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Planck 2013 Results. XXIV. Constraints on primordial non-Gaussianity

Planck Collaboration, P. A. R. Ade, N. Aghanim, C. Armitage-Caplan, M. Arnaud, M. Ashdown, F. Atrio-Barandela, J. Aumont, C. Baccigalupi, A. J. Banday, R. B. Barreiro, J. G. Bartlett, N. Bartolo, E. Battaner, K. Benabed, A. Benoît, A. Benoit-Lévy, J. -P. Bernard, M. Bersanelli, P. Bielewicz, J. Bobin, J. J. Bock, A. Bonaldi, L. Bonavera, J. R. Bond, J. Borrill, F. R. Bouchet, M. Bridges, M. Bucher, C. Burigana, R. C. Butler, J. -F. Cardoso, A. Catalano, A. Challinor, A. Chamballu, H. C. Chiang, L. -Y Chiang, P. R. Christensen, S. Church, D. L. Clements, S. Colombi, L. P. L. Colombo, F. Couchot, A. Coulais, B. P. Crill, A. Curto, F. Cuttaia, L. Danese, R. D. Davies, R. J. Davis, P. de Bernardis, A. de Rosa, G. de Zotti, J. Delabrouille, J. -M. Delouis, F. -X. Désert, J. M. Diego, H. Dole, S. Donzelli, O. Doré, M. Douspis, A. Ducout, J. Dunkley, X. Dupac, G. Efstathiou, F. Elsner, T. A. Enßlin, H. K. Eriksen, J. Fergusson, F. Finelli, O. Forni, M. Frailis, E. Franceschi, S. Galeotta, K. Ganga, M. Giard, Y. Giraud-Héraud, J. González-Nuevo, K. M. Górski, S. Gratton, A. Gregorio, A. Gruppuso, F. K. Hansen, D. Hanson, D. Harrison, A. Heavens, S. Henrot-Versillé, C. Hernández-Monteagudo, D. Herranz, S. R. Hildebrandt, E. Hivon, M. Hobson, W. A. Holmes, A. Hornstrup, W. Hovest, K. M. Huffenberger, A. H. Jaffe, T. R. Jaffe, W. C. Jones, M. Juvela, E. Keihänen, R. Keskitalo, T. S. Kisner, J. Knoche, L. Knox, M. Kunz, H. Kurki-Suonio, F. Lacasa, G. Lagache, A. Lähteenmäki, J. -M. Lamarre, A. Lasenby, R. J. Laureijs, C. R. Lawrence, J. P. Leahy, R. Leonardi, J. Lesgourgues, A. Lewis, M. Liguori, P. B. Lilje, M. Linden-Vørnle, M. López-Caniego, P. M. Lubin, J. F. Macías-Pérez, B. Maffei, D. Maino, N. Mandolesi, A. Mangilli, D. Marinucci, M. Maris, D. J. Marshall, P. G. Martin, E. Martínez-González, S. Masi, M. Massardi, S. Matarrese, F. Matthai, P. Mazzotta, P. R. Meinhold, A. Melchiorri, L. Mendes, A. Mennella, M. Migliaccio, S. Mitra, M. -A. Miville-Deschênes, A. Moneti, L. Montier, G. Morgante, D. Mortlock, A. Moss, D. Munshi, J. A. Murphy, P. Naselsky, P. Natoli, C. B. Netterfield, H. U. Nørgaard-Nielsen, F. Noviello, D. Novikov, I. Novikov, S. Osborne, C. A. Oxborrow, F. Paci, L. Pagano, F. Pajot, D. Paoletti, F. Pasian, G. Patanchon, H. V. Peiris, O. Perdereau, L. Perotto, F. Perrotta, F. Piacentini, M. Piat, E. Pierpaoli, D. Pietrobon, S. Plaszczynski, E. Pointecouteau, G. Polenta, N. Ponthieu, L. Popa, T. Poutanen, G. W. Pratt, G. Prézeau, S. Prunet, J. -L. Puget, J. P. Rachen, B. Racine, R. Rebolo, M. Reinecke, M. Remazeilles, C. Renault, A. Renzi, S. Ricciardi, T. Riller, I. Ristorcelli, G. Rocha, C. Rosset, G. Roudier, J. A. Rubiño-Martín, B. Rusholme, M. Sandri, D. Santos, G. Savini, D. Scott, M. D. Seiffert, E. P. S. Shellard, K. Smith, L. D. Spencer, J. -L. Starck, V. Stolyarov, R. Stompor, R. Sudiwala, R. Sunyaev, F. Sureau, P. Sutter, D. Sutton, A. -S. Suur-Uski, J. -F. Sygnet, J. A. Tauber, D. Tavagnacco, L. Terenzi, L. Toffolatti, M. Tomasi, M. Tristram, M. Tucci, J. Tuovinen, L. Valenziano, J. Valiviita, B. Van Tent, J. Varis, P. Vielva, F. Villa, N. Vittorio, L. A. Wade, B. D. Wandelt, M. White, S. D. M. White, D. Yvon, A. Zacchei, A. Zonca

