Ho'oleilana: An Individual Baryon Acoustic Oscillation?

Pressure waves generated in the hot plasma of the early universe become imprinted in baryon fluctuations approximately 390,000 yr after the hot Big Bang (Peebles & Yu 1970; Sunyaev & Zeldovich 1970). The remnants of these waves create a ruler that, observed across time in the evolving universe, provides constraints on the physics governing cosmic evolution (Weinberg et al. 2013; Aubourg et al. 2015). Eisenstein et al. (1998) investigated the possibility that early universe fluctuations caused by the baryon component of matter might explain structure on scales of ∼13,000 km s⁻¹ (Tully 1986; Tully et al. 1992; Broadhurst et al. 1990) and hints of baryon-induced features in the power spectrum of galaxy correlations were first announced by Percival et al. (2001). Subsequently, compelling evidence for what have come to be called baryon acoustic oscillations (BAOs) has been seen as a peak in the pairwise separations of galaxies throughout cosmic history (Cole et al. 2005; Eisenstein et al. 2005; Beutler et al. 2011; Blake et al. 2011; Ross et al. 2015; Alam et al. 2017, 2021). In all published cases, the BAO feature is a statistical imprint compounded by contributions from many locations.

Studies such as Scrimgeour et al. (2012) and Gonçalves et al. (2018) have identified the scale at which the Universe reaches 1% homogeneity as ∼70–120 h₇₅ Mpc. By logical arguments, the density fluctuations anticipated in individual BAO shells (which exist on scales larger than the homogeneity scale) can then be only a few percent of the mean matter density. So it has not been expected that individual BAOs can be discerned. It was demonstrated by Arnalte-Mur et al. (2012), though, that assuming BAOs developed out of prerecombination central dark matter concentrations identifiable today as rich clusters, the scales of associated BAO could be identified by wavelet analysis and the stacked density maps from ∼800 centers. These centers can be further studied to identify the structures that contribute most substantially to the total BAO signal.

We were not looking for BAOs. However, visual examination of maps from the Cosmicflows-4 compilation of galaxy distances (Tully et al. 2023) revealed a structure that invited further inspection. By way of introduction, the two orthogonal views in supergalactic coordinates in Figure 1 show the distribution of galaxy groups north of the Milky Way equator in this data set. ⁴ The SGY axis roughly tracks redshifts. An evident overdensity is seen at SGY ∼ 20,000 km s⁻¹, part of which is the Sloan Great Wall (Gott et al. 2005). The Center for Astrophysics Great Wall (de Lapparent et al. 1986) is seen at SGY ∼ 7000 km s⁻¹.

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Restricting the velocity range to the interval 19,000–26,000 km s⁻¹, the domain including the Sloan Great Wall, a view from the third orthogonal direction emphasizes what appears to be a ring structure, shown in the top panel of Figure 2. A reasonable by-eye fit to the structure is given by the red ring of radius 11,300 km s⁻¹ centered at SGX_c = –400 km s⁻¹, SGZ_c = 5000 km s⁻¹ in the bottom panel, which we show later in this work is also statistically justified when considering the full 3D galaxy distribution.

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This structure has been noted by Einasto et al. (2016) as the most prominent of several shell-like structures revealed in the main Sloan Digital Sky Survey (SDSS) sample. These authors looked for features around clusters and groups of varying richness and found the richer groups provided the more convincing evidence as the centers for shell-like structures. These authors considered but did not favor that any of the features they detect are related to the BAOs. As will be discussed further, we find contrary evidence that the ring seen in Figure 2 does indeed form part of a large coherent "BAO shell"—the biggest contribution to the overall BAO signal that we will report. In any event, this apparent ring structure at a distance of ∼250 Mpc from us is one of the largest structures observed in the nearby Universe to date and links together a number of hitherto disconnected components of our cosmic neighborhood. We name this remarkable structure "Ho'oleilana." ⁵

The two histograms in Figure 3 give more insight into the properties of Ho'oleilana. The top panel gives the numbers of our galaxies in annular rings about the center illustrated in the bottom panel of Figure 2, normalized by the area in each ring. There is an evident peak centered at 11,300 km s⁻¹. The lower panel shows the numbers of groups of galaxies projected into the ring with a radius spanning 10,600−12,000 km s⁻¹ as a function of systemic velocity in the cosmic microwave background (CMB) frame. Numbers peak at 23,000 ± 300 km s⁻¹. ⁶

