Spatial uniformity test

In this Section I show the ability of the new UVB model in maintaining a good spatial sampling with reduced computational requirements.

I have simulated a UV background in a box of side length L_b = 10.0h⁻¹Mpc comoving, filled with a pristine gas mixture of H and He (their respective fractions are 0.9 and 0.1). The gas is assumed to be initially neutral and with constant number density of n_gas = 1.0· 10⁻⁵cm⁻³ and temperature T₀ = 100 K. The simulation is set up with a duration of t_f = 7·10⁶yrs as in [158]. A standard grid resolution of N_c³ = 128³ cells is adopted and it is masked in three runs with three U grids of 16³ (U₁₆), 32³ (U₃₂) and 64³ (U₆₄) emitting cells respectively.

The spectrum selected for this test is the first Haardt and Madau cosmic UV background model ([109], hereafter HM96) used also in [158]. The global emissivity in the cube is maintained constant in the three runs, after calibrating the field with the values provided in HM96. I have adopted the same calibration algorithm illustrated in [158]; a brief summary of the calibration technique is provided in the following Paragraph.

The problem of calibrating the intensity of the simulated UVB raises when the assigned emissivity of the UVB must be related to the photon content of each emitted packet: it is in fact not obvious how to convert the assigned flux J_ν

erg s⁻¹cm⁻²Hz⁻¹sr⁻¹

in photons per frequency ν of the sampling packets. In the formalism of the Maselli and Ferrara, the number of monochromatic photons in a packet (N_γ), as function of the simulation parameters, can be written as:

N_γ =A N_c²t_f N_pN_cyc

J_ν

J₉₁₂, (5.11)

whereA[s⁻¹] is the calibration factor and J₉₁₂ indicates the UV background flux at the ionisation edge for hydrogen. In the HM96 J912 ∼ 0.4·10⁻²¹erg cm⁻²s⁻¹Hz⁻¹ is assumed at redshiftz ∼3.

Figure5.1:Leftpanel:temperaturefield(red-to-yellowgradient)simulatedbytheCRASHbackgroundalgorithm volumesliceofLb=10h−1 MpcwithdomainresolutionofNc=128cellsperboxside.Thesliceisplacedat thecubeandperpendiculartothezaxis.TheimagereferstoasimulationconfigurationusingtheHM96([109]) andanemittermaskof163 nodes,afteremittingNp∼5.3·106 packets.Rightpanel:temperaturefield(red-to-y gradient)simulatedbytheCRASHbackgroundalgorithmonavolumesliceofLb=10h−1 Mpcwithdomain Nc=128cellsperboxside.Thesliceisplacedatthecenterofthecubeandperpendiculartothezaxis. referstoasimulationconfigurationusingtheHM96([109])spectrumandanemittermaskof323 nodes,after Np∼5.3·106 packets.

5.2:Leftpanel:ionisationfractionofHeIII(bue-to-redgradient)simulatedbytheCRASHbackgroundalgorithmavolumesliceofLb=10h −1MpcwithdomainresolutionofNc=128cellsperboxside.Thesliceisplacedattheterofthecubeandperpendiculartothezaxis.TheimagereferstoasimulationconfigurationusingtheHM96([109])ectrumandanemittermaskof16 3nodes,afteremittingNp∼5.3·10 6packets.Rightpanel:ionisationfractionHeIII(bue-to-redgradient)simulatedbytheCRASHbackgroundalgorithmonavolumesliceofLb=10h −1MpcwithresolutionofNc=128cellsperboxside.ThesliceisplacedatthecenterofthecubeandperpendiculartothezTheimagereferstoasimulationconfigurationusingtheHM96([109])spectrumandanemittermaskof32 3nodes,emittingNp∼5.3·10 6packets.

calibration is required, as described in the following Paragraph.

As suggested by the authors, I have performed a simulation on a box empty in all the cells but the central one in which I have assigned the valuesn_gas,T = 10⁴K as representative of the ionisation equilibrium imposed by the HM96 model. The cell is chosen at the center of the box to ensure a uniform illumination in each cell face and to maximise the sampling of the radiation field created inside its volume. The number density of the cell used for the calibration is simply n_gas because the box selected for this test is uniform, and T is the reference value for the photo-ionisation of a gas compatible with the equilibrium assumptions of the HM96 model.

