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Single field slow-roll inflation with step uplift to $n_s=1$

Hao-Shi Yuan, Ze-Yu Peng, Yun-Song Piao

Abstract

The early dark energy resolution of Hubble tension seems to be suggesting a scale-invariant Harrison-Zeldovich spectrum of primordial scalar perturbation, i.e. $n_s=1$ ($|n_s-1|\sim {\cal O}(0.001)$) for $H_0\sim 73$km/s/Mpc. In this work, we propose a possibility to acquire $n_s=1$ in single field slow-roll models of inflation. In our consideration, the potential of inflaton during inflation still preserve the shape of well-known single field inflation models in deep slow-roll region, but inflation ends suddenly due to a large step of inflaton potential. In particular, we investigate the implication of our scheme for chaotic inflation and Starobinski inflation, and show how they can be compatible with the observation for $n_s=1$.

Single field slow-roll inflation with step uplift to $n_s=1$

Abstract

The early dark energy resolution of Hubble tension seems to be suggesting a scale-invariant Harrison-Zeldovich spectrum of primordial scalar perturbation, i.e. () for km/s/Mpc. In this work, we propose a possibility to acquire in single field slow-roll models of inflation. In our consideration, the potential of inflaton during inflation still preserve the shape of well-known single field inflation models in deep slow-roll region, but inflation ends suddenly due to a large step of inflaton potential. In particular, we investigate the implication of our scheme for chaotic inflation and Starobinski inflation, and show how they can be compatible with the observation for .

Paper Structure

This paper contains 7 sections, 11 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Hint of EDEs for $n_s=1$
  3. Inflation with step uplift to $n_s=1$
  4. Applications to some popular models
  5. Monomial inflation
  6. Starobinsky inflation
  7. Discussion

Figures (5)

  • Figure 1: $n_{s}-H_{0}$ relation. The data shown are sourced from Peng:2025tqt, where the full Planck data Planck:2019nip, DESI DR2 BAO data DESI:2025zgx and Type Ia supernovas dataset Patheon Scolnic:2021amr are combined for the upper panel, while the large-scale Planck 2018 data Planck:2019nip, ACT DR6AtacamaCosmologyTelescope:2025bloAtacamaCosmologyTelescope:2025nti, SPT-3G D1SPT-3G:2025bzuBalkenhol:2024sbv, together with BAO and SNIa, are responsible for the lower panel. Gray band represents the recent SH0ES result $H_{0}=73.04\pm1.04$ km/s/Mpc Riess:2021jrx, and black solid line marks $n_{s}=1$.
  • Figure 2: The slow-roll potential and its step-modified version. In both cases, inflation ended at the position where $\varepsilon_{V} = 1$, corresponding to $\phi_{e,o}$ and $\phi_{e,s}$ respectively. The perturbation mode that exited the horizon at $N_{\ast,o}\gg 60$ remains far outside the current observable Hubble radius. However, with the abrupt step-caused exit of inflation, this mode can be regarded as the CMB window, and accordingly gives $n_{s}\to 1$.
  • Figure 3: Comparison of chaotic inflation with and w/o the step. The parameters of the step are $\phi_{c}=25,\ d=0.4$. Top panels: the shapes, potential slow-roll parameters of both models. The inflation happened from $\phi_{\ast}=30.12$ to $\phi_{s}=25.20$, predicting a larger spectral index $n_{s}\simeq 0.991$. Bottom panels: the evolution of the inflaton $\phi$ and the slow-roll parameter $\varepsilon_{V},\ \eta_{V}$ with respect to efolds number $N$.
  • Figure 4: Comparison of monomial inflation with (blue) and w/o the step (red). Here we choose $n=0.1,\ \phi_{c}=14.5,\ d=0.7$. Left panel: Intuitive illustration of the effect of the step in the shape of the potential. Right panel: The evolution of slow-roll parameter $\varepsilon$ as the function of inflaton $\phi$.
  • Figure 5: Starobinsky inflation with the step.