26 We note again that Eqs. (B21) and (B22) have the same form as the decomposition formulae (2.36) and (2.37), respectively. The perturbative formulae for the perturbation of the Einstein tensor is derived by the similar manner to the case of the per- turbations of the scalar curvature. Through these formu- lae summarized above, straightforward calculations leads Eqs. (3.6)–(3.10). We have to note that to derive the formulae (3.9) with Eq. (3.10), we have to consider the general relativistic gauge-invariant perturbation theory with two infinitesimal parameters which is developed in Refs. [8, 9], as commented in the main text. ¯ G b a = ¯ g bc ¯ G$_{ac}$ (B23) [1] J. M. Bardeen, Phys. Rev. D 22 (1980), 1882. [2] H. Kodama and M. Sasaki, Prog. Theor. Phys. Suppl. No. 78 (1984), 1. [3] V. F. Mukhanov, H. A. Feldman and R. H. Branden- berger, Phys. Rep. 215 (1992), 203. [4] C. L. Bennett et al., Astrophys. J. Suppl. Ser. 148 (2003), 1. [5] E. Komatsu et al., Astrophys. J. Suppl. Ser. 180 (2009), 330. [6] V. Acquaviva, N. Bartolo, S. Matarrese, and A. Ri- otto, Nucl. Phys. B 667 (2003), 119. J. Maldacena, J. High Energy Phys, 05 (2003), 013. K. A. Malik and D. Wands, Class. Quantum Grav. 21 (2004), L65. N. Bar- tolo, S. Matarrese and A. Riotto, Phys. Rev. D 69 (2004), 043503. N. Bartolo, S. Matarrese and A. Ri- otto, J. High Energy Phys. 04 (2004), 006. D. H. Lyth and Y. Rodr´ıguez, Phys. Rev. D 71 (2005), 123508. F. Vernizzi, Phys. Rev. D 71 (2005), 061301R. [7] N. Bartolo, S. Matarrese and A. Riotto, JCAP 0401 (2004), 003; N. Bartolo, S. Matarrese and A. Riotto, Phys. Rev. Lett. 93 (2004), 231301; N. Bartolo, E. Ko- matsu, S. Matarrese and A. Riotto, Phys. Rep. 402 (2004), 103; N. Bartolo, S. Matarrese and A. Riotto, JCAP 0605 (2006), 010; N. Bartolo, S. Matarrese and A. Riotto, JCAP 0606 (2006), 024; N. Bartolo, S. Matar- rese and A. Riotto, JCAP 0701 (2007), 019; D. Nitta, E. Komatsu, N. Bartolo, S. Matarrese and A. Riotto, JCAP 0905 (2009), 014. C. Pitrou, J. P. Uzan and F. Bernardeau, Phys. Rev. D 78 (2008), 063526. L. Sen- atore, S. Tassev and M. Zaldarriaga, JCAP 0908 (2009), 031. [8] K. Nakamura, Prog. Theor. Phys. 110 (2003), 723. [9] K. Nakamura, Prog. Theor. Phys. 113 (2005), 481. [10] M. Bruni, S. Matarrese, S. Mollerach and S. Sonego, Class. Quantum Grav. 14 (1997), 2585. [11] K. Nakamura, Phys. Rev. D 74 (2006), 101301(R); K. Nakamura, Prog. Theor. Phys. 117 (2005), 17. [12] K. Nakamura, Phys. Rev. D 80 (2009), 124021. [13] K. Nakamura, Prog. Theor. Phys. 121 (2009), 1321. [14] C. Pitrou, Class. Quantum Grav. 24 (2007), 6127; ibid . 26 (2009), 065006. [15] K. Nakamura, ““Gauge” in General Relativity: – Second- order general relativistic gauge-invariant perturbation theory –“, in Lie Theory and its Applications in Physics VII ed. V. K. Dobrev et al, (Heron Press, Sofia, 2008) [16] R. K. Sachs, “Gravitational Radiation”, in Relativity, Groups and Topology ed. C. DeWitt and B. DeWitt, (New [17] J. M. Stewart and M. Walker, Proc. R. Soc. London A 341 (1974), 49; J. M. Stewart, Class. Quantum Grav. 7 (1990), 1169; Advanced General Relativity (Cambridge University Press, Cambridge, 1991). York: Gordon and Breach, 1964). [18] S. Sonego and M. Bruni, Commun. Math. Phys. 193 (1998), 209. [19] S. Matarrese, S. Mollerach and M. Bruni, Phys. Rev. D 58 (1998), 043504. [20] M. Bruni, L. Gualtieri and C. F. Sopuerta, Class. Quan- tum Grav. 20 (2003), 535. [21] C. F. Sopuerta, M. Bruni and L. Gualtieri, Phys. Rev. D 70 (2004), 064002. [22] M. Bruni and S. Sonego, Class. Quantum Grav. 16 (1999), L29. [23] S. Kobayashi and K. Nomizu, Foundations of Differen- tial Geometry, volume I , (Wiley Classic Library Edition (1996), John Wiley & Sons, Inc.) [24] R. M. Wald, General Relativity (Chicago, IL, University of Chicago Press, 1984). [25] S. Weinberg, Gravitation and Cosmology (New York: Wi- ley, 1972). [26] F. C. Mena, D. J. Mulryne and R. Tavakol, Class. Quant. Grav. 24 (2007), 2721. T. C. Lu, K. Ananda and C. Clarkson, Phys. Rev. D 77 (2008), 043523. T. C. Lu, K. Ananda, C. Clarkson and R. Maartens, JCAP 0902 (2009), 023. A. J. Christopherson, K. A. Malik and D. R. Matravers, Phys. Rev. D 79 (2009), 123523. [27] K. N. Ananda, C. Clarkson and D. Wands, Phys. Rev. D 75 (2007), 123518; B. Osano, C. Pitrou, P. Dunsby, J. P. Uzan and C. Clarkson, JCAP 0704 (2007), 003; D. Baumann, P. J. Steinhardt, K. Takahashi and K. Ichiki, Phys. Rev. D 76 (2007), 084019; N. Bartolo, S. Matarrese, A. Riotto and A. Vaihkonen, Phys. Rev. D 76 (2007), 061302; P. Martineau and R. Branden- berger, Mod. Phys. Lett. A 23 (2008), 727; R. Saito and J. Yokoyama, Phys. Rev. Lett. 102 (2009), 161101; F. Arroja, H. Assadullahi, K. Koyama and D. Wands, Phys. Rev. D 80 (2009), 123526. [28] K. Nakamura, A. Ishibashi and H. Ishihara, Phys. Rev. D 62 (2000), 101502(R); K. Nakamura and H. Ishi- hara, Phys. Rev. D 63 (2001), 127501; K. Nakamura, Class. Quantum Grav. 19 (2002), 783; Phys. Rev. D 66 (2002), 084005; K. Nakamura, Prog. Theor. Phys. 110 , (2003), 201.