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playground:here [2013/02/01 11:43]
llegodi |
playground:here [2021/12/09 16:42] (current)
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| {{:playground:vuvu.png?300|}} | {{:playground:vuvu.png?300|}} |
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| | // Fig1. The vuvuzela simulation.// |
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| **Carlson White Methodology** | **Carlson White Methodology** |
| **Implementation to the "vuvuzela"** | **Implementation to the "vuvuzela"** |
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| 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) 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. | 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. |
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| | 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. |
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| | 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). |
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| | {{:playground:004vu.png?700|}} |
| | {{:playground:005vu_copy_.png?700|}} |
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| | // Fig2. The vuvuzela simulation for the 250 Mpc/h N-body Gadget2 simulation.// |
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| | {{:playground:006vu.png?700|}} |
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| | // Fig3. The 2.56 degree wide beam (at z = 0.6)// |
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| | {{:playground:vuvuzela0-01.jpg?500 |}} |
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| | // Fig4. The Vuvuzela (at z = 0.6). Here the plot is of xy and z (spatial coordinates)// |
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| From the above, the angular diameter distance //DA// is given by the relation //DA = d'/Θ// where //Θ// is now in radians and //d' ∼ d// at low redshifts. | |
