Fractional Newtonian Cosmology

• Fractional Newtonian cosmology is a framework that deforms the standard Newtonian action using a power-law time kernel with parameter α, introducing memory effects and time-dependent friction.
• The model reproduces key cosmic epochs—including radiation, matter domination, and late-time acceleration—without manually adding a cosmological constant.
• The approach generates consistent background evolution, perturbation growth, and weak-field predictions, with observational constraints suggesting 0.8 ≤ α ≤ 1.07 for viability.

Fractional Newtonian cosmology denotes a class of non-relativistic cosmological models in which the standard Newtonian action is deformed by a power-law time kernel characterized by a single fractional parameter $\alpha$. In this framework, the dynamics acquire a memory weighting and a time-dependent friction or anti-friction term, while a suitably generalized effective potential produces cosmological background equations that are formally identical to Friedmann–Lemaître equations. Recent formulations claim that, without introducing a cosmological constant by hand, the same one-parameter deformation can reproduce radiation domination, matter domination, late-time acceleration, and, in extended constructions, a nonsingular pre-inflationary regime and a stable inflationary attractor; the limit $\alpha\to1$ recovers standard Newtonian gravity (Rasouli, 3 Mar 2026, Rasouli, 5 Mar 2026, Rasouli, 16 Mar 2026).

1. Fractional action and modified Newtonian dynamics

The basic construction starts from a deformation of the Newtonian particle action by a time-weighting kernel,

$$S_{\alpha}=\frac{1}{\Gamma(\alpha)}\int_{t'}^{\bar t}\xi(\bar t)\,\mathcal L\bigl[\mathbf r(\bar t),\dot{\mathbf r}(\bar t)\bigr]\,d\bar t,$$

with

$$\xi(\bar t)\equiv\left(\frac{\bar t-t'}{t_*}\right)^{\alpha-1}, \qquad \mathcal L=T-V_{\rm eff}, \qquad T=\tfrac12 m\dot{\mathbf r}^2.$$

Varying the action yields

$$m\ddot{\mathbf r}+m\gamma_\alpha(t)\dot{\mathbf r}=-\nabla V_{\rm eff}(\mathbf r;\alpha), \qquad \gamma_\alpha(t)=\frac{\dot\xi}{\xi}=\frac{\alpha-1}{t}.$$

The additional term $m\gamma_\alpha(t)\dot{\mathbf r}$ acts like a time-dependent linear friction for $\alpha>1$ and anti-friction for $\alpha<1$ (Rasouli, 3 Mar 2026).

In the perturbative formulation developed for the matter era, the only fractional ingredient is explicitly stated to be a power-law time kernel in the action rather than a spatial fractional Laplacian. The fluid equations then retain the standard Newtonian continuity and Poisson structure, while the Euler equation acquires the extra $\gamma_\alpha(t)$ term. In the special limit $\alpha=1$, $\alpha\to1$0 and the standard Newtonian equations are exactly recovered (Rasouli, 8 Mar 2026).

The significance of this formulation is methodological: the deformation is minimal, single-parameter, and encoded at the level of the action. This makes departures from Newtonian dynamics structurally transparent and allows the same parameter $\alpha\to1$1 to appear in background evolution, perturbation growth, and weak-field tests.

2. Conserved fractional energy and effective potential

Despite the friction-like term in the equations of motion, the theory admits a conserved quantity. Dotting the equation of motion with $\alpha\to1$2 gives

$\alpha\to1$3

so that

$\alpha\to1$4

with the fractional kinetic-memory contribution

$\alpha\to1$5

In the limit $\alpha\to1$6, $\alpha\to1$7 and ordinary conservation of $\alpha\to1$8 is recovered (Rasouli, 3 Mar 2026).

