Abstract
The purpose of this chapter is to study the competition between pressure and gravity in the evolution of perturbations in a planar stratification.
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Notes
- 1.
It is a slightly different definition from \(L_1\) given by (6.25) to avoid carrying \(\sqrt{2}\) factors in the following.
- 2.
- 3.
And not \(2 x_t\), though the slab spans between \(-x_t\) and \(+x_t\), because we have imposed boundary conditions such that the displacement vanishes at \(x=x_t\) and \(x=0\).
- 4.
Of course, one may eliminate \(k_y^2\), writing \(\omega _G^2 = \omega _y^2 - (c_a^2)'' - \omega _0^2\), but the form given here is how it naturally appears, and is such that the comparison with its cylindrical generalization (9.6) is transparent. And in the latter case, the analogue of \(k_y\) is position dependent so that we may not simplify like here.
- 5.
“he ends up with a fourth order differential equation with very complicated coefficients”.
- 6.
The dependence on \((c_a^2)'''\) is hidden in the factors \(\Omega _G' = - k_y^2 (c_a^2)' + (c_a^2)''' + \omega _0 \frac{\rho _0'}{\rho _0}\) that I have left this way to make the expression (8.57) a little bit more compact.
- 7.
It is tempting to think that the case considered here is physically motivated by the fact that in most cosmological environments dark matter dominates the density budget and the gravitational potential in which baryons evolve. But there is no reason a priori that if \(\rho _0 \ll \rho _\mathrm {0ext}\), then \(\rho _\mathrm {1ext} \ll \rho _1\). So my feeling about this model is that it is an interesting case in which, for some reason, the background fluid is stiff. Note also that in reality, in the cosmic web, the dark matter density distribution contains non linear substructures (e.g. Schneider et al. 2010) into which cold baryonic gas may fall. I do not include those in my analysis precisely because my aim is to investigate whether the gas may fragment gravitationally on its own.
- 8.
As stated above, WKB approximations are often very good approximations even beyond their strict domain of validity, so that the precautions taken here may turn out to be unnecessary. Comparison with a numerical resolution (e.g. with a shooting method Goedbloed and Poedts 2004) will give this answer.
- 9.
As in the sections about the wave equation, we are working with \(\omega ^2 \ne \omega _y^2\) and discussing singularities later.
- 10.
This is not a trivial statement per se, since in general \(e^{A} e^{B} \ne e^{A+B}\), but it is the case when A and B commute. Here the matrices \(-s A_0\) and \(s A_0\) clearly commute.
- 11.
Recall that \(\hat{\xi }_x\) is proportional to \(\psi \) so that with the boundary conditions we are going to consider shortly, we may use \(\psi \) rather than \(\hat{\xi }_x\) to determine the discrete spectrum.
- 12.
Indeed, \(\Delta < 0\) is equivalent to \(\frac{\omega ^2}{c_a^2}-k_y^2-\frac{k_\rho ^2}{4}>0\), but since \(\frac{\omega ^2}{c_a^2}-k_y^2-\frac{k_\rho ^2}{4} = \frac{\omega ^2}{c_a^2} - k_y k_\rho -(k_y - \frac{k_\rho }{2})^2\) and that \((k_y - \frac{k_\rho }{2})^2 \ge 0\), we have \(\frac{\omega ^2}{c_a^2} - k_y k_\rho >0\).
- 13.
My precaution in the formulation comes from the fact that \(\delta \) contains \(k_\rho x_t\), and the competition of the \(e^{k_y x_t}\) factor has to be assessed properly. At this point I will just say the \(k_y\) has to be small enough, leaving the rest for future work.
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Durrive, JB. (2017). Stability of Cosmic Walls. In: Baryonic Processes in the Large-Scale Structuring of the Universe. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-61881-4_8
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