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Several major features lie within this annulus. The region of high density from 7 o'clock to 10 o'clock in Figure 2 is the main component of the Sloan Great Wall (Gott et al. 2005). The Corona Borealis supercluster lies at ∼12 o'clock, the Ursa Majoris supercluster lies at ∼4 o'clock, and the Virgo–Coma supercluster lies at ∼6:30 (Einasto et al. 2001). Lesser structures include SCL 95 at ∼5 o'clock and SCL 154 at ∼10:30. The Boötes supercluster with 12 Abell clusters (Einasto et al. 2001) is found to be close to, but not quite at, the center. This supercluster contains the Abell Richness 2 cluster A1795 at V_cmb = 19,204 km s⁻¹ plus three Abell Richness 1 clusters. The projected positions of A1781, A1795, A1825, and A1831 are plotted in the bottom panel of Figure 2.

The BAOs are expected to be spherical shells, rather than just the ring that has been identified above. The galaxies in the relevant region in the Cosmicflows-4 collection are overwhelmingly contributed by the SDSS Peculiar Velocity (SDSS PV) catalog (Howlett et al. 2022). For three reasons the following statistical analysis of Ho'oleilana as a BAO feature is restricted to the elements of this catalog. This subset of the full Cosmicflows-4 collection has (1) a very well defined selection function; (2) a random, ungrouped catalog with the same selection function; and (3) an ensemble of 2 × 256 mock galaxy catalogs that also reproduce the large-scale structure, galaxy bias, and selection function of the data, but where in one-half the BAOs have been suppressed by generating the initial conditions of the simulations using a smoothed "no-wiggle" linear power spectrum (Hinton et al. 2017). Figure 4 shows the power spectra of the SDSS PV data and mocks, as well as the ratio of the two sets of simulations highlighting the BAO feature/suppression. These additional data products are essential for interpreting the significance of Ho'oleilana and fitting its size and shape and are only available for the SDSS PV portion of this catalog in the region of the Universe containing Ho'oleilana.

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We explore the consistency of Ho'oleilana emerging from the physics of the BAO in two different ways: (1) by fitting the 3D radial distribution of galaxy groups in the SDSS PV catalog with a physical model for the BAO feature and (2) by blindly searching for similar contributors to the BAO signal in the simulations and data using the wavelet-convolution method of Arnalte-Mur et al. (2012). We report on these tests in separate sections below. Overall, our findings provide evidence that Ho'oleilana is not a chance arrangement of galaxies, but instead is a part of the total BAO signal in our nearby Universe.

2.1. Modeling Ho'oleilana as a BAO Feature

For our first test of the origin of Ho'oleilana, we begin by converting the SDSS PV catalog from R.A., decl., and redshift to supergalactic Cartesian coordinates assuming a fiducial flat Lambda cold dark matter (ΛCDM) cosmological model with matter density Ω_m = 0.31 and expansion rate H₀ = 75 h₇₅ km s⁻¹ Mpc⁻¹. We use the "group" redshifts from the data and simulations as this suppresses the impact of nonlinear galaxy motions and is expected to make any large-scale coherent structures more prominent. Although each of the SDSS PV galaxies also has a measured distance from the fundamental plane, we do not use these for this analysis as their large uncertainties could "smear" out any apparent large-scale structures. Our use of "redshift-space" coordinates will do the same thing, but to a much lesser extent as the typical peculiar velocities of galaxies are much less than the typical distance uncertainties (the same reasoning is used when measuring the velocity clustering from such data; Howlett 2019). From there, we evaluate

$$\begin{eqnarray}&&{N}_{\mathrm{shell}}(r)=\displaystyle \frac{{N}_{D}(r)}{{N}_{D,\mathrm{tot}}}\displaystyle \frac{{N}_{R,\mathrm{tot}}}{{N}_{R}(r)}-1\end{eqnarray} \tag{ 1 }$$

in radial bins of width ${\rm{\Delta }}r\,=\,5\,{{\rm{h}}}_{75}^{-1}\mathrm{Mpc}$ centered on some location. N_D (r) is the number of galaxies in each radial bin centered on r, while N_R is the number of random, unclustered points. The subscript "tot" denotes the total number of galaxies in the data and random catalogs, N_D,tot = 34,059 and N_R,tot = 4 × 10⁶, respectively. N_shell is hence normalized, such that for a completely homogeneous distribution of galaxies we expect N_shell(r) ≡ 0. This normalization is verified by applying the same procedure to our simulations—〈N_shell(r)〉, computed from 256 histograms of N_shell using the same central location and then averaged in each bin, is consistent with zero. This test indicates that the selection function (for instance, when approaching the edge of the available survey data) is correctly being mitigated by our method. The standard deviation in the 256 measurements in each bin is used as our uncertainty in the real data.

