University of Chicago

The most commonly considered and familiar types of perturbations are scalar modes. These modes represent perturbations in the (energy) density of the cosmological fluid(s) at last scattering and are the only fluctuations which can form structure though gravitational instability.

Consider a single large-scale Fourier component of the fluctuation, i.e. for the photons, a single plane wave in the temperature perturbation. Over time, the temperature and gravitational potential gradients cause a bulk flow, or dipole anisotropy, of the photons. Both effects can be described by introducing an ``effective'' temperature

where is the gravitational potential. Gradients in the effective temperature always create flows from hot to cold effective temperature. Formally, both pressure and gravity act as sources of the momentum density of the fluid in a combination that is exactly the effective temperature for a relativistic fluid.

To avoid confusion, let us explicitly consider the case of adiabatic
fluctuations,
where initial perturbations to the density imply potential
fluctuations that dominate at large scales.
Here gravity overwhelms pressure in overdense regions causing matter to
flow towards density peaks initially.
Nonetheless, overdense regions are effectively *cold* initially
because photons must climb out of the potential wells they create and
hence lose energy in the process.
Though flows are established from cold to hot temperature regions on large
scales, they still go from hot to cold *effective* temperature regions.
This property is true more generally of our adiabatic assumption:
we hereafter refer only to effective temperatures to keep the argument
general.

Let us consider the *quadrupole* component of the temperature pattern
seen by an observer located in a trough of a plane wave.
The azimuthal symmetry in the problem requires that
and hence the flow is irrotational
.
Because hotter photons from the crests flow into the trough from the
directions
while cold photons surround the observer in the plane,
the quadrupole pattern seen in a
trough has an *m*=0,

structure
with angle
(see Fig. 2).
The opposite effect occurs at the crests, reversing the sign of the
quadrupole but preserving the *m*=0 nature in its local angular dependence.
The full effect is thus described by a local quadrupole modulated by a plane
wave in space, ,
where the sign denotes the fact that photons flowing into cold regions are
hot. This infall picture must be modified slightly on scales smaller than
the sound horizon where pressure plays a role (see §3.3.2),
however the essential property that the flows are parallel to and
thus generate an *m*=0 quadrupole remains true.

Fig. 2: The scalar quadrupole moment (*l*=2, *m*=0). Flows from hot (blue) regions into cold (red), produce the azimuthally symmetric pattern
*Y*_{2}^{0}.

The sense of the quadrupole moment determines the polarization pattern through
Thomson scattering.
Recall that polarized scattering peaks when the temperature varies in the
direction orthogonal to .
Consider then the tangent plane with
normal . This may be visualized in an angular ``lobe'' diagram
such as Fig. 2 as a plane which passes through the
``origin'' of the quadrupole pattern perpendicular to the line of sight.
The polarization is maximal when the hot and cold lobes of the quadrupole
are in this tangent plane, and is aligned with the component of the colder
lobe which lies in the plane.
As varies from 0 to (pole to equator) the temperature
differences in this plane increase from zero (see Fig. 3a).
The local polarization at the crest of the temperature perturbation is thus
purely in the N-S direction tapering off in amplitude toward the poles
(see Fig. 3b).
This pattern represents a pure *Q*-field on the sky whose amplitude varies
in angle as an , *m*=0 tensor or *spin-2* spherical harmonic

In different regions of space, the plane wave modulation of the quadrupole can change the sign of the polarization, but not its sense.

Fig. 3:
The transformation of quadrupole anisotropies into linear polarization. (a) The orientation
of the quadrupole moment with respect to the scatterig direction **n** determines
the sense and magnitude of the polarization. It is aligned with the cold (red, long) lobe
in the **e**_{&theta}x**e**_{&phi} tangent
plane. (b) In spherical coordinates where **n**.**k** = cos θ,
the polarization points north-south (*Q*) with magnitude varying as
sin^{2}θ for scalar fluctuations.

This pattern (Fig. 4, yellow lines) is of course not the
only logical possibility for an , *m*=0 polarization pattern.
Its rotation by is also a valid configuration (purple lines).
This represents a pure NW-SE (and by sign reversal NE-SW), or *U*-polarization
pattern. We return in §3.3 to consider the geometrical distinction
between the two patterns, the *electric* and *magnetic* modes.

Fig. 4: Polarization pattern for *l*=2, *m*=0
(note the azimuthal symmetry). The scattering of a scalar *m*=0 quadrupole
perturbation generates the electric *E* (yellow, thick lines) pattern on the sphere.
Its rotation by 45 degrees represents the orthogonal magnetic *B* (purple, thin lines)
pattern.

Animation: As the line of sight **n** changes,
the lobes of the quadrupole rotate in and out of the tangent plane. The polarization
follows the orientation of the colder (red) lobe in the tangent plane.

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