---
title: Running Vacuum Model (RVM) Overview
url: https://www.emergentmind.com/topics/running-vacuum-model-rvm
type: topic
---

# Running Vacuum Model (RVM) Overview

Searching arXiv for recent and foundational papers on the Running Vacuum Model to ground the article in the provided literature.
The Running Vacuum Model (RVM) is a class of cosmological models in which the vacuum energy density is not a rigid constant but a dynamical quantity tied to the FLRW background through the Hubble rate \(H\) and, in more general formulations, its derivatives. In the modern universe its leading variation is typically of order \(H^2\), whereas higher-order terms such as \(H^4\), \(H^6\), and related even-adiabatic structures can become relevant in the primordial universe and may drive inflation [2411.06582][2203.13757][2109.12086]. In its canonical phenomenological form the vacuum fluid keeps the de Sitter equation of state \(p_{\rm vac}=-\rho_{\rm vac}\), so the novelty lies in the running of its magnitude rather than in replacing vacuum by a generic quintessence fluid [1812.06710][2008.00523]. Across the literature, the RVM functions simultaneously as a QFT-in-curved-spacetime framework for vacuum renormalization, a phenomenological deformation of \(\Lambda\)CDM, and, in string-inspired constructions, a vehicle for inflation without an external inflaton through anomaly-induced \(H^4\) vacuum terms [2307.13130][2012.07971][2108.03998].

## 1. Definition and formal structure

At the broadest level, the RVM is defined by promoting the vacuum energy density to a function of cosmological scalars,
\[
\rho_{\rm vac}=\rho_{\rm vac}(H,\dot H,\ldots),
\]
with the dependence organized in even adiabatic orders, equivalently in even powers of \(H\) and derivative combinations compatible with general covariance [2411.06582][2109.12086]. This structure is often written schematically as a constant plus \({\cal O}(H^2,\dot H)\) terms, followed by \({\cal O}(H^4,H^2\dot H,\dot H^2,\ldots)\), and higher orders [2109.12086][2203.13757].

A representative low-energy expression is
\[
\rho_{\rm vac}(H)\simeq \rho_{\rm vac}^0+\frac{3\nu}{8\pi G_N}(H^2-H_0^2),
\]
or, in more general notation,
\[
\rho_{\rm vac}(H)= \frac{3}{8\pi G_N}\left(c_0+\nu H^2+\tilde\nu \dot H\right)+{\cal O}(H^4),
\]
with \(|\nu|\ll 1\) [2109.12086][2203.13757]. In early-universe applications one commonly uses
\[
\rho_{\Lambda}(H)=\frac{3}{\kappa^2}\left(c_0+\nu H^2+\alpha \frac{H^4}{H_I^2}\right),
\qquad \kappa^2=8\pi G,
\]
so that the \(H^4\) term dominates when \(H\sim H_I\) [1901.06638][2307.13130].

In the canonical flat-universe implementation constrained against CMB and large-scale-structure data, the vacuum sector is parameterized as
\[
\Lambda=3\nu H^2+\Lambda_0,
\qquad
\rho_\Lambda(H)=\kappa^{-2}(3\nu H^2+\Lambda_0),
\qquad
w_\Lambda=-1,
\]
with \(\nu\) the dimensionless running parameter and \(\Lambda_0\) the residual late-time vacuum contribution [1812.06710]. A non-flat generalization replaces \(H^2\) by \(H^2+K/a^2\),
\[
\Lambda=3\nu\left(H^2+\frac{K}{a^2}\right)+c_0,
\]
thereby coupling the vacuum explicitly to spatial curvature [2002.05290].

The RVM is therefore distinct from both rigid-\(\Lambda\) cosmology and generic scalar-field dark energy. In the canonical presentations, it is not a quintessence model with \(w\neq -1\); rather, it is an interacting or running vacuum fluid with \(w=-1\) and a time-dependent density [1812.06710][2002.05290]. In the QFT-renormalized treatment, however, the effective vacuum equation of state can deviate slightly from \(-1\), which allows the model to mimic quintessence-like or phantom-like behavior without introducing a fundamental dark-energy scalar [2411.06582][2203.13757].

