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Parity-Odd Power Spectra: Concise Statistics for Cosmological
Parity Violation
Drew Jamieson1,⋆Angelo Caravano2, Jiamin Hou3,4, Zachary Slepian3, & Eiichiro Komatsu1,4,5
1Max-Planck-Institut f¨ur Astrophysik, Karl-Schwarzschild-Straße 1, 85748 Garching, Germany
2Institut d’Astrophysique de Paris, UMR 7095 du CNRS et de Sorbonne Universit´e, 98 bis Bd Arago, 75014 Paris, France
3Department of Astronomy, University of Florida, Gainesville, FL 32611, USA
4Max-Planck-Institut f¨ur extraterrestrische Physik, Gießenbachstraße 1, 85748 Garching, Germany
5Ludwig-Maximilians-Universit¨at M¨unchen, Schellingstr. 4, 80799 M¨unchen, Germany
6Kavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU, WPI), University of Tokyo, Chiba 277-8582, Japan
25 June 2024
ABSTRACT
We introduce the Parity-Odd Power (POP) spectra, a novel set of observables for probing parity violation in cosmolog-
ical N-point statistics. POP spectra are derived from composite fields obtained by applying nonlinear transformations,
involving also gradients, curls, and filtering functions, to a scalar field. This compresses the parity-odd trispectrum
into a power spectrum. These new statistics offer several advantages: they are computationally fast to construct,
estimating their covariance is less demanding compared to estimating that of the full parity-odd trispectrum, and
they are simple to model theoretically. We measure the POP spectra on simulations of a scalar field with a specific
parity-odd trispectrum shape. We compare these measurements to semi-analytic theoretical calculations and find
agreement. We also explore extensions and generalizations of these parity-odd observables.
1 INTRODUCTION
Modern physics has advanced considerably by proposing theoretical symmetries and testing whether or not those symmetries
are respected or violated in nature. A crucial breakthrough for the Standard Model of particle physics occurred with the
surprising discovery of weak nuclear parity violation in 1957 (Lee & Yang 1956;Wu et al. 1957;Garwin et al. 1957). Parity is
the negation of spatial coordinates: x→ −x, and its violation implies that the laws of physics distinguish between right-handed
and left-handed chiralities. While parity is known to be maximally violated on small scales by weak nuclear interactions, the
status of parity on cosmological scales remains an open question.
Recent investigations using the method proposed in Cahn et al. (2023) and Cahn & Slepian 2023 have uncovered intriguing
evidence of cosmological parity violation in the large-scale structure (LSS) of the Universe. Notably, analyses of the BOSS
galaxy four-point correlation function detected parity violation with significance of up to 7σ(Hou et al. 2023b;Philcox 2022).
Confirmation of parity violation in the galaxy clustering would have profound implications. If the signal is primordial in origin,
it is a form of primordial non-Gaussianity and would greatly inform our early-Universe inflationary models (Bartolo et al. 2004).
Current cosmic microwave background (CMB) observations are consistent with parity conservation (Philcox 2023;Philcox &
Shiraishi 2024). However, since the CMB is measured on a 2D surface and is sensitive to different scales than the LSS, it is
still unclear what this implies for the BOSS signal.
The central challenge in the current BOSS analysis lies in robustly estimating the covariance (Cahn et al. 2023), which can
be influenced by both instrumental systematics and observational effects (Hou et al. 2023b) as well as the type of mocks used
to calibrate or compute the covariance (Hou et al. 2022a;Philcox & Ereza 2024).
Next-generation 3D spectroscopic surveys such as DESI (Aghamousa et al. 2016;Adame et al. 2024), Euclid (Amendola et al.
2018), Subaru PFS (Takada et al. 2014), Roman (Wang et al. 2022), and SPHEREx (Dor´e et al. 2014), as well as photometric
efforts such as LSST (Ivezi´c et al. 2019), will provide significantly more data than is currently available. To constrain large-scale
parity violation robustly with this wealth of data, we need analysis tools that facilitate accurate covariance estimation and
characterization of observational systematics.
For scalar fields like the primordial curvature perturbation or the matter density contrast, N-point statistics are sensitive
to parity at orders four and above, as noted in the CMB context by Shiraishi (2016) and first pointed out for 3D large-scale
structure (LSS) by Cahn et al. (2023). Measurements of these correlators have an enormous number of degrees of freedom (e.g.
the BOSS analysis of Hou et al. 2023b had 18,000). Covariance estimation for such large data vectors requires a large number of
simulated mock datasets, often combined with approximate analytic covariance templates (e.g. Hou et al. 2022b, also Slepian
©0000 The Authors
arXiv:2406.15683v1 [astro-ph.CO] 21 Jun 2024
2Jamieson et al.
& Eisenstein 2015,Xu et al. 2012). Even with statistically accurate covariance estimation, systematics in observational data
can bias both the detected signal and its significance.
A complementary approach to high-order N-point statistics is lower-order statistics, such as power spectra, that are computed
on composite fields. These composite fields are created by applying nonlinear transformations to the original scalar field.
Composite-field statistics average or compress the high-dimensional correlators down to low-dimensional correlators. As a
result, the covariances of these composite-field power spectra are less demanding to estimate, due to having fewer degrees of
freedom, while still carrying higher-order information. The trade-off involved in this approach is the loss of information through
compression. Examples of weighted, compressed analyses for bispectra and parity-even trispectra have been done before and
are known as skew-spectra and kurt-spectra (Schmittfull & Moradinezhad Dizgah 2021;Moradinezhad Dizgah et al. 2020;Hou
et al. 2023a;Munshi et al. 2022), as well as works on unweighted integrated bispectrum (Chiang et al. 2014) or three-point
correlation function (Slepian et al. 2017) and on integrated trispectrum (Gualdi et al. 2021;Gualdi & Verde 2022) or four-point
function (Sabiu et al. 2019); a Fourier-transform-based algorithm for the full four-point function is Sunseri et al. (2023). An
estimator of the CMB lensing power spectrum can also be constructed in this manner (Hu 2001).
In the current work, we introduce new methods for detecting cosmological parity violation by constructing power-spectrum-
like statistics sensitive to parity. We call our new observables parity-odd power spectra, or POP spectra. The POP spectra are
compressions of the six-dimensional four-point statistics down to one-dimensional power spectra. They are computationally
efficient to construct and their lower dimensionality alleviates the burden of full four-point covariance estimation.
Importantly, POP spectra are sensitive to soft limits of parity-violating trispectrum shapes, which can help to place strong
constraints on primordial parity violation. Moreover, being formulated in Fourier space, they are easy to interpret and can
facilitate theoretical understanding of the signal, which is crucial to distinguish it from potential observational systematics. It
is also possible to construct real-space and spherical-harmonic-space versions of these statistics. As we will demonstrate below,
constructing parity-odd statistics is nontrivial, requiring more sophisticated techniques than those required for parity-even
compressed estimators.
This work is structured as follows. In §2, we discuss the primordial parity-odd trispectrum. In §3, we present two POP
spectra constructions based on composite vector and scalar fields. In §4, we validate the estimator on simulated data, including
comparisons with semi-analytical calculations. In §5, we explore extensions and generalizations of our parity-odd observables.
Finally, we conclude and discuss prospects for measuring these parity-odd statistics on observational data in §6.
2 PARITY-VIOLATING TRISPECTRUM
Under statistical homogeneity and isotropy, the four-point correlation function is the lowest-order parity-sensitive statistic
for a single scalar field. In Fourier space, this corresponds to the trispectrum, T(k1,k2,k3), defined through the four-point
correlator,
Φ(k1)Φ(k2)Φ(k3)Φ(k4)≡(2π)3δ(3)
D(k1+k2+k3+k4)T(k1,k2,k3).(1)
The Dirac delta function imposes translational invariance by requiring that the sum of the four wave vectors vanishes.
Thus, they form a closed loop in Fourier space, and define a tetrahedron, as displayed in Fig. 1. More precisely, the four
wave vectors define an equivalence class of tetrahedra, consisting of the 24 tetrahedra produced by permuting them. These
are not all geometrically distinct, since a cyclic permutation of wave vectors within a given tetrahedron yields a congruent
tetrahedron. The three independent wave vectors, which we take to be k1,k2, and k3, uniquely determine these tetrahedra.
