A Variable-Slope Smooth-$k$ Filter for Modeling Halo Abundances with Damped and Oscillatory Power Spectra

TL;DR

This work addresses the challenge of analytically predicting halo abundances in non-cold dark matter scenarios that feature both small-scale damping and intermediate-scale oscillations, such as dark acoustic oscillations. It introduces the variable-slope smooth-$k$ (VSMK) filter, a minimal extension of the SMK that allows the effective $k$-space window slope to interpolate between two asymptotic values, thereby decoupling the suppression of low-mass halos from the oscillatory DAO imprint at intermediate scales. The authors derive the VSMK filter form, with $W_{ m VSMK}=[1+(k/k_M)^{f(k)}]^{-1}$ and $f(k)=eta_2 - (eta_2-eta_1)[1+(\mu k/k_M)^{\delta}]^{-1}$, where $eta_1$ and $eta_2$ control the small- and intermediate-scale behavior, respectively, and $eta_1$ and $eta_2$ are fixed to reproduce WDM and DAO features. Comparisons with $N$-body simulations for WDM and ETHOS DAO show that a single parameter set ($eta_1=4.8$, $eta_2=3.6$, $\mu=2.1$, $\\delta=12$, $c=3.6$) accurately captures the halo mass function across regimes and redshifts, outperforming the traditional SMK approach. The VSMK framework thus provides a unified and flexible analytic tool for modeling halo abundances in non-CDM scenarios with damped and oscillatory power spectra, with potential for constraining small-scale dark matter physics using upcoming data.

Abstract

We introduce a variable-slope smooth-$k$ (VSMK) filter within the Press-Schechter formalism to model halo mass functions derived from damped and oscillatory matter power spectra. While the standard smooth-$k$ approach successfully captures small-scale suppression effects, it intrinsically couples these to oscillatory features at intermediate scales. The VSMK filter generalizes this framework by allowing the effective logarithmic slope of the $k$-space window function to vary smoothly between two asymptotic regimes, thereby decoupling the small-scale suppression of halo abundances from the intermediate-scale oscillatory features characteristic of dark acoustic oscillations. We compare the analytic predictions obtained with the VSMK filter to $N$-body simulations for warm dark matter and ETHOS-based models, showing that a single parameter set reproduces both regimes simultaneously. The VSMK filter thus provides a unified and flexible analytic framework for modeling halo abundances in non-cold dark matter scenarios with damped and oscillatory power spectra.

Figures (5)

• Figure 1: Comparison between smooth-$k$ (SMK) and variable-slope smooth-$k$ (VSMK) filters. The SMK filter is shown for different values of the slope parameter $\beta$ and compared to a VSMK filter with $\beta_1 = 4.8$, $\beta_2 = 3.6$, $\delta = 12$, and $\mu = 1$, chosen for illustrative purposes. In all cases, $c = 1$ is also adopted for visualization only. The lower panel shows the difference $\Delta W(k,R) = W(k,R) - W_{\mathrm{SMK}(\beta=3.6)}(k,R)$ for each filter.
• Figure 2: Impact of the VSMK filter parameters on the halo mass function. Both panels show halo mass functions computed using the variable-slope smooth-$k$ (VSMK) filter with fixed $\mu = 2.1$, $\delta = 12$, and $c = 3.6$. Left: WDM-like power spectrum with fixed $\beta_2 = 3.6$ and varying $\beta_1$, illustrating that $\beta_1$ exclusively controls the small-mass slope of the HMF. Right: DAO power spectrum with simultaneous variations of $\beta_1$ and $\beta_2$, demonstrating that $\beta_1$ regulates the small-mass behavior, while $\beta_2$ controls the intermediate-mass regime where DAO-induced oscillations are present.
• Figure 3: Comparison between analytical halo mass functions and $N$-body simulations for different dark matter models. The HMFs computed using the optimal SMK filter for each model are compared to those obtained with the SMK filter optimized for the alternative model, as well as with the VSMK filter. Unless stated otherwise, $c = 3.6$. Left: WDM model with $m_{\mathrm{WDM}} = 0.25~\mathrm{keV}$ from Schaeffer2021, where the optimal SMK filter has $\beta = 4.8$ and $c = 3.3$ (grey dashed line). Right: DAO model with $h_{\mathrm{peak}} = 1$, $k_{\mathrm{peak}} = 100~h\,\mathrm{Mpc}^{-1}$ at $z=10$ from Verwohlt2024, where the optimal SMK filter has $\beta = 3.6$ and $c = 3.6$. Simulation data (empty squares) are extracted from published figures in Schaeffer2021 and Verwohlt2024. The lower panels show the relative deviations $\Delta$ with respect to the optimal SMK prediction.
• Figure 4: Comparison between analytical halo mass functions and $N$-body simulations at different redshifts for DAO models. Analytical HMFs (solid lines) computed using the general VSMK filter and the optimal SMK filter are compared to $N$-body simulations with similar cosmological parameters from Verwohlt2024Bohr2021 (open circles). Left: DAO model from Verwohlt2024, where the optimal SMK filter has $\beta = 3.6$ and $c = 3.6$. Right: DAO model from Bohr2021, where the optimal SMK filter has $\beta = 3.46$ and $c = 3.79$. Simulation data (empty circles) are extracted from published figures in Verwohlt2024 and Bohr2021. The lower panels show the relative deviations $\Delta$ with respect to the optimal SMK prediction. Transfer-function parameters are indicated within each panel.
• Figure 5: Normalized integrand of the variance derivative for WDM and DAO models. The normalized integrands of the variance derivative corresponding to large and small mass scales are presented for the two dark matter models considered in this work. Both the SMK filter (dashed lines), with $\beta = 4.8$ and $c = 3.6$ and the VSMK filter (solid lines), with $\beta_1 = 4.8$, $\beta_2 = 3.6$, $\mu = 2.1$, $\delta = 12$ and $c = 3.6$ (Section \ref{['sec:results']}) are shown for each model. Left: WDM model with $m_{\mathrm{WDM}} = 1.61\,\mathrm{keV}$. Right: DAO model with $k_{\mathrm{peak}} = 35\,h\,\mathrm{Mpc}^{-1}$ and $h_{\mathrm{peak}} = 1$. Each integrand is normalized to its own maximum.