Volume 520, September-October 2010
|Number of page(s)||22|
|Section||Cosmology (including clusters of galaxies)|
|Published online||24 September 2010|
In this appendix, we derive an approximation to the surface density and projected mass (or, equivalently, projected number) profiles for the Einasto model.
For any density model, the projected mass is
where the second equality is obtained after reversing the order of integration. Equation (A.1) is general, while Eq. (A.2) is only valid for models with finite total mass . For the Einasto model of total mass , the 3D mass profile is
where is the regularized incomplete gamma function. The ratio
determined from Eqs. (A.3) and (A.2), with Eq. (15), varies little, as seen in the right panel of Fig. A.1. We fit again a two-dimensional fourth-order polynomial in m and and find
In the interval and , Eqs. (A.5) and (A.6) are accurate to better than 1.5% everywhere (0.23% rms).
The projected mass of the Einasto model can thus be written
where again .
Writing the dimensionless mass density as
where the second equality derives from Eq. (15), we can express the ratio of dimensionless surface to space densities as
where X=R/r-2. In the range (spanned by CDM halos in the redshift range according to Gao et al. 2008) and , varies little and regularly, as seen in the left panel of Fig. A.1. We fit a two-dimensional 4th-order polynomial in m and to . We find
In the interval and , Eqs. (A.13) and (A.14) are accurate to better than 0.8% everywhere (0.12% rms).
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The dimensionless surface density can then be written as
or equivalently, with , where is the regularized incomplete gamma function and the total mass of the Einasto model:
Alternatively, the surface density profile can be, self-consistently, estimated from Eq. (A.7) by differentiation over the projected mass profile, yielding after some algebra
where is given in Eq. (A.11).
Equation (A.17) has the advantage of providing an approximation for the surface density profile that is consistent with that of the projected mass profile. This is crucial for maximum likelihood estimation of concentration (and possibly Einasto index and background level). On the other hand, the accuracy of Eq. (A.17) is about 5 times worse than that of Eq. (A.15).
For any density model, the tangential shear measured by weak lensing can be
written (e.g. Miralda-Escude 1991)
where is the mean surface density, while is the critical surface density, with c the velocity of light, and where , , and are the angular diameter distances between the observer and the source, the observer and the lens, and the lens and the source, respectively. Equation (A.18) indicates that adding a constant term to the surface density (Eq. (24)) has no effect on (this is the mass-sheet degeneracy). For the Einasto model, the tangential shear (Eq. (A.18)) is readily computed using Eqs. (A.15) with (A.14) and (A.8) with (A.6). Figure A.2 shows the subtle differences in the shear profile between the NFW and Einasto models of index m=4, 5, and 6. While the tangential shear of the four models is indistinguishable in the wide range 0.8 < R/r-2 < 10, there are potentially measurable differences at R > 10 r-2 (at 100 r-2, the NFW shear is 1.5 times greater than that for the m=5 Einasto model) and possibly at R < 0.8 r-2 (as long as the weak linear approximation assumed for the measured shear to match the expression of of Eq. (A.18) remains valid).
Dimensionless tangential shear profile for the NFW model (black) and the m=4 (red, long-dashed), 5 (green, short-dashed) and 6 (blue, dotted) Einasto models, using Eq. (A.18) with Eqs. (A.9), (A.14), (A.15), (A.6), and (A.8).
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Appendix B: Surface density and projected mass of the NFW model with lines of sight limited to a sphereIn this appendix, we derive the surface density and projected mass (or, equivalently, projected number) profiles of the NFW model, with the lines of sight restricted to a sphere (which we conveniently choose as the virial sphere) instead of extending to infinity.
In an analogous manner as the case with line-of-sight extending to infinity
the surface density at projected radius R within the sphere of radius
We now consider the case of the virial sphere: . The surface density can then be written
where Eq. (B.3) is found by inserting the NFW density profile (Eqs. (4)) into (B.1).
For the NFW model, the projected mass within the virial sphere is
where Eq. (B.7) was found by inserting Eq. (B.4) into Eq. (B.6). For , one recovers the mass within the virial sphere.
In this appendix, we illustrate the maximum likelihood calculations that we have performed.
and data points
the MLE is found by minimizing
where is the likelihood.
The probability of measuring an object (galaxy or dark matter particle)
at radius r in a spherical model of
concentration c is
where and N(R) are respectively the density and number (proportional to mass) profiles, and are respectively the minimum and maximum radii, c is the concentration, while b is the constant density background.
The probability of measuring a galaxy at projected radius R in a spherical
model of concentration c and background b is
where and are respectively the surface density and projected number (proportional to projected mass) profiles, and are respectively the minimum and maximum projected radii, c is the concentration, while is the constant surface density background.
According to Eq. (8),
the distribution of interloper line-of-sight absolute velocities,
is to first order the sum of a
Gaussian and a constant term:
where the denominator is found by ensuring (Eq. (12)), and where is the maximum considered value of (so in Figs. 6). If A and B are expressed in virial units, then the denominator of Eq. (C.4) is the surface density of particles under consideration in virial units, which we directly measure from the simulation as , where Nv is the number of particles within the virial sphere, while N is the number of particles in the radial bin (or within the full virial cone), and S is the surface of the radial bin (i.e. for the full virial cone). Hence, substituting for , we can write the probability of measuring an interloper absolute velocity as
where . Then given the respective uncertainties and in and B', we deduce the uncertainties in B and A as
For one-parameter fits, we first search on a wide linear grid of equally-spaced 11 points for , then we consider the three points with the lowest values of and create a subgrid of 11 equally-spaced points (thus typically zooming in by a factor of 5), and iterate with finer subgrids until the two values of the parameter with the highest likelihoods differ by less than 0.0001 or when the lowest decreases by less than 10-12. We then obtain the confidence interval fitting a cubic spline to the points below and above the best-fit parameter to solve for .
For two-parameter (three-parameter) fits, we first search on a wide rectangular (cuboidal) grid of equally-spaced 11 points. Then we consider the rectangle (cuboid) obtained by searching for the lowest values of , such that there are at least 3 different values for both (all three) parameters. We create a sub-grid in this rectangle (cuboid) with again ( ) points, and iterate with finer subgrids until the pair of each of the two (three) parameters with the highest two likelihoods differ by less than 0.0001 or when the lowest decreases by less than 10-12. We then obtain the contour by considering those points in parameter space for which (1.77), and then define as the minimum and maximum values for each parameter the extreme values in this contour.
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