G-step Model: Late-Time Varying G

• GSM is defined as a late-time, local-physics proposal introducing a sharp step in the effective gravitational constant (G_eff) at z≈0.01, impacting SN Ia luminosities.
• The model modifies the distance ladder by making Hubble-flow Type Ia supernovae intrinsically brighter than local calibrators, thereby altering standard candle calibrations.
• GSM implications extend to stellar evolution, planetary dynamics, and paleoclimate, with proposed tests including JWST Cepheid-SN Ia calibration and standard siren observations.

Searching arXiv for papers on the G-step Model related to the Hubble tension and varying $G$. The G-step Model (GSM) is an ultra-late-time varying-$G$ proposal for the Hubble tension in which the effective gravitational constant relevant to local astrophysical calibrators differs from that relevant to Hubble-flow Type Ia supernovae. In the cosmological usage of the term, GSM posits a sharp transition in $G_{\rm eff}$ at approximately $z_t \approx 0.01$, corresponding to a distance of roughly $40\,{\rm Mpc}$ and a lookback time of order $100$–$150\,{\rm Myr}$. Its intended purpose is to break the assumed universality of the distance ladder by making standardized SNe Ia in the Hubble flow intrinsically brighter than their locally calibrated counterparts, thereby reducing the inferred local redshift gradient without modifying the early-universe background cosmology in the manner of standard early-time solutions (Banik et al., 2024, Perivolaropoulos et al., 6 Aug 2025).

1. Definition and conceptual setting

The GSM was formulated as a late-time, local-physics response to the discrepancy between distance-ladder determinations of the Hubble constant and the lower value inferred from Planck $\Lambda$CDM. In its basic form, the model assumes that $G_{\rm eff}$ was larger than the present value $G_0$ for essentially all of cosmic and Galactic history and then underwent a sharp decrease to $G$0 at a recent epoch. The placement of the transition is deliberate: local Cepheid and TRGB calibrators lie inside the region where $G$1, while Hubble-flow SNe Ia lie outside it, so the first two rungs of the ladder and the third rung no longer share identical astrophysical calibration (Banik et al., 2024).

This structure is attractive because it targets the specific scale separation built into the contemporary distance ladder. The proposal does not primarily alter the Friedmann background; rather, it changes the astrophysics of objects used as standard candles or stellar chronometers. A later re-examination presents the same idea in explicitly phenomenological language, writing the effect as a step in $G$2 beyond $G$3 and emphasizing that the relevant quantity is the effective gravitational coupling entering stellar structure, Cepheid pulsation, and SN Ia explosion physics (Perivolaropoulos et al., 6 Aug 2025).

Within the recent literature, the model sits in a broader class of modified-local-physics explanations of the Hubble tension. The central distinction is between a sharp, universal step in $G$4 and more elaborate screened or environment-dependent modified-gravity scenarios. The former is the canonical GSM; the latter were explicitly suggested as possible alternatives if a simple universal step proves untenable (Banik et al., 2024).

2. Distance-ladder mechanism and formal parameterization

The operational mechanism of GSM is the sensitivity of standardized SN Ia luminosity to the gravitational coupling. The formulation used in both discussions adopts the scaling

$G$5

with the dependence attributed to numerical modeling summarized by Wright, Desmond, and Zhao. If Hubble-flow SNe exploded when $G$6, then their intrinsic luminosities exceed those inferred from local calibration, so their true distances are larger than assumed and the locally inferred $G$7 is biased high (Banik et al., 2024).

A phenomenological representation used in the re-examination is

$G$8

or equivalently a Heaviside-like step in $G$9. The required luminosity shift is written as

$G_{\rm eff}$0

while the earlier analysis uses the same logic in the notation

$G_{\rm eff}$1

Using $G_{\rm eff}$2 and $G_{\rm eff}$3, the re-examination quotes $G_{\rm eff}$4, notes that a naive use of $G_{\rm eff}$5 gives $G_{\rm eff}$6, and then argues that the viable $G_{\rm eff}$7 shifts once Cepheid effects and uncertainties are included. By contrast, the same paper emphasizes $G_{\rm eff}$8 as a characteristic step once systematics are propagated (Perivolaropoulos et al., 6 Aug 2025).

The original critique stresses that the transition must occur at $G_{\rm eff}$9 lookback in order to remain hidden behind local peculiar-velocity scatter rather than producing an obvious discontinuity in the SN Hubble diagram. This requirement is central because it forces the model into a regime where Solar System, stellar-evolution, and paleoclimate constraints become exceptionally sharp (Banik et al., 2024).

