Unsupervised searches for cosmological parity violation: An investigation with convolutional neural networks

Recent measurements of the 4-point correlation functions (4PCF) from spectroscopic surveys provide evidence for parity violations in the large-scale structure of the Universe. If physical in origin, this could point to exotic physics during the epoch of inflation. However, searching for parity violations in the 4PCF signal relies on a large suite of simulations to perform a rank test, or an accurate model of the 4PCF covariance to claim a detection, and this approach is incapable of extracting parity information from the higher-order $N$-point functions. In this work we present an unsupervised method which overcomes these issues, before demonstrating the approach is capable of detecting parity violations in a few toy models using convolutional neural networks. This technique is complementary to the 4-point method and could be used to discover parity violations in several upcoming surveys including DESI, Euclid, and Roman.

Figure 1

In three dimensions, a tetrahedron is the lowest order polyhedron which is sensitive to parity i.e., handedness. Top: for the purposes of this work we say a tetrahedron is right-handed if when moving the right hand to the primary vertex with the thumb pointed into the tetrahedron, the fingers curl from the shortest to longest edges connected to the primary. It is impossible to rotate a left-handed tetrahedron (left) onto its mirror image (right), which is equivalent to a parity transform up to a 180° rotation. Bottom: in contrast the left-handed triangle (left) is equivalent to rotating the right-handed triangle (right) on the right “out of the page.”

Figure 2

Schematic showing the algorithm presented in Sec. 2a. Some data, $x$, and the parity flipped data, $Px$, are both passed through a CNN denoted, $g$, such that $g(x)\in \mathbb{R}$. For every pair we define, $f(x)=g(x)-g(Px)$. The network learns to maximize the difference between $g(x)$ and $g(Px)$ during training (see Sec. 2a for more details). As shown in Sec. 2b, the expected value of $f(x)$ is exactly zero if no parity violations are present. Any deviation from zero is indicative of parity violation, while the statistical significance can be estimated from boostrapping over the validation set.

Figure 3

Left: in this example, parity-violating “fields” are generated by mapping a fixed right-handed triangle, $T$, to a random location in the field and assigning a galaxy to each vertex (see Sec. 3b1 for more details). Right: histogram of the mean of the validation set over bootstrap resamplings, $⟨f{⟩}_B$, while $N_R$ gives the number of occurrences in the resampling in each histogram bin. This is greater than zero (indicated by a dashed line) at high-confidence ($>5\sigma$), so we have detected parity violations.

Figure 4

Top left: an example of a parity-violating “field” generated by mapping left-handed and right-handed triangles (with an overabundance of one kind) to a random location in the field and placing a galaxy at each vertex. The fields have on average 72.5 triangles per field and we place on average 5 more right-handed compared to left-handed triangles per field. Given all the possible triangles that can be formed from these vertices, this represents less than a 1% relative overabundance of right-handed triangles on average. See Sec. 3b2 for more details. Top right: histogram of the mean of $f$ from bootstrap resampling the validation set. Parity-violations are detected at the $3.9\sigma$ level. Bottom left: same as top left, but generated as a null test using a procedure where there is no overabundance of left versus right handed triangles (see Sec. 3b2) for more details). Bottom right: same as above. No parity violations are detected in the data as $⟨f{⟩}_B$ is consistent with zero.

Figure 5

Same as the right hand side of Fig. 3 but for this example we use a single right-handed tetrahedron per field. We make a clear detection ($>5\sigma$) of parity violation.