---
title: Fractional Newtonian Cosmology
url: https://www.emergentmind.com/topics/fractional-newtonian-cosmology
type: topic
---

# Fractional Newtonian Cosmology

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 [2603.03113, 2603.04712, 2603.16009].

## 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 $\alpha<1$ [2603.03113].

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 $\alpha=1$, $\gamma_\alpha\to0$ and the standard Newtonian equations are exactly recovered [2603.07781].

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 $\alpha$ 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 $\dot{\mathbf r}$ gives
$$
\frac{d}{dt}\left[T+V_{\rm eff}+T_\alpha\right]=0,
$$
so that
$$
\mathcal E^{\rm mech}_{\rm eff}=T+V_{\rm eff}+T_\alpha=\text{constant},
$$
with the fractional kinetic-memory contribution
$$
T_\alpha\equiv2(\alpha-1)\int_{t_i}^{t_f}T(t)\,d(\ln t)
=2(\alpha-1)\int_{t_i}^{t_f}\tfrac12 m\dot r^2\,\frac{dt}{t}.
$$
In the limit $\alpha\to1$, $T_\alpha\to0$ and ordinary conservation of $T+V$ is recovered [2603.03113].

The effective potential is written as
$$
V_{\rm eff}(r;\alpha)=V_N(r)+V_\alpha(r),
$$
where $V_N(r)=-GmM/r$ is the usual Newtonian potential and $V_\alpha(r)\to0$ as $\alpha\to1$. In the background cosmology construction, the precise form of $V_\alpha$ is determined by demanding that the cosmological equations reproduce radiation, matter, and $\Lambda$-like eras. The simplest ansatz is summarized as
$$
V_{\rm eff}(r;\alpha)= -\,\frac{GmM}{r} + \langle \text{fractional correction depending on }\alpha\rangle.
$$
This correction is not arbitrary: it is fixed by consistency between the modified mechanics and the desired cosmological phases [2603.03113].

A later weak-field extension proposes a single unified effective potential,
$$
V_{\rm eff}(r;\alpha)
=
-\,\frac{m\mu}{r}
-\frac{m\mu A(\alpha)}{r^3}
-\frac{m\mu B(\alpha)}{r}e^{-r/\lambda},
$$
with $A(\alpha),B(\alpha)\to0$ as $\alpha\to1$ and $A(\alpha)\simeq|\alpha-1|\ell^2$ 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 [2603.16009].

## 3. Cosmological reduction and Friedmann-like equations

The cosmological model is obtained by considering a uniform sphere of radius $a(t)$ containing non-relativistic matter. A test particle on its surface obeys
$$
\ddot a+\gamma_\alpha(t)\dot a+\frac{d}{da}\Phi_{\rm eff}(a;\alpha)=0,
\qquad
\gamma_\alpha(t)=\frac{\alpha-1}{t}.
$$
Defining
$$
T=\tfrac12\dot a^2,\quad E=T+\Phi_N,\quad E_\alpha=T_\alpha+\Phi_\alpha,
$$
and setting the total conserved energy $\mathcal E=E+E_\alpha=-\tfrac12\mathcal K$, one finds
$$
H^2\equiv\left(\frac{\dot a}{a}\right)^2
=
\frac{8\pi G}{3}(\rho_m+\rho_\alpha)-\frac{\mathcal K}{a^2},
$$
with
$$
\rho_\alpha\equiv-\frac{3}{4\pi G}\frac{E_\alpha}{a^2}.
$$
The associated fractional pressure is
$$
p_\alpha=\frac{1}{4\pi G\,a^2}\frac{d}{da}\bigl(aE_\alpha\bigr),
$$
and the acceleration equation becomes
$$
\frac{\ddot a}{a}
=
-\frac{4\pi G}{3}\bigl[(\rho_m+\rho_\alpha)+3p_\alpha\bigr].
$$
Matter and fractional sectors each satisfy their own continuity equations, so that the total effective fluid obeys
$$
\dot\rho_{\rm eff}+3H(\rho_{\rm eff}+p_{\rm eff})=0,
\qquad
\rho_{\rm eff}=\rho_m+\rho_\alpha,
\qquad
p_{\rm eff}=p_\alpha.
$$
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 [2603.03113].

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 $\Phi_\alpha=0$, which isolates the effect of the time-kernel on structure growth while preserving the standard Newtonian Poisson equation [2603.07781].

