all cases by \(\nu =\nu _m+\nu _G\), irrespective of whether one or none of these parameters is zero.

### 3.2 Free parameters

Let us summarize the situation with the free parameters in the class of RVMs considered here. The basic parameters are \((\nu _m,\nu _G)\), which are associated to the anomalous matter conservation law (5) and the time evolution of the gravitational coupling, Eq. (7). Given these two parameters the Bianchi identity (3) determines the evolution of the vacuum energy density, Eq. (8), through \(\nu =\nu _m+\nu _G\). However, as we have mentioned, there is no reason to expect that all of the particle masses should have the same anomaly index, and hence we expect that the anomaly mass index \(\nu _m\) can be expressed in terms of the different particle indices \(\nu _i\) defined in Eq. (6). For example, baryons can have the index \(\nu _B\) (assumed the same for all of them), but it could be different from the index for DM particles, X, which we call \(\nu _X\). The relation between the overall anomaly index \(\nu _m\) and the specific indices \((\nu _B,\nu _X)\) will be discussed in Sect. 4.

Furthermore, as indicated, if it turns that \(a_1\ne 0\) in Eq. (10), then an additional parameter is still possible for the running of the vacuum energy in the present Universe, but we shall not take it into account since it is not necessary to illustrate the possible existence of the basic effects under study. In actual fact, in all phenomenological considerations we will assume \(a_1=0\) together with one of the following two possibilities: either i) \(\nu _m\ne 0\) with \(\nu _G=0\), or ii) \(\nu _G\ne 0\) with \(\nu _m=0\). This will suffice to parametrize the time variation of the fundamental constants that we are considering here. In such context the Bianchi identity enforces the value of \(\nu \) (the parameter that controls the running of the vacuum energy density in (11)) to be either \(\nu _m\) or \(\nu _G\), depending on whether we assume either that *G* is fixed and the matter has some anomaly conservation law, or that the matter is strictly conserved and *G* has some evolution, but not both situations at the same time. While \(\nu \) could perfectly receive simultaneous contributions from both kinds of effects—in the above-mentioned form \(\nu =\nu _m+\nu _G\)—at the moment it is not possible to individually disentangle them phenomenologically. Thus, in our numerical evaluations we will always assume the separate situations in which either \(\nu =\nu _m\) or \(\nu =\nu _G\).

From the foregoing considerations we see that the RVMs offer several possibilities for the time variation of the fundamental “constants”, all of them being connected through the evolution laws (5), (7), (8) and (11), in which there is a built-in principle for exchanging energy between matter, vacuum and the gravitational coupling in different combinations that are compatible with GR.

## 4 Time evolution of the fundamental constants in the RVM

In this section we wish to evaluate the specific impact of the running vacuum models (RVMs) on the time evolution of the fundamental constants of the standard model (SM) of particle physics and the fundamental constants of cosmology. Basically we will assess the predicted variation of the particle masses, most conspicuously the proton mass (through the proton-to-electron mass ratio \(\mu \equiv m_p/m_e\)), the QCD scale \(\Lambda _\mathrm{QCD}\), the QED fine-structure constant \(\alpha _\mathrm{em}\), the corresponding QCD coupling \(\alpha _s\), the gravitational constant *G* and the cosmological constant \(\Lambda \). We will also consider the possible implications from Grand Unified Theories (GUT’s).

### 4.1 Time variation of masses and couplings in the SM

In this paper we attribute the cosmic variation of the particle masses to the energy exchange with the cosmic vacuum according to the RVM framework outlined in the previous section, in which a possible additional variation of the gravitational constant may also concur. In order to estimate quantitatively these effects within the RVM, we take as a basis the numerical fit estimates obtained in [49, 50, 59, 60] using the known data on SNIa+BAO+H(z)+LSS+BBN+CMB – see also [48] for a review. Among these observational sources (which involve several hundreds of data points indicated in these references) we use 36 data points on the Hubble rate *H*(*z*) at different redshifts in the range \(0 < z \le 2.36\), as compiled in [77, 78], out of which 26 data points are inferred from the differential age method, whereas 10 correspond to measures obtained from the baryonic acoustic oscillation method (cf. Fig. 1). These data will play a significant role in our aim to constrain cosmological parameters because they are obtained from model-independent direct observations. In particular, we use this compilation for investigating a possible temporal evolution of the particle masses, both for baryons and dark matter.