Statistical isotropy, however, reduces the dimensionality of the trispectrum from nine to six continuous degrees of freedom. We
will choose the wave vector magnitudes k1,k2,k3, and k4as four of these continuous parameters, and refer to them as sides
of the tetrahedron. Then we take K=|k1+k2|=|k3+k4|and ˜
K=|k1+k4|=|k2+k3|as the other two parameters and
refer to these as diagonals to distinguish them from the wave vector magnitudes.
Among the set of 24 tetrahedra, there is an additional discrete degree of freedom that splits them into two groups of 12.
These are distinguished by the sign of the vector triple product k1·(k2×k3). This can be thought of as the handedness
or helicity of the tetrahedron. The triple product is positive for a right-handed configuration and negative for a left-handed
configuration. Under a parity transform, the sign of the triple product changes, so a parity transformation interchanges the
two helicities.
We could isolate the trispectrum for only right-handed configurations, and obtain the right-handed trispectrum TR(k1,k2,k3),
and similarly for the left-handed trispectrum TL(k1,k2,k3). These are not necessarily equal. If they are different, this indicates
that the trispectrum has a parity-odd component. For the modes of a real field, a parity transformation is complex conjugation,
Φ(−k) = Φ(k)∗,(2)
so the parity-even component of the trispectrum is its real part and is proportional to the sum of TR+TL. The parity-odd
MNRAS 000, 000–000 (0000)
POP Spectra 3
k1
k2˜
K
k3
k4
K
Figure 1. Here we display a Fourier-space tetrahedron representing the trispectrum configuration T(k1,k2,k3). The diagonals (in dashed
magenta) are defined as K=k1+k2and ˜
K=k1+k4. The tetrahedron is formed by joining two triangles, {k1,k2,−K}in red and
{k3,k4,K}in blue, along their shared edge.
component is its imaginary part and is proportional to the difference TR−TL:
T(k1,k2,k3) = 1
√2TR(k1,k2,k3) + TL(k1,k2,k3)+i
√2TR(k1,k2,k3)−TL(k1,k2,k3)(3)
=T+(k1,k2,k3) + iT−(k1,k2,k3).(4)
Due to isotropy, the only vectors we can use to form a parity-odd structure are k1,k2, and k3, so T−(k1,k2,k3) must be
proportional to the triple product k1·(k2×k3). We parameterize the shape of the imaginary trispectrum as (Coulton et al.
2024)
T−(k1,k2,k3) = k1·(k2×k3)τ−(k1, k2, k3, k4, K, ˜
K).(5)
The left-hand side of Eq. (1) is totally symmetric under the interchange of any of the wave vectors, and the triple product
in Eq. (5) is totally antisymmetric. These symmetries require that τ−(k1, k2, k3, k4, K, ˜
K) is totally antisymmetric under the
interchange of any two of the four wave vectors. The trispectrum cannot depend on one diagonal and not the other since
K→˜
Kunder the interchange of k2and k4. Also, under the interchange of k2and k3,Kbecomes |k1+k3|, which is not
independent, since
|k1+k3|2=−K2−˜
K2+k2
1+k2
2+k2
3+k2
4.(6)
This parameterization is not unique. We could choose any six geometrically independent quantities of the tetrahedra to
parameterize the continuous degrees of freedom. For example, we could reparameterize by replacing Kwith the angle between
k1and k2. Similarly, we can choose either parity or helicity to parameterize the discrete degree of freedom.
3 PARITY-ODD POWER SPECTRA
CMB and LSS observables are consistent with cosmic structure originating from a single primordial scalar potential, Φ(x). In
this section, we construct parity-odd power-spectrum-like observables, which we call POP spectra. These are compressions of
the six-dimensional parity-odd trispectrum down to a one-dimensional power spectrum of composite fields. First, we construct
the vector POP spectrum, defined below in Eq. (11), by cross-correlating a vector field with a pseudovector field, both derived
as composite fields quadratic in Φ. Next, we construct the scalar POP spectrum, defined below in Eq. (21), by cross-correlating
a scalar field with a pseudoscalar field. The latter is formed as a composite field that is cubic in Φ.
3.1 Vector POP Spectrum
Our goal in this subsection is to construct composite vector and pseudovector fields out of a scalar field such that the cross-
power spectrum between them is a compressed estimator of the parity-odd trispectrum. Since a vector is parity odd and a
pseudovector is parity even, their cross-correlation will be a POP spectrum. Obtaining a vector field from a scalar is trivial:
take its gradient, ∇Φ(x). Obtaining a pseudovector is non-trivial. One possibility is to convolve the field with a filtering or
smoothing function fa(x):
Φa(x) = Zd3y fa(x−y)Φ(y) (7)
=Zk
fa(k)Φ(k)eik·x,(8)
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4Jamieson et al.
where the integral notation is defined in Table 1. For two distinct filtering functions, fa(x) and fb(x), the composite vector
field Vab(x)≡Φa(x)∇Φb(x) is not the gradient of a scalar, so it also has a curl component. Therefore, we can carry out a
Helmholtz decomposition,
Vab(x) = ∇ϕab (x)−∇×Aab(x).(9)
Since Vab and the spatial derivatives are parity odd, Aab is a pseudovector. Notice that if the two fields, Φaand Φb, were
identical, Aab would vanish. Although one of these can be the original, unfiltered field. We impose that its divergence vanishes
because any possible divergence would not contribute to Vab(x). Then we solve for its modes by taking the curl of Vab(x)
and solving the resulting Poisson equation,
Aab(k) = −1
k2ik×Vab(k).(10)
In general, we are free to choose four distinct filter functions: fa(x), fb(x), fc(x), and fd(x). From the first two, we define
the composite vector field Vab(x). From the second two, we define the composite vector field Vcd(x) and then isolate the
pseudovector field Acd(x) from it. The dot product Vab(x)·Acd (x) is parity odd. As a result, we can define our first POP
spectrum as
⟨Vab(k)·Acd (k′)⟩= (2π)3δ(3)
Dk+k′Pvector(k).(11)
Explicitly, we can express the power spectrum as
Pvector(k) = 1
Nvector(k)Z
q1,q2,q3,q4
h(2π)6δ(3)
D(q1+q2−k)δ(3)
D(q3+q4+k)fa(q1)fb(q2)fc(q3)fd(q4)
×[k·(q1×q3)]2τ−(q1, q2, q3, q4, k, |q1+q4|)i.(12)
The wave vector integrals are defined in Table 1. The normalization factor Nvector(k) is somewhat arbitrary, but if chosen
poorly, the estimator will depend strongly on the geometry and resolution of the region where we sample Φ(x). A sensible
choice is
Nvector(k) = Z
q1,q2,q3,q4
(2π)6δ(3)
D(q1+q2−k)δ(3)
D(q3+q4+k)fa(q1)fb(q2)fc(q3)fd(q4) [k·(q1×q3)]2,(13)
so that Eq. (12) is a weighted average over the parity-odd trispectrum shape τ−(q1, q2, q3, q4, k, |q1+q4|).
If we choose isotropic filtering functions that depend only on the magnitudes of the qi, the vector POP spectrum is a real-
valued quantity that is non-vanishing only if the parity-odd trispectrum is non-vanishing. With isotropic filtering functions,
the parity-even part of the trispectrum does not contribute to the imaginary part of the POP spectrum. To see this, we note
that the parity-even trispectrum is invariant under qi→ −qiwhile the triple product receives a minus sign. Therefore, the
integrand is antisymmetric and the integral vanishes.1
The vector-pseudovector construction compresses the trispectrum down to a power spectrum by correlating two fields of order
Φ2. The modes of these fields represent triangles of wave vectors satisfying k+q1+q2= 0 and k′+q3+q4= 0. Computing
the power spectrum enforces that the two triangles connect along a side of equal length, k, which imposed k+k′= 0. Thus,
the two triangles form a tetrahedron (shown in Fig. 1) as the trispectrum requires. The vector POP spectrum is a weighted
average over all trispectrum configurations while holding fixed the diagonal side length k, which is the argument of this POP
spectrum. For this reason, the vector POP carries crucial information about the soft limit where the diagonal approaches zero.