The effective potential is written as

$\alpha\to1$9

where $$S_{\alpha}=\frac{1}{\Gamma(\alpha)}\int_{t'}^{\bar t}\xi(\bar t)\,\mathcal L\bigl[\mathbf r(\bar t),\dot{\mathbf r}(\bar t)\bigr]\,d\bar t,$$0 is the usual Newtonian potential and $$S_{\alpha}=\frac{1}{\Gamma(\alpha)}\int_{t'}^{\bar t}\xi(\bar t)\,\mathcal L\bigl[\mathbf r(\bar t),\dot{\mathbf r}(\bar t)\bigr]\,d\bar t,$$1 as $$S_{\alpha}=\frac{1}{\Gamma(\alpha)}\int_{t'}^{\bar t}\xi(\bar t)\,\mathcal L\bigl[\mathbf r(\bar t),\dot{\mathbf r}(\bar t)\bigr]\,d\bar t,$$2. In the background cosmology construction, the precise form of $$S_{\alpha}=\frac{1}{\Gamma(\alpha)}\int_{t'}^{\bar t}\xi(\bar t)\,\mathcal L\bigl[\mathbf r(\bar t),\dot{\mathbf r}(\bar t)\bigr]\,d\bar t,$$3 is determined by demanding that the cosmological equations reproduce radiation, matter, and $$S_{\alpha}=\frac{1}{\Gamma(\alpha)}\int_{t'}^{\bar t}\xi(\bar t)\,\mathcal L\bigl[\mathbf r(\bar t),\dot{\mathbf r}(\bar t)\bigr]\,d\bar t,$$4-like eras. The simplest ansatz is summarized as

$$S_{\alpha}=\frac{1}{\Gamma(\alpha)}\int_{t'}^{\bar t}\xi(\bar t)\,\mathcal L\bigl[\mathbf r(\bar t),\dot{\mathbf r}(\bar t)\bigr]\,d\bar t,$$5

This correction is not arbitrary: it is fixed by consistency between the modified mechanics and the desired cosmological phases (Rasouli, 3 Mar 2026).

A later weak-field extension proposes a single unified effective potential,

$$S_{\alpha}=\frac{1}{\Gamma(\alpha)}\int_{t'}^{\bar t}\xi(\bar t)\,\mathcal L\bigl[\mathbf r(\bar t),\dot{\mathbf r}(\bar t)\bigr]\,d\bar t,$$6

with $$S_{\alpha}=\frac{1}{\Gamma(\alpha)}\int_{t'}^{\bar t}\xi(\bar t)\,\mathcal L\bigl[\mathbf r(\bar t),\dot{\mathbf r}(\bar t)\bigr]\,d\bar t,$$7 as $$S_{\alpha}=\frac{1}{\Gamma(\alpha)}\int_{t'}^{\bar t}\xi(\bar t)\,\mathcal L\bigl[\mathbf r(\bar t),\dot{\mathbf r}(\bar t)\bigr]\,d\bar t,$$8 and $$S_{\alpha}=\frac{1}{\Gamma(\alpha)}\int_{t'}^{\bar t}\xi(\bar t)\,\mathcal L\bigl[\mathbf r(\bar t),\dot{\mathbf r}(\bar t)\bigr]\,d\bar t,$$9 by dimensional analysis. In that construction, the same potential is used for cosmology, perihelion precession, and light deflection, which suggests a unified phenomenological description across scales (Rasouli, 16 Mar 2026).

3. Cosmological reduction and Friedmann-like equations

The cosmological model is obtained by considering a uniform sphere of radius $$\xi(\bar t)\equiv\left(\frac{\bar t-t'}{t_*}\right)^{\alpha-1}, \qquad \mathcal L=T-V_{\rm eff}, \qquad T=\tfrac12 m\dot{\mathbf r}^2.$$0 containing non-relativistic matter. A test particle on its surface obeys

$$\xi(\bar t)\equiv\left(\frac{\bar t-t'}{t_*}\right)^{\alpha-1}, \qquad \mathcal L=T-V_{\rm eff}, \qquad T=\tfrac12 m\dot{\mathbf r}^2.$$1

