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We apply a method recently introduced to the statistical literature to directly estimate the precision matrix from an ensemble of samples drawn from a corresponding Gaussian distribution. Motivated by the observation that cosmological precision matrices are often approximately sparse, the method allows one to exploit this sparsity of the precision matrix to more quickly converge to an asymptotic rate while simultaneously providing an error model for all of the terms. Such an estimate can be used as the starting point for further regularization efforts which can improve upon the limit above, and incorporating such additional steps is straightforward within this framework. We demonstrate the technique with toy models and with an example motivated by large-scale structure two-point analysis, showing significant improvements in the rate of convergence. For the large-scale structure example, we find errors on the precision matrix which are factors of 5 smaller than for the sample precision matrix for thousands of simulations or, alternatively, convergence to the same error level with more than an order of magnitude fewer simulations.

We study ∼330 massive (M* > 109.5 M⊙), newborn spheroidal galaxies (SGs) around the epoch of peak star formation (1 < z < 3) to explore the high-redshift origin of SGs and gain insight into when and how the old stellar populations that dominate today's Universe formed. The sample is drawn from the Hubble Space Telescope (HST)/WFC3 Early-Release Science programme, which provides deep 10-filter (0.2–1.7 μm) HST imaging over one-third of the GOODS-South field. We find that the star formation episodes that built our SGs likely peaked in the redshift range 2 < z < 5 (with a median of z ∼ 3) and have decay time-scales shorter than ∼1.5 Gyr. Starburst time-scales and ages show no trend with stellar mass in the range 109.5 < M* < 1010.5 M⊙. However, the time-scales show increased scatter towards lower values (<0.3 Gyr) for M* > 1010.5 M⊙, and an age trend becomes evident in this mass regime: SGs with M* > 1011.5 M⊙ are ∼2 Gyr older than their counterparts with M* < 1010.5 M⊙. Nevertheless, a smooth downsizing trend with galaxy mass is not observed, and the large scatter in starburst ages indicates that SGs are not a particularly coeval population. Around half of the blue SGs appear not to drive their star formation via major mergers, and those that have experienced a recent major merger show only modest enhancements (∼40 per cent) in their specific star formation rates. Our empirical study indicates that processes other than major mergers (e.g. violent disc instability driven by cold streams and/or minor mergers) likely play a dominant role in building SGs, and creating a significant fraction of the old stellar populations that dominate today's Universe. [ABSTRACT FROM PUBLISHER]