Running Vacuum Model (RVM) Overview

• The Running Vacuum Model is a dynamic cosmological framework where vacuum energy density evolves as a function of the Hubble rate and its derivatives, impacting both early inflation and late-time acceleration.
• It leverages quantum field theory in curved spacetime to renormalize the vacuum energy, addressing the cosmological constant problem without invoking a traditional scalar field.
• Observational constraints tightly bound the running parameter, while extended models show promise for resolving tensions in H0, structure growth, and cosmic curvature.

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 (Peracaula et al., 2024, Peracaula, 2022, Peracaula, 2021). 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 (Zhang et al., 2018, Mavromatos et al., 2020). 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 (Mavromatos et al., 2023, Mavromatos et al., 2020, Mavromatos, 2021).

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 (Peracaula et al., 2024, Peracaula, 2021). This structure is often written schematically as a constant plus ${\cal O}(H^2,\dot H)$ terms, followed by $H^2$0, and higher orders (Peracaula, 2021, Peracaula, 2022).

A representative low-energy expression is

$H^2$1

or, in more general notation,

$H^2$2

with $H^2$3 (Peracaula, 2021, Peracaula, 2022). In early-universe applications one commonly uses

$H^2$4

so that the $H^2$5 term dominates when $H^2$6 (Basilakos et al., 2019, Mavromatos et al., 2023).

In the canonical flat-universe implementation constrained against CMB and large-scale-structure data, the vacuum sector is parameterized as

$H^2$7

with $H^2$8 the dimensionless running parameter and $H^2$9 the residual late-time vacuum contribution (Zhang et al., 2018). A non-flat generalization replaces $H^4$0 by $H^4$1,

$H^4$2

thereby coupling the vacuum explicitly to spatial curvature (Geng et al., 2020).

The RVM is therefore distinct from both rigid-$H^4$3 cosmology and generic scalar-field dark energy. In the canonical presentations, it is not a quintessence model with $H^4$4; rather, it is an interacting or running vacuum fluid with $H^4$5 and a time-dependent density (Zhang et al., 2018, Geng et al., 2020). In the QFT-renormalized treatment, however, the effective vacuum equation of state can deviate slightly from $H^4$6, which allows the model to mimic quintessence-like or phantom-like behavior without introducing a fundamental dark-energy scalar (Peracaula et al., 2024, Peracaula, 2022).

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 (Peracaula et al., 2024, Peracaula, 2021). 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 $H^4$7 (Peracaula et al., 2024, Peracaula, 2021).

For a non-minimally coupled scalar field, the renormalized vacuum energy density takes the form

$H^4$8

and the explicit low-energy expansion contains a constant-like contribution, an $H^4$9 term proportional to $H^6$0, and $H^6$1 pieces (Peracaula et al., 2024). The corresponding “RG-like” relation between two renormalization points $H^6$2 and $H^6$3 is presented as the physical running law for the vacuum sector (Peracaula et al., 2024). The crucial interpretive step is the identification of the cosmological renormalization scale with the expansion rate,

$H^6$4

with $H^6$5 as a reference epoch (Peracaula et al., 2024).

This leads to the canonical late-time law

$H^6$6

where $H^6$7 varies only logarithmically and is often approximated by a quasi-constant $H^6$8 in the observable universe (Peracaula et al., 2024). Theoretical estimates in the review literature place $H^6$9 roughly in the range $p_{\rm vac}=-\rho_{\rm vac}$0, while phenomenological fits are reported around $p_{\rm vac}=-\rho_{\rm vac}$1 and often near $p_{\rm vac}=-\rho_{\rm vac}$2 (Peracaula et al., 2024).

A major claim of the QFT-based program is that the physically relevant running vacuum is not governed by uncanceled quartic mass terms $p_{\rm vac}=-\rho_{\rm vac}$3. After combining the renormalized cosmological term with the renormalized zero-point contribution and comparing vacuum energies across scales, the harmful $p_{\rm vac}=-\rho_{\rm vac}$4 pieces cancel in the running law, leaving soft geometric contributions controlled by $p_{\rm vac}=-\rho_{\rm vac}$5, $p_{\rm vac}=-\rho_{\rm vac}$6, and higher even orders (Peracaula et al., 2024, Peracaula, 2021, Peracaula, 2022). 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 (Peracaula et al., 2024).

