Fractional Power-Law Inflation Models

• Fractional power-law inflation employs non-integer exponents in the inflaton potential, kinetic terms, or curvature actions, maintaining analytic tractability while altering ultraviolet physics.
• Different frameworks—including strongly coupled SUSY theories, non-canonical kinetic models, and fractional calculus approaches—dynamically generate these fractional power laws with distinct inflationary observables.
• The models yield unique scalar and tensor predictions, offering solutions for issues like the graceful exit, and inform potential signatures such as primordial black holes and gravitational-wave enhancements.

Fractional power-law inflation denotes a family of inflationary constructions in which a non-integer power law is central to the dynamics. In the most direct realization, the inflaton potential is a monomial $V(\phi)=\Lambda^4(\phi/\Lambda)^p$ with rational or fractional exponent $p$, but the literature also uses closely related language for non-canonical models with fractional power-law kinetic terms, for fractional-derivative or non-local cosmologies that yield power-law expansion, and for modified-gravity actions with fractional curvature exponents. Across these realizations, the shared objective is to retain the analytic tractability of power-law inflation while altering the ultraviolet origin of the inflaton sector, the tensor amplitude, or the end of inflation [(Harigaya et al., 2012); (Unnikrishnan et al., 2013); (Harigaya et al., 2014); (Oliveira et al., 17 Feb 2026)].

1. Conceptual scope and taxonomy

The phrase “fractional power-law inflation” is not attached to a single formalism. In one strand, the fractional quantity is the exponent of the inflaton potential; in another, it is the exponent of the kinetic term; in another, it is the order of a fractional integral or derivative; and in modified gravity it can refer to a non-integer curvature power. These usages are related by their emphasis on exact or approximate power-law behavior and on non-standard scaling relations for observables.

Framework	Defining structure	Representative result
Strongly coupled SUSY gauge dynamics	$V(\phi)=\Lambda^4(\phi/\Lambda)^p$	$p=2/(N+1)$ in an $SP(N)$ construction (Harigaya et al., 2012)
Dynamical chaotic inflation	$$V(\phi)\simeq \Lambda^4(\lambda_{\rm eff}|\phi|/\Lambda)^p$$	$p=4(b_{\rm eff}-b)/b_{\rm eff}$ from decoupling and dimensional transmutation (Harigaya et al., 2014)
Running kinetic term in supergravity	Effective $V(\varphi)\propto \varphi^{2m/n}$	$p=2m/n$ after canonical normalization (Takahashi, 2010)
Non-canonical power-law inflation	$\mathcal{L}(X,\phi)=X(X/M^4)^{\alpha-1}-V(\phi)$	Exact $p$0 with $p$1, $p$2 (Unnikrishnan et al., 2013)
Fractional scalar-field cosmology	Memory correction $p$3 in Friedmann and Klein–Gordon equations	$p$4 with suppressed $p$5 through $p$6 (Oliveira et al., 17 Feb 2026)
Fractional quantum cosmology	Riesz fractional Wheeler–DeWitt operator	$p$7 (Rasouli et al., 2022)
Fractional $p$8 or curvature-coupled models	$p$9 or $V(\phi)=\Lambda^4(\phi/\Lambda)^p$0	Viable departures from $V(\phi)=\Lambda^4(\phi/\Lambda)^p$1 inflation are allowed (Odintsov et al., 8 Sep 2025, Saini et al., 2023)

In this taxonomy, the monomial-potential class is the closest descendant of chaotic inflation, whereas the non-canonical, fractional-calculus, and modified-gravity classes alter the dynamical equations more radically. This suggests that the subject is best understood as a research domain defined by structure rather than by a unique Lagrangian.

2. Dynamical generation of fractional monomial potentials

A central development was the demonstration that a fractional monomial inflaton potential can be generated dynamically in strongly coupled supersymmetric gauge theory. In an $V(\phi)=\Lambda^4(\phi/\Lambda)^p$2 gauge theory with $V(\phi)=\Lambda^4(\phi/\Lambda)^p$3 fundamentals $V(\phi)=\Lambda^4(\phi/\Lambda)^p$4 and antisymmetric gauge-singlet chiral superfields $V(\phi)=\Lambda^4(\phi/\Lambda)^p$5, the tree-level superpotential is

$V(\phi)=\Lambda^4(\phi/\Lambda)^p$6

Below the Landau pole, the gauge coupling becomes strong, the theory enters an $V(\phi)=\Lambda^4(\phi/\Lambda)^p$7-confining phase at the scale $V(\phi)=\Lambda^4(\phi/\Lambda)^p$8, and dimensional transmutation generates

$V(\phi)=\Lambda^4(\phi/\Lambda)^p$9

After integrating out heavy composites and solving the quantum-deformed constraint, one obtains

$p=2/(N+1)$0

with $p=2/(N+1)$1 identified with the imaginary part of a gauge-singlet field $p=2/(N+1)$2. In this construction, both the fractional exponent and the inflationary scale arise from strong-coupling effects and the quantum-deformed moduli space; fitting the scalar amplitude gives $p=2/(N+1)$3, naturally below $p=2/(N+1)$4 (Harigaya et al., 2012).

