# Flat Patterns in Cosmic Structure

P. J. E. Peebles1
1Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544, USA
E-mail<EMAIL_ADDRESS>
###### Abstract

It is natural to wonder how far the flat pattern in the distribution of
galaxies and clusters of galaxies around the de Vaucoueurs Local Supercluster
extends, and whether there are other similarly extended flat patterns in the
large-scale structure of the universe. I present evidence of two extended flat
and thin sheet-like patterns in the distributions of galaxies and clusters
detected at redshift $z<0.021$. Sheet A contains our position and is tilted
$11^{\circ}$ from the supergalactic pole, meaning the Local Supercluster is a
moderately bent part of the more extended Sheet A. The continuation of this
sheet is detected in the disjoint sample of galaxies at redshifts
$0.021<z<0.041$ and again in the disjoint samples of galaxies and clusters of
galaxies at $0.042<z<0.085$. Sheet B is 15 Mpc from us at its closest point.
It is detected at $z<0.021$ and at $0.021<z<0.041$. These results make a
serious case for the reality of signatures of close to flat and thin extended
sheet-like patterns in cosmic structure, and an interesting challenge for the
$\Lambda$CDM cosmology.

###### keywords:

cosmology — large-scale structure of Universe — galaxies: clusters: general

††pubyear: 2023††pagerange: Flat Patterns in Cosmic Structure–References

##  1 Introduction

Vera Cooper Rubin (1951) and Gérard de Vaucouleurs (1953, 1958) pointed out
that the nearby galaxies tend to be concentrated in a flat and maybe rotating
system. I am not aware of detection of rotation, but the flattened
distribution is clearly seen and variously known as the Virgo Supercluster,
the Local Sheet (Tully, Shaya, Karachentsev, et al. 2008; McCall 2014) or,
following de Vaucouleurs, the Local Supercluster. Two questions naturally
arise. What is the detectible extent of this flat pattern with greater than
average number density? And are there other similarly extended and close to
flat patterns in cosmic structure? Important steps to addressing the first
question include the demonstrations by Zel’dovich, Einasto, and Shandarin
(1982, Fig. 5), Einasto, Corwin, Huchra, Miller, Tarenghi (1983, Fig. 1), and
Tully (1986, 1987) that clusters of galaxies tend to be close to the plane of
the Local Supercluster in a region several hundred megaparsecs across.
Einasto, Tago, Jaaniste, Einasto, and Andernach (1997) remark that “One
example of [sheets or planes] is the Supergalactic Plane, which contains the
Local Supercluster, the Coma Supercluster, the Pisces-Cetus and the Shapley
superclusters.” Of equal importance is the evidence presented by Shaver (1990,
1991) and Shaver and Pierre (1989) that radio galaxies at distances $z\lower
2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}0.02$ also tend to be at low
supergalactic latitudes. Observations along these lines are not widely
discussed but they are informative and potentially important aspects of the
large-scale nature of the universe. An analog on a smaller scale, the “plane
of satellites problem,” is more widely discussed. Considerations along this
line trace back at least to Lynden-Bell (1976), who pointed out interesting
alignments of high-velocity HI clouds and dwarf satellites of the Milky Way.
The length scales differ but the philosophy is the same: Ask what careful
examination of the world around us might reveal that is new and maybe
informative about the nature and origin of the universe.

The examination of distributions of galaxies and clusters of galaxies
presented here yields a case for detection of two flat and thin regions of
greater than average mean density, identified as exceptionally large counts of
galaxies in disk-like cells 5 Mpc thick and 150 Mpc across. One sheet is close
enough to the direction and position of the de Vaucouleurs Local Supercluster
to justify naming it the Extended Local Supercluster. The sheet is detected in
three disjoint saples of redshift, the furthest at redshift-based distance 180
to 360 Mpc. It undoubtedly is real by the test of reproducibility of detection
in separate ranges of redshift, and it is an interesting challenge to the
standard cosmology.

