Figure - available from: Journal of Physics A: Mathematical and Theoretical
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Yutsis diagram describing the coefficient of PΛ′′ in the product of the four-point functions PΛ and PΛ′ .
Source publication
Isotropic functions of positions r1, r2, . . . , rN, i.e. functions invariant under simultaneous rotations of all the coordinates, are
conveniently formed using spherical harmonics and Clebsch-Gordan coefficients. An orthonormal basis of such functions provides
a formalism suitable for analyzing isotropic distributions such as those that arise in c...
Citations
... Also, the parity-violating part of K({k i }) must be a pseudoscalar, and contains a factor of Levi-Civita symbol ϵ ijk . To utilize the properties of K({k i }) under rotations and space inversion, it is convenient to expand it in terms of a set of basis functions, known as N -point isotropic basis functions [63]. For concreteness, we show explicit formulae of 3-point and 4-point isotropic basis functions here: ...
... in which big parentheses denote Wigner's 3j symbols, Y m ℓ (k) represents spherical harmonics, and * means complex conjugation. Note that, among the subscripts of P ℓ 1 ,ℓ 2 ,(ℓ ′ ),ℓ 3 ,ℓ 4 , ℓ 1 , · · · , ℓ 4 outside the parentheses label external angular momenta (named primaries in Ref. [63]), while (ℓ ′ ) denotes the intermediate angular momentum label (named intermediates in Ref. [63]) that is subject to the triangle inequality |ℓ 1 − ℓ 2 | ≤ ℓ ′ ≤ ℓ 1 + ℓ 2 as well as |ℓ 3 − ℓ 4 | ≤ ℓ ′ ≤ ℓ 3 + ℓ 4 . Usually, K({k i }) comes from tensor structures expressed in the Cartesian basis, but the conversion to the spherical basis is straightforward. ...
... in which big parentheses denote Wigner's 3j symbols, Y m ℓ (k) represents spherical harmonics, and * means complex conjugation. Note that, among the subscripts of P ℓ 1 ,ℓ 2 ,(ℓ ′ ),ℓ 3 ,ℓ 4 , ℓ 1 , · · · , ℓ 4 outside the parentheses label external angular momenta (named primaries in Ref. [63]), while (ℓ ′ ) denotes the intermediate angular momentum label (named intermediates in Ref. [63]) that is subject to the triangle inequality |ℓ 1 − ℓ 2 | ≤ ℓ ′ ≤ ℓ 1 + ℓ 2 as well as |ℓ 3 − ℓ 4 | ≤ ℓ ′ ≤ ℓ 3 + ℓ 4 . Usually, K({k i }) comes from tensor structures expressed in the Cartesian basis, but the conversion to the spherical basis is straightforward. ...
Parity-violating interactions are ubiquitous phenomena in particle physics. If they are significant during cosmic inflation, they can leave imprints on primordial perturbations and be observed in correlation functions of galaxy surveys. Importantly, parity-violating signals in the four-point correlation functions (4PCFs) cannot be generated by Einstein gravity in the late universe on large scales, making them unique and powerful probes of high-energy physics during inflation. However, the complex structure of the 4PCF poses challenges in diagnosing the underlying properties of parity-violating interactions from observational data. In this work, we introduce a general framework that provides a streamlined pipeline directly from a particle model in inflation to galaxy 4PCFs in position space. We demonstrate this framework with a series of toy models, effective-field-theory-like models, and full models featuring tree-level exchange-type processes with chemical-potential-induced parity violation. We further showed the detection sensitivity of these models from BOSS data and highlighted potential challenges in data interpretation and model prediction.
... In particular, we will show that all the properties of interests can be derived from a formalism based on multipolar-spherical harmonics (MultiSHs) [47], which are the many-body generalization of the spherical harmonics. These have been already exploited in astrophysical studies [48,49] (we also refer to Ref. [50] for a general and in-depth treatment of the isotropic case). Indeed, a crucial result of this paper is that all the derivations and properties of the most well-known ML descriptors are directly inherited from the choice of MultiSHs as a multi-body basis, and do require just an handful of symmetries of the atomic density. ...
... , r ν ), andx := (r 1 , . . . ,r ν ) , (50) respectively. To further reduce the number of indexes, we will always imply the dimensionality of these vectors, which will be understood from the context. ...
