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Register a new userModeling Biased Tracers at the Field Level
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Added by
Marcel M. Schmittfull
Publication date
2018
fields
Physics
and research's language is
English
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In this paper we test the perturbative halo bias model at the field level. The advantage of this approach is that any analysis can be done without sample variance if the same initial conditions are used in simulations and perturbation theory calculations. We write the bias expansion in terms of modified bias operators in Eulerian space, designed such that the large bulk flows are automatically resummed and not treated perturbatively. Using these operators, the bias model accurately matches the Eulerian density of halos in N-body simulations. The mean-square model error is close to the Poisson shot noise for a wide range of halo masses and it is rather scale-independent, with scale-dependent corrections becoming relevant at the nonlinear scale. In contrast, for linear bias the mean-square model error can be higher than the Poisson prediction by factors of up to a few on large scales, and it becomes scale dependent already in the linear regime. We show that by weighting simulated halos by their mass, the mean-square error of the model can be further reduced by up to an order of magnitude, or by a factor of two when including $60%$ mass scatter. We also test the Standard Eulerian bias model using the nonlinear matter field measured from simulations and show that it leads to a larger and more scale-dependent model error than the bias expansion based on perturbation theory. These results may be of particular relevance for cosmological inference methods that use a likelihood of the biased tracer at the field level, or for initial condition and BAO reconstruction that requires a precise estimate of the large-scale potential from the biased tracer density.
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We investigate the application of Hybrid Effective Field Theory (HEFT) -- which combines a Lagrangian bias expansion with subsequent particle dynamics from $N$-body simulations -- to the modeling of $k$-Nearest Neighbor Cumulative Distribution Functions ($k{rm NN}$-${rm CDF}$s) of biased tracers of the cosmological matter field. The $k{rm NN}$-${rm CDF}$s are sensitive to all higher order connected $N$-point functions in the data, but are computationally cheap to compute. We develop the formalism to predict the $k{rm NN}$-${rm CDF}$s of discrete tracers of a continuous field from the statistics of the continuous field itself. Using this formalism, we demonstrate how $k{rm NN}$-${rm CDF}$ statistics of a set of biased tracers, such as halos or galaxies, of the cosmological matter field can be modeled given a set of low-redshift HEFT component fields and bias parameter values. These are the same ingredients needed to predict the two-point clustering. For a specific sample of halos, we show that both the two-point clustering textit{and} the $k{rm NN}$-${rm CDF}$s can be well-fit on quasi-linear scales ($gtrsim 20 h^{-1}{rm Mpc}$) by the second-order HEFT formalism with the textit{same values} of the bias parameters, implying that joint modeling of the two is possible. Finally, using a Fisher matrix analysis, we show that including $k{rm NN}$-${rm CDF}$ measurements over the range of allowed scales in the HEFT framework can improve the constraints on $sigma_8$ by roughly a factor of $3$, compared to the case where only two-point measurements are considered. Combining the statistical power of $k{rm NN}$ measurements with the modeling power of HEFT, therefore, represents an exciting prospect for extracting greater information from small-scale cosmological clustering.
It has been recognized that the observed galaxy distribution is susceptible to long-wavelength density and tidal fluctuations whose wavelengths exceed the accessible scale of a finite-volume observation, referred to as the super-sample modes. The super-sample modes modulate the growth and expansion rate of local structures, thus affecting the cosmological information encoded in the statistics of galaxy clustering data. In this paper, based on the Lagrangian perturbation theory, we develop a new formalism to systematically compute the response of a biased tracer of matter distribution to the super-sample modes at the field level. The formalism presented here reproduces the power spectrum responses that have been previously derived, and beyond the leading order, it also enables us to proceed to a higher-order calculation. As an application, we consider the statistics of the intrinsic alignments of galaxies and halos, and derive the field response of the galaxy/halo ellipticity to the super-sample modes. Possible impacts of the long-mode contributions on the covariance of the power spectra are also discussed, and the signal-to-noise ratios are estimated.
We develop an analytical forward model based on perturbation theory to predict the redshift-space galaxy overdensity at the field level given a realization of the initial conditions. We find that the residual noise between the model and simulated galaxy density has a power spectrum that is white on large scales, with size comparable to the shot noise. In the mildly nonlinear regime, we see a $k^2mu^2$ correction to the noise power spectrum, corresponding to larger noise along the line of sight and on smaller scales. The parametric form of this correction has been predicted on theoretical grounds before, and our simulations provide important confirmation of its presence. We have also modeled the galaxy velocity at the field-level and compared it against simulated galaxy velocities, finding that about $10%$ of the galaxies are responsible for half of the rms velocity residual for our simulated galaxy sample.
We present the one-loop perturbation theory for the power spectrum of the marked density field of matter and biased tracers in real- and redshift-space. The statistic has been shown to yield impressive constraints on cosmological parameters; to exploit this, we require an accurate and computationally inexpensive theoretical model. Comparison with $N$-body simulations demonstrates that linear theory fails on all scales, but inclusion of one-loop Effective Field Theory terms gives a substantial improvement, with $sim 5%$ accuracy at $z = 1$. The expansion is less convergent in redshift-space (achieving $sim 10%$ accuracy), but there are significant improvements for biased tracers due to the freedom in the bias coefficients. The large-scale theory contains non-negligible contributions from all perturbative orders; we suggest a reorganization of the theory that contains all terms relevant on large-scales, discussing both its explicit form at one-loop and structure at infinite-loop. This motivates a low-$k$ correction term, leading to a model that is sub-percent accurate on large scales, albeit with the inclusion of two (three) free coefficients in real- (redshift-)space. We further consider the effects of massive neutrinos, showing that beyond-EdS corrections to the perturbative kernels are negligible in practice. It remains to see whether the purported gains in cosmological parameters remain valid for biased tracers and can be captured by the theoretical model.
With the completion of the Planck mission, in order to continue to gather cosmological information it has become crucial to understand the Large Scale Structures (LSS) of the universe to percent accuracy. The Effective Field Theory of LSS (EFTofLSS) is a novel theoretical framework that aims to develop an analytic understanding of LSS at long distances, where inhomogeneities are small. We further develop the description of biased tracers in the EFTofLSS to account for the effect of baryonic physics and primordial non-Gaussianities, finding that new bias coefficients are required. Then, restricting to dark matter with Gaussian initial conditions, we describe the prediction of the EFTofLSS for the one-loop halo-halo and halo-matter two-point functions, and for the tree-level halo-halo-halo, matter-halo-halo and matter-matter-halo three-point functions. Several new bias coefficients are needed in the EFTofLSS, even though their contribution at a given order can be degenerate and the same parameters contribute to multiple observables. We develop a method to reduce the number of biases to an irreducible basis, and find that, at the order at which we work, seven bias parameters are enough to describe this extremely rich set of statistics. We then compare with the output of $N$-body simulations. For the lowest mass bin, we find percent level agreement up to $ksimeq 0.3,h,{rm Mpc}^{-1}$ for the one-loop two-point functions, and up to $ksimeq 0.15,h,{rm Mpc}^{-1}$ for the tree-level three-point functions, with the $k$-reach decreasing with higher mass bins. This is consistent with the theoretical estimates, and suggests that the cosmological information in LSS amenable to analytical control is much more than previously believed.
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