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Long-wavelength perturbations around homogeneous but anisotropic spacetimes Atsushi NARUKO (FRIS, Tohoku U) in collaboration with E. Komatsu (MPA) M.Yamaguchi (TITech) Long-wavelength perturbations around homogeneous but anisotropic

  1. Long-wavelength perturbations around homogeneous 
 but anisotropic spacetimes Atsushi NARUKO (FRIS, Tohoku U) in collaboration with E. Komatsu (MPA) M.Yamaguchi (TITech)

  2. Long-wavelength perturbations around homogeneous 
 but anisotropic spacetimes Going beyond δ N

  3. text book for δ N

  4. What is δ N ?? ✓ evolution of fluctuations during inflation � � � � � ∂ i Q � ∂ t Q H − 1 � � � � � � λ > H − 1 � Spa/al gradient λ = H − 1 expansion t Perturbation λ < H − 1 theory

  5. <latexit sha1_base64="totuFkPKUEqRwMXiObw8M5HCbwQ=">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</latexit> <latexit sha1_base64="totuFkPKUEqRwMXiObw8M5HCbwQ=">AEvnicbdPdatswFABgtfW2LvtLt8vdiIXB2EVxmDbxaDrWgi0TZswr4U4GFk+SUQlW8hy0+D5HfY0u91eY28z2bFhdSwfTjfEZLFUSg5S7Xr/t3a3nHu3X+w+7Dz6PGTp8+6e8+/p0mKHg04Ym6CkKnMXgaY5XEkFRIQcLsPr6Vf3oBKWRJ/0ysJU0HmMZsxSrRJBd23PiV8HFDsH7K5+hH4guQzFhNe4M/YVxDlfgRcEzwsgm7P3XergTeDfh30UD0ugr2dwo8SmgmINeUkTSd9V+pTpRmlEPR8bMUJKHXZA4TE8ZEQDrNq58q8GuTifAsUeaJNa6y/8/IiUjTlQhNpSB6kbatTNpskunZx2nOYplpiOl6oVnGsU5weUI4Ygqo5isTEKqY2SumC6I1eYc76wSCjM7THhULtLpdPwYljQRgsTm2My5HhV59cVHRbGBxw0eW3DQ4MCJw2eWPC0wVMLnjV4ZsFhg0MLnjd4bsFRgyMLjhscW9Br0GtjFJWkBI7aIklRvkwTmS5tkawo8CUouWENapJhXy5Yq+CmTJkPURYDuSaQKeNJ3OKlLib9qdnvkWgGY8g7/WL9sYFqavKdlHCVmIuXF2j4VZXPZ+XOVtyDPYK6SdbW5sP329dwMvHf7n/bd0fvewWF9c3fRS/QKvUF9AEdoAG6QB6i6Cf6hX6jP84XZ+4IJ1mXbm/Vc16gO8O5/Qe/46jg</latexit> <latexit sha1_base64="totuFkPKUEqRwMXiObw8M5HCbwQ=">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</latexit> δ N and the next ?? ✓ based on the leading order approximation of GE method ✓ a way to evaluate the (conserved) curvature perturbation 
 around homogeneous & isotropic universe 
 just by solving BG eqs. not (involved ?) perturbation eqs. � final = δ N R c � � 1. the next order in gradient expansion ?? 2. breaking homogeneity ?? 3. breaking isotropy ??

  6. ① next order in GE ✓ We have already investigated the next order in GE. PTEP 2013 (2013) arXiv:1210.6525 gauge choice : uniform N (e-folding) gauge (slicing) Sasaki & Tanaka [1998] non-linear gauge transformation : sol. in the N gauge -> sol. in the comoving gauge proper definition of non-linear curvature perturbation : R = [perturbation of “a”] + [perturbation of GW]

  7. ② breaking homogeneity ✓ maybe interesting… analysis involved ? any application ?

  8. ③ breaking isotropy Long-wavelength perturbations around homogeneous 
 but anisotropic spacetimes Atsushi NARUKO (FRIS, Tohoku U) in collaboration with E. Komatsu (MPA) M.Yamaguchi (TITech)

  9. <latexit sha1_base64="b/h0d/7cfOKw1MQxDupCgDJQSqE=">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</latexit> <latexit sha1_base64="b/h0d/7cfOKw1MQxDupCgDJQSqE=">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</latexit> <latexit sha1_base64="b/h0d/7cfOKw1MQxDupCgDJQSqE=">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</latexit> motivation -why anisotropic BG- Gibbons & Hawking 1977, Wald 1983 ✓ cosmic no-hair theorem/conjecture for inflationary universe -- Λ (c.c.) ➕ (homogeneous) matter with energy conditions → universe will be isotropized & evolve towards de Sitter = anisotropy will disappear (even if exist initially) ✓ symmetry during inflation -- time-translation symmetry in de Sitter ↔ scale inv. Ps d s 2 = − d t 2 + e 2 Ht d ~ [ & ] x 2 x → e − H λ x t → t + λ → ns -1 = O( ε ) implies the breaking of time-tr. symmetry !! → what about spatial rotational symmetry ?? O( ε ) ???

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