5. Deprojection of cluster images from gas-dynamical simulations
Having put the multiple-data Richardson-Lucy algorithm together, the next important step is to explore how this algorithm behaves, when it is applied to observed data. In particular, we have to assess the key property of the MDRL algorithm, namely the quality of the reconstructed gravitational potential achievable for a given set of input data. Therefore we have to apply the method first to input data for which we know the true gravitational potential. For this purpose we use clusters from gas-dynamical simulations kindly provided by Klaus Dolag (Dolag et al. 1999), to construct observed images for the lensing potential , the X-ray surface brightness , and the SZ-temperature decrement , reconstruct the gravitational potential , and compare it to the true gravitational potential calculated directly from the simulation data. The gas-dynamical simulations include information on the dark-matter distribution, the gas distribution and the temperature of the cluster. They were created using a GRAPE-MSPH code that combines the gravitational interaction of the dark matter component with the hydrodynamics of a gaseous component. In addition, the code includes the magnetohydrodynamic equations following the evolution of the magnetic fields.
The cluster simulations were run using a COBE-normalized CDM power spectrum with a Hubble constant km s-1 Mpc-1, and = 1.0, . The virial mass of the cluster used is , which resides in a volume of roughly . The simulations contain approximately dark-matter particles and also the same number of gas particles. The dark-matter particles have a mass of . The masses of the DM and gas particles provide an estimate for the resolution limit of the simulations. For the purpose of mimicking "observed data sets" within current observational limits, the above resolution is completely sufficient.
The gas and DM distributions of single clusters from the simulations are then used to compute the true gravitational potential of the cluster, from which then the observed lensing (Eq. 14), X-ray (Eq. 22), and SZ-data (Eq. 25) are deduced, which in turn serve as input for the MDRL algorithm. Fig. 3 shows three typical input sets created from the gas-dynamical cluster simulation of a single, very massive sample cluster. From the left to the right the lensing potential , the X-ray surface brightness in the energy band 2 keV to 12 keV, and the temperature decrement at an assumed frequency of 10 GHz are displayed. The inclination angle is fixed at . If not stated otherwise all reconstructions shown were obtained with 8 iteration steps.
Before turning to the multiple data RL-reconstruction of the potential , we have to investigate how well the algorithm works for each of the three different types of input data separately. This means we first consider only the case where is deprojected from either lensing, X-ray, or SZ-data alone, and then compare the findings.
We start by looking at different initial guesses for the gravitational potential , which are used in the first iteration cycle. Ideally the algorithm should not depend on the choice of the initial guess, therefore two extreme cases for the initial guess are tested and the results are shown in Fig. 4. On the left of Fig. 4 we use a gravitational potential obtained from the universal dark matter profile found by Navarro et al. (1996, 1997; combined NFW) as initial guess, which resembles the original profile rather closely, especially concerning the curvature of the potential. On the right of Fig. 4 we use a plane as initial guess, which makes only minimal assumptions about the potential. Fig. 4 shows the reconstructed gravitational potential after 8 iterations for lensing data only, but the results for X-ray and Sunyaev-Zel'dovich data are very similar. Comparing the reconstructed potentials in the lower panels obtained from these completely different initial guesses we clearly see that both initial guesses lead to qualitatively very similar results. The main difference is that the potential reconstructed with the NFW profile as initial guess is steeper in the most central part as opposed to the potential obtained from the plane as initial guess. This can be attributed to the fact that the smallest ellipses with coordinates close to zero are not taken into account for numerical reasons. This is due to the fact that the finite difference formula requires at least four points on the ellipse. Therefore the NFW profile when used as inital guess provides a very high curvature for this part of the potential. In addition, the potential reconstructed from the NFW profile shows less "artefacts" for large -coordinates. Even though the "bump" for large Z-coordinates and R-coordinates is less pronounced for the potential reconstructed from the NFW profile than for the one reconstructed from the plane, the behaviour for these large values of is introduced by the fact that the data field used for the integration is finite. As large -coordinates constitute the boundaries these differences are not relevant for assessing the quality of the reconstruction. For the reconstruction the behaviour in the central Mpc of the cluster is much more important. In this sense the differences found for the two choices of initial guesses are negligible.
The evolution of the reconstructed gravitational potential with the number of iteration steps n is exemplified in Fig. 5. The potential obtained from the X-ray and SZ case evolves in a qualitatively similar way. The initial guess in this case is a plane shown in the upper left panel, while the lower right panel shows the original potential obtained directly from the simulated cluster. Fig. 5 demonstrates that the algorithm converges extremely fast, even for an initial guess making only minimal a priori assumptions about the cluster potential. Already after the second iteration the potential is in the correct order of magnitude and has acquired the characteristic features of the true cluster potential. In addition, we notice that is hardly altered in the last two steps, indicating that the algorithm has converged in the sense that most of the large-scale information is recovered. The two main differences between the reconstructed potential from the last step, , and the true cluster potential is the presence of two "dents" in at ( Mpc Mpc), and several small "wiggles" at the flanks of the potential for . we found that the size of the "dents" can be correlated with the finite range of the "observed data", hinting again at the inherent problems with the finiteness of the boundaries as discussed in Sect. 4.1. The "wiggles" reflecting the property of the algorithm to fit small scale fluctuations last, are in this case probably caused by numerical discretization effects, and thus reflect an unwanted property of the algorithm. This numerical noise can be suppressed by using a smoothing procedure after every iteration step.
