The presence of the radial peculiar
velocities v is included in the statistical modelization
by rewriting the density probability 
of Eq.
(5 (click here))
as follows:
where
is the distribution function of radial
peculiar velocities depending in general
on the spatial position
with
(l,b) the direction of the line-of-sight in
galactic coordinates.
The aim is herein to estimate the calibration parameters
of the DTF relation entering into
the radial peculiar velocity estimator
of Eq (9 (click here)) (i.e. the DTF
slope
aD and DTF "zero-point" defined as 
).
The calibration sample
is constituted of field galaxies selected along the same line-of-sight
of direction (l,b) for which apparent magnitude m,
log line-width distance indicator p and redshift z
are measured. It is assumed hereafter that the sample
is described by the density probability of Eq.
(11 (click here)) and
satisfies the following hypotheses:
0) After appropriate corrections (galactic extinction,
inclination effects, ...), residual measurement errors can be neglected.
1)
is gaussian, i.e.
.
)
The distance modulus distribution function reads
with
.
3)
The distribution of radial peculiar velocities v
is a gaussian of
mean u and dispersion 
,
i.e. 
.
appears as a reasonable assumption.
Anyway, if the line-of-sight direction of the sample goes across a
physical cluster, nothing prevents us to discard
galaxies known to belong to the cluster.
Assumption 
implies that the radial peculiar velocity field
is not correlated with the distance r.
is thus ruled out if flows such
like the Great Attractor are present along the
line-of-sight. On the other hand,
is less restrictive than the pure Hubble flow hypothesis
(i.e. v=0 everywhere). First, galaxies
may have a Maxwellian agitation
of velocity dispersion .
Second,
the mean radial peculiar velocity along the line-of-sight u
is not forced to zero. It allows to mimic the
following situation. Suppose that a whole-sky sample
is calibrated in a velocity frame of reference where
sampled galaxies are not globally at rest (say
that the sample has a bulk flow 
in cartesian galactic coordinates with respect
to the velocity frame of reference).
The radial peculiar velocity of galaxies belonging
to the same line-of-sight of direction (l,b) will
be shifted by
.
Assumption in fact tolerates this
kind of situation.
Calibration of the DTF slope aD using null-correlation
approach (NCA) is based on the following remark.
For a calibration sample satisfying assumptions
,
,
and
with 
(i.e. pure Hubble flow hypothesis),
the variable X=X(aD) defined as:
is not correlated with p (see Appendix C
for proof):
where 
is the covariance
of variables p and X.
On the other hand,
a wrong value of the
DTF slope (i.e. 
) correlates p and
the random variable 
:
where
is the square of
the standard deviation
of p. The null-correlation approach consists in adopting
the value of the parameter a verifying
as the correct estimate
of the DTF slope aD:
Figure 1:
Variation of the ratio R(r*) of selected object within the observed
sample with respect to the extra cut-off in distance estimate r*
Figure 2:
Variation of the bias 
inferred by the presence
of a Maxwellian velocity agitation of dispersion 
with respect to the extra cut-off in distance estimate r*
(
)
which gives in practice:
Note that in the
case of pure Hubble flow, the NCA is a quite
robust technique for calibrating the slope of the
DTF relation. It furnishes indeed an unbiased estimate
of aD whatever the selection effects 
on m and p which affect the observed sample,
and whatever the specific shape of the theoretical
distribution function fp(p) of the variable p
(see Appendix C).
Unfortunately,
the presence of radial peculiar velocities such as
the ones assumed in with 
or
biases the NCA estimate of the slope
aD since Eq. (13 (click here)) rewrites in this case
(see Appendix C):
However, the magnitude of this bias
can be attenuated by
selecting only distant galaxies of the observed sample. It is not
surprising since the term
in Eq. (17 (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
smaller than
a given r*. It corresponds to introducing an extra selection
function 
defined as follows:
where is given Eq. (6 (click here)).
Introducing this extra selection effect does not
alterate the result obtained Eq. (13 (click here)) since
this property is insensitive to the specific shape adopted
for the
selection function on m and p.
In order to evaluate amplitude of the bias appearing Eq. (17 (click here)) and its variation with respect to the cut-off in distance estimate r*, calculations have been performed on a synthetic sample characterized as follows:
, ,
and are satisfied by the sample.
.
.
.
and in function of u,
and r* can be found
Eq. (C11).
The amplitude of these terms has to be compared with 
appearing in Eq. (14 (click here))
when a wrong value of the DTF slope is adopted.
Expression of 
in function of 
and r*
is given Eq. (C12).
Figure 1 shows variation of R(r*), i.e.
the ratio of selected objects within the observed sample,
with respect to the cut-off in distance estimate
r* (analytical expression of R(r*) is
given Eq. (C13)). On one hand,
high values of r* are required in order to minimize
as far as possible the amplitude of the bias created by
radial peculiar velocities. On the other hand,
accuracy 
of the NCA slope estimate
depends on the size of the selected subsample
(i.e. 
due to
the intrinsic statistical fluctuations affecting
the sample). Since theses two features are
indeed competitive (i.e. R(r*) decreases when r* increases,
compromise on the optimal value of r* has to be
chosen with regard to the specific
characteristics of the data sample under consideration.
