Hubble Models: Cosmic Expansion & Lensing

• Hubble models are frameworks that relate redshift, distance, and cosmic expansion through FRW-based cosmologies and effective parametrizations.
• They utilize diverse methodologies, from standard ΛCDM comparisons and modified-gravity approaches to perturbation techniques and early-dark-energy dynamics.
• In observational studies, Hubble models underpin Hubble diagram analyses and HST-driven lensing mass reconstructions, addressing challenges like the Hubble tension.

Hubble models are theoretical and observational constructions organized around the Hubble relation between redshift, distance, and cosmic expansion. In the cited literature, the expression is used in multiple technical senses: as a label for FRW-based cosmological models specified by a Hubble function $H(z)$; as a designation for modified-gravity or effective-metric constructions that reproduce a Hubble law; as a shorthand for Hubble diagrams built from standard candles or other distance indicators; and, in gravitational lensing, as Hubble Space Telescope-driven mass models and magnification maps for galaxy clusters (Chen et al., 2016, Hsu et al., 2021, Johnson et al., 2014).

1. Terminological scope

In cosmology in the narrow sense, Hubble models are specific FRW cosmological models whose expansion history $H(z)$ is confronted with data to infer the present expansion rate $H_0$ and other parameters. This usage appears explicitly for spatially flat and non-flat $\Lambda$CDM, flat constant-$w$ dark-energy models, scalar-field $\phi$CDM, generalized Chaplygin gas models, and direct parametrizations of the dimensionless Hubble function $E(z)=H(z)/H_0$ (Chen et al., 2016, Sharov et al., 2018, Koussour et al., 2023).

A distinct usage appears in alternative-gravity and nonstandard-cosmology papers, where Hubble models are dynamical constructions for the beginning, expansion, and future of the universe. In one example, Yang–Mills gravity in flat Minkowski space-time, combined with a strong cosmological principle, yields an effective metric $G_{\mu\nu}(t)=\eta_{\mu\nu}A^2(t)$ and an Okubo Hamilton–Jacobi equation from which a Hubble law follows in the non-relativistic regime (Hsu et al., 2021).

In observational lensing studies, “Hubble models” refers not to $H(z)$ at all, but to gravitational-lensing mass models and magnification maps built primarily from Hubble Space Telescope imaging, especially for the Hubble Frontier Fields. There the term denotes HST-driven reconstructions of cluster mass distributions, convergence, shear, and magnification (Priewe et al., 2016, Johnson et al., 2014).

This terminological spread suggests that the common thread is methodological rather than ontological: each usage centers on the empirical or theoretical realization of a Hubble relation, whether through background expansion, light propagation, or HST-enabled lens inversion.

2. Expansion-history models based on $H(z)$

In homogeneous and isotropic FRW cosmology, the Hubble parameter is

$H(z)$0

For flat $H(z)$1CDM, the theoretical Hubble function used in direct $H(z)$2 analyses is

$H(z)$3

whereas the non-flat version adds curvature,

$H(z)$4

Using 31 cosmic-chronometer measurements in $H(z)$5, one study found $H(z)$6 for flat $H(z)$7CDM and $H(z)$8 for non-flat $H(z)$9CDM, with a smaller AIC for the flat model (Thakur et al., 2023).

A broader $H_0$0-based comparison using 28 measurements over $H_0$1 considered four FRW models: flat $H_0$2CDM, non-flat $H_0$3CDM, flat $H_0$4CDM (XCDM/$H_0$5CDM), and flat $H_0$6CDM. The inferred Hubble constants were

$H_0$7

respectively. These values were described as more consistent with lower CMB- and BAO-based determinations than with higher local measurements, while still including the higher local values within the $H_0$8 confidence limits (Chen et al., 2016).

The dependence of such Hubble models on dataset construction is itself nontrivial. A comparison of $H_0$9CDM and generalized Chaplygin gas models using either 57 $\Lambda$0 points or only 31 differential-age points found strong dependence of inferred parameters on the chosen $\Lambda$1 compilation, traced largely to four high-redshift points with $\Lambda$2 (Sharov et al., 2018). This indicates that even within standard FRW-style Hubble modeling, the mapping from data to parameters is sensitive to the composition of the $\Lambda$3 dataset.

A separate parametrization program treats the Hubble function itself as the primary phenomenological object. One proposal writes

$\Lambda$4

and in the concrete realization

$\Lambda$5

so that

$\Lambda$6

This reduces exactly to $\Lambda$7CDM for $\Lambda$8. MCMC constraints from CC, BAO, and Pantheon+ gave $\Lambda$9, $w$0, and $w$1, indicating no strong evidence for deviations from $w$2CDM within this parametrization (Koussour et al., 2023).

3. Hubble-tension models and extended early-universe physics

A major contemporary use of Hubble models concerns extensions of $w$3CDM designed to shift the CMB-inferred value of $w$4. In this setting the operative quantity is the sound horizon at recombination,

$w$5

together with its observed angular scale $w$6. Early-universe modifications that reduce $w$7 can raise the inferred $w$8 at fixed $w$9 (Ange et al., 2023).

Three model classes are emphasized in delensing forecasts: varying fundamental constants, early dark energy, and self-interacting dark radiation. The varying-constants models introduce

$\phi$0

which alter recombination microphysics, the visibility function, and the sound horizon. Early dark energy is modeled with an axion-like potential,

$\phi$1

and effective parameters $\phi$2, $\phi$3, and $\phi$4. Interacting-dark-radiation models add a tightly coupled relativistic component, optionally with dark-matter–dark-radiation scattering characterized by $\phi$5 and $\phi$6. Fisher forecasts for a $\phi$7 CMB experiment show that delensing improves constraints on $\phi$8 by roughly $\phi$9–$E(z)=H(z)/H_0$0, with $E(z)=H(z)/H_0$1 improvements typical for viable models, and significantly improves constraints on the nonstandard parameters in the low-noise regime (Ange et al., 2023).

Interacting-radiation Hubble models are also strongly constrained by small-scale structure. In models with dark radiation and dark-matter–dark-radiation interactions, the matter power spectrum develops a break: modes entering the horizon while the interaction is active are suppressed. The eBOSS Ly$E(z)=H(z)/H_0$2 forest flux power spectrum is especially constraining because it probes $E(z)=H(z)/H_0$3, where these models predict the largest deviations. The simplest dark-radiation models, although able to improve the Hubble tension, worsen the fit to Ly$E(z)=H(z)/H_0$4 data. By contrast, models with DM–DR interactions can simultaneously address both the Hubble and $E(z)=H(z)/H_0$5 tensions (Bagherian et al., 2024).

Redshift-space distortions provide an additional filter on viable Hubble models because they directly constrain $E(z)=H(z)/H_0$6. An analysis including RSD found that RSD data prefer a smaller amplitude of perturbations, and that including RSD results in a slightly weaker upper limit on the neutrino mass than without RSD, a pattern that persists in extended models. In that framework, a varying-electron-mass model combined with non-zero neutrino mass was identified as promising for relaxing the Hubble tension and the $E(z)=H(z)/H_0$7 tension simultaneously, whereas $E(z)=H(z)/H_0$8 dark energy and free-$E(z)=H(z)/H_0$9 extensions did not provide a comparably favorable joint outcome once RSD were imposed (Toda et al., 2024).

4. Alternative dynamical constructions and effective Hubble laws

One explicit alternative program derives Hubble behavior from Yang–Mills gravity in flat space-time. The basic assumption is the strong cosmological principle,

$G_{\mu\nu}(t)=\eta_{\mu\nu}A^2(t)$0

with $G_{\mu\nu}(t)=\eta_{\mu\nu}A^2(t)$1 an effective metric arising in the geometric-optics limit. Starting from the Okubo action,

$G_{\mu\nu}(t)=\eta_{\mu\nu}A^2(t)$2

one obtains a cosmic Okubo equation,

$G_{\mu\nu}(t)=\eta_{\mu\nu}A^2(t)$3

which yields galaxy trajectories and, in the non-relativistic approximation, the usual Hubble law

$G_{\mu\nu}(t)=\eta_{\mu\nu}A^2(t)$4

Two concrete models are developed. In the HHK model, $G_{\mu\nu}(t)=\eta_{\mu\nu}A^2(t)$5, giving $G_{\mu\nu}(t)=\eta_{\mu\nu}A^2(t)$6, an initial recession speed $G_{\mu\nu}(t)=\eta_{\mu\nu}A^2(t)$7, a finite initial radius $G_{\mu\nu}(t)=\eta_{\mu\nu}A^2(t)$8, and $G_{\mu\nu}(t)=\eta_{\mu\nu}A^2(t)$9, $H(z)$0, $H(z)$1. In a second model with $H(z)$2, the non-relativistic regime yields a strict linear Hubble relation $H(z)$3, with a “silent beginning” characterized by $H(z)$4, $H(z)$5, $H(z)$6 as $H(z)$7, and $H(z)$8, $H(z)$9, $H(z)$0 as $H(z)$1. In all models with the strong cosmological principle in flat space-time, recession velocities have a maximum equal to the speed of light as measured in an inertial frame (Hsu et al., 2021).

A different effective-background construction uses second-order general-relativistic perturbation theory. There the spatial average of quadratic first-order adiabatic perturbations generates homogeneous, isotropic corrections that are absorbed into a renormalized scale factor $H(z)$2 and renormalized density $H(z)$3. This leads to two effective Hubble constants: $H(z)$4 For a background $H(z)$5, the present values are

$H(z)$6

both larger than the background value. The paper interprets $H(z)$7 as the Hubble constant relevant for optical and gravitational-wave observations, and $H(z)$8 as the one relevant for motion and dynamical evolution (Tomita, 2019).

