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C. Correlation between tex2html_wrap_inline3443 and tex2html_wrap_inline3579

In this appendix, the reliability of null-correlation approach in order to estimate the slope aD of the DTF relation is discussed. In a first step, it is shown that the non-observable variable tex2html_wrap_inline3583 is not correlated with p as long as hypotheses tex2html_wrap_inline2911 and tex2html_wrap_inline2755 are satisfied. Particularly, this property does not depend on the selection function on m and p. In a second step, the correlation between p and the observable variable X:
is investigated. The presence of radial peculiar velocities such as the ones assumed in tex2html_wrap_inline2883 indeed correlates p and X. However, it is shown that the amplitude of this effect can be weakened by selecting only distant galaxies of the sample. It is not surprising since the term tex2html_wrap_inline2977 in Eq. (C1 (click here)) becomes negligible for distances r large enough. Such a subsampling can be performed by discarding galaxies of the observed sample which have a distance estimate tex2html_wrap_inline2761 less than a given r*. It corresponds to introduce an extra selection function tex2html_wrap_inline2985 defined as follows:
Note that introducing this extra selection effect does not alterate the result tex2html_wrap_inline3615. On the other hand, a wrong value of the DTF slope (i.e. tex2html_wrap_inline3617) correlates p and tex2html_wrap_inline3621:
where tex2html_wrap_inline2935 is the square of standard deviation of p. The null-correlation approach thus consists to adopt the value of the parameter a verifying tex2html_wrap_inline2941 as the correct estimate of the slope aD of the DTF relation.

It is proven below that tex2html_wrap_inline3633 as long as assumptions tex2html_wrap_inline2911 and tex2html_wrap_inline2755 are satisfied by the sample. Rewriting the non-observable variable X0 in terms of m, p and tex2html_wrap_inline2727 gives tex2html_wrap_inline3647. In terms of tex2html_wrap_inline2707, it reads tex2html_wrap_inline3651 where tex2html_wrap_inline2827 was previously defined Eq. (9 (click here)). Since tex2html_wrap_inline3655 and tex2html_wrap_inline3657 are constants, tex2html_wrap_inline3659. The null-correlation between p and tex2html_wrap_inline2707 can be shown by replacing tex2html_wrap_inline2739 by tex2html_wrap_inline3499 and using the probability density tex2html_wrap_inline2835 of Eq. (5 (click here)) expressed in terms of p, m and tex2html_wrap_inline2707:


where the normalisation factor A2 reads:
It thus turns out that tex2html_wrap_inline3679 , which implies tex2html_wrap_inline3615. This resultgif is insensitive to the shape of the selection function tex2html_wrap_inline2733 in m and p and to the theoretical distribution function fp(p) of the variable p. The term tex2html_wrap_inline3695 entering into Eq. ({C3 (click here)) does not vanish since p is correlated with distance r. However, its amplitude can be reendered arbitrarily small by adding an extra selection effect such as the one mentioned Eq. (C2 (click here)). This feature is illustrated by the following example. It is assumed that:

The subsampling is performed by using the extra selection function tex2html_wrap_inline2985 proposed Eq. (C2 (click here)). By introducing tex2html_wrap_inline3727 such that tex2html_wrap_inline3729, tex2html_wrap_inline3731 rewrites tex2html_wrap_inline3733. Adding this extra selection function tex2html_wrap_inline2985, the probability density tex2html_wrap_inline3469 of Eq. (11 (click here)) reads:
Since the probability density tex2html_wrap_inline3739 describing the sample is fully specified, calculations of quantities of interest can be performed analytically or by using numerical simulations. Hereafter, cumbersome intermediate calculations have been deliberately omitted and we just furnish the ultimate analytical expressions of these quantities.

In order to evaluate the amplitude of the term tex2html_wrap_inline3695 entering Eq. ({C3 (click here)), tex2html_wrap_inline3743 is expanded in Taylor's series up to order 2:
It implies that the covariance may be expanded as follows:
The calculations give for the terms tex2html_wrap_inline3023 and tex2html_wrap_inline2949:
where the functions tex2html_wrap_inline3749 and tex2html_wrap_inline3751 are defined Eq. (C14 (click here)) and Eq. (C15 (click here)). For a given cut-off in distance estimate tex2html_wrap_inline3727, the contribution of tex2html_wrap_inline3023 and tex2html_wrap_inline2949 to tex2html_wrap_inline3759 has to be compared with tex2html_wrap_inline3761 appearing in Eq. (C3 (click here)) when a wrong value of the DTF slope is adopted. This term tex2html_wrap_inline3035 reads:
where the function tex2html_wrap_inline3765 is defined Eq. (C19 (click here)) and tex2html_wrap_inline3767 and tex2html_wrap_inline3769 Eqs. (C14 (click here), C15 (click here)). Finally, the last quantity of interest is the ratio tex2html_wrap_inline3771 of selected objects within the observed sample. This ratio is equal to the normalisation factor tex2html_wrap_inline3773 divided by tex2html_wrap_inline3775, i.e. the normalisation factor of the sample when no extra selection effect is present (r*=0 or tex2html_wrap_inline3779). It reads in function of the extra cut-off in distance estimate tex2html_wrap_inline3727 introduced Eq. (C8 (click here)) as follows:
where tex2html_wrap_inline3783 is defined Eq. (C18 (click here)) and the tex2html_wrap_inline3785 function is introduced Eq. (C17 (click here)).


tex2html_wrap_inline3749 and tex2html_wrap_inline3751, function of the extra cut-off tex2html_wrap_inline3727 and of an integer N, are defined as follows:

where K(N), function of an integer N, is:
tex2html_wrap_inline3785, tex2html_wrap_inline3809 and tex2html_wrap_inline3811 functions reads:
and tex2html_wrap_inline3783, function of the extra cut-off tex2html_wrap_inline3727 and of an integer N, is:
The function tex2html_wrap_inline3765, involved in the calculation of tex2html_wrap_inline3821, is defined as follows:


Figure 4: Mathewson spiral field galaxies sample versus simulated sample

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