arXiv subjects

Tobias Baldauf

Publications and source records attributed to Tobias Baldauf.

At least 19 recordsLinked to original sources

arXiv2022

The two-loop bispectrum in the effective theory of large-scale structure

We study the bispectrum of large-scale structure in the EFTofLSS including corrections up to two-loop. We derive an analytic result for the double-hard limit of the two-loop correction, and show that the UV-sensitivity can be absorbed by the same four EFT operators that renormalize the one-loop bispectrum. For the single-hard region, we employ a simplified treatment, introducing one extra EFT parameter. We compare our results to N-body simulations, and show that going from one- to two-loop extends the wavenumber range with percent-level agreement from $k \simeq 0.08$ to $0.15~h/\mathrm{Mpc}$.

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arXiv2021

The two-loop bispectrum of large-scale structure

The bispectrum is the leading non-Gaussian statistic in large-scale structure, carrying valuable information on cosmology that is complementary to the power spectrum. To access this information, we need to model the bispectrum in the weakly non-linear regime. In this work we present the first two-loop, i.e., next-to-next-to-leading order perturbative description of the bispectrum within an effective field theory (EFT) framework. Using an analytic expansion of the perturbative kernels up to $F_6$ we derive a renormalized bispectrum that is demonstrated to be independent of the UV cutoff. We show that the EFT parameters associated with the four independent second-order EFT operators known from the one-loop bispectrum are sufficient to absorb the UV sensitivity of the two-loop contributions in the double-hard region. In addition, we employ a simplified treatment of the single-hard region, introducing one extra EFT parameter at two-loop order. We compare our results to N-body simulations using the realization-based grid-PT method and find good agreement within the expected range, as well as consistent values for the EFT parameters. The two-loop terms start to become relevant at $k\approx 0.07h~\mathrm{Mpc}^{-1}$. The range of wavenumbers with percent-level agreement, independently of the shape, extends from $0.08h~\mathrm{Mpc}^{-1}$ to $0.15h~\mathrm{Mpc}^{-1}$ when going from one to two loops at $z=0$. In addition, we quantify the impact of using exact instead of Einstein-de-Sitter kernels for the one-loop bispectrum, and discuss in how far their impact can be absorbed into a shift of the EFT parameters.

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arXiv2021

Primordial non-Gaussianity without tails -- how to measure fNL with the bulk of the density PDF

We investigate the possibility to detect primordial non-Gaussianity by analysing the bulk of the probability distribution function (PDF) of late-time cosmic density fluctuations. For this purpose we devise a new method to predict the impact of general non-Gaussian initial conditions on the late-time density PDF. At redshift $z=1$ and for a smoothing scale of 30Mpc/$h$ our predictions agree with the high-resolution Quijote N-body simulations to $\sim 0.2\%$ precision. This is within cosmic variance of a $\sim 100(\mathrm{Gpc}/h)^3$ survey volume. When restricting to this 30Mpc/$h$ smoothing scale and to mildly non-linear densities ($δ[30\mathrm{Mpc}/h] \in [-0.3, 0.4]$) and also marginalizing over potential ignorance of the amplitude of the non-linear power spectrum an analysis of the PDF for such a survey volume can still measure the amplitude of different primordial bispectrum shapes to an accuracy of \smash{$Δf_{\mathrm{NL}}^{\mathrm{loc}}=\pm 7.4\ ,\ Δf_{\mathrm{NL}}^{\mathrm{equi}}=\pm 22.0\ ,\ Δf_{\mathrm{NL}}^{\mathrm{ortho}}=\pm 46.0$} . When pushing to smaller scales and assuming a joint analysis of the PDF with smoothing radii of 30Mpc/$h$ and 15Mpc/$h$ ($δ[15\mathrm{Mpc}/h] \in [-0.4, 0.5]$) this improves to \smash{$Δf_{\mathrm{NL}}^{\mathrm{loc}}=\pm 3.3\ ,\ Δf_{\mathrm{NL}}^{\mathrm{equi}}=\pm 11.0\ ,\ Δf_{\mathrm{NL}}^{\mathrm{ortho}}=\pm 17.0\ $} - even when marginalizing over the non-linear variances at both scales as two free parameters. Especially, such an analysis could simultaneously measure $f_{\mathrm{NL}}$ and the amplitude and slope of the non-linear power spectrum. However, at 15Mpc/$h$ our predictions are only accurate to $\lesssim 0.8\%$ for the considered density range. We discuss how this has to be improved in order to push to these small scales and make full use of upcoming surveys with a PDF-based analysis.