## 2. QFT in curved spacetime and renormalized vacuum running

A central line of development interprets the RVM as an outcome of QFT in curved spacetime rather than as a purely phenomenological ansatz [2411.06582][2109.12086]. In this framework one quantizes matter fields on a classical FLRW background and computes the renormalized vacuum energy-momentum tensor using adiabatic regularization and renormalization, with an off-shell subtraction at an arbitrary mass scale \(M\) [2411.06582][2109.12086].

For a non-minimally coupled scalar field, the renormalized vacuum energy density takes the form
\[
\rho_{\rm vac}(M)=\rho_\Lambda(M)+\frac{\langle T_{00}^{\delta\phi}\rangle_{\rm Ren}(M)}{a^2},
\]
and the explicit low-energy expansion contains a constant-like contribution, an \(H^2\) term proportional to \(\bar\xi\equiv \xi-1/6\), and \({\cal O}(H^4)\) pieces [2411.06582]. The corresponding “RG-like” relation between two renormalization points \(M\) and \(M_0\) is presented as the physical running law for the vacuum sector [2411.06582]. The crucial interpretive step is the identification of the cosmological renormalization scale with the expansion rate,
\[
M=H,
\]
with \(M_0=H_0\) as a reference epoch [2411.06582].

This leads to the canonical late-time law
\[
\rho_{\rm vac}(H)\simeq \rho_{\rm vac}^0+\frac{3\nu_{\rm eff}(H)}{8\pi G_N}(H^2-H_0^2),
\]
where \(\nu_{\rm eff}(H)\) varies only logarithmically and is often approximated by a quasi-constant \(\nu_{\rm eff}(H_0)\) in the observable universe [2411.06582]. Theoretical estimates in the review literature place \(\nu_{\rm eff}(H_0)\) roughly in the range \(10^{-5}-10^{-3}\), while phenomenological fits are reported around \(10^{-4}-10^{-2}\) and often near \(10^{-3}\) [2411.06582].

A major claim of the QFT-based program is that the physically relevant running vacuum is not governed by uncanceled quartic mass terms \(\sim m^4\). After combining the renormalized cosmological term with the renormalized zero-point contribution and comparing vacuum energies across scales, the harmful \(\sim m^4\) pieces cancel in the running law, leaving soft geometric contributions controlled by \(H^2\), \(H^4\), and higher even orders [2411.06582][2109.12086][2203.13757]. This is presented as a substantive softening of the usual fine-tuning formulation of the cosmological constant problem, although the same literature is explicit that renormalization theory by itself does not determine the absolute value of the vacuum energy [2411.06582].

The same QFT literature also stresses a conceptual point: in an expanding quantum background, the observed cosmological term should be regarded as an effective vacuum energy at a given epoch rather than a rigid constant valid for all times [2411.06582]. This motivates the statement that there is no rigid cosmological constant in the QFT context of an expanding FLRW universe [2411.06582].

## 3. Canonical realizations and interacting-vacuum dynamics

Several concrete RVM realizations recur in the literature.

| Realization | Vacuum law | Distinguishing feature |
|---|---|---|
| Canonical flat RVM | \(\Lambda=3\nu H^2+\Lambda_0\) | Vacuum decays into matter and radiation; \(w_\Lambda=-1\) [1812.06710] |
| Non-flat RVM | \(\Lambda=3\nu(H^2+K/a^2)+c_0\) | Curvature-vacuum degeneracy enters CMB phenomenology [2002.05290] |
| Special RVM | \(\Lambda=3\alpha H^2+3\beta H_0^4H^{-2}+\Lambda_0\) | Non-analytic background evolution; numerical treatment required [2001.05092] |

In the canonical flat model, vacuum is not separately conserved because \(\Lambda\) depends on time through \(H\). The total conservation law
\[
\nabla^\mu(T^M_{\mu\nu}+T^\Lambda_{\mu\nu})=0
\]
implies a nonzero \(\dot\rho_\Lambda\), and the paper implementing this model assumes vacuum decay into both nonrelativistic matter and radiation [1812.06710]. For each component \(l=m,r\),
\[
\dot\rho_l+3H(1+w_l)\rho_l=Q_l,
\qquad
Q_l=3\nu H(1+w_l)\rho_l,
\]
which yields the modified dilution laws
\[
\rho_m(a)=\rho_m^{(0)}a^{-3(1-\nu)},
\qquad
\rho_r(a)=\rho_r^{(0)}a^{-4(1-\nu)}.
\]
Matter and radiation therefore dilute more slowly than in \(\Lambda\)CDM because they are continuously fed by vacuum decay [1812.06710].

