Fractional Scalar-Field Cosmology Insights

• Fractional scalar-field cosmology is a framework that combines scalar fields with fractional calculus to introduce memory effects and nonlocal modifications in cosmological dynamics.
• It modifies standard equations by incorporating fractional integrals and derivatives, leading to reconstructed potentials and altered effective equations of state.
• The approach spans multiple formulations—from fractional action integrals to emergent quantum operators—offering novel insights into inflationary behavior and dark sector modeling.

Fractional scalar-field cosmology denotes a family of cosmological constructions in which scalar-field dynamics are combined with fractional calculus, most commonly through a fractional action functional, a kernel-weighted time measure, or a fractional minisuperspace kinetic operator. In the current literature, the term does not refer to a single formalism. In one line of work, the Einstein–scalar-field system on an FLRW background is modified by replacing the ordinary time integral in the action by a fractional integral of order $\mu$, which yields modified Friedmann and Klein–Gordon equations with explicit $t^{-1}$ and $t^{-2}$ terms (Rasouli et al., 2024). In another, a time-dependent kernel $\xi(t)$ multiplies the gravitational action, generating an effective fractional sector that exchanges energy–momentum with the scalar field while keeping the geometric side standard (Rasouli et al., 12 Dec 2025). A distinct minisuperspace program arises in K-essence cosmology, where a non-integer power of the scalar momentum $\Pi_\phi$ produces a fractional Wheeler–DeWitt equation in the scalar variable after quantization, with order $\beta=\frac{2\alpha}{2\alpha-1}=1+\omega_X$ (Socorro et al., 2023). Across these variants, the recurrent themes are memory-like corrections, nonlocality encoded through power-law kernels, modified effective equations of state, and scalar sectors whose potentials or quantum dynamics are constrained by the fractional structure rather than chosen freely.

1. Conceptual scope and principal formulations

Fractional scalar-field cosmology has developed along several technically distinct routes. A useful classification given in recent work distinguishes three ways to “fractionalize” cosmology: kernel-weighted action, replacing derivatives by fractional derivatives in the action or equations, and building a full fractional geometry (Rasouli et al., 12 Dec 2025). The literature represented here is concentrated mainly in the first route, with additional quantum constructions based on fractional operators in minisuperspace.

Before comparing models, it is useful to separate the dominant formulations.

Formulation	Fractional ingredient	Representative papers
Fractional action cosmology	Time integral weighted by $(t-\tau)^{\xi-1}$ or $(t-\tau)^{\mu-1}$	(Debnath et al., 2011, Rasouli et al., 2024, Micolta-Riascos et al., 2023)
Fractional Einstein–Hilbert gravity	Only the gravitational sector carries the fractional measure	(Shchigolev, 2013, Marroquín et al., 2024)
Emergent fractional quantum cosmology	Non-integer momentum powers produce fractional WDW operators	(Socorro et al., 2023, Socorro et al., 2023, Socorro et al., 2024)

In the classical fractional-action approach, the ordinary Einstein–Hilbert plus scalar-field Lagrangian is retained at the level of local field variables, but the time integral is replaced by a Riemann–Liouville-type fractional integral. One widely used form is

$$S_\mu=\frac{1}{\Gamma(\mu)}\int_0^t d\tau\,(t-\tau)^{\mu-1}L(\tau),$$

with $0<\mu\le 1$ in the FLRW Einstein–scalar setting (Rasouli et al., 2024). Another closely related form uses a weight $t^{-1}$0, with $t^{-1}$1, in Fractional Action Cosmology (Debnath et al., 2011). In both cases, the fractional structure enters through the measure of the action rather than by replacing all derivatives of $t^{-1}$2 or $t^{-1}$3 with fractional derivatives.

A second important branch fractionalizes only the gravitational sector. In “Fractional Einstein-Hilbert Action Cosmology” the matter action is kept standard while the Einstein–Hilbert minisuperspace action is integrated with a left-sided Riemann–Liouville-type fractional kernel (Shchigolev, 2013). This two-measure construction is important because it preserves the standard continuity equation for matter, unlike earlier fractional-total-action models. “Conformal and Non-Minimal Couplings in Fractional Cosmology” likewise uses a fractional variational principle for the Einstein–Hilbert action, but augments it with a scalar field and a non-minimal interaction $t^{-1}$4, producing explicit fractional “friction/memory” terms governed by a fractional parameter $t^{-1}$5 (Marroquín et al., 2024).