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arXiv2021

Precise Calibration of the One-Loop Trispectrum in the Effective Field Theory of Large Scale Structure

The Large-Scale Structure (LSS) of the universe has the potential to provide decisive answers to the remaining open questions in cosmology. Early attempts at modelling it analytically focused on using perturbation theory. However, small-scale effects introduced by gravitational collapse cannot be described perturbatively and this failure of perturbation theory is reflected even on the largest scales. The Effective Field Theory of Large Scale Structure (EFTofLSS) has emerged as a consistent method for describing LSS on large scales by introducing counterterms that account for the effects of small-scale dynamics. So far studies of the EFT have mostly focused on the two and three point functions with little attention devoted to the four point function or trispectrum. The trispectrum probes cubic interactions arising from non-linear clustering, biasing, and primordial non-Gaussianities, and constitutes a key element of the covariance matrix of the power spectrum. In this paper, we present explicit calibrations of the EFT counterterms for the one-loop trispectrum. Specifically, we find clear evidence for non-zero EFT corrections. We define two one-parameter ansätze for the counterterm of the one-loop propagator and show that they provide a good correction to the residual at scales below k~0.07 h/Mpc. We then take the amplitudes of the linear and quadratic counterkernels calculated in our previous paper on the bispectrum and use them in the remaining counterterms, establishing consistency of the counterterms in the two, three and four point function. We also show that the commonly used EdS approximation for the growth of the density fields leads to errors that are of the same magnitude as loop corrections to the trispectrum on large scales.

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arXiv2020

Non-perturbative halo clustering from cosmological density peaks

Associating the formation sites of haloes with the maxima of the smoothed linear density field, we present non-perturbative predictions for the Lagrangian and evolved halo correlation functions that are valid at all separations. In Lagrangian space, we find significant deviations from the perturbative bias calculation at small scales, in particular, a pronounced exclusion region where $ξ=-1$ for maxima of unequal height. Our predictions are in good agreement with the Lagrangian clustering of dark matter proto-haloes reconstructed from N-body simulations. Our predictions for the mean infall and velocity dispersion of haloes, which differ from the local bias expansion, show a similar level of agreement with simulations. Finally, we displace the initial density peaks according to the Zeldovich approximation in order to predict the late-time clustering of dark matter haloes. While we are able to reproduce the early evolution of this conserved set of tracers, our approximation fails at the collapse epoch (z=0) on non-linear scales r<10Mpc/h, emphasizing the need for a non-perturbative treatment of the halo displacement field.

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arXiv2020

Iterative initial condition reconstruction

Motivated by recent developments in perturbative calculations of the nonlinear evolution of large-scale structure, we present an iterative algorithm to reconstruct the initial conditions in a given volume starting from the dark matter distribution in real space. In our algorithm, objects are first moved back iteratively along estimated potential gradients, with a progressively reduced smoothing scale, until a nearly uniform catalog is obtained. The linear initial density is then estimated as the divergence of the cumulative displacement, with an optional second-order correction. This algorithm should undo nonlinear effects up to one-loop order, including the higher-order infrared resummation piece. We test the method using dark matter simulations in real space. At redshift $z=0$, we find that after eight iterations the reconstructed density is more than $95\%$ correlated with the initial density at $k\le 0.35\; h\mathrm{Mpc}^{-1}$. The reconstruction also reduces the power in the difference between reconstructed and initial fields by more than 2 orders of magnitude at $k\le 0.2\; h\mathrm{Mpc}^{-1}$, and it extends the range of scales where the full broadband shape of the power spectrum matches linear theory by a factor of 2-3. As a specific application, we consider measurements of the baryonic acoustic oscillation (BAO) scale that can be improved by reducing the degradation effects of large-scale flows. In our idealized dark matter simulations, the method improves the BAO signal-to-noise ratio by a factor of 2.7 at $z=0$ and by a factor of 2.5 at $z=0.6$, improving standard BAO reconstruction by $70\%$ at $z=0$ and $30\%$ at $z=0.6$, and matching the optimal BAO signal and signal-to-noise ratio of the linear density in the same volume. For BAO, the iterative nature of the reconstruction is the most important aspect.