The non-flat extension preserves the same interacting-vacuum logic but with the geometrical combination \(H^2+K/a^2\) in the running law [2002.05290]. Its background solution becomes
\[
H^2 = H_0^2\left\{ \frac1{\xi}\left[ \Omega_m^0\left(a^{-3\xi}-1\right)+ \Omega_r^0\left(a^{-4\xi}-1\right) \right] +\Omega_K^0\left(a^{-2}-1\right)+1 \right\},
\qquad
\xi=1-\nu,
\]
and the vacuum-to-matter/radiation source terms remain proportional to each component’s enthalpy [2002.05290].

The “special” model with both positive and negative powers of \(H\),
\[
\Lambda=3\alpha H^2+3\beta H_0^4H^{-2}+\Lambda_0,
\]
belongs to a distinct subclass. Its coupled background equations do not admit analytic solutions for \(\rho_m(a)\) and \(\rho_r(a)\), so the model is solved numerically in \(\log a\) and then implemented in CAMB and CosmoMC [2001.05092]. The paper imposes \(\alpha>0\) to avoid a negative dark-energy density in the early universe [2001.05092].

Although the interacting-vacuum picture is central in these phenomenological realizations, later review literature distinguishes it from scenarios in which matter is separately conserved and the Bianchi identities are instead satisfied through a mild running of \(G_N\) [2109.12086][2411.06582]. This distinction underlies the usual classification into fixed-\(G\) interacting vacuum models and running-\(G\) vacuum models.

## 4. Inflationary RVM, graceful exit, and the vacuumon

The RVM is frequently used as a unified description of the full cosmic sequence, from an initial de Sitter-like stage through radiation and matter domination to late-time acceleration [1901.06638][1910.01638]. In the standard inflationary implementation,
\[
\rho_{\Lambda}(H)=\frac{3}{\kappa^2}\left(c_0+\nu H^2+\alpha \frac{H^4}{H_I^2}\right),
\]
the \(H^4\) term dominates when \(H\) is large and yields a de Sitter solution
\[
H^2=\frac{(1-\nu)H_I^2}{\alpha}
\]
in the regime \(c_0/H^2\ll1\) [1901.06638].

The corresponding early-time solution can be written as
\[
H(a)=\left(\frac{1-\nu}{\alpha}\right)^{1/2}\frac{H_I}{\sqrt{D\,a^{3(1-\nu)(1+\omega_m)}+1}},
\]
which for radiation simplifies to
\[
H(a)=\left(\frac{1}{\alpha}\right)^{1/2}\frac{H_I}{\sqrt{D a^4+1}}
\]
when \(|\nu|\ll1\) [1901.06638][2008.00523]. The vacuum and radiation densities then evolve as
\[
\rho_\Lambda(a)=\frac{\rho_I}{\alpha}\frac{1}{(1+Da^4)^2},
\qquad
\rho_r(a)=\frac{\rho_I}{\alpha}\frac{Da^4}{(1+Da^4)^2},
\]
so the model provides a graceful exit from vacuum domination to radiation domination without a separate reheating stage driven by inflaton oscillations [1901.06638].

This early RVM is explicitly described as nonsingular: at \(a=0\), \(\rho_\Lambda\) is finite and \(\rho_r=0\), so the model starts from a non-singular initial de Sitter vacuum stage [1901.06638][1910.01638]. The same literature emphasizes that continuous vacuum decay into radiation can generate the large entropy of the present universe and remove the standard horizon problem [1901.06638][1910.01638].

A scalar-field representation, the “vacuumon,” is introduced as an effective classical description of the total RVM fluid [1901.06638]. The mapping is defined by
\[
\rho_{\rm tot}\equiv \rho_\phi=\frac{\dot\phi^2}{2}+U(\phi),
\qquad
p_{\rm tot}\equiv p_\phi=\frac{\dot\phi^2}{2}-U(\phi),
\]
with
\[
\dot\phi^2=-\frac{2}{\kappa^2}\dot H,
\qquad
U=\frac{3H^2}{\kappa^2}\left(1+\frac{\dot H}{3H^2}\right).
\]
For the early RVM this gives
\[
U(\phi)=\frac{H_I^2}{\alpha\kappa^2}\frac{2+\cosh^2(\kappa\phi)}{\cosh^4(\kappa\phi)}
\]
in the radiation-like case [1901.06638]. In the string-inspired stiff-matter case the analogous potential is
\[
U(\phi)=\widetilde U_0\frac{\tfrac23+\cosh^2(\kappa\phi)}{\cosh^4(\kappa\phi)},
\qquad
\widetilde U_0=\frac{9H_I^2}{\alpha\kappa^2},
\]
again with a hill-top form [2008.00523].

