In the tidal torque model used in this work, the alignment of spiral galaxies is purely due to their orientation, which in turn is related to the angular momentum correlation of neighbouring galaxies relative to the line of sight (Croft & Metzler 2000; Crittenden et al. 2001). Angular momentum correlations are mainly build up at early times during structure formation and are thus due to initial correlations (Catelan & Theuns 1996, 1997; Theuns & Catelan 1997). The correlated angular momenta result into to correlated inclination angles of neighbouring galaxies and thus ultimately into correlated ellipticities (Catelan, Kamionkowski & Blandford 2001). Assuming that the symmetry axis of the galactic disc coincides with the direction of the angular momentum $\hat{L} = \boldsymbol{L}/L$, the ellipticity can be written as the alignment of spiral galaxies is purely due to the orientation of their circular discs, which in turn is related to the angular momentum correlation of neighbouring galaxies relative to the line of sight (Croft & Metzler 2000; Crittenden et al. 2001). Angular momentum correlations are mainly build up at early times during structure formation and are thus due to initial correlations (Catelan & Theuns 1996, 1997; Theuns & Catelan 1997). The correlated angular momenta result into correlated inclination angles of neighbouring galaxies and thus ultimately into correlated ellipticities (Catelan et al. 2001). Assuming that the symmetry axis of the galactic disc coincides with the direction of the angular momentum $\hat{L} = \boldsymbol{L}/L$, the ellipticity can be written as
(17)$$\begin{eqnarray*}
\epsilon = \frac{\hat{L}^2_x - \hat{L}^2_y}{1+ \hat{L}^2_z} + 2\mathrm{i}\frac{\hat{L}_x\hat{L}_y}{1+\hat{L}^2_z}\,\, .
\end{eqnarray*}$$Angular momentum is generated by a torque exerted by the ambient large-scale structure on to the protogalactic halo, a mechanism called tidal torquing (White 1984; Barnes & Efstathiou 1987; Schaefer 2009; Stewart et al. 2013). For Gaussian random fields, the autocorrelation of angular momenta is given by (Lee & Pen 2001)
(18)$$\begin{eqnarray*}
\left\langle \hat{L}_\alpha \hat{L}_\beta \right\rangle = \frac{1}{3}\left(\frac{1+ A}{3}\delta _{\alpha \beta } - A \hat{\Phi }_{\alpha \mu }\hat{\Phi }_{\mu \beta } \right)\,\, .
\end{eqnarray*}$$The free parameter A determines the strength of the coupling between alignment and tidal torque. Since the correlation is determined by the traceless part of the shear tensor $\hat{\Phi }_{\alpha \beta }$ the resulting effect is clearly due to orientation effects only. For a Gaussian distribution $p(\hat{L}|\hat{\Phi }_{\alpha \beta })\mathrm{d}\hat{L}$ and the use of equation (17), one can express the ellipticity in terms of the tidal field
(19)$$\begin{eqnarray*}
\epsilon (\hat{\Phi }) = \frac{A}{2} \left(\hat{\Phi }_{x\alpha }\hat{\Phi }_{\alpha x} - \hat{\Phi }_{y\alpha }\hat{\Phi }_{\alpha y} -2\mathrm{i}\hat{\Phi }_{x\alpha }\hat{\Phi }_{\alpha y}\right).
\end{eqnarray*}$$Correlations in the ellipticities can thus be traced back to the four-point function of the shear field, which is given in equation (11). For keeping a correct relative normalization of the shape correlations, we scale the resulting angular ellipticity spectra $C^{\mathrm{s},II}_{ij}(\ell)$ with the squared number of spiral galaxies $n_s^2$. It is remarkable that the shapes of spiral galaxies in the quadratic alignment model are in fact sensitive to tidal shear components parallel to the line of sight; in fact, those components determine the magnitude of the alignment effect, in contrast to the alignment of elliptical galaxies in the linear alignment model or to gravitational lensing, which reflect purely the tidal shear components perpendicular to the line of sight.
Citation: Catelan et al. 2001
Citation function: Uses