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arXiv2020

Precise Calibration of the One-Loop Bispectrum in the Effective Field Theory of Large Scale Structure

The bispectrum is the leading non-Gaussian statistic in Large-Scale Structure (LSS) clustering and encodes the interactions in the underlying field. It is thus an important diagnostic for primordial non-Gaussianity and higher order galaxy biasing. In this paper we present a detailed test and calibration of the matter bispectrum counterterms in the Effective Field Theory of LSS against a suite of $N$-body simulations. We are going beyond previous studies in employing realisation based perturbation theory that allows for a significant reduction in cosmic variance error bars. This enables the measurement of the low-energy constants on large scales before two-loop corrections become relevant, around $k<0.09 h\mathrm{Mpc}^{-1}$ at $z=0$. We also go beyond previous work in using bispectrum propagator terms, i.e. correlators with linear and second order fields, to quantify the two new counterterms in isolation and to establish consistency with the power spectrum counterterm. By investigating the fully non-linear bispectrum, $B_{\mathrm{nnn}}$, as well as the terms $B_{\mathrm{n}11}$ and $B_{\mathrm{n}21}$, we find evidence for the new counterterms deviating from the shape suggested by the UV-limit of the relevant bispectrum contributions. We also show that the commonly used Einstein-de Sitter approximation for the time dependence of the tree-level bispectrum is insufficient for precise studies of the one-loop bispectrum and that it is necessary to use $Λ$CDM growth factors in order to obtain meaningful one-loop counterterm constraints. Finally, we also find evidence for small deviations in the growth factors that arise from time integration inaccuracies in the $N$-body simulations.

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arXiv2020

Projected two- and three-point statistics: Forecasts and mitigation of non-linear RSDs

The combination of two- and three-point clustering statistics of galaxies and the underlying matter distribution has the potential to break degeneracies between cosmological parameters and nuisance parameters and can lead to significantly tighter constraints on parameters describing the composition of the Universe and the dynamics of inflation. Here we investigate the relation between biases in the estimated parameters and inaccurate modelling of non-linear redshift-space distortions for the power spectrum and bispectrum of projected galaxy density fields and lensing convergence. Non-linear redshift-space distortions are one of the leading systematic uncertainties in galaxy clustering. Projections along the line of sight suppress radial modes and are thus allowing a trade-off between biases due to non-linear redshift-space distortions and statistical uncertainties. We investigate this bias-error trade-off for a CMASS-like survey with a varying number of redshift bins. Improved modelling of the non-linear redshift-space distortions allows the recovery of more radial information when controlling for biases. Not modelling non-linear redshift space distortions inflates error bars for almost all parameters by 20%. The information loss for the amplitude of local non-Gaussianities is smaller, since it is best constrained from large scales. In addition, we show empirically that one can recover more than 99% of the 3D power spectrum information if the depth of the tomographic bins is reduced to 10 $h^{-1}$Mpc.

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arXiv2020

Density reconstruction from biased tracers and its application to primordial non-Gaussianity

Large-scale Fourier modes of the cosmic density field are of great value for learning about cosmology because of their well-understood relationship to fluctuations in the early universe. However, cosmic variance generally limits the statistical precision that can be achieved when constraining model parameters using these modes as measured in galaxy surveys, and moreover, these modes are sometimes inaccessible due to observational systematics or foregrounds. For some applications, both limitations can be circumvented by reconstructing large-scale modes using the correlations they induce between smaller-scale modes of an observed tracer (such as galaxy positions). In this paper, we further develop a formalism for this reconstruction, using a quadratic estimator similar to the one used for lensing of the cosmic microwave background. We incorporate nonlinearities from gravity, nonlinear biasing, and local-type primordial non-Gaussianity, and verify that the estimator gives the expected results when applied to N-body simulations. We then carry out forecasts for several upcoming surveys, demonstrating that, when reconstructed modes are included alongside directly-observed tracer density modes, constraints on local primordial non-Gaussianity are generically tightened by tens of percents compared to standard single-tracer analyses. In certain cases, these improvements arise from cosmic variance cancellation, with reconstructed modes taking the place of modes of a separate tracer, thus enabling an effective "multitracer" approach with single-tracer observations.

