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http://dx.doi.org/10.5303/JKAS.2014.47.5.187

THE KRAMERS-HEISENBERG FORMULA AND THE GUNN-PETERSON TROUGH  

Bach, Kiehunn (Department of Astronomy, Yonsei University)
Lee, Hee-Won (Department of Astronomy and Space Science, Sejong University)
Publication Information
Journal of The Korean Astronomical Society / v.47, no.5, 2014 , pp. 187-193 More about this Journal
Abstract
Recent cosmological observations indicate that the reionized universe may have started at around z = 6, where a significant suppression around $Ly{\alpha}$ has been observed from the neutral intergalactic medium. The associated neutral hydrogen column density is expected to exceed $10^{21}cm^{-2}$, where it is very important to use the accurate scattering cross section known as the Kramers-Heisenberg formula that is obtained from the fully quantum mechanical time-dependent second order perturbation theory. We present the Kramers-Heisenberg formula and compare it with the formula introduced in a heuristic way by Peebles (1993) considering the hydrogen atom as a two-level atom, from which we find a deviation by a factor of two in the red wing region far from the line center. Adopting a representative set of cosmological parameters, we compute the Gunn-Peterson optical depths and absorption profiles. Our results are quantitatively compared with previous work by Madau & Rees (2000), who adopted the Peebles approximation in their radiative transfer problems. We find deviations up to 5 per cent in the Gunn-Peterson transmission coefficient for an accelerated expanding universe in the red off-resonance wing part with the rest wavelength ${\Delta}{\lambda}{\sim}10{\AA}$.
Keywords
cosmology: theory; intergalactic medium; quasars: absorption lines; radiative transfer; galaxies: high-redshift;
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1 Sokasian, A., Abel, T., & Hernquist, L. 2002, The Epoch of Helium Reionization, MNRAS, 332, 601   DOI   ScienceOn
2 Turnshek, D. A., & Rao, S. 1998, QSO Damped Lyalpha Absorption Systems at Low Redshift and the Giant Hydrogen Cloud Model for Damped $Ly{\alpha}$, ApJ, 500, L115   DOI
3 Wyithe, S., & Loeb, A. 2003, Reionization of Hydrogen and Helium by Early Stars and Quasars, ApJ, 586, 693   DOI
4 Weisskopf, V. F. I., & Wigner, E. 1930, Berechnung der naturlichen Linienbreite auf Grund der Diracschen Lichttheorie, Z. Phys., 63, 54   DOI
5 Zafar, T., Peroux, C., Popping, A., et al. 2013, The ESO UVES Advanced Data Products Quasar Sample. I. Dataset and New NHI Measurements of Damped Absorbers, A&A, 556, 141   DOI
6 Berestetskii, V. B., Lifshitz, E. M., & Pitaevskii, L. P. 1971, Relativistic Quantum Theory (Oxford: Pergamon Press)
7 Ahn, K., Iliev, I. T., Shapiro, P. R., Mellema, G., Koda, J., & Mao, Y. 2012, Detecting the Rise and Fall of the First Stars by Their Impact on Cosmic Reionization, ApJ, 756, L16   DOI
8 Baschek, B., & Scholz, M. 1982, Astronomy and Astrophysics, Stars and Star Clusters Vol. 2b (Berlin: Springer-Verlag)
9 Becker, R. H., Fan, X., White, R. L., et al. 2001, Evidence for Reionization at z - 6: Detection of a Gunn-Peterson Trough in a z = 6.28 Quasar, AJ, 122, 2580
10 Bromm, V., & Yoshida, N. 2011, The First Galaxies, ARA&A, 49, 373   DOI   ScienceOn
11 Ciardi, B., Stoehr, F., & White, D. M. 2004, Simulating Intergalactic Medium Reionization, MNRAS, 343, 1101
12 Ciardi, B., & Ferrara, A. 2005, The First Cosmic Structures and Their Effects, SSRv, 116, 625
13 Fan, X., Strauss, M. A., Richards, G. T., et al. 2006, A Survey of z > 5.7 Quasars in the Sloan Digital Sky Survey. IV. Discovery of Seven Additional Quasars, AJ, 131, 1203   DOI   ScienceOn
