#
Running Vacuum in the Universe: Phenomenological Status in Light of the Latest Observations, and Its Impact on the σ_{8} and H_{0} Tensions

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## Abstract

**:**

`CLASS`and the Monte Carlo sampler

`MontePython`for the statistical analysis, as well as the statistical DIC criterion to compare the running vacuum against the rigid vacuum (${\nu}_{\mathrm{eff}}=0$). On fundamental grounds, ${\nu}_{\mathrm{eff}}$ receives contributions from all the quantized matter fields in FLRW spacetime. We show that with a tiny amount of vacuum dynamics (${\nu}_{\mathrm{eff}}\ll 1$) the global fit can improve significantly with respect to the $\mathsf{\Lambda}$CDM and the mentioned tensions may subside to inconspicuous levels.

## 1. Introduction

## 2. Running Vacuum in the Universe

## 3. Type I: Running Vacuum Interacting with Dark Matter

#### 3.1. Background Equations

`CLASS`1 [102,103], where we have implemented our model.

`CLASS`solves the Einstein and Boltzmann differential equations at any value of the scale factor and, in particular, provides the functions ${\rho}_{\mathrm{ncdm}}\left(a\right)$ and ${p}_{\mathrm{ncdm}}\left(a\right)$.

`CLASS`then performs a rather artificial splitting of these quantities, as if they came from the sum of an ultra-relativistic fluid (denoted with a subscript $\nu $) and a nonrelativistic one (denoted with a subscript h),

`CLASS`, computing the ratio $r={\rho}_{h}/{\rho}_{m}$ for the whole of cosmic history. We have found that r varies smoothly from ${10}^{-7}$ at redshift $z={10}^{14}$ to ${10}^{-3}$ at $z=0$, considering a massive neutrino with mass $\sim \mathcal{O}\left(0.1\right)$ eV. In addition, r is multiplied by $\nu $ in Equation (7), so the resulting quantity is of the order $\mathcal{O}\left({10}^{-5}\right)$ at most. Therefore, we deem it natural and licit to drop this term to make things easier without any significant loss in accuracy in our calculation.

#### 3.2. Perturbation Equations

`CLASS`, using the synchronous gauge. Denoting by $\tau $ the conformal time, the perturbed (flat three-dimensional) FLRW metric in the conformal frame reads [104],

`CLASS`in order to accommodate the dynamical character of the VED, i.e., the fact that ${\overline{\rho}}_{\mathrm{vac}}^{\prime}\ne 0$. In this work, we consider adiabatic perturbations for the various matter and radiation species.

#### 3.3. Type I with Threshold

## 4. Type II: Running Vacuum with Running $\mathbf{G}$

#### 4.1. Background Equations

`CLASS`system solver [102,103]. Now, the point is that for the standard cosmological model,

`CLASS`computes H and $\dot{H}$ after computing the energy densities of the various components that fill the universe. In the model under consideration though, we cannot proceed in the same way, because we first need to compute G. Before explaining how it is still possible to solve the system in the

`CLASS`platform, it is useful to rewrite the above equations in terms of the auxiliary variable4$\phi \equiv {G}_{N}/G$. One expects $\phi \simeq 1$ at present and one may even impose this condition (see, however, below). In terms of $\phi $, the set of relevant equations read

`CLASS`to apply the finite difference method to solve the system step by step. In each of these steps

`CLASS`computes ${H}_{n+1}$ from ${H}_{n}$ and ${\dot{H}}_{n}$ (46). For the latter it takes the various energy densities and pressures. Then, we can employ the Friedmann equation to compute ${\phi}_{n+1}$,

Survey | z | Observable | Measurement | References |
---|---|---|---|---|

6dFGS+SDSS MGS | $0.122$ | ${D}_{V}({r}_{d}/{r}_{d,\mathrm{fid}})$ [Mpc] | $539\pm 17$ [Mpc] | [125] |

DR12 BOSS | 0.32 | $H{r}_{d}/({10}^{3}\mathrm{km}/\mathrm{s})$ | $11.549\pm 0.385$ | [127] |

${D}_{A}/{r}_{d}$ | $6.5986\pm 0.1337$ | |||

0.57 | $H{r}_{d}/({10}^{3}\mathrm{km}/\mathrm{s})$ | $14.021\pm 0.225$ | ||

${D}_{A}/{r}_{d}$ | $9.389\pm 0.1030$ | |||

WiggleZ | 0.44 | ${D}_{V}({r}_{d}/{r}_{d,\mathrm{fid}})$ [Mpc] | $1716.4\pm 83.1$ [Mpc] | [126] |

0.60 | ${D}_{V}({r}_{d}/{r}_{d,\mathrm{fid}})$ [Mpc] | $2220.8\pm 100.6$ [Mpc] | ||

0.73 | ${D}_{V}({r}_{d}/{r}_{d,\mathrm{fid}})$ [Mpc] | $2516.1\pm 86.1$ [Mpc] | ||

DESY3 | $0.835$ | ${D}_{M}/{r}_{d}$ | $18.92\pm 0.51$ | [128] |

eBOSS Quasar | 1.48 | ${D}_{M}/{r}_{d}$ | $30.21\pm 0.79$ | [129] |

${D}_{H}/{r}_{d}$ | $13.23\pm 0.47$ | |||

Ly$\alpha $-Forests | $2.334$ | ${D}_{M}/{r}_{d}$ | ${37.5}_{-1.1}^{+1.2}$ | [130] |

${D}_{H}/{r}_{d}$ | $8.{99}_{-0.19}^{+0.20}$ |

**Table 2.**Values of $H\left(z\right)$ from cosmic chronometers and their $1\sigma $ uncertainties, which include the contribution of statistical and systematic effects [131]. They are expressed in km/s/Mpc. We have considered the correlations between the data points marked with a *, as discussed in [131]. In some of the quoted references, the authors provide measurements obtained with two different stellar population synthesis (SPS) models. In these cases, we have employed the mean of the two central values and statistical errors. The systematic uncertainty already accounts for the choice of SPS model.