The broader program of dynamical chaotic inflation extends this mechanism beyond the original $p=2/(N+1)$5 model. The general recipe is to start from a SUSY $p=2/(N+1)$6-confining theory, couple $p=2/(N+1)$7 quark pairs to the inflaton through

$p=2/(N+1)$8

and use the fact that for $p=2/(N+1)$9 these flavors become heavy and decouple. Matching the gauge coupling across thresholds gives a field-dependent dynamical scale,

$SP(N)$0

and in the SUSY-breaking vacuum this implies

$SP(N)$1

This framework realizes a wide range of rational exponents, including $SP(N)$2 in $SP(N)$3 models, $SP(N)$4 in an $SP(N)$5 chiral DSB model, $SP(N)$6 in an $SP(N)$7 model, $SP(N)$8 in the $SP(N)$9 “3–2” model, and $$V(\phi)\simeq \Lambda^4(\lambda_{\rm eff}|\phi|/\Lambda)^p$$0 in $$V(\phi)\simeq \Lambda^4(\lambda_{\rm eff}|\phi|/\Lambda)^p$$1 with $$V(\phi)\simeq \Lambda^4(\lambda_{\rm eff}|\phi|/\Lambda)^p$$2 (Harigaya et al., 2014).

A distinct supergravity mechanism achieves the same effective structure through a running kinetic term. Imposing a shift symmetry on the composite field $$V(\phi)\simeq \Lambda^4(\lambda_{\rm eff}|\phi|/\Lambda)^p$$3, taking the superpotential

$$V(\phi)\simeq \Lambda^4(\lambda_{\rm eff}|\phi|/\Lambda)^p$$4

and using the large-field form of the Kähler metric,

$$V(\phi)\simeq \Lambda^4(\lambda_{\rm eff}|\phi|/\Lambda)^p$$5

one canonically normalizes the inflaton by $$V(\phi)\simeq \Lambda^4(\lambda_{\rm eff}|\phi|/\Lambda)^p$$6. The original monomial $$V(\phi)\simeq \Lambda^4(\lambda_{\rm eff}|\phi|/\Lambda)^p$$7 is then converted into an effective fractional power-law potential,

$$V(\phi)\simeq \Lambda^4(\lambda_{\rm eff}|\phi|/\Lambda)^p$$8

After inflation the kinetic term becomes canonical in a different variable and the potential steepens from $$V(\phi)\simeq \Lambda^4(\lambda_{\rm eff}|\phi|/\Lambda)^p$$9 during slow roll to $p=4(b_{\rm eff}-b)/b_{\rm eff}$0 in the oscillatory phase (Takahashi, 2010).

3. Background dynamics and inflationary observables

For monomial large-field models with

$p=4(b_{\rm eff}-b)/b_{\rm eff}$1

the standard slow-roll expressions give

$p=4(b_{\rm eff}-b)/b_{\rm eff}$2

with $p=4(b_{\rm eff}-b)/b_{\rm eff}$3. In the strongly coupled $p=4(b_{\rm eff}-b)/b_{\rm eff}$4 realization this becomes

$p=4(b_{\rm eff}-b)/b_{\rm eff}$5

for $p=4(b_{\rm eff}-b)/b_{\rm eff}$6, while $p=4(b_{\rm eff}-b)/b_{\rm eff}$7 for $p=4(b_{\rm eff}-b)/b_{\rm eff}$8 (Harigaya et al., 2012). In the more general dynamical chaotic inflation analysis, $p=4(b_{\rm eff}-b)/b_{\rm eff}$9 yields $V(\varphi)\propto \varphi^{2m/n}$0 and $V(\varphi)\propto \varphi^{2m/n}$1, while the scalar amplitude fixes $V(\varphi)\propto \varphi^{2m/n}$2, $V(\varphi)\propto \varphi^{2m/n}$3, and $V(\varphi)\propto \varphi^{2m/n}$4 (Harigaya et al., 2014). The same observables arise in the supergravity running-kinetic construction, where $V(\varphi)\propto \varphi^{2m/n}$5 and, for $V(\varphi)\propto \varphi^{2m/n}$6, examples include $V(\varphi)\propto \varphi^{2m/n}$7 with $V(\varphi)\propto \varphi^{2m/n}$8 and $V(\varphi)\propto \varphi^{2m/n}$9 with $p=2m/n$0 (Takahashi, 2010).

