arXiv subjects

J. R. Fergusson

Publications and source records attributed to J. R. Fergusson.

15 recordsLinked to original sources

arXiv2016

Separable projection integrals for higher-order correlators of the cosmic microwave sky: Acceleration by factors exceeding 100

We present a case study describing efforts to optimise and modernise "Modal", the simulation and analysis pipeline used by the Planck satellite experiment for constraining general non-Gaussian models of the early universe via the bispectrum (or three-point correlator) of the cosmic microwave background radiation. We focus on one particular element of the code: the projection of bispectra from the end of inflation to the spherical shell at decoupling, which defines the CMB we observe today. This code involves a three-dimensional inner product between two functions, one of which requires an integral, on a non-rectangular domain containing a sparse grid. We show that by employing separable methods this calculation can be reduced to a one-dimensional summation plus two integrations, reducing the overall dimensionality from four to three. The introduction of separable functions also solves the issue of the non-rectangular sparse grid. This separable method can become unstable in certain cases and so the slower non-separable integral must be calculated instead. We present a discussion of the optimisation of both approaches. We show significant speed-ups of ~100x, arising from a combination of algorithmic improvements and architecture-aware optimisations targeted at improving thread and vectorisation behaviour. The resulting MPI/OpenMP hybrid code is capable of executing on clusters containing processors and/or coprocessors, with strong-scaling efficiency of 98.6% on up to 16 nodes. We find that a single coprocessor outperforms two processor sockets by a factor of 1.3x and that running the same code across a combination of both microarchitectures improves performance-per-node by a factor of 3.38x. By making bispectrum calculations competitive with those for the power spectrum (or two-point correlator) we are now able to consider joint analysis for cosmological science exploitation of new data.

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arXiv2015

Using inpainting to construct accurate cut-sky CMB estimators

The direct evaluation of manifestly optimal, cut-sky CMB power spectrum and bispectrum estimators is numerically very costly, due to the presence of inverse-covariance filtering operations. This justifies the investigation of alternative approaches. In this work, we mostly focus on an inpainting algorithm that was introduced in recent CMB analyses to cure cut-sky suboptimalities of bispectrum estimators. First, we show that inpainting can equally be applied to the problem of unbiased estimation of power spectra. We then compare the performance of a novel inpainted CMB temperature power spectrum estimator to the popular apodised pseudo-$C_l$ (PCL) method and demonstrate, both numerically and with analytic arguments, that inpainted power spectrum estimates significantly outperform PCL estimates. Finally, we study the case of cut-sky bispectrum estimators, comparing the performance of three different approaches: inpainting, apodisation and a novel low-l leaning scheme. Providing an analytic argument why the local shape is typically most affected we mainly focus on local type non-Gaussianity. Our results show that inpainting allows to achieve optimality also for bispectrum estimation, but interestingly also demonstrate that appropriate apodisation, in conjunction with low-l cleaning, can lead to comparable accuracy.

astro-ph.CO↗

arXiv2015

Polyspectra searches for sharp oscillatory features in cosmic microwave sky data

We undertake a thorough search for signatures of sharp oscillatory features in the WMAP9 power spectrum and bispectrum as well as in the Planck power spectrum. For the first time, we carry out searches in both the power spectrum and bispectrum simultaneously, employing well-defined look-elsewhere statistics to assess significances in a rigorous manner. Developing efficient methods to scan power spectrum likelihoods for oscillatory features, we present results for the phenomenological bare sine and cosine modulations, allowing validation against existing Planck Likelihood surveys, as well as templates that include the correct sharp feature scaling. In particular, we study degeneracies between feature and cosmological parameters. For frequencies beyond the scale set by the acoustic peaks, the dependencies are realised through uninteresting adjustments of the comoving distance to last scattering. Hence, it is sufficient to keep cosmological parameters fixed and employ fast Gaussian approximations to the likelihood as a function of the feature model amplitude. In cases where results can be compared to the literature, our method shows excellent agreement. We supplement results from the Planck Likelihood with an analysis based on the Planck SMICA component separation map that, working on the assumption that the component separation algorithm is reliable, allows for the inclusion of a larger sky fraction. In principle, this allows us to place the most stringent constraints to date on the amplitudes of feature models in the temperature power spectrum. Invoking the WMAP bispectrum, we perform a combined power spectrum and bispectrum survey. We use and slightly generalise statistics developed in previous work to reliably judge the significance of large feature model amplitude estimates. We conclude that our results are entirely consistent with a featureless realisation of a Gaussian CMB.