Defining

$$\xi(\bar t)\equiv\left(\frac{\bar t-t'}{t_*}\right)^{\alpha-1}, \qquad \mathcal L=T-V_{\rm eff}, \qquad T=\tfrac12 m\dot{\mathbf r}^2.$$2

and setting the total conserved energy $$\xi(\bar t)\equiv\left(\frac{\bar t-t'}{t_*}\right)^{\alpha-1}, \qquad \mathcal L=T-V_{\rm eff}, \qquad T=\tfrac12 m\dot{\mathbf r}^2.$$3, one finds

$$\xi(\bar t)\equiv\left(\frac{\bar t-t'}{t_*}\right)^{\alpha-1}, \qquad \mathcal L=T-V_{\rm eff}, \qquad T=\tfrac12 m\dot{\mathbf r}^2.$$4

with

$$\xi(\bar t)\equiv\left(\frac{\bar t-t'}{t_*}\right)^{\alpha-1}, \qquad \mathcal L=T-V_{\rm eff}, \qquad T=\tfrac12 m\dot{\mathbf r}^2.$$5

The associated fractional pressure is

$$\xi(\bar t)\equiv\left(\frac{\bar t-t'}{t_*}\right)^{\alpha-1}, \qquad \mathcal L=T-V_{\rm eff}, \qquad T=\tfrac12 m\dot{\mathbf r}^2.$$6

and the acceleration equation becomes

$$\xi(\bar t)\equiv\left(\frac{\bar t-t'}{t_*}\right)^{\alpha-1}, \qquad \mathcal L=T-V_{\rm eff}, \qquad T=\tfrac12 m\dot{\mathbf r}^2.$$7

Matter and fractional sectors each satisfy their own continuity equations, so that the total effective fluid obeys

$$\xi(\bar t)\equiv\left(\frac{\bar t-t'}{t_*}\right)^{\alpha-1}, \qquad \mathcal L=T-V_{\rm eff}, \qquad T=\tfrac12 m\dot{\mathbf r}^2.$$8

These equations are formally identical to relativistic Friedmann–Lemaître equations with curvature, matter, radiation, and an effective dark-energy sector sourced by the fractional terms (Rasouli, 3 Mar 2026).

The framework therefore does not merely perturb Newtonian dynamics locally. It recasts a coarse-grained Newtonian system into an effective cosmology in which the fractional sector behaves as an emergent source component. In the matter-dominated perturbative analysis, one may consistently set $$\xi(\bar t)\equiv\left(\frac{\bar t-t'}{t_*}\right)^{\alpha-1}, \qquad \mathcal L=T-V_{\rm eff}, \qquad T=\tfrac12 m\dot{\mathbf r}^2.$$9, which isolates the effect of the time-kernel on structure growth while preserving the standard Newtonian Poisson equation (Rasouli, 8 Mar 2026).

4. Cosmic epochs: inflation, standard eras, and late acceleration

At the background level, the model reproduces the standard power-law exponents for radiation and matter:

$$m\ddot{\mathbf r}+m\gamma_\alpha(t)\dot{\mathbf r}=-\nabla V_{\rm eff}(\mathbf r;\alpha), \qquad \gamma_\alpha(t)=\frac{\dot\xi}{\xi}=\frac{\alpha-1}{t}.$$0

The amplitudes depend on $$m\ddot{\mathbf r}+m\gamma_\alpha(t)\dot{\mathbf r}=-\nabla V_{\rm eff}(\mathbf r;\alpha), \qquad \gamma_\alpha(t)=\frac{\dot\xi}{\xi}=\frac{\alpha-1}{t}.$$1, but the exponents remain exactly those of $$m\ddot{\mathbf r}+m\gamma_\alpha(t)\dot{\mathbf r}=-\nabla V_{\rm eff}(\mathbf r;\alpha), \qquad \gamma_\alpha(t)=\frac{\dot\xi}{\xi}=\frac{\alpha-1}{t}.$$2CDM. For late times the theory admits an asymptotic de Sitter attractor,

$$m\ddot{\mathbf r}+m\gamma_\alpha(t)\dot{\mathbf r}=-\nabla V_{\rm eff}(\mathbf r;\alpha), \qquad \gamma_\alpha(t)=\frac{\dot\xi}{\xi}=\frac{\alpha-1}{t}.$$3

so that for $$m\ddot{\mathbf r}+m\gamma_\alpha(t)\dot{\mathbf r}=-\nabla V_{\rm eff}(\mathbf r;\alpha), \qquad \gamma_\alpha(t)=\frac{\dot\xi}{\xi}=\frac{\alpha-1}{t}.$$4 one recovers effectively

$$m\ddot{\mathbf r}+m\gamma_\alpha(t)\dot{\mathbf r}=-\nabla V_{\rm eff}(\mathbf r;\alpha), \qquad \gamma_\alpha(t)=\frac{\dot\xi}{\xi}=\frac{\alpha-1}{t}.$$5