*H*(

*z*) for both the \(\Lambda \)CDM model (dashed line) and the RVM (solid one). The difference between the two is small, of course, because the RVM Hubble function (9) differs only mildly from the standard one owing to the parameters \(\nu _m\), \(\nu _G\) being small in absolute value. However, the small differences are perfectly visible in Fig. 1 and are sufficient to improve significantly the overall fit to the SNIa+BAO+H(z)+LSS+BBN+CMB data points. According to [49], the data show a preference (at the level of \({\gtrsim }3\sigma \)) for a dynamical vacuum of the form (11) rather than the rigid vacuum (\(\nu =0\)) of the \(\Lambda \)CDM case, for which \(\rho _{\Lambda }=\hbox {const}.\) The obtained fit values of the vacuum parameter \(\nu \) stay in the ballpark of \(10^{-3}\) [48, 59, 60], and therefore we can use this order of magnitude determination as a characteristic input in our estimate of the variation of the fundamental constants. The predicted mass drift rates are indicated in Figs. 2 and 3, and in the following we explain their origin.

*X*, respectively. Assuming that the mass non-conservation law in Eq. (5) is to be attributed to the change of the mass of the particles – cf. Eq. (6) – the relative total time variation of the mass density associated to such mass anomaly can be estimated as follows:

*r.h.s.*can be written

*l.h.s*of Eq. (20) we can rewrite it in the following convenient way:

As far as the lepton index \(\nu _L\) is concerned, it cannot have any significant effect in the above Eq. (25) since it is suppressed by the lepton mass rate in the Universe. Such equation can actually be checked experimentally, for \(\nu _m\) is related to the running of the vacuum energy density (e.g. \(\nu _m=\nu \) when \(\nu _G=0\), as explained in Sect. 3) and \(\nu \) can be fitted from cosmological observations based on SNIa+BAO+H(z)+LSS+BBN+CMB data [48, 59, 60], and it is found to be of order \(10^{-4}{-}10^{-3}\), whereas \(\nu _B\) can be determined from astrophysical and lab experiments usually aimed at determining the time evolution of the ratio \(\mu =m_p/m_e\) [28, 29, 41]. Thus, if the Eq. (21) – or equivalently (25) – must be fulfilled, we can indeed check if the DM part \(\nu _{X}\) (which, of course, cannot be measured individually) plays a significant role in it.

^{4}Both of them, however, lead to \(\nu _X\sim \nu _m\sim 10^{-3}\) via Eq. (25), what clearly points to the crucial role of the DM contribution to explain the bulk of the mass drift rate in the Universe (cf. Fig. 4).

### 4.2 Time variation of *G* and \(\Lambda \)

*G*decreases with the redshift and therefore

*G*behaves as an asymptotically free coupling, that is to say,

*G*decreases in the past, which is the epoch where the Hubble rate (with natural dimension of energy) is higher. This is of course already obvious from Eq. (7) for \(\nu _G>0\). Moreover, from the plot on the right in Fig. 5 we learn that the rhythm of variation of

*G*slows down with the cosmic expansion, i.e. the rate of change is larger in the past.

*G*at any given redshift. The upper curve in Fig. 6 corresponds to \(\nu _G=0.0005\) whereas the lower one to \(\nu _G=0.001\). The shaded area gives the prediction of \(\Delta G/G\) for the values of \(\nu _G\) comprised in that interval for each

*z*. Worth noticing is that if we extend the domain of applicability of Eq. (7) up to the BBN epoch (\(z=z_N\sim 10^9\)), we find that the relative variation of

*G*at the BBN time as compared to the present is \((G(z_N)-G_0)/G_0\simeq -0.06\), which is less than \(10\%\) in absolute value and hence compatible with the current BBN bounds [28, 29].

*H*. As we shall see in Sect. 5, there are alternatives models that may lead to Eq. (31) using ad hoc assumptions on the interaction between DE and DM. By the same token the expression of

*G*directly in terms of

*H*is closer to the spirit of the RVM since it can be derived from the RG formalism of QFT in curved spacetime [45, 46, 58].

