In this appendix, the reliability of null-correlation approach
in order to estimate the slope a^{D} 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 a^{D} of the DTF relation.
It is proven below that
as long
as assumptions and are satisfied
by the sample. Rewriting the non-observable variable X_{0} 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 A_{2} 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 f_{p}(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:
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:
where
is defined Eq. (C18 (click here)) and
the
function is introduced Eq. (C17 (click here)).
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
functions reads:
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