astro-ph.CO↗

arXiv2015

Combining power spectrum and bispectrum measurements to detect oscillatory features

The simplest inflationary models present us with few observable parameters to discriminate between them. A detection of features in the spectra of primordial density perturbations could provide valuable insights and lead to stringent tests of models of the early Universe. So far, searches for oscillatory features have not produced statistically significant results. In this work we consider a combined search for features in the power spectrum and bispectrum. We show that possible dependencies between the estimates of feature model amplitudes based on the two- and three-point correlators are largely statistically independent under the assumption of the null hypothesis of a nearly Gaussian featureless CMB. Building on this conclusion we propose an optimal amplitude estimator for a combined search and study the look-elsewhere effect in feature model surveys. In particular we construct analytic models for the distribution of amplitude estimates that allow for a reliable assessment of the significance of potential findings. We also propose a well-behaved integrated statistic that is designed to detect evidence for models exhibiting features at multiple frequencies.

astro-ph.CO↗

arXiv2014

Efficient optimal non-Gaussian CMB estimators with polarisation

In this paper we demonstrate an efficient method for including both CMB temperature and polarisation data in optimal non-Gaussian estimators. The method relies on orthogonalising the multipoles of the temperature and polarisation maps and results in a reduction by a factor of over 3 the terms required to calculate the estimator. The method is illustrated with the modal method applied to bispectrum estimation via the CMB with the trispectrum included as an appendix. However, the method is quite general and can be applied to any optimal bispectrum or trispectrum estimator including the KSW, binned and wavelet approaches. It would also be applicable to any situation where multiple data sets with known correlations are being considered.

astro-ph.CO↗

arXiv2012

The CMB Bispectrum

We use a separable mode expansion estimator with WMAP data to estimate the bispectrum for all the primary families of non-Gaussian models. We review the late-time mode expansion estimator methodology which can be applied to any non-separable primordial and CMB bispectrum model, and we demonstrate how the method can be used to reconstruct the CMB bispectrum from an observational map. We extend the previous validation of the general estimator using local map simulations. We apply the estimator to the coadded WMAP 5-year data, reconstructing the WMAP bispectrum using $l<500$ multipoles and $n=31$ orthonormal 3D eigenmodes. We constrain all popular nearly scale-invariant models, ensuring that the theoretical bispectrum is well-described by a convergent mode expansion. Constraints from the local model $ \fnl=54.4\pm 29.4$ and the equilateral model $\fnl=143.5\pm 151.2$ ($\Fnl = 25.1\pm 26.4$) are consistent with previously published results. (Here, we use a nonlinearity parameter $\Fnl$ normalised to the local case, to allow more direct comparison between different models.) Notable new constraints from our method include those for the constant model $\Fnl = 35.1 \pm 27.4 $, the flattened model $\Fnl = 35.4\pm 29.2$, and warm inflation $\Fnl = 10.3\pm 27.2$. We investigate feature models surveying a wide parameter range in both the scale and phase, and we find no significant evidence of non-Gaussianity in the models surveyed. We propose a measure $\barFnl$ for the total integrated bispectrum and find that the measured value is consistent with the null hypothesis that CMB anisotropies obey Gaussian statistics. We argue that this general bispectrum survey with the WMAP data represents the best evidence for Gaussianity to date and we discuss future prospects, notably from the Planck satellite.

astro-ph.CO↗

arXiv2012

Optimal Polyspectra Estimation

We show that full inverse covariance weighting can be naturally incorporated into modal estimation methods making them optimal for the CMB power spectrum, bispectrum and trispectrum, as well as in other 3D applications. Modal estimation methods are highly efficient requiring inversion of only an nmax x nmax covariance matrix, where nmax is the limited number of modes needed to describe both the theoretical model and the noise which affects it (in contrast to the full lmax^2 x lmax^2 inverse covariance weighting). A WMAP resolution implementation for modal bispectrum estimation is discussed and shown to yield optimal constraints for local and other models at lmax =500 and nmax <= 50. We also discuss further improvements by characterising and preconditioning the covariance matrix using a single bispectrum variance shape induced by the noise and mask. We describe optimal modal estimation in general terms applicable to polyspectra of any order in a formalism that can be extended beyond the isotropic case. We use the example of anisotropic noise in bispectrum estimation to show the completeness of the isotropic modal basis and the necessity of subtracting a linear estimator term for optimality, removing cross-terms in the variance. We briefly discuss applications to the CMB power spectrum and trispectrum, as well as large-scale structure.

