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Spectral Theory

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Baryonic Processes in the Large-Scale Structuring of the Universe

Part of the book series: Springer Theses ((Springer Theses))

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Abstract

Equilibrium configurations such as those discussed in the previous chapter may be stable or unstable.

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Notes

  1. 1.

    These variables are referred to as ‘primitive’ to contrast with the other variable, \(\vec {\xi }\), that we will later use instead, and which is, in a sense, more ‘sophisticated’.

  2. 2.

    Omitting surface terms, the discussion of which is out of my scope here. For their analysis in the stellar case, see Cox (1980) and Smeyers and Van Hoolst (2010) for instance.

  3. 3.

    More precisely, this is called a Klein-Gordon equation, in which the Jeans wavenumber (cf. below) acts as the mass parameter. Physically, it is interesting to thus see the competition between gravitational attraction of the background and pressure (quantified by the Jeans wavenumber) as an inertia of the perturbation (like mass in the particle physics context), and hence acoustic waves behave differently than in the absence of gravity.

  4. 4.

    Cf. Eq. (7.22) for \(\vec {v}_1\), which comes from the definition of \(\vec {\xi }\), and then respectively (7.30) for \(\rho _1\), (7.31) for \(p_1\), (7.32) for \(\vec {B}_1\) and (7.34) for \(\vec {g}_1\), which come from the physics governing the system, i.e. from the conservation laws.

  5. 5.

    Note that we are only considering linear stability here, with infinitesimal displacements, so that we do not consider possible finite jumps from various local minima of the potential.

  6. 6.

    To be complete, this expression contains too, cf. Sect. 7.1.3.2.

  7. 7.

    It is so because we are dealing with a background state that is at rest. When adding flow, as introduced in Chap. 10, each of these three waves splits up into two, one backward and one forward. And also, it can be shown that because we are adopting a Lagrangian description, in terms of the displacement vector \(\vec {\xi }\), we are not taking into account a seventh wave, namely the Eulerian entropy wave, but which is marginal (\(\omega ^2=0\)) in this context. For more details see Sect. 5.2 of Goedbloed and Poedts (2004) and Sect. 13.1.3 of Goedbloed et al. (2010).

  8. 8.

    Finally, on the self-adjointness of the force operator, and thus on the reality of \(\omega ^2\), it is interesting to note that some stability studies do report complex-valued \(\omega \)s (e.g. Freundlich et al. 2014b, who analyze self-gravitating, non-rotatingfilaments, linearizing the system of perturbed equations in primitive variables). The origin of this complexity, whether physical or an artefact, in those cases is unclear at this point, but may reside in the chosen formal approach and the assumptions made.

  9. 9.

    In this process, the other two components of \(\vec {\xi }\) can be expressed as functions of \(\xi _x\) only, so that we may, if needed, recover the full expression of \(\vec {\xi }\) once the equation on \(\xi _x\) is solved. Therefore no information is lost, so this can really be seen as a reformulation of the vector problem.

  10. 10.

    To be precise, in this article the author takes gravity into account, but in the Cowling approximation only, the discussion of which is left for the next chapter.

  11. 11.

    As the following of the manuscript will show, determining the nature of the singularities will be somewhat subtle.

  12. 12.

    This ordering also justifies the terminology ‘slow’ and ‘fast’ for the magneto-acoustic waves.

  13. 13.

    This is at the origin of the terminology ‘Sturmian’ mentioned.

  14. 14.

    To be a little more precise, because his discussion happens in the context of stellar pulsations, Cox (1980) shows that \(|\Phi _1| \propto \left[ k^2 + \frac{l(l+1)}{r^2}\right] ^{-1}\), where k is the radial wavenumber and l is the spherical harmonic order, so that in spherical systems, what is meant by ‘high order mode’ is large k and/or large l.

  15. 15.

    Therefore we do not expect WKB dispersion relations to reveal instabilities in the directions of stratification.

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  1. Institut d’Astrophysique Spatiale, Université Paris-Sud, Orsay, France

    Jean-Baptiste Durrive

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Durrive, JB. (2017). Spectral Theory. 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_7

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