---
title: Effective Running Hubble Constant
url: https://www.emergentmind.com/topics/effective-running-hubble-constant
type: topic
---

# Effective Running Hubble Constant

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 [2011.02858; 2408.01410].

## 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 \(w_{\rm eff}(z)\) is not the true one, the inferred \(H_0\) becomes redshift dependent. The corresponding flat-\(\Lambda\)CDM null diagnostic is
\[
\mathcal{H}_0(z)\equiv \frac{H(z)}{\sqrt{1-\Omega_{m0}+\Omega_{m0}(1+z)^3}},
\]
which should be constant if flat \(\Lambda\)CDM is correct [2011.02858].

A second, closely related construction rewrites a nonstandard late-time background in explicitly \(\Lambda\)CDM-like form,
\[
H(z)=\mathcal{H}_0(z)\sqrt{\Omega_{0m}(1+z)^3 + 1-\Omega_{0m}},
\]
with
\[
\mathcal{H}_0(z) = H_0 \sqrt{ \frac{\Omega_{0m}(1+z)^3+\Omega_{de}(z)} {\Omega_{0m}(1+z)^3+1-\Omega_{0m} } }.
\]
Here the running is generated by the departure of \(\Omega_{de}(z)\) from a constant vacuum term, so \(\mathcal H_0(z)\) measures the mismatch between the true expansion law and the reference \(\Lambda\)CDM denominator [2411.07060].

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 \(H(z)\) to \(H_0\).

## 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 \(0<z<2.26\) split the sample into equally populated three-bin and four-bin subsamples and estimated \(H_0\) in each bin under flat \(\Lambda\)CDM and flat \(w_0w_a\)CDM. The inferred binwise values were fit by
\[
H_0(z)=\tilde{H}_0(1+z)^{-\alpha},
\]
with \(\alpha\sim 10^{-2}\). The reconstructed trend is decreasing with redshift, but the no-evolution case \(\alpha=0\) remains allowed at about \(1.2\sigma\) to \(2.0\sigma\). Extrapolating the fit to \(z=1100\) yields values consistent within \(1\sigma\) with Planck for both cosmological models and for both binning schemes [2205.07033].

A later 40-bin Pantheon analysis recast the effect in terms of a redshift-dependent \(\mathcal H_0(z)\) and compared three cases: a constant \(\Lambda\)CDM line, a power-law running law,
\[
\mathcal{H}_0(z)=H_0(1+z)^{-\alpha}, \qquad \alpha=0.016\pm0.009,
\]
and an evolutionary dark-energy model driven by bulk viscosity. In that model the extra parameter is
\[
\Omega_* \equiv \frac{3\chi \bar{\xi}}{c^2 H_0},
\]
with best fit
\[
\Omega_* = 0.280 \pm 0.117,
\qquad
w_{de} = -0.867 \pm 0.056,
\qquad
j_0 = 0.937.
\]
The inferred \(\mathcal H_0(z)\) decreases slowly with redshift, and the reduced chi-square values were reported as
\[
\chi^2_{\rm red} = 2.066 \quad (\text{PL}), \qquad 2.095 \quad (\text{EDE}), \qquad 2.176 \quad (\Lambda\text{CDM}),
\]
so the ranking mildly favors a running form over a fixed one [2411.07060].

These supernova reconstructions established the basic empirical motif of the subject: a mild, low-significance, but recurrent downward trend in the \(H_0\) 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,
\[
\mathcal{H}_0(z)=H_0\,\frac{E(z)}{E(z)^{\Lambda\mathrm{CDM}},
\]
so \(\Lambda\)CDM corresponds to \(\mathcal H_0(z)=H_0\) exactly, whereas non-\(\Lambda\)CDM models generate redshift dependence [2506.04162].

For \(w_0w_a\)-type dynamics, the low-redshift slope satisfies
\[
\mathcal{H}'^{\,w_0w_a}_{0}(z=0) = \frac{3}{2}H_0(1-\Omega_{m0})(1+w_0).
\]
This gives the proposed sign criterion: increasing \(\mathcal H_0(z)\) with redshift indicates quintessence-like behavior, while decreasing \(\mathcal H_0(z)\) indicates phantom-like behavior. The matter density affects features such as extrema, but the sign of the low-\(z\) trend is set by the dark-energy sector, not by the normalization \(H_0\) [2506.04162].

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 \(\Lambda\)CDM. The binned Pantheon sample nevertheless allowed a discrimination of dark-energy nature at least at the \(1\sigma\) level via the fit of \(\mathcal H_0(z)\) [2506.04162].

## 4. Realizations in modified gravity and nonstandard vacuum dynamics

Several theoretical frameworks generate an effective running Hubble constant by modifying the relation between \(H(z)\) and a reference \(\Lambda\)CDM background. In Jordan-frame \(f(R)\) gravity, with scalar degree of freedom \(\phi=df/dR\), the modified Friedmann equation leads to
\[
H_0^{\mathrm{eff}}(z)\equiv \frac{H(z)}{E^{\Lambda\mathrm{CDM}}(z)}.
\]
With the ansatz
\[
\phi(z)=\phi_0(1+z)^{2\alpha},
\]
one obtains
\[
H_0^{\mathrm{eff}}(z)\sim (1+z)^{-\alpha}.
\]
Using \(\phi_0=1-10^{-7}\) and matching the effective Hubble constant to local and CMB values gives
\[
\alpha = 1.1\times 10^{-2},
\]
which is consistent at \(1\sigma\) with values inferred from binned Pantheon analyses. The construction is explicitly low-redshift and relies on a slowly varying potential that mimics dark energy [2408.01410].

