Unification of polynomial and exponential cosmological attractors

Abstract: We introduce a family of simple $α$-attractor models that can interpolate between exponential and polynomial cosmological attractors. By varying the interpolation parameter $μ$ in these models, one can scan a wide range of values of the spectral index $n_{s}$ matching any combination of CMB and DESI data.

• The paper presents a unified framework that interpolates between polynomial and exponential attractors using a deformation parameter μ to bridge distinct inflationary regimes.
• It employs both analytical and numerical methods to derive precise predictions for the scalar spectral index (nₛ) and tensor-to-scalar ratio (r) across different model limits.
• The model robustly reconciles observational tensions by tuning μ, enabling a seamless fit to both CMB and DESI data through customizable inflationary dynamics.

Summary and Theoretical Context

The paper "Unification of polynomial and exponential cosmological attractors" (2607.07684) presents a unified framework of inflationary models that systematically interpolate between the well-studied classes of polynomial and exponential (specifically $\alpha$-) cosmological attractors. These models are pivotal in early universe cosmology as they underlie the dynamics of inflation and generate precise predictions for the scalar spectral index ($n_s$) and the tensor-to-scalar ratio ($r$)—the key observables for distinguishing among inflationary scenarios in light of the latest CMB and large-scale structure datasets.

Model Construction: Interpolating Potentials

The authors begin by formally defining two prominent families of inflationary attractors:

• Exponential $\alpha$-attractors: Characterized by plateau potentials which approach their asymptote exponentially as the inflaton field increases. The archetype takes the form $$V(\varphi)\approx V_0 - 2\sqrt{6\alpha}V'_0 e^{-\sqrt{2/3\alpha}\,\varphi}$$ at large field, leading to universal predictions $n_s = 1-2/N$ and $r = 12\alpha/N^2$. For $\alpha=1$, this coincides with both Starobinsky and Higgs inflation.
• Polynomial attractors: Potentials typically behave as $V(\phi)=V_0(1-C/\phi^k)$ or $V_k(\phi)=V_0\,\phi^k/(\phi^k+\mu^k)$. In well-controlled limits (small $n_s$0), these deliver $n_s$1 and $n_s$2 for the classes considered.

By introducing a deformation parameter $n_s$3 and embedding the polynomial potentials within the non-canonical field geometry of $n_s$4-attractors (disk or half-plane moduli spaces), the authors construct a family that transitions between the two regimes as $n_s$5 is varied.

Figure 1: Potential $n_s$6 with $n_s$7, $n_s$8, and $n_s$9, demonstrating smooth interpolation from polynomial to exponential plateau forms as $r$0 increases.

This interpolation is not trivial: for small $r$1 the inflationary trajectory and observables are governed by the polynomial regime, whereas at large $r$2 the standard $r$3-attractor behavior is restored. The point at which the transition occurs depends parametrically on $r$4, $r$5, $r$6, and the number of e-folds $r$7.

Analytical and Numerical Predictions

The analytic discourse is supplemented by explicit calculations for $r$8 and $r$9 in both limits, and by stating the mathematical criteria under which the last $\alpha$0 e-folds of inflation (necessary for observational phenomenology) occur in either the small- or large-field regime. The field excursion $\alpha$1 at the onset of the final $\alpha$2 e-foldings is shown to scale as $\alpha$3, guaranteeing that for small enough $\alpha$4, inflation is polynomial-dominated.

Crucially, the authors performed detailed numerical studies that validate the predicted bifurcation: for $\alpha$5, $\alpha$6, $\alpha$7:

• For $\alpha$8: $\alpha$9 (canonical $$V(\varphi)\approx V_0 - 2\sqrt{6\alpha}V'_0 e^{-\sqrt{2/3\alpha}\,\varphi}$$0-attractor prediction).
• For $$V(\varphi)\approx V_0 - 2\sqrt{6\alpha}V'_0 e^{-\sqrt{2/3\alpha}\,\varphi}$$1: $$V(\varphi)\approx V_0 - 2\sqrt{6\alpha}V'_0 e^{-\sqrt{2/3\alpha}\,\varphi}$$2 (polynomial attractor regime).

This double-attractor structure is general, holding across modified potentials (e.g., those with well-regularized minima) and different values of $$V(\varphi)\approx V_0 - 2\sqrt{6\alpha}V'_0 e^{-\sqrt{2/3\alpha}\,\varphi}$$3 and $$V(\varphi)\approx V_0 - 2\sqrt{6\alpha}V'_0 e^{-\sqrt{2/3\alpha}\,\varphi}$$4.

Figure 2: Regularized potential for $$V(\varphi)\approx V_0 - 2\sqrt{6\alpha}V'_0 e^{-\sqrt{2/3\alpha}\,\varphi}$$5, $$V(\varphi)\approx V_0 - 2\sqrt{6\alpha}V'_0 e^{-\sqrt{2/3\alpha}\,\varphi}$$6 and varying $$V(\varphi)\approx V_0 - 2\sqrt{6\alpha}V'_0 e^{-\sqrt{2/3\alpha}\,\varphi}$$7; as $$V(\varphi)\approx V_0 - 2\sqrt{6\alpha}V'_0 e^{-\sqrt{2/3\alpha}\,\varphi}$$8 grows, the minimum shifts and the potential transitions to exponential plateau behavior.

Phenomenological Implications and Data Compatibility

A key motivation is the empirical tension between CMB and recent DESI DR2 measurements, which suggest a higher spectral index ($$V(\varphi)\approx V_0 - 2\sqrt{6\alpha}V'_0 e^{-\sqrt{2/3\alpha}\,\varphi}$$9) incompatible with standard $n_s = 1-2/N$0CDM analyses. Previous inflationary potentials fail to fully span the range required to simultaneously fit both datasets.

The constructed polyattractor class robustly accommodates this empirical range. By smoothly tuning $n_s = 1-2/N$1, all intermediate values $n_s = 1-2/N$2 can be realized, covering $n_s = 1-2/N$3 for $n_s = 1-2/N$4. This feature enables a model-agnostic matching of current and anticipated high-precision cosmological data. The tensor-to-scalar ratio $n_s = 1-2/N$5 can also be fully configured within observational bounds by tuning $n_s = 1-2/N$6 and $n_s = 1-2/N$7.

Theoretical and Future Directions

The presented framework demonstrates that the rich landscape of inflationary attractors can be continuously traversed via a single parameter, deeply linking previously distinct classes of potentials. This highlights the redundancy and universality inherent in inflationary phenomenology—an essential consideration for attempts at reconstructing the inflaton potential from observations.

Practically, these results have significant consequences for the interpretation of upcoming CMB Stage-IV and large-scale structure surveys: broader classes of inflationary models must be included within parametrizations for consistency analysis and model selection.

Theoretically, the natural embedding in $n_s = 1-2/N$8 supergravity and moduli space geometry paves the way for deeper studies in high-energy completions of inflation. Additionally, exploring quantum corrections and reheating dynamics within the polyattractor scenario remain promising avenues.

Conclusion

This work rigorously formulates and analyzes a continuous family of inflationary models—polyattractors—that bridge polynomial and exponential attractors via an interpolation parameter $n_s = 1-2/N$9. The resulting models achieve complete phenomenological coverage for critical observables such as $r = 12\alpha/N^2$0 and $r = 12\alpha/N^2$1, aligning with both CMB and DESI datasets and thereby addressing outstanding tension in the inflationary paradigm. Their construction, embedding in supergravity, and predictive versatility consolidate the attractor landscape and will play an indispensable role in further theoretical developments and data confrontation in early universe cosmology.