astro-ph.CO↗

arXiv2011

Rapid Separable Analysis of Higher Order Correlators in Large Scale Structure

We present an efficient separable approach to the estimation and reconstruction of the bispectrum and the trispectrum from observational (or simulated) large scale structure data. This is developed from general CMB (poly-)spectra methods which exploit the fact that the bispectrum and trispectrum in the literature can be represented by a separable mode expansion which converges rapidly (with $n_\textrm{max}={\cal{O}}(30)$ terms). With an effective grid resolution $l_\textrm{max}$ (number of particles/grid points $N=l_\textrm{max}^3$), we present a bispectrum estimator which requires only ${\cal O}(n_\textrm{max} \times l_\textrm{max}^3)$ operations, along with a corresponding method for direct bispectrum reconstruction. This method is extended to the trispectrum revealing an estimator which requires only ${\cal O}(n_\textrm{max}^{4/3} \times l_\textrm{max}^3)$ operations. The complexity in calculating the trispectrum in this method is now involved in the original decomposition and orthogonalisation process which need only be performed once for each model. However, for non-diagonal trispectra these processes present little extra difficulty and may be performed in ${\cal O}(l_\textrm{max}^4)$ operations. A discussion of how the methodology may be applied to the quadspectrum is also given. An efficient algorithm for the generation of arbitrary nonGaussian initial conditions for use in N-body codes using this separable approach is described. This prescription allows for the production of nonGaussian initial conditions for arbitrary bispectra and trispectra. A brief outline of the key issues involved in parameter estimation, particularly in the non-linear regime, is also given.

astro-ph.CO↗

arXiv2011

Universal Non-Gaussian Initial Conditions for N-body Simulations

In this paper we present the implementation of an efficient formalism for the generation of arbitrary non-Gaussian initial conditions for use in N-body simulations. The methodology involves the use of a separable modal approach for decomposing a primordial bispectrum or trispectrum. This approach allows for the far more efficient generation of the non-Gaussian initial conditions already described in the literature, as well as the generation for the first time of non-separable bispectra and the special class of diagonal-free trispectra. The modal approach also allows for the reconstruction of the spectra from given realisations, a fact which is exploited to provide an accurate consistency check of the simulations.

astro-ph.CO↗

arXiv2010

Optimal Trispectrum Estimators and WMAP Constraints

We present an implementation of an optimal CMB trispectrum estimator which accounts for anisotropic noise and incomplete sky coverage. We use a general separable mode expansion which can and has been applied to constrain both primordial and late-time models. We validate our methods on large angular scales using known analytic results in the Sachs-Wolfe limit. We present the first near-optimal trispectrum constraints from WMAP data on the cubic term of local model inflation $ g_{\rm NL} = (1.6 \pm 7.0)\times 10^5$, for the equilateral model $t_{\rm NL}^{\rm{equil}}=(-3.11\pm 7.5)\times 10^6 $ and for the constant model $t_{\rm NL}^{\rm{const}}=(-1.33\pm 3.62)$. These results, particularly the equilateral constraint, are relevant to a number of well-motivated models (such as DBI and K-inflation) with closely correlated trispectrum shapes. We also use the trispectrum signal predicted for cosmic strings to provide a conservative upper limit on the string tension $Gμ\le 1.1\times 10^{-6}$ (at 95% confidence), which is largely background and model independent. All these new trispectrum results are consistent with a Gaussian Universe. We discuss the importance of constraining general classes of trispectra using these methods and the prospects for higher precision with the Planck satellite.

astro-ph.CO↗

arXiv2010

General CMB and Primordial Trispectrum Estimation

We present trispectrum estimation methods which can be applied to general non-separable primordial and CMB trispectra. We present a general optimal estimator for the connected part of the trispectrum, for which we derive a quadratic term to incorporate the effects of inhomogeneous noise and masking. We describe a general algorithm for creating simulated maps with given arbitrary (and independent) power spectra, bispectra and trispectra. We propose a universal definition of the trispectrum parameter $T_{NL}$, so that the integrated bispectrum on the observational domain can be consistently compared between theoretical models. We define a shape function for the primordial trispectrum, together with a shape correlator and a useful parametrisation for visualizing the trispectrum. We derive separable analytic CMB solutions in the large-angle limit for constant and local models. We present separable mode decompositions which can be used to describe any primordial or CMB bispectra on their respective wavenumber or multipole domains. By extracting coefficients of these separable basis functions from an observational map, we are able to present an efficient estimator for any given theoretical model with a nonseparable trispectrum. The estimator has two manifestations, comparing the theoretical and observed coefficients at either primordial or late times. These mode decomposition methods are numerically tractable with order $l^5$ operations for the CMB estimator and approximately order $l^6$ for the general primordial estimator (reducing to order $l^3$ in both cases for a special class of models). We also demonstrate how the trispectrum can be reconstructed from observational maps using these methods.