A related metric-\(f(R)\) model supplements the scalar sector with dark energy decaying into dark matter,
\[
\dot{\rho}_m + 3H\rho_m = \bar{H}\rho_{de}, \qquad \dot{\rho}_{de} = -\bar{H}\rho_{de},
\]
and defines an effective Hubble diagnostic from the ratio of the modified background to \(\Lambda\)CDM. After imposing \(j_0=1\), the model is reduced to one extra parameter, \(\gamma\), and fitted to the 40-bin Pantheon sample with \(H_0=73.5\) km s\(^{-1}\) Mpc\(^{-1}\) and \(\Omega_m^0=0.298\) fixed. The best fit is
\[
\gamma = 0.0162 \pm 0.0091,
\]
with
\[
\chi^2 = 78.51,\qquad \chi^2_{\rm red}=2.01,
\]
slightly better than both the power-law and \(\Lambda\)CDM fits. However, the high-redshift extrapolation approaches only \(\mathcal H(x)\to 72.35\), so the model only weakly alleviates the Hubble tension and does not reproduce the Planck value at recombination [2506.13288].

In frame-dependent dark energy, the relevant quantity is instead a proper-time expansion rate,
\[
H_{\rm eff}(T)\equiv \frac{1}{\tilde a(T)}\frac{d\tilde a(T)}{dT},
\qquad
\tilde a(T)=\frac{a(t)\,\Phi(t)}{\Phi(0)}.
\]
This gives
\[
H_{\rm eff}(t)=\frac{\dot a(t)}{a(t)}+\frac{\dot\Phi(t)}{\Phi(t)},
\]
so a late-time deviation in \(\Phi\) 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 [1905.08228].

By contrast, running-vacuum models use the Hubble rate as the renormalization scale of the vacuum sector,
\[
\Lambda=\Lambda(H), \qquad \rho_{\rm vac}=\rho_{\rm vac}(H,\dot H,\ddot H,\ldots),
\]
with mild late-time running of order \(H^2\) and early-time inflation driven by \(H^4\). In this framework, “running Hubble constant” refers to the role of \(H\) as the physical scale controlling vacuum evolution, not to a directly reconstructed \(H_0(z)\) [2503.01041].

## 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 \(D_A(z)/r_d\) and \(D_H(z)/r_d\), so at low redshift they are especially close to constraining the product
\[
H_0 r_d.
\]
This implies that a higher \(H_0\) requires a lower sound horizon \(r_d\). 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 \(H_0\) must be accompanied by a modification of early-Universe physics that changes \(r_d\), for example through dark radiation or very early dark energy [1711.01051].

Cosmic-chronometer analyses illustrate the complementary point that model dependence alone does not establish a genuine running \(H_0\). Using 31 \(H(z)\) measurements over \(0\le z\le 2.42\) from differential ages of passively evolving early-type galaxies, one study compared only flat and non-flat \(\Lambda\)CDM backgrounds. The marginalized values were
\[
H_0 = 68.7 \pm 3.1 \ \mathrm{km\,s^{-1}\,Mpc^{-1}}
\]
for flat \(\Lambda\)CDM and
\[
H_0 = 72.2 \pm 4 \ \mathrm{km\,s^{-1}\,Mpc^{-1}}
\]
for non-flat \(\Lambda\)CDM, with AIC favoring the flat model. That work explicitly did not define an effective Hubble constant or a redshift-dependent running \(H_0(z)\); its result is better interpreted as model dependence of inferred \(H_0\), not evidence that a running Hubble constant is required [2301.06140].

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

## 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
\[
H_0=73.3 \pm 2.1 \ \text{(stat)} \pm 3.5 \ \text{(sys)}\ \mathrm{km\,s^{-1}\,Mpc^{-1}},
\]
and the paper explicitly treats this as an observationally inferred effective \(H_0\) for the nearby Universe rather than as a model-independent global constant [2408.03660].

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 \(\sim 2\%-3\%\), and the inferred shifts are
\[
\Delta H_0^{\rm SN\,Ia}\approx 0.5 \ \mathrm{km\,s^{-1}\,Mpc^{-1}},
\qquad
\Delta H_0^{\rm SBF}\approx 1.1 \ \mathrm{km\,s^{-1}\,Mpc^{-1}},
\]
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 [2311.00215].

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
\[
H_0 = 66.2(77)\ \mathrm{km\,s^{-1}\,Mpc^{-1}},
\]
presented as the most probable or recommended Hubble constant value. This is effective only in the sense of being an evaluated consensus estimate [1506.02978].

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 \(H_0\), but a departure from the single-number interpretation of the present expansion rate when data are analyzed across different redshifts, models, or environments.

Source: https://www.emergentmind.com/topics/effective-running-hubble-constant