Scale-Invariant Vacuum (SIV) in Cosmology

• SIV is a theoretical framework that extends general relativity using Weyl integrable geometry, where the vacuum is assumed to be scale invariant.
• It modifies cosmological dynamics by introducing additional terms in the FLRW equations, affecting galaxy rotation curves and potentially unifying dark matter and dark energy phenomena.
• The approach also addresses early-universe challenges such as Big-Bang Nucleosynthesis and the Lithium-7 problem, while provoking debates over the appropriate gauge and symmetry scaling methods.

Scale-Invariant Vacuum (SIV) denotes, in the contemporary cosmology literature, a scale-covariant extension of gravitation in which the macroscopic empty space is taken to be scale invariant and is described in Weyl Integrable Geometry (WIG), with a conformal scale factor $\lambda$ and an exact Weyl connection $\kappa_\mu=-\partial_\mu\ln\lambda$ (Maeder et al., 2023). In a distinct but related usage in scale-invariant quantum field theory, a scale-invariant vacuum is a vacuum state of a classically scale-free theory in which spontaneous breaking of global Weyl symmetry generates physical mass scales such as the Planck scale or the electroweak scale (Ferreira et al., 2016). Within the gravitational SIV paradigm, the same geometric structure has been applied to FLRW cosmology, weak-field dynamics, galaxy rotation curves, MOND-like phenomenology, structure growth, Big-Bang Nucleosynthesis (BBN), and recent attempts to address the Lithium-7 problem; the most recent BBN analysis uses SIV as a guiding framework but concludes that Reparametrization Invariant Symmetry Scaling (RISS) is the more appropriate description during the BBN epoch (Gueorguiev, 12 Sep 2025).

1. Definition, scope, and competing usages

In the gravitational literature, SIV is formulated as an extension to standard Einstein General Relativity in WIG, with the guiding assumption that the macroscopic empty space is scale invariant, homogeneous, and isotropic (Gueorguiev et al., 2023). The basic conformal mapping is

$g'_{\mu\nu}=\lambda^2 g_{\mu\nu},$

with the GR-frame metric denoted by a prime and the scale-invariant metric unprimed. The corresponding metrical connection is

$\kappa_\mu=-\partial_\mu\ln\lambda,$

and integrability implies $\partial_\nu\kappa_\mu=\partial_\mu\kappa_\nu$ (Maeder et al., 2020).

The same expression, “scale-invariant vacuum,” is also used in quantum field theory to denote vacua of classically scale-invariant models. In one strand, cosmological expansion dilutes the conserved Weyl current and dynamically drives its kernel $K$ to a constant, thereby fixing scalar vacuum expectation values and generating the Planck scale (Ferreira et al., 2016). In another strand, a classically scale-invariant Higgs sector acquires a vacuum by Coleman–Weinberg dimensional transmutation, with enhanced Higgs self-interactions and a radiatively generated electroweak scale (Endo et al., 2015). These usages share the absence of input mass scales at the classical level, but they do not coincide with the WIG-based SIV cosmology.

This terminological overlap is a recurrent source of confusion. A plausible implication is that “SIV” should be read contextually: in cosmology it usually refers to the Weyl-integrable gravitational paradigm, whereas in high-energy theory it often denotes the vacuum structure of classically scale-free models.

2. Geometric formulation in Weyl Integrable Geometry

The gravitational SIV framework is built on WIG and Dirac’s co-tensor formalism (Maeder et al., 2023). Under a local scale transformation, a quantity $Y$ of scale weight $n$ obeys

$Y'=\lambda^n Y.$

For a co-scalar $S$ of power $\kappa_\mu=-\partial_\mu\ln\lambda$0, the co-covariant derivative is

$\kappa_\mu=-\partial_\mu\ln\lambda$1

The curvature scalar in WIG is

$\kappa_\mu=-\partial_\mu\ln\lambda$2

(Maeder et al., 2023).

The field equations may be written in scale-covariant form as

$\kappa_\mu=-\partial_\mu\ln\lambda$3

with $\kappa_\mu=-\partial_\mu\ln\lambda$4 preserving scale covariance (Maeder et al., 2023). In the weak-field regime, the geodesic equation acquires additional scale-connection terms relative to GR: $\kappa_\mu=-\partial_\mu\ln\lambda$5 (Maeder, 2023).

The SIV gauge is fixed by requiring that the vacuum be scale-invariant, homogeneous, and isotropic. For $\kappa_\mu=-\partial_\mu\ln\lambda$6 this yields

$\kappa_\mu=-\partial_\mu\ln\lambda$7

with solution

$\kappa_\mu=-\partial_\mu\ln\lambda$8

(Gueorguiev et al., 2023). This gauge choice is central to later SIV cosmology and to the weak-field term proportional to velocity.