The strength and properties of any features in N_shell(r) will clearly depend on the assumed origin, so we explore a range of different possible central locations. Based on our initial finding of Ho'oleilana in the 2D distribution of data seen in Figure 1, we choose a grid of 4225 nearby possible coordinates for the presumed center. In supergalactic X, Y, and Z coordinates these span from $[-56,208,26]\,{{\rm{h}}}_{75}^{-1}\,\mathrm{Mpc}$ to $[8,272,90]\,{{\rm{h}}}_{75}^{-1}\,\mathrm{Mpc}$ in bins of width $[5,2.5,5]\,{{\rm{h}}}_{75}^{-1}\,\mathrm{Mpc}$, respectively. ⁷

Figure 5 shows the normalized radial distribution of groups at the central location where the supposed BAO "bump" is most prominent, [SGX, SGY, SGZ] = [–24, 242, 58] $\,{h}_{75}^{-1}\,$ Mpc. This same figure shows our best-fitting model of the feature that is used to define the detection significance as elaborated on below.

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2.1.1. BAO Model

The detection significance is derived from fitting a model with and without a BAO feature to N_shell and comparing the difference in best-fitting χ² between the two. We generate our model based on the work of Eisenstein et al. (1998, 2007) and Slepian & Eisenstein (2016), modeling the histogram of N_shell as a mass profile in configuration space generated from Fourier/Hankel transforming a smooth, "no-wiggle" transfer function describing the central overdensity (T_nw(k)) and a BAO "wiggle"-only transfer function with nonlinear damping (T_w (k)). We model this damping as an additional Gaussian smoothing with a free parameter Σ_nl to account for evolution in the positions of galaxies since the time the BAO was frozen-in and z ≈ 0.07.

Our model also needs to include flexibility in the radial distance of the feature from the central location, its amplitude (or, equivalently, linear galaxy bias), and the possibility that the entire region we consider is over- or underdense compared to the full SDSS PV data. We do this by including three free parameters, α, B, and N₀ for these three characteristics. Finally, in our tests we also found that the relative amplitudes of Ho'oleilana and the central overdensity could not be well described with a single amplitude parameter—the measured values of N_shell for small r and near the peak of Ho'oleilana itself are larger than would be predicted by just our linear theory model leading to a poor fit. We attribute this mainly to nonlinear clustering and galaxy bias and to account for this effect we included an additional linear term N₁ on top of the constant N₀, slightly increasing the order of the polynomial used to marginalize over the overall shape of the mass profile. This same procedure is used in standard BAO fitting, where a cubic or quartic polynomial is typically used to marginalize over the shape of the correlation function rather than attempting to model nonlinear effects, which ensures the constraints on the BAO peak position itself are insensitive to the broadband shape of the clustering (e.g., Ross et al. 2015; Hinton et al. 2020; Alam et al. 2021).

Overall, our BAO model hence contains five free parameters and can be summarized as

$$\begin{eqnarray}&&{N}_{\mathrm{shell}}^{\mathrm{model}}(r)={{Br}}^{2}{\int }_{0}^{\infty }\displaystyle \frac{{k}^{2}{dk}}{2{\pi }^{2}}T(k){j}_{0}(\alpha {kr})+{N}_{1}r+{N}_{0},\end{eqnarray} \tag{ 2 }$$

where

$$\begin{eqnarray}&&T(k)={T}_{w}(k){e}^{-1/2{k}^{2}{{\rm{\Sigma }}}_{\mathrm{nl}}^{2}}+{T}_{\mathrm{nw}}(k)\end{eqnarray} \tag{ 3 }$$

and j₀(x) is the zeroth-order spherical Bessel function. The model in the absence of BAOs can be recognized from the above expressions as the case where we set T(k) = T_nw(k) or, equivalently, Σ_nl → ∞ and fix α = 1. The model without BAO hence has three free parameters and should only reproduce the central overdensity.

Modeling the "wiggle" and "no-wiggle" transfer functions is not quite trivial. First, one is required to assume a "template" cosmological model to create a BAO feature that can then be dilated by the value of α. To allow for comparison/combination later with the Planck Collaboration et al. (2020) constraints on the sound horizon, we adopt a template cosmology close to the Planck Collaboration et al. (2020) best fit. Parameter specifications are given in Table 1.