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arXiv2019

Primordial Non-Gaussianity

Our current understanding of the Universe is established through the pristine measurements of structure in the cosmic microwave background (CMB) and the distribution and shapes of galaxies tracing the large scale structure (LSS) of the Universe. One key ingredient that underlies cosmological observables is that the field that sources the observed structure is assumed to be initially Gaussian with high precision. Nevertheless, a minimal deviation from Gaussianityis perhaps the most robust theoretical prediction of models that explain the observed Universe; itis necessarily present even in the simplest scenarios. In addition, most inflationary models produce far higher levels of non-Gaussianity. Since non-Gaussianity directly probes the dynamics in the early Universe, a detection would present a monumental discovery in cosmology, providing clues about physics at energy scales as high as the GUT scale.

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arXiv2018

Cubic Halo Bias in Eulerian and Lagrangian Space

Predictions of the next-to-leading order, i.e. one-loop, halo power spectra depend on local and non-local bias parameters up to cubic order. The linear bias parameter can be estimated from the large scale limit of the halo-matter power spectrum, and the second order bias parameters from the large scale, tree-level, bispectrum. Cubic operators would naturally be quantified using the tree-level trispectrum. As the latter is computationally expensive, we extent the quadratic field method proposed in Schmittfull et al. 2014 to cubic fields in order to estimate cubic bias parameters. We cross-correlate a basis set of cubic bias operators with the halo field and express the result in terms of the cross-spectra of these operators in order to cancel cosmic variance. We obtain significant detections of local and non-local cubic bias parameters, which are partially in tension with predictions based on local Lagrangian bias schemes. We directly measure the Lagrangian bias parameters of the protohaloes associated with our halo sample and clearly detect a non-local quadratic term in Lagrangian space. We do not find a clear detection of non-local cubic Lagrangian terms for low mass bins, but there is some mild evidence for their presence for the highest mass bin. While the method presented here focuses on cubic bias parameters, the approach could also be applied to quantifications of cubic primordial non-Gaussianity.

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arXiv2017

Cosmological Perturbation Theory Using the FFTLog: Formalism and Connection to QFT Loop Integrals

We present a new method for calculating loops in cosmological perturbation theory. This method is based on approximating a $Λ$CDM-like cosmology as a finite sum of complex power-law universes. The decomposition is naturally achieved using an FFTLog algorithm. For power-law cosmologies, all loop integrals are formally equivalent to loop integrals of massless quantum field theory. These integrals have analytic solutions in terms of generalized hypergeometric functions. We provide explicit formulae for the one-loop and the two-loop power spectrum and the one-loop bispectrum. A chief advantage of our approach is that the difficult part of the calculation is cosmology independent, need be done only once, and can be recycled for any relevant predictions. Evaluation of standard loop diagrams then boils down to a simple matrix multiplication. We demonstrate the promise of this method for applications to higher multiplicity/loop correlation functions.

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arXiv2017

Verifying the consistency relation for the scale-dependent bias from local primordial non-Gaussianity

We measure the large-scale bias of dark matter halos in simulations with non-Gaussian initial conditions of the local type, and compare this bias to the response of the mass function to a change in the primordial amplitude of fluctuations. The two are found to be consistent, as expected from physical arguments, for three halo-finder algorithms which use different Spherical Overdensity (SO) and Friends-of-Friends (FoF) methods. On the other hand, we find that the commonly used prediction for universal mass functions, that the scale-dependent bias is proportional to the first-order Gaussian Lagrangian bias, does not yield a good agreement with the measurements. For all halo finders, high-mass halos show a non-Gaussian bias suppressed by 10-15% relative to the universal mass function prediction. For SO halos, this deviation changes sign at low masses, where the non-Gaussian bias becomes larger than the universal prediction.

