Signature of cosmic string wakes in the CMB polarization

We calculate a signature of cosmic strings in the polarization of the cosmic microwave background. We find that ionization in the wakes behind moving strings gives rise to extra polarization in a set of rectangular patches in the sky whose length distribution is scale-invariant. The length of an individual patch is set by the comoving Hubble radius at the time the string is perturbing the cosmic microwave background. The polarization signal is largest for string wakes produced at the earliest post-recombination time, and for an alignment in which the photons cross the wake close to the time the wake is created. The maximal amplitude of the polarization relative to the temperature quadrupole is set by the overdensity of free electrons inside a wake which depends on the ionization fraction $f$ inside the wake. For a cosmic string wake coming from an idealized string segment, the signal can be as high as $0.06\mu \mathrm{K}$ in degree scale polarization for a string at high redshift (near recombination) and a string tension $\mu$ given by $G\mu ={10}^{-7}$.

Figure 1

Sketch of the mechanism by which a wake behind a moving string is generated. Consider a string perpendicular to the plane of the graph moving straight downward. From the point of view of the frame in which the string is at rest, matter is moving upwards, as indicated with the arrows in the left panel. From the point of view of an observer sitting behind the string (relative to the string motion) matter flowing past the string receives a velocity kick towards the plane determined by the direction of the string and the velocity vector (right panel). This velocity kick towards the plane leads to a wedge-shaped region behind the string with twice the background density (the shaded region in the right panel).

Figure 2

Sketch of the geometry of a cosmic string wake. The wake is the wedge-shaped region shown. The string is moving in horizontal direction towards the right, and is located along the line where the wake has vanishing thickness. The horizontal dimension of the wake is determined by the string velocity and the Hubble time when the wake is created (time $t_i$). The size of the wake in direction of the string is set by the curvature radius of the cosmic string network and is in our toy model taken to be the Hubble scale. The thickness of the wake perpendicular to the plane spanned by the tangent vector of the string and the string velocity vector changes as we move away from the tip of the string. The average wake thickness is indicated by the short vertical line with arrow signs in both directions.

Figure 3

The polarization signal of a single wake which is taken to be perpendicular to the line of sight between us and the center of the string segment. The tip of the wake (position of the string at the time the wake is laid down) is the vertical edge of the rectangle, and the string velocity vector is pointing horizontally to the left The length and direction of the arrows indicate the magnitude and orientation of the polarization vector. We have assumed that the variation of the quadrupole vector across the plane of the wake is negligible. Note that the angle between the velocity vector of the string and the CMB quadrupole is random. We have assumed that the quadrupole vector is at an angle relative to the plane of the wake. This determines the direction of the polarization vector we have drawn. The orientation we have chosen in this figure corresponds to an almost pure B-mode.

Figure 4

The Q-mode polarization signal of a single wake which is taken to be perpendicular to the line of sight between us and the center of the string segment, superimposed on the Gaussian noise signal which is expected to dominate the total power spectrum. The string signal is multiplied by a factor of 100 to render it visible by eye. The tip of the wake (position of the string at the time the wake is laid down) is the vertical bright edge of the right side, and the string velocity vector is pointing horizontally to the left. At the final string position the wake thickness vanishes and there is no polarization discontinuity line. We have assumed that the variation of the quadrupole vector across the plane of the wake is negligible and that the quadrupole vector is at an angle relative to the plane of the wake such that the Q-mode picks up half of the polarization power.