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Register a new userCreating Jackknife and Bootstrap estimates of the covariance matrix for the two-point correlation function
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Added by
Faizan Gohar Mohammad
Publication date
2021
fields
Physics
and research's language is
English
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We present correction terms that allow delete-one Jackknife and Bootstrap methods to be used to recover unbiased estimates of the data covariance matrix of the two-point correlation function $xileft(mathbf{r}right)$. We demonstrate the accuracy and precision of this new method using a large set of 1000 QUIJOTE simulations that each cover a comoving volume of $1rm{left[h^{-1}Gpcright]^3}$. The corrected resampling techniques accurately recover the correct amplitude and structure of the data covariance matrix as represented by its principal components. Our corrections for the internal resampling methods are shown to be robust against the intrinsic clustering of the cosmological tracers both in real- and redshift space using two snapshots at $z=0$ and $z=1$ that mimic two samples with significantly different clustering. We also analyse two different slicing of the simulation volume into $n_{rm sv}=64$ or $125$ sub-samples and show that the main impact of different $n_{rm sv}$ is on the structure of the covariance matrix due to the limited number of independent internal realisations that can be made given a fixed $n_{rm sv}$.
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We present a fast and robust alternative method to compute covariance matrix in case of cosmology studies. Our method is based on the jackknife resampling applied on simulation mock catalogues. Using a set of 600 BOSS DR11 mock catalogues as a reference, we find that the jackknife technique gives a similar galaxy clustering covariance matrix estimate by requiring a smaller number of mocks. A comparison of convergence rates show that $sim$7 times fewer simulations are needed to get a similar accuracy on variance. We expect this technique to be applied in any analysis where the number of available N-body simulations is low.
All estimators of the two-point correlation function are based on a random catalogue, a set of points with no intrinsic clustering following the selection function of a survey. High-accuracy estimates require the use of large random catalogues, which imply a high computational cost. We propose to replace the standard random catalogues by glass-like point distributions or glass catalogues, which are characterized by a power spectrum $P(k)propto k^4$ and exhibit significantly less power than a Poisson distribution with the same number of points on scales larger than the mean inter-particle separation. We show that these distributions can be obtained by iteratively applying the technique of Zeldovich reconstruction commonly used in studies of baryon acoustic oscillations (BAO). We provide a modified version of the widely used Landy-Szalay estimator of the correlation function adapted to the use of glass catalogues and compare its performance with the results obtained using random samples. Our results show that glass-like samples do not add any bias with respect to the results obtained using Poisson distributions. On scales larger than the mean inter-particle separation of the glass catalogues, the modified estimator leads to a significant reduction of the variance of the Legendre multipoles $xi_ell(s)$ with respect to the standard Landy-Szalay results with the same number of points. The size of the glass catalogue required to achieve a given accuracy in the correlation function is significantly smaller than when using random samples. Even considering the small additional cost of constructing the glass catalogues, their use could help to drastically reduce the computational cost of configuration-space clustering analysis of future surveys while maintaining high-accuracy requirements.
We perform theoretical and numerical studies of the full relativistic two-point galaxy correlation function, considering the linear-order scalar and tensor perturbation contributions and the wide-angle effects. Using the gauge-invariant relativistic description of galaxy clustering and accounting for the contributions at the observer position, we demonstrate that the complete theoretical expression is devoid of any long-mode contributions from scalar or tensor perturbations and it lacks the infrared divergences in agreement with the equivalence principle. By showing that the gravitational potential contribution to the correlation function converges in the infrared, our study justifies an IR cut-off $(k_{text{IR}} leq H_0)$ in computing the gravitational potential contribution. Using the full gauge-invariant expression, we numerically compute the galaxy two-point correlation function and study the individual contributions in the conformal Newtonian gauge. We find that the terms at the observer position such as the coordinate lapses and the observer velocity (missing in the standard formalism) dominate over the other relativistic contributions in the conformal Newtonian gauge such as the source velocity, the gravitational potential, the integrated Sachs-Wolf effect, the Shapiro time-delay and the lensing convergence. Compared to the standard Newtonian theoretical predictions that consider only the density fluctuation and redshift-space distortions, the relativistic effects in galaxy clustering result in a few percent-level systematic errors beyond the scale of the baryonic acoustic oscillation. Our theoretical and numerical study provides a comprehensive understanding of the relativistic effects in the galaxy two-point correlation function, as it proves the validity of the theoretical prediction and accounts for effects that are often neglected in its numerical evaluation.
Context: The cross-covariance of solar oscillations observed at pairs of points on the solar surface is a fundamental ingredient in time-distance helioseismology. Wave travel times are extracted from the cross-covariance function and are used to infer the physical conditions in the solar interior. Aims: Understanding the statistics of the two-point cross-covariance function is a necessary step towards optimizing the measurement of travel times. Methods: By modeling stochastic solar oscillations, we evaluate the variance of the cross-covariance function as function of time-lag and distance between the two points. Results: We show that the variance of the cross-covariance is independent of both time-lag and distance in the far field, i.e., when they are large compared to the coherence scales of the solar oscillations. Conclusions: The constant noise level for the cross-covariance means that the signal-to-noise ratio for the cross-covariance is proportional to the amplitude of the expectation value of the cross-covariance. This observation is important for planning data analysis efforts.
The two-point correlation function of the galaxy distribution is a key cosmological observable that allows us to constrain the dynamical and geometrical state of our Universe. To measure the correlation function we need to know both the galaxy positions and the expected galaxy density field. The expected field is commonly specified using a Monte-Carlo sampling of the volume covered by the survey and, to minimize additional sampling errors, this random catalog has to be much larger than the data catalog. Correlation function estimators compare data-data pair counts to data-random and random-random pair counts, where random-random pairs usually dominate the computational cost. Future redshift surveys will deliver spectroscopic catalogs of tens of millions of galaxies. Given the large number of random objects required to guarantee sub-percent accuracy, it is of paramount importance to improve the efficiency of the algorithm without degrading its precision. We show both analytically and numerically that splitting the random catalog into a number of subcatalogs of the same size as the data catalog when calculating random-random pairs, and excluding pairs across different subcatalogs provides the optimal error at fixed computational cost. For a random catalog fifty times larger than the data catalog, this reduces the computation time by a factor of more than ten without affecting estimator variance or bias.
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