Causal Anisotropic Stress

  Stress perturbations are fundamental to seeded models of structure formation because causality combined with energy-momentum conservation forbids perturbations in the energy or momentum density until matter has had the opportunity to move around inside the horizon (see e.g. [30]). Isotropic stress, or pressure, only arises for scalar perturbations and have been considered in detail by [27]. Anisotropic stress perturbations can also come in vector and tensor types and it is their effect that we wish to study here. Combined they cover the full range of possibilities available to causally seeded models such as defects.

We impose two constraints on the anisotropic stress seeds: causality and scaling. Causality implies that correlations in the stresses must vanish outside the horizon. Anisotropic stresses represent spatial derivatives of the momentum density and hence vanish as k² for $k\eta \ll 1$.Scaling requires that the fundamental scale is set by the current horizon so that evolutionary effects are a function of $x=k\eta$. A convenient form that satisfies these criteria is [27,31]

$$4\pi G a^2 \pi_s^{(m)} = A^{(m)}\eta^{-1/2} f_B(x) \, ,$$ (78)

with  

$$f_B^{}(x) = {6 \over B_2^{2} - B_1^{2}} \left[ {\sin(B_1 x) \over (B_1 x)} -{\sin(B_2 x) \over (B_2 x)} \right] \, ,$$ (79)

with 0 < (B₁,B₂) < 1. We caution the reader that though convenient and complete, this choice of basis is not optimal for representing the currently popular set of defect models. It suffices for our purposes here since we only wish to illustrate general properties of the anisotropy formation process.

Assuming B₁ > B₂, B₁ controls the characteristic time after horizon crossing that the stresses are generated, i.e. the peak in f_B scales as $k\eta_c \equiv x_c \propto B_1^{-1}$ (see Fig. 7). B₂ controls the rate of decline of the source at late times. In the general case, the seed may be a sum of different pairs of (B₁,B₂) which may also differ between scalar, vector, and tensor components.

  

Figure 7: Metric fluctuations from scaling anisotropic stress seeds sources. The same anisotropic stress seed (bold solid lines $a^2 \pi_s \propto f_B/x$) produce qualitatively different scalar (short-dashed), vector (long-dashed), and tensor (solid) metric perturbations. As discussed in the text the behavior scales with the characteristic time of the source $x_c \propto B_1^{-1}$. The left panel (a) shows a source which begins to decay as soon as causally permitted (B₁=1) and the right panel (b) the effect of delaying the decay (B₁=0.2). We have displayed the results here for a photon-dominated universe for simplicity.

Postscript: (a)(b)