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CMB lensing bispectrum: Assessing analytical predictions against full-sky lensing simulations

Toshiya Namikawa, Benjamin Bose, François R. Bouchet, Ryuichi Takahashi, and Atsushi Taruya
Phys. Rev. D 99, 063511 – Published 12 March 2019
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  • INTRODUCTION
  • CMB LENSING BISPECTRUM
  • CMB LENSING BISPECTRUM IN SIMULATIONS
  • RESULTS
  • SIGNAL OF MODIFIED GRAVITY
  • SUMMARY
  • ACKNOWLEDGMENTS
  • APPENDICES
    • References

    Abstract

    Cosmic microwave background (CMB) lensing is an integrated effect whose kernel is greater than half the peak value in the range 1<z<5. Measuring this effect offers a powerful tool to probe the large-scale structure of the Universe at high redshifts. With the increasing precision of ongoing CMB surveys, other statistics than the lensing power spectrum, in particular the lensing bispectrum, will be measured at high statistical significance. This will provide ways to improve the constraints on cosmological models and lift degeneracies. Following up on an earlier paper, we test analytical predictions of the CMB lensing bispectrum against full-sky lensing simulations, and we discuss their validity and limitation in detail. The tree-level prediction of perturbation theory agrees with the simulation only up to ℓ∼200, but the one-loop order allows capturing the simulation results up to ℓ∼600. We also show that analytical predictions based on fitting formulas for the matter bispectrum agree reasonably well with simulation results, although the precision of the agreement depends on the configurations and scales considered. For instance, the agreement is at the 10% level for the equilateral configuration at multipoles up to ℓ∼2000, but the difference in the squeezed limit increases to more than a factor of 2 at ℓ∼2000. This discrepancy appears to come from limitations in the fitting formula of the matter bispectrum. We also find that the analytical prediction for the post-Born correction to the bispectrum is in good agreement with the simulation. We conclude by discussing the bispectrum prediction in some theories of modified gravity.

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    • Received 19 January 2019

    DOI:https://doi.org/10.1103/PhysRevD.99.063511

    © 2019 American Physical Society

    Physics Subject Headings (PhySH)

    1. Research Areas
    Alternative gravity theoriesCosmic microwave backgroundGravitational lenses
    Gravitation, Cosmology & Astrophysics

    Authors & Affiliations

    Toshiya Namikawa1,2, Benjamin Bose3,4, François R. Bouchet5, Ryuichi Takahashi6, and Atsushi Taruya7,8

    • 1Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom
    • 2Leung Center for Cosmology and Particle Astrophysics, National Taiwan University, Taipei, 10617, Taiwan
    • 3Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan
    • 4Departement de Physique Theorique, Universite de Geneve, 24 quai Ernest Ansermet, 1211 Geneve 4, Switzerland
    • 5Institut d’Astrophysique de Paris, UMR 7095, Sorbonne Université & CNRS, 75014 Paris, France
    • 6Faculty of Science and Technology, Hirosaki University, 3 Bunkyo-cho, Hirosaki, Aomori 036-8561, Japan
    • 7Center for Gravitational Physics, Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan
    • 8Kavli Institute for the Physics and Mathematics of the Universe, The University of Tokyo Institutes for Advanced Study (UTIAS), The University of Tokyo, Chiba 277-8583, Japan

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    Issue

    Vol. 99, Iss. 6 — 15 March 2019

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    Images

    • Figure 1
      Figure 1

      Reduced bispectra, as computed from Eqs. (8) and (11), in various configurations: equilateral (top left), folded (top right), squeezed (bottom left) and isosceles (bottom right). The dashed lines show the contribution from the matter bispectrum alone [Eq. (8)]. The dotted-dashed lines show the binned bispectrum defined in Eq. (13) with 20 multipole bins between ℓ=1 and 2048.