During the calibration run,CRASHcounts all the photons absorbed in the cell and derives the average hydrogen ionisation rate Γ_HI in the central cell. This is done first by collecting the absorbed photons per spectrum frequency and per species, and after by solving the integrals in Formula 3.2. Once the photo-ionisation rate is known per each species, A is obtained as the best calibration parameter reproducing the value of Γ_HI andx_HI, indicated in HM96 as Γ_HI = 1.2·10⁻¹²s⁻¹, i.e. calibrating the photo-ionisation equilibrium of the hydrogen. I found a calibration value of A= 1.48·10⁻⁵²s⁻¹; this value is compatible with the value obtained in [158] with a similar simulation set-up but the old model in use.

I now proceed to show the results of this test. Because the RT is performed on a uniform medium, once the box is in equilibrium with the UVB after a time t_eq, I expect to see a low spatial scatter in the values of the tracked UV intensity and also in the gas ionisation fractions and temperature.

The time t_eq is determined comparing the volume averaged values of all the physical quantities (i.e. ionisation fractions and temperature) and selecting teq after which the values in the successive output differ by 10⁻⁵.

As expected, I found the quality of the spatial sampling is increased with higher resolu-tions of the maskU but the large number of emitters in the runU64results in a prohibitive slowdown of the code performances. This is due to a combination of factors. First, the number of emitters is increased and then the loop over the array of EMITTER POINTS re-quires more CPU time. Second, the number of emitters near the faces of the cube is higher and the number of packets immediately escaping the box (i.e. with large energy content) is significant. With the adoption of periodic boundary conditions and N_cyc = 10 (see [158]), each packets re-entering the box, performs a large number of cell-crossings and then the code rapidly slows down in its performances.

Finally, because the HM96 background is very efficient in rapidly ionising the box with the assumedngas, the increased transparency of the medium implies that more CPU time is needed to complete a simulation.

After a series of numerical experiments balancing N_p and the resolution of the grid U, I found that a combination of 32³ emission nodes and a number of Np = 179 packets per node is sufficient to create an equilibrium configuration within t_eq = 10³yrs. This is also

consistent with similar numerical experiments performed by Maselli and Ferrara.

The differences in the new model reside in the quality of the obtained spatial resolution as visually illustrated by a slice of the temperature map in Figure 5.1, taken at the equilib-rium time. The left and right panels of this Figure refer to the simulation set-up reported above but with different resolutions of the mask U. The left panel refers to the case U₁₆ and reports the temperature field (red-to-yellow gradient) after a total emission of 5.3·10⁶ packets. The right panel instead reports the case with U₃₂ nodes after the emission of the same number of total packets.

The field in the right panel results visually more homogeneous with less regular patterns thanks to the increased number of equispaced emitters in the volume. The simulated UVB creates then a more uniform temperature background on a constant gas configuration, as expected. The finer grid in fact allows more cell-crossings for each packet and the bubbles created around each emitter overlap faster, reducing the spatial inhomogeneities.

The situation is better illustrated in Figure 5.2 reporting the ionisation fraction of HeIII

on the same slice. From the tests performed on various Str¨omgren spheres (see Chapter 4), I know that the patterns of the HeIII bubbles are sharper than the temperature ones; this is clearly shown in the left panel. The right panel instead appears remarkably regular, but on the box borders where the background is still not well established. Note however that the same trend is uniformly present in all the box sides, showing a high degree of symmetry of the generated background field. To solve the problem at the borders, moreN_p emissions are needed, as confirmed by the successive ionised snapshots. On the other hand, because the problematic cells are regularly confined to the borders of the simulated box, in many applications they could be easily rejected from the data, before doing a statistical analysis.

As final remark I want to point out that this test can be used also to constraint the average MC noise affecting any realistic simulation involving a cosmic web. After selecting an appropriate size for U and a value of Np which guarantees the MC convergence, it is possible to run a simulation with a constant number density equal to the mean gas number density of the cosmic web and to provide an average estimate of the MC noise in the uniform case. This estimate can then be subtracted to the fluctuations found in the real simulation once the best grid resolution has been established.

A realistic application of this technique is provided in the following Sections.