If the trispectrum has no explicit dependence on the diagonals and the filtering functions depend only on the magnitudes of
the wave vectors in Fourier space, we can compute the angular integrals in Eqs. (12–13) analytically. Then the expression for
the vector POP spectrum simplifies (as detailed in Appendix A) to
Pvector(k) = π4
2Nvector(k)Z
q1,q2,q3,q4hq1
q2
Θsin2θ12sin2θ12
q3
q4
Θsin2θ34sin2θ34 fa(q1)fb(q2)fc(q3)fd(q4)
×τ−(q1, q2, q3, q4)i,(14)
and
Nvector(k) = π4
2Z
q1,q2,q3,q4
q1
q2
Θsin2θ12sin2θ12
q3
q4
Θsin2θ34sin2θ34 fa(q1)fb(q2)fc(q3)fd(q4).(15)
1The sign of the kin the Dirac delta functions does not matter, since Pvector(k) is a function of only the magnitude of k.
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POP Spectra 5
Notation Definition
ZkZd3k
(2π)3
Z
q1,...,qnZq1
... Zqn
ZqZ∞
0
q2dq
2π2
Z
q1,...,qnZq1
... Zqn
Table 1. Here, we list our Fourier-space integral notation and conventions. The bold symbols indicate wave vectors and the integral is
over all of Fourier space. The nonbold symbols indicate the wave vector magnitude.
The radial Fourier integrals are defined in Table 1. Here, Θ(x) is the Heaviside function,
Θ(x) =
0,if x < 0
1
2,if x= 0
1,if x > 0
,(16)
and θ12 the angle between q1and kin the first triangle, so
sin2θ12 = 1 −q2
1+k2−q2
2
2q1k2
,(17)
and similarly for θ34, with 1, 2 →3, 4. Notice that if qi,qjand kfail to satisfy the triangle inequalities |qj−qj| ≤ k≤qi+qj,
then the right-hand sides in Eqs. (17) would be negative and the corresponding sine would be purely imaginary. Thus, the
Heaviside functions in Eqs. (14–15) enforce the triangle inequalities.
As a final remark, note that the pseudovector defined in Eq. (10) is not unique. For example, one could alternatively take
the following curl,
Bab(x) = ∇Φa(x)×∇Φb(x).(18)
Using this instead of Aab simply multiplies the vector POP spectrum Pvector(k) by a factor of −k2.
3.2 Scalar POP Spectrum
We can construct another POP spectrum by cross-correlating a scalar field with a pseudoscalar field. The construction of a
pseudoscalar, similar to that of the pseudovector from the previous subsection, is nontrivial and requires convolving the scalar
field with filtering functions. In this case, we first construct the pseudovector given in Eq. (18). We have the freedom to further
smooth this pseudovector by another filtering function, with modes
Bbcd(Q) = −fd(Q)Z
q3,q4
(2π)3δ(3)
D(q3+q4−Q)fb(q3)fc(q4) (q3×q4)Φ(q3)Φ(q4).(19)
Then, we construct the triple product field
Ψabcd(x) = ∇Φa(x)·Bbcd (x),(20)
which is a pseudoscalar. Its cross-power spectrum with Φ(k) is a POP spectrum,
⟨Φ(k)Ψabcd(k′)⟩= (2π)3δ(3)
Dk+k′Pscalar(k).(21)
The scalar POP spectrum is given by
Pscalar(k) = 1
Nscalar(k)Z
q2,q3,q4,Qh(2π)6δ(3)
D(k+q2+Q)δ(3)
D(q3+q4−Q)fa(q2)fb(q3)fc(q4)fd(Q)
×[Q·(q2×q4)]2τ−(k, q2, q3, q4, Q, |q2+q3|)i,(22)
MNRAS 000, 000–000 (0000)
6Jamieson et al.
with normalization
Nscalar(k) = Z
q2,q3,q3,Q
(2π)6δ(3)
D(k+q2+Q)δ(3)
D(q3+q4−Q)fa(q2)fb(q3)fc(q4)fd(Q) [Q·(q2×q4)]2.(23)
As with the vector-pseudovector construction, we choose the normalization so that the POP spectrum is a weighted average
over the imaginary trispectrum. In this case, we have constructed the compressed four-point statistic by cross-correlating a
pseudoscalar of order Φ3with the original potential Φ. The resulting power spectrum is an average over the trispectrum tetra-
hedra, fixing the magnitude of one of the wave vector sides. The average integrates over the diagonal Qthis time (corresponding
to the kin Eqs. 11–15), and the scalar POP spectrum is a function of only the fixed wave vector magnitude k, a side length
of the tetrahedron. This property makes the scalar POP particularly useful for probing the soft limit where one of the side
lengths approaches zero.
The side length that we label khere was labelled q1for the vector POP spectrum in Eqs. (12–15), so in the following
equations a subscript 1 will refer to this side length. Again, if the trispectrum does not depend on the diagonals and the
filtering functions depend only on the magnitudes of the wave vectors, we can compute the angular integrals analytically, and
the expression simplifies (as detailed in Appendix A) to
Pscalar(k) = π4
2kNscalar(k)Z
q2,q3,q3,Q hq2Θsin2θ21sin2θ21
q4
q3
Θsin2θ43sin2θ43 fa(q2)fb(q3)fc(q4)fd(Q)
×τ−(k, q2, q3, q4)i,(24)
and
Nscalar(k) = π4
2kZ
q2,q3,q3,Q
q2Θsin2θ21sin2θ21
q4
q3
Θsin2θ43sin2θ43 fa(q2)fb(q3)fc(q4)fd(Q).(25)
Here, we evaluate sin2θ21 and sin2θ43 using
sin2θ21 = 1 −q2
2+Q2−k2
2q2Q2
,(26)
sin2θ43 = 1 −q2
4+Q2−q2
3
2q4Q2
.(27)
3.3 Constructing the Estimators
To construct the vector POP spectrum from a scalar field Φ(x) on a discrete grid, we start by taking its Fourier transform,
Φ(q). Then, we make four copies of these modes and rescale them by the filter functions {fa(q), fb(q), fc(q), fd(q)}, obtaining
{Φa(q),Φb(q),Φc(q), Φd(q)}. Multiplying the modes of Φb(q) and Φd(q) by iqand then inverse Fourier transforming yields
the modes of ∇Φb(x) and ∇Φd(x). We also inverse Fourier transform the other two fields, Φaand Φcand, now in position
space, form the products Vab (x) = Φa(x)∇Φb(x) and Vcd(x) = Φc(x)∇Φd(x). Next, we Fourier transform both of these and,
from Vcd(q), solve for the modes of Acd (q) using Eq. (10). The unbinned vector POP spectrum estimator is
Pvector(k) = 1
VboxNvector (k)X
|q|=k
Vab(q)·A∗
cd(q),(28)
where the sum is over all modes with equal wave vector magnitudes on the grid.
The normalization is computed by initializing four grids of modes corresponding to the filter functions: fa(q), fb(q), fc(q),
and fd(q). From here, obtaining the normalization corresponding to Eq. (13) is somewhat complicated. When constructing
the power spectrum, we get one triple product from the estimator and another from the parity-odd trispectrum. Overall, the
power spectrum involves the square of the triple product. To get a squared triple product in the normalization factor, we need
to compute the Hessian of the field fa(x) and construct the composite tensor field with components
Fij
ab(x) = fa(x)∇i∇jfb(x).(29)
After Fourier transforming the Hessian of the filter function fields, we take the curl of one of the two components,
Jij
ab(q)≡ϵimn ikmFnj
ab (q),(30)
where repeated indices imply summation. Following the same steps for fc(x) and fd(x), the desired normalization is
Nvector(k) = −X
|q|=k
Jij
ab(q)hJij
cd(q)i†
.(31)
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POP Spectra 7
The dagger †denotes the Hermitian conjugate of the matrix. We then bin the estimator in wave number bins of width ∆k,
Pvector(k|∆k) = 1
Nvector(k|∆k)X
q∈k
Pvector(q)Nvector(q),(32)
where
Nvector(k|∆k) = X
q∈k
Nvector(q).(33)
Here, the sum over q∈kmeans summing over all q∈[k−∆k/2, k + ∆k/2).