A different use of fractional power laws appears in non-canonical power-law inflation. There the action is

$p=2m/n$1

with $p=2m/n$2, and exact power-law expansion $p=2m/n$3 is obtained for the inverse power-law potential

$p=2m/n$4

The sound speed is constant,

$p=2m/n$5

and exact power-law spectra give

$p=2m/n$6

The equilateral bispectrum amplitude is

$p=2m/n$7

Using the Planck $p=2m/n$8 WP $p=2m/n$9 BAO bounds

$\mathcal{L}(X,\phi)=X(X/M^4)^{\alpha-1}-V(\phi)$0

the canonical case $\mathcal{L}(X,\phi)=X(X/M^4)^{\alpha-1}-V(\phi)$1 is excluded because it predicts $\mathcal{L}(X,\phi)=X(X/M^4)^{\alpha-1}-V(\phi)$2 when $\mathcal{L}(X,\phi)=X(X/M^4)^{\alpha-1}-V(\phi)$3, whereas $\mathcal{L}(X,\phi)=X(X/M^4)^{\alpha-1}-V(\phi)$4 satisfies all bounds. A representative choice $\mathcal{L}(X,\phi)=X(X/M^4)^{\alpha-1}-V(\phi)$5, $\mathcal{L}(X,\phi)=X(X/M^4)^{\alpha-1}-V(\phi)$6 gives $\mathcal{L}(X,\phi)=X(X/M^4)^{\alpha-1}-V(\phi)$7, $\mathcal{L}(X,\phi)=X(X/M^4)^{\alpha-1}-V(\phi)$8, and $\mathcal{L}(X,\phi)=X(X/M^4)^{\alpha-1}-V(\phi)$9 (Unnikrishnan et al., 2013).

That model also addresses the graceful-exit problem of exact power-law expansion by replacing the pure inverse power with the two-branch potential

$p$00

For $p$01 one recovers the PLI branch, while for $p$02 the potential behaves as $p$03. In the benchmark $p$04, $p$05, inflation ends at $p$06, sixty e-folds earlier the field is at $p$07, and the second branch yields $p$08, $p$09 for $p$10 (Unnikrishnan et al., 2013).

4. Fractional calculus, memory effects, and quantum cosmology

Fractional scalar-field cosmology modifies the gravitational dynamics directly by replacing the local action with a fractional one. In the Riemann–Liouville formulation, the non-local memory is encoded in

$p$11

and the corresponding derivative introduces a correction proportional to $p$12. Varying the fractional action yields, in $p$13 dimensions and for $p$14,

$p$15

$p$16

$p$17

For the ansatz

$p$18

the Hubble slow-roll parameters remain

$p$19

so the leading scalar tilt stays

$p$20

while the tensor-to-scalar ratio is modified to

$p$21

with $p$22 as $p$23 and $p$24 for $p$25. The potential reconstructed from the background is exponential,

$p$26

In four dimensions, observationally favored values $p$27 give $p$28 and $p$29, and the dynamical-systems analysis shows that the fractional power-law solutions are stable inflationary attractors over the viable parameter range (Oliveira et al., 17 Feb 2026).

Fractional quantum cosmology implements non-locality earlier, at the minisuperspace level, by replacing the standard Wheeler–DeWitt Laplacian with a one-dimensional Riesz fractional operator of order $p$30. In a flat, compact FLRW minisuperspace with constant potential $p$31, the standard equation

$p$32

becomes

$p$33

A WKB treatment yields a fractional Hamilton–Jacobi constraint and a power-law solution

$p$34

Defining an effective fractal dimension

$p$35

one may rewrite the result as

$p$36

In the standard integer-order limit one instead obtains de Sitter expansion $p$37. The exponent $p$38 in the fractional theory depends only on the fractal dimension $p$39, not on the inflationary energy scale $p$40. Inflation requires $p$41, equivalently $p$42, and obtaining roughly $p$43 e-folds over a Planck-time interval leads to $p$44 or $p$45 (Rasouli et al., 2022).

5. Modified gravity and curvature-coupled extensions

In Jordan-frame $p$46 gravity, fractional power-law inflation refers to models in which the curvature correction carries a non-integer exponent,

$p$47

The field equations can be written entirely in the Jordan frame, and the slow-roll quantities are expressed through the Hubble slow-roll parameters $p$48, $p$49, and $p$50, or equivalently through

$p$51

For $p$52 in the large-$p$53 regime,

$p$54

and the formalism yields analytic expressions for $p$55 and $p$56. A central claim of this framework is that “power-law” $p$57 inflation is not tied to “power-law” scale-factor evolution: the standard treatment that imposes $p$58 a priori is rejected in favor of slow roll near a de Sitter point. Within this approach, the model is viable for small negative $p$59: Planck gives $p$60, ACT gives $p$61, and the example $p$62 yields $p$63 and $p$64 (Odintsov et al., 8 Sep 2025).

