The plan is to use the Carlson and White (2012) approach to remapping a simulated cube to make a“pencil beam” cuboid from which we will cut the vuvuzela volume that will be observed in, for example, the LADUMA survey planned for the MeerKAT.
Why a vuvuzela?
Neutral hydrogen (HI) emits photons at a very specific frequency (around 21cm in wavelength, a result of a hyperfine transition that flips the spin of an electron relative to the proton in hydrogen). As one moves further away form earth, the photons become redshifted due to the cosmological expansion of space. When we observe HI in distant galaxies, the HI photons have been redshifted to longer wavelengths (lower frequencies). The field of view of a radio telescope scales inversely with the frequency of observations. At 21cm (1.4GHhz), the field of view of the MeerKAT observation is about 1 square degree. At 700MHz (the frequency at which HI is observed in galaxies at redshift z~1) the field of view is more like 4 square degrees.
Fig1. The vuvuzela simulation.
Carlson White Methodology
J. Carlson and M. White published a paper in 2010 (ApjS. 190:311-314) in which they ouline a method of taking a canonical unit cube of an N-body cosmological simulation and remapping it into a desired cuboid cosmological simulation of the same volume. In their paper they illustrate that the remapping can be thought of as dividing the unit cube into individual regions which are then translated by integer offsets and put back together (much like puzzle pieces) along periodic baundaries. Their algorythm produces a bijective remapping of the canonical unit cube i.e. the remapping is one-to-one (each point/simulation particle in the unit cube appears once and is remapped to a single point in the remapped cuboid). This then allows it to preserve structure and volume. The goal of the remapping is to provide a map (bijective) from the canonical unit cube (with coordinates x ∈ [0;1]) to the desired cuboid of dimensions L1xL2xL3, where the Li represent the lengths of the remapped cuboid in units of the unit cube. The lengths Li can be chosen from a descrete list generated by another algorythm written by Carlson-White 2010 which uses a brute force search for 3×3 invertible matrices which produce the lattice vectors required for the remapping. The implementation of both codes is simple and outlined at http://mwhite.berkeley.edu/BoxRemap.
Implementation to the “vuvuzela”
The plan was to take an existing dark matter simulation cube and then remap this into a pencil beam cuboid using the Carlson-White methodology. The beam of the deep HI line (as invisioned to be observed by MeerKAT) is then cut out of the pencil beam cuboid. The relations governing the cutting of the beam were those relating comoving distance (cosmological distance which does not change with the expansion of the universe) to the redshift of the 21 cm HI line. The algorythm for generating Li's was used to find a suitable remapping of our unit cube. Once the remapping was done we proceded with cutting the vuvuzela by first investigating the low redshift limit (z « 1) where cosmological expansion of the universe is approximated to be v = H0d (Hubble expansion) where v is the expansion rate (in km/s) and H0 is the Hubble constant and d is our distance in units of Mpc/h (h comes from H0 = 100h). This then lead to the familiar redshift-distance relation z ≈ H0d/c , c is the speed of light. The beam diameter Θ(z) at redshift z goes as 1x(1+z)^2 degrees where our beam has a width of 1 degree at the telescope. Our input data (canonical unit cube of random points, initially) is in the format of spatial coordinates (x,y,z) and one can use these to find the redshift at a z-coordinate of w i.e z = z(z-coordinate) = z(w) and thus Θ(z) = 1x(1+z(w))^2 degrees.
From the above, the angular diameter distance DA is given by the relation DA = d'/Θ where Θ is now in radians (with d' as the actual beam width) and d' ∼ d at low redshifts. Thus d ≈ Θ(z)DA and DA ≈ cz/100h Mpc and the telescope beam will include all points which satisfy (x^2 + y^2) < (DA)^2 where DA is now in units of our unit cube length. This allowed for our algorythm (for cutting the beam shape) to be refined through trial and error so that it could later be used on real data.
The very same methodology was then followed but with the redshift-distance relation now determined through the use of Ned Wright's cosmology calculator (implemented in the Python programming language). The input data was also switched from a random N-body simulation to a Gadget2 simulation where the canonocal unit cube was of length 250 Mpc/h. The beam was simulated out to redshift 0.6. This was initially due to the lack of a simulation of sufficient size such that it would result in a 4 degree beam at z = 1. Cosmological calculations were then done and the results showed that our maximum redshift would be 0.6 and thus a beam width at this redshift is 2.56 degrees (Θ(z) = 1x(1+z)^2 degrees).
Fig2. The vuvuzela simulation for the 250 Mpc/h N-body Gadget2 simulation.
Fig3. The 2.56 degree wide beam (at z = 0.6)
Fig4. The Vuvuzela (at z = 0.6). Here the plot is of xy and z (spatial coordinates)