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The Intrinsic Bispectrum of the CMB

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The Intrinsic Bispectrum of the Cosmic Microwave Background

Part of the book series: Springer Theses ((Springer Theses))

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Abstract

The non-Gaussianity of the cosmic microwave background is a powerful probe of the physics of the early Universe. However, we do not expect all of the observed non-Gaussianity to be of primordial origin. The late-time non-linear effects described in the previous chapters will generate some degree of non-Gaussianity even in the absence of a primordial signal. This results in the emergence of an intrinsic CMB bispectrum, the topic of this chapter, which can bias the primordial signal and is interesting in its own. We first derive the formula needed to compute the intrinsic bispectrum starting from the second-order transfer functions, which is now fully implemented in our numerical code, SONG, and discuss the most popular templates for the primordial bispectrum: local, equilateral and orthogonal (Sect. 6.2). We then set to compute the signal-to-noise ratio of the intrinsic bispectrum and the bias that its presence induces in the measurements of the primordial non-Gaussianity. To do so, we adopt a Fisher matrix approach (Sect. 6.3), and find that the amplitude of the temperature intrinsic bispectrum is beyond the sensitivity of the Planck CMB survey, with a signal-to-noise ratio of \(\sim \)1/3 and biases smaller than the error bars (Sect. 6.4). We conclude the chapter with a number of numerical and analytical checks on SONG ’s results ( Sect. 6.5). These include extensive convergence tests on the most important numerical parameters in SONG and a successful comparison which the well-known analytical limit for the squeezed configurations of the bispectrum.

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Notes

  1. 1.

    Note that from now on we shall omit writing the time dependence. This does not create ambiguity as the transfer functions \(\mathcal {T} \) are always evaluated today, \(\tau _{\small {\text {0}}} \), and the potentials \(\varPhi \) at the initial time \(\tau _\text {in} \).

  2. 2.

    Note that we have also assumed that the Dirac delta function does not depend on \(\varvec{\hat{k}_3}\,\); we shall prove this point later in the comment to Eq. 6.30

  3. 3.

    The adjective “angle-averaged” comes from the fact that, using Eq. A.36, \(\,B_{{\ell _1} {\ell _2} {\ell _3}} \,\) can be written as

    figure a
  4. 4.

    More precisely, our uncertainties are about \(15\!-\!20\,\%\) smaller than Planck’s. The reason is that the error budget in Planck’s analysis includes uncertainties from more subtle effects such as incomplete foreground removal. By setting \(f_\text {sky}=0.74\,\) in our Fisher matrix estimator, we obtain a percent-level match.

  5. 5.

    Cyril Pitrou, private communication (2013).

  6. 6.

    Note that the inclusion of the non-scalar modes should not affect the \(S/N\) of the primordial templates, because we assume that the vector and tensor modes vanish at first order. However, we can see from the Fisher matrix in Eq. 6.84 that there are differences of the order \(5\,\%\) for the local template. The reason for this discrepancy is purely numerical: in order to compute the intrinsic bispectrum for the \(m\ne 0\) modes we have adopted a different \(\ell \)-grid that contains only configurations where \({\ell _1} +{\ell _2} +{\ell _3} \) is even, as the bispectrum formula (Eq. 6.36) vanishes otherwise. The local template is the most affected one by this slightly worse grid because it is very peaked for squeezed configurations.

References

  1. Alishahiha M, Silverstein E, Tong D (2004) DBI in the sky: Non-Gaussianity from inflation with a speed limit. Phys Rev D 70(12):123505. doi:10.1103/PhysRevD.70.123505. arXiv:hep-th/0404084

  2. Bartolo N, Matarrese S, Riotto A (2004) Evolution of second-order cosmological perturbations and non-Gaussianity. J Cosmol Astropart Phys 1:003. doi:10.1088/1475-7516/2004/01/003. arXiv:astro-ph/0309692

    Google Scholar 

  3. Bartolo N, Matarrese S, Riotto A (2004b) Gauge-invariant temperature anisotropies and primordial non-Gaussianity. Phy Rev Lett 93(23):231301. doi:10.1103/PhysRevLett.93.231301. arXiv:astro-ph/0407505

  4. Bartolo N, Matarrese S, Riotto A (2012) Non-gaussianity in the cosmic microwave background anisotropies at recombination in the squeezed limit. J Cosmol Astropart Phys 2:017. doi:10.1088/1475-7516/2012/02/017. arXiv:1109.2043

    Google Scholar 

  5. Bennett CL, Larson D, Weiland JL, Jarosik N, Hinshaw G et al (2013) Nine-year Wilkinson Microwave Anisotropy Probe (WMAP) observations: final maps and results. ApJS 208:20. doi:10.1088/0067-0049/208/2/20. arXiv:1212.5225

