Fluctuations in the cosmic microwave background. I. Form factors and their calculation in the synchronous gauge

Physical Review D

Volumn 64, Issue 12, 2001, Pages

  Weinberg, Steven ^a  

^a UNIVERSITY OF TEXAS AT AUSTIN   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

ACOUSTICS; ARTICLE; CALCULATION; COSMIC RADIATION; DIPOLE; EVOLUTION; HYDRODYNAMICS; MATHEMATICAL ANALYSIS; MICROWAVE RADIATION; MODEL; OSCILLATION; TIME;

EID: 85038299493     PISSN: 05562821     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (39)

1. S. Weinberg, following paper, Phys. Rev. D 64, 123512 (2001).
2. E.R. Harrison, Phys. Rev. D 1, 2726 (1970);
3. P.J.E. Peebles and J.T. Yu, Astrophys. J. 162, 815 (1970);
4. Ya.B. Zel'dovich, Astron. Astrophys. 5, 84 (1970).
5. S. Hawking, Phys. Lett. 115B, 295 (1982);
6. A.A. Starobinsky, Phys. Lett. ibid. 117B, 175 (1982);
7. A. Guth and S.-Y. Pi, Phys. Rev. Lett. 49, 1110 (1982);
8. J.M. Bardeen, P.J. Steinhardt, and M.S. Turner, Phys. Rev. D 28, 679 (1983);
9. W. Fischler, B. Ratra, and L. Susskind, Nucl. Phys. B259, 730 (1985).
10. W. Hu, U. Seljak, M. White, and M. Zaldarriaga, Phys. Rev. D 57, 3290 (1998), and earlier references cited therein; for analysis of recent observations, see J.R. Bond et al., astro-ph/0011378.
11. Peebles and Yu [2]; J.R. Bond and G. Efstathiou, Astrophys. J. Lett. 285, L45 (1984);
12. Mon. Not. R. Astron. Soc. 226, 655 (1987);
13. A.G. Doroshkevich, Sov. Astron. Lett. 14, 125 (1988);
14. F. Atrio-Barandela, A.G. Doroshkevich, and A.A. Klypin, Astrophys. J. 378, 1 (1991);
15. P. Naselsky and I. Novikov, Astrophys. J. ibid. 413, 14 (1993);
16. H.E. Jørgensen, E. Kotok, P. Naselsky, and I. Novikov, Astron. Astrophys. 294, 639 (1995);
17. C.-P. Ma and E. Bertschinger, Astrophys. J. 455, 7 (1995);
18. for comments on some of these articles, see Ref. [6]. The most up-to-date and comprehensive calculation is that of W. Hu and N. Sugiyama, Astrophys. J. 444, 489 (1995);
19. W. Hu and N. Sugiyama, Astrophys. J. 471, 542 (1996), but they do not collect their results into a single formula for the temperature shift, so it is not easy to compare their results with those of the present paper.
20. 2, which had been omitted by Naselsky and Novikov, but omitted the factors of (1 + ξ) that had been included by Naselsky and Novikov, and also included a spurious term in the analog of G(k) [the 1 in the numerator of the second line of their Eq. (4)]. This paper gave differential equations for the time development of the perturbations, but did not explain how they were used to calculate the temperature fluctuation. A few years later the formula given by Naselsky and Novikov was repeated by Jørgensen, Kotok, Naselsky, and Novikov [5], and the differential equations on which the formula was based were given. However, once again the derivation of the formula from these equations was not explained, and an overly restrictive lower bound on k was given for the validity of the formula, that k must be larger than the inverse conformal time. If this condition were really necessary, then the formula would not be applicable at the first Doppler peak. We will see in Sec. III that the lower bound on k is actually less restrictive, and in particular disappears for ξ ≪ 1.
21. S. Weinberg, Gravitation and Cosmology (Wiley, New York, 1972), Eq. (15.10.13). In conformity with common present notation, the symbol R(t) used in this book for the Robertson-Walker scale factor has been replaced in the present paper with a(t).
22. Weinberg [7], Eq. (15.10.52).
23. Weinberg [7], Eq. (15.10.51).
24. Weinberg [7], Eq. (15.10.53). A factor T was missing in the second term in the curly brackets in Eq. (15.10.53), and has been supplied here.
25. These formulas are obtained by comparing the acoustic damping rate calculated by N. Kaiser, Mon. Not. R. Astron. Soc. 202, 1169 (1983)
26. with the damping rate calculated for arbitrary values of η and χ by S. Weinberg, Astrophys. J. 168, 175 (1971),
27. Eq. (4.15); the latter article also gives values for χ and η, repeated in Ref. [7]: it gives the same value of χ and a value for η that is 3/4 the value quoted in Eq. (55), but these results were based on calculations of L.H. Thomas, Quarterly J. Math. 1, 239 (1930),
28. that assumed isotropic scattering and ignored photon polarization. [The same value for η had been given by C. Misner, Astrophys. J. 151, 431 (1968).].
29. Kaiser's results are calculated using the correct differential cross section for Thomson scattering and take photon polarization into account, and therefore supersede the earlier value quoted for η. As late as 1995, the wrong value of the damping rate was still being used, for instance by Hu and Sugiyama [5], but the correct rate was used by Hu and White, Astrophys. J. 479, 568 (1997).
30. J. A. Peacock, Cosmological Physics (Cambridge University Press, Cambridge, England, 1999), p. 591.
31. The derivation is given, for instance, in Weinberg [7], Eq. (15.9.13). The presence of the second term on the left-hand side has as a consequence the well known decay ∝ 1/a(t) of the peculiar velocities of nonrelativistic free particles. The factor 1/a(t) multiplying the gradient of the potential enters again to convert a derivative with respect to comoving coordinates into a derivative with respect to coordinates that measure proper distances.
32. Hu and Sugiyama [5]; Hu and White [11].
33. A. Dimitropoulos and L.P. Grishchuk, gr-qc/0010087.
34. See, e.g., P.H. Frampton, Y.J. Ng, and R. Rohm, Mod. Phys. Lett. A 13, 2541 (1998).
35. Hu and White [11].
36. R.K. Sachs and A.M. Wolfe, Astrophys. J. 1, 73 (1967).
37. Weinberg [7], Eqs. (15.10.29) and (15.1.19). [A misprint has been corrected here: the equals sign in the first line of Eq. (15.10.29) has been changed to a minus sign.]
38. L.A. Kofman and A.A. Starobinskii, Sov. Astron. Lett. 11, 271 (1985).
39. The 1/l dependence was found in Ref. [20], but without consideration of a possible interference between this effect and the Doppler shift and intrinsic temperature shift.