3. Solar, terrestrial, and orbital consequences

A principal criticism of GSM is that a universal change in $z_t \approx 0.01$0 affects far more than supernovae. The sharpest version of the argument uses the classic Teller scaling for Sun-like stars,

$z_t \approx 0.01$1

so that a percent-level change in $z_t \approx 0.01$2 implies a much larger change in solar luminosity. In the 2024 analysis, a $z_t \approx 0.01$3–$z_t \approx 0.01$4 change in $z_t \approx 0.01$5 corresponds to a $z_t \approx 0.01$6 change in solar luminosity; even the milder alternative $z_t \approx 0.01$7 is described there as still producing large climatic and evolutionary effects (Banik et al., 2024).

Combining stellar-luminosity and orbital scalings yields a strong terrestrial constraint. If the Earth’s orbital angular momentum is conserved, then

$z_t \approx 0.01$8

so a $z_t \approx 0.01$9 drop in $40\,{\rm Mpc}$0 implies an abrupt $40\,{\rm Mpc}$1 increase in year length. The same critique further derives

$40\,{\rm Mpc}$2

and argues that a $40\,{\rm Mpc}$3 decrease in $40\,{\rm Mpc}$4 would lower Earth’s equilibrium temperature by more than $40\,{\rm Mpc}$5, corresponding to a cooling of $40\,{\rm Mpc}$6. This is then taken to imply a high likelihood of a Snowball Earth episode, yet no such global glaciation is reported there for the last $40\,{\rm Mpc}$7 (Banik et al., 2024).

The same paper extends the argument to Earth–Moon dynamics. Since the Moon’s orbital radius also scales as $40\,{\rm Mpc}$8, tidal stress scales as $40\,{\rm Mpc}$9, implying a sharp change in tidal torque and therefore in the secular evolution of the number of days per year. The authors state that geochronometry and cyclostratigraphy show a broadly continuous evolution of day count and do not exhibit the predicted discontinuities (Banik et al., 2024).

These objections are not merely auxiliary. In the anti-GSM reading, they show that a universal step in $100$0 cannot be confined to the distance ladder; it propagates immediately into planetary climate, orbital dynamics, and tidal evolution, all of which retain empirical memory over the same $100$1–$100$2 interval that the model must occupy.

4. Stellar evolution, chronometry, and cosmological age tests

The GSM also feeds directly into stellar evolution because higher historical $100$3 accelerates nuclear burning. The 2024 critique summarizes the consequence as follows: instead of having consumed about half of its hydrogen fuel, the Sun would have consumed about two-thirds. This would spoil the agreement between helioseismic ages and the meteoritic age of the Solar System, quoted as $100$4, because the present solar structure would then correspond to a more evolved star than in constant-$100$5 models (Banik et al., 2024).

The same logic extends to Galactic and cosmological chronometers. If stars evolve faster when $100$6 is larger, then ages inferred with constant-$100$7 stellar models are overestimated. The 2024 paper argues that recalibration in a GSM-like universe would make the oldest stars younger by $100$8, creating an “age gap” in which the Universe is $100$9 old but no local stars survive from the first $150\,{\rm Myr}$0 of cosmic history. The paper describes this as inconsistent with the standard concordance between the age of the Universe and the ages of the oldest Galactic stars (Banik et al., 2024).

Cosmic chronometers are presented as a further difficulty. Since differential galaxy ages are used to infer $150\,{\rm Myr}$1, any systematic bias in stellar aging propagates into the expansion history inferred from passive galaxies. The critique argues that this would produce a mismatch between chronometer-based $150\,{\rm Myr}$2 and the Planck $150\,{\rm Myr}$3CDM expansion history that GSM seeks to preserve (Banik et al., 2024).

The same paper additionally states that a sharp, universal step in $150\,{\rm Myr}$4 would alter stellar mass-to-light ratios and affect multiple distance indicators differently, including Cepheids, TRGB stars, surface-brightness-fluctuation methods, and SN Ia calibrations. It further suggests that this would disrupt galaxy scaling relations such as the radial acceleration relation, Tully–Fisher, Faber–Jackson, and the Fundamental Plane, especially for samples straddling the proposed transition radius (Banik et al., 2024).