z | $\mathit{H}\left(\mathit{z}\right)$ [km/s/Mpc] | References |
---|---|---|

$0.07$ | $69.0\pm 19.6$ | [132] |

$0.09$ | $69.0\pm 12.0$ | [133] |

$0.12$ | $68.6\pm 26.2$ | [132] |

$0.17$ | $83.0\pm 8.0$ | [134] |

0.1791 * | $77.72\pm 6.01$ | [135] |

0.1993 * | $77.79\pm 6.83$ | [135] |

$0.2$ | $72.9\pm 29.6$ | [132] |

$0.27$ | $77.0\pm 14.0$ | [134] |

$0.28$ | $88.8\pm 36.6$ | [132] |

0.3519 * | $85.45\pm 15.75$ | [135] |

0.3802 * | $86.17\pm 14.61$ | [136] |

$0.4$ | $95.0\pm 17.0$ | [134] |

0.4004 * | $79.90\pm 11.38$ | [136] |

0.4247 * | $90.39\pm 12.76$ | [136] |

0.4497 * | $96.24\pm 14.38$ | [136] |

$0.47$ | $89.0\pm 49.6$ | [137] |

0.4783 * | $83.74\pm 10.18$ | [136] |

$0.48$ | $97.0\pm 62.0$ | [138] |

0.5929 * | $106.80\pm 15.06$ | [135] |

0.6797 * | $94.875\pm 10.600$ | [135] |

$0.75$ | $89.0\pm 49.6$ | [139] |

0.7812 * | $96.27\pm 12.72$ | [135] |

0.8754 * | $124.70\pm 17.13$ | [135] |

$0.88$ | $90.0\pm 40.0$ | [138] |

$0.9$ | $117.0\pm 23.0$ | [134] |

1.037 * | $133.35\pm 18.12$ | [135] |

$1.3$ | $168.0\pm 17.0$ | [134] |

1.363 * | $163.95\pm 34.61$ | [140] |

$1.43$ | $177.0\pm 18.0$ | [134] |

$1.53$ | $140.0\pm 14.0$ | [134] |

$1.75$ | $202.0\pm 40.0$ | [134] |

1.965 * | $191.10\pm 51.91$ | [140] |

**Table 3.**Published values of $f\left(z\right){\sigma}_{8}\left(z\right)$; see the quoted references and text in Section 5.

Survey | z | $\mathit{f}\left(\mathit{z}\right){\mathit{\sigma}}_{8}\left(\mathit{z}\right)$ | References |
---|---|---|---|

ALFALFA | $0.013$ | $0.46\pm 0.06$ | [141] |

6dFGS+SDSS | $0.035$ | $0.338\pm 0.027$ | [142] |

GAMA | $0.18$ | $0.29\pm 0.10$ | [143] |

$0.38$ | $0.44\pm 0.06$ | [144] | |

WiggleZ | $0.22$ | $0.42\pm 0.07$ | [145] |

$0.41$ | $0.45\pm 0.04$ | ||

$0.60$ | $0.43\pm 0.04$ | ||

$0.78$ | $0.38\pm 0.04$ | ||

DR12 BOSS | $0.32$ | $0.427\pm 0.056$ | [127] |

$0.57$ | $0.426\pm 0.029$ | ||

VIPERS | $0.60$ | $0.49\pm 0.12$ | [146] |

$0.86$ | $0.46\pm 0.09$ | ||

VVDS | $0.77$ | $0.49\pm 0.18$ | [147,148] |

FastSound | $1.36$ | $0.482\pm 0.116$ | [149] |

eBOSS Quasar | $1.48$ | $0.462\pm 0.045$ | [129] |

#### 4.2. Perturbation Equations

`CLASS`computing platform is more difficult than the background part (which was already nontrivial), and certainly much more involved than the one carried out for the type-I variant (viz. the one with vacuum–CDM interaction). We use again the synchronous gauge, but in this case the gauge unfortunately does not fix $\delta {\rho}_{\mathrm{vac}}=0$, in contrast to the situation in Section 3.2, so we have to keep the contribution of the vacuum perturbation in our equations.

`CLASS`computes $g=\left[(\overline{p}+\overline{\rho})\theta \right]$ but not its derivative. If we knew how to write ${\left[(\overline{p}+\overline{\rho})\theta \right]}^{\prime}$ in terms of $\delta \phi $, $\delta {\phi}^{\prime}$, g, and other accessible quantities, then we could obtain the differential equation for $\delta \phi $ from (64) and implement it in

`CLASS`without defining explicitly $\delta {\rho}_{\mathrm{vac}}$, just incorporating the effect of the vacuum perturbation directly into the equations. This is actually possible. Let us start by differentiating (65). We obtain:

`CLASS`does not provide it to us we need to evaluate it by adding some supplementary piece in the code. It is possible to obtain ${g}^{\prime}$ upon differentiating (61) and combining it with (62). The result reads

`CLASS`. In addition, ${h}^{\prime}$ is provided by

`CLASS`. Now, we only need to substitute ${g}^{\prime}$ from (68) into (67), and substitute the resulting expression for $\delta {\rho}_{\mathrm{vac}}^{\prime}$ into (64). In doing so, we finally obtain the equation for the $\delta \phi $ perturbation:

`CLASS`.

