16 research outputs found
Properties and use of CMB power spectrum likelihoods
Fast robust methods for calculating likelihoods from CMB observations on
small scales generally rely on approximations based on a set of power spectrum
estimators and their covariances. We investigate the optimality of these
approximation, how accurate the covariance needs to be, and how to estimate the
covariance from simulations. For a simple case with azimuthal symmetry we
compare optimality of hybrid pseudo-C_l CMB power spectrum estimators with the
exact result, indicating that the loss of information is not negligible, but
neither is it enough to have a large effect on standard parameter constraints.
We then discuss the number of samples required to estimate the covariance from
simulations, with and without a good analytic approximation, and assess the use
of shrinkage estimators. Finally we discuss how to combine an approximate
high-ell likelihood with a more exact low-ell harmonic-space likelihood as a
practical method for accurate likelihood calculation on all scales.Comment: 15 pages, 11 figures; updated to match version accepted by PR
Likelihood Analysis of CMB Temperature and Polarization Power Spectra
Microwave background temperature and polarization observations are a powerful
way to constrain cosmological parameters if the likelihood function can be
calculated accurately. The temperature and polarization fields are correlated,
partial sky coverage correlates power spectrum estimators at different ell, and
the likelihood function for a theory spectrum given a set of observed
estimators is non-Gaussian. An accurate analysis must model all these
properties. Most existing likelihood approximations are good enough for a
temperature-only analysis, however they cannot reliably handle a
temperature-polarization correlations. We give a new general approximation
applicable for correlated Gaussian fields observed on part of the sky. The
approximation models the non-Gaussian form exactly in the ideal full-sky limit
and is fast to evaluate using a pre-computed covariance matrix and set of power
spectrum estimators. We show with simulations that it is good enough to obtain
correct results at ell >~ 30 where an exact calculation becomes impossible. We
also show that some Gaussian approximations give reliable parameter constraints
even though they do not capture the shape of the likelihood function at each
ell accurately. Finally we test the approximations on simulations with
realistically anisotropic noise and asymmetric foreground mask.Comment: 35 pages, corrected over-generalization in eq A9 and missing
symmetrization factors in Eqs. A13,A14 - no results change
On the assumption of Gaussianity for cosmological two-point statistics and parameter dependent covariance matrices
In this brief paper we revisit the Fisher information content of cosmological
power spectra or two-point functions of Gaussian fields in order to comment on
the assumption of Gaussian estimators and the use of parameter-dependent
covariance matrices for parameter inference in the context of precision
cosmology. Even though the assumption of a Gaussian likelihood is motivated by
the central limit theorem, we discuss that it leads to Fisher information
content that violates the Cram\'er-Rao bound if used consistently, owing to
independent but artificial information from the parameter-dependent covariance
matrix. At any fixed multipole, this artificial term is shown to become
dominant in the case of a large number of correlated fields. While the
distribution of the estimators does indeed tend to a Gaussian with a large
number of modes, it is shown, however, that its Fisher information content does
not, in the sense that their covariance matrix never carries independent
information content, precisely because of the non-Gaussian shape of the
distribution. In this light, we discuss the use of parameter-dependent
covariance matrices with Gaussian likelihoods for parameter inference from
two-point statistics. As a rule of thumb, Gaussian likelihoods should always be
used with a covariance matrix fixed in parameter space, since only this
guarantees that conservative information content is assigned to the
observables, and at the same time prevents biases appearing.Comment: Matches version published in A&
Lensed CMB temperature and polarization maps from the Millennium Simulation
We have constructed the first all-sky CMB temperature and polarization lensed
maps based on a high-resolution cosmological N-body simulation, the Millennium
Simulation (MS). We have exploited the lensing potential map obtained using a
map-making procedure (Carbone et al. 2008) which integrates along the
line-of-sight the MS dark matter distribution by stacking and randomizing the
simulation boxes up to , and which semi-analytically supplies the
large-scale power in the angular lensing potential that is not correctly