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 (Peracaula et al., 2024). This motivates the statement that there is no rigid cosmological constant in the QFT context of an expanding FLRW universe (Peracaula et al., 2024).

3. Canonical realizations and interacting-vacuum dynamics

Several concrete RVM realizations recur in the literature.

Realization	Vacuum law	Distinguishing feature
Canonical flat RVM	$p_{\rm vac}=-\rho_{\rm vac}$7	Vacuum decays into matter and radiation; $p_{\rm vac}=-\rho_{\rm vac}$8 (Zhang et al., 2018)
Non-flat RVM	$p_{\rm vac}=-\rho_{\rm vac}$9	Curvature-vacuum degeneracy enters CMB phenomenology (Geng et al., 2020)
Special RVM	$\Lambda$0	Non-analytic background evolution; numerical treatment required (Geng et al., 2020)

In the canonical flat model, vacuum is not separately conserved because $\Lambda$1 depends on time through $\Lambda$2. The total conservation law

$\Lambda$3

implies a nonzero $\Lambda$4, and the paper implementing this model assumes vacuum decay into both nonrelativistic matter and radiation (Zhang et al., 2018). For each component $\Lambda$5,

$\Lambda$6

which yields the modified dilution laws

$\Lambda$7

Matter and radiation therefore dilute more slowly than in $\Lambda$8CDM because they are continuously fed by vacuum decay (Zhang et al., 2018).

The non-flat extension preserves the same interacting-vacuum logic but with the geometrical combination $\Lambda$9 in the running law (Geng et al., 2020). Its background solution becomes

$H^4$0

and the vacuum-to-matter/radiation source terms remain proportional to each component’s enthalpy (Geng et al., 2020).

The “special” model with both positive and negative powers of $H^4$1,

$H^4$2

belongs to a distinct subclass. Its coupled background equations do not admit analytic solutions for $H^4$3 and $H^4$4, so the model is solved numerically in $H^4$5 and then implemented in CAMB and CosmoMC (Geng et al., 2020). The paper imposes $H^4$6 to avoid a negative dark-energy density in the early universe (Geng et al., 2020).

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 $H^4$7 (Peracaula, 2021, Peracaula et al., 2024). This distinction underlies the usual classification into fixed-$H^4$8 interacting vacuum models and running-$H^4$9 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 (Basilakos et al., 2019, Sola et al., 2019). In the standard inflationary implementation,

$\rho_{\rm vac}=\rho_{\rm vac}(H,\dot H,\ldots),$0

the $\rho_{\rm vac}=\rho_{\rm vac}(H,\dot H,\ldots),$1 term dominates when $\rho_{\rm vac}=\rho_{\rm vac}(H,\dot H,\ldots),$2 is large and yields a de Sitter solution

$\rho_{\rm vac}=\rho_{\rm vac}(H,\dot H,\ldots),$3

in the regime $\rho_{\rm vac}=\rho_{\rm vac}(H,\dot H,\ldots),$4 (Basilakos et al., 2019).

The corresponding early-time solution can be written as

$\rho_{\rm vac}=\rho_{\rm vac}(H,\dot H,\ldots),$5

which for radiation simplifies to

$\rho_{\rm vac}=\rho_{\rm vac}(H,\dot H,\ldots),$6

when $\rho_{\rm vac}=\rho_{\rm vac}(H,\dot H,\ldots),$7 (Basilakos et al., 2019, Mavromatos et al., 2020). The vacuum and radiation densities then evolve as

$\rho_{\rm vac}=\rho_{\rm vac}(H,\dot H,\ldots),$8

so the model provides a graceful exit from vacuum domination to radiation domination without a separate reheating stage driven by inflaton oscillations (Basilakos et al., 2019).

This early RVM is explicitly described as nonsingular: at $\rho_{\rm vac}=\rho_{\rm vac}(H,\dot H,\ldots),$9, $H$0 is finite and $H$1, so the model starts from a non-singular initial de Sitter vacuum stage (Basilakos et al., 2019, Sola et al., 2019). 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 (Basilakos et al., 2019, Sola et al., 2019).

A scalar-field representation, the “vacuumon,” is introduced as an effective classical description of the total RVM fluid (Basilakos et al., 2019). The mapping is defined by

$H$2

with

$H$3

For the early RVM this gives

$H$4

in the radiation-like case (Basilakos et al., 2019). In the string-inspired stiff-matter case the analogous potential is

$H$5

again with a hill-top form (Mavromatos et al., 2020).