## 5. Data and Methodology

**SNIa**: We consider the data from the so-called ‘Pantheon+’ compilation [151], which contains the apparent magnitudes and redshifts associated with 1701 light curves obtained from 1550 SNIa in the redshift range $0.001\le z\le 2.26$. See Section 2.2 of [152] for details of the theoretical formulae employed to take into account these data points. Interestingly, the new Pantheon+ compilation also includes the 77 light curves from the 42 SNIa in the host galaxies employed by the SH0ES team in their analysis [153,154]. The distance to the host galaxies has been measured with calibrated Cepheids. The inclusion of these luminosity distances in our dataset will be made clear by adding the label “+SH0ES”. They break the existing full degeneracy between ${H}_{0}$ and the absolute magnitude of SNIa, M, when only SNIa are considered in the analysis. The SH0ES calibration of the supernovae in conjunction with the cosmic distance ladder leads to larger preferred values of the Hubble parameter of $73.04\pm 1.04$ km/s/Mpc [153]. This large value, as compared to Planck’s measurement ($67.36\pm 0.54$ km/s/Mpc, obtained from the TT,TE,EE+lowE+lensing data [155]), is at the root of the $\sim 5\sigma $${H}_{0}$ tension.

**BAO**: We employ 13 data points on isotropic and anisotropic BAO estimators. See Table 1 for the exact values and the corresponding references.

**Cosmic chronometers**: In our analyses, we use 32 data points on the Hubble parameter $H\left({z}_{i}\right)$ measured with the differential age technique [156]. They span the redshift range $0.07\le z\le 1.965$. We provide the complete list of data points and the corresponding references in Table 2. We have considered the effect of the known correlations between the various data points, as explained in [131]; see also Table 2 and its caption. The covariance matrix has been computed using the script provided in the following link6.

**LSS**: Fifteen large-scale structure (LSS) data points between $0.01\lesssim z\lesssim 1.5$, embodied in the observable $f\left({z}_{i}\right){\sigma}_{8}\left({z}_{i}\right)$, which is known as the weighted linear growth rate, with $f\left(z\right)$ being the so-called growth factor and ${\sigma}_{8}\left(z\right)$ the root-mean-square mass fluctuations on the ${R}_{8}=8{h}^{-1}$ Mpc scale. See Table 3 for the complete list of data points and the corresponding references. We can take advantage of the relation $f\left(z\right){\sigma}_{8}\left(z\right)=-(1+z)\frac{d{\sigma}_{8}\left(z\right)}{dz}$ to compute this quantity. The function ${\sigma}_{8}\left(z\right)$ involves the matter power spectrum, which is computed numerically by our modified version of the Einstein–Boltzmann code

`CLASS`. It is important to note that this way of computing $f\left(z\right){\sigma}_{8}\left(z\right)$ can only be used provided that we are in the linear regime, since in this case, and in our models, the matter density contrast can be written as ${\delta}_{m}(a,k)=D\left(a\right)F\left(k\right)$, where the dependence on the scale factor and the comoving wave number k is factored out. The term $D\left(a\right)$ is known as the growth function and $F\left(k\right)$ encodes the initial conditions7.

**CMB**: For the cosmic microwave background data, we utilize the full Planck 2018 TT,TE,EE+lowE likelihood [15]. This incorporates the information of the CMB temperature and polarization power spectra, and their cross-correlation. We refer to this dataset simply as “CMB”. We also test separately the Planck 2018 TT+lowE likelihood, which does not include the effect of the high-ℓ multipoles of the CMB polarization spectrum, in order to check the impact of this particular dataset on our fitting results. It is also useful to compare with our previous analyses [85], in which only this type of CMB data were used. In our fitting scenarios, we indicate the removal of the high-ℓ CMB polarization data from the complete CMB likelihood with the label “CMB (No pol.)”.

**Baseline**: In our Baseline dataset, we consider the string SNIa+BAO+$H\left(z\right)$+LSS+CMB. Note that here we do not include the SH0ES data.**Baseline+SH0ES**: The Baseline dataset is in this case complemented with the apparent magnitudes of the SNIa in the host galaxies and their distance moduli employed by SH0ES.**Baseline (No pol.)**: The same as in the Baseline case, but now removing the high-ℓ polarization data from the CMB likelihood. That is to say, we have replaced the “CMB” dataset with “CMB (No pol.)”.**Baseline (No pol.)+SH0ES**: The same as in “Baseline (No pol.)”, but including also the data from SH0ES.

**Table 4.**Mean values with $68\%$ confidence intervals obtained from our fitting analysis of our Baseline dataset, composed by the string SNIa+BAO+$H\left(z\right)$+LSS+CMB. We display the values of the different cosmological parameters: the Hubble parameter (${H}_{0}$), the reduced baryon and CDM density parameters (${\omega}_{\mathrm{b}}\equiv {\mathsf{\Omega}}_{\mathrm{b}}^{0}{h}^{2}$ and ${\omega}_{\mathrm{cdm}}\equiv {\mathsf{\Omega}}_{\mathrm{cdm}}^{0}{h}^{2}$, respectively, with ${\mathsf{\Omega}}_{i}^{0}\equiv 8\pi {G}_{N}{\rho}_{i}^{0}/3{H}_{0}^{2}$), the current nonrelativistic matter density parameter (${\mathsf{\Omega}}_{\mathrm{m}}^{0}$), the equation of state of the vacuum/DE fluid (${w}_{0}$), the effective parameter of the running vacuum (${\nu}_{\mathrm{eff}}$) (see (22) and (55)), the initial and current values of the variable $\phi \equiv {G}_{N}/G$, the optical depth to reionization (${\tau}_{\mathrm{reio}}$), the amplitude and spectral index of the primordial power spectrum (${A}_{s}$ and ${n}_{s}$, respectively), the absolute magnitude of SNIa (M), the rms mass fluctuations at 8${h}^{-1}$ Mpc scale at present time (${\sigma}_{8}$), the derived parameter ${S}_{8}\equiv {\sigma}_{8}\sqrt{{\mathsf{\Omega}}_{\mathrm{m}}^{0}/0.3}$, and the comoving sound horizon at the drag epoch (${r}_{d}$). We also show the incremental value of DIC with respect to the $\mathsf{\Lambda}$CDM, denoted $\Delta $DIC.