The formulation of descriptors of the local chemical environment, enabling the construction of machine-learning models, is usually obtained by studying the properties of the expansion coefficients of a neighborhood density. In this work, we show that all the transformation properties of the descriptors and their behaviour under rotation, inversion and complex conjugation, are derived from the choice of the basis over which the density is expanded. Furthermore, crucially they are independent from the explicit mathematical form of the neighborhood density. In particular, we show that all the descriptors investigated, can be obtained by an expansion in multipolar spherical harmonics, which constitutes the core of this work, and which is introduced and analysed in great detail. By exploiting the orthogonality and the transformation rules of the multipolar spherical harmonics, we show that several formulations are simplified, such as the one needed to obtain the SOAP kernel and its properties. We close this work by applying our framework to several multi-body descriptors available in literature, providing an in-depth analysis of their main properties, as made clear from the vantage viewpoint of a basis-centered approach.
... 1 Efficient methods for computing this exist (e.g. [17,18]; using mathematical tools developed in [19]), which have facilitated parity measurements from the SDSS-III Baryon Oscillation Spectroscopic Survey (BOSS) galaxy catalogue containing O(10 6 ) galaxies [20,21]. ...
... where δ is the galaxy overdensity, 3 and P ℓ 1 ℓ 2 ℓ 3 is an angular basis function relative to one vertex of the galaxy tetrahedron (as defined in [19]; see also the BiPoSH basis [63]). The r i parameters give the (discretized) tetrahedron side lengths, while ℓ i index the Fourier complement of the internal angle, with odd ℓ 1 + ℓ 2 + ℓ 3 encoding parity-violation. ...
Recent works have uncovered an excess signal in the parity-odd four-point correlation function measured from the Baryon Oscillation Spectroscopic Survey (BOSS) galaxy catalogue. If physical in origin, this could indicate new parity-breaking processes in inflation. At heart, these studies compare the observed four-point correlator with the distribution obtained from parity-conserving mock galaxy surveys; if the simulations underestimate the covariance of the data, noise fluctuations may be misinterpreted as a signal. To test this, we reanalyse the BOSS CMASS parity-odd dataset with the noise distribution model using the newly developed GLAM-Uchuu suite of mocks. These comprise full N-body simulations that follow the evolution of 20003 dark matter particles and represent a significant upgrade compared with the formerly used MultiDark-Patchy mocks, which were based on an alternative (non N-body) gravity solver. We find no significant evidence for parity-violation (with a baseline detection significance of 1.0σ), suggesting that the former signal (2.9σ with our data cuts) could be caused by an underestimation of the covariance in MultiDark-Patchy. The significant differences between results obtained with the two sets of BOSS-calibrated galaxy catalogues (whose covariances differ at the 10−20% level) showcase the heightened sensitivity of beyond-two-point analyses to nonlinear effects and indicate that previous constraints may suffer from large systematic uncertainties.
This article is part of the discussion meeting issue ‘Challenging the standard cosmological model’.
... We will call this term the inverse decay trispectrum. Although this term contains both parityodd and even contributions, we can always project the result onto the isotropic basis functions of [60] to single out the parity-odd modes. ...
... The fact that the integral (6.5) is a convolution is more apparent if we write the scalar triple product 5 (the first term in the integrand of 6.5) in a different form. We may rewrite the scalar triple product in terms of the three-argument isotropic basis functions P ℓ 1 ℓ 2 ℓ 3 of [60], in particular the lowest-lying parity odd one: ...
Recent measurements of the galaxy 4-Point Correlation Function (4PCF) have seemingly detected non-zero parity-odd modes at high significance. Since gravity, the primary driver of galaxy formation and evolution is parity-even, any parity violation, if genuine, is likely to have been produced by some new parity-violating mechanism in the early Universe. Here we investigate an inflationary model with a Chern-Simons interaction between an axion and a U(1) gauge field, where the axion itself is the inflaton field. Evaluating the trispectrum (Fourier-space analog of the 4PCF) of the primordial curvature perturbations is an involved calculation with very high-dimensional loop integrals. We demonstrate how to simplify these integrals and perform all angular integrations analytically by reducing the integrals to convolutions and exploiting the Convolution Theorem. This leaves us with low-dimensional radial integrals that are much more amenable to efficient numerical evaluation. This paper is the first in a series in which we will use these results to compute the full late-time 4PCF for axion inflation, thence enabling constraints from upcoming 3D spectroscopic surveys such as Dark Energy Spectroscopic Instrument (DESI), Euclid, or Roman.
... Hence, functions that are invariant under simultaneous rotation of all of their unit-vector arguments are particularly useful. In [1], these isotropic basis functions were developed, and their importance and other related previous work is more fully outlined there. The idea of using them to parameterize N-Point Correlation Functions, quantifying the spatial clustering of the distribution of galaxies, was presented in [2], with a detection of the even-parity, "connected" (due to nonlinear gravitational evolution) 4-Point Correlation Function (4PCF) in [3]. ...
... Another work [7] also found evidence for parity violation in CMASS, using the same covariance matrix template. 1 This latter work also used the isotropic basis functions to perform calculations of an inflationary model, and subsequent works have also done so, e.g. [8]. ...