The results presented here have in common the large length scales associated
with great walls of galaxies (Huchra and Geller, 1982; Gott, Jurić, Schlegel,
et al. 2005; Lietzen, Tempel, Liivamägi et al. 2016; and references therein)
and Large Quasar Groups (Clowes, Harris, Raghunathan, et al. 2013; Friday,
Clowes, Williger 2022, and references therein). The cosmic web is said to
contain sheets of objects, but the sheets typically are curved. The sheets to
be considered here are modeled after the Local Supercluster: thin, extended,
and quite close to flat. And the method of detection is meant to be applied
equally well to observations and to numerical simulations of cosmic structure.

The redshifts of objects in this study are not large, $z\lower
2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}0.1$, so I ignore space curvature
and other relativistic effects and use the low distance conversion of redshift
$z$ to distance $r=cz/H_{\rm o}$, with the Hubble parameter

$H_{\rm o}=70\hbox{ km s}^{-1}\hbox{ Mpc}^{-1}.$ (1)

This is not always a safe approximation for nearby galaxies, but in the
central part of the de Vaucouleurs Local Supercluster, at distances $\lower
2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}10$ Mpc, many of the galaxies have
measured or sensible inferences of distances. For objects that are further
away distances based on redshifts seem adequate.

The probe or filter used in this study for the discovery of extended flat
concentrations of galaxies is explained in Section 2. A test by its
application to the distribution of galaxies in the Nearby Universe, at
distances less than about 10 Mpc, is reported in Section 3. The probe shows
that the distribution of galaxies in the Local Supercluster is quite flat, as
we already new. Section 4 presents results of applications of the probe to the
distributions of galaxies and clusters of galaxies at distances less than
redshift $z_{\rm max}=0.021$, or about 90 Mpc, a reasonable outer bound of the
Local Universe. The application yields evidence of two similarly thin and flat
sheet-like patterns in the spatial distributions of galaxies and clusters.
Section 4.3 presents results of exploration of the natures and contents of
these two candidates for true flat sheets. The results of applications of the
probe at greater distances, which reveal evidence of continuations of the two
sheets, are presented in Section 5. Concluding remarks are in Section 6.

Figure 1: Objects are counted in layers of disk cells with radius $r_{\perp}$
and thickness $\Delta h$. The axis passes through the disk centers normal to
the disks. The stack of these disks define a truncated cylinder. The positions
$h_{j}$ of the disk midplane are measured from the origin, our position.

##  2 A Probe for Planar Patterns

The probe, or filter, used to detect close to flat patterns in the
distributions of objects is illustrated in Figure 1. Objects are counted in
stacks of cells, each cell a disk with radius $r_{\perp}$ and thickness
$\Delta h$. A stacked array of $2N+1$ adjacent disks centered on and normal to
an axis passing through the origin defines a cylinder truncated by planes
normal to the axis. The distance measured along the axis from the origin to
the centerplane of the $j^{\rm th}$ disk is $h_{j}=j\times\Delta h$, with
$j=0,\pm 1,\pm 2$, to $\pm N$. The origin is at the centerplane of the $j=0$
cell. The two ends of the cylinder, at distance from the origin

$h_{\rm max}=(N+1/2)\Delta h,$ (2)

fit in a sphere of radius $R$. At maximum volume of the cylinder given the
radius of the sphere the dimensions are

$R=\sqrt{3}h_{\rm max},\quad r_{\perp}=\sqrt{2}h_{\rm max}.$ (3)

The applications presented here use $N=10$ for 21 disk cells. The ratio of the
disk cell diameter to the disk thickness is then

$2r_{\perp}/\Delta h=2^{3/2}(N+1/2)=29.7.$ (4)

The large aspect ratio, 30:1, means the filter is sensitive to quite flat and
thin patterns in cosmic structure.