In order to gain a better understanding on how the algorithm converges for the three different types of input data, it is instructive to look at the integrals 27a to 27c. For fixed these integrals determine the multiplicative factors that advance to via Eq. 29, and, as already mentioned, good convergence requires that these integrals approach unity. The values of , , and after the last step of Fig. 5 are plotted in Fig. 6. We see that the convergence after 8 iterations is already excellent over the full range of values for all integrals. The X-ray and SZ integrals and both overestimate unity by the same amount and feature a similar shape, reflecting the fact that they have a similar dependence on the potential , whereas the lensing integral deviates more strongly from unity to both larger and smaller values, indicating a different convergence behaviour. The main difference is the fact that the integral for small values of R and large values of Z is below unity thus lowering the potential in this range.
We finish the discussion of the reconstruction from single data sources by comparing in Fig. 7 the true, original gravitational potential to the results of the reconstructions using the lensing potential, the X-ray surface brightness and the SZ temperature decrement alone as input data for the MDRL algorithm. By this the amount of information on the 3-dim. structure can be determined that is already present in each of the single data sets. Looking at the surface plots of Fig. 7 one sees that all three types of input data give qualitatively very similar results. The lensing reconstruction is superimposed by numerical noise, which is also present in the X-ray and SZ case, albeit much less pronounced.
A more detailed comparison of the three reconstructed and the reference potential is possible if surface cuts such as in Fig. 8 are studied. For the cuts through the central part of the cluster we generally see a good agreement of the three reconstructions with the original potential. The agreement becomes worse for larger radial coordinates as displayed in the right panel of Fig. 8 for the cut through . We also notice that the difference between the X-ray and the SZ case is negligible, reflecting their very similar dependence on the gravitational potential.
Compared with the lensing potential both, the X-ray and the SZ data give a very good reconstruction of the inner parts of the potential ( Mpc) which is especially true for the cut along the Z-axis, where the match is nearly perfect. The numerical noise in the lensing potential is more pronounced than in the X-ray and the SZ-case. More importantly, in contrast to the X-ray and SZ-case the curvature and the overall shape of the lensing reconstruction is closer to the true potential even for larger radial coordinates Mpc.
Finally, we are in a position to combine all input data sets for a true multiple-data reconstruction , thus allowing a better reconstruction of the gravitational potential . For the present example we chose to combine all three data sets with weight factors of each. The results of the reconstruction are shown in Fig. 9. In the upper panel we compare two cuts of the result of the reconstruction obtained after 8 iterations with the original potential and the reconstructions computed for the single data sets, respectively. Especially for the cut we do see an improvement over the use of just one single data set: The combined reconstruction is more reliable even for values of Mpc. This leads to better results when compared to the reconstructions obtained from X-ray and SZ data alone. The full surface plots of the original potential and the combined reconstruction demonstrate that the multiple data set reconstruction is able to recover all important features.
At this point it is worthwile to assess the quality of the reconstruction in a quantitative way. For this purpose it is instructive to look at the relative errors between the original gravitational potential and the reconstructed one , which is computed as . In order to compute the relative error we have to shift the potentials in such a way that their respective central parts are on the same level as only the gradient of the potential has a physical meaning. The result for the inner part of the potential, i.e. Mpc and Mpc, is displayed in Fig. 10. In the different panels of this figure the relative errors between the original and the reconstructed potential for lensing data (upper left panel), X-ray data (upper right panel), and Sunyaev-Zel'dovich data (lower left panel) are shown; in addition, the result for the combination of all three data types is given in the lower right panel.
For all four reconstructions we see that the deviation over large parts of the potential is less than . When looking at the lensing reconstruction in more detail, we note that in this case the zone with an error margin of less than is relatively wide, especially in the Z-direction. As already noted before, both, the X-ray and the Sunyaev-Zel'dovich reconstruction, show very similar features, which is also reflected in Fig. 10. Both cases give excellent reconstructions in the center with coordinate values of less than Mpc, but the quality of the reconstruction in the outer parts is not as good as in the lensing case. This confirms the theoretical expectation, that the data from X-ray and SZ measurements, which have their main contributions coming from the cluster core, are less affected by projection effects, nicely complementing the weak lensing data, which is only sensitive to the gravitating matter.
When looking at the results of the combined reconstruction an improvement over the single data reconstructions is obvious. Here the region with error margins of less than is the largest.
Depending on the quality of the data at hand and a priori knowledge of the possibly different levels of noise present in the data, it is possible to adjust the weighting of the different data sets. This is also useful when one is interested in a limited region of the cluster which might be represented more accurately by a certain observable, like e.g. the cluster center which significantly contributes to the X-ray and SZ data. In summary, the combination of data sets can be expected to give improved results, with the astronomer being able to control the reconstruction process by means of the weight factors.
© European Southern Observatory (ESO) 2000
Online publication: January 29, 2001