... The left-hand side above is the most common form for the "generating function" of the Legendre polynomials, which themselves form an isotropic basis set for functions of two unit vectors (i.e.r 1 andr 2 ; since x is the dot product it is invariant under simultaneous rotation ofr 1 andr 2 ). Since the isotropic basis functions of [1] are essentially generalizations of the Legendre polynomials to more than two unit vectors, it seems desirable to have a generating function for them as well. Motivated by work proving the addition theorem for regular solid harmonics using the plane wave expansion [12,13], we do so here. ...
Recently isotropic basis functions of N unit vector arguments were presented; these are of significant use in measuring the N-point correlation functions (NPCFs) of galaxy clustering. Here we develop the generating function for these basis functions—i.e. that function which, expanded in a power series, has as its angular part the isotropic functions. We show that this can be developed using basic properties of the plane wave. A main use of the generating function is as an efficient route to obtaining the Cartesian basis expressions for the isotropic functions. We show that the methods here enable computing difficult overlap integrals of multiple spherical Bessel functions, and we also give related expansions of the Dirac Delta function into the isotropic basis. Finally, we outline how the Cartesian expressions for the isotropic basis functions might be used to enable a faster NPCF algorithm on the CPU.
... Moreover, no evidence of parity violation in the 4PCF in the CMB [16][17][18] and in the public SDSS Data Release 16 Lyman-α forest data [19] has been found. As noted in [20] (see also [21,22]), the 4PCF [23] represents the lowest order statistics capable of exhibiting parity violation within the scalar sector, for example, through the use of correlators of the gauge-invariant [24,25] curvature perturbation ζ(t, x) [26]. ...
... [27][28][29]). For example, precise algorithms for performing such analyses were developed in [21], based on innovative methods [22,[30][31][32] for 4PCF. Other examples include the introduction of novel observables for studying parity violation in LSS analyses, such as Parity-Odd Power Spectra [33] in which the parity-odd trispectrum is compressed into simpler statistics computationally fast to construct 1 . ...
... • We are computing the scalar trispectrum because it is the lowest scalar correlator capable of exhibiting parity violation signatures [22]. ...
Recently, possible hints of parity violation have been observed in the connected galaxy four-point correlation function. Although the true origin of the signal from the analysis has been debated, should they have a physical origin, they might point to primordial non-Gaussianity and would be evidence of new physics. In this work, we examine the single-field slow-roll model of inflation within chiral scalar-tensor theories of modified gravity. These theories extend the Chern-Simons one by including parity-violating operators containing first and second derivatives of the non-minimally coupled scalar (inflaton) field. This model is capable of imprinting parity-violating signatures in late-time observables, such as the galaxy four-point correlation function. We perform an analysis of the graviton-mediated scalar trispectrum of the gauge-invariant curvature perturbation . We estimate that for a set of parameters of the theory it is possible to produce a signal-to-noise ratio for the parity-violating part of the trispectrum of order one without introducing modifications to the single-field slow-roll setup. Even if the signal found in the analysis turns out to be spurious or if no parity violation is ever detected in the galaxy four-point correlation function, our analysis can be used to constrain the free parameters of these theories.
... In the context of galaxy surveys, [21] proposed using the galaxy 4PCF to test parity violation ( [22] suggested this prospect for the Cosmic Microwave Background (CMB)). The approach of [21] exploits the isotropic basis functions developed in [23] and efficient NPCF measurement algorithms built on them [24,25]. [26] applied this idea to Sloan Digital Sky Survey Baryon Oscillation Spectroscopic Survey (SDSS BOSS) data [27], resulting in a 7 detection of the parity-odd 4PCF for the larger, higher-redshift CMASS sample and a 3 detection for the smaller, lower-redshift LOWZ sample. ...
... In this section, we illustrate the BAO signal in the parity-odd 4PCF. We will first review the 4PCF in the isotropic basis of [23]. Then we will compute an example detection significance of the BAO signal using a suite of simulations with a toy model parity-violating 4PCF encoded in them. ...
... A direct measurement of the 4PCF in Eq. (5) is computationally challenging. To accelerate the process, we decompose the 4PCF into a basis of isotropic basis functions P ℓ 1 ℓ 2 ℓ 3 [23] that capture its angular behavior about one galaxy (the "primary';), times radial coefficients that capture its dependence on tetrahedron side lengths 1 , 2 , 3 from that primary. The isotropic basis [23] has rotational invariance and can well capture the isotropic information in the large-scale distribution of galaxies. ...
Measurements of the galaxy 4-Point Correlation Function (4PCF) from theSloan Digital Sky Survey Baryon Oscillation Spectroscopic Survey (SDSS BOSS) have recently found strong statistical evidence for parity violation. If this signal is of genuine physical origin, it must stem from beyond-Standard Model physics, most likely during the very early Universe, prior to decoupling (z1,020). Since the Baryon Acoustic Oscillation (BAO) features imprint at decoupling, they are expected in the parity-odd galaxy 4PCF, and so detecting them would be an additional piece of evidence that the signal is genuine. We demonstrate in a toy parity-violating model how the BAO imprint on the parity-odd 4PCF. We then outline how to perform a model-independent search for BAO in the odd 4PCF, desirable since, if the signal is real, we may not know for some time what model of e.g. inflation is producing it. If BAO are detected in the parity-odd sector, they can be used as a standard ruler as is already done in the 2PCF and 3PCF. We derive a simple formula relating the expected precision on the BAO scale to the overall parity-odd detection significance. Pursuing BAO in the odd 4PCF of future redshift surveys such as DESI, Euclid, Spherex, and Roman will be a valuable additional avenue to determine if parity violation in the distribution of galaxies is of genuine cosmological origin.
... Each value of the flattened index a = 1, . . . , N dof corresponds to one six-tuple ((β i ), (l i )), In this section, we will review the definition and key properties of E from [1,2,4,34]. ...
... Reproducing results from[1,2] In this section we will define the χ 2 statistic, and reproduce the ∼ 7σ and ∼ 3σ results from[1,2]. This section mostly reviews results from previous papers, especially[1,2,4,34], but is included to establish consistency between our pipeline and previous results, and to make our paper self-contained.10 However, LOWZE3 does require angular weights depending on the seeing, due to the incorrect application of the CMASS star-galaxy separation[32].11 ...
Recent studies have found evidence for parity violation in the BOSS spectroscopic galaxy survey, with statistical significance as high as 7σ. These analyses assess the significance of the parity-odd four-point correlation function (4PCF) with a statistic called X ². This statistic is biased if the parity-even eight-point correlation function (8PCF) of the data differs from the mock catalogs. We construct new statistics X ² ×, X ² null that separate the parity violation signal from the 8PCF bias term, allowing them to be jointly constrained. Applying these statistics to BOSS, we find that the parity violation signal ranges from 0 to 2.5σ depending on analysis choices, whereas the 8PCF bias term is ~ 6σ. We conclude that there is no compelling evidence for parity violation in BOSS. Our new statistics can be used to search for parity violation in future surveys, such as DESI, without 8PCF biases.
... The possibility of parity violation in the galaxy distribution is made perhaps more interesting given a reported preference in Planck 2018 CMB polarization data for a nonzero value of the cosmic birefringence angle [5], which also hints at cosmological parity-violating physics [6] The possibility to seek parity violation in galaxy clustering was suggested briefly in Refs. [7,8], and precise algorithms to carry out such searches were developed in Ref. [9], capitalizing upon novel techniques [10][11][12][13] for the more general 4PCF. The implementation with BOSS data and evidence for parity breaking in Refs. ...
Recent searches for parity breaking in the galaxy four-point correlation function, as well as the prospects for greatly improved sensitivity to parity breaking in forthcoming surveys, motivate the search for physical mechanisms that could produce such a signal. Here we show that a parity-violating galaxy four-point correlation function may be induced by lensing by a chiral gravitational-wave background. We estimate the amplitude of a signal that would be detectable with a current galaxy survey, taking into account constraints to the primordial gravitational-wave-background amplitude. We find that this mechanism is unlikely to produce a signal large enough to be seen with a galaxy survey but note that it may come within reach with future 21cm observations.
... 3 Reproducing results from [1,2] In this section we will define the χ 2 statistic, and reproduce the ∼ 7σ and ∼ 3σ results from [1,2]. This section mostly reviews results from previous papers, especially [1,2,4,34], but is included to establish consistency between our pipeline and previous results, and to make our paper self-contained. ...
... Each value of the flattened index a = 1, . . . , N dof corresponds to one six-tuple ((β i ), (l i )), In this section, we will review the definition and key properties of E from [1,2,4,34]. ...
Recent studies have found evidence for parity violation in the BOSS spectroscopic galaxy survey, with statistical significance as high as . These analyses assess the significance of the parity-odd four-point correlation function (4PCF) with a statistic called . This statistic is biased if the parity-even eight-point correlation function (8PCF) of the data differs from the mock catalogs. We construct new statistics , that separate the parity violation signal from the 8PCF bias term, allowing them to be jointly constrained. Applying these statistics to BOSS, we find that the parity violation signal ranges from 0 to depending on analysis choices, whereas the 8PCF bias term is . We conclude that there is no compelling evidence for parity violation in BOSS. Our new statistics can be used to search for parity violation in future surveys, such as DESI, without 8PCF biases.