The vacuumon, however, is repeatedly described as a classical background field rather than a fully fledged quantum field [1901.06638][2008.00523]. The scalar representation is therefore not physically equivalent to Starobinsky inflation or to a conventional inflaton model, even though the potentials can be mapped at the classical level [1901.06638]. In the swampland analysis, the vacuumon potential can satisfy relevant swampland and weak-gravity criteria in suitable regimes, but the paper concludes that this does not rehabilitate it as a standard slow-roll inflaton; the inflationary mechanism remains the running vacuum itself, especially the \(\sim H^4\) term [2008.00523].

The QFT-based review literature introduces an additional nuance: in one explicit sixth-order adiabatic calculation, the first constant-\(H\) inflationary contribution appears as
\[
\rho_{\rm vac}^{\rm inf}\sim C_{\rm inf}H^6,
\]
with a graceful-exit solution
\[
H(\hat a)=H_I(1+\hat a^8)^{-1/4},
\qquad
\rho_{\rm vac}(\hat a)=\rho_I(1+\hat a^8)^{-3/2},
\qquad
\rho_r=\hat a^8\rho_{\rm vac}(\hat a),
\]
so the early RVM sector is not limited to a single \(H^4\) realization across all derivations [2411.06582].

## 5. String-inspired and anomaly-driven RVMs

A major extension embeds the RVM in a string-inspired gravitational theory with torsion, Kalb-Ramond axions, and gravitational Chern-Simons anomalies [2012.07971][2105.02659][2108.03998][2306.08064]. In this construction the bosonic massless gravitational multiplet of string theory contains the graviton, the dilaton, and the antisymmetric Kalb-Ramond field \(B_{\mu\nu}\); in four dimensions the dual of the Kalb-Ramond field strength behaves as an axion-like field \(b(x)\) [2307.13130].

After integrating out the torsion three-form and setting the dilaton to zero, the effective action acquires a Chern-Simons modified-gravity structure with a \(b\,R\widetilde R\) coupling [2307.13130][2108.03998]. Primordial chiral gravitational waves then generate a nonzero expectation value of the gravitational Pontryagin density, and the anomaly condensate produces an effective vacuum energy containing both \(H^2\) and \(H^4\) pieces [2012.07971][2105.02659][2108.03998]. One explicit inflationary expression quoted in this literature is
\[
\rho_{\rm total}\simeq 3\kappa^{-4}\Big[-1.65\times 10^{-3}(\kappa H)^2+\frac{\sqrt2}{3}|\overline b(0)|\kappa\times 5.86\times 10^6(\kappa H)^4\Big]>0,
\]
so the \(H^4\) term dominates and drives inflation without an external inflaton [2008.00523][2105.02659].

These models interpret the KR axion as a slow-roll background but not as the inflaton itself [2108.03998]. During a pre-inflationary epoch it behaves as stiff matter with equation of state \(w=+1\), and in one formulation the KR+gCS sector first realizes a “phantom vacuum” state with \(p=-\rho\) and \(\rho<0\), which then transmutes into the positive-energy RVM vacuum once the condensate contribution dominates [2105.02659]. The same framework is also used to motivate late-time dynamical dark energy, axionic dark matter, and anomaly-induced baryogenesis through leptogenesis [2012.07971][2108.03998].

