The Harrison-Zeldovich attractor: From Planck to ACT

TL;DR

The paper addresses the tension between ACT's hints toward a Harrison-Zeldovich spectrum and Planck-era inflationary models by introducing nonminimal derivative coupling (NDC) as a mechanism to modulate gravitational friction during inflation. This Horndeski-based approach shifts the horizon-exit field value, allowing the predicted scalar spectral index $n_s$ and tensor-to-scalar ratio $r$ to move toward a near scale-invariant value $n_s \approx 1$ without changing the potential $V(φ)$. Models such as monomial, α-attractor E-model, and quartic hilltop can be made compatible with current data, whereas natural inflation remains challenging unless parameters imply large field excursions. Overall, the NDC framework provides a flexible route to realize a Harrison-Zeldovich attractor and adapt inflationary predictions to evolving observational constraints.

Abstract

In the era of Planck cosmology, the inflationary paradigm is best fitted towards the cosmological attractor scenarios, including the induced inflation, universal attractors, conformal attractors, and special attractors that are cataloged as $ξ$-models and $α$-models. The recent hint from the ACT results pushes the scalar spectral index closer to the scale-invariant Harrison-Zeldovich spectrum, calling for a theoretical paradigm shift towards a Harrison-Zeldovich attractor, which is difficult to realize in the standard single-field slow-roll inflationary scenario. In this Letter, we achieve the Harrison-Zeldovich attractor scenario via nonminimal derivative coupling, attracting the monomial inflation, hilltop inflation, and $α$-attractor E-model towards the Harrison-Zeldovich spectrum.

Figures (3)

• Figure 1: Schematic illustration of the effect of a high or low friction region on the field value $\phi_\ast$.
• Figure 2: The evolution of $\phi$ as a function of $N$ for the monomial potential with $p=1/10$ in the cases of $\lambda=0$, $\lambda<0$, and $\lambda>0$.
• Figure 3: The theoretical predictions in the $n_s-r$ plane for typical inflation models with the help of NDC. The yellow-shaded regions show the constraints from Planck and BICEP/Keck (Planck–BK18). The green-shaded regions denote the joint analysis of Planck and ACT, including CMB lensing and BAO measurements from DESI, together with BICEP/Keck (Planck–ACT–LB–BK18). The red-shaded region indicates the credible interval for $n_s$ inferred from the Planck-SPT-ACT dataset within the AdS-EDE extension of $\Lambda$CDM. The dashed lines and open markers indicate predictions for $n_s$ and $r$ in the minimal coupling case for monomial potential (blue), $\alpha$-attractor E-model (red), quartic hilltop potential (green), and natural inflation (purple). As an illustrative example for the monomial potential, we adopt $q=1/10$ and $M=0.01 M_{\rm Pl}$. For one parameter set we choose $\lambda\Lambda^4M_{\rm Pl}^{-2}=-0.94$ with $\phi_{\rm c}/M_{\rm Pl}=\{4, 6,9,14 \}$ (blue dots); for the other we choose $\lambda\Lambda^4M_{\rm Pl}^{-2}=2$ with $\phi_{\rm c}/M_{\rm Pl}=\{2, 2.5, 2.8, 3 \}$ (blue squares). For the $\alpha$-attractor E-model we adopt $\alpha=3$ and $\lambda\Lambda^4M_{\rm Pl}^{-2} = -1$ with $\phi_{\rm c}/M_{\rm Pl}=\{5, 6, 7, 8, 9, 10, 11.5, 15 \}$ (red dots). For the quartic hilltop potential we adopt $\mu=18M_{\rm Pl}$ and $\lambda\Lambda^4M_{\rm Pl}^{-2} = -1$ with $\phi_{\rm c}/M_{\rm Pl}=\{ 12.5, 11, 9.5, 8, 6.5, 5, 3\}$ (green dots). For the natural inflation we adopt $f=5.6M_{\rm Pl}$ (purple dots) and $f=6.2 M_{\rm Pl}$ (purple squares) with $\lambda\Lambda^4M_{\rm Pl}^{-2} = -0.5$ and $\phi_{\rm c}/M_{\rm Pl}=\{ 11, 10, 9, 8, 7, 6, 5, 4, 3\}$. The monomial potential, $\alpha$-attractor E-model, and quartic hilltop potential can be attracted towards the Harrison-Zeldovich spectrum, while the natural inflation seems unlikely.