### 4.3 Time evolution of the fine-structure constant

Motivated by a possible indication of a variation (decrease) in the fine-structure constant at high redshift, as well as a possible spatial variation (see [82] and the references therein, as well as the reviews [28, 29, 30]), we will address here this topic from the point of view of the implications of the running vacuum energy density throughout the cosmic history. We have mentioned before that in the electroweak sector of the SM is not possible to establish a connection between the cosmological evolution of the weak and electromagnetic couplings to the particle masses because there is no analog in this sector of the QCD scale parameter \(\Lambda _\mathrm{QCD}\). Notwithstanding, it is still possible to relate the electroweak couplings to \(\Lambda _\mathrm{QCD}\) itself in an indirect way if we use the hypothesis of Grand Unification of the SM couplings at a very high energy scale. We will focus here on the fine-structure constant \(\alpha _\mathrm{em}=e^2/4\pi \) and its correlated time evolution with the strong coupling counterpart \(\alpha _{s}=g_s^2/4\pi \), and ultimately with the time evolution of \(\Lambda _\mathrm{QCD}\) and \(\mu =m_p/m_e\).

*z*. Such variation is possible if we have a consistent theoretical framework supporting this possibility, such as the RVM picture described in Sect. 3. Each of the couplings \(\alpha _i=g^2_i/4\pi \) is a function of the running scale \(\mu _R\), and they follow the standard (1-loop) running laws

*Z*-boson mass scale \(\mu _R = M_Z\), where both \(\alpha _{em}\) and \(\alpha _s\) are known with precision, one obtains

*z*at a fixed value of \(\mu _R\), and \(\alpha _\mathrm{em}^0(z)\) is its current value (\(z=0\)). Since \(\nu _B\) is a small parameter, related to the fitted value \(\nu _m\sim 10^{-3}\) [59, 60, 61] through (25), we can estimate the relative variation of the electromagnetic fine-structure constant with the redshift as follows:

Compilation of recent direct measurements of the fine-structure constant obtained by different spectrographic methods. For details of these methods, see the references cited above

| \( \Delta \alpha / \alpha (ppm) \) | Ref. |
---|---|---|

1.08 | \( 4.3 \pm 3.4\) | [87] |

1.14 | \( -7.5 \pm 5.5\) | [88] |

1.15 | \( -0.1 \pm 1.8\) | [89] |

1.15 | \( 0.5 \pm 2.4\) | [90] |

1.34 | \( -0.7 \pm 6.6\) | [88] |

1.58 | \( -1.5 \pm 2.6\) | [91] |

1.66 | \( -4.7 \pm 5.3\) | [87] |

1.69 | \( 1.3 \pm 2.6\) | [92] |

1.74 | \( -7.9 \pm 6.2\) | [87] |

1.80 | \( -6.4 \pm 7.2\) | [88] |

1.84 | \( 5.7 \pm 2.7\) | [89] |

Defining \(\Delta \alpha _\mathrm{em}/\alpha _\mathrm{em}=3\nu _\mathrm{em}\) by analogy with Eq. (24), we learn that the effective running index of the em coupling is some 30 times smaller than that of the baryonic index and with opposite sign, in other words \(\nu _\mathrm{em}\simeq -0.03\,\nu _B\) (up to logarithmic evolution with the redshift).

What is the possible impact of the RVM here? It is remarkable that the above mentioned results, whether from astrophysical or lab measurements, can be accounted for (in order of magnitude) within the RVM in combination with the GUT hypothesis. Indeed, we can see that the theoretical RVM prediction falls right within the order of magnitude of the typical measurements quoted in Table 1 and in Fig. 8, provided \(\nu _B\) lies in the range from \(10^{-4}\) to \(10^{-5}\). This follows from Eq. (38), which, roughly speaking, says that the RVM prediction is of order \(\Delta \alpha _\mathrm{em}/\alpha _\mathrm{em}\sim -0.09\nu _B\) up to log corrections in the redshift. More precisely, in Fig. 8 we have superimposed the exact theoretical prediction \(\alpha _\mathrm{em}(z)\) according to Eq. (38). We can see that, notwithstanding the sizable error bars, the trend of the measurements in Table 1 suggests a decrease of \(\alpha _\mathrm{em}\) with the redshift (as there are more points compatible with \(\Delta \alpha _\mathrm{em}<0\) than points compatible with \(\Delta \alpha _\mathrm{em}>0\)). This behavior has been previously noted in the literature [82] and is roughly in accordance with our theoretical curves in Fig. 8. But of course we need more precise measurements to confirm the real tendency of the data, as the errors are still too large and no firm conclusion is currently possible.

