Fergusson, J. R. | Search NASA

DOE OSTI2016

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 scenarios and so the slower non-separable integral must be calculated instead. We present a discussion of the optimisation of both approaches. We demonstrate significant speed-ups of ≈100×, 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 1.3× and that running the same code across a combination of both microarchitectures improves performance-per-node by a factor of 3.38×. 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.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗

DOE OSTI2012

The CMB bispectrum

We use a separable mode expansion estimator with WMAP7 data to estimate the bispectrum for all the primary families of non-Gaussian models, including non-scaling feature (periodic) models, the flat (trans-Planckian) model, DBI and ghost inflation, as well as previously constrained simple cases. 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 7-year V and W channel maps, reconstructing the WMAP bispectrum using l < 500 multipoles and n = 50 orthonormal 3D eigenmodes; both the mode expansion parameters and the reconstructed 3D WMAP bispectrum are plotted. 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 f{sub NL} = 20.31±27.64 and the equilateral model f{sub NL} = 10.19±127.38 (F{sub NL} = 1.90±23.79) are consistent with previously published results. (Here, we use a nonlinearity parameter F{sub NL} 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 F{sub NL} = 7.82±24.57, the flat model F{sub NL} = 7.31±26.22, and warm inflation F{sub NL} = 2.10±25.83. We investigate feature models, which break scale invariance, surveying a wide parameter range for both the scale and phase (scanning for feature models with an effective period l* > 150). We find no significant evidence of non-Gaussianity for all cases well-described by the given eigenmodes. In the overall non-Gaussian analysis, we find one anomalous mode n = 33 with a 3.39σ amplitude which could give rise to an oscillatory model signal with l* ≤ 150. We propose a measure F-bar {sub NL} 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 test of Gaussianity to date and we discuss future prospects with higher precision and resolution, notably from the Planck satellite.

79 ASTRONOMY AND ASTROPHYSICS↗