A related but distinct generalization of Starobinsky inflation begins from

$p$65

After conformal transformation, the Einstein-frame scalar potential becomes

$p$66

For $p$67 this reduces to the usual Starobinsky plateau. Numerical parameter estimation using MODECODE, CAMB, and COSMOMC, with Planck 2018, BICEP3, BAO, and Pantheon likelihoods, gives the marginalized $p$68 C.L. constraints

$p$69

together with derived values

$p$70

This indicates that current observations allow a small deviation from Starobinsky inflation and that $p$71 remains allowed at $p$72 (Saini et al., 2023).

Fractional monomial potentials also reappear in scalar inflation coupled to the Gauss–Bonnet invariant. In that setting the potential is

$p$73

and the action includes the term $p$74. With a coupling

$p$75

the condition

$p$76

can trigger a transient ultra slow-roll regime. Numerically, this corresponds to $p$77 and $p$78, producing an enhancement of the curvature spectrum from the standard $p$79 level to a peak of order $p$80. At CMB scales the model can still match $p$81 and $p$82; explicit examples give $p$83 for $p$84, $p$85 for $p$86, and $p$87 for $p$88. The enhanced small-scale power produces primordial black holes in asteroid-mass, Earth-mass, and stellar-mass windows, together with induced gravitational-wave peaks $p$89 at $p$90 (Ashrafzadeh et al., 2023).

6. Phenomenology, misconceptions, and outstanding issues

A common phenomenological pattern is that fractional power laws alter the relation between scalar tilt and tensor amplitude. In monomial large-field models, decreasing the effective exponent $p$91 lowers $p$92 while keeping $p$93 within the observationally relevant range; in the non-canonical PLI model, the suppression instead comes through the reduced sound speed $p$94; and in fractional scalar-field cosmology the memory factor $p$95 reduces tensors while leaving $p$96 unchanged at leading order [(Harigaya et al., 2012); (Unnikrishnan et al., 2013); (Harigaya et al., 2014); (Oliveira et al., 17 Feb 2026)].

A frequent conceptual misunderstanding is to treat all of these constructions as variants of one model. The literature instead separates at least four mechanisms: fractional monomial potentials generated by strong dynamics or running kinetic terms; exact PLI from a fractional kinetic Lagrangian; non-local or fractional-derivative cosmologies; and modified-gravity models with fractional curvature exponents. A second misconception concerns $p$97 gravity: according to the Jordan-frame analysis, “power-law” $p$98 inflation need not imply a power-law scale factor, because inflation may emerge as a quasi-de Sitter slow-roll phase rather than from an imposed $p$99 ansatz (Odintsov et al., 8 Sep 2025).

Several technical issues remain model-dependent. Supergravity embeddings rely on an approximate shift symmetry to control the $V(\phi)=\Lambda^4(\phi/\Lambda)^p$00-problem; in dynamical chaotic inflation, loop-induced Kähler corrections motivate $V(\phi)=\Lambda^4(\phi/\Lambda)^p$01, and the ultraviolet origin of the shift symmetry is deferred to a more fundamental theory such as string axions. Exact power-law inflation requires a graceful-exit mechanism, which in the non-canonical model is supplied by a two-branch potential. Fractional scalar-field cosmology identifies reheating and graceful exit as requiring either a dynamical $V(\phi)=\Lambda^4(\phi/\Lambda)^p$02 returning to unity after inflation or a deformation of the exponential potential. In Gauss–Bonnet models, viability depends on keeping $V(\phi)=\Lambda^4(\phi/\Lambda)^p$03, $V(\phi)=\Lambda^4(\phi/\Lambda)^p$04, $V(\phi)=\Lambda^4(\phi/\Lambda)^p$05, and $V(\phi)=\Lambda^4(\phi/\Lambda)^p$06 positive throughout the background evolution [(Harigaya et al., 2014); (Unnikrishnan et al., 2013); (Oliveira et al., 17 Feb 2026); (Ashrafzadeh et al., 2023)].

Taken together, these results show that fractional power-law inflation is not a single hypothesis but a structured set of inflationary strategies. In some versions the fractional exponent is a calculable output of strong gauge dynamics; in others it is a consequence of non-canonical kinetic structure, of non-local memory terms, or of curvature-sector deformations. The topic therefore occupies an intersection of inflationary phenomenology, supersymmetric model building, non-local cosmology, and modified gravity, with each branch retaining the central power-law motif while redefining its dynamical origin [(Takahashi, 2010); (Harigaya et al., 2012); (Harigaya et al., 2014); (Saini et al., 2023)].