A later refinement argues that approximately de Sitter eras also generate logarithmic corrections of the form
\[
H^n\ln(H^2),\qquad n\in 2\mathbb Z^+,
\]
with special emphasis on the late-time term
\[
\delta\rho_{\rm RVM}^{\rm QG}\propto d_1H^2\ln(H^2)
\]
from graviton loops [2307.13130]. In the supergravity-based realization, the coefficient of the logarithmic term is linked to a primordial supersymmetry-breaking scale \(f\), and the proceedings text argues that these corrections may contribute to the alleviation of the \(H_0\) and structure-growth tensions [2307.13130]. A later phenomenological analysis of the stringy running vacuum model, formulated through an effective action
\[
S=-\int d^4x\sqrt{-g}\left[c_0+R\left(c_1+c_2\ln\left(\frac{R}{R_0}\right)\right)\right]+S_m,
\]
finds strong preference for the model only when SH0ES information is included, with the late-time phenomenology driven mainly by a renormalized cosmological gravitational coupling \(d=G_N/G(z=0)\) and an extremely small running parameter \(|\epsilon|\lesssim {\cal O}(10^{-7})\) [2305.15774].

These string-inspired models are therefore more microscopic than the canonical phenomenological RVM, but they also introduce additional assumptions. The promotion of de Sitter \(\ln\Lambda\) terms to general \(R^n\ln R\) structures is explicitly described as conjectural in the late-time stringy literature [2305.15774], and the microscopic transition from cosmological \(G\) to local \(G_N\) remains open [2305.15774].

## 6. Perturbations, observational constraints, and model assessment

The canonical RVM has been confronted directly with high-precision observables. In the flat model
\[
\Lambda=3\nu H^2+\Lambda_0,
\]
the perturbation analysis is performed in the synchronous gauge, with modified density-contrast and velocity-divergence equations
\[
\dot\delta_l=\cdots-\frac{aQ_l}{\rho_l}\delta_l,
\qquad
\dot\theta_l=\cdots-\frac{aQ_l}{\rho_l}\theta_l,
\]
so positive \(\nu\) damps both \(\delta_l\) and \(\theta_l\) and suppresses structure growth [1812.06710]. CAMB was modified to incorporate the altered background scalings and perturbation equations, and the parameter space was sampled with CosmoMC using Planck 2015 CMB temperature and polarization, Planck 2015 lensing, BAO, matter power spectrum, and CFHTLenS weak-lensing data [1812.06710].

The main quantitative result is a very tight bound,
\[
\nu \le 1.54\times 10^{-4}\qquad (68\%~{\rm C.L.}),
\]
with best fit
\[
\nu=1.19\times 10^{-4},
\qquad
\chi^2_{\rm RVM}=13968.8,
\qquad
\chi^2_{\Lambda{\rm CDM}}=13969.8.
\]
Although the RVM gives a slightly smaller \(\chi^2\), the paper states that the improvement is not statistically significant and that the model cannot be distinguished from \(\Lambda\)CDM within \(1\sigma\) [1812.06710]. Physically, this means that vacuum decay into matter and radiation through
\[
Q_l=3\nu H(1+w_l)\rho_l
\]
must be minuscule over cosmic history [1812.06710].

The non-flat extension yields a similar conclusion. Using Planck 2018 CMB, BAO, JLA supernovae, weak lensing, and \(f\sigma_8\) data, the paper finds
\[
\nu\lesssim 1.39\times 10^{-4}\qquad (68\%~{\rm C.L.}),
\qquad
|\Omega_K|\le {\cal O}(10^{-2})\qquad (95\%~{\rm C.L.}),
\]
with only a marginal best-fit improvement over non-flat \(\Lambda\)CDM [2002.05290]. The analysis emphasizes a geometrical degeneracy in the CMB between \(\nu\) and \(\Omega_K\), although positive \(\nu\) produces a more distinctive suppression of the TT spectrum [2002.05290].

The special model with
\[
\Lambda=3\alpha H^2+3\beta H_0^4H^{-2}+\Lambda_0
\]
is also found to lie very close to \(\Lambda\)CDM. The reported constraints are
\[
\alpha<2.83\times 10^{-4},
\qquad
\beta=(-0.2^{+3.9}_{-4.5})\times 10^{-4}
\qquad (95\%~{\rm C.L.}),
\]
with \(\chi^2_{\rm best-fit}=2543.259\) versus \(2546.662\) for \(\Lambda\)CDM [2001.05092]. The paper concludes that the model fits the data comparably to \(\Lambda\)CDM but does not provide evidence for nonzero running [2001.05092].

