# Planck 2013 results: XXII. Constraints on inflation

Jussi Varis, Jussi Tuovinen, Planck Collaboration

Research output: Contribution to journalArticleScientificpeer-review

583 Citations (Scopus)

### Abstract

We analyse the implications of the Planck data for cosmic inflation. The Planck nominal mission temperature anisotropy measurements, combined with the WMAP large-angle polarization, constrain the scalar spectral index to be ns = 0.9603 ± 0.0073, ruling out exact scale invariance at over 5{\sigma}.Planck establishes an upper bound on the tensor-to-scalar ratio of r<0.11 (95% CL). The Planck data thus shrink the space of allowed standard inflationary models, preferring potentials with V''<0. Exponential potential models, the simplest hybrid inflationary models, and monomial potential models of degree n = 2 do not provide a good fit to the data. Planck does not find statistically significant running of the scalar spectral index, obtaining dns/ dlnk = - 0.0134 ± 0.0090. We verify these conclusions through a numerical analysis, which makes no slow-roll approximation, and carry out a Bayesian parameter estimation and model-selection analysis for a number of inflationary models including monomial, natural, and hilltop potentials. For each model, we present the Planck constraints on the parameters of the potential and explore several possibilities for the post-inflationary entropy generation epoch, thus obtaining nontrivial data-driven constraints. We also present a direct reconstruction of the observable range of the inflaton potential. Unless a quartic term is allowed in the potential, we find results consistent with second-order slow-roll predictions. We also investigate whether the primordial power spectrum contains any features. We find that models with a parameterized oscillatory feature improve the fit by $\triangle{\chi^2}_{eff}\approx 10$; however, Bayesian evidence does not prefer these models. We constrain several single-field inflation models with generalized Lagrangians by combining power spectrum data with Planck bounds on fNL. Planck constrains with unprecedented accuracy the amplitude and possible correlation (with the adiabatic mode) of non-decaying isocurvature fluctuations. The fractional primordial contributions of cold dark matter (CDM) isocurvature modes of the types expected in the curvaton and axion scenarios have upper bounds of 0.25% and 3.9% (95% CL), respectively. In models with arbitrarily correlated CDM or neutrino isocurvature modes, an anticorrelated isocurvature component can improve the ${\chi^2}_{eff}$ by approximately 4 as a result of slightly lowering the theoretical prediction for the $\ell \lesssim 40$ multipoles relative to the higher multipoles. Nonetheless, the data are consistent with adiabatic initial conditions.
Original language English A22 42 Astronomy and Astrophysics 571 https://doi.org/10.1051/0004-6361/201321569 Published - 2014 A1 Journal article-refereed

### Fingerprint

inflation
scalars
multipoles
power spectra
dark matter
predictions
triangles
prediction
numerical analysis
invariance
entropy
neutrinos
time measurement
anisotropy
tensors
polarization
approximation

### Keywords

• inflation
• early Universe

### Cite this

Varis, Jussi ; Tuovinen, Jussi ; Planck Collaboration. / Planck 2013 results : XXII. Constraints on inflation. In: Astronomy and Astrophysics. 2014 ; Vol. 571.
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abstract = "We analyse the implications of the Planck data for cosmic inflation. The Planck nominal mission temperature anisotropy measurements, combined with the WMAP large-angle polarization, constrain the scalar spectral index to be ns = 0.9603 ± 0.0073, ruling out exact scale invariance at over 5{\sigma}.Planck establishes an upper bound on the tensor-to-scalar ratio of r<0.11 (95{\%} CL). The Planck data thus shrink the space of allowed standard inflationary models, preferring potentials with V''<0. Exponential potential models, the simplest hybrid inflationary models, and monomial potential models of degree n = 2 do not provide a good fit to the data. Planck does not find statistically significant running of the scalar spectral index, obtaining dns/ dlnk = - 0.0134 ± 0.0090. We verify these conclusions through a numerical analysis, which makes no slow-roll approximation, and carry out a Bayesian parameter estimation and model-selection analysis for a number of inflationary models including monomial, natural, and hilltop potentials. For each model, we present the Planck constraints on the parameters of the potential and explore several possibilities for the post-inflationary entropy generation epoch, thus obtaining nontrivial data-driven constraints. We also present a direct reconstruction of the observable range of the inflaton potential. Unless a quartic term is allowed in the potential, we find results consistent with second-order slow-roll predictions. We also investigate whether the primordial power spectrum contains any features. We find that models with a parameterized oscillatory feature improve the fit by $\triangle{\chi^2}_{eff}\approx 10$; however, Bayesian evidence does not prefer these models. We constrain several single-field inflation models with generalized Lagrangians by combining power spectrum data with Planck bounds on fNL. Planck constrains with unprecedented accuracy the amplitude and possible correlation (with the adiabatic mode) of non-decaying isocurvature fluctuations. The fractional primordial contributions of cold dark matter (CDM) isocurvature modes of the types expected in the curvaton and axion scenarios have upper bounds of 0.25{\%} and 3.9{\%} (95{\%} CL), respectively. In models with arbitrarily correlated CDM or neutrino isocurvature modes, an anticorrelated isocurvature component can improve the ${\chi^2}_{eff}$ by approximately 4 as a result of slightly lowering the theoretical prediction for the $\ell \lesssim 40$ multipoles relative to the higher multipoles. Nonetheless, the data are consistent with adiabatic initial conditions.",
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Planck 2013 results : XXII. Constraints on inflation. / Varis, Jussi; Tuovinen, Jussi; Planck Collaboration.