Table 1. A Summary of the Cosmological Parameters Used in Creating Our BAO Model Template, and for Converting Our Constraints on the BAO Model Parameter α to Cosmological Constraints

Parameter	Template	No-baryon Template
Ωb h2	0.0224	0.001
Ωcdm h2	0.1199	0.1412
H0(km s−1 Mpc−1)	67.51	67.51
ns	0.9653	0.9653
As	2.25 × 10−9	2.25 × 10−9
Neff	3.046	3.046
∑Mν (eV)	0.06	0.06

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Note that this cosmology is not the same as that used to convert our catalog redshifts to distance (nor does it have to be), but it is important that the template cosmology be the one used to compute any cosmological constraints from our fit to α. Second, the presence of baryons also introduces Silk damping effects in the transfer function, suppressing it on small scales, as well as adding the BAOs. One cannot simply then take a numerical transfer function evaluated for a cosmology with and without baryons and difference the two to extract the wiggles. Eisenstein et al. (1998) provide fitting formulae for the smooth transfer function T_EH(k) that could be used, although the approximations used therein for the sound horizon and equality scales are considered not quite accurate enough for modern BAO analyses (Anderson et al. 2014). Instead, we take a hybrid approach and compute both numerical and Eisenstein et al. (1998) smooth transfer functions for our fiducial cosmology and a second "no-baryon" cosmology (where Ω_b is reduced by a factor of ∼20 as also shown in Table 1). We then compute the "no-wiggle" and "wiggle" transfer functions as

$$\begin{eqnarray}&&{T}_{\mathrm{nw}}(k)={T}_{\mathrm{EH}}(k)\displaystyle \frac{{T}_{\mathrm{no}\,\mathrm{baryon}}(k)}{{T}_{\mathrm{EH},\mathrm{no}\,\mathrm{baryon}}(k)},\quad {T}_{w}(k)=T(k)-{T}_{\mathrm{nw}}(k),\end{eqnarray} \tag{ 4 }$$

which effectively corrects the Eisenstein et al. (1998) smooth transfer function T_EH(k) so that its form is more representative of the broadband shape of the numerical transfer function before subtracting the two.

We perform our fits with both BAO and no-BAO models restricting to scales $40\,{{\rm{h}}}_{75}^{-1}\,{\rm{Mpc}}\lt r\lt 250\,{{\rm{h}}}_{75}^{-1}\,{\rm{Mpc}}$, avoiding nonlinearities at the core of the central overdensity. We perform a full Markov Chain Monte Carlo (MCMC) fit using the dynesty sampler (Speagle 2020) at each proposed center from Section 2.1 and obtain the best fit and errors on the five model parameters.

2.1.2. Results

From the χ² difference between our two models, we conclude that Ho'oleilana, centered at the aforementioned coordinates, is detected at greater than 6σ significance compared to the expectations of a random field of galaxies, and there is excellent agreement between the data and physical BAO model. Of all the possible centers we tested, 716/4225 and 127/4225 result in greater than 4σ and 5σ detections, respectively. We caution, however, that our choice of centers was conditioned on our visual identification of a feature in the data and so the relative fraction of 4σ and 5σ centers should not be interpreted in the usual way chance events are interpreted.

Of the five free parameters we fit for, B, N₀, and N₁ are nuisance parameters and so are not commented on further in this work. For the others we perform an average of the posteriors at each proposed center with greater than 3.25σ significance (of which there are 1661). This threshold was chosen as it is close to the turnover point in a histogram of the BAO strengths across all the centers we tested. Combining fits in this way allows us to propagate the uncertainty in the central location into our constraints on the radius and damping within the BAO model. From just the fit using our most likely center for Ho'oleilana (see Figure 5), we find α = 0.87 ± 0.01, while averaging over all centers above our threshold gives $\alpha ={0.88}_{-0.09}^{+0.06}$, indicating that a substantial portion of our error budget comes from the uncertainty in the central location.