The computation of the scalar POP spectrum follows similar steps. In this case, we compute the gradients ∇Φa(x), ∇Φb(x)
and ∇Φc(x). We use the latter two to form Bbc(x) according to Eq. (18). We Fourier transform this and then multiply by the
filtering function modes fd(q), resulting in the modes of Bbcd (q). We then inverse Fourier transform this, and its dot product
with ∇Φa(x) yields Ψabcd(x). The unbinned scalar POP spectrum is
Pscalar(k) = 1
VboxNscalar (k)X
|q|=k
Φ(q) [Ψabcd(q)]∗.(34)
Constructing the normalization is again nontrivial due to the squared triple product factor. In this case, we require the
Hessians of the field fa(x), fb(x), and fc(x). From the latter two, we construct the tensor field Fij
bc (x) as in Eq. (29). By
Fourier transforming all components and multiplying the modes by the filter function fd(q), we obtain Fij
bcd(q)≡fd(q)Fij
bc (q).
Inverse Fourier transforming this field and multiplying by the Hessian of fa(x) gives
Gij
abcd(x) = ϵimn ϵjpq Fmp
bcd (x)∇n∇qfa(x).(35)
Fourier transforming this, the scalar POP spectrum normalization is
Nscalar(k) = −X
|q|=k
kikjGij
abcd(q).(36)
The binned estimator is
Pscalar(k|∆k) = 1
Nscalar(k|∆k)X
q∈k
Pscalar(q)Nscalar (q),(37)
where
Nscalar(k|∆k) = X
q∈k
Nscalar(q).(38)
If we choose filter functions that only select wave vector magnitude shells of width ∆k, then both POP spectra coincide and
equal the binned trispectrum estimator in Eq. (A30). This fact illustrates the computational advantage of our POP spectra.
Evaluating all the bins of one POP spectrum is computationally equivalent to constructing a one-dimensional slice of the
full trispectrum. The tradeoff is that we have significantly compressed the information of the full trispectrum. However, the
presence of two estimators, each carrying complementary information from the parity-odd trispectrum, partially compensates
for this reduction in information. We also have the freedom to choose the filter functions, which can be optimized. The extent
of information loss could be further mitigated by considering extensions and generalizations of the POP spectra, which we
explore in §5.
The algorithms that we presented above for constructing these estimators are efficient since the most costly operations are
the Discrete Fourier Transforms (DFTs), which scale as Nlog N, with Nthe number of grid points. All other operations are
point-wise multiplications, which scale as N. Ignoring the normalization factors, the vector POP spectrum requires 15 and
the scalar POP spectrum requires 17 3D DFTs. Constructing the estimators without the normalization factor takes about 6
seconds for the vector and 8 seconds for the scalar POP spectrum running on a single node with 128 cores. The normalization
factors are more expensive to compute, but these only need to be computed once and can be saved and reused since they are
not data-dependent.
4 VALIDATION OF THE ESTIMATORS
4.1 Random Parity-Violating Realizations
To validate our POP estimators, we generate a set of random primordial curvature fields that include a specific parity-violating
trispectrum. We start by generating the modes of a Gaussian random field primordial curvature perturbation ζG(k),
ζG(k) = sVBox π2As
k3k
kpns−1
N1(k) + iN2(k),(39)
MNRAS 000, 000–000 (0000)
8Jamieson et al.
10−21
10−19
k9
1T−
0
10
n1
0
10
n2
0
10
n3
0
10
n4
0
10
N
0 250 500 750 1000 1250 1500 1750
Tetrahedron number
0
10
˜
N
Figure 2. Here we show the trispectrum in the thin bin limit (top panel) and the binning scheme (lower six panels). The shape of the
trispectrum template from Eq. (42) appears in the top panel. The bottom six panels indicate the binning scheme with ni=ki/kF,N=
K/kF, and ˜
N=˜
K/kF. Here, kFis the fundamental mode of a box interpreted as having length 4 Gpc h−1. We impose k1< k2< k3< k4.
We have used 10 k-bins of width kF.
where N1and N2are drawn from a standard normal distribution, Asis the amplitude of the primordial power spectrum, nsis
the spectral tilt, and kp= 0.05 Mpc−1is the pivot scale. We then transform this Gaussian random field into a non-Gaussian
field in real space as (Coulton et al. 2024)
ζ(x) = ζG(x) + g−∇ζ[α]
G(x)·h∇ζ[β]
G(x)×∇ζ[γ]
G(x)i.(40)
The coefficient g−controls the amplitude of the primordial parity-odd trispectrum, and the modes of the fields in the triple
product are
ζ[α]
G(k) = kαζG(k).(41)
The leading order imaginary part of the trispectrum for this template is given by
T−(k1,k2,k3) = g−k1·(k2×k3)2π2As3kα−4+ns
1kβ−4+ns
2kγ−4+ns
3k0
4∓23 signed permutations.(42)
The 24 terms in the parenthesis are the permutations of {k1, k2, k3, k4}. Even permutations (with an even number of transpo-
sitions) get a positive sign, and odd permutations get a negative sign in the sum. We have included the k0
4factor to emphasize
that every term has one of the four wave vector magnitudes raised to the exponent zero.
The trispectrum is scale invariant if α+β+γ=−3. To preserve the large-scale power spectrum we also restrict these
exponents to be less than or equal to zero. We choose α=−2, β=−1, and γ= 0, which is the same template used in
Coulton et al. (2024). For simplicity, we choose As= 2 ×10−9,ns= 1, and g−=±106. In Fig. 2, we display the shape of this
trispectrum.
Choosing a binning scheme where k1< k2< k3< k4, the trispectrum shape function is dominated by the terms
τ−(k1, k2, k3, k4)≃g−(2π2As)3
k5
1k4
21
k3
3−1
k3
4.(43)
This shape peaks when the wave vectors k1,k2, and k3are parallel so that k4=k1+k2+k3. The diagonals for this configuration
are K=k1+k2and ˜
K=k4−k1. However, this configuration is insensitive to parity, since the triple product vanishes when
any two wave vectors are colinear. The trispectrum peaks on a tetrahedron shape that deviates from colinearity with diagonals
K < k1+k2and ˜
K > k4−k1that maximize the product of the right-hand side of Eq. (43) and the triple product k1·(k2×k3).
This peak analysis is true for any scale-invariant trispectrum of the form in Eq. (42), although the particular diagonals that
maximize the trispectrum depend on the values of the template exponents. These parity-odd trispectrum templates diverge in
the soft limit k1→0.
MNRAS 000, 000–000 (0000)
POP Spectra 9
−2
−1
0
1
2
106×k2Pvector [Mpc10 h−10]
g−= +106
Theory
non-Gaussian
Gaussian
g−=−106
Vector-Pseudovector Parity-odd Power Spectrum
106×k2
2P(+)
vector −P(−)
vector
10−210−1
k[Mpc−1h]
−0.5
0.0
0.5
Frac.res.
10−210−1
k[Mpc−1h]
10−210−1
k[Mpc−1h]
Figure 3. The vector POP spectrum of the primordial potential defined in Eq. (40). In the left panels, the amplitude of the primordial
trispectrum is g−= 106, while in the middle panels g−=−106. We used the same underlying Gaussian realizations for both signs of
g−. The right panels show the variance-suppressed estimator, taking the difference between results with positive and negative g−. The
black dashed curve is the expected signal, computed from Eqs. (14–15). The bottom row shows fractional residuals of the simulated data
with respect to the semi-analytical calculation. Gray data points show the estimator on a Gaussian random field. These scatter around
zero and are consistent with a vanishing parity-violating signal. All data points have error bars obtained by bootstrap averaging over the
64 simulations, resampling with replacement 105times. In the right-most panel, the sample variance cancellation is not perfect. This is
because the POP spectrum involves four weakly non-Gaussian fields.