3. Cosmological dynamics and early-universe applications

With the SIV gauge imposed, the FLRW equations acquire extra terms proportional to $\kappa_\mu=-\partial_\mu\ln\lambda$9 and the explicit cosmological-constant contribution drops out of the background equations (Gueorguiev et al., 4 Feb 2025). In one common form,

$g'_{\mu\nu}=\lambda^2 g_{\mu\nu},$0

$g'_{\mu\nu}=\lambda^2 g_{\mu\nu},$1

(Maeder et al., 2023). For flat matter-dominated models,

$g'_{\mu\nu}=\lambda^2 g_{\mu\nu},$2

(Gueorguiev et al., 2023).

SIV has also been connected to inflation by identifying the SIV scalar clock with an inflationary scalar. In this construction,

$g'_{\mu\nu}=\lambda^2 g_{\mu\nu},$3

and the inflation condition is satisfied for $g'_{\mu\nu}=\lambda^2 g_{\mu\nu},$4 and $g'_{\mu\nu}=\lambda^2 g_{\mu\nu},$5 (Maeder et al., 2021). The same review literature states that a graceful exit occurs at

$g'_{\mu\nu}=\lambda^2 g_{\mu\nu},$6

(Gueorguiev et al., 2023). The published summaries do not provide $g'_{\mu\nu}=\lambda^2 g_{\mu\nu},$7 or $g'_{\mu\nu}=\lambda^2 g_{\mu\nu},$8, and this omission is itself one of the open technical gaps.

Matter content plays a limiting role. The SIV literature explicitly states that scale-invariant effects are rapidly reduced from $g'_{\mu\nu}=\lambda^2 g_{\mu\nu},$9 to $\kappa_\mu=-\partial_\mu\ln\lambda,$0 and are forbidden for densities above $\kappa_\mu=-\partial_\mu\ln\lambda,$1 (Maeder et al., 2021). This is one reason why early-universe applications have remained more delicate than late-time weak-field applications.

4. Weak-field dynamics, MOND-like behavior, and the dark sector

In the weak-field, slow-motion limit, SIV predicts a Newton-like equation with an additional acceleration parallel to the velocity: $\kappa_\mu=-\partial_\mu\ln\lambda,$2 (Maeder et al., 2023). In current time units this becomes

$\kappa_\mu=-\partial_\mu\ln\lambda,$3

where the extra term is small locally because $\kappa_\mu=-\partial_\mu\ln\lambda,$4 is of order the age of the Universe (Maeder et al., 2023).

Applied to circular orbits and galactic systems, this modification yields a closed relation between the observed acceleration $\kappa_\mu=-\partial_\mu\ln\lambda,$5 and the baryonic Newtonian acceleration $\kappa_\mu=-\partial_\mu\ln\lambda,$6: $\kappa_\mu=-\partial_\mu\ln\lambda,$7 (Maeder et al., 2020). The same analysis reports that SIV reproduces the observed Radial Acceleration Relation for $\kappa_\mu=-\partial_\mu\ln\lambda,$8 and predicts a horizontal asymptote

$\kappa_\mu=-\partial_\mu\ln\lambda,$9

at very low accelerations, calibrated as

$\partial_\nu\kappa_\mu=\partial_\mu\kappa_\nu$0

(Maeder et al., 2020). This asymptote is used to account for dwarf spheroidal data, whereas MOND in its deep form predicts $\partial_\nu\kappa_\mu=\partial_\mu\kappa_\nu$1 as $\partial_\nu\kappa_\mu=\partial_\mu\kappa_\nu$2 (Maeder et al., 2020).

The relation to MOND is treated explicitly in later work: MOND is described as a peculiar case of SIV when $\partial_\nu\kappa_\mu=\partial_\mu\kappa_\nu$3 is effectively constant over the relevant dynamical timescale, an approximation said to be valid to within $\partial_\nu\kappa_\mu=\partial_\mu\kappa_\nu$4 over the last $\partial_\nu\kappa_\mu=\partial_\mu\kappa_\nu$5 Myr (Maeder, 2023). In that limit SIV yields the deep-MOND form

$\partial_\nu\kappa_\mu=\partial_\mu\kappa_\nu$6

but with an acceleration scale that is not universal: $\partial_\nu\kappa_\mu=\partial_\mu\kappa_\nu$7 (Maeder, 2023). Subsequent work makes this redshift dependence explicit through

$\partial_\nu\kappa_\mu=\partial_\mu\kappa_\nu$8

and argues that present data are statistically compatible with a weak redshift dependence but do not yet establish it clearly (Gueorguiev, 2024).

A broader cosmological interpretation is that SIV attempts to explain both dark-energy-like and dark-matter-like phenomenology geometrically: the scale connection contributes to the background expansion and also modifies weak-field dynamics without adding new particles (Gueorguiev et al., 4 Feb 2025).