Our combined chain also gives ${{\rm{\Sigma }}}_{\mathrm{nl}}={12.8}_{-5.8}^{+0.8}\,{{\rm{h}}}_{75}^{-1}\,\mathrm{Mpc}$. Under the condition that it is a remnant of the BAO, our constraint on Σ_nl implies Ho'oleilana has undergone additional Gaussian smoothing due to the bulk motions of galaxies in the intervening years between z ≃ 1060 (when the BAO feature was first frozen-in) and the observed z ≈ 0.07, with a standard deviation of $\sim 13\,{{\rm{h}}}_{75}^{-1}\,\mathrm{Mpc}$. This value of Σ_nl can be compared to the expected dispersion from linear theory. Given we are considering only relative separations from a fixed central point, rather than pairwise separations between objects, we expect this value to be comparable to the dispersion predicted to arise from the linear-scale velocities of galaxies. At z = 0 for a ΛCDM cosmological model assuming General Relativity, this is given by

$$\begin{eqnarray}&&\displaystyle \frac{{\sigma }_{v}}{{H}_{0}}=\displaystyle \frac{{{\rm{\Omega }}}_{m}^{0.55}}{\sqrt{6}\pi }{\left({\int }_{0}^{\infty }{P}_{\mathrm{lin}}(k){dk}\right)}^{1/2}\approx 4.2\,{{\rm{h}}}_{75}^{-1}\,\mathrm{Mpc},\end{eqnarray} \tag{ 5 }$$

where the latter approximation is obtained using our template cosmology. ⁸ One would expect nonlinear evolution and redshift-space distortions to increase this value somewhat. We consider our measurement consistent with our expectations of the typical distances traversed by individual galaxies from their formation to the present day. Our recovered value for α can also be further interpreted, but this is done in Section 3.

2.2. Prevalence in Simulations

To quantify the true significance of Ho'oleilana without conditioning on our visual identification we make use of our simulations with and without BAOs. The key questions are (1) whether Ho'oleilana could have been identified without an a priori knowledge of such a structure in our nearby Universe and (2) to what extent would we find similar features in simulations with large-scale structure, but where we know there are no BAOs.

To answer these questions, we turn to a variant of the algorithm developed in Arnalte-Mur et al. (2012), implemented in the publicly available BAO "centerfinder" code of Brown et al. (2021). Our algorithm first uses a set of data and the random catalog to compute the overdensity field on a grid of cell size $5\,{{\rm{h}}}_{75}^{-1}\,\mathrm{Mpc}$. It then convolves this field with a "BAO wavelet," a normalized, spherically symmetric function that has a shape expected of a BAO feature. The wavelet has two free parameters, r_BAO and s_BAO, which set the radius and width of the BAO feature. The result is, for each cell in the gridded overdensity field, a weight W_r,s that describes its likelihood as the center of a BAO-like feature with the given radius and width.

In order to determine whether or not a given field is evidence of the presence of BAOs, Arnalte-Mur et al. (2012) proposed to average the values of W_r,s over a subset of likely central locations (in their case halos, galaxy groups, or galaxies likely to be found in the centers of large clusters), to find ${B}_{r,s}={\sum }_{{N}_{\mathrm{subset}}}{W}_{r,s}/{N}_{\mathrm{subset}}$, claiming a positive value for this coefficient indicates a positive detection of BAOs. We do the same, computing W_r,s for each cell on our grid that contains at least one galaxy. We then construct B_r,s by summing over all these cells (weighted by the number of galaxies in that cell).

Figure 6 shows the results of this procedure for a wide variety of wavelet radii and widths. There is evidence that, on average, positive values of B_r,s can be found preferentially in the BAO simulations at a scale around ${r}_{\mathrm{BAO}}=135\,{{\rm{h}}}_{75}^{-1}\,\mathrm{Mpc}$. This corresponds well with the expected BAO radius given the cosmology used to generate the simulations ($132.4\,{{\rm{h}}}_{75}^{-1}\,\mathrm{Mpc}$) and indicates that the algorithm can be used to identify BAOs and the main contributors to the BAOs. However, it is worth highlighting that the significance is relatively weak—in an ideal scenario one would use a highly complete sample of galaxies to compute W_r,s before summing this quantity only over objects close to the expected BAO centers. However, we suspect that our use of the SDSS PV sample (consisting only of ellipticals likely to be found in dense clusters) weakens the fluctuations in W_r,s for larger radii and hence also weakens the signal in B_r,s .