TL;DR

Planck 2013’s NG analysis delivers the most precise CMB constraints to date on primordial NG, finding f_NL^local=2.7±5.8, f_NL^equil=-42±75, and f_NL^ortho=-25±39, consistent with Gaussian initial conditions. The study employs three optimal bispectrum estimators (KSW, binned, modal) with extensive cross-validation, foreground tests, and component-separation consistency, while also reconstructing the full CMB bispectrum and probing a wide range of non-standard shapes. It robustly accounts for secondary contributions like ISW-lensing and point-source NG, and provides initial trispectrum bounds (τ_NL<2800 at 95% CL). The results place stringent constraints on inflationary mechanisms, notably bounding the sound speed (c_s≥0.02) and curvaton decay fraction (r_D≥0.15), while disfavouring ekpyrotic scenarios, thereby reinforcing the standard single-field slow-roll paradigm. Overall, Planck’s NG results offer powerful, model-discriminating tests of early-Universe physics and set benchmarks for future CMB and large-scale-structure analyses.

Abstract

The Planck nominal mission cosmic microwave background (CMB) maps yield unprecedented constraints on primordial non-Gaussianity (NG). Using three optimal bispectrum estimators, separable template-fitting (KSW), binned, and modal, we obtain consistent values for the primordial local, equilateral, and orthogonal bispectrum amplitudes, quoting as our final result fNL^local= 2.7+/-5.8, fNL^equil= -42+/-75, and fNL^ortho= -25+\-39 (68% CL statistical). NG is detected in the data; using skew-C_l statistics we find a nonzero bispectrum from residual point sources, and the ISW-lensing bispectrum at a level expected in the LambdaCDM scenario. The results are based on comprehensive cross-validation of these estimators on Gaussian and non-Gaussian simulations, are stable across component separation techniques, pass an extensive suite of tests, and are confirmed by skew-C_l, wavelet bispectrum and Minkowski functional estimators. Beyond estimates of individual shape amplitudes, we present model-independent, 3-dimensional reconstructions of the Planck CMB bispectrum and thus derive constraints on early-Universe scenarios that generate primordial NG, including general single-field models of inflation, excited initial states (non-Bunch-Davies vacua), and directionally-dependent vector models. We provide an initial survey of scale-dependent feature and resonance models. These results bound both general single-field and multi-field model parameter ranges, such as the speed of sound, c_s \geq 0.02 (95% CL), in an effective field theory parametrization, and the curvaton decay fraction r_D \geq 0.15 (95% CL). The Planck data significantly limit the viable parameter space of the ekpyrotic/cyclic scenarios. The amplitude of the 4-point function in the local model tauNL < 2800 (95% CL). These constraints represent the highest precision tests to date of physical mechanisms for the origin of cosmic structure.

Planck 2013 Results. XXIV. Constraints on primordial non-Gaussianity

TL;DR

Planck 2013’s NG analysis delivers the most precise CMB constraints to date on primordial NG, finding f_NL^local=2.7±5.8, f_NL^equil=-42±75, and f_NL^ortho=-25±39, consistent with Gaussian initial conditions. The study employs three optimal bispectrum estimators (KSW, binned, modal) with extensive cross-validation, foreground tests, and component-separation consistency, while also reconstructing the full CMB bispectrum and probing a wide range of non-standard shapes. It robustly accounts for secondary contributions like ISW-lensing and point-source NG, and provides initial trispectrum bounds (τ_NL<2800 at 95% CL). The results place stringent constraints on inflationary mechanisms, notably bounding the sound speed (c_s≥0.02) and curvaton decay fraction (r_D≥0.15), while disfavouring ekpyrotic scenarios, thereby reinforcing the standard single-field slow-roll paradigm. Overall, Planck’s NG results offer powerful, model-discriminating tests of early-Universe physics and set benchmarks for future CMB and large-scale-structure analyses.

