Dynamically generated tilt of isocurvature fluctuations

TL;DR

The work addresses the tension between observing isocurvature fluctuations and the stringent CMB constraints by showing that a scalar spectator with a nontrivial potential naturally produces a blue-tilted isocurvature spectrum during inflation when its effective mass $m_{ m eff}^2 = V''(\phi_0)$ is near $H_I^2$. The authors develop a dynamical mechanism where the condensate spends most of inflation near the slow-roll boundary, yielding a tilt that scales as $\left.\frac{d\log P_{\delta}}{d\log k}\right|_{k_*} \approx 2\alpha(N_*)$, with $\alpha(N)$ governed by $\frac{d\alpha}{dN} = -\kappa \alpha^2$ and $\kappa = \frac{V'''(\phi_0)V'(\phi_0)}{V''(\phi_0)^2}$. They further show that if the scalar is long-lived, the tilt and abundance become attractor-like, enabling a predictive relation between potential parameters, particularly for a quartic self-interaction which can account for all dark matter for a wide mass range. By analyzing the relic abundance and density perturbations through radiation domination, the paper maps the viable $m$–$\lambda$ parameter space and discusses cosmological constraints, highlighting a viable scenario with distinctive small-scale isocurvature signatures that avoid large-scale CMB bounds. The results offer a concrete, testable path for dark matter models with blue-tilted isocurvature spectra sourced by nontrivial scalar potentials during inflation.

Abstract

Light scalar fields acquire isocurvature fluctuations during inflation. While these fluctuations could lead to interesting observable signatures at small scales, they are strongly constrained on large scales by cosmic microwave background observations. When the mass of the scalar is much lighter than the inflationary Hubble scale, $m\ll H_I$, the spectrum of these fluctuations is flat. Meanwhile, if $m\gg H_I$, the fluctuations are suppressed. A blue-tilted isocurvature spectrum which exhibits enhanced structure on small scales but avoids observational constraints on large scales therefore requires a coincidence of scales $m\sim H_I$ for a free massive scalar. In this Letter, we show that if a scalar field possesses a nontrivial potential, its inflationary dynamics naturally cause this condition to be satisfied, and so a blue-tilted spectrum is generically expected for a large class of potentials. Specifically, if its potential $V$ exhibits a region which satisfies the slow-roll condition $V''<3H_I^2$, the scalar condensate will spend most of inflation close to the boundary of this region, so that its effective mass is typically close to $H_I$. The resulting blue tilt is inversely proportional to the number of $e$-folds of inflation prior to horizon crossing. If the scalar is long-lived, this mechanism leads to an attractor prediction for its relic abundance, which is insensitive to initial conditions of the scalar. In particular, a scalar field with quartic self-interactions can achieve the correct abundance to constitute all of the dark matter for a wide range of masses. We compute the relationship between the mass and self-coupling of quartic dark matter predicted by this mechanism.

Figures (2)

• Figure 1: Left: Evolution of $\alpha$ [see Eq. (\ref{['eq:alpha']})] as a function of $e$-folds $N$ during inflation. We take the inflationary Hubble scale to be $H_I=10^{12}\,\mathrm{GeV}$ in this plot. The spectator begins with $\alpha_i\gg1$ at the start of inflation $N_i=-80$. Initially, $\alpha(N)$ oscillates and decreases in magnitude exponentially. Once $\alpha<1$, it decreases as $1/N$. Dashed lines show the analytic approximation in Eq. (\ref{['eq:alpha_est']}). We show evolutions for two choices of potential $V(\phi)\sim\phi^p$. The blue curve has $\kappa=\mathcal{O}(1)$, while the orange has small $\kappa$. In accordance with Eq. (\ref{['eq:alpha_est']}), the latter results in much larger values of $\alpha$. Right: Primordial isocurvature spectrum for scalar DM with same potentials as in left plot. Dashed lines show the analytic approximation in Eq. (\ref{['eq:Pdel']}). In both numerical and analytic curves, we include the constant factor from late-time evolution [see Eq. (\ref{['eq:primordial']})], which for quartic DM with mass $m=10\,\mathrm{eV}$ applies to the left of the grey dotted line [see Eq. (\ref{['eq:sep_bound']})]. The tilt of the spectrum is related to $\alpha$ at horizon crossing $N_*$ (shown on top axis), as in Eq. (\ref{['eq:tilt']}). For $\kappa=\mathcal{O}(1)$, this generically results in a small blue tilt. For smaller $\kappa$, as in the orange curve, the tilt can be much stronger. In grey, we show constraints on the primordial isocurvature spectrum from observations of the CMB Buckley_2025, Lyman-$\alpha$ forest Bird_2011Graham_2024, and ultrafaint dwarf galaxies Graham_2024. In order to avoid these constraints, the normalizations of the potentials are fixed at $\lambda=10^{-9}$ [as in Eq. (\ref{['eq:quartic']})] for the blue curve and $\Lambda=4.4\times10^{11}\,\mathrm{GeV}$ [as in Eq. (\ref{['eq:potential']})] for the orange curve. Note that the total matter power spectrum will be the sum of the isocurvature and adiabatic fluctuations [see Eq. (\ref{['eq:primordial']})].
• Figure 2: Parameter space for inflationary production of scalar DM with a quartic potential [see Eq. (\ref{['eq:quartic']})]. The blue line indicates parameters which produce the correct DM relic abundance for $H_I=10^{10}\,\mathrm{GeV}$ and $N_\mathrm{tot}=80$ (assuming instantaneous reheating). Colored regions indicate constraints on this production scenario, including: overproduction of isocurvature fluctuations (see right plot of Fig. \ref{['fig:Pdel']}), violation of $N_\mathrm{eff}$ constraints at the time of BBN, and transplanckian initial conditions $\phi_{0,i}>M_\mathrm{pl}$. (Note that all parameters below the blue line are also constrained due to overclosure of the universe.) Dotted lines show these curves for $H_I=10^{12}\,\mathrm{GeV}$. In grey, we show late-time constraints on self-interacting dark matter (SIDM) and disruption of structure formation Budker_2023.