A separate branch of work argues that combined running of the vacuum sector and the gravitational coupling can ease cosmological tensions more substantially. In the mini-review treatment of interacting-vacuum and running-\(G\) models, type-I interacting vacuum improves the growth tension but not \(H_0\), whereas type-II models with mild running of \(G_{\rm eff}\) can alleviate both tensions simultaneously, with indicative values
\[
H_0=70.93^{+0.93}_{-0.87}\ {\rm km/s/Mpc},
\qquad
\sigma_8=0.794^{+0.013}_{-0.012},
\qquad
S_8=0.761^{+0.018}_{-0.017}
\]
for the favored type-II case [2109.12086][2203.13757].

Composite dark-energy extensions go further. The \(\Lambda X\)CDM and \(wX\)CDM scenarios treat dark energy as a mixed fluid built from running vacuum and an additional component \(X\), with the latter interpreted as “phantom matter” when \(w_X\lesssim -1\), \(\rho_X<0\), and \(p_X>0\) [2410.20382]. In the simplified \(wX\)CDM version, the relevant free parameters are \((z_t,w_X,w_Y)\), and the paper reports
\[
z_t=1.46^{+0.02}_{-0.01},
\qquad
w_X=-1.16^{+0.13}_{-0.16},
\qquad
w_Y=-0.90\pm0.03,
\]
together with \(\chi^2_{\rm min}=4107.62\) against \(4166.76\) for \(\Lambda\)CDM and a \(3.3\sigma\) preference for quintessence-like behavior around the present epoch [2410.20382]. The same work stresses, however, that the outcome depends sensitively on whether 2D or 3D BAO are used [2410.20382].

The overall observational picture is therefore split. The minimal canonical RVM is tightly constrained to lie very near \(\Lambda\)CDM [1812.06710][2002.05290][2001.05092]. More elaborate running-vacuum frameworks—especially those allowing running \(G\), composite dark sectors, or logarithmic corrections—are presented as more effective in alleviating the \(H_0\) and growth tensions [2109.12086][2410.20382][2305.15774]. This suggests that late-time phenomenology is highly model-dependent within the broader running-vacuum family.

## 7. Conceptual issues, misconceptions, and open problems

A recurring misconception is to identify the RVM with generic quintessence. In the canonical literature this is incorrect: the vacuum fluid is kept at
\[
w_\Lambda=-1,
\]
and the dynamics comes from the running of \(\rho_\Lambda\), not from a non-vacuum equation of state [1812.06710][2002.05290][2008.00523]. A related misconception is to equate the vacuumon with a fundamental inflaton; the vacuumon is explicitly described as an effective classical representation of the RVM background and not as a full quantum scalar field [1901.06638][2008.00523].

Another central issue is statistical evidence. The flat and non-flat canonical RVM fits show that nonzero running is allowed only at the level \(\nu\lesssim 10^{-4}\), and the reported \(\chi^2\) improvements over \(\Lambda\)CDM are not statistically significant [1812.06710][2002.05290]. This means that the simplest RVM is observationally viable but currently almost indistinguishable from the concordance model.

On the theoretical side, the QFT-in-curved-spacetime program argues strongly that the RVM is not an arbitrary deformation of \(\Lambda\)CDM but an effective description of renormalized vacuum energy in a dynamical background [2411.06582][2109.12086][2203.13757]. Even there, however, important assumptions remain. The identification \(M=H\) is treated as physically motivated but still interpretive [2411.06582]. The absolute value of the vacuum energy cannot be computed from renormalization theory alone [2411.06582]. The explicit derivations are usually carried out for prototype scalar fields and then generalized conceptually to realistic particle content [2109.12086][2203.13757].

String-inspired RVMs add a further layer of uncertainty. They provide a microscopic rationale for an \(H^4\) inflationary term, but late-time logarithmic corrections and running-gravity effects are only partially derived and in some places explicitly conjectural [2307.13130][2305.15774]. The same applies to proposed links with primordial-black-hole dark matter and induced gravitational-wave spectra in multi-axion extensions [2306.08064].

Despite these open issues, the RVM occupies a distinctive position in contemporary cosmology. It offers a framework in which inflationary vacuum enhancement, late-time dark-energy dynamics, and part of the cosmological constant problem are treated within a single vacuum-based language [2411.06582][2203.13757]. Its minimal realizations are forced by data to remain close to \(\Lambda\)CDM, while its extended realizations remain an active arena for addressing cosmological tensions, anomaly-driven inflation, and QFT-based vacuum renormalization [1812.06710][2410.20382][2307.13130].

Source: https://www.emergentmind.com/topics/running-vacuum-model-rvm