The intuition that this arrangement might be an interesting probe is drawn
from the evidence that the Local Supercluster is quite flat and that radio
galaxies and clusters of galaxies tend to be close to the extended plane of
the Local Supercluster (e.g. Einasto et al. 1983; Tully 1987; Shaver 1990). As
mentioned above this probe, or filter, is not sensitive to curved sheets of
galaxies in great walls (e.g. Lietzen et al. 2016 and references therein), or
to the configurations of Large Quasar Groups (e.g. Clowes, Harris,
Raghunathan, et al. 2013 and references therein), unless these configurations
are flat.

The advantages of this filter are that the judgement of significance of
detection is simplified by the rigidity of the definition, and that it would
be easy to compare results of application of the filter to observations and to
numerical simulations of cosmic structure in a theory such as $\Lambda$CDM. It
must be noted, however, that despite the claim of rigidity I do not hesitate
to postulate “bends” that offer intuitively reasonable continuations of sheets
detected in disjoint ranges of redshift.

A convenient disc cell thickness for the search for flat patterns in the Local
Universe is $\Delta h=5$ Mpc. With $N=10$ the separation of midplanes of the
disks on each end of the cylinder is 100 Mpc, and the full length of the
cylinder is $2h_{\rm max}=105$ Mpc. The cylinder has radius $r_{\perp}=74.2$
Mpc, which fits in a sphere of radius $90.9$ Mpc. This radius, with the Hubble
parameter in equation (1) is redshift

$z_{\rm max}=0.0212$ (5)

or $z_{\rm max}=0.021$ for short. This bound is a convenient limiting redshift
of the Local Universe. The ratio of disk cell diameter to thickness, 30:1, is
about right for the search for patterns as flat as the Local Supercluster.

At given orientation of the axis of this cylindrical arrangement, objects are
sampled only inside the cylinder, not all that are in the sphere. Different
objects in the sphere enter the counts in disks in cylinders with different
orientations. Data would be more completely sampled if the disks closer to the
origin had larger radii, but trials indicate that taking account of the
different disk areas is not a useful complication. More important for this
exploratory survey is the uniformity of sampling by the disk cells.

The search for close to plane patterns in the second redshift bin takes
account of the smaller number densities of objects in the catalogs used here
by doubling dimensions to redshift bounds $0.021<z<0.042$, disk cell thickness
$\Delta h=10$ Mpc., and cylinder radius $r_{\perp}=149$ Mpc. This preserves
the ratio 30:1 of cell diameter to thickness. We need disjoint samples and
uniform sampling of the space covered by the disk cells. This is done by
replacing the solid cylinder used for the Local Universe by a hollow cylinder,
a straight tube. The inner radius of the tube is $r_{\perp}^{\rm min}=R/2$,
which removes the data from the closer redshift bin. It follows from equation
(3) that the inner radius of the tube is

$r_{\perp}^{\rm inner}=\sqrt{3/8}r_{\perp}.$ (6)

The inner tube radius is 0.6124 times the outer tube radius.111In the first
draft circulated on the arXiv I erroneously took the inner radius of the tube
to be half the outer radius, not 0.61 times the outer radius. This allowed
inclusion of a small slice of data from the next nearer redshift bin.
Correcting this error changed all figures in the second and third redshift
bins in the first draft. The changes are small, mostly difficult to see,
except in the fourth redshift bin, where the sample becomes too small for
interesting analysis. This means the volume sampled in the second redshift
sample is five eights of the volume of the solid cylinder.

The cell used for the third redshift bin, $0.042<z<0.085$ doubles all
dimensions again.

An alternative approach uses a cylinder of solid disk cells applied to
positions of objects in the distance range $R/2$ to $R$. The problem with this
is that the disk cells at distance less than half the sphere radius have
smaller mean counts than in cells further out. Defining interestingly large
counts is complicated in this situation. Again, the uniformity of disjoint
sampling is preferable.