Since we are dealing with a scale-invariant primordial potential, the box length is arbitrary, but we interpret it as LBox =
4 Gpc h−1so that the scales involved are relevant for CMB and LSS. We generate 64 random pairs of these non-Gaussian
potential fields on a grid of size Ngrid = 5123. The pairs have the same underlying Gaussian realization but opposite signs
for their nonlinear terms. Subtracting the POP spectra measured on these pairs suppresses cosmic variance from the purely
Gaussian part of the fields. As can be seen in Fig 3, the cosmic variance is not perfectly cancelled. This is because the POP
spectrum is a compression of the four-point correlator of weakly non-Gaussian fields.
There is a considerable amount of freedom in our estimators since we can choose any set of filtering functions. For the
vector POP spectrum, fa(k) must be distinct from fb(k), and fc(k) must be distinct from fd(k), otherwise Pvector(k) vanishes
identically. For the scalar POP spectrum, fa(k), fb(k), and fc(k) must all be distinct for Pscalar (k) not to vanish. Here, we
choose
fa(k) = k2Θ(k−kmin)Θ(kmax −k),(44)
fc(k) = k−2Θ(k−kmin)Θ(kmax −k),(45)
fb(k) = fd(k) = Θ(k−kmin)Θ(kmax −k).(46)
Each filtering function involves the same scale cuts, kmin = 5 ×10−3Mpc−1hand kmax = 2 ×10−1Mpc−1h, which selects a
thick spherical shell of modes. As long as the fundamental mode in the box is less than kmin, and the Nyquist mode is greater
than kmax, the box contains the full range of modes and the estimator should be independent of the box size and resolution.
However, the discreteness of the grid is noticeable on large scales if kmin is very close to the fundamental mode.
The autocorrelation of the nonlinear term in Eq. (40) will introduce corrections to the power spectrum. These corrections
are of the order ∼g2
−A3
sand are subdominant compared to the Gaussian power spectrum for our template at the scales we
consider. The non-Gaussian corrections to the power spectrum grow at small scales and increase the power of the Nyquist
mode, k= 0.4 Mpc−1h, in our simulated boxes by about 1%. After imposing the scale cuts in our POP spectra the corrections
to the power spectrum are far below percent level for the modes we analyze. These corrections to the power spectrum can be
removed by further rescaling the non-Gaussian field (Coulton et al. 2024). However, corrections to the power spectrum could
be a physical effect. The (inflationary) mechanisms that generate the primordial trispectrum can also affect the shape of the
MNRAS 000, 000–000 (0000)
10 Jamieson et al.
−0.50
−0.25
0.00
0.25
0.50
106×k3Pscalar [Mpc9h−9]
g−= +106
Theory
non-Gaussian
Gaussian g−=−106
Scalar-Pseudoscalar Parity-odd Power Spectrum
106×k3
2P(+)
scalar −P(−)
scalar
10−210−1
k[Mpc−1h]
−0.1
0.0
0.1
Frac.res.
10−210−1
k[Mpc−1h]
10−210−1
k[Mpc−1h]
Figure 4. Same as Fig. 3but for the scalar POP spectrum. In this case, the signal grows in the limit k→0, showing sensitivity to the
soft limit of the trispectrum as one of the tetrahedron wave vector side lengths becomes small. In contrast, Fig. 3demonstrates decreasing
sensitivity in the diagonal soft limit of the vector POP spectrum because the specific trispectrum template that we simulate does not peak
in this limit. The behavior of the residual in the right-most panel is again (as in Fig. 3) due to the additional sample variance contributed
by the weakly non-Gaussian nature of the fields.
power spectrum. This template does not introduce a primordial bispectrum because all contributions would involve an odd
number of Gaussian fields, resulting in a vanishing expectation value.
We have omitted many complications of observational survey data in this analysis. Cosmological surveys target observables
that are biased tracers of the underlying matter field, which has undergone nonlinear evolution. Accurate modelling will require
the linear matter transfer function, the bias expansion, and possibly the non-linearity of gravitational clustering depending
on the scales considered. Surveys observe objects in redshift space where isotropy does not hold. Observational analyses must
also model the survey mask, selection function, and shot noise. We will investigate the impact of these aspects of observational
data on the POP spectra in future work.
4.2 Comparison with Analytical Computation
In Fig. 3, we display the vector POP spectrum compared with the theoretical expectation obtained by numerically evaluating
the integrals in Eqs. (14–15). The left panel shows the results for 64 simulations with g−= +106. The middle panel shows
results for the same underlying Gaussian random fields but with g−=−106. We significantly reduce cosmic variance by
computing the difference between these two estimators, as illustrated in the right panel of Fig. 3. We estimated the error bars
by bootstrap averaging over the estimators constructed for the 64 realizations, resampling 105times with replacement.
We obtained the semi-analytic result in Fig. 3by evaluating Eq. (14) on a discretized lattice for the radial integrals over all
qiand k=|q1+q2|. Specifically, for a given bin, we construct a three-dimensional grid of values for {q1, q2, k}, with q1and q2
spaced uniformly between kmin and kmax and kspanning the width of the power spectrum bin. We then mask out regions that
violate the triangle inequalities. Next, we construct a second grid for {q3, q4, k}, masking the non-triangular regions. Finally,
we integrate Eq. (14) term by term. For each permutation term in the trispectrum of Eq. (42), we construct the integrand for
q1and q2, and compute the discretized integral using Romberg integration with 28+ 1 sample points. We do the same for q3
and q4, except we include explicit factors of kappearing in Eq. (42) in this grid. We then multiply the results from the two
grids. Finally, we compute the discretized integral of the diagonal kusing Romberg integration with 27+ 1 sample points.
The procedure for integrating Eq. (15) is the same, omitting the factors from the primordial trispectrum template shape.
We parallelize this calculation by computing multiple bins simultaneously. From Fig. 3, we see excellent agreement with the
semi-analytic calculation of the expected signal. This agreement demonstrates that we accurately recover the expected signal
from the trispectrum shape injected into the data.
In Fig. 4, we display similar results for the scalar POP spectrum. The theoretical calculation of Eqs. (24–25) differs from the
MNRAS 000, 000–000 (0000)
POP Spectra 11
Figure 5. On the left panel, we display the correlation matrix, Eq. (47), for the vector-pseudovector (upper triangle) and scalar-
pseudoscalar (lower triangle) estimators. The correlation matrix is dominated by the diagonal, with off-diagonal contributions less than
10%. On the right panels, we display histograms of the χ2values from 64 random non-Gaussian fields compared with the distribution
from Gaussian realizations in gray. The peak of the dashed, analytical χ2curve coincides with the number of bins analyzed.
vector case in that kin the triangle {k, q2, Q}is integrated over the estimator’s bin width spanning the range between kmin and
kmax and the filter function fd(Q) applies to the diagonal Q. Again, we find excellent agreement between our simulated data
and the expected shape of Pscalar (k). The bias at low kis due to the discreteness of the grid, compared with the continuous-limit
integrals computed in the semi-analytic prediction. This bias is not noticeable for the vector estimator in Fig. 3due to the
signal vanishing at low k.
As a final remark note that, unlike the vector POP spectrum, the scalar POP spectrum increases as kapproaches zero. This
occurs because the argument kof the scalar POP spectrum corresponds to the side length of the tetrahedron. Our specific
trispectrum template, Eq. (42), diverges as one side length goes to zero, so the scalar POP spectrum also diverges in this limit.
If we chose a template that peaks in the soft limit where a diagonal approaches zero, the vector POP spectrum would increase
as its argument approaches zero. We could also construct a template that diverges in the soft side length and soft diagonal
limits. Then the vector and scalar POP spectra would increase as their arguments go to zero. This demonstrates that the
k→0 limits of our POP spectra encode crucial and complementary information about the soft limits of the trispectrum.