sampled by the N-body simulation. The lensed sky has been obtained by properly
modifying the latest version of the LensPix code (Lewis 2005) to account for
the MS structures. We have also produced all-sky lensed maps of the so-called
and potentials, which are directly related to the electric
and magnetic types of polarization. The angular power spectra of the simulated
lensed temperature and polarization maps agree well with semi-analytic
estimates up to , while on smaller scales we find a slight excess of
power which we interpret as being due to non-linear clustering in the MS. We
also observe how non-linear lensing power in the polarised CMB is transferred
to large angular scales by suitably misaligned modes in the CMB and the lensing
potential. This work is relevant in view of the future CMB probes, as a way to
analyse the lensed sky and disentangle the contribution from primordial
gravitational waves.Comment: 13 pages, 11 figures, comments added, MNRAS in pres
Detecting relic gravitational waves in the CMB: A statistical bias
Analyzing the imprint of relic gravitational waves (RGWs) on the cosmic
microwave background (CMB) power spectra provides a way to determine the signal
of RGWs. In this Letter, we discuss a statistical bias, which could exist in
the data analysis and has the tendency to overlook the RGWs. We also explain
why this bias exists, and how to avoid it.Comment: 4 pages, 1 figur
TEASING: a fast and accurate approximation for the low multipole likelihood of the Cosmic Microwave Background temperature
We explore the low-l likelihood of the angular spectrum C(l) of masked CMB
temperature maps using an adaptive importance sampler. We find that, in spite
of a partial sky coverage, the likelihood distribution of each C(l) closely
follows an inverse gamma distribution. Our exploration is accurate enough to
measure the inverse gamma parameters along with the correlation between
multipoles. Those quantities are used to build an approximation of the joint
posterior distribution of the low-l likelihood. The accuracy of the proposed
approximation is established using both statistical criteria and a mock
cosmological parameter fit. When applied to the WMAP5 data set, this
approximation yields cosmological parameter estimates at the same level of
accuracy as the best current techniques but with very significant speed gains.Comment: 10 pages, 10 figures, submitted to MNRA
Planck 2013 results. XXII. Constraints on inflation
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_: 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 V00 < 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=dln k = 0:0134 _ 0:0090. We verify these conclusions through a numerical analysis, which makes no slowroll 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 __2 e_ _ 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 _2 e_ by approximately 4 as a result of slightly lowering the theoretical prediction for the ` <_ 40 multipoles relative to the higher multipoles. Nonetheless, the data are consistent with adiabatic initial conditions
Planck 2015 results. XIII. Cosmological parameters
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
Planck 2015 results. XX. Constraints on inflation
We present the implications for cosmic inflation of the Planck measurements of the cosmic microwave background (CMB) anisotropies in both temperature and polarization based on the full Planck survey. The Planck full mission temperature data and a first release of polarization data on large angular scales measure the spectral index of curvature perturbations to be n s = 0.968 ± 0.006 and tightly constrain its scale dependence to dn s /dlnk = â0.003 ± 0.007 when combined with the Planck lensing likelihood. When the high-â polarization data is included, the results are consistent and uncertainties are reduced. The upper bound on the tensor-to-scalar ratio is r 0.002 <0.11 (95% CL), consistent with the B-mode polarization constraint r<0.12 (95% CL) obtained from a joint BICEP2/Keck Array and Planck analysis. These results imply that V(Ï)âÏ 2 and natural inflation are now disfavoured compared to models predicting a smaller tensor-to-scalar ratio, such as R 2 inflation. Three independent methods reconstructing the primordial power spectrum are investigated. The Planck data are consistent with adiabatic primordial perturbations. We investigate inflationary models producing an anisotropic modulation of the primordial curvature power spectrum as well as generalized models of inflation not governed by a scalar field with a canonical kinetic term. The 2015 results are consistent with the 2013 analysis based on the nominal mission data
Identification and mapping of Algerian island vegetation using high-resolution images (Pléiades and SPOT 6/7) and random forest modeling
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