5. Re-examination and contested viability

A 2025 re-examination disputes much of the foregoing constraint set and argues that GSM remains viable once uncertainties and alternative scalings are treated more conservatively. Its first major revision concerns stellar luminosity: instead of the Teller relation $150\,{\rm Myr}$5, it emphasizes modern stellar modeling that indicates

$150\,{\rm Myr}$6

citing Adams and Davis et al. Under this assumption, a $150\,{\rm Myr}$7 change in $150\,{\rm Myr}$8 implies about a $150\,{\rm Myr}$9 rather than $\Lambda$0 luminosity change, and the paper treats solar-age exclusions as model-dependent rather than decisive (Perivolaropoulos et al., 6 Aug 2025).

The same re-examination rejects the rigid-Earth interpretation of paleorotation. It argues that, on geological timescales and especially $\Lambda$1 ago, Earth should be treated as fluid-like, with

$\Lambda$2

Since the orbital period also scales as $\Lambda$3, the ratio $\Lambda$4 remains constant across the transition. On that reading, the absence of a jump in days per year is not a valid objection (Perivolaropoulos et al., 6 Aug 2025).

Paleoclimate is similarly reinterpreted. Replacing the earlier blackbody-like argument with a weaker effective scaling,

$\Lambda$5

the paper notes that global mean temperature over the last $\Lambda$6 shows a net cooling of about $\Lambda$7, from roughly $\Lambda$8 to $\Lambda$9. It then infers $G_{\rm eff}$0, i.e. a $G_{\rm eff}$1 decrease in $G_{\rm eff}$2, and presents this as consistent with the GSM scale rather than contradictory to it (Perivolaropoulos et al., 6 Aug 2025).

The re-examination also revisits external constraints. It quotes CMB-based bounds in the form

$G_{\rm eff}$3

with a $G_{\rm eff}$4 interval $G_{\rm eff}$5, and argues that this overlaps with the range needed once SN Ia systematics are included. It further states that Cepheid–TRGB comparisons allow $G_{\rm eff}$6 variations up to $G_{\rm eff}$7 at $G_{\rm eff}$8, that cosmic-chronometer measurements are themselves $G_{\rm eff}$9-dependent through $G_0$0, and that none of these observables currently provides a decisive exclusion (Perivolaropoulos et al., 6 Aug 2025).

6. Present status, related models, and proposed tests

The contemporary status of GSM is therefore explicitly controversial. One line of argument concludes that a simple, universal, unscreened step in $G_0$1 is in severe tension with Earth’s climate history, year-length evolution, helioseismic and meteoritic ages, the oldest stellar populations, and the consistency of distance indicators; on this basis it treats the model as essentially ruled out and suggests that any viable varying-$G_0$2 explanation would need environmental dependence or screening (Banik et al., 2024). The opposing line argues that the strongest exclusions rely on outdated stellar scalings, rigid-body assumptions for the Earth, over-tight interpretations of distance-indicator comparisons, and insufficient treatment of systematic uncertainty; on that basis it maintains that GSM remains a viable candidate (Perivolaropoulos et al., 6 Aug 2025).

This divide has sharpened the distinction between the GSM proper and screened modified-gravity alternatives. The 2024 critique points to the models of Desmond and Sakstein as examples in which environmental screening changes effective gravity for Cepheid hosts relative to anchor galaxies without generating the full set of Solar System and stellar-evolution pathologies attributed to a universal step (Banik et al., 2024). The 2025 re-examination, by contrast, proposes a late first-order phase transition in scalar-tensor gravity as a possible microphysical realization of a local gravitational “bubble” of size $G_0$3 (Perivolaropoulos et al., 6 Aug 2025).

The most direct proposed tests lie in the transition region itself. The re-examination calls for JWST Cepheid–SN Ia calibration in the $G_0$4–$G_0$5 window, where GSM predicts a shift in SN Ia absolute magnitude of $G_0$6; TRGB cross-calibration over the same range, because TRGB stars have a different $G_0$7-dependence; standard sirens from future gravitational-wave facilities; tighter asteroseismic constraints on the exponent in $G_0$8; and improved paleoclimate and paleorotation reconstructions across the $G_0$9–$G$00 interval (Perivolaropoulos et al., 6 Aug 2025). In that sense, the model remains diagnostically useful even where its ultimate viability is disputed: it converts the Hubble tension into a set of sharply localized predictions about the astrophysics of the nearby Universe.