Running vacuum against the H_0 and σ_8 tensions
Joan Sola, Adria Gomez-Valent, Javier de Cruz Perez, Cristian Moreno-Pulido
eepl draft
Running vacuum against the H and σ tensions Joan Sol`a Peracaula a , Adri`a G´omez-Valent b , Javier de Cruz P´erez a and Cristian Moreno-Pulido aa Departament de F´ısica Qu`antica i Astrof´ısica, and Institute of Cosmos Sciences, Universitat de Barcelona,Av. Diagonal 647, E-08028 Barcelona, Catalonia, Spain b Institut f¨ur Theoretische Physik, Ruprecht-Karls-Universit¨at Heidelberg, Philosophenweg 16, D-69120 Heidelberg,Germany
PACS – Cosmology
PACS – Dark Energy
Abstract –The cosmological term, Λ, was introduced 104 years ago by Einstein in his gravitationalfield equations. Whether Λ is a rigid quantity or a dynamical variable in cosmology has been amatter of debate for many years, especially after the introduction of the general notion of darkenergy (DE). Λ is associated to the vacuum energy density, ρ vac , and one may expect that itevolves slowly with the cosmological expansion. Herein we present a devoted study testing thispossibility using the promising class of running vacuum models (RVM’s). We use a large stringSNIa+BAO+ H ( z )+LSS+CMB of modern cosmological data, in which for the first time the CMBpart involves the full Planck 2018 likelihood for these models. We test the dependence of theresults on the threshold redshift z ∗ at which the vacuum dynamics is activated in the recent pastand find positive signals up to ∼ . σ for z ∗ (cid:39)
1. The RVM’s prove very competitive against thestandard ΛCDM model and give a handle for solving the σ tension and alleviating the H one. Introduction. –
Despite Einstein’s original formula-tion [1], in which the cosmological term Λ is treated asa strict constant in the gravitational field equations, theidea that Λ (and its associated vacuum energy density ρ vac ) can be a dynamical quantity should be most natu-ral in the context of an expanding universe. This pointof view has led to the notion of dynamical dark energy(DDE) in its multifarious forms [2, 3]. Herein, however,we stick to the notion of dynamical vacuum energy (DVE)as the ultimate cause of DDE. Despite the fact that ρ vac has long been associated with the so-called cosmologicalconstant problem [4, 5], which involves severe fine-tuningof the parameters, such a conundrum actually underliesall of the DE models known up to date, with no exception[5]. In addition, recent calculations of ρ vac in the con-text of quantum field theory (QFT) in curved spacetimehave brought new light into this problem [6] and suggestthat if the vacuum energy density (VED) is renormalizedusing an appropriate regularization procedure, it evolvesin a mild way as a series of powers of the Hubble rate H and its cosmic time derivatives: ρ vac ( H, ˙ H, ... ), denoted ρ vac ( H ) for short. This fact was long foreseen from generalrenormalization group arguments which led to the notionof running vacuum models (RVM’s), see the reviews [5, 7]and references therein. For the current universe, the lea- ding VED term is constant but the next-to-leading one isdynamical, specifically it evolves as a power ∼ H with asmall coefficient | ν | (cid:28)
1. For the early universe, terms oforder ∼ H or higher appear and these can trigger infla-tion [7–9]. It is remarkable that the fourth power H canbe motivated within the context of string theory calcula-tions at low energy (meaning near the Planck scale) [10],what reveals a distinctive mechanism of inflation differentfrom that of Starobinsky inflation [11], for example. See[6, 12] for a detailed discussion. Here, however, we willconcentrate on the post-inflationary universe, where onlythe leading power ∼ H is involved in the dynamics of ρ vac . A variety of phenomenological analyses have sup-ported this possibility in recent years and within differentapproaches [13–19].In this Letter, we present a devoted study of theclass of RVM’s based on a large and updated stringSNIa+BAO+ H ( z )+LSS+CMB of modern cosmologicalobservations, in which for the first time the CMB partinvolves the full Planck 2018 likelihood. We also test thepotential dependence of the results on the threshold red-shift z ∗ at which the DVE becomes activated in the recentpast. We find that different RVM’s prove very helpful toalleviate the persisting tensions between the concordanceΛCDM model and the structure formation data (the so-p-1 a r X i v : . [ a s t r o - ph . C O ] F e b . Sol`a Peracaula et al. Baseline
Parameter ΛCDM type I RRVM type I RRVM thr . type II RRVM H (km/s/Mpc) 68 . +0 . − . . +0 . − . . +0 . − . . +1 . − . ω b . +0 . − . . +0 . − . . +0 . − . . +0 . − . ω dm . +0 . − . . +0 . − . . +0 . − . . +0 . − . ν eff - . +0 . − . . +0 . − . . ± . ϕ ini - - - . +0 . − . ϕ - - - . +0 . − . τ reio . +0 . − . . +0 . − . . +0 . − . . ± . n s . +0 . − . . +0 . − . . ± .
