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Second order gravitational effects on CMB temperature anisotropy in Λ dominated flat universes

Kenji Tomita and Kaiki Taro Inoue
Phys. Rev. D 77, 103522 – Published 27 May 2008
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  • INTRODUCTION
  • GENERAL RELATIVISTIC SECOND ORDER…
  • COSMOLOGICAL NEWTONIAN APPROXIMATION
  • TEMPERATURE ANISOTROPY
  • BEHAVIORS OF FIRST ORDER AND SECOND…
  • CONCLUDING REMARKS
  • ACKNOWLEDGEMENTS
  • APPENDICES
    • References

    Abstract

    We study second order gravitational effects of local inhomogeneities on the cosmic microwave background radiation in flat universes with matter and a cosmological constant Λ. We find that the general relativistic correction to the Newtonian approximation is negligible at second order provided that the size of the inhomogeneous region is sufficiently smaller than the horizon scale. For a spherically symmetric top-hat type quasilinear perturbation, the first order temperature fluctuation corresponding to the linear integrated Sachs-Wolfe effect is enhanced (suppressed) by the second order one for a compensated void (lump). As a function of redshift of the local inhomogeneity, the second order temperature fluctuations due to evolution of the gravitational potential have a peak before the matter-Λ equality epoch for a fixed comoving size and a density contrast. The second order gravitational effects from local quasilinear inhomogeneities at a redshift z∼1 may significantly affect the cosmic microwave background.

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    • Received 8 December 2007

    DOI:https://doi.org/10.1103/PhysRevD.77.103522

    ©2008 American Physical Society

    Authors & Affiliations

    Kenji Tomita

    • Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan

    Kaiki Taro Inoue

    • Department of Science and Engineering, Kinki University, Higashi-Osaka, 577-8502, Japan

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    Issue

    Vol. 77, Iss. 10 — 15 May 2008

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    Images

    • Figure 1
      Figure 1
      Time dependent terms ζ1′ and −ζ2′ in second order temperature fluctuations as a function of redshift z. Ωm0 denotes the present value for the matter density parameter.Reuse & Permissions
    • Figure 2
      Figure 2
      A light path in the spherical model.Reuse & Permissions
    • Figure 3
      Figure 3
      The matter density contrast for a top-hat type spherical void.Reuse & Permissions
    • Figure 4
      Figure 4
      The potential function −F(r) for a top-hat type compensated lump with c=−0.1 and r1=0.1.Reuse & Permissions
    • Figure 5
      Figure 5
      The first and second order temperature fluctuations as functions of the matter density parameter at present Ωm0 for photons passing through the center of a compensated spherical void at z∼0. The matter density contrast at the center is (ϵm)c=−0.3 and r0=0.09H0−1, r1=0.1H0−1, where H0 is the Hubble constant. The dashed and dashed-dotted curves denote (ΔT(1)/T)loc and (ΔT(2)/T)loc, respectively. The solid curve represents the total temperature fluctuation (ΔT(1)/T)loc+(ΔT(2)/T)loc.Reuse & Permissions
    • Figure 6
      Figure 6
      For a compensated spherical lump. The parameters are the same as in Fig. 5. The matter density contrast is (ϵm)c=0.3.Reuse & Permissions
    • Figure 7
      Figure 7
      The ratio between the second and first order temperature fluctuations U for a light path passing through the center as a function of the redshift z of the void/lump. The solid, dashed, and dashed-dotted curves correspond to flat models with Ωm0=0.9, 0.6, and 0.3, respectively. The width of the wall is chosen to be 20% of the outer radius, i.e., (r1−r0)/r1=0.2. It turns out that the dependence on the ratio between the width and the outer radius is not prominent.Reuse & Permissions
    • Figure 8
      Figure 8
      The first order and second order temperature fluctuations as a function of the redshift z of a void/lump with wall width (r1−r0)/r1=0.2, a density contrast parameter c, and density parameters (Ωm0,ΩΛ0)=(0.27,0.73). The solid and dashed curves denote 0.05(ΔT(2)/T)loc(z)/(ΔT(2)/T)loc(0) and (ΔT(1)/T)loc(z)/(ΔT(1)/T)loc(0), respectively.Reuse & Permissions
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