Effective Running Hubble Constant

• The effective running Hubble constant is a redshift-dependent parameter that re-expresses observed expansion within a ΛCDM framework, highlighting deviations due to assumed effective equations of state.
• Empirical analyses using Type Ia supernovae show a mild downward trend in inferred H₀ with increasing redshift, offering a diagnostic for dark-energy dynamics and potential tension resolution.
• The parameter emerges in various contexts—including modified gravity, dark-energy models, and local environmental studies—emphasizing its role in revealing model dependencies beyond a single H₀ value.

An effective running Hubble constant is a redshift-dependent quantity, usually written as $\mathcal H_0(z)$ or $H_0^{\mathrm{eff}}(z)$, obtained when the observed expansion history is re-expressed relative to a fiducial $\Lambda$CDM form. In this usage, the FLRW integration constant $H_0$ remains constant by definition; what “runs” is the value inferred from finite-redshift data, binned reconstructions, or modified dynamics. The literature does not use the term in a single uniform way: in some works it is a null test of flat $\Lambda$CDM, in others a phenomenological descriptor of Type Ia supernova binning results, and in others an emergent quantity in modified gravity or non-equilibrium dark-energy models (Krishnan et al., 2020, Schiavone et al., 2024).

1. Conceptual definition and formal constructions

Within FLRW cosmology, one formulation starts from

$$H_0 = H(z)\exp\!\left[-\frac{3}{2}\int_0^z \frac{1+w_{\rm eff}(z')}{1+z'}\,dz'\right].$$

This makes clear that $H_0$ is a constant of integration, but also that any inferred value of $H_0$ depends on the assumed effective equation of state used to propagate data from redshift $z$ to $z=0$. If the assumed $H_0^{\mathrm{eff}}(z)$0 is not the true one, the inferred $H_0^{\mathrm{eff}}(z)$1 becomes redshift dependent. The corresponding flat-$H_0^{\mathrm{eff}}(z)$2CDM null diagnostic is

$H_0^{\mathrm{eff}}(z)$3

which should be constant if flat $H_0^{\mathrm{eff}}(z)$4CDM is correct (Krishnan et al., 2020).

A second, closely related construction rewrites a nonstandard late-time background in explicitly $H_0^{\mathrm{eff}}(z)$5CDM-like form,

$H_0^{\mathrm{eff}}(z)$6

with

$H_0^{\mathrm{eff}}(z)$7

Here the running is generated by the departure of $H_0^{\mathrm{eff}}(z)$8 from a constant vacuum term, so $H_0^{\mathrm{eff}}(z)$9 measures the mismatch between the true expansion law and the reference $\Lambda$0CDM denominator (Montani et al., 2024).

These two definitions share the same operational meaning: the “running Hubble constant” is not a literal time-varying fundamental constant, but an effective parameter encoding model dependence in the mapping from $\Lambda$1 to $\Lambda$2.

2. Reconstruction from redshift-binned Type Ia supernovae

The most explicit empirical reconstructions use the Pantheon Type Ia supernova sample. One analysis of 1048 spectroscopically confirmed SNe Ia over $\Lambda$3 split the sample into equally populated three-bin and four-bin subsamples and estimated $\Lambda$4 in each bin under flat $\Lambda$5CDM and flat $\Lambda$6CDM. The inferred binwise values were fit by

$\Lambda$7

with $\Lambda$8. The reconstructed trend is decreasing with redshift, but the no-evolution case $\Lambda$9 remains allowed at about $H_0$0 to $H_0$1. Extrapolating the fit to $H_0$2 yields values consistent within $H_0$3 with Planck for both cosmological models and for both binning schemes (Schiavone et al., 2022).

A later 40-bin Pantheon analysis recast the effect in terms of a redshift-dependent $H_0$4 and compared three cases: a constant $H_0$5CDM line, a power-law running law,

$H_0$6

and an evolutionary dark-energy model driven by bulk viscosity. In that model the extra parameter is

$H_0$7

with best fit

$H_0$8

The inferred $H_0$9 decreases slowly with redshift, and the reduced chi-square values were reported as

$\Lambda$0

so the ranking mildly favors a running form over a fixed one (Montani et al., 2024).

These supernova reconstructions established the basic empirical motif of the subject: a mild, low-significance, but recurrent downward trend in the $\Lambda$1 inferred from progressively higher-redshift bins.