Baseline | |||||
---|---|---|---|---|---|

Parameter | $\mathsf{\Lambda}$CDM | Type-I RRVM | Type-I RRVM_{thr.} | Type-II RRVM | XCDM |

${H}_{0}$ (km/s/Mpc) | $68.27\pm 0.35$ | $68.22\pm 0.47$ | $67.65\pm 0.38$ | $68.12\pm 0.97$ | $67.49\pm 0.56$ |

${\omega}_{\mathrm{b}}$ | $0.02251\pm 0.00013$ | $0.02253\pm 0.00015$ | $0.02252\pm 0.00013$ | $0.02247\pm 0.00020$ | $0.02258\pm 0.00013$ |

${\omega}_{\mathrm{cdm}}$ | $0.11803\pm 0.00078$ | $0.11807\pm 0.00078$ | $0.1248\pm 0.0019$ | $0.1181\pm 0.0011$ | $0.11712\pm 0.00094$ |

${\mathsf{\Omega}}_{\mathrm{m}}^{0}$ | $0.3029\pm 0.0045$ | $0.3036\pm 0.0056$ | $0.3235\pm 0.0071$ | $0.3032\pm 0.0089$ | $0.3082\pm 0.0055$ |

${w}_{0}$ | $-1$ | $-1$ | $-1$ | $-1$ | $-0.962\pm 0.022$ |

${\nu}_{\mathrm{eff}}$ | - | $0.00006\pm 0.00030$ | $0.0227\pm 0.0055$ | $-0.00008\pm 0.00035$ | - |

${\phi}_{\mathrm{ini}}$ | - | - | - | $1.006\pm 0.024$ | - |

$\phi \left(0\right)$ | - | - | - | $1.008\pm 0.028$ | - |

${\tau}_{\mathrm{reio}}$ | $0.0512\pm 0.0073$ | $0.0511\pm 0.0080$ | $0.0601\pm 0.0082$ | $0.0505\pm 0.0078$ | $0.0546\pm 0.0077$ |

$ln\left({10}^{10}{\mathrm{A}}_{\mathrm{s}}\right)$ | $3.033\pm 0.015$ | $3.032\pm 0.016$ | $3.053\pm 0.017$ | $3.031\pm 0.016$ | $3.038\pm 0.016$ |

${n}_{\mathrm{s}}$ | $0.9698\pm 0.0035$ | $0.9701\pm 0.0038$ | $0.9707\pm 0.0035$ | $0.9681\pm 0.0069$ | $0.9722\pm 0.0038$ |

M | $-19.415\pm 0.010$ | $-19.416\pm 0.014$ | $-19.429\pm 0.011$ | $-19.420\pm 0.030$ | $-19.432\pm 0.014$ |

${\sigma}_{8}$ | $0.8003\pm 0.0064$ | $0.799\pm 0.011$ | $0.7733\pm 0.0092$ | $0.801\pm 0.010$ | $0.7885\pm 0.0093$ |

${S}_{8}$ | $0.804\pm 0.010$ | $0.803\pm 0.011$ | $0.803\pm 0.010$ | $0.805\pm 0.015$ | $0.802\pm 0.011$ |

${r}_{\mathrm{d}}$ (Mpc) | $147.46\pm 0.21$ | $147.47\pm 0.25$ | $147.44\pm 0.21$ | $147.9\pm 1.9$ | $147.62\pm 0.23$ |

$\Delta DIC$ | - | −2.04 | +15.34 | −4.18 | $+1.74$ |

`CLASS`[102,103], which is now equipped with the additional features that we have briefly described in the previous sections. We explore and put constraints on the parameter spaces of our models with Markov chain Monte Carlo (MCMC) analyses. More specifically, we make use of the Metropolis–Hastings algorithm [166,167], which is already implemented in the Monte Carlo sampler

`MontePython`9 [168,169]. We stop the MCMC when the Gelman–Rubin convergence statistic is $R-1<0.02$ [170,171], and analyze the converged chains with the Python code

`GetDist`10 [172] to compute the mean values of the cosmological parameters, their confidence intervals, and the posterior distributions.

**Table 5.**Same as in Table 4, but adding the information from SH0ES to our Baseline dataset.

Baseline +SH0ES | |||||
---|---|---|---|---|---|

Parameter | $\mathsf{\Lambda}$CDM | Type-I RRVM | Type-I RRVM_{thr.} | Type-II RRVM | XCDM |

${H}_{0}$ (km/s/Mpc) | $68.82\pm 0.33$ | $69.17\pm 0.43$ | $68.33\pm 0.35$ | $70.79\pm 0.69$ | $68.67\pm 0.50$ |

${\omega}_{\mathrm{b}}$ | $0.02264\pm 0.00013$ | $0.02253\pm 0.00015$ | $0.02266\pm 0.00013$ | $0.02281\pm 0.00017$ | $0.02265\pm 0.00013$ |

${\omega}_{\mathrm{cdm}}$ | $0.11697\pm 0.00073$ | $0.11685\pm 0.00075$ | $0.01227\pm 0.0018$ | $0.1178\pm 0.0011$ | $0.11679\pm 0.00089$ |

${\mathsf{\Omega}}_{\mathrm{m}}^{0}$ | $0.2961\pm 0.0041$ | $0.2928\pm 0.0049$ | $0.3128\pm 0.0064$ | $0.2808\pm 0.0058$ | $0.2971\pm 0.0047$ |

${w}_{0}$ | $-1$ | $-1$ | $-1$ | $-1$ | $-0.993\pm 0.020$ |

${\nu}_{\mathrm{eff}}$ | - | $-0.00037\pm 0.00029$ | $0.0197\pm 0.0055$ | $-0.00003\pm 0.00033$ | - |

