--- title: Fractional Power-Law Inflation Models url: https://www.emergentmind.com/topics/fractional-power-law-inflation type: topic --- # Fractional Power-Law Inflation Models 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 [1211.6241; 1305.5260; 1407.3084; 2602.15991]. ## 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 [1211.6241] | | 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 [1407.3084] | | Running kinetic term in supergravity | Effective \(V(\varphi)\propto \varphi^{2m/n}\) | \(p=2m/n\) after canonical normalization [1006.2801] | | Non-canonical power-law inflation | \(\mathcal{L}(X,\phi)=X(X/M^4)^{\alpha-1}-V(\phi)\) | Exact \(a(t)\propto t^q\) with \(V(\phi)\propto \phi^{-s}\), \(s=2\alpha/(\alpha-1)\) [1305.5260] | | Fractional scalar-field cosmology | Memory correction \((1-\alpha)/t\) in Friedmann and Klein–Gordon equations | \(n_s=1-2/m\) with suppressed \(r\) through \(\Xi(\alpha;n,m)\) [2602.15991] | | Fractional quantum cosmology | Riesz fractional Wheeler–DeWitt operator | \(a(t)\propto t^{2/(D-2)}\) [2210.00909] | | Fractional \(F(R)\) or curvature-coupled models | \(F(R)=R+\beta R^n\) or \(R+R^\beta/(6M^2)\) | Viable departures from \(R^2\) inflation are allowed [2509.06251; 2305.00682] | 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 \(SP(N)\) gauge theory with \(2(N+2)\) fundamentals \(Q^I\) and antisymmetric gauge-singlet chiral superfields \(\mathcal{Z}_{IJ}\), the tree-level superpotential is \[ W_{\rm tree}=\frac12\,\lambda_{IJ}\,\mathcal{Z}_{IJ}\,Q^I Q^J . \] Below the Landau pole, the gauge coupling becomes strong, the theory enters an \(s\)-confining phase at the scale \(\Lambda\), and dimensional transmutation generates \[ \Lambda \sim M_{\rm Pl}\,\exp\!\Bigl[-\frac{8\pi^2}{b\,g^2(M_{\rm Pl})}\Bigr]. \] After integrating out heavy composites and solving the quantum-deformed constraint, one obtains \[ V(\phi)=\Lambda^4\Bigl(\frac{\phi}{\Lambda}\Bigr)^p, \qquad p=\frac{2}{N+1}, \] with \(\phi\) identified with the imaginary part of a gauge-singlet field \(S\). 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 \(\Lambda\sim 10^{15}\,\mathrm{GeV}\), naturally below \(M_{\rm Pl}\) [1211.6241]. The broader program of dynamical chaotic inflation extends this mechanism beyond the original \(SP(N)\) model. The general recipe is to start from a SUSY \(s\)-confining theory, couple \(N_d\) quark pairs to the inflaton through \[ W \supset \sum_{i=1}^{N_d}\lambda_i \phi Q_i \bar Q_i, \] and use the fact that for \(|\phi|\gg \Lambda/\lambda\) these flavors become heavy and decouple. Matching the gauge coupling across thresholds gives a field-dependent dynamical scale, \[ \Lambda_{\rm eff}(\phi)=\Lambda\prod_{i=1}^{N_d}\Bigl(\frac{\lambda_i\phi}{\Lambda}\Bigr)^{(b_{\rm eff}-b)/b_{\rm eff}}, \] and in the SUSY-breaking vacuum this implies \[ V(\phi)\simeq \Lambda^4\Bigl(\frac{\lambda_{\rm eff}|\phi|}{\Lambda}\Bigr)^p, \qquad p=\frac{4(b_{\rm eff}-b)}{b_{\rm eff}}. \] This framework realizes a wide range of rational exponents, including \(p=2/(N_c+1)\) in \(SP(N_c)\) models, \(p=14/11\) in an \(SO(10)\) chiral DSB model, \(p=16/13\) in an \(SU(5)\) model, \(p=8/7\) in the \(SU(3)\times SU(2)\) “3–2” model, and \(p=1\) in \(SU(N_c)\) with \(N_f=N_c+1\) [1407.3084]. A distinct supergravity mechanism achieves the same effective structure through a running kinetic term. Imposing a shift symmetry on the composite field \(\phi^n\), taking the superpotential \[ W=\lambda X\phi^m, \] and using the large-field form of the Kähler metric, \[ K_{\phi\phi^\dagger}\simeq n^2|\phi|^{2n-2}, \] one canonically normalizes the inflaton by \(\varphi\simeq \phi^n/n\). The original monomial \(V(\phi)\propto |\phi|^{2m}\) is then converted into an effective fractional power-law potential, \[ V(\varphi)\propto \varphi^p, \qquad p=\frac{2m}{n}. \] After inflation the kinetic term becomes canonical in a different variable and the potential steepens from \(p=2m/n\) during slow roll to \(p_{\rm low}=2m\) in the oscillatory phase [1006.2801]. ## 3. Background dynamics and inflationary observables For monomial large-field