Peak Patch and WebSky

[ Note this page is under construction ]

The Peak Patch and WebSky simulations are a highly efficient tool for modelling the large-scale structure of the universe.

The Peak Patch simulations

The Peak Patch simulations^[1,2,3] model the distribution of dark matter (DM) in the universe by mapping out catalogues of DM halos. While observations do support halos of DM surrounding most galaxies and clusters, these structures do not have well defined boundaries, so in simulations we must make a choice of where we define the edge of the halo. Typically what is done is to choose the virial radius within which DM particles are gravitationally bound. In [math]\displaystyle{ \Lambda }[/math]CDM cosmology, this corresponds to a halo with an average density about [math]\displaystyle{ \Delta_c=179 }[/math] times the mean matter density of the universe. Since this is subject to changes in the model parameters, a factor [math]\displaystyle{ \Delta_c=200 }[/math] is conventionally used, so we do the same.

For the most up-to date description of the Peak Patch algorithm for identifying DM halos, see arXiv:1810.07727 ^[4].

For a summary of the Peak Patch algorithm:

1. Generate [math]\displaystyle{ \delta_L(\mathbf{x}) }[/math] field: as initial conditions, Peak Patch uses a linear-theory matter overdensity field [math]\displaystyle{ \delta_L(\mathbf{x}) }[/math], which is the matter overdensity [math]\displaystyle{ \delta(\mathbf{x}) = \rho_m(\mathbf{x},t) / \bar{\rho}_m(t) - 1 }[/math] at a time [math]\displaystyle{ t }[/math] sufficiently early in the history of the universe that linear theory is a suitable approximation to the dynamics that formed it. [math]\displaystyle{ \delta_L(\mathbf{x}) }[/math] can either by read from a file or generated from a power spectrum (which describe the number of structures of different sizes that we observe in the universe).
2. Smooth and find peaks: the [math]\displaystyle{ \delta_L(\mathbf{x}) }[/math] field is then smoothed at a series of scales by convolving with top hat functions (e.g. see section 3.2 of this review article). This removes any fluctuations smaller than about the top-hat function's cutoff radius [math]\displaystyle{ R_\mathrm{th} }[/math] allowing us to isolate just structures in [math]\displaystyle{ \delta_L(\mathbf{x}) }[/math] larger than [math]\displaystyle{ R_\mathrm{th} }[/math]. Peaks in each filtered density field are found, these are candidate sites where dark matter halos may form.
3. Ellipsoidal collapse: dark matter halos are gravitationally bound collections of dark matter, and therefore virialised. To determine if a given peak will collapse to form a virialised halo, we model it as a uniform-density sphere of radius [math]\displaystyle{ R_\mathrm{th} }[/math] (the top-hat filter radius at which the peak was identified), and allow it to collapse ellipsoidally (e.g. see section 3.3 of this review article) subject to the local strain of the [math]\displaystyle{ \delta_L(\mathbf{x}) }[/math] field. If the collapse has enough time to reach a density contrast characteristic of virialised structure by time [math]\displaystyle{ t }[/math], it is saved as a candidate halo of radius [math]\displaystyle{ R_\mathrm{th} }[/math], or mass [math]\displaystyle{ M = \bar{\rho}_{m}(t_0) \frac{4}{3} \pi R_\mathrm{th}^3 }[/math].

   Peak Patch can either produce single-redshift simulations, where all halos collapse until the same time [math]\displaystyle{ t }[/math], or light cone simulations where each halo is evolved only until the age it appears to be at relative to a specified observer (so [math]\displaystyle{ t }[/math] is the light travel time to the halo). The latter is ideal for making sky maps.

4. Merging and exclusion: because halo candidates are found at multiple filter scales, overlaps are inevitable. So a merging and exclusion algorithm is run with any halo entirely inside another being tossed out and halos overlapping regions partially allotted to each halo to avoid double counting mass.
5. Displacement to final state halos: up to this point, we have been working in Lagrangian space, essentially determining what mass in the initial conditions fields will form halos. But as matter collapses into halos, these halos will also move relative to one another. Most of the highly nonlinear dynamics occurs within DM halos, so the displacement of the halos themselves can be accomplished with 2nd order Lagrangian perturbation theory.

The DM halo catalogues generated by Peak Patch have been validated by [math]\displaystyle{ N }[/math]-body simulations^[4], achieving similar results with much greater computational efficiency. Peak Patch has performed favourably compared to other halo-finders^[5,6,7].

Peak Patch is additionally a particularly powerful tool for studying beyond-standard-model (BSM) cosmologies because the simulation has built-in support for varying [math]\displaystyle{ \Lambda }[/math]CDM model parameters and primordial non-Gaussianities of a number of forms informed by a range of early-universe physics phenomena. Because it employs only approximate dynamics, complete knowledge of the BSM equations of motion is not required, allowing us to probe wider parameter space.

Coupled with WebSky, Peak Patch light-cone runs can be used to generate mock sky maps for observatories and have been used extensively by ACT in recent years to make CMB foreground mock maps. The WebSky algorithm is summarised in the next section.

The WebSky simulations

The WebSky simulations produce mock sky maps in a range of observables. Mock sky maps are simulated maps of the sky that are statistically analogous to maps of the sky made by observatories.

[ discuss model ]

[ discuss different response functions ]

[ link to paper ]

Notes

1. WebSky has two pronunciations wɛbskaɪ and wɛbskiː which are understood to exist in superposition [math]\displaystyle{ | \text{WEB} \rangle \otimes \left( \frac{ | \text{sky} \rangle + | \text{skee} \rangle }{ \sqrt{2} } \right) }[/math].