${\phi}_{\mathrm{ini}}$ | - | - | - | $0.949\pm 0.016$ | - |

$\phi \left(0\right)$ | - | - | - | $0.950\pm 0.021$ | - |

${\tau}_{\mathrm{reio}}$ | $0.0533\pm 0.0074$ | $0.0501\pm 0.0079$ | $0.{0617}_{-0.0095}^{+0.0081}$ | $0.0523\pm 0.0077$ | $0.0539\pm 0.0078$ |

$ln\left({10}^{10}{\mathrm{A}}_{\mathrm{s}}\right)$ | $3.035\pm 0.015$ | $3.031\pm 0.016$ | $3.{053}_{-0.019}^{+0.017}$ | $3.041\pm 0.016$ | $3.036\pm 0.016$ |

${n}_{\mathrm{s}}$ | $0.9726\pm 0.0035$ | $0.9705\pm 0.0037$ | $0.9736\pm 0.0034$ | $0.9824\pm 0.0058$ | $0.9730\pm 0.0037$ |

M | $-19.3989\pm 0.0096$ | $-19.390\pm 0.012$ | $-19.410\pm 0.010$ | $-19.339\pm 0.021$ | $-19.402\pm 0.012$ |

${\sigma}_{8}$ | $0.7978\pm 0.0064$ | $0.808\pm 0.011$ | $0.7747\pm 0.0093$ | $0.807\pm 0.010$ | $0.7955\pm 0.0089$ |

${S}_{8}$ | $0.7927\pm 0.0094$ | $0.799\pm 0.011$ | $0.7910\pm 0.0098$ | $0.781\pm 0.013$ | $0.801\pm 0.010$ |

${r}_{\mathrm{d}}$ (Mpc) | $147.59\pm 0.21$ | $147.44\pm 0.25$ | $147.60\pm 0.21$ | $143.3\pm 1.4$ | $147.63\pm 0.23$ |

$\Delta \mathrm{DIC}$ | - | $-0.64$ | $+10.94$ | $+6.58$ | $-1.92$ |

**Table 6.**Same as in Table 4, but without including the high-ℓ CMB polarization data from Planck in our combined dataset.

Baseline (No pol.) | |||||
---|---|---|---|---|---|

Parameter | $\mathsf{\Lambda}$CDM | Type-I RRVM | Type-I RRVM_{thr.} | Type-II RRVM | XCDM |

${H}_{0}$ (km/s/Mpc) | $68.29\pm 0.38$ | $68.10\pm 0.48$ | $67.66\pm 0.41$ | $68.8\pm 1.2$ | $67.31\pm 0.56$ |

${\omega}_{\mathrm{b}}$ | $0.02228\pm 0.00019$ | $0.02235\pm 0.00022$ | $0.02231\pm 0.00019$ | $0.02242\pm 0.00026$ | $0.02239\pm 0.00020$ |

${\omega}_{\mathrm{cdm}}$ | $0.11746\pm 0.00085$ | $0.11744\pm 0.00086$ | $0.1242\pm 0.0019$ | $0.1166\pm 0.0016$ | $0.1160\pm 0.0011$ |

${\mathsf{\Omega}}_{\mathrm{m}}^{0}$ | $0.3011\pm 0.0048$ | $0.3029\pm 0.0056$ | $0.3215\pm 0.0072$ | $0.294\pm 0.011$ | $0.3068\pm 0.0055$ |

${w}_{0}$ | $-1$ | $-1$ | $-1$ | $-1$ | $-0.948\pm 0.022$ |

${\nu}_{\mathrm{eff}}$ | - | $0.00025\pm 0.00038$ | $0.0223\pm 0.0056$ | $0.00028\pm 0.00043$ | - |

${\phi}_{\mathrm{ini}}$ | - | - | - | $0.982\pm 0.030$ | - |

$\phi \left(0\right)$ | - | - | - | $0.976\pm 0.035$ | - |

${\tau}_{\mathrm{reio}}$ | $0.{0489}_{-0.0076}^{+0.0084}$ | $0.0508\pm 0.0083$ | $0.0581\pm 0.0082$ | $0.0508\pm 0.0083$ | $0.0540\pm 0.0080$ |

$ln\left({10}^{10}{\mathrm{A}}_{\mathrm{s}}\right)$ | $3.{026}_{-0.016}^{+0.018}$ | $3.028\pm 0.016$ | $3.047\pm 0.017$ | $3.030\pm 0.017$ | $3.034\pm 0.016$ |

${n}_{\mathrm{s}}$ | $0.9695\pm 0.0037$ | $0.9712\pm 0.0045$ | $0.9703\pm 0.0037$ | $0.9754\pm 0.0087$ | $0.9736\pm 0.0041$ |

M | $-19.415\pm 0.0011$ | $-19.420\pm 0.014$ | $-19.429\pm 0.012$ | $-19.397\pm 0.038$ | $-19.436\pm 0.0014$ |

${\sigma}_{8}$ | $0.7965\pm 0.0069$ | $0.790\pm 0.013$ | $0.7710\pm 0.0094$ | $0.792\pm 0.012$ | $0.7799\pm 0.0098$ |

${S}_{8}$ | $0.798\pm 0.011$ | $0.793\pm 0.013$ | $0.798\pm 0.011$ | $0.783\pm 0.020$ | $0.793\pm 0.012$ |

${r}_{\mathrm{d}}$ (Mpc) | $147.86\pm 0.30$ | $148.00\pm 0.35$ | $147.81\pm 0.30$ | $146.5\pm 2.4$ | $148.15\pm 0.33$ |

$\Delta \mathrm{DIC}$ | - | $-1.84$ | $+14.54$ | $-3.06$ | $+3.82$ |

**Table 7.**Same as in Table 4, but removing the high-ℓ polarization data from Planck and including the information provided by SH0ES.