A third branch appears in minisuperspace quantization. In “Quantum fractionary cosmology: K-essence theory” the scalar-field Hamiltonian contains $t^{-1}$6; after quantization, this becomes a derivative of non-integer order in $t^{-1}$7, so the Wheeler–DeWitt equation is fractional in the scalar sector (Socorro et al., 2023). The same mechanism was extended to Bianchi I anisotropic cosmology (Socorro et al., 2023) and to a noncommutative flat FRW model (Socorro et al., 2024).

This dispersion of formalisms suggests that “fractional scalar-field cosmology” is better understood as a research area organized around fractional measures, fractional operators, or emergent fractional quantum equations, rather than as a single canonical theory.

2. Classical FLRW scalar cosmology from fractional actions

In the FLRW Einstein–scalar system with a fractional action functional, the basic modification is encoded in the field equations. For a spatially flat universe, “Fractional Scalar Field Cosmology” gives the fractional Friedmann and Klein–Gordon equations

$t^{-1}$8

$t^{-1}$9

$t^{-2}$0

with the modified continuity equation

$t^{-2}$1

These equations reduce to the standard flat FLRW + scalar equations when $t^{-2}$2 (Rasouli et al., 2024).

A closely related but structurally distinct implementation appears in Fractional Action Cosmology. There the modified Friedmann equations are

$t^{-2}$3

$t^{-2}$4

with $t^{-2}$5 the fractional time variable, and an interaction term between matter and dark energy can be included through

$t^{-2}$6

In this framework, the scalar-field Lagrangians themselves are standard; the fractional structure enters only via the background equations and the modified matter scaling $t^{-2}$7 (Debnath et al., 2011).

The fractional Einstein–Hilbert approach modifies the gravitational sector while preserving ordinary matter conservation. In the spatially flat FRW case, the resulting equations are

$t^{-2}$8

$t^{-2}$9

while the continuity equation retains its usual form,

$\xi(t)$0

The same construction admits an “effective cosmological term” $\xi(t)$1, with

$\xi(t)$2

which is interpreted as kinematical induction by the Hubble parameter (Shchigolev, 2013).

A non-minimally coupled scalar field in fractional cosmology produces a further layer of structure. With a coupling $\xi(t)$3, the modified Friedmann equation becomes

$\xi(t)$4

and the Klein–Gordon equation becomes

$\xi(t)$5

The term $\xi(t)$6 is explicitly identified as a purely fractional “friction/memory” contribution (Marroquín et al., 2024).

These classical constructions agree on one point: the fractional measure does not simply perturb the standard equations by a small constant; it introduces a time-dependent deformation of the Hubble friction, continuity law, and effective gravitational coupling. That is the core mechanism by which fractional effects alter scalar-field cosmology.

3. Scalar sectors, reconstructed potentials, and exact solutions

A major difference from standard scalar cosmology is that, in several fractional formulations, the scalar potential is not arbitrary. “Fractional Scalar Field Cosmology” stresses that, unlike the corresponding standard models, one cannot use any arbitrary scalar potentials, because there are three independent fractional field equations for the three unknowns $\xi(t)$7. In that model, $\xi(t)$8 with $\xi(t)$9 is inconsistent, so the potential is determined by solving the fractional dynamics rather than postulated freely (Rasouli et al., 2024).

The same work derives an exact Riccati-type equation for the Hubble parameter,

$\Pi_\phi$0

with solution

$\Pi_\phi$1

From this one reconstructs

$\Pi_\phi$2

$\Pi_\phi$3

$\Pi_\phi$4

This gives a parametric family $\Pi_\phi$5 for each $\Pi_\phi$6 and integration constant $\Pi_\phi$7 (Rasouli et al., 2024).

Fractional Action Cosmology takes a different reconstruction strategy. Given a prescribed expansion history—emergent, logamediate, or intermediate—the modified Friedmann equations are used to solve algebraically for $\Pi_\phi$8 and $\Pi_\phi$9 for standard dark-energy fields: quintessence, phantom, tachyon, k-essence, DBI-essence, hessence, dilaton, and Yang–Mills condensate (Debnath et al., 2011). The resulting qualitative behaviors depend strongly on both the background scenario and the field model. For example, in the emergent scenario, the quintessence/phantom potential initially decays with $\beta=\frac{2\alpha}{2\alpha-1}=1+\omega_X$0 and then starts increasing, while k-essence and DBI potentials decrease with $\beta=\frac{2\alpha}{2\alpha-1}=1+\omega_X$1; in the logamediate scenario, the DBI potential increases with $\beta=\frac{2\alpha}{2\alpha-1}=1+\omega_X$2, whereas k-essence decreases; in the intermediate scenario, the phantom tachyon potential depends sensitively on the spatial curvature (Debnath et al., 2011).