Independent of other details, the following remarks should be emphasized. First, despite it is not possible to be more precise concerning the best fit value for \(\nu _B\), in all cases the measurements in Table 1 indicate a maximum effect of 1–10 ppm, i.e. \(|\Delta \alpha _\mathrm{em}/\alpha _\mathrm{em}|\) at the level of \(10^{-6}\) to \(10^{-5}\). This can be accommodated in the RVM framework since \(\nu _B\) describes the effect from only the baryonic component in Eq. (25). Such component should be naturally smaller than the value of the total index \(\nu _m\sim 10^{-3}\) fitted on the basis of the overall analysis involving the SNIa+BAO+H(z)+LSS+BBN+CMB observables [49, 59, 60]. Second, the correct order of magnitude for \(\nu _B\), which we have obtained from the direct \(\Delta \alpha _\mathrm{em}/\alpha _\mathrm{em}\) observations (viz. \(\nu _B\sim 10^{-4}{-}10^{-5}\)) does coincide with the result inferred from our previous considerations on the alternative observable \(\Delta \mu /\mu \) in Sect. 4.1. Put another way, we could have input the value of \(\nu _B\) needed to explain the typical measurements of the observable \(\Delta \mu /\mu \) and we would have naturally predicted the typical range of values of \(\Delta \alpha _\mathrm{em}/\alpha _\mathrm{em}\) derived from direct observations, and vice versa. As noted previously, this is because the RVM in combination with the GUT framework neatly predicts the relation \(\nu _\mathrm{em}\simeq -0.03\,\nu _B\) (up to a logarithmic correction with the redshift).

In the light of the above results, the baryonic index \(\nu _B\) in Eq. (25) is definitely subdominant as compared to the dark matter one, \(|\nu _B|\ll |\nu _X|\), and hence \(\nu _X\) must be of order of the total matter index \(\nu _m\sim 10^{-3}\) fitted from the overall cosmological observations [49, 59, 60]. In other words, we find once more that it must be the DM component that provides the bulk of the contribution to the time variation of masses in the Universe. This fact was not obvious a priori, and is not necessarily related to the overwhelming abundance of DM as compared to baryons, for the large amounts of DM could simply remain passive and not evolve at all throughout the cosmic expansion. If the best fit value of the total mass variation index \(\nu _m\) would have been, say of order \(10^{-5}\), Eq. (25) could have been naturally fulfilled with \(\nu _X\ll \nu _B\sim 10^{-4}\) and this would still be compatible with the measurements in Table 1. However, the fact that the value of \(\nu _m\) (obtained from the overall cosmological fit to the data within the RVM [59, 60, 61]) comes out significantly larger than the baryonic index \(\nu _B\) has nontrivial consequences and provides an independent hint of the need for (time evolving) dark matter. Taking into account that we have been able to infer this same conclusion from the analysis of the two independent observables \(\Delta \mu /\mu \) and \(\Delta \alpha _\mathrm{em}/\alpha _\mathrm{em}\), which become correlated in this theoretical framework, does reinforce the RVM scenario and places the contribution from the DM component to the forefront of our considerations concerning the total mass drift rate in the Universe [42].

## 5 Alternative dynamical vacuum models interacting with matter

*Q*denotes the background energy source between dark matter and vacuum energy (or in general some dark energy source). From the previous equations we see that for \(Q > 0\) the matter energy density increases whereas the vacuum energy density decreases, and hence the energy flows from vacuum to matter, and vice versa for

*Q*featuring the opposite sign. In other words, for \(Q>0\) the vacuum is decaying into matter whereas for \(Q<0\) the matter decays into vacuum. Of these two options the naturally preferred one, at least from the point of view of the second principle of thermodynamics, should be the first one. Let us also mention that it is usually assumed that the vacuum decays only into DM [61]. This effect has little quantitative implications for the present analysis since we have seen that the baryonic component is essentially conserved (\(\xi _B\ll \xi _X\), see Sect. 4.1), and therefore we shall not consider this correction here. For more details, see [48].

*Q*; see e.g. [94, 95, 96]. For illustration, let us consider two frequently discussed phenomenological expressions in the literature, namely a source proportional to \(\rho _{\Lambda }\) in the form \(Q = q_{\Lambda }H \rho _{\Lambda }\) (hereafter called “\(q_{\Lambda }\)-model”), and a source proportional to the matter density, \(Q = q_m H \rho _{m}\) (“\(q_m\)-model”), where \(q_{\Lambda }\) and \(q_m\) are small dimensionless parameters (\(|q_{\Lambda }|,|q_m|\ll 1\)). Considering \(Q = q_{\Lambda }H \rho _{\Lambda }\), we trivially find from (40) the vacuum evolution law

*m*(

*z*) decreases with

*z*, and hence increases with the expansion, if \(q_m>0\) (corresponding to a situation of decay of vacuum into matter), and decreases with the expansion if \(q_m<0\) (when matter decays into vacuum). In contradistinction to the \(q_{\Lambda }\)-model, we note that the masses remain now always positive irrespective of the sign of \(q_m\).