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The bottom panel in Figure 6 shows the same algorithm applied to the data. There is an excess of B_r,s at somewhat larger radii around ${r}_{\mathrm{BAO}}=160\,{{\rm{h}}}_{75}^{-1}\,\mathrm{Mpc}$, although this result is also only weakly significant—it is possible to find similar values of B_r,s at these radii and widths even within simulations without BAOs. In this same panel we identify with a green cross the values ${r}_{\mathrm{BAO}}=155\,{{\rm{h}}}_{75}^{-1}\,\mathrm{Mpc}$ and ${s}_{\mathrm{BAO}}=27\,{{\rm{h}}}_{75}^{-1}\,\mathrm{Mpc}$, which result in the largest single value for W_r,s for all BAO wavelets and grid cells we consider. This wavelet also returns a positive value of B_r,s —which is not guaranteed, given B_r,s is the mean across all ∼23,000 cells in our overdensity grid.

Figure 7 shows a histogram of all the values of W_r,s we return (i.e., the value for each grid cell in our catalog containing at least one galaxy), for this choice of wavelet, for both the data and no-BAO simulations. There is a significant excess of grid cells with large weights. The most prominent excess corresponds closely to the center of Ho'oleilana identified in Section 2.1.1. Our conclusion is that, using the independent algorithm of Arnalte-Mur et al. (2012) and without conditioning on our a priori visual inspection, there is evidence of BAO-like features in the SDSS PV catalog, and that the strongest contributor to these features is Ho'oleilana.

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To answer the second of our proposed questions, we similarly identify the largest single BAO-like feature (the grid cell with the largest W_r,s ) within our 256 no-BAO simulations. We then generate radial profiles N_shell(r) centered at each of these 256 grid centers and fit the apparent BAO feature using our model from Section 2.1.1. The aim is to compare the strength of these BAO-like features (in simulations that we know have heavily suppressed primordial BAO) to that inferred for Ho'oleilana.

Figure 8 shows this comparison, as histograms of the χ² difference between the BAO model and a straight line fit for each of the no-BAO mocks, in the first instance, allowing for freedom in the values of r_BAO and s_BAO and, then, restricting to the radius and scale ${r}_{\mathrm{BAO}}=155\,{{\rm{h}}}_{75}^{-1}\,\mathrm{Mpc}$ and ${s}_{\mathrm{BAO}}=27\,{{\rm{h}}}_{75}^{-1}\,\mathrm{Mpc}$ identified previously. Of the 256 simulations we test, two of them have features of any radius or width that are more significant than Ho'oleilana. We can hence conclude that the probability of Ho'oleilana being a chance alignment, and not associated with the primordial BAO, is <1%. None of the simulations we test have a feature with the same radius and width as Ho'oleilana that is as significant. These tests provide reasonably strong evidence that Ho'oleilana is not a chance occurrence, although the possibility cannot be ruled out completely.

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Under the assumption that Ho'oleilana is a significant contributor to the true, full BAO signal, and that its properties are representative of the translationally averaged BAO, we can use its radius to extract cosmological information. The parameter α fit in Section 2.1.1 is proportional to the ratio between the size and distance to the BAO, and so encodes this information. From Eisenstein et al. (2005)

$$\begin{eqnarray}&&\alpha =\displaystyle \frac{{D}_{v}}{{r}_{\mathrm{drag}}}\displaystyle \frac{{r}_{\mathrm{drag}}^{\mathrm{fid}}}{{D}_{v}^{\mathrm{fid}}},\end{eqnarray} \tag{ 6 }$$

where D_v is a measure of the distance to the center of the BAO, and r_drag is its radius, or more formally, the size of the sound horizon at the baryon-drag epoch after the photons and baryons have decoupled and the baryon inertia has subsided. The superscript "fid" denotes these quantities in our fiducial cosmology. D_v is the "volume averaged" distance, which is a hybrid of two parts angular diameter distance D_A and one part Hubble parameter H(z) arising from the spherical symmetry of the BAO,

$$\begin{eqnarray}&&\displaystyle \frac{{D}_{v}(z)}{{r}_{\mathrm{drag}}}={\left(\displaystyle \frac{{cz}}{H(z){r}_{\mathrm{drag}}}\right)}^{1/3}{\left(\displaystyle \frac{1+z}{2{\sin }^{-1}({r}_{\mathrm{drag}}/2{D}_{A}(z))}\right)}^{2/3}.\end{eqnarray} \tag{ 7 }$$

This expression differs slightly from the conventional one (Eisenstein et al. 2005) as we have avoided using the small-angle approximation given the low redshift of Ho'oleilana.