14 Fan, X., Carilli, C. L., & Keating, B. 2006, Observational Constraints on Cosmic Reionization, ARA&A, 44, 415   DOI   ScienceOn
15 Fan, X. 2012, Observations of the First Light and the Epoch of Reionization, RAA, 12, 865
16 Ferland, G. 2001, Hazy, A Brief Introduction to Cloudy 94.00
17 Gunn, J. E., & Peterson, B. A. 1965, On the Density of Neutral Hydrogen in Intergalactic Space, ApJ, 142, 1633
18 Gnedin, N. Y. 2000, Cosmological Reionization by Stellar Sources, ApJ, 535, 530   DOI   ScienceOn
19 Gavrila, M. 1967, Elastic Scattering of Photons by a Hydrogen Atom, Phys. Rev., 163, 147   DOI
20 Gnedin, N. Y., & Ostriker, J. P. 1997, Reionization of the Universe and the Early Production of Metals, ApJ, 486, 581   DOI   ScienceOn
21 Iliev, I. T., Mellema, G., Pen, U.-L., Merz, H., Shapiro, P. R., & Alvarez, M. A. 2006, Simulating Cosmic Reionization at Large Scales - I. The Geometry of Reionization, MNRAS, 369, 1625   DOI   ScienceOn
22 Karzas, W. J., & Latter, R. 1961, Electron Radiative Transitions in a Coulomb Field, ApJS, 6, 167   DOI
23 Kramers, H. A., & Heisenberg, W. 1925, Uber die Streuung von Strahlung durch Atome, Z. Phys., 31, 681   DOI
24 Lee, H.-W. 2003, Asymmetric Deviation of the Scattering cross Section around $Ly{\alpha}$ by Atomic Hydrogen, ApJ, 594, 637   DOI
25 Loeb, A., & Rybicki, G. B. 1999, Scattered Lyalpha Radiation around Sources before Cosmological Reionization, ApJ, 524, L527   DOI
26 Loeb, A., & Barkana, R. 2001, The Reionization of the Universe by the First Stars and Quasars, ARA&A, 39, 19   DOI   ScienceOn
27 Madau, P., Haardt, F., & Rees, M. J. 1999, Radiative Transfer in a Clumpy Universe. III. The Nature of Cosmological Ionizing Sources, ApJ, 514, 648   DOI
28 Madau, P., & Rees, M. J. 2000, The Earliest Luminous Sources and the Damping Wing of the Gunn-Peterson Trough, ApJ, 542, L69   DOI
29 Miralda-Escude, J., Haehnelt, M., & Rees, M. J. 2000, Reionization of the Inhomogeneous Universe, ApJ, 530, 1   DOI
30 Miralda-Escude, J. 1998, Reionization of the Intergalactic Medium and the Damping Wing of the Gunn-Peterson Trough, ApJ, 501, 15   DOI   ScienceOn
31 Peebles, P. J. E. 1993, Principles of Physical Cosmology (Princeton: Princeton University Press)
32 Morales, M. F., & Wyithe, S. 2010, Reionization and Cosmology with 21-cm Fluctuations, ARA&A, 48, 127   DOI   ScienceOn
33 Noterdaeme, P., Petitjean, P., Carithers, W. C., et al. 2012, Column Density Distribution and Cosmological Mass Density of neutral gas: Sloan Digital Sky Survey-III Data Release 9, A&A, 547, 1
34 Ota, K., Iye, M., Kashikawa, N., et al. 2010, Lyalpha Emitters at z = 7 in the Subaru/XMM-Newton Deep Survey Field: Photometric Candidates and Luminosity Functions, ApJ, 722, 803   DOI
35 Pentericci, L., Fontana, A., Vanzella, E., Castellano, M., et al. 2011, Spectroscopic Confirmation of z - 7 Lyman Break Galaxies: Probing the Earliest Galaxies and the Epoch of Reionization, ApJ, 743, 132   DOI
36 Razoumov, A. O., Norman, M. L., Abel, T., & Scott, D. 2002, Cosmological Hydrogen Reionization with Three-Dimensional Radiative Transfer, ApJ, 572, 695   DOI
37 Sakurai, J. J. 1967, Advanced Quantum Mechanics (Reading, Massachusetts: Addison-Wesley Publishing Company)
38 Scheuer, P. A. G. 1965, A Sensitive Test for the Presence of Atomic Hydrogen in Intergalactic Space, Nature, 207, 963
39 Heitler, W. 1954, Quantum Theory of Radiation (Oxford: Oxford University Press)
40 Bethe, H. A., & Salpeter, E. E. 1957, Quantum Mechanics of One and Two Electron Atoms (New York: Academic Press Inc.)