In the $\beta=\frac{2\alpha}{2\alpha-1}=1+\omega_X$3-dimensional kernel-weighted Sáez–Ballester model, the scalar sector is similarly generated rather than assumed. The modified Friedmann and Klein–Gordon equations are

$\beta=\frac{2\alpha}{2\alpha-1}=1+\omega_X$4

$\beta=\frac{2\alpha}{2\alpha-1}=1+\omega_X$5

together with a modified continuity equation

$\beta=\frac{2\alpha}{2\alpha-1}=1+\omega_X$6

In that framework there are three independent equations for the three unknowns $\beta=\frac{2\alpha}{2\alpha-1}=1+\omega_X$7, so the potential can be derived rather than specified ad hoc (Rasouli et al., 12 Dec 2025).

The same principle reappears in recent power-law inflation. In “Power-Law Inflation in $\beta=\frac{2\alpha}{2\alpha-1}=1+\omega_X$8-Dimensional Fractional Scalar Field Cosmology” the exact power-law ansatz

$\beta=\frac{2\alpha}{2\alpha-1}=1+\omega_X$9

is consistent only when the fractional parameter satisfies

$(t-\tau)^{\xi-1}$0

The scalar potential then follows self-consistently as an exponential rather than being chosen independently (Oliveira et al., 17 Feb 2026).

A further example is fractional Einstein–Gauss–Bonnet scalar cosmology. There the logarithmic ansatz

$(t-\tau)^{\xi-1}$1

and the scaling ansätze

$(t-\tau)^{\xi-1}$2

lead to $(t-\tau)^{\xi-1}$3, $(t-\tau)^{\xi-1}$4, an exponential potential,

$(t-\tau)^{\xi-1}$5

and an exponential Gauss–Bonnet coupling

$(t-\tau)^{\xi-1}$6

Here again the scalar potential and coupling are reconstructed from the fractional dynamics rather than selected by hand (Micolta-Riascos et al., 2024).

4. K-essence, minisuperspace quantization, and emergent fractional Wheeler–DeWitt equations

The most explicit realization of fractional scalar-field cosmology at the quantum level appears in K-essence minisuperspace models. In the flat FLRW case with $(t-\tau)^{\xi-1}$7, one begins from the reduced Lagrangian

$(t-\tau)^{\xi-1}$8

which follows from the K-essence choice $(t-\tau)^{\xi-1}$9 (Socorro et al., 2023). The scalar behaves as an effective barotropic fluid with

$(t-\tau)^{\mu-1}$0

The Hamiltonian contains a non-integer power of the scalar momentum,

$(t-\tau)^{\mu-1}$1

and this is the source of the fractional quantum dynamics. Promoting

$(t-\tau)^{\mu-1}$2

the Wheeler–DeWitt equation acquires a fractional derivative in $(t-\tau)^{\mu-1}$3,

$(t-\tau)^{\mu-1}$4

with the scalar derivative treated in the Caputo sense in (Socorro et al., 2023) and in the Riemann–Liouville sense in the noncommutative extension (Socorro et al., 2024).

The order of the fractional derivative is directly tied to the equation of state: $(t-\tau)^{\mu-1}$5 Hence

$(t-\tau)^{\mu-1}$6

$(t-\tau)^{\mu-1}$7

Radiation gives $(t-\tau)^{\mu-1}$8, stiff matter gives $(t-\tau)^{\mu-1}$9, dust gives $$S_\mu=\frac{1}{\Gamma(\mu)}\int_0^t d\tau\,(t-\tau)^{\mu-1}L(\tau),$$0, and inflation-like regimes with $$S_\mu=\frac{1}{\Gamma(\mu)}\int_0^t d\tau\,(t-\tau)^{\mu-1}L(\tau),$$1 give $$S_\mu=\frac{1}{\Gamma(\mu)}\int_0^t d\tau\,(t-\tau)^{\mu-1}L(\tau),$$2 (Socorro et al., 2023). This is one of the clearest quantitative links between a cosmological epoch and a fractional operator in the literature.

