Probing Parity-Violation with the Four-Point Correlation Function of BOSS Galaxies

Parity-violating physics in the early Universe can leave detectable traces in late-time observables. Whilst vector- and tensor-type parity-violation can be observed in the $B$-modes of the cosmic microwave background, scalar-type signatures are visible only in the four-point correlation function (4PCF) and beyond. This work presents a blind test for parity-violation in the 4PCF of the BOSS CMASS sample, considering galaxy separations in the range $[20,160]h^{-1}\mathrm{Mpc}$. The parity-odd 4PCF contains no contributions from standard $\Lambda$CDM physics and can be efficiently measured using recently developed estimators. Data are analyzed using both a non-parametric rank test (comparing the BOSS 4PCFs to those of realistic simulations) and a compressed $\chi^2$ analysis, with the former avoiding the assumption of a Gaussian likelihood. These find similar results, with the rank test giving a detection probability of $99.6\%$ ($2.9\sigma$). This provides significant evidence for parity-violation, either from cosmological sources or systematics. We perform a number of systematic tests: although these do not reveal any observational artefacts, we cannot exclude the possibility that our detection is caused by the simulations not faithfully representing the statistical properties of the BOSS data. Our measurements can be used to constrain physical models of parity-violation. As an example, we consider a coupling between the inflaton and a $U(1)$ gauge field and place bounds on the latter's energy density, which are several orders of magnitude stronger than those previously reported. Upcoming probes such as DESI and Euclid will reveal whether our detection of parity-violation is due to new physics, and strengthen the bounds on a variety of models.Comment: 30 pages, 11 figures, accepted by Phys. Rev. D. Code available at https://github.com/oliverphilcox/Parity-Odd-4PC

Optimal Estimation of the Binned Mask-Free Power Spectrum, Bispectrum, and Trispectrum on the Full Sky: Scalar Edition

We derive optimal estimators for the two-, three-, and four-point correlators of statistically isotropic scalar fields defined on the sphere, such as the Cosmic Microwave Background temperature fluctuations, allowing for arbitrary (linear) masking and inpainting schemes. In each case, we give the optimal unwindowed estimator (obtained via a maximum-likelihood prescription, with an associated Fisher deconvolution matrix), and an idealized form, and pay close attention to their efficient computation. For the trispectrum, we include both parity-even and parity-odd contributions, as allowed by symmetry. The estimators can include arbitrary weighting of the data (and remain unbiased), but are shown to be optimal in the limit of inverse-covariance weighting and Gaussian statistics. The normalization of the estimators is computed via Monte Carlo methods, with the rate-limiting steps (involving spherical harmonic transforms) scaling linearly with the number of bins. An accompanying code package, PolyBin, implements these estimators in Python, and we demonstrate the estimators' efficacy via a suite of validation tests.Comment: 33 pages, 10 figures, code available at https://github.com/oliverphilcox/PolyBin. Accepted by Phys. Rev.

Do the CMB Temperature Fluctuations Conserve Parity?

Observations of the Cosmic Microwave Background (CMB) have cemented the notion that the large-scale Universe is both statistically homogeneous and isotropic. But is it invariant also under reflections? To probe this we require parity-sensitive statistics: for scalar observables, the simplest is the trispectrum. We make the first measurements of the parity-odd scalar CMB, focusing on the large-scale ($2<\ell<510$) temperature anisotropies measured by Planck. This is facilitated by new quasi-maximum-likelihood estimators for binned correlators, which account for mask convolution and leakage between even- and odd-parity components, and achieve ideal variances within $\approx 20\%$. We perform a blind test for parity violation by comparing a $\chi^2$ statistic from Planck to theoretical expectations, using two suites of simulations to account for the possible likelihood non-Gaussianity and residual foregrounds. We find consistency at the $\approx 0.4\sigma$ level, yielding no evidence for novel early-Universe phenomena. The measured trispectra allow for a wealth of new physics to be constrained; here, we use them to constrain eight primordial models, including Ghost Inflation, Cosmological Collider scenarios, and Chern-Simons gauge fields. We find no signatures of new physics, with a maximal detection significance of $2.0\sigma$. Our results also indicate that the recent parity excesses seen in the BOSS galaxy survey are not primordial in origin, given that the CMB dataset contains roughly $250\times$ more primordial modes, and is far easier to interpret, given the linear physics, Gaussian statistics, and accurate mocks. Tighter CMB constraints can be wrought by including smaller scales and adding polarization data.Comment: 7+13 pages, 4+5 figures, accepted by Phys. Rev. Lett. Code available at https://github.com/oliverphilcox/PolyBin/tree/main/planck_publi

Could Sample Variance be Responsible for the Parity-Violating Signal Seen in the BOSS Galaxy Survey?