astro-ph.CO↗

arXiv2010

Primordial non-Gaussianity and Bispectrum Measurements in the Cosmic Microwave Background and Large-Scale Structure

The most direct probe of non-Gaussian initial conditions has come from bispectrum measurements of temperature fluctuations in the Cosmic Microwave Background and of the matter and galaxy distribution at large scales. Such bispectrum estimators are expected to continue to provide the best constraints on the non-Gaussian parameters in future observations. We review and compare the theoretical and observational problems, current results and future prospects for the detection of a non-vanishing primordial component in the bispectrum of the Cosmic Microwave Background and large-scale structure, and the relation to specific predictions from different inflationary models.

astro-ph.CO↗

arXiv2009

General CMB and Primordial Bispectrum Estimation I: Mode Expansion, Map-Making and Measures of f_NL

We present a detailed implementation of two bispectrum estimation methods which can be applied to general non-separable primordial and CMB bispectra. The method exploits bispectrum mode decompositions on the domain of allowed wavenumber or multipole values. Concrete mode examples constructed from symmetrised tetrahedral polynomials are given, demonstrating rapid convergence for known bispectra. We use these modes to generate simulated CMB maps of high resolution (l > 2000) given an arbitrary primordial power spectrum and bispectrum or an arbitrary late-time CMB angular power spectrum and bispectrum. By extracting coefficients for the same separable basis functions from an observational map, we are able to present an efficient and general f_NL estimator for a given theoretical model. The estimator has two versions comparing theoretical and observed coefficients at either primordial or late times, thus encompassing a wider range of models, including secondary anisotropies, lensing and cosmic strings. We provide examples and validation of both f_NL estimation methods by direct comparison with simulations in a WMAP-realistic context. In addition, we show how the full bispectrum can be extracted from observational maps using these mode expansions, irrespective of the theoretical model under study. We also propose a universal definition of the bispectrum parameter F_NL for more consistent comparison between theoretical models. We obtain WMAP5 estimates of f_NL for the equilateral model from both our primordial and late-time estimators which are consistent with each other, as well as with results already published in the literature. These general bispectrum estimation methods should prove useful for the analysis of nonGaussianity in the Planck satellite data, as well as in other contexts.

astro-ph.CO↗

arXiv2009

The shape of primordial non-Gaussianity and the CMB bispectrum

We present a set of formalisms for comparing, evolving and constraining primordial non-Gaussian models through the CMB bispectrum. We describe improved methods for efficient computation of the full CMB bispectrum for any general (non-separable) primordial bispectrum, incorporating a flat sky approximation and a new cubic interpolation. We review all the primordial non-Gaussian models in the present literature and calculate the CMB bispectrum up to l <2000 for each different model. This allows us to determine the observational independence of these models by calculating the cross-correlation of their CMB bispectra. We are able to identify several distinct classes of primordial shapes - including equilateral, local, warm, flat and feature (non-scale invariant) - which should be distinguishable given a significant detection of CMB non-Gaussianity. We demonstrate that a simple shape correlator provides a fast and reliable method for determining whether or not CMB shapes are well correlated. We use an eigenmode decomposition of the primordial shape to characterise and understand model independence. Finally, we advocate a standardised normalisation method for $f_{NL}$ based on the shape autocorrelator, so that observational limits and errors can be consistently compared for different models.

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arXiv2007

Primordial non-Gaussianity and the CMB bispectrum

We present a new formalism, together with efficient numerical methods, to directly calculate the CMB bispectrum today from a given primordial bispectrum using the full linear radiation transfer functions. Unlike previous analyses which have assumed simple separable ansatze for the bispectrum, this work applies to a primordial bispectrum of almost arbitrary functional form, for which there may have been both horizon-crossing and superhorizon contributions. We employ adaptive methods on a hierarchical triangular grid and we establish their accuracy by direct comparison with an exact analytic solution, valid on large angular scales. We demonstrate that we can calculate the full CMB bispectrum to greater than 1% precision out to multipoles l<1800 on reasonable computational timescales. We plot the bispectrum for both the superhorizon ('local') and horizon-crossing ('equilateral') asymptotic limits, illustrating its oscillatory nature which is analogous to the CMB power spectrum.

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