3. Diagnostic use for dark-energy phenomenology

A subsequent development treated the effective running Hubble constant as a diagnostic of dark-energy nature rather than only a fit function. In this framework,

$\Lambda$2

so $\Lambda$3CDM corresponds to $\Lambda$4 exactly, whereas non-$\Lambda$5CDM models generate redshift dependence (Fazzari et al., 4 Jun 2025).

For $\Lambda$6-type dynamics, the low-redshift slope satisfies

$\Lambda$7

This gives the proposed sign criterion: increasing $\Lambda$8 with redshift indicates quintessence-like behavior, while decreasing $\Lambda$9 indicates phantom-like behavior. The matter density affects features such as extrema, but the sign of the low-$$H_0 = H(z)\exp\!\left[-\frac{3}{2}\int_0^z \frac{1+w_{\rm eff}(z')}{1+z'}\,dz'\right].$$0 trend is set by the dark-energy sector, not by the normalization $$H_0 = H(z)\exp\!\left[-\frac{3}{2}\int_0^z \frac{1+w_{\rm eff}(z')}{1+z'}\,dz'\right].$$1 (Fazzari et al., 4 Jun 2025).

This diagnostic was applied to two 20-bin SNe Ia datasets: a Pantheon-bin sample and a Master-bin sample combining DES, PantheonPlus, Pantheon, and JLA without duplicated supernovae. The phenomenological power-law model was statistically favored for both datasets. At the same time, the data did not indicate that the studied evolving dark-energy models are favored with respect to $$H_0 = H(z)\exp\!\left[-\frac{3}{2}\int_0^z \frac{1+w_{\rm eff}(z')}{1+z'}\,dz'\right].$$2CDM. The binned Pantheon sample nevertheless allowed a discrimination of dark-energy nature at least at the $$H_0 = H(z)\exp\!\left[-\frac{3}{2}\int_0^z \frac{1+w_{\rm eff}(z')}{1+z'}\,dz'\right].$$3 level via the fit of $$H_0 = H(z)\exp\!\left[-\frac{3}{2}\int_0^z \frac{1+w_{\rm eff}(z')}{1+z'}\,dz'\right].$$4 (Fazzari et al., 4 Jun 2025).

4. Realizations in modified gravity and nonstandard vacuum dynamics

Several theoretical frameworks generate an effective running Hubble constant by modifying the relation between $$H_0 = H(z)\exp\!\left[-\frac{3}{2}\int_0^z \frac{1+w_{\rm eff}(z')}{1+z'}\,dz'\right].$$5 and a reference $$H_0 = H(z)\exp\!\left[-\frac{3}{2}\int_0^z \frac{1+w_{\rm eff}(z')}{1+z'}\,dz'\right].$$6CDM background. In Jordan-frame $$H_0 = H(z)\exp\!\left[-\frac{3}{2}\int_0^z \frac{1+w_{\rm eff}(z')}{1+z'}\,dz'\right].$$7 gravity, with scalar degree of freedom $$H_0 = H(z)\exp\!\left[-\frac{3}{2}\int_0^z \frac{1+w_{\rm eff}(z')}{1+z'}\,dz'\right].$$8, the modified Friedmann equation leads to

$$H_0 = H(z)\exp\!\left[-\frac{3}{2}\int_0^z \frac{1+w_{\rm eff}(z')}{1+z'}\,dz'\right].$$9

With the ansatz

$H_0$0

one obtains

$H_0$1

Using $H_0$2 and matching the effective Hubble constant to local and CMB values gives

$H_0$3

which is consistent at $H_0$4 with values inferred from binned Pantheon analyses. The construction is explicitly low-redshift and relies on a slowly varying potential that mimics dark energy (Schiavone et al., 2024).

A related metric-$H_0$5 model supplements the scalar sector with dark energy decaying into dark matter,

$H_0$6

and defines an effective Hubble diagnostic from the ratio of the modified background to $H_0$7CDM. After imposing $H_0$8, the model is reduced to one extra parameter, $H_0$9, and fitted to the 40-bin Pantheon sample with $H_0$0 km s$H_0$1 Mpc$H_0$2 and $H_0$3 fixed. The best fit is

$H_0$4

with

$H_0$5

slightly better than both the power-law and $H_0$6CDM fits. However, the high-redshift extrapolation approaches only $H_0$7, so the model only weakly alleviates the Hubble tension and does not reproduce the Planck value at recombination (Montani et al., 16 Jun 2025).