Recent works have uncovered an excess signal in the parity-odd four-point correlation function measured from the BOSS spectroscopic galaxy survey. If physical in origin, this could indicate evidence for new parity-breaking processes in the scalar sector, most likely from inflation. At heart, these studies compare the observed four-point correlator to 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 + LOWZ parity-odd dataset with the noise distribution modeled using the newly developed GLAM-Uchuu suite of mocks. These comprise full N-body simulations that follow the evolution of $2000^3$ dark matter particles in a $\Lambda$CDM universe, and represent a significant upgrade compared to the formerly MultiDark-Patchy mocks, which were based on an alternative (non N-body) gravity solver. We find no significant evidence for parity-violation in the BOSS dataset (with a baseline detection significance of $1.4\sigma$), suggesting that the former signal ($>3.5\sigma$ 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 catalogs showcases the heightened sensitivity of beyond-two-point analyses to the treatment of non-linear effects and indicates that previous constraints may suffer from large systematic uncertainties.Comment: 7 pages, 4 figures, submitted to Phys. Rev.

Efficient Computation of NN-point Correlation Functions in DD Dimensions

We present efficient algorithms for computing the $N$-point correlation functions (NPCFs) of random fields in arbitrary $D$-dimensional homogeneous and isotropic spaces. Such statistics appear throughout the physical sciences, and provide a natural tool to describe a range of stochastic processes. Typically, NPCF estimators have $\mathcal{O}(n^N)$ complexity (for a data set containing $n$ particles); their application is thus computationally infeasible unless $N$ is small. By projecting onto a suitably-defined angular basis, we show that the estimators can be written in separable form, with complexity $\mathcal{O}(n^2)$, or $\mathcal{O}(n_{\rm g}\log n_{\rm g})$ if evaluated using a Fast Fourier Transform on a grid of size $n_{\rm g}$. Our decomposition is built upon the $D$-dimensional hyperspherical harmonics; these form a complete basis on the $(D-1)$-sphere and are intrinsically related to angular momentum operators. Concatenation of $(N-1)$ such harmonics gives states of definite combined angular momentum, forming a natural separable basis for the NPCF. In particular, isotropic correlation functions require only states with zero combined angular momentum. We provide explicit expressions for the NPCF estimators as applied to both discrete and gridded data, and discuss a number of applications within cosmology and fluid dynamics. The efficiency of such estimators will allow higher-order correlators to become a standard tool in the analysis of random fields.Comment: 12 pages, 2 figures, submitted to PNAS. Comments welcome

Testing Graviton Parity and Gaussianity with Planck T-, E- and B-mode Bispectra

Many inflationary theories predict a non-Gaussian spectrum of primordial tensor perturbations, sourced from non-standard vacuum fluctuations, modified general relativity or new particles such as gauge fields. Several such models also predict a chiral spectrum in which one polarization state dominates. In this work, we place constraints on the non-Gaussianity and parity properties of primordial gravitational waves utilizing the Planck PR4 temperature and polarization dataset. Using recently developed quasi-optimal bispectrum estimators, we compute binned parity-even and parity-odd bispectra for all combinations of CMB T-, E- and B-modes with $2\leq \ell<500$, and perform both blind tests, sensitive to arbitrary three-point functions, and targeted analyses of a well-motivated equilateral gravitational wave template (sourced by gauge fields), with amplitude $f_{\rm NL}^{ttt}$. This is the first time B-modes have been included in primordial non-Gaussianity analyses; they are found to strengthen constraints on the parity-even sector by $\simeq 30\%$ and dominate the parity-odd bounds, without inducing bias. We report no detection of non-Gaussianity (of either parity), with the template amplitude constrained to $f_{\rm NL}^{ttt}=900\pm 700$ (stable with respect to a number of analysis variations), compared to $1300\pm1200$ in Planck 2018. The methods applied herein can be reapplied to upcoming CMB datasets such as LiteBIRD, with the inclusion of B-modes poised to dramatically improve future bounds on tensor non-Gaussianity.Comment: 20 pages, 9 figures, accepted by Phys. Rev.