The variations of term , i.e. the contribution
of a Maxwellian agitation of velocity dispersion
to the bias of Eq.
(19 (click here)), are illustrated Fig. 2.
Influence of on the NCA estimate
of the DTF slope aD given Eq.
(15 (click here)) is obtained by comparing
with the contribution
of to the covariance
of Eq. (14 (click here)).
For example, if the mean radial peculiar velocity
along the line-of-sight u is zero,
the magnitude of the bias on the NCA estimate of aD
created by a velocity field of dispersion
km s-1 is greater than
for r*=0,
falls to
for 
(i.e. 
),
equalizes
for 
(i.e. 
)
and finally falls below
for 
(i.e. 
).
Figure 2 reveals two important features.
-
If nearby galaxies are not discarded from the observed sample,
the presence of a Maxwellian velocity agitation for
galaxies contaminates strongly the NCA estimate of the DTF
slope aD (a velocity dispersion of
500 km s-1
induces a bias on aD of magnitude
).
-
This bias can be rendered arbitrarily small by selecting
only distant galaxies by means of the extra cut-off in
distance estimate r*.
Figure 3:
Variation of the bias inferred by the presence
of a constant velocity u along the line-of-sight
with respect to the extra cut-off in distance estimate r*
Figure 3 shows the variations of , i.e.
the dominant term of the bias entering Eq.
(19 (click here)) due to the presence of a mean
radial peculiar velocity along the line-of-sight u,
with respect to r*.
A careful analysis of Fig. 3 leads to the three
following remarks.
The term decreases a function of r*
less rapidly as than .
For comparable values of and
at r*=0, say for
u = -750 km s-1
and 
km s-1,
bias falls to
at r* =10 (to
be compared with
for bias),
to at r* =20
( for ) and finally to
at r* =60
(
for ).
It thus turns out that a particular attention
has to be paid in priority to the presence of
constant velocity fields.
As the existence of constant velocities strongly biases
the NCA estimate of aD (at r*=0,
for
u = -500 km s-1), discarding nearby
galaxies by means of distance estimate selection appears
as a quite crucial step. Note however that the situation
is not so stringent for samples affected by bulk flow.
Since the bias is antisymmetric with respect
to u (see Fig. 3), the bias on the NCA estimate of aD
cancels in average if the opposite line-of-sight
direction is also considered. By scanning the sky by line-of-sight
directions, this interesting symmetry allows indeed to
detect bulk flows already at the level of the calibration step.
Finally, some upper bounds on the aD bias created by
large-scale coherent velocity fields can be extracted from
analysis of Fig. 3. Suppose that the line-of-sight
of the calibration sample points toward the direction
of a Great Attractor, located at a distance estimate of
Mpc and creating back-side infall velocities,
say of amplitude
u = -500 km s-1 at
Mpc and slowly decreasing at larger distances.
The bias on the NCA estimate of aD will be necessarily
smaller than the bias for
u = -500 km s-1
(i.e. at r*=50, 
). This property
is closely related to the subsampling in distance estimate
allowed by the null-correlation approach. The peculiar
velocity field, whatever its specific form, becomes
negligible compared to the mean Hubble flow as long
as the cut-off in distance estimate r* is large enough.
In this case the null-correlation approach furnishes unbiased
estimate of the DTF slope aD.
Assuming that the DTF slope aD has been correctly calibrated
by means of the technique presented in Sect.
3.1 (click here) or others calibration procedures,
the null-correlation approach is herein used for
calibrating the remaining calibration parameters entering
into the radial peculiar velocity estimator
of Eq. (9 (click here)).
For this purpose,
this equation is
rewritten as follows:
where 
depends on
H0,
bD and 
.
For a calibration sample satisfying assumptions
,
,
and
,
the radial peculiar velocity estimator
is not correlated with p (see Appendix B
for proof):
On the other hand,
a wrong value of the B* parameter
(i.e. 
)
correlates p and the random variable
:
which does not vanish since
selection effects on apparent magnitude m correlate
variables p and (i.e. for
selected galaxies, observable p increases in average
with the distance estimate ).
Assuming that the DTF slope aD has been correctly
calibrated, the NCA estimate of the "zero-point" B*
is then defined as:
which gives in practice:
NCA estimate of B* is clearly robust. It is
insensitive to observational selection effects
on m and p, specific shape of the
luminosity function fp(p), constant velocity
field and Maxwellian agitation of galaxies (see Appendix B).
Note however that presence of
non-constant large scale velocity fields (such GA flow
for example) biases NCA estimate of B*. Unfortunatly
the subsampling procedure in distance estimate previously
proposed is not efficient for dealing with
this kind of biases since the contribution of
the peculiar velocity v to 
entering Eq.
(24 (click here)) does not decrease with the distance r.
If one want to express the distance estimator
of Eq. (6 (click here)) in true distance units (Mpc),
the Hubble constant H0 has to be estimated (i.e.
).
For this purpose, estimates of the parameters B*
--by using NCA calibration for example--,
DTF zero-point bD
--using primary distance indicators--
and DTF intrinsic dispersion
--in galaxies clusters for example--
are required (i.e. 
).
If B* is estimated with an error of 
,
bD and with errors of
and 
respectively,
the dominant term of the relative error on the H0 estimate
reads:
