We present a framework for self-consistent cosmological analyses of the full-shape anisotropic bispectrum, including the quadrupole $(\ell=2)$ and hexadecapole $(\ell=4)$ moments. This features a novel window-free algorithm for extracting the latter quantities from data, derived using a maximum-likelihood prescription. Furthermore, we introduce a theoretical model for the bispectrum multipoles (which does not introduce new free parameters), and test both aspects of the pipeline on several high-fidelity mocks, including the PT Challenge suite of gigantic cumulative volume. This establishes that the systematic error is significantly below the statistical threshold, both for the measurement and modeling. As a realistic example, we extract the large-scale bispectrum multipoles from BOSS DR12 and analyze them in combination with the power spectrum data. Assuming a minimal $Λ$CDM model, with a BBN prior on the baryon density and a \textit{Planck} prior on $n_s$, we can extract the remaining cosmological parameters directly from the clustering data. The inclusion of the unwindowed higher-order $(\ell>0)$ large-scale bispectrum multipoles is found to moderately improve one-dimensional cosmological parameter posteriors (at the $5\%-10\%$ level), though these multipoles are detected only in three out of four BOSS data segments at $\approx 5σ$. Combining information from the power spectrum and bispectrum multipoles, the real space power spectrum, and the post-reconstructed BAO data, we find $H_0 = 68.2\pm 0.8~\mathrm{km}\,\mathrm{s}^{-1}\mathrm{Mpc}^{-1}$, $Ω_m =0.33\pm 0.01$ and $σ_8 = 0.736\pm 0.033$ (the tightest yet found in perturbative full-shape analyses). Our estimate of the growth parameter $S_8=0.77\pm 0.04$ agrees with both weak lensing and CMB results.
We present the first high significance detection ($4.1σ$) of the Baryon Acoustic Oscillations (BAO) feature in the galaxy bispectrum of the twelfth data release (DR12) of the Baryon Oscillation Spectroscopic Survey (BOSS) CMASS sample ($0.43 \leq z \leq 0.7$). We measured the scale dilation parameter, $α$, using the power spectrum, bispectrum, and both simultaneously for DR12, plus 2048 MultiDark-PATCHY mocks in the North and South Galactic Caps (NGC and SGC, respectively), and the volume weighted averages of those two samples (N+SGC). The fitting to the mocks validated our analysis pipeline, …
We present a novel method to compress galaxy clustering three-point statistics and apply it to redshift space galaxy bispectrum monopole measurements from BOSS DR12 CMASS data considering a $k$-space range of $0.03-0.12\,h/\mathrm{Mpc}$. The method consists in binning together bispectra evaluated at sets of wave-numbers forming closed triangles with similar geometrical properties: the area, the cosine of the largest angle and the ratio between the cosines of the remaining two angles. This enables us to increase the number of bispectrum measurements for example by a factor of $23$ over the stan…
Above the equality scale the galaxy bispectrum will be a key probe for measuring primordial non-Gaussianity which can help differentiate between different inflationary models and other theories of the early universe. On these scales a variety of relativistic effects come into play once the galaxy number-count fluctuation is projected onto our past lightcone. By decomposing the Fourier-space bispectrum into invariant multipoles about the observer's line of sight we examine in detail how the relativistic effects contribute to these. We show how to perform this decomposition analytically, which i…
The redshift space anisotropy of the bispectrum is generally quantified using multipole moments. The possibility of measuring these multipoles in any survey depends on the level of statistical fluctuations. We present a formalism to compute the statistical fluctuations in the measurement of bispectrum multipoles for galaxy surveys. We consider specifications of a {\it Euclid} like galaxy survey and present two quantities: the signal-to-noise ratio (SNR) which quantifies the detectability of a multipole, and the rank correlation which quantifies the correlation in measurement errors between any…
Recently, the power spectrum (PS) multipoles using the Baryon Oscillation Spectroscopic Survey (BOSS) Data Release 12 (DR12) sample are analyzed \cite{160703150}. The based model for the analysis is the so-called TNS quasi-linear model and the analysis provides the multipoles up to the hexadecapole \cite{TNS}. Thus, one might be able to recover the real-space linear matter PS by using the combinations of multipoles to investigate the cosmology \cite{0407214}. We provide the analytic form of the ratio of quadrupole (hexadecapole) to monopole moments of the quasi-linear PS including the Fingers-…
We apply two compression methods to the galaxy power spectrum monopole/quadrupole and bispectrum monopole measurements from the BOSS DR12 CMASS sample. Both methods reduce the dimension of the original data-vector to the number of cosmological parameters considered, using the Karhunen-Loève algorithm with an analytic covariance model. In the first case, we infer the posterior through MCMC sampling from the likelihood of the compressed data-vector (MC-KL). The second, faster option, works by first Gaussianising and then orthogonalising the parameter space before the compression; in this option …
As a generalization of our previous work [Phys. Rev. D 95 043528 (2017)], in which an analytic model for the galaxy bispectrum in redshift space was developed on the basis of the halo approach, we here investigate its higher multipoles that have not been known so far. The redshift-space bispectrum includes the two variables $ω$ and $ϕ$ for the line-of-sight direction, and the higher multipole bispectra are defined by the coefficients in the expansion of the redshift-space bispectrum using the spherical harmonics. We find 6 new nonvanishing components out of $25$ total components up to $\ell=4$…
Redshift-space 21-cm bispectrum multipoles as an SKA-era gravity test in the post-reionization Universe
Indian Statistical Institute（印度统计研究所）；McGill University（麦吉尔大学）；Ciela—Montreal Institute for Astrophysical Data Analysis and Machine Learning（Ciela-蒙特利尔天体物理数据分析与机器学习研究所）
We present an analytic formula for the galaxy bispectrum in redshift space on the basis of the halo approach description with the halo occupation distribution of central galaxies and satellite galaxies. This work is an extension of a previous work on the galaxy power spectrum, which illuminated the significant contribution of satellite galaxies to the higher multipole spectrum through the non-linear redshift space distortions of their random motions. Behaviors of the multipoles of the bispectrum are compared with results of numerical simulations assuming a halo occupation distribution of the L…
We present a full $Λ$CDM analysis of the BOSS DR12 dataset, including information from the power spectrum multipoles, the real-space power spectrum, the reconstructed power spectrum and the bispectrum monopole. This is the first analysis to feature a complete treatment of the galaxy bispectrum, including a consistent theoretical model and without large-scale cuts. Unlike previous works, the statistics are measured using window-free estimators: this greatly reduces computational costs by removing the need to window-convolve the theory model. Our pipeline is tested using a suite of high-resoluti…
We develop a possibility of generating tensor non-Gaussianity in a kind of anisotropic inflation, where a $U(1)$ gauge field is kinetically coupled to a spectator scalar field. Owing to this coupling, the coherent mode of the electric field appears and softly breaks the isotropy of the Universe. We compute the bispectrum of linearly-polarized tensor perturbations sourced by the gauge field and find that it is strongly red-tilted and has distinctive statistical anisotropies including higher-order multipole moments. Interestingly, the tensor bispectra with the specific combinations of linear pol…
The redshift-space bispectrum (three point statistics) of galaxies depends on the expansion rate, the growth rate, and geometry of the Universe, and hence can be used to measure key cosmological parameters. In a homogeneous Universe the bispectrum is a function of five variables and unlike its two point statistics counterpart -- the power spectrum, which is a function of only two variables -- is difficult to analyse unless the information is somehow reduced. The most commonly considered reduction schemes rely on computing angular integrals over possible orientations of the bispectrum triangle,…
We measure and analyse the bispectrum of the final, Data Release 12, galaxy sample provided by the Baryon Oscillation Spectroscopic Survey, splitting by selection algorithm into LOWZ and CMASS galaxies. The LOWZ sample contains 361\,762 galaxies with an effective redshift of $z_{\rm LOWZ}=0.32$, and the CMASS sample 777\,202 galaxies with an effective redshift of $z_{\rm CMASS}=0.57$. Combining the power spectrum, measured relative to the line-of-sight, with the spherically averaged bispectrum, we are able to constrain the product of the growth of structure parameter, $f$, and the amplitude of…
One of the cornerstones of general relativity is the equivalence principle. However, the validity of the equivalence principle has only been established on solar system scales for standard matter fields; this result cannot be assumed to hold for the non-standard matter fields that dominate the gravitational dynamics on cosmological scales. Here we show how the equivalence principle may be tested on cosmological scales for non-standard matter fields using the odd multipoles of the galaxy cross-power spectrum and bispectrum. This test makes use of the imprint on the galaxy cross-power spectrum a…
The anisotropy of the redshift space bispectrum contains a wealth of cosmological information. This anisotropy depends on the orientation of three vectors ${\bf k_1,k_2,k_3}$ with respect to the line of sight. Here we have decomposed the redshift space bispectrum in spherical harmonics which completely quantify this anisotropy. To illustrate this we consider linear redshift space distortion of the bispectrum arising from primordial non-Gaussianity. In the plane parallel approximation only the first four even $\ell$ multipoles have non-zero values, and we present explicit analytical expressions…
We analyse the anisotropic clustering of the Baryon Oscillation Spectroscopic Survey (BOSS) CMASS Data Release 11 sample, which consists of $690 827$ galaxies in the redshift range $0.43 < z < 0.70$ and has a sky coverage of $8 498$ deg$^2$ corresponding to an effective volume of $\sim6\,\rm{Gpc}^3$. We fit the Fourier space statistics, the power spectrum and bispectrum monopoles to measure the linear and quadratic bias parameters, $b_1$ and $b_2$, for a non-linear non-local bias model, the growth of structure parameter $f$ and the amplitude of dark matter density fluctuations parametrised by …
We make use of recent developments in the analysis of galaxy redshift surveys to present an easy to use matrix-based analysis framework for the galaxy power spectrum multipoles, including wide-angle effects and the survey window function. We employ this framework to derive the deconvolved power spectrum multipoles of 6dFGS DR3, BOSS DR12 and the eBOSS DR16 quasar sample. As an alternative to the standard analysis, the deconvolved power spectrum multipoles can be used to perform a data analysis agnostic of survey specific aspects, like the window function. We show that in the case of the BOSS d…
We examine the cosmological implications of the measurements of the linear growth rate of cosmological structure obtained in a companion paper from the power spectrum and bispectrum monopoles of the Sloan Digital Sky Survey III Baryon Oscillation Spectroscopic Survey Data, Release 11, CMASS galaxies. This measurement was of $f^{0.43}σ_8$, where $σ_8$ is the amplitude of dark matter density fluctuations, and $f$ is the linear growth rate, at the effective redshift of the survey, $z_{\rm eff}=0.57$. In conjunction with Cosmic Microwave Background (CMB) data, interesting constraints can be placed…
The galaxy bispectrum, measuring excess clustering of galaxy triplets, offers a probe of dark energy via baryon acoustic oscillations (BAOs). However up to now it has been severely underused due to the combinatorically explosive number of triangles. Here we exploit interference in the bispectrum to identify triangles that amplify BAOs. This approach reduces the computational cost of estimating covariance matrices, offers an improvement in BAO constraints equivalent to lengthening BOSS by 30%, and simplifies adding bispectrum BAO information to future large-scale redshift survey analyses.
We present the analysis of the halo bispectrum in redshift-space in terms of its multipoles, monopole, quadrupole and hexadecapole, measured from a large set of simulations. We fit such measurements with a tree-level model in perturbation theory that depends on linear and nonlinear bias parameters as well as on the growth rate $f$ of density fluctuations. The likelihood analysis takes advantage of a very large set of mock catalogs, enabling a robust estimation of the covariance properties for all multipoles. We compare the numerical estimate of the covariance matrix to its Gaussian prediction …
Interacting dark energy models have been suggested as alternatives to the standard cosmological model, $Λ$CDM. We focus on a phenomenologically interesting class of dark scattering models that is characterised by pure momentum exchange between dark energy and dark matter. This model extends the parameter space with respect to $Λ$CDM by two parameters, $w$ and $A$, which define the dark energy equation of state and the strength of the coupling between dark energy and dark matter, respectively. In order to test non-standard cosmologies with Stage-IV galaxy clustering surveys, it is crucial to mo…
We forecast the benefits induced by adding the bispectrum anisotropic signal to the standard, two- and three-point, clustering statistics analysis. In particular, we forecast cosmological parameter constraints including the bispectrum higher multipoles terms together with the galaxy power spectrum (monopole plus quadrupole) and isotropic bispectrum (monopole) data vectors. To do so, an analytical covariance matrix model is presented. This template is carefully calibrated on well-known terms of a numerical covariance matrix estimated from a set of simulated galaxy catalogues. After testing the …
In order to best improve constraints on cosmological parameters and on models of modified gravity using current and future galaxy surveys it is necessary maximally exploit the available data. As redshift-space distortions mean statistical translation invariance is broken for galaxy observations, this will require measurement of the monopole, quadrupole and hexadecapole of not just the galaxy power spectrum, but also the galaxy bispectrum. A recent (2015) paper by Scoccimarro demonstrated how the standard bispectrum estimator may be expressed in terms of Fast Fourier Transforms (FFTs) to afford…
We use redshift-space galaxy clustering data from the BOSS survey to constrain local primordial non-Gaussianity (LPNG). This is of particular importance due to the consistency relations, which imply that a detection of LPNG would rule out all single-field inflationary models. Our constraints are based on the consistently analyzed redshift-space galaxy power spectra and bispectra, extracted from the public BOSS data with optimal window-free estimators. We use a complete perturbation theory model including all one-loop power spectrum corrections generated by LPNG. Our constraint on the amplitude…
The anisotrpy of the redshift space bispectrum $B^s(\mathbf{k_1},\mathbf{k_2},\mathbf{k_3})$, which contains a wealth of cosmological information, is completely quantified using multipole moments $\bar{B}^m_{\ell}(k_1,μ,t)$ where $k_1$, the length of the largest side, and $(μ,t)$ respectively quantify the size and shape of the triangle $(\mathbf{k_1},\mathbf{k_2},\mathbf{k_3})$. We present analytical expressions for all the multipoles which are predicted to be non-zero ($\ell \le 8, m \le 6$ ) at second order perturbation theory. The multipoles also depend on $β_1,b_1$ and $γ_2$, which quantif…
We present the first detection of the integrated trispectrum ($\mathit{i}$-trispectrum) monopole and quadrupoles signal from BOSS CMASS NGC DR12. Extending the FKP estimators formalism to the Fourier transform of the four-point correlation function, we test shot-noise subtraction, Gaussianity of the i-trispectrum data-vector, significance of the detection and similarity between the signal from the data and from the galaxy mock catalogues used to numerically estimate the covariance matrix. Using scales corresponding to modes from minimum $k_\mathrm{min}=0.03\,h/\mathrm{Mpc}$ to maximum $k_\math…
We use analytic calculations of the post-recombination gravitational effects of cosmic strings to estimate the resulting CMB power spectrum, bispectrum and trispectrum. We place a particular emphasis on multipole regimes relevant for forthcoming CMB experiments, notably the Planck satellite. These calculations use a flat sky approximation, generalising previous work by integrating string contributions from last scattering to the present day, finding the dominant contributions to the correlators for multipoles l > 50. We find a well-behaved shape for the string bispectrum (without divergences) …
We study the information content of the angle-averaged (monopole) redshift space galaxy bispectrum. The main novelty of our approach is the use of a systematic tree-level perturbation theory model that includes galaxy bias, IR resummation, and also accounts for nonlinear redshift space distortions, binning, and projection effects. We analyze data from the PT challenge simulations, whose cumulative volume of 566 $h^{-3}$Gpc$^3$ allows for a precise comparison to theoretical predictions. Fitting the power spectrum and bispectrum of our simulated data, and varying all necessary cosmological and n…
We study the cosmological information content of the redshift-space galaxy bispectrum monopole at one-loop order in perturbation theory. We incorporate all effects necessary for comparison to data: fourth-order galaxy bias, infrared resummation (accounting for the non-linear evolution of baryon acoustic oscillations), ultraviolet counterterms, non-linear redshift-space distortions, stochastic contributions, projection, and binning effects. The model is implemented using FFTLog, and validated with the PT Challenge suite of $N$-body simulations, whose large volume allows for high-precision tests…
The most commonly used estimators of the anisotropic galaxy power spectrum employ Fast Fourier transforms, and rely on a specific choice of the line-of-sight that breaks the symmetry between the galaxy pair. This leads to wide-angle effects, including the presence of odd power spectrum multipoles like the dipole ($\ell = 1$) and octopole ($\ell = 3$). In Fourier-space these wide-angle effects also couple to the survey window function. We present a self-consistent framework extending the commonly used window function treatment to include the wide-angle effects. We show that our framework can su…
To fully extract cosmological information from nonlinear galaxy distribution in redshift space, it is essential to include higher-order statistics beyond the two-point correlation function. In this paper, we propose a new decomposition formalism for computing the anisotropic bispectrum in redshift space and for measuring it from galaxy samples. Our formalism uses tri-polar spherical harmonic decomposition with zero total angular momentum to compress the 3D modes distribution in the redshift-space bispectrum. This approach preserves three fundamental properties of the Universe: statistical homo…
We investigate a statistical anisotropy on the Cosmic Microwave Background (CMB) bispectrum, which can be generated from the primordial non-Gaussianity induced by quantum fluctuations of a vector field. We find new configurations in the multipole space of the CMB bispectrum given by $\ell_1 = \ell_2 + \ell_3 + 2, |\ell_2 - \ell_3| - 2$ and their permutations, which violate the rotational invariance, such as an off-diagonal configuration in the CMB power spectrum. We also find that in a model presented by Yokoyama and Soda (2008), the amplitude of the statistically anisotropic bispectrum in the…
The Bispectrum of Intrinsic Alignments: II. Precision Comparison Against Dark Matter Simulations
Utrecht University（乌得勒支大学）；Max-Planck-Institut für Astrophysik（马克斯·普朗克天体物理学研究所）；The University of Tokyo（东京大学）；Universität Bonn（波恩大学）；Leiden University（莱顿大学）；Ruđer Bošković Institute（鲁德博什科维奇研究所）；University of Cambridge（剑桥大学）
We extend the modal decomposition method, previously applied to compress the information in the real-space bispectrum, to the anisotropic redshift-space galaxy bispectrum. In the modal method approach, the bispectrum is expanded on a basis of smooth functions of triangles and their orientations, such that a set of modal expansion coefficients can capture the information in the bispectrum. We assume a reference survey and compute Fisher forecasts for the compressed modal bispectrum and two other basis decompositions of the redshift-space bispectrum in the literature, one based on (single) spher…
We propose an (optimal) estimator for a CMB bispectrum containing logarithmically spaced oscillations. There is tremendous theoretical interest in such bispectra, and they are predicted by a plethora of models, including axion monodromy models of inflation and initial state modifications. The number of resolved logarithmical oscillations in the bispectrum is limited due to the discrete resolution of the multipole bispectrum. We derive a simple relation between the maximum number of resolved oscillations and the frequency. We investigate several ways to factorize the primordial bispectrum, and …