## 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:
$$
a(t)=A_r(\alpha)\,t^{1/2},
\qquad
a(t)=A_m(\alpha)\,t^{2/3}.
$$
The amplitudes depend on $\alpha$, but the exponents remain exactly those of $\Lambda$CDM. For late times the theory admits an asymptotic de Sitter attractor,
$$
a(t)\simeq t^{(1-\alpha)/2}\exp[\kappa(\alpha)t],
$$
so that for $t\gg1$ one recovers effectively
$$
a(t)\to e^{H_\alpha t},
\qquad
H_\alpha=\kappa(\alpha),
\qquad
\Lambda_{\rm eff}(\alpha)=3\kappa^2(\alpha).
$$
In this sense, a $\Lambda$CDM-like background emerges from the fractional sector without inserting a cosmological constant by hand [2603.03113].

An early-universe extension constructs a fractional potential of the form
$$
\Phi_\alpha(a)=-\Phi_0\,\frac{a^2}{1+(a/a_c)^p},
$$
with fractional force
$$
\mathcal F(a)=\mathcal F_0\left[\frac{1}{1+x}-\frac{px}{2(1+x)^2}\right],
\qquad
x=(a/a_c)^p,
\qquad
\mathcal F_0=2\Phi_0.
$$
The modified acceleration law can then be written as
$$
\frac{\ddot a}{a}
=
-\,\gamma_\alpha(t)\frac{\dot a}{a}
-\frac{4\pi G}{3}\rho_m(a)
+\mathcal F(a).
$$
A transition time between pre-inflation and inflation is defined by the balance of friction and fractional force,
$$
t_*\simeq\frac{|\alpha-1|}{2\sqrt{\mathcal F_0}},
$$
while the inflationary solution near the attractor has
$$
H_{\rm inf}(t)\simeq\sqrt{\mathcal F_0}-\frac{\alpha-1}{2t}.
$$
For $\alpha>1$, 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 $e$-folds and $\alpha$; requiring $N\simeq50\!-\!60$ is reported to be compatible with $|\alpha-1|\ll1$ [2603.04712].

The same early-universe construction also provides a graceful exit. In the post-inflationary regime,
$$
a(t)\propto t^{2/p},
$$
and in the special case $p=4$ one obtains an exact radiation law,
$$
a(t)=a_r\,t^{1/2},
\qquad
a_r=a_c\left(-\frac{4\mathcal F_0}{2\alpha-3}\right)^{1/4},
$$
provided $\alpha<3/2$. 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 [2603.04712, 2603.16009].

## 5. Density perturbations and empirical constraints

Linear perturbations in the matter-dominated era are derived using the fluid-flow approach. Writing
$$
\rho(\mathbf x,t)=\bar\rho(t)[1+\delta(\mathbf x,t)],
\qquad
\mathbf v=Ha\mathbf x+\mathbf u,
\qquad
\Phi_N=\bar\Phi+\phi(\mathbf x,t),
$$
the first-order equations are
$$
\dot\delta+\frac{1}{a}\nabla_x\!\cdot\mathbf u=0,
$$
$$
\dot{\mathbf u}+(H+\gamma_\alpha)\mathbf u=-\frac{1}{a}\nabla_x\phi,
$$
$$
\nabla_x^2\phi=4\pi G a^2\bar\rho_m\,\delta.
$$
Eliminating $\mathbf u$ and $\phi$ yields the modified growth equation
$$
\ddot\delta+\bigl[2H(t)+\gamma_\alpha(t)\bigr]\dot\delta-4\pi G\bar\rho_m(t)\delta=0,
$$
so the only formal change with respect to the standard Newtonian growth equation is $2H\mapsto2H+\gamma_\alpha$ [2603.07781].

For the matter-era background with $\Phi_\alpha=0$,
$$
a(t)=a_i\,t^{2/3},
\qquad
H(t)=\frac{2}{3t},
\qquad
\bar\rho_m(t)\propto t^{-2},
$$
the perturbation equation becomes
$$
\ddot\delta+\frac{3\alpha+1}{3t}\dot\delta
+2\frac{3\alpha-4}{3t^2}\delta=0.
$$
Using $\delta\propto t^p$ gives
$$
p_{\pm}=
\frac{-\,(3\alpha-2)\pm\sqrt{(3\alpha-14)^2-96}}{6}.
$$
Real exponents require
$$
\alpha\le\frac{14-\sqrt{96}}{3}\approx1.4007
\quad\text{or}\quad
\alpha\ge\frac{14+\sqrt{96}}{3}\approx7.933,
$$
and the physically relevant window is taken to be $0<\alpha\lesssim1.4007$. In that interval, $p_-<0$ is always a decaying mode, while $p_+>0$ for $0<\alpha<4/3$ gives a growing mode. At $\alpha=1$ one recovers the standard result
$$
p_+=2/3,
\qquad
p_-=-1.
$$
No dynamical instability is reported in the physically relevant parameter space [2603.07781].