Baseline (No pol.) +SH0ES | |||||
---|---|---|---|---|---|

Parameter | $\mathsf{\Lambda}$CDM | Type-I RRVM | Type-I RRVM_{thr.} | Type-II RRVM | XCDM |

${H}_{0}$ (km/s/Mpc) | $68.94\pm 0.37$ | $69.10\pm 0.44$ | $68.48\pm 0.39$ | $71.69\pm 0.80$ | $68.61\pm 0.51$ |

${\omega}_{\mathrm{b}}$ | $0.02247\pm 0.00018$ | $0.02240\pm 0.00022$ | $0.02251\pm 0.00018$ | $0.02280\pm 0.00024$ | $0.02252\pm 0.00019$ |

${\omega}_{\mathrm{dm}}$ | $0.11630\pm 0.00083$ | $0.11632\pm 0.00083$ | $0.1220\pm 0.0019$ | $0.1160\pm 0.0015$ | $0.1157\pm 0.0010$ |

${\mathsf{\Omega}}_{\mathrm{m}}^{0}$ | $0.2933\pm 0.0045$ | $0.2919\pm 0.0062$ | $0.3095\pm 0.0067$ | $0.2702\pm 0.0068$ | $0.2950\pm 0.0048$ |

${w}_{0}$ | $-1$ | $-1$ | $-1$ | $-1$ | $-0.981\pm 0.021$ |

${\nu}_{\mathrm{eff}}$ | - | $-0.00022\pm 0.00036$ | $0.0193\pm 0.0055$ | $0.00048\pm 0.00040$ | - |

${\phi}_{\mathrm{ini}}$ | - | - | - | $0.{919}_{-0.022}^{+0.019}$ | - |

$\phi \left(0\right)$ | - | - | - | $0.{908}_{-0.028}^{+0.025}$ | - |

${\tau}_{\mathrm{reio}}$ | $0.0512\pm 0.0074$ | $0.0494\pm 0.0084$ | $0.{0595}_{-0.0092}^{+0.0082}$ | $0.0528\pm 0.0085$ | $0.0533\pm 0.0079$ |

$ln\left({10}^{10}{\mathrm{A}}_{\mathrm{s}}\right)$ | $3.029\pm 0.016$ | $3.027\pm 0.017$ | $3.{047}_{-0.019}^{+0.017}$ | $3.041\pm 0.017$ | $3.032\pm 0.016$ |

${n}_{\mathrm{s}}$ | $0.9728\pm 0.0036$ | $0.9715\pm 0.0044$ | $0.9739\pm 0.0037$ | $0.9915\pm 0.0070$ | $0.9744\pm 0.0041$ |

M | $-19.396\pm 0.011$ | $-19.392\pm 0013$ | $-19.406\pm 0.011$ | $-19.311\pm 0.024$ | $-19.403\pm 0.013$ |

${\sigma}_{8}$ | $0.7939\pm 0.0068$ | $0.801\pm 0.014$ | $0.7719\pm 0.0094$ | $0.794\pm 0.012$ | $0.7876\pm 0.0096$ |

${S}_{8}$ | $0.785\pm 0.010$ | $0.790\pm 0.014$ | $0.784\pm 0.010$ | $0.754\pm 0.017$ | $0.781\pm 0.011$ |

${r}_{\mathrm{d}}$ (Mpc) | $147.97\pm 0.30$ | $147.85\pm 0.85$ | $147.92\pm 0.30$ | $141.3\pm 1.6$ | $148.08\pm 0.32$ |

$\Delta \mathrm{DIC}$ | - | $-0.10$ | $+10.06$ | $+13.78$ | $-0.96$ |

## 6. Discussion of the Results

`CLASS`(see, however, below). We refer the reader to Table 4, Table 5, Table 6 and Table 7 for the detailed list of the fitting results. In particular, we would like to mention that the results quoted in the last two tables (namely, Table 6 and Table 7, where the CMB data are used without polarizations) are perfectly compatible within error bars (both in order of magnitude and sign) with the results obtained in our previous analysis [85].

**Figure 1.**Contour plots at $1\sigma $ and $2\sigma $ c.l. in the ${\sigma}_{8}-{H}_{0}$, ${S}_{8}-{H}_{0}$, and ${\tilde{S}}_{8}-{H}_{0}$ planes and their corresponding one-dimensional posteriors, obtained from the fit of the various models to the Baseline dataset (cf. Section 5). The parameter ${\tilde{S}}_{8}\equiv {S}_{8}/\sqrt{\phi \left(0\right)}$ can only differ from the standard ${S}_{8}$ in the type-II RRVM; see the main text of Section 6 and [85,119]. The type-I RRVM${}_{\mathrm{thr}.}$ can explain a value of ${\sigma}_{8}\sim 0.78$, much smaller than in the other models. This is accompanied by a $4.1\sigma $ evidence for a non-zero value of the RVM parameter ${\nu}_{\mathrm{eff}}$; see Table 4. We find in all cases similar values of ${\tilde{S}}_{8}$ and ${H}_{0}$ to those found in $\mathsf{\Lambda}$CDM, but the type-II RRVM has a much wider posterior for this parameter, and hence this model can accommodate a larger Hubble constant. See also the comments in the main text.