038 0 . +0 . − . σ . ± .
007 0 . +0 . − . . +0 . − . . +0 . − . S . ± .
011 0 . +0 . − . . +0 . − . . +0 . − . r s (Mpc) 147 . +0 . − . . +0 . − . . ± .
30 146 . +2 . − . χ - -2.70 +13.82 -4.59 Table 1: The mean values and 68.3% confidence limits for the models under study using our Baseline dataset, which is almostthe same as the one employed in [17], with few changes: (i) for the eBOSS survey we have replaced the data from [20] withthe one from [21]; (ii) the LyF data have been updated, replacing [22] with [23]; (iii) finally, we have replaced the two fσ datapoints [24, 25] with the one provided in [26]. We display the fitting values for the usual parameters, to wit: H , the reduceddensity parameter for baryons ( w b = Ω b h ) and CDM ( w dm = Ω dm h ), with Ω i = 8 πG N ρ i / H and h the reduced Hubbleconstant, the reionization optical depth τ reio , the spectral index n s and the current matter density rms fluctuations withinspheres of radius 8 h − Mpc, i.e. σ . We have also included a couple of useful derived parameters, namely: the sound horizon atthe baryon drag epoch r s and S ≡ σ (cid:112) Ω m / .
3. For all the RRVM’s we show ν eff , and for the type II one we also report theinitial and current values of ϕ , ϕ ini and ϕ , respectively. Finally, we also provide the corresponding values of χ and ∆DIC. called σ tension) and the mismatch between the localvalues of the Hubble parameter and those derived fromthe CMB [27] (the H tension). These tensions are welldescribed in the literature, see e.g. the reviews [28, 29].Many models in the market try to address them, see e.g.Ref. [30] and the long list of references therein.In the current (fully updated) study we find significantsignals of DVE (using z ∗ (cid:39)
1) at ∼ . σ c.l., which can beenhanced up to ∼ . σ . Finally, we show that the RVM’sprovide an overall fit to the cosmological data which iscomparable or significantly better than in the ΛCDM case,as confirmed by calculating the relative Deviance Informa-tion Criterion (DIC) differences obtained form the MonteCarlo chains of our numerical analysis. Running vacuum Universe. –
As indicated, thetotal vacuum part of the energy-momentum tensor, T vac µν ,can be appropriately renormalized into a finite quantitywhich depends on the Hubble rate H and its time deriva-tives [6]. The corresponding 00-component defines thevacuum energy density (VED), ρ vac ( H ). Let us denote by ρ ≡ ρ vac ( H ) = Λ / (8 πG N ) ( G N being Newton’s cons-tant) the current value of the latter, with H today’s valueof the Hubble parameter and Λ the measured cosmologicalconstant term. We define two types of DVE scenarios. Intype I scenario the vacuum is in interaction with matter,whereas in type II matter is conserved at the expense ofan exchange between the vacuum and a slowly evolving gravitational coupling G ( H ). The combined cosmological‘running’ of these quantities insures the accomplishmentof the Bianchi identity (and the local conservation law).Let us therefore consider a generic cosmological frame-work described by the spatially flat Friedmann-Lemaˆıtre-Robertson-Walker (FLRW) metric. The vacuum energydensity in the RVM can be written in the form [5, 7]: ρ vac ( H ) = 38 πG N (cid:16) c + νH + ˜ ν ˙ H (cid:17) + O ( H ) , (1)in which the O ( H ) terms will be neglected for the physicsof the post-inflationary epoch. The above generic struc-ture can be motivated from the aforementioned explicitQFT calculations on a FLRW background [6]. The ad-ditive constant c is fixed by the boundary condition ρ vac ( H ) = ρ . Notice that the two dynamical com-ponents H and ˙ H are dimensionally homogeneous and,in principle, independent. Their associated (dimension-less) coefficients ν and ˜ ν encode the dynamics of the vac-uum at low energy and we naturally expect | ν, ˜ ν | (cid:28) ν in QFT indicates that it is of order10 − at most [31]. In the calculation of [6] these coeffi-cients are expected to be of order ∼ M X /m (cid:28)