*Q*. In contrast, in the RVM case the form of

*Q*is not taken as a mere phenomenological ansatz, as it is theoretically determined. In fact, it can be derived from the dynamical vacuum equation (11), which leads to Eq. (8) when the model is explicitly solved. For example, take \(G=\) const. (\(\nu _G=0\)) in which case the local conservation equation for the RVM can be put in the form (39) with \(Q=-\dot{\rho }_{\Lambda }\). We may compute explicitly the time derivative of \(\rho _{\Lambda }\) and reexpress the result in terms of the redshift using Eq. (1). We find

Recall that ultimately the behavior of the RVM stems from the dynamical vacuum equation (11), which follows from the RG-flow in QFT in curved spacetime [45, 46, 47]. As mentioned in Sect. 3, Eq. (11) can be extended with higher powers of *H* so as to include inflation in a single unified theory describing the cosmic evolution from the early Universe until the current one [47, 52, 53, 54, 55, 56, 57].

From the foregoing discussion it is clear that different dynamical vacuum energy models exist for describing the possible time variation of the fundamental constants in a framework which is consistent with GR. Some of these models are more phenomenological, but in the RVM case there is a more concrete theoretical motivation for the dynamical vacuum structure. It suggests that a slow change in the fundamental “constants” can be theoretically motivated and cannot be ruled out at present. The subject is therefore worthwhile being further investigated in the light of new data, as it can reveal new clues to fundamental physics.

## 6 Discussion and conclusions

In this paper we have addressed the running vacuum models (RVM) of the cosmic evolution and the possible implications they could have in explaining the reported hints of the time variation of the so-called fundamental constants of Nature, such as masses, coupling constants etc, including the gravitational coupling *G* and the \(\Lambda \)-term in Einstein’s equations. The impact from the RVM on this issue stems from the cosmological energy exchange between vacuum, matter and the possible interplay with the Newtonian coupling *G* and the vacuum energy density \(\rho _{\Lambda }=\Lambda /8\pi G\). Because the possible cosmological running of these quantities is controlled by the Hubble parameter, the RVM predicts that the associated rhythm of change, i.e. the drift rate of the fundamental constants should naturally be as moderate as dictated by the expansion rate of the Universe at any given instant of the cosmic history. On this basis it is possible to compute the time evolution of the vacuum energy density and the corresponding change of the gravitational coupling and the particle masses. Combining this scenario with the GUT hypothesis we have obtained a prediction for the time evolution of the fine-structure constant correlated with the time evolution of the proton mass, or more precisely the proton-to-electron mass ratio \(\mu =m_p/m_e\).

Taking into account that the small, but non-vanishing, rate of change of the vacuum energy density has been fitted to the cosmological data at a rather significant confidence level—see the recent studies [59, 60, 61, 62]—and bearing in mind that such vacuum rate of change impinges on the corresponding variation of the particle masses, we conclude that the mass variation must be essentially supported by the mass drift rate of the dark matter (DM) particles (since the time evolution of the baryonic component is found to be some two orders of magnitude smaller). This can be interpreted as an indirect alternative hint of the need for DM. If in the future the precision of these experiments further improves we might well find ourselves on the verge of measuring these subtle effects and perhaps be in a position to check if they can be explained within the kind of theoretical running vacuum models that we have studied here.

## Footnotes

- 1.
- 2.
- 3.
- 4.
We note that despite the fact that stable leptons (essentially electrons) do not contribute in any significant way to the r.h.s. of Eqs. (21) and (25), the relative variation \(\Delta m_e/m_e\) could be as big as \(\Delta m_p/m_p\), in principle. This option must be kept in mind when considering the total time variation of quantities involving a ratio of baryon and lepton masses, such as \(\mu =m_p/m_e\).

## Notes

### Acknowledgements

HF and JS are both grateful to the Institute for Advanced Study at the Nanyang Technological University in Singapore for hospitality and support while this work was being accomplished. JS has been supported in part by FPA2013-46570 (MICINN), CSD2007-00042 (CPAN), 2014-SGR-104 (Generalitat de Catalunya) and MDM-2014-0369 (ICCUB). RCN acknowledges financial support from CAPES Foundation Grant No. 13222/13-9 and is grateful for the hospitality at the Dept. FQA, Universitat de Barcelona.

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