To test the validity of our approach, we first perform a similar analysis on our simulated catalogs with BAOs. As was done for the no-BAO simulations, we identify the highest weighted central locations across the same range of r_BAO and s_BAO used previously in each of our 256 simulations (i.e., in each cell of Figure 6). We then measure N_shell for each center and fit BAO and no-BAO models. Figure 9 shows a composite of a random subset of our detected features in simulations with B_r,s > 0, scaled by the best-fit value of α to make the BAOs more prominent. We also plot the average best-fit BAO model and expected BAO radius given the input cosmology of the simulation.

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In addition, in Figure 9 we also show histogram of these BAO fits, again restricting to mocks with B_r,s > 0, but also normalizing by the number of s_BAO bins at each r_BAO. ⁹ The average α value is close to 1, indicating no bias in our method for identifying and fitting the individual BAO contributors. However, the uncertainty from the mocks both using only detections within a single mock realization, or compared across realizations, is slightly larger than that found from our fit to Ho'oleilana. This may simply be a result of the fact that Ho'oleilana is an exceptionally strong feature, even within mocks containing BAOs. Or it may indicate an additional contribution that should be included in our error budget from sample variance. We hence elect to provide constraints and draw conclusions using both our fitted error on α from Ho'oleilana ($\alpha ={0.88}_{-0.09}^{+0.06}$) and using the uncertainty derived from the scatter in the mocks (α = 0.88 ± 0.14) but fixed to our most likely center for Ho'oleilana.

Turning back to the data, we convert our values of α to a distance ratio D_v /r_drag using our fiducial cosmology. A substantial fraction of our uncertainty on α comes from marginalizing over the possible centers—similarly we then have a range of possible fiducial distances to the center of Ho'oleilana, which translates to uncertain values for our redshift and fiducial distance ratio of $z\,=\,{0.068}_{-0.007}^{+0.003}$ and ${D}_{v}^{\mathrm{fid}}/{r}_{\mathrm{drag}}^{\mathrm{fid}}={1.99}_{-0.20}^{+0.07}$, respectively. Propagating this constraint properly alongside the constraint on α we recover ${D}_{v}/{r}_{\mathrm{drag}}={1.63}_{-0.08}^{+0.07}$. ¹⁰ In the case where we fix the central location of Ho'oleilana but use the mock scatter as our error on α, we have a fixed ${D}_{v}^{\mathrm{fid}}/{r}_{\mathrm{drag}}^{\mathrm{fid}}=1.88$ and find D_v /r_drag = 1.66 ± 0.26.

Finally, we adopt a prior on the sound horizon r_drag = 147.13 ± 0.26 Mpc from Planck Collaboration et al. (2020) and fit our combined posterior for Ω_m and H₀. Although we have effectively allowed the redshift to vary in incorporating all our proposed centers, the general low redshift of Ho'oleilana makes it an almost pure probe of H₀, with little constraining power on Ω_m . This is reflected in our final constraints, shown in Figure 10, where Ω_m is unconstrained by our methodology. We find ${H}_{0}={76.9}_{-4.8}^{+8.2}$ km s⁻¹ Mpc⁻¹ and ${H}_{0}={74.7}_{-9.7}^{+12.4}$ km s⁻¹ Mpc⁻¹ using the uncertainties of α from Ho'oleilana only and from the distribution of our simulations, respectively. For the former, the majority of the uncertainty on H₀ comes from the expected variance in N_shell as measured from the mocks (reflected in the error bars in Figure 5) and our uncertainty in the central location/redshift. Our constraints on the expansion rate of the Universe and comparisons to other measurements, including statistical BAOs, are shown in Figures 10 and 11. Being a single feature rather than a statistical average, the errors bars from Ho'oleilana are larger than those from other large-scale structure surveys; however, they are still constraining enough to provide a preference for larger expansion rates.

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Given the presence of Ho'oleilana, an interesting question is whether the clustering of the full SDSS PV sample contains any evidence of statistical BAOs. The power spectrum measurements presented in Figure 4 were analyzed using the BAO fitting code Barry (Hinton et al. 2020) taking into account the survey window function; however, unfortunately, we found no significant statistical BAO detection. Nonetheless, statistical BAOs have been detected in both the 6dF and SDSS Main galaxy surveys (Beutler et al. 2011; Ross et al. 2015), which cover an effective volume only a few times larger than the catalog we analyze here. The latter of these also covers the redshift range 0.07 < z < 0.20 and so partially overlaps with our sample. It is worth noting that the SDSS results of Ross et al. (2015) relied on BAO reconstruction to enable their detection. As such, further investigation of the SDSS PV sample, using both the correlation function and BAO reconstruction, is warranted.