In this sense, a $$m\ddot{\mathbf r}+m\gamma_\alpha(t)\dot{\mathbf r}=-\nabla V_{\rm eff}(\mathbf r;\alpha), \qquad \gamma_\alpha(t)=\frac{\dot\xi}{\xi}=\frac{\alpha-1}{t}.$$6CDM-like background emerges from the fractional sector without inserting a cosmological constant by hand (Rasouli, 3 Mar 2026).

An early-universe extension constructs a fractional potential of the form

$$m\ddot{\mathbf r}+m\gamma_\alpha(t)\dot{\mathbf r}=-\nabla V_{\rm eff}(\mathbf r;\alpha), \qquad \gamma_\alpha(t)=\frac{\dot\xi}{\xi}=\frac{\alpha-1}{t}.$$7

with fractional force

$$m\ddot{\mathbf r}+m\gamma_\alpha(t)\dot{\mathbf r}=-\nabla V_{\rm eff}(\mathbf r;\alpha), \qquad \gamma_\alpha(t)=\frac{\dot\xi}{\xi}=\frac{\alpha-1}{t}.$$8

The modified acceleration law can then be written as

$$m\ddot{\mathbf r}+m\gamma_\alpha(t)\dot{\mathbf r}=-\nabla V_{\rm eff}(\mathbf r;\alpha), \qquad \gamma_\alpha(t)=\frac{\dot\xi}{\xi}=\frac{\alpha-1}{t}.$$9

A transition time between pre-inflation and inflation is defined by the balance of friction and fractional force,

$m\gamma_\alpha(t)\dot{\mathbf r}$0

while the inflationary solution near the attractor has

$m\gamma_\alpha(t)\dot{\mathbf r}$1

For $m\gamma_\alpha(t)\dot{\mathbf r}$2, linearization around the de Sitter background gives a stable node. The fractional force vanishes near the end of inflation, and the small separation between its zero and the end of inflation yields a relation between the number of $m\gamma_\alpha(t)\dot{\mathbf r}$3-folds and $m\gamma_\alpha(t)\dot{\mathbf r}$4; requiring $m\gamma_\alpha(t)\dot{\mathbf r}$5 is reported to be compatible with $m\gamma_\alpha(t)\dot{\mathbf r}$6 (Rasouli, 5 Mar 2026).

The same early-universe construction also provides a graceful exit. In the post-inflationary regime,

$m\gamma_\alpha(t)\dot{\mathbf r}$7

and in the special case $m\gamma_\alpha(t)\dot{\mathbf r}$8 one obtains an exact radiation law,

$m\gamma_\alpha(t)\dot{\mathbf r}$9

provided $\alpha>1$0. A later summary of the framework states that a single potential reproduces the full sequence of cosmic evolution, from a nonsingular pre-inflationary phase and stable inflationary attractor to radiation domination, matter domination, and present accelerated expansion (Rasouli, 5 Mar 2026, Rasouli, 16 Mar 2026).

5. Density perturbations and empirical constraints

Linear perturbations in the matter-dominated era are derived using the fluid-flow approach. Writing

$\alpha>1$1

the first-order equations are

$\alpha>1$2

$\alpha>1$3

$\alpha>1$4

Eliminating $\alpha>1$5 and $\alpha>1$6 yields the modified growth equation

$\alpha>1$7

so the only formal change with respect to the standard Newtonian growth equation is $\alpha>1$8 (Rasouli, 8 Mar 2026).

For the matter-era background with $\alpha>1$9,

$\alpha<1$0

the perturbation equation becomes

$\alpha<1$1

Using $\alpha<1$2 gives

$\alpha<1$3

Real exponents require

$\alpha<1$4

and the physically relevant window is taken to be $\alpha<1$5. In that interval, $\alpha<1$6 is always a decaying mode, while $\alpha<1$7 for $\alpha<1$8 gives a growing mode. At $\alpha<1$9 one recovers the standard result

$\gamma_\alpha(t)$0

No dynamical instability is reported in the physically relevant parameter space (Rasouli, 8 Mar 2026).

