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# C. Correlation between and

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 is not correlated with p as long as hypotheses and 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 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 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 less than a given r*. It corresponds to introduce an extra selection function defined as follows:

Note that introducing this extra selection effect does not alterate the result . On the other hand, a wrong value of the DTF slope (i.e. ) correlates p and :

where is the square of standard deviation of p. The null-correlation approach thus consists to adopt the value of the parameter a verifying as the correct estimate of the slope aD of the DTF relation.

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

where the normalisation factor A2 reads:

It thus turns out that , which implies . This result is insensitive to the shape of the selection function in m and p and to the theoretical distribution function fp(p) of the variable p. The term 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:

• Hypotheses , and are satisfied by the sample.
• The distribution function of p is a gaussian of mean p0 and dispersion (i.e. ).
• The selection effects in observation restrict to a cut-off in apparent magnitude (i.e. where is the Heaveside distribution function, if and otherwise).
The subsampling is performed by using the extra selection function proposed Eq. (C2 (click here)). By introducing such that , rewrites . Adding this extra selection function , the probability density of Eq. (11 (click here)) reads:

Since the probability density 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 entering Eq. ({C3 (click here)), 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 and :

where the functions and are defined Eq. (C14 (click here)) and Eq. (C15 (click here)). For a given cut-off in distance estimate , the contribution of and to has to be compared with appearing in Eq. (C3 (click here)) when a wrong value of the DTF slope is adopted. This term reads:

where the function is defined Eq. (C19 (click here)) and and Eqs. (C14 (click here), C15 (click here)). Finally, the last quantity of interest is the ratio of selected objects within the observed sample. This ratio is equal to the normalisation factor divided by , i.e. the normalisation factor of the sample when no extra selection effect is present (r*=0 or ). It reads in function of the extra cut-off in distance estimate introduced Eq. (C8 (click here)) as follows:

NOTATIONS:

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

where K(N), function of an integer N, is:

and , function of the extra cut-off and of an integer N, is:

The function , involved in the calculation of , is defined as follows:

Figure 4: Mathewson spiral field galaxies sample versus simulated sample

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