Enhancing cosmological constraints with nonlinear tanh transformations of Hermite-Gaussian Derivative fields

Abstract

A key goal in large-scale structure analysis is to extract multi-scale information to improve cosmological parameter constraints. In particular, higher-order derivative fields are especially valuable as they capture the geometric and topological information of the cosmic web that is highly sensitive to cosmological parameters. Traditional derivative-based methods, such as finite-difference or Fourier approaches, suffer from noise amplification at small scales and cannot stably capture multi-scale features. We present a robust two-step framework: first, stable multi-scale arbitrary-order derivatives are obtained via Hermite-Gaussian convolutional filters that suppress small-scale noise; second, a tanh nonlinear transformation compresses extreme density contrasts and enhances the visibility of cosmic web structures. Using the Quijote simulations, we show that combining multi-scale first-order spectra yields improvements of 1.2-3.0 times across all seven cosmological parameters, while multi-order spectra at a fixed scale provide 1.3-2.9 times gains. The most comprehensive combination achieves nominal gains of 2.0-5.3 times. Our method offers a robust approach to extracting additional cosmological information for future surveys.

Figures (11)

• Figure 1: The functional forms of $K_n(x/\sigma) = H_n(x/\sigma)\exp(-x^2/\sigma^2)$ for different orders $n = (0, 1, 2, 3)$. These kernel functions exhibit distinct oscillatory and smoothing characteristics across the normalized coordinate $x/\sigma$. The $n = 0$ case (blue solid line) shows a smooth, non-oscillatory Gaussian profile, while higher-order kernels($n\ge 1$), display increasing numbers of zero-crossings and oscillatory behavior, which is characteristic of Hermite polynomial-based filters. These kernels are generalized to 3D and used in our Hermite convolution framework to extract cosmological information from the density field.
• Figure 2: Projected halo density fields from the high-resolution AbacusSummit simulation over the redshift slice $0.475\le z \le 0.538$ ,covering a sky area of $40^\circ\times 40^\circ$. Top row: Raw halo density field $\delta+1$. Middle row: First-order Hermite-Gaussian (HG) convolutional field $\delta^{3}_{1}$ ($\sigma = 3\ h^{-1}\mathrm{Mpc}$). Bottom row: tanh-transformed HG (HG-tanh) field $\widetilde{\delta}^{3}_{1}$ ($\alpha = 60$). The right column shows the corresponding field value histograms. The HG operation acts as a spatial differentiator that enhances structural gradients, sharpening filaments and cluster boundaries. The subsequent tanh transformation compresses extreme values, further improving the visibility of intermediate-density structures and the connectivity of the cosmic web.
• Figure 3: Power spectrum comparison of different field transformations applied to the Quijote fiducial simulation density field, all using a fixed smoothing scale of $\sigma = 10 h^{-1}\text{Mpc}$. The blue curve shows the raw density field power spectrum $P(k)$. The purple curve corresponds to the HG convolutional magnitude field $\delta^{10}_{1}$, while the orange curve shows the HG-tanh magnitude field $\widetilde{\delta}^{10}_{1}(\alpha = 600)$. Each power spectrum is normalized by its mean value $\langle P(k) \rangle$, which is obtained by averaging $P(k)$ over all $k$-bins.
• Figure 4: Correlation matrices comparing the power spectrum of the raw density field $P(k)$ with those of the HG-tanh magnitude fields. The notation $H^\sigma_{J}(k)$ denotes the power spectrum of the field $\widetilde{\delta}^\sigma_{J}$. The analysis covers $k < 0.5~h \mathrm{Mpc}^{-1}$ with 59 $k$-bins. The HG-tanh magnitude fields' spectra exhibit enhanced diagonal dominance compared to the standard power spectrum, particularly on small scales, indicating reduced mode coupling effects, which allows for more information to be extracted from small scales.
• Figure 5: Numerical logarithmic derivatives with respect to cosmological parameters of the standard power sepctrum $P(k)$ and the transformed field's power spectrum $H^{\sigma}_J(k)$. We show one representative component for each order of the Hermite convolution, for the smoothing scale $\sigma = 10h^{-1}\mathrm{Mpc}$ case.