In this appendix, we calculate the
expected amplitude of the velocity quadrupole generated by density
fluctuations external to the *IRAS* sample (i.e., outside of *R*
= 12,800 km s^{-1}), and internal to it, that are due to the
difference between the true density field and the noisy, smoothed
estimation of the density field we have from the *IRAS* redshift
survey. The *IRAS*-excluded zone is another potential source
of quadrupole error, but it is filled in by interpolation from regions
above and below the excluded zone
(Yahil et al. 1991),
a procedure that agrees well with a multipole interpolation procedure based
on spherical harmonics, at least for the 10° wide *IRAS* zone of
avoidance
(Lahav et al. 1994).

B1. QUADRUPOLE INDUCED BY
FLUCTUATIONS BEYOND *IRAS* VOLUME

We express peculiar velocity in terms of a potential function
(**r**),
such that the radial component of the velocity field is given
by *u*(*r*) =
- /
*r*. We
will isolate the quadrupole component of this potential and calculate its
angle-averaged rms contribution.

The contribution to
from
material at distances greater than *R* is given by

(B1) |

Here
is the *mass*, not the galaxy, density fluctuation. We now expand the
denominator in the integrand in terms of spherical harmonics (e.g.,
Jackson 1976,
eq. [3.70]) and isolate the quadrupole term to obtain

(B2) |

where
is the solid angle. Taking the radial component of the quadrupole velocity
*u*_{Q}
= -_{Q} /
*r*, squaring,
and averaging over the solid angle gives, after several steps of
algebra,

(B3) |

where the last step follows from the orthonormality of
the *Y*_{lm}, and for convenience we have defined the
five complex coefficients

(B4) |

The expectation value of
|*C*_{2m}|^{2} is
independent of *m*, so when we take the expectation value of
equation (B3), we can replace the sum
with 5 × <*C*_{20}^{2}>:

(B5) |

Using the definition of *Y*_{20} in terms of the
second Legendre polynomial *P*_{2} in equations (B4) and
(B5) gives

(B6) |

Expressing the correlation function
as the
Fourier transform of the power spectrum *P*(*k*)
(e.g., SW,
eq. [46]) allows the integrals over **r**_{1} and
**r**_{2} to separate. This yields

(B7) |

where the kernel is given by

(B8) |

and *j*_{n} is the *n*th order spherical Bessel
function. A comparison of equations (B7) and (B8) with equations (37)
and (38) of
SW
allows us to recast our result as

(B9) |

for the expected rms quadrupole velocity on a sphere due to mass
density fluctuations at distances greater than *R*, expressed as
a fraction of Hubble flow.
Here _{R}^{2}
is the variance in the mass overdensity within spheres of radius *R*.
As mentioned in the text, this gives a fractional quadrupole of the order
of 1%-2% for a variety of *COBE*-normalized power spectra.

**B2. EFFECTS OF WIENER FILTERING AND SHOT NOISE**

The Wiener filter operates on the Fourier
transform of the *IRAS* density field. The final density field differs
from the true density field for two reasons: the discreteness of the
galaxy distribution gives rise to shot noise, and the Wiener filter,
while suppressing shot noise, also suppresses the density field itself.
We calculate the contribution to the quadrupole from both effects.

Let _{T}(**k**) represent
the true Fourier component of the underlying (noiseless) density field at
wavevector **k**; the
quantity with which we calculate the velocity field is the Wiener-filtered
noisy image, whose Fourier modes are given by

(B10) |

where the Wiener filter itself is (e.g., Zaroubi et al. 1995)

(B11) |

and *P*(*k*) is set a priori; we used a functional fit to the
*IRAS* power spectrum found by
Fisher et al. (1993).
The noise term in the denominator of the Wiener filter
is independent of *k* (cf.
Fisher et al. 1993;
SW, Section 5.3);
however, it is dependent
on the density of galaxies, which is a decreasing function of distance in
the flux-limited *IRAS* sample. As explained in
Sigad et al. (1997),
we therefore calculate a *series* of
Wiener-filtered density fields for different noise levels and interpolate
between them to find the appropriate density field at any given
distance.

We wish to calculate the quadrupole due to
the *error* in the derived density field, i.e., that due to the
difference between equation (B10)
and _{T}(**k**). If
we expand the density field in equation (B4) into its Fourier
components, substitute this difference
for each component, and square the result, we find the rms contribution to
*u*_{Q} due to the Wiener filter:

(B12) |

This rather horrific expression can be simplified by multiplying out the
term in braces, realizing that the cross terms vanish and
that <_{T}(**k**_{1})
_{T}(**k**_{2})> =
(2)^{3}
*P*_{T}(*k*)
_{D}(**k**_{1} - **k**_{2}), where
*P*_{T}(*k*) is the *true* underlying power
spectrum, not necessarily the same as that assumed in equation (B11). We
then get two terms, one depending on the power
spectrum and the other due to shot noise. For the first term, the integrals
over **r**_{1} and **r**_{2} separate to give

(B13) |

where the new window function is given by

(B14) |

(cf. eq. [B8]). We integrate from the outer volume of our peculiar
velocity sample, *R*_{1} = 3000 km s^{-1}, to *R*
= 12,800 km s^{-1}; at smaller radii, the contribution to
the quadrupole goes like *r*^{-2}, not *r*, and this is
not included in our modeling of the quadrupole (eq. [19]). The
contribution to the quadrupole from this term is
between 1.5% and 3%, depending on which model we take for the true power
spectrum. This is pleasingly close to the value we find for the real
universe. The mock catalogs have a power spectrum set by the observed
*IRAS* power spectrum (of course, with a cutoff at *k* < 2
/*L*)
and thus give a somewhat smaller contribution to this integral, about 1%.

Let us now calculate the shot noise contribution to the quadrupole. It is given by

(B15) |

Notice now the dependence on
, not
;
here we will make no reference to a *COBE*-normalized power spectrum.
The Fourier modes are calculated in a box of side *L* = 25,600
km s^{-1} and therefore are uncorrelated
for *k* > 2
/*L*.
Thus, we can write the product of the two shot noise terms as a Dirac delta
function:

(B16) |

the expression for
<^{2}(**k**)> comes from
Fisher et al. (1993).
When we insert equation (B16) into equation (B15), the latter simplifies
dramatically. The integrals over **r**_{1}
and **r**_{2} now separate, giving

(B17) |

where the shot noise window function looks very similar to what we have seen before:

(B18) |

Notice that unlike the previous calculation, this result is independent of
the true power spectrum. If we calculate this using the
observed *IRAS* selection function, integrating from 3000 to 12,800
km s^{-1}, we find an rms quadrupole
of *r*^{-1}
<*u*^{2}_{Q, shot}(*r*)>^{1/2} =
1.7%.

We conclude that the 3.3% quadrupole found
for the real data can be understood as a combination of the three effects
discussed here: power on scales larger than the *IRAS* sample, the
Wiener suppression factor, and shot noise; the Wiener suppression factor is
the dominant one of the three. For the mock catalogs, we still do
not understand completely why the measured residual quadrupole (< 1%)
is smaller than we have calculated (~ 2%).