A phenomenological anisotropic program instead treats the Hubble constant as a direction-dependent effective quantity generated by line-of-sight electron-density gradients in an intergalactic plasma. In that framework the effective $H(z)$9 is proportional to the line-of-sight-averaged electron density $H(z)$00. All-sky variation in HST Key Project determinations, with weighted mean $H(z)$01, standard deviation $H(z)$02, and range $H(z)$03 to $H(z)$04 over 76 sky regions, is modeled using a rim model, an auto-gravitating model, a Voronoi-diagrams model, and a 2MASS-based astronomer’s model with a Lane–Emden $H(z)$05 electron profile (Zaninetti, 2014).

5. Hubble diagrams, frame dependence, and high-redshift tracers

The Hubble diagram is the observational relation between distance indicator and redshift, usually expressed as luminosity distance $H(z)$06 or distance modulus

$H(z)$07

In modified gravity, however, identical practical Hubble diagrams need not imply physical equivalence of the underlying models. In $H(z)$08 gravity, the Jordan and Einstein frames are related by a conformal transformation, but if the time-variation of particle masses in the Einstein frame is properly included, then the Hubble diagram derived operationally from Type Ia supernova surveys does not distinguish the two frames. The paper argues that comparison of the rates of changes in Hubble diagrams, rather than the Hubble diagrams alone, could in principle differentiate them (Rashidi, 2017).

Gamma-ray bursts extend Hubble-diagram construction to much higher redshift. Using empirical correlations between directly observed GRB quantities and intrinsic luminosity or radiated energy, an estimated and a calibrated GRB Hubble diagram were constructed, the latter using local regression on a 307-SN Ia sample to calibrate GRBs with $H(z)$09. The paper reports that correlation parameters depend only weakly on the cosmological model, that the estimated and calibrated GRB Hubble diagrams are fully statistically consistent for the common GRBs, and that the Amati-based Hubble diagram extends to $H(z)$10. In that analysis, the calibrated GRB Hubble diagram was then used to constrain a quintessential cosmological model and derive likelihood values of $H(z)$11 and $H(z)$12 (Demianski et al., 2010).

Exact space-time studies further show that the relation between averaged geometry and observed Hubble diagrams depends on the existence of a statistical homogeneity scale. In statistically homogeneous but anisotropic universes, ray-tracing and the large-scale-averaged anisotropic cosmological model yield closely agreeing Hubble diagrams whenever a statistical homogeneity scale exists. In cosmologies without such a scale, the Hubble diagrams inferred from the averaged model can differ considerably from those actually constructed by observers in the space-time (Anton et al., 2024). This suggests that the interpretation of Hubble diagrams depends not only on local light propagation but also on the validity of the averaging scheme used to represent the large-scale universe.

6. Hubble Space Telescope lens models

In strong-lensing work on the Hubble Frontier Fields, Hubble models are gravitational-lensing mass models and magnification maps of galaxy clusters built primarily from HST imaging. Parametric models constructed with LENSTOOL represent cluster-scale and galaxy-scale halos with PIEMD components and are constrained by the locations and redshifts of multiple images. For the six Frontier Fields clusters, such models were used to derive mass maps, magnification maps, shear maps, and deflection fields, and the resulting products were released through MAST. The work also found that photometric redshift estimates of lensed galaxies are generally in excellent agreement with spectroscopic redshifts, but that relaxed redshift priors may cause the complexity of large-scale structure needed to account for the lensing signal to be underestimated (Johnson et al., 2014).

The fundamental lensing observable in these Hubble-based lens models is the magnification,

$H(z)$13

where $H(z)$14 is convergence and $H(z)$15 is shear. Comparison of the public HFF mass models for Abell 2744 and MACS J0416 showed that model-to-model dispersion grows strongly with magnification: about $H(z)$16 at common low magnifications $H(z)$17, rising to about $H(z)$18 at rare high magnifications $H(z)$19. MACS J0416 shows smaller dispersions than Abell 2744 for $H(z)$20, and magnification maps based on different lens inversion techniques typically differ by more than their quoted statistical errors. The same analysis concluded that generalized mass-sheet degeneracy is not broken in Abell 2744, even with HFF constraints (Priewe et al., 2016).

These lensing studies establish a separate technical meaning of Hubble models: not models of the cosmic expansion rate, but Hubble-imaging-driven reconstructions of cluster mass and magnification. A plausible implication is that, in current usage, the term has become context-dependent. In background cosmology it denotes models of $H(z)$21 and $H(z)$22; in strong lensing it denotes HST-based mass inversions; and in alternative cosmology it can denote explicit dynamical mechanisms proposed to generate or reinterpret Hubble-law behavior.