Issue 
A&A
Volume 578, June 2015



Article Number  A10  
Number of page(s)  12  
Section  Cosmology (including clusters of galaxies)  
DOI  https://doi.org/10.1051/00046361/201424456  
Published online  22 May 2015 
Online material
Appendix A: CosmicPy package
Listing 1
Example of 3D SFB and tomographic Fisher matrix computations using CosmicPy. 

Open with DEXTER 
CosmicPy is an interactive Python package that allows for simple cosmological computations. Designed to be modular, welldocumented and easily extensible, this package aims to be a convenient tool for forecasting cosmological parameter constraints for different probes and different statistics. Currently, the package includes basic functionalities such as cosmological distances and matter power spectra (based on Eisenstein & Hu 1998; and Smith et al. 2003), and facilities for computing tomographic (using the Limber approximation) and 3D SFB power spectra for galaxy clustering and the associated Fisher matrices.
Listing 1 illustrates how CosmicPy can be used to easily compute the 3D SFB Fisher matrix, extract the figure of merit, and generate the associated corner plot similar to Fig. 3.
The full documentation of the package and a number of tutorials demonstrating how to use the different functionalities and reproduce the results of this paper is provided at the CosmicPy webpage: http://cosmicpy.github.io
Although CosmicPy is primarily written in Python for code readability, it also includes a simple interface to C/C++, allowing critical parts of the codes to have a fast C++ implementation as well as enabling existing codes to be easily interfaced with CosmicPy.
Contributions to the package are very welcome and can be in the form of feedback, requests for additional features, documentation, or even code contributions. This is made simple through the GitHub hosting of the project at https://github.com/cosmicpy/cosmicpy
Appendix B: Computing the SFB covariance matrix
Performing a Fisher analysis requires computing the SFB covariance matrix, and more importantly, computing the inverse of this matrix. This last step can be quite challenging as the covariance of the spherical FourierBessel coefficients is a continuous quantity C_{ℓ}(k,k′). Two approaches can be considered to define a covariance matrix in this situation: (i) only using the diagonal covariance C_{ℓ}(k_{i},k_{i}) at discrete points k_{i} (advocated by Nicola et al. 2014); or (ii) binning C_{ℓ}(k,k′) into bins of size Δ_{k}. However, by neglecting the correlation between neighbouring wavenumbers, the first approach overestimates the information content if the interval between wavenumbers is too small, while the second approach would lose information for bins of increasing size and become numerically challenging to invert for bins too small. Another problem is to select the largest scale k_{min} to include in the covariance matrix. Indeed, C_{ℓ}(k,k) becomes extremely small and numerically challenging to compute for very small k, but small wavenumbers can still potentially contribute to the Fisher information. A careful study is necessary to select a k_{min} that does not lose information.
Instead, using the k_{ln} sampling defined by Eq. (23)naturally introduces a minimum wavenumber and a discrete sampling of scales that preserves all the information. As an added benefit, this approach yields numerically invertible covariance matrices in practice for sensible choices of the boundary condition r_{max}. Indeed, as , the choice of cutoff radius sets the fineness of the C_{ℓ}(n,n′) matrix and affects its condition number. However, we find that the Fisher information remains largely unaffected by varying r_{max} above a certain distance because cutting the very end of the galaxy distribution has little effect. In practice, we have arbitrarily set r_{max} to the comoving distance at which φ(r) reaches 10^{5} of its maximum value. This choice has proven stable in all situations considered in this work. The robustness of our computation of the Fisher matrix with respect to the choice of r_{max} is illustrated in Fig. B.1, where we show the contributions of each angular mode to the Fisher matrix element . Our empirical choice for r_{max} in this case is 5420 h^{1} Mpc, but the results are not affected by increasing r_{max} to 5700 h^{1} Mpc even more.
Fig. B.1
Contribution to the SFB Fisher matrix element as a function of angular mode, computed with different values of r_{max}. The excellent agreement between the two curves shows that our computation of the Fisher matrix is robust to our arbitrary choice of r_{max}. 

Open with DEXTER 
Appendix C: Deriving the spherical FourierBessel shot noise power spectrum
Here, we derive the expression of the shot noise by discretising the survey in cells that either contain one or zero galaxies (Peebles 1980). This method was used in Heavens et al. (2006) to yield the expression of the shot noise in the case of 3D cosmic shear. We considered a point process defined on small cells c, each of which contains n_{c} = 0 or 1 depending on whether the cell contains a galaxy or not: (C.1)where δ_{c}(r) = 1 if r is within the cell c, 0 otherwise, and where n_{c} fulfils (Peebles 1980) (C.2)where Δ_{c} is the volume of cell c and is the average number density of galaxies of the survey at distance r_{c}. Furthermore, the crossterm for c ≠ d is (C.3)The SFB expansion of the density field can now be expressed as a sum over small cells c: (C.4)\newpage\noindentFrom this expression, we can derive the twopoint correlation function of this field: In the last equation, the first term for c = d contains the shot noise contribution and the second term contains the monopole contribution and the correlation function of the density fluctuations. Returning to continuous integration by decreasing the volume of cells Δ_{c}, we have (C.7)Therefore, in this expression, we recognize three terms:

the shot noise contribution, onlyfor l = l′ and m = m′:(C.8)

the contribution from the power spectrum, only for l = l′ and m = m′: (C.10)
© ESO, 2015
Current usage metrics show cumulative count of Article Views (fulltext article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 4896 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.