4.3 Correlation Matrix and χ2Test
The POP spectra integrate over many configurations to obtain a power spectrum from a trispectrum. For this reason, we may
worry that the covariance matrices of the POP spectra have complicated structures with large off-diagonal contributions. To
test this, we generate 1024 realizations of random Gaussian fields, construct the vector and scalar POP spectra estimators,
and compute the covariance matrix for each,
C(ki, kj) = ⟨P(ki)P(kj)⟩ − ⟨P(ki)⟩⟨P(kj)⟩,(47)
where kiindicates the ith wave number bin, and Pcan be either Pvector(k) or Pscalar (k). In Fig. 5we display the correlation
matrices,
R(ki, kj) = C(ki, kj)
pC(ki, ki)C(kj, kj),(48)
for the vector POP spectrum in the upper-right triangle and the scalar POP spectrum in the lower-left triangle. We find the
correlation matrix is dominated by its diagonal for both estimators, with off-diagonal contributions less than 10%.
We assess the sensitivity of our compressed estimators by computing the χ2from the inverse of the covariance matrix,
χ2=X
ij
P(ki)C−1(ki, kj)P(kj).(49)
We compute the χ2values for all 64 of our parity-violating non-Gaussian fields with positive amplitude for the primordial
trispectrum and their underlying Gaussian fields. Fig. 5displays histograms of the χ2values. The scalar POP spectrum exhibits
greater sensitivity for detecting primordial parity violation than the vector POP spectrum. We can understand this from the
MNRAS 000, 000–000 (0000)
12 Jamieson et al.
form of primordial non-Gaussianity in Eq.(40) that leads to the shape of the trispectrum in Eq. (42). This shape peaks as k1,
the argument of the scalar POP spectrum, goes to zero. We see this in the low-kbehaviour in Fig. 4.
The sensitivity depends on the choice of filtering kernels and the shape of the primordial signal. Different parity-odd trispec-
trum templates will generally yield different χ2distributions with more or less sensitivity. In principle, we can find the optimal
filter functions to maximize the signal-to-noise ratio of the POP power spectra for a given shape of the underlying primordial
trispectrum, as was done for the CMB lensing estimator (Hu 2001), but we do not explore that in this work. Since the vector
and scalar POP spectra contain complementary information, it would be best to do a joint analysis of both simultaneously.
This joint analysis requires the joint covariance, which will have off-diagonal contributions between the vector and scalar POP
spectra bins. We leave the analysis of the joint covariance matrix to future work.
5 EXTENSIONS AND GENERALIZATIONS
The examples of parity-odd power spectra explored above are the simplest examples of a more general class of constructions.
In this section, we outline some of these generalizations.
5.1 Connection to the Polarization Basis
Before exploring some generalizations of our POP spectra, we first write down an alternative and equivalent definition of
the vector POP spectrum based on polarization vectors, which will be useful later. As described in Sec. 2, parity violation is
closely related to right- and left-handed helicities. Here we will demonstrate the relationship between our vector POP spectrum
and these helicities. To do so, let us take two vector fields Vab and Vcd. From these, we can cross-correlate their Cartesian
components as
⟨Vi
ab(k)Vj
cd(k′)⟩= (2π)3δ(3)
D(k+k′)Pij (k).(50)
Assuming isotropy, we can decompose this power spectrum matrix into its trace, its traceless symmetric, and its antisymmetric
parts,
Pij (k) = 1
3δij P∥(k) + kikj
k2−1
3δij P⊥(k) + iϵijk kk
kP−(k).(51)
Assuming parity invariance, only the first two terms are present. These are the longitudinal and transverse parts of the
parity-even power spectrum matrix. The final term is the parity-odd part of the power spectrum matrix.
One can choose left-/right-handed polarization vectors and form a complete basis {e∥(k),eR(k),eL(k)}in Fourier space,
Vab(k) = Vab,∥(k)e∥(k) + Vab,R(k)eR(k) + Vab,L(k)eL(k).(52)
The transverse polarization vectors are defined as eigenvectors of the curl operator,
ik×eR/L(k)≡ ±keR/L(k),(53)
and e∥(k)≡ik/k. These quantities transform into one another after a parity transformation, as eR/L(−k) = ±ieL/R(k).
Therefore, one can define the eigenvectors of the parity operator as
e±(k) = 1
√2eR(k)±ieL(k).(54)
Using this alternative basis, the vector field is decomposed as
Vab(k) = Vab,∥(k)e∥(k) + Vab,−(k)e+(k) + Vab,+(k)e−(k).(55)
The ±labels of the coefficients are the opposite of the basis labels because the vector must be parity odd, so the coefficients
transform with the opposite sign compared to the transverse basis vectors. We can then project the power spectrum matrix
onto the polarization vectors,
PRR/LL(k) = Pij (k)ei
R/L(k)ej
R/L(−k),(56)
or
P+−(k) = Pij (k)ei
+(k)ej
−(−k).(57)
It is straightforward to show that
P−(k) = −k
2Pvector(k) (58)
=1
2PRR(k)−PLL (k)(59)
=P+−(k),(60)
where Pvector(k) is the vector POP spectrum.
MNRAS 000, 000–000 (0000)
POP Spectra 13
5.2 Higher-Order Derivatives: Angular Dependence
The vector and scalar POP spectra that we have considered thus far are minimal in the sense that they involve the minimum
number of spatial derivatives required to form the parity-odd triple product. We are free to add any number of spatial derivatives
to these constructions, forming higher-order, tensor POP spectra. For example, we consider adding an additional derivative to
the vector and pseudovector fields from the vector POP spectrum, as
Hij
1ab(x) = ∇iΦa(x)∇jΦb(x),(61)
Hij
2cd(x) = ϵikl ∇j∇kΦc(x)∇lΦd(x).(62)
The POP spectrum formed by contracting the indices of these two fields ⟨Hij
1ab(k)Hij
2cd(k′)⟩has a similar form to the vector
POP spectrum in Eq. (12), the difference being that the integrand would now contain an additional factor of q1·q3. Therefore,
this POP tensor spectrum probes a different form of angular dependence in the parity-odd trispectrum.
We can define higher-rank tensor POP spectra by contracting ⟨Hli
1ab(k)Hlj
2cd(k′)⟩. We can project this onto the basis tensors
eij
R/L(k) = ei
R/L(k)ej
R/L(k),(63)
where ei
R/L(k) are components of the polarization vectors, defined in Eq. (53). This decomposition gives the right-handed and
left-handed components of the transverse tensor spectra, and the difference between these is a POP spectrum. This kind of
construction can be easily generalized to higher-derivative and higher-rank tensor POP spectra. One example of this is Shim
et al. (2024). Similar constructions for the scalar POP spectra are also possible.
5.3 Higher-Order Constructions: Parity-Odd Bispectra and Beyond
The scalar bispectrum is defined by the three-point correlation function in Fourier space through
⟨Φ(k1)Φ(k2)Φ(k3)⟩= (2π)3δ(3)
D(k1+k2+k3)B(k1, k2, k3).(64)
The Dirac delta function enforces statistical translational invariance and makes the bispectrum a function over two-dimensional,
closed triangles. Since in three dimensions, we can rotate a parity-transformed two-dimensional triangle back to its initial
configuration, the bispectrum is insensitive to parity under statistical isotropy.
We construct the parity-odd bispectrum that compresses the parity-odd four-point function in a similar way to the vector
POP spectra,
⟨Aab(k1)·ik2Φ(k2)Φ(k3)⟩= (2π)3δ(3)
D(k1+k2+k3)Bvector(k1, k2, k3).(65)
In this case, we have used the gradient of the scalar rather than the composite vector field. With the composite vector field,
this construction would involve five scalar fields corresponding to a compression of the parity-odd five-point function.
Similarly, the pseudoscalar requires a product of three scalar fields for its constructions, so the parity-odd bispectrum of one
composite pseudoscalar with two scalar fields compresses the parity-odd five-point statistics. Additionally, we can correlate the
vector-pseudovector inner product with two pseudoscalar fields or correlate three pseudoscalar fields. These constructions are
of order six in the underlying scalar field, compressing the parity-odd six-point function. Thus, Eq. (65) is the only form for a
parity-odd compression of the trispectrum to a bispectrum.