Separation of variables,

$$S_\mu=\frac{1}{\Gamma(\mu)}\int_0^t d\tau\,(t-\tau)^{\mu-1}L(\tau),$$3

produces a fractional scalar equation of the form

$$S_\mu=\frac{1}{\Gamma(\mu)}\int_0^t d\tau\,(t-\tau)^{\mu-1}L(\tau),$$4

For $$S_\mu=\frac{1}{\Gamma(\mu)}\int_0^t d\tau\,(t-\tau)^{\mu-1}L(\tau),$$5 and $$S_\mu=\frac{1}{\Gamma(\mu)}\int_0^t d\tau\,(t-\tau)^{\mu-1}L(\tau),$$6, the solutions are Mittag–Leffler functions,

$$S_\mu=\frac{1}{\Gamma(\mu)}\int_0^t d\tau\,(t-\tau)^{\mu-1}L(\tau),$$7

and reduce to ordinary trigonometric functions at $$S_\mu=\frac{1}{\Gamma(\mu)}\int_0^t d\tau\,(t-\tau)^{\mu-1}L(\tau),$$8, since $$S_\mu=\frac{1}{\Gamma(\mu)}\int_0^t d\tau\,(t-\tau)^{\mu-1}L(\tau),$$9 (Socorro et al., 2023).

The anisotropic Bianchi I generalization preserves the same relation

$0<\mu\le 1$0

and again yields a fractional differential equation for the scalar field in the Wheeler–DeWitt problem. There the factor-ordering problem is introduced in the variables $0<\mu\le 1$1 and their corresponding momenta, and the scalar equation is solved using a fractional power series expansion (Socorro et al., 2023).

The noncommutative extension deforms the minisuperspace algebra by

$0<\mu\le 1$2

implemented through a Bopp shift

$0<\mu\le 1$3

In that case the scalar equation becomes a complex fractional differential equation,

$0<\mu\le 1$4

while the gravitational part remains essentially unchanged. The resulting probability density develops new structure in the scalar-field direction and shifts “back in the direction of the scale factor,” which the authors interpret as causing classical evolution to arise earlier than in the commutative case (Socorro et al., 2024).

These K-essence models are distinctive because the fractional operator is not postulated in the action. It emerges from quantizing a noncanonical kinetic theory whose Hamiltonian already contains non-integer momentum powers.

5. Dynamical systems, exact cosmological histories, and stability

Dynamical systems methods have become central in determining whether fractional scalar-field cosmologies possess viable late-time or inflationary attractors. In “Revisiting Fractional Cosmology” the flat FLRW background with a minimally coupled scalar field and exponential potential

$0<\mu\le 1$5

is rewritten in terms of

$0<\mu\le 1$6

leading to an autonomous system in the $0<\mu\le 1$7 subspace (Micolta-Riascos et al., 2023). The late-time critical points depend strongly on the fractional order $0<\mu\le 1$8, and the canonical-scalar points $0<\mu\le 1$9 and $t^{-1}$00 are numerically shown to be late-time attractors for physically relevant parameter ranges, while in the phantom case $t^{-1}$01 and $t^{-1}$02 are always saddles (Micolta-Riascos et al., 2023).

The same paper extends the analysis to Bianchi I. There the anisotropy variable $t^{-1}$03 decouples from the $t^{-1}$04 subsystem, and all late-time attractors are isotropic, since the stable points satisfy $t^{-1}$05. This suggests that the fractional scalar-field dynamics isotropizes anisotropic cosmologies at late times (Micolta-Riascos et al., 2023).

In the $t^{-1}$06-dimensional kernel-weighted Sáez–Ballester model, exact analytical solutions are obtained for arbitrary $t^{-1}$07 and $t^{-1}$08. Combining the modified equations yields

$t^{-1}$09

with exact solution

$t^{-1}$10

The corresponding scale factor, effective density, and effective equation of state interpolate between an early inflation-like era, transient radiation-like and matter-like phases, and a late-time accelerated phase. The same work derives the first-order perturbation equations, including a fractional Mukhanov–Sasaki equation with $t^{-1}$11, and emphasizes that no new propagating degrees of freedom appear purely due to the kernel (Rasouli et al., 12 Dec 2025).

Recent power-law inflation sharpens the dynamical picture. In the $t^{-1}$12-dimensional fractional scalar-field model, the autonomous variables

$t^{-1}$13

satisfy the constraint

$t^{-1}$14

The exact power-law inflationary solution corresponds to a critical point with

$t^{-1}$15

and linearization of the reduced system yields a physical Jacobian $t^{-1}$16. Stability requires $t^{-1}$17 and $t^{-1}$18, which are satisfied in the observationally viable region. The paper explicitly states that the fractional power-law solutions form stable inflationary attractors over the viable parameter range (Oliveira et al., 17 Feb 2026).

