The Power Spectrum, Bias Evolution, and the Spatial Three‐Point Correlation Function

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Abstract
We calculate perturbatively the normalized spatial skewness, S3, and full three-point correlation function (3PCF), ζ, induced by gravitational instability of Gaussian primordial fluctuations for a biased tracer-mass distribution in flat and open cold dark matter (CDM) models. We take into account the explicit dependence on cosmological parameters, the shape and evolution of the CDM power spectrum, and we allow the bias to be nonlinear and/or evolving in time, using an extension of Fry's bias evolution model. We derive a scale-dependent, leading-order correction to the standard perturbative expression for S3 in the case of nonlinear biasing, as defined for the unsmoothed galaxy and dark-matter fields, and find that this correction becomes large when probing positive effective power-spectrum indices, i.e., scales above 100 h-1 Mpc for reasonable CDM models. This term implies that the inferred nonlinear-bias parameter, as usually defined in terms of the smoothed density fields, might in general depend on the chosen smoothing scale and could allow better constraints on both the linear- and nonlinear-bias parameters on the basis of skewness measurements alone (or at least distinguish between the smoothed and unsmoothed bias pictures), if S3 could be measured over very large scales. In general, we find that the dependence of S3 on the biasing scheme can substantially outweigh that on the adopted cosmology, with linear and nonlinear bias separately giving rise to distinct signatures in the skewness, but degenerate ones in combination. We demonstrate that the normalized 3PCF, Q, is an ill-behaved quantity, and speculate that reported discrepancies between perturbative and N-body predictions for Q may arise in part from systematic errors associated with the poor choice of normalization. To avoid this problem we investigate QV, the variance-normalized 3PCF. The configuration dependence of QV shows similarly strong sensitivities to the bias scheme as S3, but also exhibits significant dependence on the form of the CDM power spectrum. Though the degeneracy of S3 with respect to the cosmological parameters and constant linear- and nonlinear-bias parameters can be broken by the full configuration dependence of QV, neither statistic can distinguish well between evolving and nonevolving bias scenarios, since an evolving bias is found to effectively mimic a smaller but constant bias. We show that this can be resolved, in principle, by considering the redshift dependence of ζ, which can also yield direct constraints on Ω0 and the epoch of galaxy formation.
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