The vacuumon, however, is repeatedly described as a classical background field rather than a fully fledged quantum field (Basilakos et al., 2019, Mavromatos et al., 2020). 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 (Basilakos et al., 2019). 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 $H$6 term (Mavromatos et al., 2020).

The QFT-based review literature introduces an additional nuance: in one explicit sixth-order adiabatic calculation, the first constant-$H$7 inflationary contribution appears as

$H$8

with a graceful-exit solution

$H$9

so the early RVM sector is not limited to a single ${\cal O}(H^2,\dot H)$0 realization across all derivations (Peracaula et al., 2024).

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 (Mavromatos et al., 2020, Mavromatos et al., 2021, Mavromatos, 2021, Mavromatos, 2023). In this construction the bosonic massless gravitational multiplet of string theory contains the graviton, the dilaton, and the antisymmetric Kalb-Ramond field ${\cal O}(H^2,\dot H)$1; in four dimensions the dual of the Kalb-Ramond field strength behaves as an axion-like field ${\cal O}(H^2,\dot H)$2 (Mavromatos et al., 2023).

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 ${\cal O}(H^2,\dot H)$3 coupling (Mavromatos et al., 2023, Mavromatos, 2021). 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 ${\cal O}(H^2,\dot H)$4 and ${\cal O}(H^2,\dot H)$5 pieces (Mavromatos et al., 2020, Mavromatos et al., 2021, Mavromatos, 2021). One explicit inflationary expression quoted in this literature is

${\cal O}(H^2,\dot H)$6

so the ${\cal O}(H^2,\dot H)$7 term dominates and drives inflation without an external inflaton (Mavromatos et al., 2020, Mavromatos et al., 2021).

These models interpret the KR axion as a slow-roll background but not as the inflaton itself (Mavromatos, 2021). During a pre-inflationary epoch it behaves as stiff matter with equation of state ${\cal O}(H^2,\dot H)$8, and in one formulation the KR+gCS sector first realizes a “phantom vacuum” state with ${\cal O}(H^2,\dot H)$9 and $H^2$00, which then transmutes into the positive-energy RVM vacuum once the condensate contribution dominates (Mavromatos et al., 2021). The same framework is also used to motivate late-time dynamical dark energy, axionic dark matter, and anomaly-induced baryogenesis through leptogenesis (Mavromatos et al., 2020, Mavromatos, 2021).

A later refinement argues that approximately de Sitter eras also generate logarithmic corrections of the form

$H^2$01

with special emphasis on the late-time term

$H^2$02

from graviton loops (Mavromatos et al., 2023). In the supergravity-based realization, the coefficient of the logarithmic term is linked to a primordial supersymmetry-breaking scale $H^2$03, and the proceedings text argues that these corrections may contribute to the alleviation of the $H^2$04 and structure-growth tensions (Mavromatos et al., 2023). A later phenomenological analysis of the stringy running vacuum model, formulated through an effective action

$H^2$05

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 $H^2$06 and an extremely small running parameter $H^2$07 (Gómez-Valent et al., 2023).

These string-inspired models are therefore more microscopic than the canonical phenomenological RVM, but they also introduce additional assumptions. The promotion of de Sitter $H^2$08 terms to general $H^2$09 structures is explicitly described as conjectural in the late-time stringy literature (Gómez-Valent et al., 2023), and the microscopic transition from cosmological $H^2$10 to local $H^2$11 remains open (Gómez-Valent et al., 2023).

6. Perturbations, observational constraints, and model assessment

The canonical RVM has been confronted directly with high-precision observables. In the flat model

$H^2$12

the perturbation analysis is performed in the synchronous gauge, with modified density-contrast and velocity-divergence equations

$H^2$13

so positive $H^2$14 damps both $H^2$15 and $H^2$16 and suppresses structure growth (Zhang et al., 2018). 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 (Zhang et al., 2018).

The main quantitative result is a very tight bound,

$H^2$17

with best fit

$H^2$18

Although the RVM gives a slightly smaller $H^2$19, the paper states that the improvement is not statistically significant and that the model cannot be distinguished from $H^2$20CDM within $H^2$21 (Zhang et al., 2018). Physically, this means that vacuum decay into matter and radiation through

$H^2$22

must be minuscule over cosmic history (Zhang et al., 2018).