Fractional Einstein–Gauss–Bonnet scalar cosmology reaches a similar conclusion by perturbing around the exact scaling solution through

$t^{-1}$19

leading to a linear dynamical system with eigenvalues

$t^{-1}$20

The best-fit parameters lie in a sink region, so the scaling solution is dynamically stable (Micolta-Riascos et al., 2024).

This shared emphasis on attractors matters conceptually. In standard scalar cosmology, many solutions exist but only a subset are dynamically selected. The fractional literature increasingly shows that the physically relevant solutions are not merely exact, but stable under perturbations in the associated phase space.

6. Observational status, phenomenology, and open issues

The observational situation differs sharply across formulations. Some works remain formal or semiclassical, while others have begun to confront data directly.

In the minimal-coupling fractional cosmology analyzed in “Conformal and Non-Minimal Couplings in Fractional Cosmology,” the Hubble function from the exact fractional solution is confronted with Type Ia supernovae, cosmic chronometers, and the combined SNe Ia + OHD dataset. The best-fit joint values are

$t^{-1}$21

with a current deceleration parameter $t^{-1}$22 and a current fractional dark-energy density parameter $t^{-1}$23 (Marroquín et al., 2024). The paper notes that the non-minimal scalar-field case has not yet been systematically fitted to data.

The most detailed late-time data analysis in the present corpus is the fractional Einstein–Gauss–Bonnet scalar model. Using cosmic chronometers, type Ia supernovae, strong gravitational lensing, and supermassive black hole shadows, the estimated values are

$t^{-1}$24

consistent with

$t^{-1}$25

and implying an age of the Universe

$t^{-1}$26

at $t^{-1}$27 CL (Micolta-Riascos et al., 2024). In that model the scalar field plus fractional and Gauss–Bonnet contributions are interpreted as mimicking the dark sector, with the scalar behaving as effective dark matter and the geometric corrections as effective dark energy (Micolta-Riascos et al., 2024).

Inflationary phenomenology has also advanced rapidly. The $t^{-1}$28-dimensional fractional power-law inflation model argues that introducing a fractional order $t^{-1}$29 suppresses the tensor-to-scalar ratio $t^{-1}$30 while keeping $t^{-1}$31 essentially unchanged. For observationally favored values $t^{-1}$32-$t^{-1}$33 in four dimensions, the model gives

$t^{-1}$34

which is presented as bringing power-law inflation into agreement with current CMB data (Oliveira et al., 17 Feb 2026). The same paper emphasizes that the model yields clear targets for forthcoming CMB polarization measurements.

A different observationally oriented line reconstructs Fractional Holographic Dark Energy through quintessence, K-essence, dilaton, Yang–Mills condensate, DBI-essence, and tachyonic fields. For the fractional parameter range $t^{-1}$35, all effective field configurations asymptotically approach $t^{-1}$36CDM behavior in the far-future limit $t^{-1}$37, and the EoS avoids entering the phantom divide $t^{-1}$38, which the authors identify as a common issue in standard scalar field models without fractional dynamics, such as K-essence (Bidlan et al., 1 Feb 2025).

Several open issues recur across the literature. First, many classical formulations are minisuperspace theories with a fractional time measure, not fully covariant fractional gravity theories. This leaves the fundamental origin of the fractional parameter $t^{-1}$39, $t^{-1}$40, or $t^{-1}$41 unsettled (Rasouli et al., 2024). Second, perturbation theory is only partially developed: the $t^{-1}$42-dimensional kernel-weighted model constructs first-order perturbations and a fractional Mukhanov–Sasaki equation (Rasouli et al., 12 Dec 2025), and the inflation paper derives perturbative observables (Oliveira et al., 17 Feb 2026), but most late-time scalar-field models remain at the background level. Third, reheating, graceful exit, and full likelihood analyses are still sparse in the inflationary sector (Oliveira et al., 17 Feb 2026). Fourth, the relation between different kinds of fractionality—fractional action measures, fractional Wheeler–DeWitt operators, and fractional holographic densities—remains conceptually open (Rasouli et al., 2024).

A plausible implication is that the field has now split into two complementary programs. One treats fractionality as an effective classical deformation of the cosmological action, yielding modified expansion laws, reconstructed potentials, and attractor solutions. The other treats fractionality as an emergent quantum property of minisuperspace, especially in K-essence cosmology, where the scalar field satisfies a fractional Wheeler–DeWitt equation. Both have established nontrivial exact solutions; only the first has begun to accumulate direct observational constraints.