1, where m Pl (cid:39) . × GeV is the Planck mass and M X is of or-der of a typical Grand Unified Theory (GUT) scale, timesa multiplicity factor accounting for the number of heavyparticles in the GUT. We will be particularly interested inp-2unning vacuum against the H and σ tensions Baseline + H thr . type II RRVM H (km/s/Mpc) 68 . +0 . − . . +0 . − . . +0 . − . . +0 . − . ω b . +0 . − . . +0 . − . . +0 . − . . +0 . − . ω dm . +0 . − . . +0 . − . . +0 . − . . +0 . − . ν eff - − . +0 . − . . +0 . − . . +0 . − . ϕ ini - - - . +0 . − . ϕ - - - . +0 . − . τ reio . +0 . − . . +0 . − . . +0 . − . . ± . n s . +0 . − . . ± . . +0 . − . . +0 . − . σ . ± .
007 0 . ± .
013 0 . ± .
010 0 . +0 . − . S . +0 . − . . ± .
013 0 . ± .
011 0 . +0 . − . r s (Mpc) 147 . +0 . − . . +0 . − . . +0 . − . . +1 . − . χ - -2.36 +10.88 +5.52 Table 2: Same as in Table 1, but also considering the prior on H = (73 . ± .
4) km/s/Mpc from SH0ES [32]. the RVM density obtained from the choice ˜ ν = ν/
2. Asa result, ρ vac ( H ) = 3 / (8 πG N ) (cid:104) c + ν (cid:16) H + ˙ H (cid:17)(cid:105) . Wewill call this form of the VED the ‘RRVM’ since it reali-zes the generic RVM density (1) through the Ricci scalar R = 12 H + 6 ˙ H , namely ρ vac ( H ) = 38 πG N (cid:16) c + ν R (cid:17) ≡ ρ vac ( R ) . (2)Such a RRVM implementation has the advantage that itgives a safe path to the early epochs of the cosmologicalevolution since in the radiation dominated era we have R /H (cid:28)
1, and hence we do not generate any conflictwith the BBN nor with any other feature of the modernuniverse. Of course, early on the RVM has its own mecha-nism for inflation (as we have already mentioned), but weshall not address these aspects here, see [5, 7–9, 12].
Type I RRVM.
Friedmann’s equation and the accele-ration equation relate H and ˙ H with the energy densitiesand pressures for the different species involved, and read3 H = 8 πG N ( ρ m + ρ ncdm + ρ γ + ρ vac ( H )) , (3)3 H + 2 ˙ H = − πG N ( p ncdm + p γ + p vac ( H )) . (4)The total nonrelativistic matter density is the sum of thecold dark matter (CDM) component and the baryonic one: ρ m = ρ dm + ρ b . The contributions of massive and masslessneutrinos are included in ρ ncdm (‘ncdm’ means non-CDM).Therefore the total (relativistic and nonrelativistic) mat-ter density is ρ t = ρ m + ρ γ + ρ ncdm . Similarly, the totalmatter pressure reads p t = p ncdm + p γ (with p γ = (1 / ρ γ ).We note that there is a transfer of energy from the relati-vistic neutrinos to the nonrelativistic ones along the wholecosmic history, and hence it is not possible (in an accu-rate analysis) to make a clear-cut separation between the two. Our procedure adapts to our own modified versionof the system solver CLASS [33]. The latter solves thecoupled system of Einstein’s and Boltzmann’s differentialequations for any value of the scale factor and, in par-ticular, provides the functions ρ h = ρ ncdm − p ncdm and ρ ν = 3 p ncdm for the nonrelativistic and relativistic neutri-nos, respectively. This allows to compute the combination R /
12 = H + (1 /
2) ˙ H appearing in (2) in terms of theenergy densities and pressures using (3) and (4): R = 8 πG N ( ρ m + 4 ρ vac + ρ h ) . (5)Notice that the photon contribution cancels exactly in thisexpression and hence ρ vac from (2) remains much smallerthan the photon density in the radiation epoch, entailingno alteration of the thermal history. While neutrinos donot behave as pure radiation for the aforementioned rea-sons, one can check numerically (using CLASS ) that theratio r ≡ ρ h /ρ m is very small throughout the entire cos-mic history up to our time (remaining always below 10 − ).Thus, we can neglect it in (2) and we can solve for the vac-uum density as a function of the scale factor a as follows: ρ vac ( a ) = ρ + ν − ν ) ( ρ m ( a ) − ρ m ) , (6)where ‘ 0’ (used as subscript or superscript) always refersto current quantities. For a = 1 (today’s universe) weconfirm the correct normalization: ρ vac ( a = 1) = ρ .Needless to say, ρ m ( a ) is not just ∼ a − since the vacuumis exchanging energy with matter here. This is obviousfrom the fact that the CDM exchanges energy with thevacuum (making it dynamical):˙ ρ dm + 3 Hρ dm = − ˙ ρ vac . (7)Baryons do not interact with the vacuum, which implies˙ ρ b + 3 Hρ b = 0, and as a result the total matter contribu-p-3. Sol`a Peracaula et al. tion ( ρ m ) satisfies the same local conservation law as thatin Eq. (7): ˙ ρ m + 3 Hρ m = − ˙ ρ vac . Using it with (6) we find˙ ρ m + 3 Hξρ m = 0, where we have defined ξ ≡ − ν − ν . Since ν is small, it is convenient to encode the deviations withrespect to the standard model in terms of the effectiveparameter ν eff ≡ ν/ ξ = 1 − ν eff + O (cid:0) ν (cid:1) . (8)It is straightforward to find the expression for the matterdensities: ρ m ( a ) = ρ m a − ξ , ρ dm ( a ) = ρ m a − ξ − ρ b a − . (9)They recover the ΛCDM form for ξ = 1 ( ν eff = 0). Thesmall departure is precisely what gives allowance for a milddynamical vacuum evolution: ρ vac ( a ) = ρ + (cid:18) ξ − (cid:19) ρ m (cid:0) a − ξ − (cid:1) . (10)The vacuum becomes rigid only for ξ = 1 ( ν eff = 0). Type II RRVM.
For type II models matter is con-served (no exchange with vacuum), but the vacuum canstill evolve provided the gravitational coupling also evolves(very mildly) with the expansion: G = G ( H ). Followingthe notation of [17], let us define (just for convenience) anauxiliary variable ϕ = G N /G – in the manner of a Brans-Dicke field, without being really so. Notice that ϕ (cid:54) = 1in the cosmological domain, but remains very close to it,see Tables 1 and 2. The departure from 1 is logarithmic,similarly as in [17]. Friedman’s equation in this case takesthe form 3 H = πG N ϕ [ ρ t + ρ vac ( H )]. Using (2) we find3 H = 8 πG N ϕ (cid:20) ρ t + C + 3 ν πG N (2 H + ˙ H ) (cid:21) , (11)with C = 3 c / (8 πG N ). The Bianchi identity dictates thecorrelation between the dynamics of ϕ and that of ρ vac :˙ ϕϕ = ˙ ρ vac ρ t + ρ vac , (12)where ρ t is as before the total matter energy density and ρ vac adopts exactly the same form as in (2). Using theseequations one can show that the approximate behavior ofthe VED in the present time is (recall that | ν eff | (cid:28) ρ vac ( a ) = C (1 + 4 ν eff ) + ν eff ρ m a − + O ( ν ) . (13)Again, for ν eff = 0 the VED is constant, but otherwiseit shows a moderate dynamics of O ( ν eff ) as in the typeI case (10). Here, however, the exact solution must befound numerically. One can also show that the behaviorof ρ vac ( a ) in the radiation dominated epoch is also of theform (13), except that the constant additive term can becompletely neglected. It follows that ρ vac ( a ) (cid:28) ρ r ( a ) = ρ r a − for a (cid:28) ϕ ( a ) ∝ a − (cid:15) ≈ − (cid:15) ln a in the current epoch (with 0 < (cid:15) (cid:28) ν eff ), thusconfirming the very mild (logarithmic) evolution of G . Threshold redshift scenario for type I models. –
One possibility that has been explored in the literature indifferent type of models is to admit that the dynamics ofvacuum is relatively recent (see e.g. [34]). This means tostudy the consequences of keeping deactivated the inter-action between the vacuum energy density and the CDMfor most of cosmic history until the late universe when theDE becomes apparent. We denote the threshold value ofthe scale factor when the activation takes places by a ∗ .According to this scenario the VED was constant prior to a = a ∗ and it just started to evolve for a > a ∗ . While ρ vac is a continuous function, its derivative is not since wemimic such situation through a Heaviside step functionΘ( a − a ∗ ). If we would have a microscopic description ofthe phenomenon it should not be necessary to assume sucha sudden (finite) discontinuity. However, a Θ-function de-scription will be enough for our purposes. Therefore, weassume that in the range a < a ∗ (hence for z > z ∗ ) wehave ρ dm ( a ) = ρ dm ( a ∗ ) (cid:18) aa ∗ (cid:19) − ,ρ vac ( a ) = ρ vac ( a ∗ ) = const. ( a < a ∗ ) , (14)where ρ dm ( a ∗ ) and ρ vac ( a ∗ ) are computed from (9) and(10), respectively. In the complementary range, instead,i.e. for a > a ∗ (0 < z < z ∗ ) near our time, the originalequations (9) and (10) hold good.Notice that the above threshold procedure is motivatedspecially within type I models in order to preserve thecanonical evolution law for the matter energy density whenthe redshift is sufficiently high. In fact, the threshold red-shift value need not be very large and as we shall see inthe next section, if fixed by optimization it turns out tobe of order z ∗ (cid:39)
1. Above it ( z > z ∗ ) the matter densityevolves as in the ΛCDM and in addition ρ vac remains cons-tant. Its dynamics is only triggered at (and below) z ∗ . Animportant consequence of such threshold is that the cos-mological physics during the CMB epoch (at z (cid:39) Cosmological perturbations. –
So far so good forthe background cosmological equations in the presenceof dynamical vacuum. However, an accurate descriptionof the large scale structure (LSS) formation data is alsoof paramount importance, all the more if we take intoaccount that one of the aforementioned ΛCDM tensions(the σ one) stems from it. Allowing for some evolutionof the vacuum can be the clue to solve the σ tensionsince such dynamics affects nontrivially the cosmologicalperturbations [14]. We consider the perturbed, spatiallyflat, FLRW metric ds = − dt + ( δ ij + h ij ) dx i dx j , inwhich h ij stands for the metric fluctuations. These fluc-tuations are coupled to the matter density perturbationsp-4unning vacuum against the H and σ tensions Fig. 1: Theoretical curves of f ( z ) σ ( z ) for the various models and the data points employed in our analysis, in two differentredshift windows. To generate this plot we have used the central values of the cosmological parameters shown in Table 1. Thetype I running vacuum model with threshold redshift z ∗ (cid:39) σ tension. δ m = δρ m /ρ m . We shall refrain from providing detailsof this rather technical part, which will be deferred foran expanded presentation elsewhere. However, the readercan check e.g. [13–17] for the basic discussion of the RVMperturbations equations. The difference is that here wehave implemented the full perturbations analysis in thecontext of the Einstein-Boltzmann code CLASS [33] (inthe synchronous gauge [35]). Let us nonetheless mentiona few basic perturbations equations which have a moredirect bearing on the actual fitting analysis presented inour tables and figures. Since baryons do not interact withthe time-evolving VED the perturbed conservation equa-tions are not directly affected. However, the correspondingequation for CDM is modified in the following way:˙ δ dm + ˙ h − ˙ ρ vac ρ dm δ dm = 0 , (15)with h = h ii denoting the trace of h ij . We remark thatthe term ˙ ρ vac is nonvanishing for these models and affectsthe fluctuations of CDM in a way which obviously pro-duces a departure from the ΛCDM. The above equationis, of course, coupled with the metric fluctuations and thecombined system must be solved numerically.The analysis of the linear LSS regime is performed withthe help of the weighted linear growth f ( z ) σ ( z ), where f ( z ) is the growth factor and σ ( z ) is the rms mass fluc-tuation amplitude on scales of R = 8 h − Mpc at redshift z . The quantity σ ( z ) is directly provided by CLASS andthe calculation of f ( a ) (with z = a − − (cid:126)k denotes thecomoving wave vector and (cid:126)k/a the physical one, at sub-horizon scales its modulus (square) satisfies k /a (cid:29) H .If, in addition, we are in the linear regime the matterdensity contrast can be written as δ m ( a, (cid:126)k ) = D ( a ) F ( (cid:126)k )[3, 36], where the dependence on (cid:126)k factors out. The pro-perties of F ( (cid:126)k ) are determined by the initial conditionsand D ( a ) is called the growth function. The relation be-tween the matter power spectrum and the