models with \[ V(\phi)=V_0\Bigl(\frac{\phi}{\Lambda}\Bigr)^p, \] the standard slow-roll expressions give \[ n_s=1-\frac{p+2}{2N_e}, \qquad r=\frac{4p}{N_e}, \] with \(\phi_N\simeq \sqrt{2pN_e}\,M_{\rm Pl}\). In the strongly coupled \(SP(N)\) realization this becomes \[ p=\frac{2}{N+1}, \qquad r\simeq \frac{0.16}{N+1} \] for \(N_e\simeq 50\), while \(n_s\simeq 0.97\text{--}0.98\) for \(01\), and exact power-law expansion \(a(t)\propto t^q\) is obtained for the inverse power-law potential \[ V(\phi)=V_0\Bigl(\frac{\phi}{p}\Bigr)^{-s}, \qquad s=\frac{2\alpha}{\alpha-1}. \] The sound speed is constant, \[ c_s^2=\frac{1}{2\alpha-1}, \] and exact power-law spectra give \[ n_s-1=-\frac{2}{q-1}, \qquad n_T=-\frac{2}{q-1}, \qquad r\simeq \frac{16}{q\sqrt{2\alpha-1}}. \] The equilateral bispectrum amplitude is \[ f_{\rm NL}^{\rm equil}\simeq -0.57(\alpha-1). \] Using the Planck \(+\) WP \(+\) BAO bounds \[ n_s=0.9603\pm0.0073,\qquad r<0.11,\qquad f_{\rm NL}^{\rm equil}=-42\pm75, \] the canonical case \(\alpha=1\) is excluded because it predicts \(r=16/q\gtrsim 0.16\) when \(n_s\simeq 0.96\), whereas \(\alpha\gtrsim 2\) satisfies all bounds. A representative choice \(\alpha=6\), \(q\simeq 50\) gives \(n_s\simeq 0.96\), \(r\simeq 0.096\), and \(f_{\rm NL}^{\rm equil}\simeq -2.8\) [1305.5260]. That model also addresses the graceful-exit problem of exact power-law expansion by replacing the pure inverse power with the two-branch potential \[ V(\phi)=V_0\Bigl[\Bigl(\frac{\phi}{p}\Bigr)^{-s/2}-\Bigl(\frac{\phi}{p}\Bigr)^{s/2}\Bigr]^2. \] For \(\phip\) the potential behaves as \(\phi^{+s}\). In the benchmark \(\alpha=6\), \(q=50\), inflation ends at \(\phi_e\simeq 0.91\,p\), sixty e-folds earlier the field is at \(\phi_{60}\simeq 0.24\,p\), and the second branch yields \(n_s\simeq 0.965\), \(r\simeq 0.04\) for \(N=60\) [1305.5260]. ## 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 \[ I_0^\alpha[f](t)=\frac1{\Gamma(\alpha)}\int_0^t(t-\tau)^{\alpha-1}f(\tau)\,d\tau, \qquad 0<\alpha\le 1, \] and the corresponding derivative introduces a correction proportional to \((1-\alpha)/t\). Varying the fractional action yields, in \(n\) dimensions and for \(K=0\), \[ H^2+\frac{1-\alpha}{t}H =\frac{16\pi G}{n(n-1)}\Bigl(\frac12\dot\phi^2+V(\phi)\Bigr), \] \[ \dot H+H^2+\frac{1-\alpha}{t}H =-\frac{8\pi G}{n(n-1)}\bigl(\dot\phi^2-V(\phi)\bigr), \] \[ \ddot\phi+\Bigl(nH+\frac{1-\alpha}{t}\Bigr)\dot\phi+\frac{dV}{d\phi}=0. \] For the ansatz \[ a(t)=a_0\Bigl(\frac{t}{t_0}\Bigr)^m, \qquad \phi(t)=\phi_0+\beta\ln\Bigl(\frac{t}{t_0}\Bigr), \qquad m>1, \] the Hubble slow-roll parameters remain \[ \epsilon_H=\eta_H=\frac1m, \] so the leading scalar tilt stays \[ n_s=1-\frac{2}{m}, \] while the tensor-to-scalar ratio is modified to \[ r=\frac{16}{m}\,\Xi(\alpha;n,m), \] with \(\Xi(\alpha;n,m)\to 1\) as \(\alpha\to 1\) and \(\Xi<1\) for \(\alpha<1\). The potential reconstructed from the background is exponential, \[ V(\phi)=V_0\exp[-\lambda(\phi-\phi_0)]. \] In four dimensions, observationally favored values \(\alpha\simeq 0.8\text{--}0.9\) give \(n_s\simeq 0.965\) and \(r\lesssim 0.04\), and the dynamical-systems analysis shows that the fractional power-law solutions are stable inflationary attractors over the viable parameter range [2602.15991]. 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 \(1<\alpha\le 2\). In a flat, compact FLRW minisuperspace with constant potential \(V(\phi)=V_0\), the standard equation \[ \left(\frac{d^2}{da^2}+16\pi^2L^2 a^4\right)\Psi(a)=0 \] becomes \[ \Bigl[(-\hbar^2 d^2/da^2)^{\alpha/2}+16\pi^2L^2 a^4\Bigr]\Psi(a)=0. \] A WKB treatment yields a fractional Hamilton–Jacobi constraint and a power-law solution \[ a(t)\propto t^p, \qquad p=\frac1{2-4/\alpha}. \] Defining an effective fractal dimension \[ D=1+\frac{2}{\alpha}, \] one may rewrite the result as \[ a(t)\propto t^{2/(D-2)}. \] In the standard integer-order limit one instead obtains de Sitter expansion \(a(t)\propto \exp(t/L)\). The exponent \(p\) in the fractional theory depends only on the fractal dimension \(D\), not on the inflationary energy scale \(V_0\). Inflation requires \(2