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Revista de Matemática Teoría y Aplicaciones
versão impressa ISSN 1409-2433
Rev. Mat vol.21 no.2 San José Jul./Dez. 2014
Lensing properties of the einasto profile in terms of the meijer G function
Propiedades del efecto lente para el perfil de einasto en términos de la función G de meijer
Abstract
In N-body simulations of cold dark matter, it has been found that three-parameter models, particularly the Einasto profile, yield better fits to a wide range of dark matter haloes than two parameter models like the Navarro-Frenk-White profile. Recently, the analytical properties of the Einasto profile has been studied, allowing closed expressions for its surface mass density and lensing properties in terms of the Fox H and Meijer G functions, using a Mellin transform formalism. These expressions are valid for all values of the Einasto index in terms of the Fox H function, and valid for integer and half-integer values of Einasto index in terms of the Meijer G function. In this paper, we derive expressions for lensing properties of the Einasto profile for all rational values of the Einasto index in terms of theMeijer G function. Equivalency between these expressions and other recent results is also discussed
Keywords: cosmology; dark matter; Meijer G function.
Resumen
En simulaciones de N-cuerpos de materia oscura fría, se ha encontrado que modelos de tres parámetros, particularmente el perfil de Einasto, ofrece mejores ajustes para un amplio rango de halos de materia oscura que los modelos de dos parámetros como el perfil Navarro-Frenk-White. Recientemente, las propiedades analíticas del perfil de Einasto han sido estudiadas, lográdose expresiones cerradas para su densidad de masa superficial y propiedades de lente gravitational en términos de la función H de Fox, usando el formalismo de la transformada de Mellin. Estas expresiones son válidas para todos los valores del índice de Einasto en términos de la función H de Fox, y válidos para valores enteros y semi-enteros del índice de Einasto en términos de la función G de Meijer. En este artículo, se determinan expresiones para las propiedades de lente gravitational del perfil de Einasto para todos los valores racionales del índice de Einasto en términos de la función G de Meijer. La equivalencia entre estas expresiones y otros resultados recientes también es discutida.
Palabras clave: cosmología; materia oscura; función G de Meijer.
Mathematics Subject Classification: 33C60; 33E20.
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References
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[2] Baes, M.; Gentile, G. (2011) “Analytical expressions for the deprojected Sérsic model”, A & A, 525: A136+ (1–7), arXiv:1009.4713. [ Links ]
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[4] Bartelmann, M.; Schneider P. (2001) “Weak gravitational lensing”, Physics Reports 340(4-5): 291–472. [ Links ]
[5] Binney, J.; Tremaine, S. (1987) Galactic Dynamics. Princeton University Press, Princeton NJ. [ Links ]
[6] Cardone, V.F.; Piedipalumbo, E.; Tortora, C. (2005) “Spherical galaxy models with power-law logarithmic slope”, MNRAS 358: 1325–1336. [ Links ]
[7] Chemin, L.; de Blok, W.J.G.; Mamon, G.A. (2011) “Improved modeling of the mass distribution of disk galaxies by the Einasto halo model”, ApJ 142: 109–124. [ Links ]
[8] de Blok,W. J. G. (2010) “The core-cusp problem”, Advances in Astronomy, 2010: 1–14. [ Links ]
[9] Dhar, B.K.; Williams, L.L.R. (2011) “Surface brightness and intrinsic luminosity of ellipticals”, MNRAS, 427: 204–244. [ Links ]
[10] Dutton, A. and Maccio, A. (2014) “Cold dark matter haloes in the Planck era: evolution of structural parameters for Einasto and NFW profiles”, to appear in MNRAS, arXiv:1402.7073. [ Links ]
[11] Einasto, J. (1965) “On the construction of a composite model for the Galaxy and on the determination of the system of Galactic parameters”, Trudy Inst. Astroz. Alma-Ata 17(1): 87–100. [ Links ]
[12] Fikioris, G. (2007) “Mellin transformmethod for integral evaluation: introduction and applications to electromagnetics” Synthesis Lectures on Computational Electromagnetics 2(1): 1–67 [ Links ]
[13] Fox, C. (1961) “The G and H functions as symmetrical Fourier kernels”, Transactions of the American Mathematical Society 98: 395–429. [ Links ]
[14] Marichev, O.I. (1982) Handbook of Integral Transforms of Higher Transcendental Functions: Theory and Algorithmic Tables. Ellis Horwood, Chichester. [ Links ]
[15] Meijer, C. S. (1936) “Uber Whittakersche bezw. Besselsche Funktionen und deren Produkte”, Nieuw Archief voor Wiskunde 18: 10–39. [ Links ]
[16] Meneghetti, M. et al. (2014) “The MUSIC of CLASH: predictions on the concentration-mass relation”, arXiv:1404.1384. [ Links ]
[17] Narayan, R.; Bartelmann, M. (1995) “Lectures on gravitational lensing”, en: A. Dekel & J. P. Ostriker (Eds.) Formation of Structure in the Universe (Proceedings of the 1995 JerusalemWinter School), Cambridge University Press, London. [ Links ]
[18] Navarro, J.F.; Frenk, C.S.; White, S.D.M. (1996) “The structure of cold dark matter halos”, ApJ 462: 563–575. [ Links ]
[19] Navarro, J.F.; Frenk, C.S.; White, S.D.M. (1997) “A universal density profile from hierarchical clustering”, MNRAS 490: 493–508. [ Links ]
[20] Navarro, J.F., et al. (2010) “The diversity and similarity of simulated cold dark matter haloes”, MNRAS 402: 21–34. [ Links ]
[21] Retana-Montenegro, E.; Van Hese, E.; Gentile, G.; Baes, M.; Frutos-Alfaro, F. (2012) “Analytical properties of Einasto dark matter haloes”, A & A, 540: A70 (10p). (arXiv:1202.5242). [ Links ]
[22] Schneider, P.; Ehlers, J.; Falco, E. E. (1992) Gravitational Lenses. Springer, Berlin. [ Links ]
[23] Spergel, D.N., et al. (2003) “First year Wilkinson microwave anisotropy probe (WMAP) observations: Determination of cosmological parameters”, Astrophys.J.Suppl.148: 175–194. [ Links ]
[24] Novak, G.S., et al. (2012) “On galaxies and homology”, MNRAS, 424: 635–648. [ Links ]
* Escuela de Física, Universidad de Costa Rica, San Pedro 11501, Costa Rica. E-mail: edwin@fisica.ucr.ac.cr
† Escuela de Física, Universidad de Costa Rica, San Pedro 11501, Costa Rica. E-mail: frutos@fisica.ucr.ac.cr
Received: 22/Feb/2012; Revised: 4/Jun/2014; Accepted: 5/Jun/2014
