Dolgov's Power-Law Cosmology and VSL Models

• Dolgov's power-law cosmology is a model where the scale factor evolves as a power of cosmic time and the speed of light varies with expansion.
• It modifies the standard Robertson–Walker metric to derive generalized redshift and luminosity distance formulas that challenge conventional cosmology.
• Empirical tests using SNIa, BAO, and CMB data show that while some variants mimic dark energy effects, they fail to consistently match all observational constraints compared to ΛCDM.

Dolgov's power-law cosmology is a class of cosmological models in which the scale factor evolves as a power of cosmic time, $a(t) = (t/t_0)^\mu$, with particular attention to extensions in which the speed of light varies as a function of the scale factor, $c(a) = c_0 a^{-\zeta}$. Originating in the works of Dolgov and collaborators, this approach probes whether cosmic acceleration and certain horizon-scale puzzles can be addressed without recourse to dark energy or inflation, by exploiting non-standard kinematics and redefining the role of fundamental "constants" within Robertson–Walker geometry. The empirical and theoretical consequences of these models have been extensively analyzed with respect to observational data, including Type Ia supernovae (SNIa), baryon acoustic oscillations (BAO), and cosmic microwave background (CMB) scales.

1. Mathematical Framework: Power Law Expansion and Varying Speed of Light

In Dolgov's model, the cosmic scale factor follows a power law,

$a(t) = \left(\frac{t}{t_0}\right)^\mu,$

where $t_0$ is the present cosmic time and $\mu$ is a dimensionless exponent. Generalizations allow the speed of light to scale with the scale factor as $c(a) = c_0 a^{-\zeta}$, introducing the "Barrow–Dolgov" VSL extension. The Robertson–Walker metric accordingly becomes

$$ds^2 = c^2(a)\,dt^2 - a^2(t)\big(dr^2 + r^2 d\Omega^2\big),\quad c(a) = c_0 a^{-\zeta}.$$

A key mathematical feature is that the propagation of light through such a spacetime entails modified redshift–distance relationships. The deduction yields a generalized Lemaître redshift formula,

$1 + z = a^{-(1+\zeta)},$

which reduces to the standard $1+z = a^{-1}$ for constant $c$.

The luminosity distance in this formalism acquires the structure

$c(a) = c_0 a^{-\zeta}$0

where $c(a) = c_0 a^{-\zeta}$1 and $c(a) = c_0 a^{-\zeta}$2 is a correction factor accounting for “yardstick–matching” due to local calibration choices.

2. Confrontation with Observational Data

Analysis of SNIa, BAO, and CMB data determines the viability of the power-law and VSL cosmologies. The Pantheon compilation of Type Ia supernovae—1048 objects spanning $c(a) = c_0 a^{-\zeta}$3—was fit using the $c(a) = c_0 a^{-\zeta}$4 parametrization and found to indicate a degeneracy along $c(a) = c_0 a^{-\zeta}$5, with supernova data alone providing a narrow constraint (width $c(a) = c_0 a^{-\zeta}$6) along this locus (Nguyen, 9 Jan 2026). Several notable cases include:

Model	$c(a) = c_0 a^{-\zeta}$7	$c(a) = c_0 a^{-\zeta}$8	$c(a) = c_0 a^{-\zeta}$9 (Pantheon)
EdS	2/3	0	$a(t) = \left(\frac{t}{t_0}\right)^\mu,$0
Dolgov ($a(t) = \left(\frac{t}{t_0}\right)^\mu,$1const)	$a(t) = \left(\frac{t}{t_0}\right)^\mu,$21.52	0	$a(t) = \left(\frac{t}{t_0}\right)^\mu,$3
Kolb's Coasting	1	0	$a(t) = \left(\frac{t}{t_0}\right)^\mu,$4
SIG	2/3	1/2	$a(t) = \left(\frac{t}{t_0}\right)^\mu,$5
$a(t) = \left(\frac{t}{t_0}\right)^\mu,$6CDM	--	--	$a(t) = \left(\frac{t}{t_0}\right)^\mu,$7

Self-invariant models satisfying $a(t) = \left(\frac{t}{t_0}\right)^\mu,$8—including those based on scale-invariant gravity (SIG) and Kolb's coasting universe—marginally outperform $a(t) = \left(\frac{t}{t_0}\right)^\mu,$9CDM by $t_0$0. However, this apparent success is not robust when additional probes are included.