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arXiv2016

Phenomenology of BAO evolution from Lagrangian to Eulerian Space

The baryon acoustic oscillation (BAO) feature provides an important distance scale for the measurement of the expansion history of the Universe. Theoretical models of the BAO in the distribution of biased tracers of the large scale structure usually rely on an initially linear BAO. With aid of N-body simulations, we demonstrate that the BAO in the initial (Lagrangian) halo 2-point function is significantly sharper than in the linear matter distribution, in agreement with peak theory. Using this approach, we delineate the scale-dependence induced by the higher-derivative and velocity bias before assessing how much of the initial BAO enhancement survives until the collapse epoch. Finally, we discuss the extent to which the velocity or gravity bias, which is also imprinted in the displacement field of halos, affects the contrast of the BAO obtained with a reconstruction.

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arXiv2016

Precision measurement of the local bias of dark matter halos

We present accurate measurements of the linear, quadratic, and cubic local bias of dark matter halos, using curved "separate universe" N-body simulations which effectively incorporate an infinite-wavelength overdensity. This can be seen as an exact implementation of the peak-background split argument. We compare the results with the linear and quadratic bias measured from the halo-matter power spectrum and bispectrum, and find good agreement. On the other hand, the standard peak-background split applied to the Sheth & Tormen (1999) and Tinker et al. (2008) halo mass functions matches the measured linear bias parameter only at the level of 10%. The prediction from the excursion set-peaks approach performs much better, which can be attributed to the stochastic moving barrier employed in the excursion set-peaks prediction. We also provide convenient fitting formulas for the nonlinear bias parameters $b_2(b_1)$ and $b_3(b_1)$, which work well over a range of redshifts.

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arXiv2016

The Effective Field Theory of Large Scale Structure at Two Loops: the apparent scale dependence of the speed of sound

We study the Effective Field Theory of Large Scale Structure for cosmic density and momentum fields. We show that the finite part of the two-loop calculation and its counterterms introduce an apparent scale dependence for the leading order parameter $c_\text{s}^2$ of the EFT starting at k=0.1 h/Mpc. These terms limit the range over which one can trust the one-loop EFT calculation at the 1 % level to k<0.1 h/Mpc at redshift z=0. We construct a well motivated one parameter ansatz to fix the relative size of the one- and two-loop counterterms using their high-k sensitivity. Although this one parameter model is a very restrictive choice for the counterterms, it explains the apparent scale dependence of $c_\text{s}^2$ seen in simulations. It is also able to capture the scale dependence of the density power spectrum up to k$\approx$ 0.3 h/Mpc at the 1 % level at redshift $z=0$. Considering a simple scheme for the resummation of large scale motions, we find that the two loop calculation reduces the need for this IR-resummation at k<0.2 h/Mpc. Finally, we extend our calculation to momentum statistics and show that the same one parameter model can also describe density-momentum and momentum-momentum statistics.

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arXiv2016

LSS constraints with controlled theoretical uncertainties

Forecasts and analyses of cosmological observations often rely on the assumption of a perfect theoretical model over a defined range of scales. We explore how model uncertainties and nuisance parameters in perturbative models of the matter and galaxy spectra affect constraints on neutrino mass and primordial non-Gaussianities. We provide a consistent treatment of theoretical errors and argue that their inclusion is a necessary step to obtain realistic cosmological constraints. We find that galaxy surveys up to high redshifts will allow a detection of the minimal neutrino mass and local non-Gaussianity of order unity, but improving the constraints on equilateral non-Gaussianity beyond the CMB limits will be challenging. We argue that similar considerations apply to analyses where theoretical models are based on simulations.

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arXiv2016

Equivalence Principle and the Baryon Acoustic Peak

We study the dominant effect of a long wavelength density perturbation $δ(λ_L)$ on short distance physics. In the non-relativistic limit, the result is a uniform acceleration, fixed by the equivalence principle, and typically has no effect on statistical averages due to translational invariance. This same reasoning has been formalized to obtain a "consistency condition" on the cosmological correlation functions. In the presence of a feature, such as the acoustic peak at $l_{\rm BAO}$, this naive expectation breaks down for $λ_L<l_{\rm BAO}$. We calculate a universal piece of the three-point correlation function in this regime. The same effect is shown to underlie the spread of the acoustic peak, and is calculable to all orders in the long modes. This can be used to improve the result of perturbative calculations - a technique known as "infra-red resummation" - and is explicitly applied to the one-loop calculation of power spectrum. Finally, the success of BAO reconstruction schemes is argued to be another empirical evidence for the validity of the results.

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