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Register a new userPlanck intermediate results. XVI. Profile likelihoods for cosmological parameters
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Stephane Plaszczynski
Publication date
2013
fields
Physics
and research's language is
English
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We explore the 2013 Planck likelihood function with a high-precision multi-dimensional minimizer (Minuit). This allows a refinement of the Lambda-cdm best-fit solution with respect to previously-released results, and the construction of frequentist confidence intervals using profile likelihoods. The agreement with the cosmological results from the Bayesian framework is excellent, demonstrating the robustness of the Planck results to the statistical methodology. We investigate the inclusion of neutrino masses, where more significant differences may appear due to the non-Gaussian nature of the posterior mass distribution. By applying the Feldman--Cousins prescription, we again obtain results very similar to those of the Bayesian methodology. However, the profile-likelihood analysis of the CMB combination (Planck+WP+highL) reveals a minimum well within the unphysical negative-mass region. We show that inclusion of the Planck CMB-lensing information regularizes this issue, and provide a robust frequentist upper limit $M_ u < 0.26 eV$ ($95%$ confidence) from the CMB+lensing+BAO data combination.
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We present the first results based on Planck measurements of the CMB temperature and lensing-potential power spectra. The Planck spectra at high multipoles are extremely well described by the standard spatially-flat six-parameter LCDM cosmology. In this model Planck data determine the cosmological parameters to high precision. We find a low value of the Hubble constant, H0=67.3+/-1.2 km/s/Mpc and a high value of the matter density parameter, Omega_m=0.315+/-0.017 (+/-1 sigma errors) in excellent agreement with constraints from baryon acoustic oscillation (BAO) surveys. Including curvature, we find that the Universe is consistent with spatial flatness to percent-level precision using Planck CMB data alone. We present results from an analysis of extensions to the standard cosmology, using astrophysical data sets in addition to Planck and high-resolution CMB data. None of these models are favoured significantly over standard LCDM. The deviation of the scalar spectral index from unity is insensitive to the addition of tensor modes and to changes in the matter content of the Universe. We find a 95% upper limit of r<0.11 on the tensor-to-scalar ratio. There is no evidence for additional neutrino-like relativistic particles. Using BAO and CMB data, we find N_eff=3.30+/-0.27 for the effective number of relativistic degrees of freedom, and an upper limit of 0.23 eV for the summed neutrino mass. Our results are in excellent agreement with big bang nucleosynthesis and the standard value of N_eff=3.046. We find no evidence for dynamical dark energy. Despite the success of the standard LCDM model, this cosmology does not provide a good fit to the CMB power spectrum at low multipoles, as noted previously by the WMAP team. While not of decisive significance, this is an anomaly in an otherwise self-consistent analysis of the Planck temperature data.
We present results based on full-mission Planck observations of temperature and polarization anisotropies of the CMB. These data are consistent with the six-parameter inflationary LCDM cosmology. From the Planck temperature and lensing data, for this cosmology we find a Hubble constant, H0= (67.8 +/- 0.9) km/s/Mpc, a matter density parameter Omega_m = 0.308 +/- 0.012 and a scalar spectral index with n_s = 0.968 +/- 0.006. (We quote 68% errors on measured parameters and 95% limits on other parameters.) Combined with Planck temperature and lensing data, Planck LFI polarization measurements lead to a reionization optical depth of tau = 0.066 +/- 0.016. Combining Planck with other astrophysical data we find N_ eff = 3.15 +/- 0.23 for the effective number of relativistic degrees of freedom and the sum of neutrino masses is constrained to < 0.23 eV. Spatial curvature is found to be |Omega_K| < 0.005. For LCDM we find a limit on the tensor-to-scalar ratio of r <0.11 consistent with the B-mode constraints from an analysis of BICEP2, Keck Array, and Planck (BKP) data. Adding the BKP data leads to a tighter constraint of r < 0.09. We find no evidence for isocurvature perturbations or cosmic defects. The equation of state of dark energy is constrained to w = -1.006 +/- 0.045. Standard big bang nucleosynthesis predictions for the Planck LCDM cosmology are in excellent agreement with observations. We investigate annihilating dark matter and deviations from standard recombination, finding no evidence for new physics. The Planck results for base LCDM are in agreement with BAO data and with the JLA SNe sample. However the amplitude of the fluctuations is found to be higher than inferred from rich cluster counts and weak gravitational lensing. Apart from these tensions, the base LCDM cosmology provides an excellent description of the Planck CMB observations and many other astrophysical data sets.