    Google Scholar 

  6. Blas D, Lesgourgues J, Tram T (2011) The Cosmic Linear Anisotropy Solving System (CLASS). Part II: Approximation schemes. J Cosmol Astropart Phys 7:034. doi:10.1088/1475-7516/2011/07/034. arXiv:1104.2933

    Google Scholar 

  7. Boubekeur L, Creminelli P, D’Amico G, Noreña J, Vernizzi F (2009) Sachs-Wolfe at second order: the CMB bispectrum on large angular scales. J Cosmol Astropart Phys 8:029. doi:10.1088/1475-7516/2009/08/029. arXiv:0906.0980

    Google Scholar 

  8. Creminelli P, Zaldarriaga M (2004) CMB 3-point functions generated by nonlinearities at recombination. Phys Rev D 70(8):083532. doi:10.1103/PhysRevD.70.083532. arXiv:astro-ph/0405428

  9. Creminelli P, Nicolis A, Senatore L, Tegmark M, Zaldarriaga M (2006) Limits on non-Gaussianities from WMAP data. J Cosmol Astropart Phys 5:004. doi:10.1088/1475-7516/2006/05/004. arXiv:astro-ph/0509029

    Google Scholar 

  10. Creminelli P, Pitrou C, Vernizzi F (2011) The CMB bispectrum in the squeezed limit. J Cosmol Astropart Phys 11:025. doi:10.1088/1475-7516/2011/11/025. arXiv:1109.1822

    Google Scholar 

  11. DLMF (2013) NIST digital library of mathematical functions. http://dlmf.nist.gov/, Release 1.0.6 of 2013-05-06. http://dlmf.nist.gov/, online companion to [40]

  12. Enqvist K, Sloth MS (2002) Adiabatic CMB perturbations in pre-Big-Bang string cosmology. Nucl Phy B 626:395–409. doi:10.1016/S0550-3213(02)00043-3. arXiv:hep-ph/0109214

    Google Scholar 

  13. Fergusson JR, Shellard EPS (2007) Primordial non-Gaussianity and the CMB bispectrum. Phys Rev D 76(8):083523. doi:10.1103/PhysRevD.76.083523. arXiv:astro-ph/0612713

  14. Fergusson JR, Shellard EPS (2009) Shape of primordial non-Gaussianity and the CMB bispectrum. Phys Rev D 80(4):043510. doi:10.1103/PhysRevD.80.043510. arXiv:0812.3413

  15. Fergusson JR, Liguori M, Shellard EPS (2012) The CMB bispectrum. J Cosmol Astropart Phys 12:032. doi:10.1088/1475-7516/2012/12/032. arXiv:1006.1642

    Google Scholar 

  16. Galassi M, Davies J, Theiler J, Gough B, Jungman G, Alken P, Booth M, Rossi F (2009) GNU Scientific Library (GSL) Reference Manual, 3rd edn. Network Theory Ltd., Cambridge. http://www.gnu.org/software/gsl/

  17. Gangui A, Lucchin F, Matarrese S, Mollerach S (1994) The three-point correlation function of the cosmic microwave background in inflationary models. ApJ 430:447–457. doi:10.1086/174421. arXiv:astro-ph/9312033

    Google Scholar 

  18. Hanson D, Smith KM, Challinor A, Liguori M (2009) CMB lensing and primordial non-Gaussianity. Phys Rev D 80(8):083004. doi:10.1103/PhysRevD.80.083004. arXiv:0905.4732

  19. Hinshaw G, Larson D, Komatsu E, Spergel DN, Bennett CL, Dunkley J, Nolta MR, Halpern M, Hill RS, Odegard N, Page L, Smith KM, Weiland JL, Gold B, Jarosik N, Kogut A, Limon M, Meyer SS, Tucker GS, Wollack E, Wright EL (2013) Nine-year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Cosmological Parameter Results. ApJS 208:19. doi:10.1088/0067-0049/208/2/19. arXiv:1212.5226

    Google Scholar 

  20. Hu W, White M (1997) CMB anisotropies: total angular momentum method. Phys Rev D 56:596–615. doi:10.1103/PhysRevD.56.596. arXiv:astro-ph/9702170

    Google Scholar 

  21. Huang Z, Vernizzi F (2013) Cosmic microwave background bispectrum from recombination. Phys Rev Lett 110(101):303. doi:10.1103/PhysRevLett.110.101303, http://link.aps.org/doi/10.1103/PhysRevLett.110.101303

  22. Khatri R, Wandelt BD (2009) Crinkles in the last scattering surface: Non-Gaussianity from inhomogeneous recombination. Phys. Rev. D 79(2):023501. doi:10.1103/PhysRevD.79.023501. arXiv:0810.4370