We use the full bispectrum of spherical needlets applied to the WMAP data of the cosmic microwave background as an estimator for the primordial non-Gaussianity parameter f_NL. We use needlet scales up to l_max=1000 and the KQ75 galactic cut and find f_NL=84 +/- 40 corrected for point source bias. We also introduce a set of consistency tests to validate our results against the possible influence of foreground residuals or systematic errors. In particular, fluctuations in the value of f_NL obtained from different frequency channels, different masks and different multipoles are tested against sim…
We describe a fast, optimal estimator for measuring angular bispectra between two correlated weakly non-Gaussian fields ($Y$ and $Z$) from observational datasets, based on a separable modal bispectrum expansion. Our methodology is applicable to (1) any shape of the input theoretical bispectrum templates (factorizable or not), (2) both even and odd $\ell_1 + \ell_2 + \ell_3$ multipole domains and (3) both amplitude ($f_{\rm NL})$ bispectrum estimation and full bispectrum reconstruction, considering either joint estimation of ($YYY$, $ZZZ$, $YYZ$ and $ZZY$) shapes, or independent estimation of a…
We present the first detection of a correlation between the Lyman-$α$ forest and cosmic microwave background (CMB) lensing. For each Lyman-$α$ forest in SDSS-III/BOSS DR12, we correlate the one-dimensional power spectrum with the CMB lensing convergence on the same line of sight from Planck. This measurement constitutes a position-dependent power spectrum, or a squeezed bispectrum, and quantifies the non-linear response of the Lyman-$α$ forest power spectrum to a large-scale overdensity. The signal is measured at 5~$σ$ and is consistent with the $Λ$CDM expectation. We measure the linear bias o…
We study two key issues militating against the use of the anisotropic three-point correlation function (3PCF) for cosmological parameter inference: difficulties with its computational estimation and high-dimensionality. We show how high-dimensionality may be reduced significantly by multipole decompositions of all angular dependence. This allows deriving the full expression for the multipole moments of the anisotropic 3PCF and its covariance matrix in a basis where the dimensionality reduces from nine to two at each multipole in the plane-parallel limit. We use 2D FFTLog formalism to show how …
We address the amount of information in the non-Gaussian regime of weak lensing surveys by modelling all relevant covariances of the power spectra and bispectra, using 1000 ray-tracing simulation realizations for a Lambda-CDM model and an analytical halo model. We develop a formalism to describe the covariance matrices of power spectra and bispectra of all triangle configurations. In addition to the known contributions which extend up to six-point correlation functions, we propose a new contribution `the halo sample variance (HSV)' arising from the coupling of the lensing Fourier modes with la…
We present a new method for consistent, joint analysis of the pre- and post-reconstruction two-point functions of the BOSS survey. The post-reconstruction correlation function is used to accurately measure the distance-redshift relation and expansion history, while the pre-reconstruction power spectrum multipoles constrain the broad-band shape and the rate-of-growth of large-scale structure. Our technique uses Lagrangian perturbation theory to self-consistently work at the level of two-point functions, i.e.\ directly with the measured data, without approximating the constraints with summary st…
We present an all-sky formalism for the Cosmic Microwave Background (CMB) bispectrum induced by the primordial non-Gaussianities not only in scalar but also in vector and tensor fluctuations. We find that the bispectrum can be formed in an explicitly rationally invariant way by taking into account the angular and polarization dependences of the vector and tensor modes. To demonstrate this and present how to use our formalism, we consider a specific example of the correlation between two scalars and a graviton as the source of non-Gaussianity. As a result, we show that the CMB reduced bispectru…
Preference for evolving dark energy in light of the galaxy bispectrum
University of Science and Technology of China（中国科学技术大学）；CAS Key Laboratory for Research in Galaxies and Cosmology, School of Astronomy and Space Science, University of Science and Technology of China（中国科学院星系与宇宙学重点实验室，中国科学技术大学天文与空间科学学院）；Deep Space Exploration Laboratory（深空探测实验室）；Laboratoire Univers & Particules de Montpellier, CNRS & Université de Montpellier（蒙彼利埃宇宙与粒子实验室，法国国家科学研究中心与蒙彼利埃大学）；ETH Zürich（苏黎世联邦理工学院）；Università degli Studi di Milano（米兰大学）