In: Astronomy and Astrophysics, Vol. 571, A22, 2014.

Research output: Contribution to journalArticleScientificpeer-review

TY - JOUR

T1 - Planck 2013 results

T2 - XXII. Constraints on inflation

AU - Varis, Jussi

AU - Tuovinen, Jussi

AU - Planck Collaboration

N1 - Project code: 18080

PY - 2014

Y1 - 2014

N2 - We analyse the implications of the Planck data for cosmic inflation. The Planck nominal mission temperature anisotropy measurements, combined with the WMAP large-angle polarization, constrain the scalar spectral index to be ns = 0.9603 ± 0.0073, ruling out exact scale invariance at over 5{\sigma}.Planck establishes an upper bound on the tensor-to-scalar ratio of r<0.11 (95% CL). The Planck data thus shrink the space of allowed standard inflationary models, preferring potentials with V''<0. Exponential potential models, the simplest hybrid inflationary models, and monomial potential models of degree n = 2 do not provide a good fit to the data. Planck does not find statistically significant running of the scalar spectral index, obtaining dns/ dlnk = - 0.0134 ± 0.0090. We verify these conclusions through a numerical analysis, which makes no slow-roll approximation, and carry out a Bayesian parameter estimation and model-selection analysis for a number of inflationary models including monomial, natural, and hilltop potentials. For each model, we present the Planck constraints on the parameters of the potential and explore several possibilities for the post-inflationary entropy generation epoch, thus obtaining nontrivial data-driven constraints. We also present a direct reconstruction of the observable range of the inflaton potential. Unless a quartic term is allowed in the potential, we find results consistent with second-order slow-roll predictions. We also investigate whether the primordial power spectrum contains any features. We find that models with a parameterized oscillatory feature improve the fit by $\triangle{\chi^2}_{eff}\approx 10$; however, Bayesian evidence does not prefer these models. We constrain several single-field inflation models with generalized Lagrangians by combining power spectrum data with Planck bounds on fNL. Planck constrains with unprecedented accuracy the amplitude and possible correlation (with the adiabatic mode) of non-decaying isocurvature fluctuations. The fractional primordial contributions of cold dark matter (CDM) isocurvature modes of the types expected in the curvaton and axion scenarios have upper bounds of 0.25% and 3.9% (95% CL), respectively. In models with arbitrarily correlated CDM or neutrino isocurvature modes, an anticorrelated isocurvature component can improve the ${\chi^2}_{eff}$ by approximately 4 as a result of slightly lowering the theoretical prediction for the $\ell \lesssim 40$ multipoles relative to the higher multipoles. Nonetheless, the data are consistent with adiabatic initial conditions.

AB - We analyse the implications of the Planck data for cosmic inflation. The Planck nominal mission temperature anisotropy measurements, combined with the WMAP large-angle polarization, constrain the scalar spectral index to be ns = 0.9603 ± 0.0073, ruling out exact scale invariance at over 5{\sigma}.Planck establishes an upper bound on the tensor-to-scalar ratio of r<0.11 (95% CL). The Planck data thus shrink the space of allowed standard inflationary models, preferring potentials with V''<0. Exponential potential models, the simplest hybrid inflationary models, and monomial potential models of degree n = 2 do not provide a good fit to the data. Planck does not find statistically significant running of the scalar spectral index, obtaining dns/ dlnk = - 0.0134 ± 0.0090. We verify these conclusions through a numerical analysis, which makes no slow-roll approximation, and carry out a Bayesian parameter estimation and model-selection analysis for a number of inflationary models including monomial, natural, and hilltop potentials. For each model, we present the Planck constraints on the parameters of the potential and explore several possibilities for the post-inflationary entropy generation epoch, thus obtaining nontrivial data-driven constraints. We also present a direct reconstruction of the observable range of the inflaton potential. Unless a quartic term is allowed in the potential, we find results consistent with second-order slow-roll predictions. We also investigate whether the primordial power spectrum contains any features. We find that models with a parameterized oscillatory feature improve the fit by $\triangle{\chi^2}_{eff}\approx 10$; however, Bayesian evidence does not prefer these models. We constrain several single-field inflation models with generalized Lagrangians by combining power spectrum data with Planck bounds on fNL. Planck constrains with unprecedented accuracy the amplitude and possible correlation (with the adiabatic mode) of non-decaying isocurvature fluctuations. The fractional primordial contributions of cold dark matter (CDM) isocurvature modes of the types expected in the curvaton and axion scenarios have upper bounds of 0.25% and 3.9% (95% CL), respectively. In models with arbitrarily correlated CDM or neutrino isocurvature modes, an anticorrelated isocurvature component can improve the ${\chi^2}_{eff}$ by approximately 4 as a result of slightly lowering the theoretical prediction for the $\ell \lesssim 40$ multipoles relative to the higher multipoles. Nonetheless, the data are consistent with adiabatic initial conditions.

KW - inflation

KW - early Universe

U2 - 10.1051/0004-6361/201321569

DO - 10.1051/0004-6361/201321569

M3 - Article

VL - 571

JO - Astronomy and Astrophysics

JF - Astronomy and Astrophysics

SN - 0004-6361

M1 - A22

ER -