The non-flat extension yields a similar conclusion. Using Planck 2018 CMB, BAO, JLA supernovae, weak lensing, and $H^2$23 data, the paper finds

$H^2$24

with only a marginal best-fit improvement over non-flat $H^2$25CDM (Geng et al., 2020). The analysis emphasizes a geometrical degeneracy in the CMB between $H^2$26 and $H^2$27, although positive $H^2$28 produces a more distinctive suppression of the TT spectrum (Geng et al., 2020).

The special model with

$H^2$29

is also found to lie very close to $H^2$30CDM. The reported constraints are

$H^2$31

with $H^2$32 versus $H^2$33 for $H^2$34CDM (Geng et al., 2020). The paper concludes that the model fits the data comparably to $H^2$35CDM but does not provide evidence for nonzero running (Geng et al., 2020).

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-$H^2$36 models, type-I interacting vacuum improves the growth tension but not $H^2$37, whereas type-II models with mild running of $H^2$38 can alleviate both tensions simultaneously, with indicative values

$H^2$39

for the favored type-II case (Peracaula, 2021, Peracaula, 2022).

Composite dark-energy extensions go further. The $H^2$40CDM and $H^2$41CDM scenarios treat dark energy as a mixed fluid built from running vacuum and an additional component $H^2$42, with the latter interpreted as “phantom matter” when $H^2$43, $H^2$44, and $H^2$45 (Peracaula, 2024). In the simplified $H^2$46CDM version, the relevant free parameters are $H^2$47, and the paper reports

$H^2$48

together with $H^2$49 against $H^2$50 for $H^2$51CDM and a $H^2$52 preference for quintessence-like behavior around the present epoch (Peracaula, 2024). The same work stresses, however, that the outcome depends sensitively on whether 2D or 3D BAO are used (Peracaula, 2024).

The overall observational picture is therefore split. The minimal canonical RVM is tightly constrained to lie very near $H^2$53CDM (Zhang et al., 2018, Geng et al., 2020, Geng et al., 2020). More elaborate running-vacuum frameworks—especially those allowing running $H^2$54, composite dark sectors, or logarithmic corrections—are presented as more effective in alleviating the $H^2$55 and growth tensions (Peracaula, 2021, Peracaula, 2024, Gómez-Valent et al., 2023). 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

$H^2$56

and the dynamics comes from the running of $H^2$57, not from a non-vacuum equation of state (Zhang et al., 2018, Geng et al., 2020, Mavromatos et al., 2020). 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 (Basilakos et al., 2019, Mavromatos et al., 2020).

Another central issue is statistical evidence. The flat and non-flat canonical RVM fits show that nonzero running is allowed only at the level $H^2$58, and the reported $H^2$59 improvements over $H^2$60CDM are not statistically significant (Zhang et al., 2018, Geng et al., 2020). 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 $H^2$61CDM but an effective description of renormalized vacuum energy in a dynamical background (Peracaula et al., 2024, Peracaula, 2021, Peracaula, 2022). Even there, however, important assumptions remain. The identification $H^2$62 is treated as physically motivated but still interpretive (Peracaula et al., 2024). The absolute value of the vacuum energy cannot be computed from renormalization theory alone (Peracaula et al., 2024). The explicit derivations are usually carried out for prototype scalar fields and then generalized conceptually to realistic particle content (Peracaula, 2021, Peracaula, 2022).

String-inspired RVMs add a further layer of uncertainty. They provide a microscopic rationale for an $H^2$63 inflationary term, but late-time logarithmic corrections and running-gravity effects are only partially derived and in some places explicitly conjectural (Mavromatos et al., 2023, Gómez-Valent et al., 2023). The same applies to proposed links with primordial-black-hole dark matter and induced gravitational-wave spectra in multi-axion extensions (Mavromatos, 2023).

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 (Peracaula et al., 2024, Peracaula, 2022). Its minimal realizations are forced by data to remain close to $H^2$64CDM, while its extended realizations remain an active arena for addressing cosmological tensions, anomaly-driven inflation, and QFT-based vacuum renormalization (Zhang et al., 2018, Peracaula, 2024, Mavromatos et al., 2023).