  23. Komatsu E (2002) The pursuit of non-gaussian fluctuations in the cosmic microwave background. PhD thesis, Tohoku University. arXiv:astro-ph/0206039

  24. Komatsu E, Spergel DN (2001) Acoustic signatures in the primary microwave background bispectrum. Phys Rev D 63(6):063002. doi:10.1103/PhysRevD.63.063002. arXiv:astro-ph/0005036

  25. Komatsu E et al (2011) Seven-year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: cosmological interpretation. ApJS 192:18. doi:10.1088/0067-0049/192/2/18. arXiv:1001.4538

    Google Scholar 

  26. Koyama K, Pettinari GW, Mizuno S, Fidler C (2014) Orthogonal non-gaussianity in dbi galileon: prospect for planck polarization and post-planck experiments. Class Quantum Gravity 31(12):125,003. http://stacks.iop.org/0264-9381/31/i=12/a=125003

    Google Scholar 

  27. Lesgourgues J (2011) The Cosmic Linear Anisotropy Solving System (CLASS) I: overview. arXiv:1104.2932

  28. Lewis A (2012) The full squeezed CMB bispectrum from inflation. J Cosmology Astropart Phys 6:023. doi:10.1088/1475-7516/2012/06/023. arXiv:1204.5018

    Google Scholar 

  29. Lewis A, Challinor A (2006) Weak gravitational lensing of the CMB Phys Rep 429:1–65. doi:10.1016/j.physrep.2006.03.002. arXiv:astro-ph/0601594

    Google Scholar 

  30. Lewis A, Challinor A, Lasenby A (2000) Efficient computation of CMB anisotropies in closed FRW models. Astrophys J 538:473–476. arXiv:astro-ph/9911177

  31. Lewis A, Challinor A, Hanson D (2011) The shape of the CMB lensing bispectrum. J Cosmology Astropart Phys 3:018. doi:10.1088/1475-7516/2011/03/018. arXiv:1101.2234

    Google Scholar 

  32. Linde A, Mukhanov V (1997) Non-Gaussian isocurvature perturbations from inflation. Phys Rev D 56:535. doi:10.1103/PhysRevD.56.R535. arXiv:astro-ph/9610219

    Google Scholar 

  33. Lyth DH, Wands D (2002) Generating the curvature perturbation without an inflaton. Phy Lett B 524:5–14. doi:10.1016/S0370-2693(01)01366-1. arXiv:hep-ph/0110002

    Google Scholar 

  34. Mehrem R (2011) The plane wave expansion, infinite integrals and identities involving spherical bessel functions. Appl Math Comput 217(12):5360–5365. doi:10.1016/j.amc.2010.12.004. http://www.sciencedirect.com/science/article/pii/S0096300310011914

    Google Scholar 

  35. Mollerach S, Gangui A, Lucchin F, Matarrese S (1995) Contribution to the three-point function of the cosmic microwave background from the rees-sciama effect. ApJ 453:1. doi:10.1086/176363. arXiv:astro-ph/9503115

    Google Scholar 

  36. Moroi T, Takahashi T (2001) Effects of cosmological moduli fields on cosmic microwave background. Phy Lett B 522:215–221. doi:10.1016/S0370-2693(01)01295-3. arXiv:hep-ph/0110096

    Google Scholar 

  37. Moroi T, Takahashi T (2002) Erratum to: Effects of cosmological moduli fields on cosmic microwave background. Phys Lett B 539:303–303. doi:10.1016/S0370-2693(02)02070-1 (Phys Lett B 522 (2001) 215)

  38. Munshi D, Souradeep T, Starobinsky AA (1995) Skewness of cosmic microwave background temperature fluctuations due to nonlinear gravitational instability. ApJ 454:552. doi:10.1086/176508. arXiv:astro-ph/9501100

    Google Scholar 

  39. Nitta D, Komatsu E, Bartolo N, Matarrese S, Riotto A (2009) CMB anisotropies at second order III: bispectrum from products of the first-order perturbations. J Cosmol Astropart Phys 5:14. doi:10.1088/1475-7516/2009/05/014. arXiv:0903.0894

    Google Scholar 

  40. Pettinari GW, Fidler C, Crittenden R, Koyama K, Wands D (2013) The intrinsic bispectrum of the cosmic microwave background. J Cosmol Astropart Phys 4:003. doi:10.1088/1475-7516/2013/04/003. arXiv:1302.0832

    Google Scholar 

  41. Pettinari GW, Fidler C, Crittenden R, Koyama K, Wands D (2015) Efficient computation of the Cosmic Microwave Background bispectrum in CLASS. In preparation

    Google Scholar 

  42. Pitrou C (2011) CMBquick: Spectrum and bispectrum of Cosmic Microwave Background (CMB). Astrophys Source Code Libr. http://www2.iap.fr/users/pitrou/cmbquick.htm. arXiv:1109.009

