3. Power spectrum analysis of the reconstructed maps
3.1. Noise statistical properties of the reconstructed maps
It is clear that the mass reconstruction process does not produce any boundary effects (which is settled by definition). The only boundary effect, slightly detectable on the figure, is a larger level of noise at the edge of the field, due to the change in the finite difference scheme at that position (see Appendix B). The noise due to the intrinsic ellipticities of the galaxies is clearly visible at small scales.
Since the least method used to reconstruct the convergence is a local process, it is unlikely that noise propagates on scales larger than the pixel size 4.
Fig. 3 shows the power spectrum analysis of 60 reconstructed mass maps in the case of two different cosmological models (cases (a) and (b)) and (cases (c) and (d)). Fig. 3a and c show the noise free power spectrum (solid lines) the power spectrum measured on the reconstructed maps with gal/arcmin2 (dotted lines) and for gal/arcmin2 (dashed lines). The plateau for the latter two cases is the consequence of the intrinsic ellipticities of the galaxies: at small scales, the power is dominated by the ellipticity of the galaxies, thus tends to be constant, generally much higher than the signal. Figs. 3b and d show the difference of the power spectrum measurements on the reconstructed maps with the noise-free power spectrum; thin dotted line is for gal/arcmin2 and thin dashed line for gal/arcmin2. For visibility, error bars for scales larger than arcmin have been dropped.
In 3D space, these angular scales correspond approximately to scales from 1 to 30 Mpc. We leave for later studies the problem of inverting the measured projected to the 3D one. This aspect has already been explored at large angular scale by Seljak (1997).
As can be inferred from the power spectra in Fig. 3, the noise has a flat power spectrum, characteristic of a white noise process. Fig. 2 illustrates the fact that the noise is independent of the underlying field and follows a Gaussian distribution. The noise model introduced by Kaiser (1998) based on the weak lensing approximation is now compared to the noise found in our simulations. The weak lensing approximation applied to Eq. (10) gives a local shear estimate , where is the true shear and , the mean intrinsic ellipticities of galaxies in the superpixel p. Since the noise components are assumed to be spatially uncorrelated, the statistical properties of the noise are,
where , is the Kronecker symbol and is the variance of one component of the intrinsic ellipticities in one superpixel. The shear and the intrinsic ellipticities of the galaxies are uncorrelated in the weak lensing approximation. The measured power (on the noisy mass maps) can then be expressed only in terms of the true power (the one we want to estimate) and the power spectrum of the noise (Eq. (13)). If denotes the Fourier transform of the measured convergence, its power spectrum is given by
This equation is only valid for a compact survey, where is the mean number density of the galaxies per superpixel, and is the mean value of over the survey. For a sparse survey, the first term in Eq. (14) is changed into a convolution term, but the noise contribution to the observed power spectrum remains independent of that power spectrum. A convenient way to estimate is to take
where is the number of superpixels and the number of galaxies in the superpixel p. This is nothing else but the variance of the observed ellipticities of the selected galaxies 5.
The thin dashed and dotted straight lines on Fig. 3 correspond to the expected noise power spectra (for gal /arcmin2 and gal/arcmin2) according to the Kaiser's model given by Eq. (14). It perfectly fits the noise part of the mass reconstructed with noisy data, whatever the cosmological model and the noise level. This is true even for the open cosmological models for which stronger non-linearities could have produced a stronger coupling with the noise. The fact that the noise component is pure white noise with an amplitude in agreement with the theoretical prediction is a remarkable result since the full non-linear equations were used, and it shows that the weak lensing approximation can be safely used to remove the noise component and to get an unbiased estimate of the power spectrum, down to the smallest scales considered here.
The behavior at scales smaller than our pixel size remains partly an open issue for two reasons: first, due to the smaller number of galaxies, the convergence of the reconstruction process as well as the stability of the noise properties has to be investigated. Second, at small scales the gravitational distortion is larger than only a few percent, and it can go up to infinity on the critical lines. Therefore, estimating the variance of the galaxies intrinsic ellipticity distribution, arcs and arclets should be removed. This issue can only be addressed in high resolution simulations like those performed by Jain et al. (1998).
3.2. Power spectrum cosmic variance
Although the estimate of the power spectrum described above is unbiased, the cosmic variance has also to be explored to come up with an optimal observational strategy.
In estimating the cosmic variance of the power spectrum, Gaussian statistics is usually assumed. This hypothesis is tested comparing the cosmic variance assuming Gaussian statistics to that of the simulated mass maps and that reconstructed using the two different level of noise defined before.
In the case of Gaussian statistics high order moment are related to the second order moments via the following relations:
which means that physically, the frequencies of a Gaussian field are not coupled, and that the moment at a given frequency is only determined by the power at that frequency.
We consider a compact survey of size , for which the number of modes available at a frequency is maximum. Thus, following Feldman et al. (1994) and Kaiser (1998), the cosmic variance of is given by the square of the measured signal (which depends on ), divided by the number of independent modes used to determine it, in the k-annulus . Simple modes count gives
Note that in this hypothesis the cosmic variance is independent from the amplitude of the fluctuations. This is not the case when the non-linear couplings are taken into account as it can be seen in Fig. 4. It shows the cosmic variance for flat ((a), (c), (e)) and open (((b), (d), (f)) cosmological models. (a) and (b) correspond to a degree survey with , (c) and (d) with , and (e) and (f) a degree survey with . On each plot, the thin solid line shows the Gaussian cosmic variance, the thick solid line shows the true cosmic variance without noise, the dotted line the noisy maps with gal/arcmin2 and the dashed line with gal/arcmin2. The vertical axis gives the error on the power spectrum measured at a given scale.
The departure from Gaussianity appears for scales below , the effect is however more important in the open case model for which non linearities are stronger. Open models ((b), (d), (f)) give almost the same features as for the flat models (((a), (c), (e)), although the cosmic variance is smaller. This is clearly a consequence of a higher power spectrum signal at low scales for these models, which is visible when comparing Fig. 3 (a) (flat) and (c) (open). Thus, as expected, the intrinsic shape of the power spectrum affects the cosmic variance (Kaiser 1998).
Going deep in redshift (by comparing (a) and (c), or (b) and (d) for the open case) clearly improves the cosmic variance at small scale, since the gravitational distortion is stronger. However this stronger distortion does not improve the large scale power estimation because the cosmic variance at these scales only depends on the whole volume survey.
If shorter wave vectors are observed, (for a degree survey), which is visualized on (e) and (f), a gain of 2 is reached at all scales, as a direct consequence of global increase of the number of modes in Eq. (17). For small scales, from the point of view of the statistics it is in fact equivalent to observe deep over a small area than to observe shallower over a large area (which reflects the fact that the dashed lines in (e) and (f) (gal/arcmin2) are almost the same as the dotted line in (a) and (b) (gal/arcmin2). But on the other hand, the shallow large survey gives a better estimate of the power at large scales than the deep survey. Since these two observational strategies require the same total exposure time it is clear that wide shallow surveys are better than small deep surveys. As it will be shown in Sect. 4, this remains true for the high order moments in real space. Moreover, deep surveys show more and more distant galaxies for which the redshift distribution is more uncertain.
© European Southern Observatory (ESO) 1999
Online publication: December 22, 1998