13 Thus in equation (31) we would get the factor Now using the symmetries of Clebsch-Gordon coefficients given by equation (8.5.34) from [48], we get C$_{2}$ ( ℓ, ℓ$_{1}$, ℓ$_{2}$ ) = ( − i ) $^{ℓ}$Π 2 $_{ℓ}$Π$_{ℓ}$$_{1}$ Π$_{ℓ}$$_{2}$$_{−}$$_{ℓ}$ β$_{ℓ}$$_{1}$$_{−}$$_{ℓ,ℓ}$$_{2}$$_{−}$$_{ℓ}$ β$_{ℓ}$$_{1}$$_{ℓ}$$_{2}$ ( − 1) L + ℓ$_{1}$ + ℓ$_{2}$ √$_{2}$$_{ℓ}$$_{Π}$$_{ℓ}$$_{1}$$_{−}$$_{ℓ}$ C ( ℓ$_{1}$ − ℓ )0 ℓ 0 ℓ$_{1}$ 0 C ℓ$_{2}$ 0 ℓ 0( ℓ$_{2}$ − ℓ )0 C ( ℓ$_{1}$ − ℓ )( ℓ$_{1}$ − ℓ ) ( ℓ$_{2}$ − ℓ )( ℓ$_{2}$ − ℓ ) LL . (D2) C ( ℓ$_{1}$ − ℓ )0 ℓ 0 ℓ$_{1}$ 0 = C ( ℓ$_{1}$ − ℓ )0 ℓ$_{1}$ 0 ℓ 0 = ( − 1) $^{ℓ}$ℓ$_{1}$ ! ℓ !( ℓ$_{1}$ − ℓ )! [ (2 ℓ )!(2 ℓ$_{1}$ − 2 ℓ + 1)! (2 ℓ$_{1}$ + 1)! ] 1 2 . (D3) C ℓ$_{2}$ 0 ℓ 0( ℓ$_{2}$ − ℓ )0 = ( − 1) ℓ Π$_{ℓ}$$_{2}$ Π$_{ℓ}$$_{2}$$_{−}$$_{ℓ}$ C ( ℓ$_{2}$ − ℓ )0 ℓ$_{2}$ 0 ℓ 0 = Π$_{ℓ}$$_{2}$ Π$_{ℓ}$$_{2}$$_{−}$$_{ℓ}$ ℓ$_{2}$ ! ℓ !( ℓ$_{2}$ − ℓ )! [ (2 ℓ )!(2 ℓ$_{2}$ − 2 ℓ + 1)! (2 ℓ$_{2}$ + 1)! ] 1 2 . (D4) C ( ℓ$_{1}$ − ℓ )( ℓ$_{1}$ − ℓ ) ( ℓ$_{2}$ − ℓ )( ℓ$_{2}$ − ℓ ) LL = δ$_{ℓ}$$_{2}$$_{−}$$_{ℓ}$$_{+}$$_{L,ℓ}$$_{1}$$_{−}$$_{ℓ}$ = δ$_{ℓ}$$_{2}$$_{+}$$_{L,ℓ}$$_{1}$ . (D5) C$_{2}$ ( ℓ, ℓ$_{1}$, ℓ$_{2}$ ) = ( − 1) $^{ℓ}$1 + $^{ℓ}$2 2 √$_{2}$$_{ℓ}$ (2 ℓ )! ℓ$_{1}$ ! ℓ$_{2}$ ! ( ℓ !) $^{2}$( ℓ$_{1}$ − ℓ )!( ℓ$_{2}$ − ℓ )! [ (2 ℓ$_{1}$ − 2 ℓ + 1)!(2 ℓ$_{2}$ − 2 ℓ + 1)! (2 ℓ$_{1}$ + 1)!(2 ℓ$_{2}$ + 1)! ] 1 2 Π$_{ℓℓℓ}$$_{1}$$_{ℓ}$$_{2}$ Π$_{ℓ}$$_{1}$$_{−}$$_{ℓ}$ √ ℓ$_{1}$ + ℓ$_{2}$ − 2 ℓ + 1 ℓ$_{1}$ + ℓ$_{2}$ + 1 . (D6) Similarly Using equation (8.5.37) from [48] we get Thus Substituting this in equation (31), the evolution equation becomes where ℓ, ℓ$_{n}$ >> | ℓ$_{n}$ − ℓ | , L with n = 1 , 2. ˜ Δ LM ℓ$_{1}$ $_{,ℓ$_{2}$}$( k, τ ) = ∑ ℓ C$_{2}$ ( ℓ, ℓ$_{1}$, ℓ$_{2}$ ) j$_{ℓ}$ ( k Δ τ ) ˜ Δ LM ℓ$_{1}$ − ℓ,ℓ$_{2}$ − $_{ℓ}$( k, τ$_{s}$ ) δ$_{L,ℓ}$$_{1}$$_{−}$$_{ℓ}$$_{2}$ , (D7) [7] F. K. Hansen, A. J. Banday, K. M. Gorski, H. K. Eriksen, and P. B. Lilje, Astrophysical Journal 704 , 1448 (2009). [1] E. Komatsu, J. Dunkley, M. R. Nolta, C. L. Bennett, B. Gold, G. Hinshaw, N. Jarosik, D. Larson, M. Limon, L. Page, et al., Astrophysical Journal 180 , 330 (2009). [2] D. N. Spergel, R. Bean, O. Dore, M. R. Nolta, C. L. Ben- nett, J. Dunkley, G. Hinshaw, N. Jarosik, E. Komatsu, L. Page, et al., Astrophysical Journal 170 , 377 (2007). [3] M. R. Nolta, J. Dunkley, R. S. Hill, G. Hinshaw, E. Ko- matsu, D. Larson, L. Page, D. N. Spergel, C. L. Bennett, B. Gold, et al., Astrophysical Journal 180 , 296 (2009). [4] C. L. Bennett, R. S. Hill, G. Hinshaw, D. Larson, K. M. Smith, J. Dunkley, B. Gold, M. Halpern, N.Jarosik, A. Kogut, et al., arXiv:1001.4758v1 (2010). [5] A. de Oliveira-Costa, M. Tegmark, M. Zaldarriaga, and A. Hamilton, Physical Review D 69 , 063516 (2004). [6] C. J. Copi, D. Huterer, D. J. Schwarz, and G. D. Stark- man, Physical Review D 75 , 023507 (2007). [14] M. Frommert and T. A. Ensslin, Mon.Not.Roy.Astron. [13] A. Rakic and D. J. Schwarz, Physical Review D 75 , 103002 (2007). [12] J. Magueijo and R. D. Sorkin, Mon.Not.Roy.Astron.Soc. 377 , L39 (2007). [11] M. Tegmark, A. de Oliveira-Costa, and A. J. S. Hamilton, Physical Review D 68 , 123523 (2003). [10] K. Land and J. Magueijo, Physical Review Letters 95 , 071301 (2005). [9] J. Hoftuft, H. K. Eriksen, A. J. Banday, K. M. Gorski, F. K. Hansen, and P. B. Lilje, Astrophysical Journal 699 , 985 (2009). [8] H. K. Eriksen, A. J. Banday, K. M. Gorski, F. K. Hansen, and P. B. Lilje, Astrophysical Journal 660 , L81 (2007).