We present cosmological parameter results from the final full-mission Planck measurements of the CMB anisotropies. We find good consistency with the standard spatially-flat 6-parameter $Lambda$CDM cosmology having a power-law spectrum of adiabatic scalar perturbations (denoted base $Lambda$CDM in this paper), from polarization, temperature, and lensing, separately and in combination. A combined analysis gives dark matter density $Omega_c h^2 = 0.120pm 0.001$, baryon density $Omega_b h^2 = 0.0224pm 0.0001$, scalar spectral index $n_s = 0.965pm 0.004$, and optical depth $tau = 0.054pm 0.007$ (in this abstract we quote $68,%$ confidence regions on measured parameters and $95,%$ on upper limits). The angular acoustic scale is measured to $0.03,%$ precision, with $100theta_*=1.0411pm 0.0003$. These results are only weakly dependent on the cosmological model and remain stable, with somewhat increased errors, in many commonly considered extensions. Assuming the base-$Lambda$CDM cosmology, the inferred late-Universe parameters are: Hubble constant $H_0 = (67.4pm 0.5)$km/s/Mpc; matter density parameter $Omega_m = 0.315pm 0.007$; and matter fluctuation amplitude $sigma_8 = 0.811pm 0.006$. We find no compelling evidence for extensions to the base-$Lambda$CDM model. Combining with BAO we constrain the effective extra relativistic degrees of freedom to be $N_{rm eff} = 2.99pm 0.17$, and the neutrino mass is tightly constrained to $sum m_ u< 0.12$eV. The CMB spectra continue to prefer higher lensing amplitudes than predicted in base -$Lambda$CDM at over $2,sigma$, which pulls some parameters that affect the lensing amplitude away from the base-$Lambda$CDM model; however, this is not supported by the lensing reconstruction or (in models that also change the background geometry) BAO data. (Abridged)
This paper presents the Planck 2015 likelihoods, statistical descriptions of the 2-point correlations of CMB data, using the hybrid approach employed previously: pixel-based at $ell<30$ and a Gaussian approximation to the distribution of spectra at higher $ell$. The main improvements are the use of more and better processed data and of Planck polarization data, and more detailed foreground and instrumental models, allowing further checks and enhanced immunity to systematics. Progress in foreground modelling enables a larger sky fraction. Improvements in processing and instrumental models further reduce uncertainties. For temperature, we perform an analysis of end-to-end instrumental simulations fed into the data processing pipeline; this does not reveal biases from residual instrumental systematics. The $Lambda$CDM cosmological model continues to offer a very good fit to Planck data. The slope of primordial scalar fluctuations, $n_s$, is confirmed smaller than unity at more than 5{sigma} from Planck alone. We further validate robustness against specific extensions to the baseline cosmology. E.g., the effective number of neutrino species remains compatible with the canonical value of 3.046. This first detailed analysis of Planck polarization concentrates on E modes. At low $ell$ we use temperature at all frequencies and a subset of polarization. The frequency range improves CMB-foreground separation. Within the baseline model this requires a reionization optical depth $tau=0.078pm0.019$, significantly lower than without high-frequency data for explicit dust monitoring. At high $ell$ we detect residual errors in E, typically O($mu$K$^2$); we recommend temperature alone as the high-$ell$ baseline. Nevertheless, Planck high-$ell$ polarization allows a separate determination of $Lambda$CDM parameters consistent with those from temperature alone.
This paper describes the 2018 Planck CMB likelihoods, following a hybrid approach similar to the 2015 one, with different approximations at low and high multipoles, and implementing several methodological and analysis refinements. With more realistic simulations, and better correction and modelling of systematics, we can now make full use of the High Frequency Instrument polarization data. The low-multipole 100x143 GHz EE cross-spectrum constrains the reionization optical-depth parameter $tau$ to better than 15% (in combination with with the other low- and high-$ell$ likelihoods). We also update the 2015 baseline low-$ell$ joint TEB likelihood based on the Low Frequency Instrument data, which provides a weaker $tau$ constraint. At high multipoles, a better model of the temperature-to-polarization leakage and corrections for the effective calibrations of the polarization channels (polarization efficiency or PE) allow us to fully use the polarization spectra, improving the constraints on the $Lambda$CDM parameters by 20 to 30% compared to TT-only constraints. Tests on the modelling of the polarization demonstrate good consistency, with some residual modelling uncertainties, the accuracy of the PE modelling being the main limitation. Using our various tests, simulations, and comparison between different high-$ell$ implementations, we estimate the consistency of the results to be better than the 0.5$sigma$ level. Minor curiosities already present before (differences between $ell$<800 and $ell$>800 parameters or the preference for more smoothing of the $C_ell$ peaks) are shown to be driven by the TT power spectrum and are not significantly modified by the inclusion of polarization. Overall, the legacy Planck CMB likelihoods provide a robust tool for constraining the cosmological model and represent a reference for future CMB observations. (Abridged)
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