5. Big-Bang Nucleosynthesis, Lithium-7, and the shift toward RISS

BBN has become the most technically demanding early-universe application of SIV. An earlier study implemented the SIV background through analytic relations such as

$\partial_\nu\kappa_\mu=\partial_\mu\kappa_\nu$9

with radiation obeying $K$0 and matter $K$1 (Gueorguiev et al., 2023). In that framework, matching $K$2 and D/H required lower matter content than in standard cosmology, with reported successful fits around $K$3 and $K$4, while the Lithium-7 overproduction remained (Gueorguiev et al., 2023).

The 2025 analysis on the Lithium-7 problem recasts the BBN modification more explicitly in terms of conformal scaling by a nearly constant factor $K$5 during BBN (Gueorguiev, 12 Sep 2025). The core transformation is

$K$6

supplemented by microphysical scalings

$K$7

A key SIV-motivated constraint used in one of the fits is

$K$8

(Gueorguiev, 12 Sep 2025). The implementation modifies radiation and matter contributions to the Friedmann equation, rescales forward rates as

$K$9

and rescales reverse-rate factors according to

$Y$0

(Gueorguiev, 12 Sep 2025).

The physical claim of that work is that resolving the $Y$1Li problem requires a departure from local thermal equilibrium during BBN, so that matter and radiation scale differently with respect to $Y$2 (Gueorguiev, 12 Sep 2025). Using the publicly available PRIMAT code, the paper reports a baseline PRIMAT value $Y$3Li/H $Y$4 for $Y$5, an unbroken SIV-like fit with $Y$6, $Y$7, and $Y$8Li/H $Y$9, a partially broken SIV fit with $n$0, $n$1, and $n$2Li/H $n$3, and a RISS-favored fit with $n$4, $n$5, $n$6, $n$7, and $n$8Li/H $n$9 (Gueorguiev, 12 Sep 2025). The same paper states that the best-performing fits give $Y'=\lambda^n Y.$0 when only $Y'=\lambda^n Y.$1He, D/H, and $Y'=\lambda^n Y.$2He/D are used, and $Y'=\lambda^n Y.$3 when $Y'=\lambda^n Y.$4Li/H is included (Gueorguiev, 12 Sep 2025).

The conceptual conclusion is not that SIV is directly vindicated in its standard gauge form. Rather, the paper argues that SIV serves as the conceptual and algebraic scaffolding, while the preferred BBN description is RISS because the successful SIV-guided fits tend to prefer $Y'=\lambda^n Y.$5, whereas the standard SIV gauge would give $Y'=\lambda^n Y.$6 in the early Universe (Gueorguiev, 12 Sep 2025). This is one of the clearest internal controversies in the recent literature.

6. Alternative scale-invariant vacua, criticisms, and open problems

Outside the WIG-based cosmological program, the phrase “scale-invariant vacuum” continues to denote vacuum structure in classically scale-free field theories. In one influential treatment, the conserved Weyl current has kernel

$Y'=\lambda^n Y.$7

and cosmological expansion drives $Y'=\lambda^n Y.$8, thereby generating the Planck mass and supporting inflation along the kernel surface (Ferreira et al., 2016). In another, a classically scale-invariant scalar dark-matter model produces electroweak symmetry breaking through the Coleman–Weinberg mechanism, with a pseudo-dilaton mass generated radiatively and perturbative stability tested up to the Planck scale (Kim et al., 2022). A still earlier study showed that scale-invariant theories can accommodate a small positive cosmological constant, at the cost of a mass relation and a metastable false vacuum, with the dilaton mass generated at two-loop level (Foot et al., 2010).

These uses are not equivalent to the SIV gravitational paradigm, but they show that the term “vacuum scale invariance” spans both geometry and quantum vacuum structure. This suggests that some confusion in the literature is semantic rather than substantive.

Within the cosmological SIV program itself, several open questions are stated explicitly in the source material. The major ones include full CMB and BAO analyses in a consistent SIV or RISS framework, a detailed perturbative treatment of inflationary observables, fuller neutrino-decoupling and QED corrections in BBN, stronger lensing and structure-growth tests, and cross-validation of modified BBN calculations with other numerical codes (Gueorguiev et al., 2023). The recent Hubble-tension study argues that SIV can convey recombination-era conditions to the present without tension and gives best agreement around $Y'=\lambda^n Y.$9, but it also describes its CMB acoustic-scale treatment as preliminary (Courbin et al., 4 Feb 2026).

Taken together, the published record presents SIV as a geometrically unified program in which scale covariance of the vacuum modifies both cosmological expansion and weak-field dynamics, while also revealing internal tensions about gauge choice, early-universe thermodynamics, and the proper interpretation of $S$0 during BBN. The strongest late-time claims concern the RAR and dwarf spheroidals (Maeder et al., 2020), whereas the strongest early-universe claim to date is that the Lithium-7 problem may be addressed only after moving from standard SIV gauge intuition toward a reparametrization-invariant framework, namely RISS (Gueorguiev, 12 Sep 2025).