Although Ho'oleilana was identified as a 2D feature, a significant component of the signal comes from the foreground part of the 3D shell. (The backside far edge falls slightly beyond the z = 0.1 limit of our sample.) Remarkably, the Coma Cluster, the Center for Astrophysics Great Wall (de Lapparent et al. 1986), and structure coursing up to the Hercules supercluster (Einasto et al. 2001; Shapley 1934) lie along the foreground surface of the posited BAO phenomenon. The famous Boötes Void (Kirshner et al. 1981) lies within the embrace of Ho'oleilana. Near the center of Ho'oleilana is the Boötes supercluster (Einasto et al. 2001), presumed to be the manifestation of the matter concentration that gave birth to the BAO (Weinberg et al. 2013). In detail, the domain of the central supercluster about the dominant A1795 is diffused over $\sim 50{h}_{75}^{-1}$ Mpc around the geometric center of the BAO shell.

The extent of Ho'oleilana is revealed in an accompanying video in the animated Figure 12. The cosmography of Ho'oleilana is further explored in the interactive Figure 13. The Cosmicflows-4 galaxy groups that lie within the Ho'oleilana shell are colored red in the interactive Figure 14. In each of these displays, the galaxy groups are located in 3D by their systemic velocities, i.e., in redshift space.

The significance of the detection of Ho'oleilana, its shape, its relation to other previously known structures in the local Universe, and the prominence of the feature compared to the expectations of both a random field of galaxies and simulations with large-scale structure but suppressed BAO, strongly suggest Ho'oleilana is itself a part of the BAO feature rather than a chance alignment.

Marginalizing over the uncertainty in the central position, width, and amplitude, we are able to extract a measurement of the ratio of the distance to the center of Ho'oleilana relative to its size predicted by early Universe physics, ${D}_{v}={1.63}_{-0.08}^{+0.07}\,{r}_{\mathrm{drag}}$. Fixing to the most likely central location, but using scatter in BAO simulations to infer the error, we find D_v = 1.66 ± 0.26 r_drag. Given the low redshift of the feature and adopting a value of the BAO radius r_drag expected from a Planck Collaboration et al. (2020) cosmological model, these distances can be almost directly converted to constraints on the Hubble constant, ${H}_{0}={76.9}_{-4.8}^{+8.2}$ km s⁻¹ Mpc⁻¹ and ${H}_{0}={74.7}_{-9.7}^{+12.4}$ km s⁻¹ Mpc⁻¹, respectively. These values are more consistent with what is found from other direct local Universe probes (H₀ = 69.8 ± 0.8 ± 1.7 km s⁻¹ Mpc⁻¹ (Freedman et al. 2020); H₀ = 73.1 ± 1.0 km s⁻¹ Mpc⁻¹ (Riess et al. 2022); 74.6 ± 3.0 km s⁻¹ Mpc⁻¹ (Tully et al. 2023)) rather than the value of H₀ = 67.4 ± 0.5 km s⁻¹ Mpc⁻¹ inferred from propagating the early Universe constraints (Planck Collaboration et al. 2020). By implication, if Ho'oleilana is representative of the statistical population of BAOs, additional late-time physics may be required to increase the expansion rate of the Universe toward the present day. Future deeper data, such as that from the Dark Energy Spectroscopic Instrument (DESI Collaboration et al. 2023) or the 4MOST Hemisphere Survey (Taylor et al. 2023) may allow for further validation of Ho'oleilana or the detection of similar structures elsewhere in the nearby Universe.

We give thanks to collaborators in the assembly of the Cosmicflows-4 catalog of galaxy distances and velocities, with special thanks to Hélène Courtois, Ehsan Kourkchi, and Khaled Said. A perceptive referee challenged us to justify our quantitative uncertainties. This paper was completed while R.B.T. attended and benefited from discussions with Nick Kaiser (sadly, Nick Kaiser passed away on 2023 June 13) and others at the workshop The Cosmic Web: Connecting Galaxies to Cosmology at High and Low Redshift at the Kavli Institute of Theoretical Physics, University of California, Santa Barbara. Funding for the Cosmicflows project has been provided by the US National Science Foundation grant AST09-08846, the National Aeronautics and Space Administration grant NNX12AE70G, and multiple awards to support observations with the Hubble Space Telescope through the Space Telescope Science Institute. CH acknowledges support from the Australian Government through the Australian Research Council's Laureate Fellowship and Discovery Project funding schemes (projects FL180100168 and DP220101395).