**Figure 2.**Same as in Figure 1, but using the Baseline+SH0ES dataset (cf. Section 5). The inclusion of the data from SH0ES shifts the one-dimensional posterior of ${H}_{0}$ towards ${H}_{0}\sim 71$ km/s/Mpc in the type-II RRVM, a region that is still allowed by the Baseline dataset, cf. Figure 1. Remarkably, the small values of ${\sigma}_{8}$ found in the type-I RRVM${}_{\mathrm{thr}.}$ remain stable, and no important differences between the models are found regarding the value of ${\tilde{S}}_{8}$. The lower value of ${S}_{8}$ obtained in the type-II RRVM is due to the fact that this parameter does not account for the $2.4\sigma $ departure of $\phi \left(0\right)$ from 1, $\phi \left(0\right)=0.950\pm 0.021$; see the caption of Table 4.

**Figure 3.**Same as in Figure 2, but removing the high-ℓ polarization data from Planck, i.e., considering the Baseline (No pol.)+SH0ES dataset (cf. Section 5). Again, as in the other fitting analyses, the value of ${\sigma}_{8}$ is kept small in the type-I RRVM${}_{\mathrm{thr}.}$. The absence of CMB polarization data allows for even smaller values of $\phi \left(0\right)$ in the type-II RRVM, $\phi \left(0\right)=0.{908}_{-0.028}^{+0.025}$, which is now $3.5\sigma $ below the GR value $\phi \left(0\right)=1$. This explains the large value of ${H}_{0}\sim 72$ km/s/Mpc, which basically renders the Hubble tension insignificant, below the $1\sigma $ c.l.

**Figure 4.**Theoretical curves of $f\left(z\right){\sigma}_{8}\left(z\right)$ for the various models together with the observational data points listed in Table 3. We have employed the central values of the Baseline fitting analysis (cf. Table 4). The type-I RRVM${}_{\mathrm{thr}}$ has the ability to solve the ${\sigma}_{8}$ tension by suppressing the clustering at $z<1$.

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Additional Tables

`MontePython`is not always very precise; see footnote 10 in [169].

**Table A1.**Detailed breakdown of the different ${\chi}^{2}$ contributions from each observable in our Baseline dataset with the cosmological parameters reported in Table 4. We call the contribution that contains the correlations between the BAO and LSS datasets BAO-$f{\sigma}_{8}$ (correl.) (see the references of Table 1), whereas the uncorrelated contributions are simply called BAO and $f{\sigma}_{8}$.

Baseline | |||||
---|---|---|---|---|---|

Experiment | $\mathsf{\Lambda}$CDM | Type-I RRVM | Type-I RRVM_{thr.} | Type-II RRVM | XCDM |

CMB | $2770.70$ | $2771.04$ | $2770.14$ | $2770.48$ | $2773.68$ |

SNIa | $1405.49$ | $1405.39$ | $1403.24$ | $1405.64$ | $1402.82$ |

$f{\sigma}_{8}$ | $17.15$ | $16.92$ | $8.29$ | $17.14$ | $15.08$ |

BAO-$f{\sigma}_{8}$ (correl.) | $19.96$ | $19.92$ | $14.54$ | $19.92$ | $17.91$ |

$H\left(z\right)$ | $13.16$ | $13.18$ | $13.33$ | $13.30$ | $13.33$ |

BAO | $10.94$ | $10.97$ | $10.70$ | $10.91$ | $10.65$ |

${\chi}_{\mathrm{total}}^{2}$ | $4237.40$ | $4237.42$ | $4220.24$ | $4237.39$ | $4233.48$ |

Baseline+SH0ES | |||||
---|---|---|---|---|---|

Experiment | $\mathsf{\Lambda}$CDM | Type-I RRVM | Type-I RRVM_{thr.} | Type-II RRVM | XCDM |

CMB | $2774.02$ | $2771.46$ | $2774.02$ | $2777.50$ | $2774.72$ |

SNIa | $1490.46$ | $1488.22$ | $1491.62$ | $1474.38$ | $1490.65$ |

$f{\sigma}_{8}$ | $15.33$ | $16.92$ | $8.21$ | $17.27$ | $14.92$ |

BAO-$f{\sigma}_{8}$ (correl.) | $19.82$ | $21.68$ | $13.35$ | $20.48$ | $19.21$ |

$H\left(z\right)$ | $12.97$ | $12.85$ | $13.07$ | $12.47$ | $13.00$ |

BAO | $10.77$ | $10.77$ | $10.43$ | $10.76$ | $10.69$ |

${\chi}_{\mathrm{total}}^{2}$ | $4323.38$ | $4321.91$ | $4310.71$ | $4312.85$ | $4323.19$ |

Baseline (No pol.) | |||||
---|---|---|---|---|---|

Experiment | $\mathsf{\Lambda}$CDM | Type-I RRVM | Type-I RRVM_{thr.} | Type-II RRVM | XCDM |

CMB | $1184.03$ | $1185.16$ | $1183.39$ | $1184.93$ | $1186.79$ |

SNIa | $1405.84$ | $1405.51$ | $1403.41$ | $1405.60$ | $1402.65$ |

$f{\sigma}_{8}$ | $16.03$ | $14.99$ | $8.27$ | $15.35$ | $13.38$ |

BAO-$f{\sigma}_{8}$ (correl.) | $19.44$ | $19.12$ | $14.12$ | $19.20$ | $16.99$ |

$H\left(z\right)$ | $13.20$ | $13.29$ | $13.35$ | $12.85$ | $13.38$ |

BAO | $10.91$ | $10.97$ | $10.65$ | $10.96$ | $10.47$ |

${\chi}_{\mathrm{total}}^{2}$ | $2649.45$ | $2649.04$ | $2633.19$ | $2648.89$ | $2643.66$ |

Baseline (No pol.) + SH0ES | |||||
---|---|---|---|---|---|

Experiment | $\mathsf{\Lambda}$CDM | Type-I RRVM | Type-I RRVM_{thr.} | Type-II RRVM | XCDM |