density con-trast reads P m ( a, (cid:126)k ) = C (cid:104) δ m ( a, (cid:126)k ) δ ∗ m ( a, (cid:126)k ) (cid:105) ≡ D ( a ) P ( (cid:126)k ), where C is a constant and P ( (cid:126)k ) = C (cid:104) F ( (cid:126)k ) F ∗ ( (cid:126)k ) (cid:105) is theprimordial power spectrum (determined from the theoryof inflation). Since neither F ( (cid:126)k ) nor P ( (cid:126)k ) depend on a ,the linear growth f ( a ) = d ln δ m ( a, (cid:126)k ) /d ln a is given by f ( a ) = d ln D ( a ) /d ln a , and ultimately by f ( a ) = d ln P / m ( a, (cid:126)k ) d ln a = a P m ( a, (cid:126)k ) dP m ( a, (cid:126)k ) da . (16)It follows that we may extract the (observationally mea-sured) linear growth function f ( a ) directly from the mat-ter power spectrum P m ( a, (cid:126)k ), which is computed numeri-cally by CLASS for all values of a and (cid:126)k (assuming adiabaticinitial conditions). This allows us to compare theory andobservation for the important LSS part. Fitting results and discussion. –
To compare theRRVM’s (types I and II) with the ΛCDM, we have defineda joint likelihood function L . The overall fitting resultsare reported in Tables 1 and 2. The used data sets arethe same as those described in detail in Ref. [17], exceptthe updated values pointed out in the caption of Table 1.Assuming Gaussian errors, the total χ to be minimizedin our case is given by χ = χ + χ + χ H + χ σ + χ . (17)The above χ terms are defined in the standard way fromthe data including the covariance matrices [3]. In parti-cular, the χ H part may contain or not the local H valuemeasured by Riess et al. [32] depending on the setup in-dicated in the tables (apart from the cosmic chronometerdata employed also in [17]). The local determination of H (which is around 4 σ away from the corresponding Planck2018 value based on the CMB) is the origin of the so-called H tension [28,29]. Taking into account that the RRVM’sof type I and II have one and two more parameters, res-pectively, as compared to the ΛCDM, a fairer model com-parison is achieved by computing the differences betweenthe Deviance Information Criterion [37], of the ΛCDMmodel and the RRVM’s: ∆DIC = DIC ΛCDM − DIC
RRVM .p-5. Sol`a Peracaula et al.
Fig. 2: 1 σ and 2 σ contours in the H - σ , S , ˜ S planes and the corresponding one-dimensional posteriors for the ΛCDM andthe RRVM’s under study, obtained from the fitting analyses with our Baseline+ H data set. The type II model manifestlyalleviates the H tension without spoiling the σ one (even if phrased through the alternative parameters S or ˜ S , see text),whereas the type I model with threshold redshift z ∗ (cid:39) These differences will be (and in fact are) positive if theRRVM’s fit better the overall data than the ΛCDM. TheDIC is defined as DIC = χ ( θ ) + 2 p D . (18)Here p D = χ − χ ( θ ) is the effective number of parame-ters of the model, and χ and θ the mean of the overall χ distribution and the parameters, respectively. The DICis a good approximation to the exact Bayesian approachand works optimal if the posterior distributions are suf-ficiently Gaussian. To obtain the posterior distributionsand corresponding constraints for the various dataset com-binations we have used the Monte Carlo cosmological pa-rameter inference code Montepython [38] in combinationwith the mentioned Einstein-Boltzmann code
CLASS [33].The value of DIC can be computed directly from theMarkov chains generated with
MontePython . For va-lues +5 < ∆DIC < +10 we would conclude strong evi-dence of the RRVM’s as compared to the ΛCDM, and for∆DIC > +10 the evidence is very strong. Such is thecase when we use a threshold redshift z ∗ (cid:39) − < ∆DIC < − χ ) than theΛCDM, similar to e.g. coupled dark energy [39]. Quite ob-viously, the effect of the threshold can be very importantand indicates that a mild dynamics of the vacuum is verymuch welcome, especially if it is activated at around the very epoch when the vacuum dominance appears, namelyat around z (cid:39)