Abstract

The Planck nominal mission cosmic microwave background (CMB) maps yield unprecedented constraints on primordial non-Gaussianity (NG). Using three optimal bispectrum estimators, separable template-fitting (KSW), binned, and modal, we obtain consistent values for the primordial local, equilateral, and orthogonal bispectrum amplitudes, quoting as our final result fNL^local= 2.7+/-5.8, fNL^equil= -42+/-75, and fNL^ortho= -25+\-39 (68% CL statistical). NG is detected in the data; using skew-C_l statistics we find a nonzero bispectrum from residual point sources, and the ISW-lensing bispectrum at a level expected in the LambdaCDM scenario. The results are based on comprehensive cross-validation of these estimators on Gaussian and non-Gaussian simulations, are stable across component separation techniques, pass an extensive suite of tests, and are confirmed by skew-C_l, wavelet bispectrum and Minkowski functional estimators. Beyond estimates of individual shape amplitudes, we present model-independent, 3-dimensional reconstructions of the Planck CMB bispectrum and thus derive constraints on early-Universe scenarios that generate primordial NG, including general single-field models of inflation, excited initial states (non-Bunch-Davies vacua), and directionally-dependent vector models. We provide an initial survey of scale-dependent feature and resonance models. These results bound both general single-field and multi-field model parameter ranges, such as the speed of sound, c_s \geq 0.02 (95% CL), in an effective field theory parametrization, and the curvaton decay fraction r_D \geq 0.15 (95% CL). The Planck data significantly limit the viable parameter space of the ekpyrotic/cyclic scenarios. The amplitude of the 4-point function in the local model tauNL < 2800 (95% CL). These constraints represent the highest precision tests to date of physical mechanisms for the origin of cosmic structure.

Paper Structure

This paper contains 72 sections, 138 equations, 26 figures, 26 tables.

Table of Contents

  1. Introduction
  2. Inflationary models for primordial non-Gaussianity
  3. General single-field models of inflation
  4. Multi-field models
  5. Non-standard models giving rise to alternative specific forms of NG
  6. Higher-order non-Gaussianity: the trispectrum
  7. Statistical estimation of the CMB bispectrum
  8. The CMB angular bispectrum
  9. CMB bispectrum estimators
  10. The KSW estimator
  11. The Skew-$C_{\ell}$ Extension
  12. Separable Modal Methodology
  13. Binned Bispectrum
  14. Wavelet $f_{\rm NL}$ estimator
  15. Wiener filtering
  16. ...and 57 more sections

Figures (26)

  • Figure 1: Permitted observational domain of Eq. (\ref{['eq:tetrapydl']}) for the CMB bispectrum $b_{\ell_1\ell_2\ell_3}$. Allowed multipole values $(\ell_1,\ell_2,\ell_3)$ lie inside the shaded "tetrapyd" region (tetrahedron$+$pyramid), satisfying both the triangle condition and the experimental resolution $\ell <L$$\equiv$$\ell_\textrm{max}$.
  • Figure 2: The binned skew-$C_\ell$ statistics from the SMICA map for (a) ISW-lensing and (b) Poisson point sources. Theoretical curves are not fitted to the data shown, but are plotted with the amplitude (the only free parameter) determined from the KSW technique. The Poisson point-source foreground is clearly detected, and the ISW-lensing skew-spectrum is evident and consistent with the overall $2.6\sigma$ detection. $b_{\textrm{ps}}$ is the Poisson point-source amplitude in dimensionless units of $10^{-29}$, and $f_{\textrm{NL}}^{\textrm{ISW-L}}$ is the ISW-lensing amplitude in units of that expected from the Planck best-fit cosmology. Error bars come from covariance estimates from 1000 simulated maps, and the points are mildly correlated.
  • Figure 3: Map-by-map comparison of the results from the different estimators for local (top), equilateral (centre), and orthogonal (bottom) $f_\mathrm{NL}$ for the set of masked non-Gaussian simulations described in Sect. \ref{['Sec_valid_mask']}, assuming the shapes to be independent. The horizontal solid line is the average value of all maps for KSW, and the dashed and dotted horizontal lines correspond to $1\sigma$ and $2\sigma$ deviations, respectively.
  • Figure 4: Map-by-map comparison of the results from the different estimators for local (top), equilateral (centre), and orthogonal (bottom) $f_\mathrm{NL}$ for 99 maps from a set of realistic lensed simulations passed through the SMICA pipeline, described in Sect. \ref{['Sec_valid_compsep']}, assuming the shapes to be independent. The horizontal solid line is the average value of the maps for KSW, and the dashed and dotted horizontal lines correspond to $1\sigma$ and $2\sigma$ deviations, respectively.
  • Figure 5: Binned skew-$C_\ell$ statistics from the SMICA map for (a) local, (b) equilateral, and (c) orthogonal. Theoretical curves are not fitted to the data shown, but are plotted with the amplitude (the only free parameter) determined from the KSW technique. There is no evidence for detection of primordial NG. Error bars are derived from the covariance of estimates from 1000 simulations. There are mild correlations between data points in all figures, but very strong correlations at high $\ell$ in the local case, reaching correlation coefficients $r>0.99$ for $\ell>1750$.
  • ...and 21 more figures