6 CONCLUSION
Exploring parity violation on large cosmological scales will provide crucial insights into the fundamental symmetries governing
the origins and evolution of the Universe. The search for a parity-odd signal within galaxy surveys presents a key challenge
in observational cosmology, especially in light of recent hints of a parity-odd signal in the four-point correlation function of
galaxies (Hou et al. 2023b;Philcox 2022), both of which used the isotropic basis functions of Cahn & Slepian (2023), the
method proposed in Cahn et al. (2023), and the covariance template of Hou et al. (2022b). Direct measurements of four-point
correlations pose several technical challenges related to the high dimensionality of the data vector and the resulting size of the
covariance matrix, making it difficult to interpret the signal and assess its statistical significance. Motivated by these challenges,
we developed a novel set of statistics for parity violation.
The new observables are two-point correlators calculated between composite fields derived via nonlinear transformations
of the original scalar field. These constructions compress the trispectrum information into a power spectrum, resulting in a
significant simplification compared to the full four-point correlation function. The advantages of this approach are two-fold.
First, the computation of a two-point function is faster because of the significant dimensionality reduction of the data vector;
our computation on a 5123grid takes fewer than 10 seconds for each spectrum on 128 cores. This reduction facilitates a more
efficient and interpretable analysis of the underlying parity-odd signal and its covariance. Second, formulating the estimators
in Fourier space enables efficient characterization of the scale dependence of the signal and its soft limits.
We defined two sets of estimators: one based on correlating a vector and a pseudovector field and another correlating a
MNRAS 000, 000–000 (0000)
14 Jamieson et al.
scalar and a pseudoscalar field, both derived from the original scalar field. These two estimators are distinct compressions
of the trispectrum and thus carry complementary information on the parity-odd signal. In particular, they are sensitive to
different soft limits of the trispectrum. To empirically validate the effectiveness of these estimators, we tested them on a set
of mock simulations with an injected parity-odd four-point correlation function. On these mocks, we compared the result of
the estimator with semi-analytical theoretical calculations. The agreement between the data and our theoretical calculations
demonstrates that our estimators robustly capture the injected parity-odd signal.
Beyond validation, we investigated the sensitivity of our new estimators by computing the correlation matrix on Gaussian
realizations. We found that different bins of our estimator are predominantly uncorrelated. Furthermore, a χ2-squared test
on simulated parity-violating mocks highlighted the substantial statistical significance of both estimators, with the scalar
variant exhibiting enhanced sensitivity compared to its vector counterpart. This sensitivity is specific to the choice of the
primordial parity-violating trispectrum template and filtering functions. In particular, we did not attempt to optimize the
filtering functions by maximizing signal-to-noise. Such an optimization could dramatically increase the significance. We plan
to explore this in future work.
Applying our estimator to realistic survey data requires several developments. Surveys observe galaxies within a specific
survey mask and selection function. We must incorporate these into our estimator to remove the spurious clustering signals
they would otherwise produce. Moreover, galaxies are discrete tracers, and this results in shot noise that should be characterized
and subtracted from the POP spectra. We leave these developments for future work. After overcoming these challenges, we
expect the POP estimators will emerge as an essential tool in the search for parity violation in current and future CMB and
galaxy surveys.
ACKNOWLEDGEMENTS
We thank Toshiki Kurita, Fabian Schmidt, Azadeh Moradinezhad Dizgah, Robert Cahn, Donghui Jeong, and Ue-Li Pen for
helpful discussions. AC acknowledges funding support from the Initiative Physique des Infinis (IPI), a research training program
of the Idex SUPER at Sorbonne Universit´e. This work was supported in part by the Excellence Cluster ORIGINS, which is
funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy:
Grant No. EXC-2094 - 390783311. This work has also received funding from the European Union’s Horizon 2020 research and
innovation programme under the Marie Sk lodowska-Curie grant agreement no. 101007633. The Kavli IPMU is supported, in
part, by World Premier International Research Center Initiative (WPI), MEXT, Japan. ZS thanks the Max Planck Institute
for Astrophysics for hospitality during some of the period during which this work was performed.
APPENDIX A: THE BINNED PARITY-ODD TRISPECTRUM AND THE BINNED POP SPECTRA
Measuring the parity-odd trispectrum from data requires choosing a binning scheme for the tetrahedral configurations of wave
vectors. To form a bin of nearby tetrahedra, we specify the side lengths k1,k2,k3k4, and the diagonal lengths Kand ˜
K. We
then select the set of wave vectors satisfying
k1+k2+K= 0 ,(A1)
k3+k4−K= 0 ,(A2)
k1+k4+˜
K= 0 .(A3)
We define the six-dimensional binning scheme through the integral
Zbin ≡Zq1∈k1Zq2∈k2Zq3∈k2Zq4∈k4
Θ4|q1+q2| − K2−∆k2Θ4|q1+q4| − ˜
K2−∆k2.(A4)
The integral notation is defined in Table A1. The Heaviside functions enforce K−∆k/2≤ |q1+q2| ≤ K+ ∆k/2 and
˜
K−∆k/2≤ |q1+q4| ≤ ˜
K+ ∆k/2, which defines the binning scheme for the diagonals.
We isolate the binned parity-odd trispectrum by integrating the product of the triple product and the four-point expectation
value over the bin,
¯τ−(k1, k2, k3, k4, K, ˜
K) = N−1
τ−Zbin −iq1·(q2×q3)DΦ(q1)Φ(q2)Φ(q3)Φ(q4)E(A5)
=N−1
τ−Zbin
(2π)3δ(3)
D(q1+q2+q3+q4)q1·(q2×q3)2τ−q1, q2, q3, q4,|q1+q2|,|q1+q4|.(A6)
Here, Nτ−is a normalization factor, which we will define later. The minus sign in the first line is for convenience, and we could
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POP Spectra 15
Notation Definition
Zq∈kZk+∆k/2
k−∆k/2
k2dk
2π2
ZˆqZdΩˆq
4π
Zq∈kZˆqZq∈k
Z6D-bin Zq1∈k1Zq2∈k2Zq3∈k2Zq4∈k4ZQ∈KZ˜
Q∈˜
K
Z5D-bin Zq1∈k1Zq2∈k2Zq3∈k3Zq4∈k4ZQ∈K
Z5D-bin Zq1∈k1Zq2∈k2Zq3∈k3Zq4∈k4ZQ∈K
X
LiMi
∞
X
L1=0
L1
X
M1=−L1
∞
X
L2=0
L2
X
M2=−L2
∞
X
L3=0
L3
X
M3=−L3
Table A1. Fourier-space integral and harmonic-space summation notation. The notation q∈kindicates a wave vector magnitude
belonging to a bin centered on kwith width ∆k. The integral over dΩˆqis the solid angle integral over the azimuthal and polar angles
associated with the wave vector qin polar coordinates.
absorb it into the normalization factor. We split the Dirac delta function as follows:
(2π)3δ(3)
D(q1+q2+q3+q4) = (2π)9Z
Q,˜
Q
δ(3)
D(q1+q2+Q)δ(3)
D(q3+q4−Q)δ(3)
Dq1+q4+˜
Q.(A7)
The right-hand side of this expression imposes the condition that the tetrahedron consists of two triangles, {q1,q2,−Q}and
{q3,q4,Q}, which are joined along their common side of length Q. The vectors q1and q4are additionally constrained to form
a triangle {q1,q4,−˜
Q}. After substituting Eq. (A7) into Eq. (A6), the Heaviside functions limit the integrations over Qand
˜
Q. Then, with the integral notation in Table A1, the binned parity-odd trispectrum is given by
¯τ−(k1, k2, k3, k4, K, ˜
K) = N−1
τ−Z6D-bin h(2π)9δ(3)
D(q1+q2+Q)δ(3)
D(q3+q4−Q)δ(3)
Dq1+q4+˜
Q
×q1·(q2×q3)2τ−(q1, q2, q3, q4, Q, ˜
Q)i.(A8)
Unfortunately, the full six-dimensional trispectrum is difficult to analyze directly. The first two Dirac delta functions factor
into integrals over two triangles, but the third couples these together, making the whole integral not factorizable. For this reason,
we extend the ˜
Qbin to be unrestricted, ˜
Q∈[0,∞), which fully averages over the second diagonal. An alternative approach is
to implement isotropic basis functions, which discretize the angular dependence through spherical harmonic expansions (Cahn
& Slepian 2023), but we will not pursue that here.