Candidates for planar patterns of larger than average density are discovered
as exceptionally large counts found in directions of the cylinder or tube of
disk cells chosen isotropically at random (to the extend that the random
number generator can be trusted). The count, direction, and midplane distance
$h_{j}=j\times\Delta h$ of the cell is recorded when the count in the cell
exceeds any previously found. Perhaps the cell found this way contains a flat
and thin sheet with systematically larger than average number density, but
this procedure produces candidates for flat configurations even if there are
none. The prime evidence of reality is the detection of continuations of
sheets in disjoint ranges of redshift. This is demonstrated in examples to be
presented.

The direction of the axis that produces a large count in a disk cell, maybe an
indication that the cell contains a flat sheet, is stated to the nearest
degree (with an exception to be mentioned). This is because discreteness noise
in the sample produces discrete directions at maximum cell count or close to
it, typically spread in direction over a degree or so. I have experimented
with allowing the zero-point of $h$ to be shifted from the origin, thus
broadening the search for maximum count. In trials this addition to the
parameters to be scanned yields larger counts in scattered values of the shift
and direction. The choice of the best shift requires specifying the direction
more closely than a degree, however, which I doubt is meaningful. So the zero-
point of $h$ is fixed at the origin.

By convention the axis of the cylinder of disks points to the north galactic
hemisphere. This means the directions of interestingly large counts that are
close to each other but on opposite sides of the galactic equator appear far
apart in a map of directions confined to the north hemisphere. The working
solution is presented in Section 4.1.

The mean count of objects in disk $j$ averaged over random orientations is a
measure of the mean number density at distances in the range from $h_{j}$ to
$r_{\perp}$. The convention of north-pointing axes means the mean densities at
$h_{j}$ and $-h_{j}$ usually differ. This convention makes the mean counts as
functions of $h_{j}$ a useful indication of the degree of departures from
homogeneity of a sample of objects, though one that would be difficult to
translate to density as a function of spatial position.

The application of this filter with 21 disk cells and the largest sample
considered here, $\sim 8000$ galaxies, using basic fortran 90 with no attempt
to improve efficiency of computation, is repeated a million times in six
minutes by my 5 year old desktop computer.

Figure 2: Counts of galaxies closer than 9.1 Mpc in disk cells 0.5 Mpc thick
with radius 7.4 Mpc along the same axis in each range of luminosity. The
dotted histograms show mean counts.

##  3 The Nearby Universe

The probe is tested by application to the positions of galaxies closer than
about 10 Mpc, where the concentration of galaxies to the plane of the de
Vaucouleurs Local Supercluster is familiar. Figure 2 is based on the NASA
NEARGALCAT or Nearby Universe (NU) version of the Karachentsev, Makarov, and
Kaisina (2013) Updated Nearby Galaxy Catalog.

The histograms in Figure 2 are based on counts of galaxies closer than 9.1
Mpc. At this distance the NU contains 53 galaxies more luminous than
$M_{B}<-19$ and 243 NU galaxies with absolute magnitudes $-19<M_{B}<-15$. The
catalog contains 461 galaxies with $-15<M_{B}<-11$, which I use, though I
appreciate that many more of these faint galaxies likely are not yet
cataloged.

The counts are in disk cells 0.5 Mpc thick with diameter $2r_{\perp}=14.8$
Mpc. The cells are thirty times as broad as they are thick. This is a test for
quite thin patterns.

Figure 2 shows counts as functions of distance $h_{j}=0.25\times j$ Mpc along
an axis common to the three luminosity classes. The axis was chosen to get
close to maxium counts in all three luminosity bounds. The normal to the disk
cells is directed to galactic and supergalactic coordinates

$l=48^{\circ},\ b=1^{\circ},\ SGL=277^{\circ},\ SGB=85^{\circ}.$ (7)

As explained in Section 2 the angles are given to the nearest degree; shot
noise makes more precise values meaningless. To this precision the counts in
the central disk cell, $h_{j}=0$, are close to the largest counts found by
adjusting the direction of the axis separately for each luminosity interval.

The direction in equation (7) is tilted $\sim 5^{\circ}$ from the
supergalactic pole SGB = 90∘. When the axis is directed to SGB = 90∘ the peak
count of galaxies is 8 of the most luminous ones, fewer than the 13 when the
pole is in the direction in equation (7). In the intermediate luminosity
interval the peak count is 27 at SGB = 90∘, fewer than the 34 for the tilted
pole. In the lowest luminosity interval there are 77 galaxies in the central
cell when the axis directed to SGB = 90∘, again fewer than the 93 for equation
(7). The differences are modest but systematic. By this measure the galaxy
distribution in the Nearby Universe is tilted by about five degrees from the
plane that de Vaucouleurs and colleagues established (as defined by the
NASA/IPAC Coordinate Calculator). This is impressively close considering the
far more limited data available then.

The histograms plotted as dotted lines in Figure 2 are the averages over
directions of the counts in the disk cells as a function of distance $h_{j}$
from the origin. By convention the axis points to the north galactic plane,
and we see that there are more of each luminosity class in the galactic north.
The peaks in the dotted histograms at $h_{j}=0$ might be caused in part by
greater completeness at closest samples, and maybe in part by the prominent
concentration of galaxies to low supergalactic latitudes.

The conclusion from this application is that the probe for flat thin sheet-
like patterns in cosmic structure yields a reasonable result in a known
situation.

##  4 The Local Universe

The Local Universe is taken to be the redshift range

$0<z<0.021.$ (8)

With the Hubble parameter in equation (1) this is distance less than about 91
Mpc. The upper bounds on the redshift and distance are not round numbers
because they are determined by a convenient disk cell thickness for the search
for traces of flat patterns in the distributions of objects in the Local
Universe. The cell dimensions are

$\Delta h=5\hbox{ Mpc},\ r_{\perp}=74.25\hbox{ Mpc},\
\hbox{diameter/thickness}\sim 30,$ (9)

for 21 disk cells.

###  4.1 Two Sheets of Galaxies

The search for flat concentrations of objects is simplest to interpret when
applied to samples that cover most of the sky. The main galaxy sample used
here is the 2MRS catalog of galaxy positions and redshifts from the NASA
TWOMASSRSC catalog of galaxies detected in the 2MASS survey selected by
apparent magnitude $K_{s}<11.75$ at $2.2\mu$ (Skrutskie, Cutri, Stiening, et
al. 2006), with redshifts and morphological types from Huchra, Macri, Masters,
et al. (2012). To reduce radial gradients in the number density in a sample
with redshift bound $z<z_{\rm max}$ I use the apparent magnitude bound

$K_{s}<11.75+5\log_{10}(z/z_{\rm max}).$ (10)

The condition in equation (8) that the redshift is positive removes some
Nearby Universe galaxies. Other NU galaxies are in this sample, with distances
computed from redshifts, but they are few enough that the distance errors are
not likely to influence the search for sheet-like patterns on distance scales
an order of magnitude larger.

I have not found evidence that obscuration in the plane of the Milky Way has
interfered with the searches for flat patterns. The situation likely would
different in the search for planar patterns of lower than average number
density.

Figure 3: Directions of axes in whic h the count in a cell of 2MRS galaxies at
$z<0.021$ exceeds 500, at the dots, and 600, at the filled black regions. A
position below the curved line is equivalent to a position above the line with
the distance $h$ reckoned in the opposite direction.

Figure 3 shows the result of a search across the sky and at redshifts less
than 0.021 for exceptionally large counts in cells with the dimensions in
equation (9). This is based on the 8540 2MRS galaxies at redshifts
$0<z<0.021$. An angular position of the axis below the curved line in the
figure is equivalent to a position above the line belonging to the same axis
with the disk cell distances $h$ labeled in the opposite direction. The
dividing line in the figure is $90^{\circ}$ from the de Vaucouleurs
supergalactic pole at $l=47.37^{\circ}$, $b=6.32^{\circ}$ in standard galactic
coordinates. This arrangement was motivated by the small clump of points in
the figure at negative galactic latitude and longitude similar to that of the
two prominent clumps at positive latitude. An equivalent figure that shows all
directions at positive latitudes puts the small clump at longitude about
$180^{\circ}$ away from the prominent clumps, which is undesirable.