Observational constraints sharpen this picture. Using the Sachs–Wolfe relation and the growth factor ratio
$$
\mathcal I(\alpha)=
\frac{\delta^{(g)}(t_0,\alpha)}{\delta^{(g)}(t_0,\alpha=1)}
=
(1+z_{\rm dec})^{\frac32[p_+(\alpha)-2/3]},
$$
a conservative requirement $\mathcal I(\alpha)\gtrsim0.3$ yields
$$
\alpha\lesssim1.07.
$$
Combined with background-level and theoretical consistency across epochs, this gives the quoted viable window
$$
0.8\lesssim\alpha\lesssim1.07.
$$
The same study emphasizes that the background evolution may closely mimic $\Lambda$CDM while density perturbations carry a distinct fractional signature [2603.07781].

Other analyses produce bounds at different levels of stringency. A rough background estimate from the present-day equation of state gives $|\,\alpha-1\,|\lesssim0.2$, while more refined fits including growth of structure and weak-field tests are summarized as pushing this to $|\,\alpha-1\,|\lesssim0.01$ [2603.03113]. In the weak-field letter, background-only fits to Type-Ia supernovae, BAO, and CMB distance priors yield $|\,\alpha-1\,|<0.2$ at $95\%$ C.L., whereas Mercury’s perihelion precession gives $|\,\alpha-1\,|\sim2\times10^{-6}$, and light-deflection consistency is obtained with $B(\alpha)\sim1$ and $|\,\alpha-1\,|\sim\varepsilon_g\sim2\times10^{-6}$ [2603.16009]. 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 $D$ rather than a time-kernel. Gauss’s law and Poisson’s equation are generalized to $D$ dimensions, the point-mass field scales as
$$
g(r)=2\,\pi^{1-D/2}\Gamma\!\left(\frac D2\right)\frac{Gm}{r^{D-1}},
$$
and a variable local dimension $D(w)$, with $w=r/l_0$, is empirically extracted from the radial acceleration relation. In the simple spherical examples discussed there, $D(w)\approx3$ for $w\ll1$ and $D(w)\to2$ for $w\gg1$, 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 $D(t)$; the paper explicitly states that a full treatment would require embedding the model in a relativistic setting [2003.05784].

A relativistic extension of the fractional-dimension program introduces a weighted Hilbert action and, for a purely temporal weight
$$
v_t(t)=\frac{\pi^{\alpha_t/2}}{\Gamma(\alpha_t/2)}\,t^{\alpha_t-1},
\qquad
0<\alpha_t\le1,
$$
derives extended Friedmann equations of the form
$$
H^2+H\frac{\dot v}{v}
=
\frac{8\pi G}{3}\rho-\frac{\kappa}{a^2}+\frac{\Lambda}{3},
$$
$$
\frac{\ddot a}{a}
+\frac12\left(H\frac{\dot v}{v}+\frac{\ddot v}{v}\right)
=
-\frac{4\pi G}{3}(\rho+3p)+\frac{\Lambda}{3}.
$$
The numerical solutions summarized for representative $\alpha_t$ show that smaller $\alpha_t$ slightly slow late-time acceleration but do not eliminate the need for a cosmological constant of order $\Omega_{\Lambda0}\sim0.7$; the conclusion is that time-fractionality alone does not replace dark energy in that relativistic fractional-dimension setting [2109.02855].

This contrasts with a different relativistic fractional-calculus construction based on Caputo derivatives of order $\mu$, where modified Friedmann equations contain an extra $(1-\mu)H/t$ term, late-time acceleration occurs for $\mu>2$, and a joint analysis of cosmic chronometers and Pantheon supernovae gives
$$
\mu=2.839^{+0.117}_{-0.193},
\qquad
h=0.692^{+0.019}_{-0.018},
\qquad
\Omega_m=0.228^{+0.035}_{-0.040},
$$
with a best-fit $\chi^2\simeq78.7$ and an older universe than in standard $\Lambda$CDM [2207.00878].

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 $\alpha$ can generate relativistic-like background evolution, viable perturbation growth, and weak-field phenomenology while reducing smoothly to Newtonian gravity as $\alpha\to1$. 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.

Source: https://www.emergentmind.com/topics/fractional-newtonian-cosmology