Web-Halo Model (WHM): Accurate non-linear matter power spectrum predictions without free parameters

Abstract

We present a parameter-free variant of the halo model that significantly improves the precision of matter clustering predictions, particularly in the challenging 1-halo to 2-halo transition regime, where standard halo models often fail. Unlike HMcode-2020, which relies on 12 phenomenological parameters, our approach achieves comparable or superior accuracy without any free fitting parameters. This new web-halo model (WHM) extends the traditional halo model by incorporating structures that have collapsed along two dimensions (filaments) and one dimension (sheets), in addition to haloes, and combines these with 1-loop Lagrangian Perturbation Theory (1$\ell$-LPT) in a consistent framework. We show that WHM matches N-body simulation power spectra within the precision of state-of-the-art emulators at the 2-halo to 1-halo transition regime at all redshifts. Specifically, the WHM achieves better than 2\% accuracy up to scales of $k = 0.4\, h\,\mathrm{Mpc}^{-1}$, $0.7\, h\,\mathrm{Mpc}^{-1}$, and $1.3\, h\,\mathrm{Mpc}^{-1}$ at redshifts $z = 0.0$, $0.8$, and $1.5$, respectively, for both the baccoemu and EuclidEmu2 emulators, across their full $w_0w_a\mathrm{CDM} + \sum m_ν$ cosmological parameter space. This marks a substantial improvement over 1$\ell$-LPT and HMcode-2020, the latter of which performs similarly at low redshift but deteriorates at higher redshifts despite its 12 tuned parameters. We publicly release WHM as WHMcode, integrated into existing HMcode implementations for CAMB and CLASS.

Figures (12)

• Figure 1: We show the resulting $z=0$ density fields of a toy 2D N-body simulation illustrating the logic behind the WHM workhorse equations \ref{['eq:whm-whm-powersum']}-\ref{['eq:whm-whm-1h']}. The left panel depicts the field assuming ballistic particle trajectories (a single timestep, ZA). The resulting power spectrum would be described by tree-level LPT. The middle panel displays the fully evolved field, but with haloes removed and their mass redistributed into filaments. Its power spectrum would be described by adding a 1-filament term compensated by the LPT filament profiles to the 'LPT order', yielding the 'filament order' (note that there is no sheet order in 2D). Finally, the right panel shows the exact fully evolved field including haloes. Within our WHM, its power spectrum would be described by the 'halo order' that adds to the filament order a 1-halo term compensated by the power these haloes had contributed when they were filaments.
• Figure 2: This shows the mass functions for sheets, filaments and haloes in equations \ref{['eq:ing-mf-gen']} and \ref{['eq:ing-mf-numbers']} following Shen_2006, but with $a=0.707$ and hence readjusted normalisation $A$. These are compared to the halo mass functions corresponding to spherical collapse from Press_1974, our baseline, ellipsoidal collapse Sheth_1999 halo mass function, and the fit to FoF haloes from Warren2006ApJ.
• Figure 3: Shown are the squared, angle-averaged window functions for various objects that share the same mass, corresponding to a homogeneous sphere of radius $R=10\,h^{-1}\mathrm{Mpc}$ and constant density $\bar{\rho}$ (black solid). First, it collapses into a sheet with overdensity $\Delta_\mathrm{v}^{1/3}$ (blue solid), then into a filament with overdensity $\Delta_\mathrm{v}^{2/3}$ (green solid), before finally creating a virialised NFW halo with overdensity $\Delta_\mathrm{v}$ (red solid). The blue and green dotted lines represent 'infinitely thin' sheets and filaments, corresponding to 1D and 2D Dirac delta functions $\delta_\mathrm{D}$, i.e., the limit where collapsed dimension(s) have zero extent ($b=0$, $a=0$, respectively). Their multiplication with the squared NFW halo window corresponds to the 'thin web' case introduced in the main textThe dash-dotted lines correspond to the approximations \ref{['eq:ing-dpwf-sheet']} and \ref{['eq:ing-dpwf-filament']} obtained from multiplying the 'inf. thin' limits with a tophat accounting for the non-zero extent in the collapsed dimension(s).As shown by the difference plot in the lower panel, they agree with the exact result within $0.005$, well below the difference towards the 'thin web' case.
• Figure 4: We show the web-halo model (WHM) results breaking down the halo (red), halo+filament (green) and the full halo+filament+sheet (blue) contributions compared to PT only (grey), the original halo model without window compensation (dashed black), and the reference non-linear power spectrum obtained from baccoemu for the 'Narya' cosmology. Each column shows the theory predictions at different redshifts. The top row uses linear PT, while the bottom row shows our baseline Lagrangian PT setup. The magenta dotted and dashed lines correspond to the inverse displacement field variance $\sigma_\mathrm{v}^{-1}$ and the inverse non-linear scale $r_\mathrm{nl}^{-1}$ defined in equations \ref{['eq:ing-pt-sigmav']} and \ref{['eq:ing-pt-rnl']}, respectively. Grey bands indicate the 1% and 5% accuracy regions, and orange bands the FLAMINGO baryonic feedback uncertainty. The coloured uncertainty bands within the WHM predictions cover the baseline assumption of either homogeneous or infinitely thin (but filtered by the same-mass halo profile) filaments and sheets, for details see section \ref{['sec:ing-dp-f']}.
• Figure 5: We compare our web-halo model (WHM) predictions using the baseline Sheth_2001 halo mass function (third row) or the one from Warren2006ApJ (fourth row) for 100 random cosmologies covering the full baccoemu range against Lagrangian PT only (first row), the original halo model (HM) without window compensation (second row), HMcode-2020 (fifth row) and the reference non-linear power spectrum from baccoemu. Each column shows results at a different redshift and lines are colour-coded by $\sigma_8$. Again, magenta dotted and dashed lines correspond to the inverse displacement field variance $\sigma_\mathrm{v}^{-1}$ and the inverse non-linear scale $r_\mathrm{nl}^{-1}$, respectively. Orange bands indicate the FLAMINGO baryonic feedback uncertainty, horizontal dotted and dashed lines the maximum baccoemu uncertainty (3%) and the $2\sigma$HMcode-2020 uncertainty (5%), respectively.