CMB | $1186.28$ | $1185.07$ | $1186.32$ | $1189.76$ | $1187.75$ |

SNIa | $1490.20$ | $1489.18$ | $1490.80$ | $1469.30$ | $1490.24$ |

$f{\sigma}_{8}$ | $14.15$ | $15.17$ | $8.16$ | $15.07$ | $13.10$ |

BAO-$f{\sigma}_{8}$ (correl.) | $20.49$ | $21.50$ | $14.07$ | $19.25$ | $19.26$ |

$H\left(z\right)$ | $12.97$ | $12.91$ | $12.91$ | $12.79$ | $13.01$ |

BAO | $10.86$ | $10.86$ | $10.48$ | $10.86$ | $10.64$ |

${\chi}_{\mathrm{total}}^{2}$ | $2734.95$ | $2734.69$ | $2722.74$ | $2717.20$ | $2734.00$ |

## Notes

1 | https://lesgourg.github.io/class_public/class.html, accessed on 22 May 2023. |

2 | In practice this means that we have first fitted the value of ${z}_{*}$ as one more free parameter in our analysis. Subsequently, we have assumed that the threshold point remains fixed at that point; see also [107,108,109,110] for a binned/tomographic approach to the DE. In our case, we have just one threshold whose existence might be motivated by QFT calculations [30,31]. |

3 | If (dark) matter is not conserved but G remains constant, we retrieve of course our previous scenario (16). In general, we may expect a mixture of both situations, but we shall refrain from dealing with the general case since it would introduce extra parameters; see, however, [111,112] for additional discussions that can be relevant for studies on the possible variation in the fundamental constants of nature. |

4 | It should be clear that $\phi $ is not a dynamical degree of freedom, in contradistinction to Brans–Dicke-type theories of gravitation [113], and therefore $\phi $ does not mediate any sort of long-range interaction that should be subdued by screening mechanisms. |

5 | Let us emphasize that Equation (56) is valid only in the MDE, and we have also pointed out that $\phi \to $ const. in the DE epoch. This means that G becomes more and more rigid when it transits from the MDE to the DE epoch, and therefore the actual limits on ${\nu}_{\mathrm{eff}}$ are weaker than those that we have roughly estimated. This works to our benefit of course. In fact, a detailed calculation would require computing $\phi $ in the DE epoch, but it proves unnecessary once we have shown that even in the most unfavorable case (i.e., when $\phi $ evolves more rapidly than it actually does in the DE epoch) the obtained limits on ${\nu}_{\mathrm{eff}}$ are nonetheless preserved by our fits. Notice that type-I models are totally unaffected by these limits, since G is in this case constant, so ${\nu}_{\mathrm{eff}}$ can be, in principle, larger for them. |

6 | https://gitlab.com/mmoresco/CCcovariance/-/blob/master/examples/CC_covariance.ipynb, accessed on 22 May 2023. |

7 | While it is common to rescale the measured values of $f{\sigma}_{8}$ by a factor $\frac{H\left(z\right){D}_{A}\left(z\right)}{\tilde{H}\left(z\right){\tilde{D}}_{A}\left(z\right)}$ to account for the Alcock–Paczynski (AP) effect [157] (in which the tildes denote the quantities computed in the fiducial cosmology employed by the galaxy surveys), there does not not seem to exist a general consensus on the exact correction to apply; see, e.g., [158] and references therein. In this sense, the above formula should be considered as just a rough estimate. We have checked that the AP-rescaling introduces negligible shifts in our fitting results, a conclusion that is well in accordance with previous analyses in the literature [38,39,158]. For this reason, we have opted to not include this correction in our work. |

8 | This region is also preferred by late-time dynamical DE models when fitted to a very wide variety of background data that are independent from the direct cosmic distance ladder and CMB, ${H}_{0}=69.8\pm 1.3$ km/s/Mpc [161]. See [162,163,164,165] for measurements of ${H}_{0}$ more in accordance with SH0ES obtained also with the tip of the red giant branch method. |

9 | https://baudren.github.io/montepython.html, accessed on 22 May 2023. |

10 | https://getdist.readthedocs.io/en/latest/, accessed on 22 May 2023. |

11 | See Section 3.3 for the practical implementation. |

12 | |

13 | |

14 | Noticeably, the central values of ${r}_{d}$, ${H}_{0}$, and the absolute magnitude of SNIa, M, obtained for the type-II RRVM when the CMB polarization data are excluded in the fitting analysis are in very good agreement with the model-independent measurements from low-z data reported in [152], which are also independent from the main drivers of the ${H}_{0}$ tension. For the Hubble constant, these authors find ${H}_{0}=71.6\pm 3.1$ km/s/Mpc. However, these measurements still have large uncertainties and cannot arbitrate the Hubble tension yet; see also [179]. |

15 |

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**MDPI and ACS Style**

Solà Peracaula, J.; Gómez-Valent, A.; de Cruz Pérez, J.; Moreno-Pulido, C.
Running Vacuum in the Universe: Phenomenological Status in Light of the Latest Observations, and Its Impact on the *σ*_{8} and *H*_{0} Tensions. *Universe* **2023**, *9*, 262.
https://doi.org/10.3390/universe9060262

**AMA Style**

Solà Peracaula J, Gómez-Valent A, de Cruz Pérez J, Moreno-Pulido C.
Running Vacuum in the Universe: Phenomenological Status in Light of the Latest Observations, and Its Impact on the *σ*_{8} and *H*_{0} Tensions. *Universe*. 2023; 9(6):262.
https://doi.org/10.3390/universe9060262

**Chicago/Turabian Style**

Solà Peracaula, Joan, Adrià Gómez-Valent, Javier de Cruz Pérez, and Cristian Moreno-Pulido.
2023. "Running Vacuum in the Universe: Phenomenological Status in Light of the Latest Observations, and Its Impact on the *σ*_{8} and *H*_{0} Tensions" *Universe* 9, no. 6: 262.
https://doi.org/10.3390/universe9060262