In frame-dependent dark energy, the relevant quantity is instead a proper-time expansion rate,

$H_0$8

This gives

$H_0$9

so a late-time deviation in $z$0 raises or lowers the locally inferred expansion rate relative to the FRW value. The model can increase the local Hubble constant relative to the CMB-inferred one, though its BAO prediction can be somewhat high (Adler, 2019).

By contrast, running-vacuum models use the Hubble rate as the renormalization scale of the vacuum sector,

$z$1

with mild late-time running of order $z$2 and early-time inflation driven by $z$3. In this framework, “running Hubble constant” refers to the role of $z$4 as the physical scale controlling vacuum evolution, not to a directly reconstructed $z$5 (Peracaula et al., 2 Mar 2025).

5. BAO, cosmic chronometers, and the limits of a purely late-time interpretation

The strongest restriction on late-time running interpretations comes from anisotropic BAO. BAO observables constrain combinations such as $z$6 and $z$7, so at low redshift they are especially close to constraining the product

$z$8

This implies that a higher $z$9 requires a lower sound horizon $z=0$0. The conclusion is that the Hubble-tension problem cannot be treated as a purely late-time effect once anisotropic BAO are included: any successful upward shift in $z=0$1 must be accompanied by a modification of early-Universe physics that changes $z=0$2, for example through dark radiation or very early dark energy (Evslin et al., 2017).

Cosmic-chronometer analyses illustrate the complementary point that model dependence alone does not establish a genuine running $z=0$3. Using 31 $z=0$4 measurements over $z=0$5 from differential ages of passively evolving early-type galaxies, one study compared only flat and non-flat $z=0$6CDM backgrounds. The marginalized values were

$z=0$7

for flat $z=0$8CDM and

$z=0$9

for non-flat $H_0^{\mathrm{eff}}(z)$00CDM, with AIC favoring the flat model. That work explicitly did not define an effective Hubble constant or a redshift-dependent running $H_0^{\mathrm{eff}}(z)$01; its result is better interpreted as model dependence of inferred $H_0^{\mathrm{eff}}(z)$02, not evidence that a running Hubble constant is required (Thakur et al., 2023).

A plausible implication is that two issues must be kept separate: finite-redshift inference can produce a redshift-dependent $H_0^{\mathrm{eff}}(z)$03, but BAO can still require the deeper resolution of the tension to involve the early-Universe ruler $H_0^{\mathrm{eff}}(z)$04.

6. Broader uses of “effective” and persistent terminological ambiguities

The term “effective Hubble constant” is also used in several papers in ways that do not denote redshift running. In a Tully–Fisher analysis of the Cosmicflows-4 catalogue, the quantity is a local low-redshift expansion rate inferred after jointly fitting the Tully–Fisher relation and a peculiar-velocity model. The headline result is

$H_0^{\mathrm{eff}}(z)$05

and the paper explicitly treats this as an observationally inferred effective $H_0^{\mathrm{eff}}(z)$06 for the nearby Universe rather than as a model-independent global constant (Boubel et al., 2024).

A different environmental use appears in the effective description of Laniakea as a triaxially expanding ellipsoid. The induced line-of-sight-dependent distance corrections are of order $H_0^{\mathrm{eff}}(z)$07, and the inferred shifts are

$H_0^{\mathrm{eff}}(z)$08

which seemingly worsen the Hubble tension rather than relieve it. Here the “effective” quantity is a local environmental bias in the Hubble flow, not a redshift-running cosmological parameter (Giani et al., 2023).

A still broader statistical meaning appears in data-evaluation work that aggregates heterogeneous measurements into a recommended value. Using USNDP procedures, one such study obtained

$H_0^{\mathrm{eff}}(z)$09

presented as the most probable or recommended Hubble constant value. This is effective only in the sense of being an evaluated consensus estimate (Pritychenko, 2015).

This suggests that the phrase “effective running Hubble constant” has become a family resemblance term rather than a uniquely standardized object. Its precise content depends on whether the underlying problem is redshift-binned inference, late-time modified dynamics, local flow corrections, or statistical synthesis. The common thread is not a literal time-varying $H_0^{\mathrm{eff}}(z)$10, but a departure from the single-number interpretation of the present expansion rate when data are analyzed across different redshifts, models, or environments.