More examples of structure formation in the Lemaître-Tolman model

Physical Review D - Particles, Fields, Gravitation and Cosmology

Volumn 69, Issue 2, 2004, Pages

  Krasiński, Andrzej ^a   Hellaby, Charles ^b  

^a NICOLAUS COPERNICUS ASTRONOMICAL CENTER   (Poland)

^b UNIVERSITY OF CAPE TOWN   (South Africa)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 1342346476     PISSN: 15507998     EISSN: 15502368     Source Type: Journal    
DOI: 10.1103/PhysRevD.69.023502     Document Type: Article

References (38)

1. A. Krasiński and C. Hellaby, Phys. Rev. D 65, 023501 (2001), paper I.
2. G. Lemaître, Ann. Soc. Sci. Bruxelles, Ser. 1 A53, 51 (1933)
3. G. Lemaîtrereprinted in Gen. Relativ. Gravit. 29, 641 (1997).
4. R.C. Tolman, Proc. Natl. Acad. Sci. U.S.A. 20, 169 (1934)
5. R.C. Tolmanreprinted in Gen. Relativ. Gravit. 29, 935 (1997).
6. A. Krasiński, Inhomogeneous Cosmological Models (Cambridge University, Cambridge, England, 1997).
7. More correctly, the elliptic, parabolic, and hyperbolic conditions are (Formula presented) (Formula presented) and (Formula presented) respectively, as both E and M go to zero at an origin.
8. One of the definitions of (Formula presented) used in astronomy is: The size (Formula presented) of a galaxy cluster is that radius at which the average density within (Formula presented) equals a specified multiple of the background density (Formula presented) (e.g. 200 8). With this definition, the procedure of determining (Formula presented) is different from the one presented in Eqs. (3.5)–(3.6).
9. A proper matching of first and second fundamental forms would have matched masses, energies and ages, and be a comoving surface (Formula presented) If our “compensation” procedure were executed at each time, the resulting surface would not be properly matched, nor would it be comoving.
10. J.F. Navarro, C.S. Frenk, and S.D.M. White, Astrophys. J. 462, 563 (1996).
11. J.F. Navarro, C.S. Frenk, and S.D.M. White, Mon. Not. R. Astron. Soc. 275, 720 (1995).
12. J.F. Navarro, C.S. Frenk, and S.D.M. White, Astrophys. J. 490, 493 (1997).
13. Note that the relation between mass and density in the LT model, Eq. (2.3), is very different from the flat-space relation (Formula presented) We will carry over to relativity the equation (Formula presented) resulting from the Newtonian relation with the hope that it is a good approximation at low densities and for small radii from the center. However, a completely self-consistent approach would require re-interpretation of all the relevant astronomical observations against the LT model.
14. This profile is meant for large (Formula presented) only. However, all our numerical codes cover (Formula presented) so we added (Formula presented) to avoid modifying the codes.
15. The LT model is general enough to describe a situation where the final profile is also just a bump on a perfectly homogeneous background, but this picture does not fit well with the popular image of mass being accreted onto the initial condensation. The “bump on a smooth background” at (Formula presented) would imply that the whole infinite background adjusted its density to the central bump, which would imply propagating the density wave to an infinite distance in a finite time.
16. In this way, we also avoid a numerical difficulty: the volume that is numerically monitored must be finite. The “condensation surrounded by a rarefaction” model has the advantage that the region outside the rarefaction remains strictly Friedmannian for all time and it is enough to monitor it only at its edge. In the “bump on a smooth background” model, the velocity perturbation spreads out to infinity; only the mass density at (Formula presented) is Friedmannian outside the bump (and even this homogeneity gets destroyed by evolution, both backward and forward in time—it is only momentary).
17. T. Padmanabhan, Cosmology and Astrophysics Through Problems (Cambridge University Press, Cambridge, England, 1996).
18. M. White and J.D. Cohn, astro-ph/0203120.
19. W. Hu and S. Dodelson, Annu. Rev. Astron. Astrophys. 40, 171 (2002).
20. E.L. Wright, see http://www.astro.ucla.edu/(Formula presented)wright/ CMB-DT.html
21. W. Hu, see http://background.uchicago.edu/(Formula presented)whu/physics/ tour.html
22. W. Hu, see http://background.uchicago.edu/(Formula presented)whu/araa/ node4.html
23. P.K.S. Dunsby, Class. Quantum Grav. 14, 3391 (1997).
24. The velocity used here is the emitting fluid velocity relative to the normals to the constant temperature surfaces.
25. We checked whether this profile would encounter a black hole horizon (Formula presented) The mass within radius R is (Formula presented)so while the function (Formula presented) could in principle have a root, it does not have one for reasonable astronomical parameter values.
26. M. Markevitch, A. Vikhlinin, W.R. Forman, and C.L. Sarazin, Astrophys. J. 527, 545 (1999).
27. See http://www.maplesoft.com
28. See http://www.mathworks.com
29. These will be useful in modelling the formation of black holes.
30. C. Hellaby and K. Lake, Astrophys. J. 290, 381 (1985)
31. Astrophys. J.C. HellabyK. Lake300, 461(E) (1985).
32. H. Sato, in General Relativity and Gravitation, edited by B. Bertotti, F. de Felice, and A. Pascolini (Reidel, Dordrecht, 1984), p. 289.
33. N. Mustapha and C. Hellaby, Gen. Relativ. Gravit. 33, 455 (2001).
34. Indeed, all physical quantities calculated are invariant under the transformation (Formula presented)const.
35. C. Hellaby, Class. Quantum Grav. 4, 635 (1987).
36. C. Hellaby and K. Lake, Astrophys. J. 282, 1 (1984).
37. A. Barnes, J. Phys. A 3, 653 (1970).
38. If we allow (Formula presented) to diverge, then we need (Formula presented) to diverge faster, and the argument is pretty much the same.