When BAO and CMB data are incorporated, the power-law models face significant difficulties. For SNIa and BAO alone, best-fit exponents are $t_0$1 (SNIa$t_0$2BAO), but these same exponents lead to a divergent sound horizon at recombination, failing to simultaneously reproduce both BAO and CMB acoustic scales (Tutusaus et al., 2016, Shafer, 2015). Empirically matching the observed CMB peak by treating the “ruler” scale as a free parameter fails to resolve this inconsistency: no single $t_0$3 allows joint agreement with BAO and CMB data. Model comparison metrics (AICc, BIC, Bayes factor) strongly favor $t_0$4CDM over all power-law cosmologies for the joint data.

3. Physical Consequences of the $t_0$5 Locus

The special locus $t_0$6 features several distinctive kinematic and geometric properties (Nguyen, 9 Jan 2026):

• For large $t_0$7, $t_0$8, mimicking supernova dimming customarily attributed to late-time acceleration without invoking a cosmological constant.
• The kinematic law $t_0$9 ensures the comoving particle and event horizons both diverge,

$\mu$0

thus providing a kinematic solution to the horizon problem distinct from inflation.

• The simple proportionality $\mu$1 is characteristic of these solutions and not present in $\mu$2CDM.

4. Theoretical Implications: Generalized Cosmological Principle and Conformal Structure

For $\mu$3, the Hubble radius $\mu$4. The “visible” universe’s size then always scales with expansion, and the metric itself is self-invariant under time rescaling. This extends the cosmological principle from spatial to temporal homogeneity, eliminating any privileged cosmic epoch and allowing a reformulation of the underlying metric in conformally flat terms: $\mu$5 which is conformal to Minkowski space and makes the absence of cosmological horizons manifest.

5. Relation to Alternative Models and Comparison with Standard Cosmology

Kolb’s coasting universe ($\mu$6) and the scale-invariant gravity case ($\mu$7) both reside on $\mu$8. Kolb's model requires an exotic "K-matter" in general relativity, but within VSL cosmology arises as a specific realization of the kinematic law $\mu$9.

While models on this locus outperform $c(a) = c_0 a^{-\zeta}$0CDM in SNIa-only and some low-redshift BAO fits, they are systematically ruled out by combined analyses that include the CMB acoustic peak due to the divergence of the sound horizon for $c(a) = c_0 a^{-\zeta}$1 or the persistent mismatch in derived scales even when the horizon is empirically treated as a free parameter (Tutusaus et al., 2016, Shafer, 2015). This quantitative failure secures the observational dominance of $c(a) = c_0 a^{-\zeta}$2CDM.

6. Statistical Model Comparison and Empirical Constraints

Quantitative comparisons using corrected Akaike (AICc), Bayesian (BIC) information criteria, and Bayes factors indicate that while SNIa or BAO data analyzed separately can be reproduced by power-law cosmology with $c(a) = c_0 a^{-\zeta}$3 (SNIa) or $c(a) = c_0 a^{-\zeta}$4 (BAO), the combination of low-redshift and CMB probes yields

$c(a) = c_0 a^{-\zeta}$5

strongly disfavoring power-law models. For example, the odds against power-law versus $c(a) = c_0 a^{-\zeta}$6CDM reach $c(a) = c_0 a^{-\zeta}$7 (JLA+BAO) to $c(a) = c_0 a^{-\zeta}$8 (Union2.1+BAO) (Shafer, 2015).

The empirical fit to the Pantheon SNIa catalog crucially reveals that $c(a) = c_0 a^{-\zeta}$9 is the unique locus providing an optimal fit under the VSL extension, but this success is not preserved when the full range of cosmological datasets is included (Nguyen, 9 Jan 2026, Tutusaus et al., 2016).

7. Summary of Empirical Status and Conceptual Impact

Dolgov’s power-law cosmology, particularly in its VSL generalization, provides a technically consistent framework with instructive geometric and kinematic innovations. It enables cosmic acceleration without dark energy, resolves the horizon problem without inflation, generalizes homogeneity to the time domain, and brings forth a novel conformally flat cosmological metric.

Nevertheless, the inability to conform to BAO and especially CMB acoustic scale constraints—in particular the sound horizon divergence for $$ds^2 = c^2(a)\,dt^2 - a^2(t)\big(dr^2 + r^2 d\Omega^2\big),\quad c(a) = c_0 a^{-\zeta}.$$0—renders all such models incompatible with the full suite of cosmological observations. The empirical evidence very strongly favors the $$ds^2 = c^2(a)\,dt^2 - a^2(t)\big(dr^2 + r^2 d\Omega^2\big),\quad c(a) = c_0 a^{-\zeta}.$$1CDM concordance model over any variant of power-law or VSL power-law cosmology (Nguyen, 9 Jan 2026, Tutusaus et al., 2016, Shafer, 2015).