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    • Figure 2
      Figure 2

      Equilateral (top left), folded (top right), squeezed (bottom left) and isosceles (bottom right) configurations of the reduced bispectra measured from simulation (solid points) compared with theoretical models using either the SC (blue) or GM (red) fitting formulas. The top panels show the bispectra with the simulation points in black, the middle panels show the fractional differences, Δb/b of the simulations points, colored according to the reference model they are compared to, and the bottom panels rather show difference normalized by the simulation uncertainties, Δb/σ.

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    • Figure 3
      Figure 3

      Same as Fig. 2 but employing 40 multipole bins instead of 20.

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    • Figure 4
      Figure 4

      Same as Fig. 2 but for the contributions only from the post-Born effect. We average over the 10 realizations of the post-Born simulation.

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    • Figure 5
      Figure 5

      Equilateral (left) and squeezed (right) LSS contributions to CMB lensing bispectrum measured from simulation (black points) compared with tree (orange), one-loop (green) and Gil-Marin fitting formula (red) predictions. The solid lines depict the unbinned result while the dotted-dashed lines depict the binned results using 20 linearly spaced bins in the range 1≤ℓ≤2048.

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    • Figure 6
      Figure 6

      Ratio of equilateral LSS contribution to the CMB lensing bispectrum in the DGP model (left) and f(R) gravity (right) to the GR prediction for various theoretical predictions. For DGP we assume Ωrc=0.438, and for f(R) we assume |f¯R0|=2.5×10−6.

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    • Figure 7
      Figure 7

      The reduced bispectrum obtained from the simple non-Gaussian simulation compares with the analytic formula. The binned analytic bispectrum is obtained from Eq. (a7). The simulation results are shown for two cases with different values of the numerical parameter, Nside (see text). Note that in the lower panel we take the difference between the analytic and simulated bispectrum and then divide it with the 1σ simulation error.

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    • Figure 8
      Figure 8

      The dependence of the analytic prediction of the reduced bispectrum on the fitting formula of the nonlinear matter power spectrum. We show the analytic bispectrum using the original HALOHIT formula derived by Ref. [30] (blue, S02) compared with that using our fiducial fitting formula (red, T12) [31]. The GM fitting formula is used in both cases to compute the modified F2 kernel.

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    • Figure 9
      Figure 9

      Contours showing the ratio of the nonlinear prediction for the LSS contribution to the CMB lensing bispectrum (GM on the left and one-loop on the right) to the tree-level prediction. In both contours ℓ2=500, with ℓ32=ℓ12+ℓ22−ℓ2ℓ1cos(θ) by the triangle condition.

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    • Figure 10
      Figure 10

      Equilateral (left) and squeezed (right) kernel of Eq. (8) as a function of redshift. Each curve denotes a different multipole ℓ=ℓ2=ℓ3. For the equilateral case ℓ1=ℓ and for the squeezed case ℓ1=50. The solid lines show the GM fitting formula prediction while the dashed lines show the tree-level prediction.

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    • Figure 11
      Figure 11

      Matter bispectrum comparisons at z=0.541 for equilateral (left, k1=k2=k3=k) and squeezed (right, k3χ∼50, k=k1=k2) cases. The x-axis is the wave number k times the comoving distance χ, representing a multipole ℓ. The squeezed vector magnitude quoted is the weighted average of the smallest bin, k3=0.021h  Mpc−1, where we take 20 linearly spaced bins from 1≤k3χ≤2000. The errors quoted in the figure are the variance over six realizations of the simulations.

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    • Figure 12
      Figure 12

      Same as Fig. 11 but at z=0.990.

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    • Figure 13
      Figure 13

      Difference of the bispectrum measured from simulations with different resolutions. We plot the fractional difference, Δb/b≡bNside=8192/bNside=4096−1, where bNside=4096 and bNside=8192 are measured bispectra obtained from the simulations of Nside=4096 and 8192, respectively.

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    • Figure 14
      Figure 14

      Squeezed bispectrum, bℓ1ℓℓ, with varying the multipole bin for ℓ1.

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    • Figure 15
      Figure 15

      Fractional difference of squeezed bispectrum, Δb/b, with varying the number of multipole bins (red denotes 20 bins; blue, 40 bins; green, 60 bins).

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