## 5. Deprojection of cluster images from gas-dynamical simulationsHaving 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 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 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
The evolution of the reconstructed gravitational potential
with the number of iteration steps
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
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 Finally, we are in a position to combine all input data sets for a
true
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 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 |