1. To be more precise, the vacuum dom-inance in the ΛCDM starts at around z (cid:39) .
3. There-fore, these results suggest that if the vacuum starts to beslightly dynamical at an earlier point which is ‘close’ (inredshift terms) to the transition from deceleration to ac-celeration ( z (cid:39) . H ( z )+LSS+CMB data becomesextraordinarily significant on statistical terms. Before thetransition point, physics can remain basically unalteredwith respect to the standard ΛCDM model, but the vac-uum dynamics allows to suppress an exceeding amount ofLSS in the universe, leading to a better description of the f ( z ) σ ( z ) data set. It is not just that the total χ is13 to 18 units smaller as compared to the ΛCDM in thepresence of the threshold z ∗ (cf. Tables 1 and 2), but thefact that the information criteria (which take into accountthe penalty to be paid by the RRVM’s for having moreparameters) still decides very strongly in its favor. In theabsence of the H prior [32], type II RRVM performs abit better than the ΛCDM (cf. Table 1), but the improve-ment is not sufficient. Occam’s razor penalizes the mo-del for having two additional parameters than ΛCDM andleads to a moderately negative evidence against it. Whenwe include the prior, however, we get a strong evidence inits favor (∆DIC (cid:38) +5, cf. Table 2), since this model canaccommodate higher values of the Hubble parameter andhence loosen the H tension. This is similar to what wefound in [17] for Brans-Dicke cosmology with Λ (cid:54) = 0.p-6unning vacuum against the H and σ tensionsFinally, we want to remark a few things about theRRVM’s under study, in connection with the cosmolo-gical tensions, cf. Tables 1 and 2, and the contours inFig. 2: (i) the only model capable of alleviating the H tension is RRVM of type II; (ii) the values of S in allRRVM’s are perfectly compatible with recent weak lens-ing and galaxy clustering measurements [40]. For type II arelated observable analogous to (but different from) S ispossible: ˜ S ≡ S / √ ϕ . It is connected with the time vari-ation of G = G N /ϕ and can be viewed also as a rescalingΩ m → Ω m /ϕ in the effective Friedmann’s equation fortype II models, see Ref. [17]. We show the correspondingcountours in Fig. 2; (iii) Quite remarkable is the fact thatthe value of σ is significantly lower in the type I RRVM thr . to the point that the σ tension can be fully accounted for.We have checked that this feature is shared by the moregeneral RVM class (1) using the same threshold redshift. Conclusions. –
We find significant evidence that amild dynamics of the cosmic vacuum would be helpful todescribe the overall cosmological observations as comparedto the standard cosmological model with a rigid Λ-term.For type I models the level of evidence is very stronglysupported by the DIC criterion provided there exists athreshold redshift z ∗ (cid:39) σ tension is renderedvirtually nonexistent ( (cid:46) . σ ) [40]. The H tension, ho-wever, can only be improved within the type II model withvariable G . But then both tensions can be dealt with ata time, the H remaining at ∼ . σ [32] and the σ oneat ∼ . σ (or at only ∼ . σ if stated in terms of S )[40]. The simultaneous alleviation of the two tensions isremarkable and is highly supported by the DIC criterion. ∗ ∗ ∗ JSP, JdCP and CMP are partially supported by MINECO(Spain), SGR (Generalitat de Catalunya) and MDM (IC-CUB). AGV is funded by DFG (Germany). JdCP andCMP are also supported by FPI and FI fellowships, res-pectively. JSP also acknowledges the COST AssociationAction QG-MM. We thank H. Gil-Mar´ın for discussions.
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