The five-dimensional binned trispectrum, with notation from Table A1, is
¯τ−(k1, k2, k3, k4, K) = N−1
τ−Z5D-bin
(2π)6δ(3)
D(q1+q2+Q)δ(3)
D(q3+q4−Q)q1·(q2×q3)2τ−q1, q2, q3, q4, Q, |q1+q4|,
(A9)
The third Dirac delta function in Eq. (A8) enforced ˜
Q=|q1+q4|. We choose the somewhat arbitrary normalization factor
Nτ−to be
Nτ−=Z5D-bin
(2π)9δ(3)
D(q1+q2+Q)δ(3)
D(q3+q4−Q)q1·(q2×q3)2,(A10)
so Eq. (A9) is the weighted average of τ−(q1, q2, q3, q4, Q, |q1+q4|) within the bin.
Expressing both Dirac delta functions as the Fourier transform of plane waves enables us to compute most angular integrals
analytically. The calculation simplifies greatly by noticing that we can rewrite the triple product as q1·(q2×q3) = −Q·(q1×q3),
which is more symmetric between the two triangles forming the tetrahedron. However, if the trispectrum depends on the
diagonals, the angular dependence on q1·q4may prevent us from analytically computing all angular integrals. From here on,
we will restrict to the case where the trispectrum has no explicit dependence on the diagonals.
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16 Jamieson et al.
The spherical harmonic expansion of the squared triple product is
[Q·(q1×q3)]2=−6(4π)3(Qq1q3)2X
Li,Mi
DM1M2M3
L1L2L3YL1M1(ˆq1)YL2M2(ˆq3)YL3M3(ˆ
Q),(A11)
where the coefficients are
DM1M2M3
L1L2L3=L1L2L3
M1M2M3
L1L2L3
1 1 1
1 1 1
3
Y
i=1 r2Li+ 1
4π1 1 Li
0 0 0 .(A12)
Here, the 2 by 3 matrices are Wigner 3-jsymbols, and the 3 by 3 matrix is a Wigner 9-jsymbol. The form of the coefficients
in the product restricts Li∈ {0,2}.
The trispectrum estimator is an integral over tetrahedra formed by two triangles: {q1,q2,−Q}and {q3,q4,Q}. By taking
one vector from each triangle and the common vector shared by both triangles in the triple product, we preserve the symmetry
between the two pairs (q1,q2) and (q3,q4), greatly simplifying the calculation. Each triangle contributes a Dirac delta function
and a YLM from a vector that is not ±Q. Expressing the Dirac delta functions as Fourier transforms of plane waves allows us
to compute all angular integrals,
Z
q1∈k1,q2∈k2
(2π)3δ(3)
D(q1+q2−Q)YL1M1(ˆq1) = YL1M1(ˆ
Q)Z
q1∈k1,q2∈k2
4πZ∞
0
dx x2jL1(q1x)j0(q2x)jL1(Qx) (A13)
=YL1M1(ˆ
Q)Z
q1∈k1,q2∈k2
IL10L1(q1, q2, Q),(A14)
and similarly,
Z
q3∈k3,q4∈k4
(2π)3δ(3)
D(q3+q4+Q)YL2M2(ˆq3) = (−1)L2YL2M2(ˆ
Q)Z
q3∈k3,q4∈k4
4πZ∞
0
dx x2jL2(q3x)j0(q4x)jL2(Qx) (A15)
= (−1)L2YL2M2(ˆ
Q)Z
q3∈k3,q4∈k4
IL20L2(q3, q4, Q).(A16)
Since L2∈ {0,2}, we have (−1)L2= 1, so we can drop this factor. Both triangles contribute factors to the integrand that
have the same form, illustrating the symmetric roles of the two triangles that form the tetrahedron. The triple-spherical Bessel
integrals are given by
IL10L1(q1, q2, Q)≡4πZ∞
0
dx x2jL1(q1x)j0(q2x)jL1(Qx) (A17)
=π2
q1q2QΘsin2θ12PL1cos θ12 .(A18)
Here, θ12 is the angle between q1and Q. As discussed in the main text, the Heaviside function enforces the triangle inequalities
that ensure q1,q2, and −Qcan form a closed triangle. PL1is the Lth
1Legendre polynomial. The only relevant ones will be
P0(x) = 1 , P2(x) = 1
23x2−1.(A19)
At this point, three YLiMiproducts remain, all with the argument ˆ
Q. Integrating over ˆ
Qgives Gaunt’s integral (divided by
4π),2
GM1M2M3
L1L2L3=Zˆ
Q
YL1M1(ˆ
Q)YL2M2(ˆ
Q)YL3M3(ˆ
Q) (A20)
=L1L2L3
M1M2M3L1L2L3
0 0 0 3
Y
i=1 r2Li+ 1
4π.(A21)
The form of the trispectrum after computing all angular integrals is
¯τ−(k1, k2, k3, k4, K) = −6N−1
τ−Z5D-bin
(Qq1q3)2X
LiMi
DM1M2M3
L1L2L3GM1M2M3
L1L2L3IL10L1(q1, q2, Q)IL20L2(q3, q4, Q)τ−(q1, q2, q3, q4),
(A22)
2https://dlmf.nist.gov/34.3, Eq. 34.3.22
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POP Spectra 17
where the summation notation is defined in Table A1. Using the orthogonality relation for the 3-jsymbol3
L1
X
M1=−L1
L2
X
M2=−L2
L3
X
M3=−L3L1L2L3
M1M2M32
= 1 ,(A23)
we can perform the sums over all Mi. Since neither triple-spherical Bessel integral involves L3, we can also sum over it, which
gives
¯τ−(k1, k2, k3, k4, K) = 6 N−1
τ−Z5D-bin
(Qq1q3)2X
L1,L2
HL1L2IL10L1(q1, q2, Q)IL20L2(q3, q4, Q)τ−(q1, q2, q3, q4),(A24)
where the coefficients now simplify considerably:
HL1L2=−X
L3L1L2L3
0 0 0
L1L2L3
1 1 1
1 1 1
3
Y
i=1
(2Li+ 1) 1 1 Li
0 0 0 (A25)
=iL1+L2
27 .(A26)
The two remaining sums factorize,
¯τ−(k1, k2, k3, k4, K) = 2
9N−1
τ−Z5D-bin
(Qq1q3)2hX
L1
iL1IL10L1(q1, q2, Q)ihX
L2
iL2IL20L2(q3, q4, Q)iτ−(q1, q2, q3, q4).(A27)
Each sum only has two terms, so we can sum them explicitly and expand the Legendre polynomials,
¯τ−(k1, k2, k3, k4, K) = π4
2N−1
τ−Z5D-bin
q1q3
q2q4
Θsin2θ12sin2θ12 Θsin2θ34 sin2θ34 τ−(q1, q2, q3, q4),(A28)
where we set τ−= 1 to obtain the normalization factor,
Nτ−=π4
2Z5D-bin
q1q3
q2q4
Θsin2θ12sin2θ12 Θsin2θ34 sin2θ34 .(A29)
Instead of restricting the integration bounds, we could define filter functions with Heaviside functions that impose the
binning. Then, the binned parity-odd trispectrum takes the form,
¯τ−(k1, k2, k3, k4, K) = π4
2N−1
τ−Z
q1,q2,q3,q4,Q hq1q3
q2q4
Θsin2θ12sin2θ12 Θsin2θ34 sin2θ34
×fa(q1)fb(q2)fc(q3)fd(q4)fe(Q)τ−(q1, q2, q3, q4)i.(A30)
By taking all but feto be generic filter functions, rather than just Heaviside functions selecting spherical shells, and taking
the thin-bin limit of fe, this has the same form as the vector POP spectrum in Eq. (14). Similarly, by taking all but fato be
generic filter functions and taking the thin-bin limit of fa, this has the same form as the scalar POP spectrum in Eq. (24).
Thus, computing a POP spectrum is equivalent to computing a one-dimensional subset of bins for the full five-dimensional
trispectrum.
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