  43. Pitrou C, Bernardeau F, Uzan JP (2010) The y-sky: diffuse spectral distortions of the cosmic microwave background. J Cosmol Astropart Phys 7:019. doi:10.1088/1475-7516/2010/07/019. arXiv:0912.3655

    Google Scholar 

  44. Pitrou C, Uzan J, Bernardeau F (2010) The cosmic microwave background bispectrum from the non-linear evolution of the cosmological perturbations. J Cosmol Astropart Phys 7:3. doi:10.1088/1475-7516/2010/07/003. arXiv:1003.0481

    Google Scholar 

  45. Planck Collaboration (2014a) Planck 2013 results. XVI. Cosmological parameters. A&A 571:A16. doi:10.1051/0004-6361/201321554. arXiv:1303.5084

  46. Planck Collaboration (2014b) Planck 2013 results. XXIV. Constraints on primordial non-Gaussianity. A&A 571:A24. doi:10.1051/0004-6361/201321591. arXiv:1303.5076

  47. Pogosian L, Corasaniti PS, Stephan-Otto C, Crittenden R, Nichol R (2005) Tracking dark energy with the integrated Sachs-Wolfe effect: Short and long-term predictions. Phys Rev D 72(10):103519. doi:10.1103/PhysRevD.72.103519. arXiv:astro-ph/0506396

  48. Rees MJ, Sciama DW (1968) Large-scale Density Inhomogeneities in the Universe. Nature 217:511–516. doi:10.1038/217511a0

    Google Scholar 

  49. Renaux-Petel S (2011) Orthogonal non-Gaussianities from Dirac-Born-Infeld Galileon inflation Orthogonal non-Gaussianities from Dirac-Born-Infeld Galileon inflation. Class Quantum Gravity 28(24):249601. doi:10.1088/0264-9381/28/24/249601. arXiv:1105.6366

    Google Scholar 

  50. Senatore L, Tassev S, Zaldarriaga M (2009) Non-gaussianities from perturbing recombination. J Cosmol Astropart Phys 9:38. doi:10.1088/1475-7516/2009/09/038. arXiv:0812.3658

    Google Scholar 

  51. Senatore L, Smith KM, Zaldarriaga M (2010) Non-Gaussianities in single field inflation and their optimal limits from the WMAP 5-year data. J Cosmol Astropart Phys 1:028. doi:10.1088/1475-7516/2010/01/028. arXiv:0905.3746

    Google Scholar 

  52. Serra P, Cooray A (2008) Impact of secondary non-Gaussianities on the search for primordial non-Gaussianity with CMB maps. Phys Rev D 77(10):107305. doi:10.1103/PhysRevD.77.107305. arXiv:0801.3276

  53. Shiraishi M, Nitta D, Yokoyama S, Ichiki K, Takahashi K (2011) CMB Bispectrum from Primordial Scalar, Vector and Tensor Non-Gaussianities. Prog Theor Phy 125:795–813. arXiv:1012.1079

    Google Scholar 

  54. Silverstein E, Tong D (2004) Scalar speed limits and cosmology: acceleration from D-cceleration. Phys Rev D 70(10):103505. doi:10.1103/PhysRevD.70.103505. arXiv:hep-th/0310221

  55. Smidt J, Amblard A, Byrnes CT, Cooray A, Heavens A, Munshi D (2010) CMB contraints on primordial non-Gaussianity from the bispectrum (f\(_{NL}\)) and a new consistency test of single-field inflation. Phys Rev D 81(12):123007. doi:10.1103/PhysRevD.81.123007. arXiv:1004.1409

  56. Smith KM, Zaldarriaga M (2011) Algorithms for bispectra: forecasting, optimal analysis and simulation. MNRAS 417:2–19. doi:10.1111/j.1365-2966.2010.18175.x. arXiv:astro-ph/0612571

    Google Scholar 

  57. Vandevender WH, Haskell KH (1982) The slatec mathematical subroutine library. SIGNUM Newsletter 17:16. http://www.netlib.org/slatec/index.html

    Google Scholar 

  58. Verde L, Wang L, Heavens AF, Kamionkowski M (2000) Large-scale structure, the cosmic microwave background and primordial non-Gaussianity. MNRAS 313:141–147. doi:10.1046/j.1365-8711.2000.03191.x. arXiv:astro-ph/9906301

    Google Scholar 

  59. Wolfram S (1991) Mathematica: a system for doing mathematics by computer. Wolfram Research

    Google Scholar 

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  1. Astronomy Centre, University of Sussex, Brighton, UK

    Guido Walter Pettinari

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Pettinari, G.W. (2016). The Intrinsic Bispectrum of the CMB. In: The Intrinsic Bispectrum of the Cosmic Microwave Background. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-21882-3_6

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