Observational constraints sharpen this picture. Using the Sachs–Wolfe relation and the growth factor ratio

$\gamma_\alpha(t)$1

a conservative requirement $\gamma_\alpha(t)$2 yields

$\gamma_\alpha(t)$3

Combined with background-level and theoretical consistency across epochs, this gives the quoted viable window

$\gamma_\alpha(t)$4

The same study emphasizes that the background evolution may closely mimic $\gamma_\alpha(t)$5CDM while density perturbations carry a distinct fractional signature (Rasouli, 8 Mar 2026).

Other analyses produce bounds at different levels of stringency. A rough background estimate from the present-day equation of state gives $\gamma_\alpha(t)$6, while more refined fits including growth of structure and weak-field tests are summarized as pushing this to $\gamma_\alpha(t)$7 (Rasouli, 3 Mar 2026). In the weak-field letter, background-only fits to Type-Ia supernovae, BAO, and CMB distance priors yield $\gamma_\alpha(t)$8 at $\gamma_\alpha(t)$9 C.L., whereas Mercury’s perihelion precession gives $\alpha=1$0, and light-deflection consistency is obtained with $\alpha=1$1 and $\alpha=1$2 (Rasouli, 16 Mar 2026). This suggests that perturbations and Solar-System tests are substantially more restrictive than background geometry alone.

6. Relation to fractional-dimension and relativistic approaches

Fractional Newtonian cosmology is not identical to all other fractional-gravity programs. In Newtonian Fractional-Dimension Gravity, the deformation is introduced through a non-integer spatial dimension $\alpha=1$3 rather than a time-kernel. Gauss’s law and Poisson’s equation are generalized to $\alpha=1$4 dimensions, the point-mass field scales as

$\alpha=1$5

and a variable local dimension $\alpha=1$6, with $\alpha=1$7, is empirically extracted from the radial acceleration relation. In the simple spherical examples discussed there, $\alpha=1$8 for $\alpha=1$9 and $\alpha\to1$00 for $\alpha\to1$01, reproducing deep-MOND behavior. A Newtonian cosmology is only sketched in that framework through generalized continuity, Euler, and Poisson equations and a Friedmann-like first integral with time-dependent $\alpha\to1$02; the paper explicitly states that a full treatment would require embedding the model in a relativistic setting (Varieschi, 2020).

A relativistic extension of the fractional-dimension program introduces a weighted Hilbert action and, for a purely temporal weight

$\alpha\to1$03

derives extended Friedmann equations of the form

$\alpha\to1$04

$\alpha\to1$05

The numerical solutions summarized for representative $\alpha\to1$06 show that smaller $\alpha\to1$07 slightly slow late-time acceleration but do not eliminate the need for a cosmological constant of order $\alpha\to1$08; the conclusion is that time-fractionality alone does not replace dark energy in that relativistic fractional-dimension setting (Varieschi, 2021).

This contrasts with a different relativistic fractional-calculus construction based on Caputo derivatives of order $\alpha\to1$09, where modified Friedmann equations contain an extra $\alpha\to1$10 term, late-time acceleration occurs for $\alpha\to1$11, and a joint analysis of cosmic chronometers and Pantheon supernovae gives

$\alpha\to1$12

with a best-fit $\alpha\to1$13 and an older universe than in standard $\alpha\to1$14CDM (García-Aspeitia et al., 2022).

Taken together, these lines of work indicate that “fractional cosmology” is a family of non-equivalent constructions. In the specific fractional Newtonian program based on a time-kernel and effective potential, the central claim is that a single deformation parameter $\alpha\to1$15 can generate relativistic-like background evolution, viable perturbation growth, and weak-field phenomenology while reducing smoothly to Newtonian gravity as $\alpha\to1$16. A plausible implication is that the empirical viability of the framework depends less on fractionalization in the abstract than on the precise mechanism chosen—time-kernel, fractional dimension, or relativistic fractional derivative—and on whether perturbative and local tests can be satisfied simultaneously.