Sampling a warm distribution from a cold sheet

We use the math that we have developed in the last sections to sample a cosmological density field which includes a thermal smoothing due to the warmth of the primordial velocity distribution: This can be done, in principle, as a pure post-processing step in any cosmological simulation. We are able to reconstruct the thermal smoothing from a single cold sheet. As a proof of concept we show this here for the case of a two dimensional Zel’dovich approximation. For this setup this can be done relatively easily and quickly, since we only have to sample a four dimensional phase space (instead of a six-dimensional one). However, everything generalizes straightforwardly to six phase space dimensions and it can also be done with the cold sheet of an evolved particle distribution. We leave this open for possible future investigations.

We set up the 2d Zel’dovich approximation for a boxsize ofL= 10 Mpc/hand a power-spectrum that has the same two-point correlation function like anm_x = 100 eV warm dark matter case. We use an Einstein - de Sitter universe and use D = a = 1., but increase the amplitude of the power spectrum by a factor 1.7 so that there are already are some shell crossings. The full setup can be seen in Figure 6.5 (which we will discuss later).

Note that this setup is very unphysical, but that does not matter for this proof of concept.

Mathematically it has most of the properties that an evolved cosmological simulation has.

From a (Lagrangian) grid of particles we reconstruct the sheet interpolating functionsx(q)

Figure 6.4: Density fields of a 2d Zel’dovich approximation (see text) for different sheets.

Top left: the central “cold” sheet p^∗ = (0,0)^T. Top right: the average of two sheets that are displaced in the x-direction of initial velocity space by p^∗ = (−1 km s⁻¹,0)^T and p^∗ = (1 km s⁻¹,0)^T. Bottom left: analogue, but using displacements in the y-direction p^∗ = (0,−1 km s⁻¹)^T and p^∗ = (0,1 km s⁻¹)^T. With the procedures described in this section we are able to reconstruct these sheets from the cold sheet in the top left panel without any additional information. Bottom right: The density field of a warm distribution, as reconstructed by the sum of 13² different pseudo-sheets (see text). The density field of the warm distribution can be reconstructed from the dynamical information of a single cold sheet.

6.2 Reconstructing sheets that are displaced in velocity space 179 and v(q) and also their derivatives Dxq(q) and Dvq(q). In the top left panel of Figure 6.4 we show the density field that can be obtained from resampling this cold sheet.

Additionally we resample the density field for a set of different p^∗-sheets. To get an intuition of how these look and whether the reconstruction behaves properly we show the sum of two sheets that are displaced in initial x-velocity space (p^∗ = (−1 km s⁻¹,0)^T and p^∗ = (1 km s⁻¹,0)^T) in the top right panel of Figure 6.5 and for two sheets that are displaced iny-velocity space in the bottom left panel. Note that the value of 1 km s⁻¹ is very close to the typical thermal velocities of a 100 eV particle which we calculate as p

hv²i ≈980 m s⁻¹ (compare Figure 2.2 and use p

hv²i ≈3.6v0). Therefore the displacements of these sheets are comparable to the actual thermal displacements of typical particles in such a WDM cosmology. Note that far away from caustics the sheets can almost not be distinguished, but close to caustics the difference is quite dramatic. As a related effect the approximation of the stream-density as ρ₀|detDxq|⁻¹ works remarkably well in most of the volume, but breaks down in caustics (where the approximation goes to infinity, whereas the true density of the stream is limited by the thermal softening) - see White & Vogelsberger (2009) for more details.

We assume a 2d Gaussian velocity distribution for the initial velocities f(p) = 1

2πσexp

−|p|² 2σ²

(6.34) and use σ = 1km/s. Note that this does not correspond to the velocity distribution of the thermal relic, but we only make rough estimates here anyways. (Typical velocities should match within a factor of two with the 100eV relic.) We then calculate the density field of 13² sheets which range uniformly from −3σ to 3σ in each velocity coordinate. We determine as the total density

ρ_tot =X

px

X

py

ρ_pf(p_x, p_y)∆p² (6.35) where ∆p= 0.5σ is the spacing of the velocity grid. We display the result in the bottom right panel of Figure 6.4 and in Figure 6.5. Note that the fuzziness of these images is not due to resolution problems (the resolution is the same as in the other panels of Figure 6.4), but purely due to the thermal broadening. In this Zel’dovich case it appears like there is almost everywhere a relatively similar smoothing. That is so, since Dvq (which determines the width of caustics) has at least in order of magnitude similar values at the caustics. However, this is not expected to be the case in an actual cosmological setup. As was show in section 2.6.1, the thermal smoothing in caustics ranges easily over 9 orders of magnitude. However, possibly the first and largest caustics behave similarly to what can be seen in the bottom right panel of Figure 6.4.

Note that this way of re-sampling the velocity distribution by reconstructing pseudo-sheets is actually quite inefficient. It would require sampling a 6d phase space distribution in the case of a full cosmological setup. However, this is not at all necessary. Instead it would be faster (and also more accurate) to sample the 3d central cold sheet, but deposit

Figure 6.5: The density field of a warm sheet in the Zel’dovich approximation. Even for this relatively warm dark matter model of 100eV, the thermal smoothing does not alter large scale structures significantly, but only introduces a “fuzziness” into the density field.

(The fuzziness is only due to the thermal broadening, but not due to resolution issues).

6.3 On the treatment of hot distribution functions (like neutrinos) 181 a Gaussian ellipsoid for each sampled pseudo-particle - instead ofδ-function-like particles.

That ellipsoid is given by a singular value decomposition of the matrix A from equation (6.33)

A=USV^T (6.36)

the singular valuessi are the semi-major axes and their directions are given by the column vectors u_i.

Now this operation of depositing a large set of ellipsoids with different alignments is an operation which would require some time³ to be implemented efficiently. However, if an efficient implementation is found, it will be straightforward to make simulations of warm dark matter which include a thermal smoothing.

We conclude thatthe density field of a warm distribution function can be reconstructed from a single cold sheet. This works in one dimension (as can be seen in many Figures throughout this thesis), but it also works in higher dimensional cases. This opens up possibilities for understanding the implications of warm distribution functions as a pure post-processing step of cold simulations. Further, it opens up the possibility of warm dark matter simulations which do not operate in the cold limit, but instead actively model the smoothing of the density field due to the thermal velocities. The additional requirement is a convolution with an ellipsoidal kernel that depends on the Lagrangian coordinate q.

However, The effect of thermal velocities is expected to be small, so it is not clear whether this would be worth the effort. In the next section, we discuss the possibility of applying the presented method in the context of the modelling of the cosmic neutrino background.

6.3 On the treatment of hot distribution functions