In the figure directions of axes for which a count in a cell is 500 or more
galaxies are plotted as dots, and the solid black areas show directions in
which the count in a disk cell is 600 or more. (This is from $3\times 10^{4}$
randomly chosen directions. In larger samples the points merge, making it
difficult to see the solid black areas that are closer to the maximum counts.)

The sky survey in Figure 3 reveals two directions in which the counts of
galaxies are particularly large in one of the thin cells in the distance range
$-10\leq j\leq 10$. The fiducial normals in the directions at which the cell
counts are close to maximum, and the ranges of distance from the origin
contained by the cells with the maximum counts, are

$\displaystyle\hbox{ Sheet
A${}_{1}$:}\,l=55^{\circ},\,b=14^{\circ},\,-2.5<h<2.5\hbox{ Mpc},$
$\displaystyle\hbox{ Sheet
B${}_{1}$:}\,l=55^{\circ},\,b=46^{\circ},\,-17.5<h<-12.5\hbox{ Mpc}.$ (11)

The subscripts are meant to indicate that these are based on the data in the
first redshift sample, or bin.

The small group of dots in Figure 3 at similar longitude and negative latitude
show a third candidate sheet, but one with maximum cell counts of about 500
galaxies, while the maximum counts in the two named sheets are above 600. By
this measure Sheets A1 and B1 are uniquely prominent in the Local Universe, at
$z<0.021$. Bear in mind that these are based on counts in thin disk cells,
aspect ratio 30:1.

Figure 4: Counts of 2MRS galaxies at redshift $z<0.021$ in disk cells as
functions of distance along the axis in the direction of the normal of Sheet
A1, plotted in black, and of Sheet B1, plotted in red. The dashed histogram
shows counts averaged over orientation.

Figure 4 shows counts of 2MRS galaxies closer than $z=0.021$ in disk cells
with normals in the directions in equation (11), as functions of the distance
$h_{j}$ of the cell centerface from the origin. The histogram plotted as the
solid black line shows counts along the fiducial axis of Sheet A1, and the
histogram plotted in red shows counts along the direction of Sheet B1. The
histograms look similar, with pronounced peaks. The maximum count in the cell
that contains Sheet A1 is 605 galaxies and the counts in the two adjacent
cells, 5 Mpc away, are 357 and 319. In Sheet B1 the maximum count is 620, and
the counts in the adjacent cells are 300 and 442. It is notable that the
galaxy counts at the two maxima are in cells that are just 5 Mpc thick and
thirty times that wide, and that the counts are considerably smaller in the
neighbouring cells centered just 5 Mpc away.

The normal of Sheet A1 is $15^{\circ}$ from the normal of the Nearby Universe
in equation (7), and $11^{\circ}$ from the de Vaucouleurs supergalactic pole.
When the axis is directed to the supergalactic pole the peak count, still at
$j=0$, is 444, appreciably down from 605 for Sheet A1. This is evidence of
reality of the tilt of Sheet A1 from the supergalactic pole, by a modest
amount.

The planes of the Nearby Universe and Sheet A1 both pass through the origin,
our position, in directions that differ by just $15^{\circ}$. This means the
Local Supercluster can be taken to be a moderately bent part of the more
extended Sheet A1, a configuration that can be termed the Extended Local
Supercluster. This is not new. Einasto et al. (1983) found the alignment of
clusters with the plane of the Local Supercluster at similar distances, and
Shaver (1990) demonstrated a similar alignment of radio galaxies. But the
greater numbers of 2MRS galaxies with measured redshifts allows a more
accurate demonstration of a remarkably broad, flat, and thin region in which
galaxies tend to be distinctly more numerous than elsewhere.

When the axis of the disk cells is aligned with the normal of Sheet B1 the
maximum count (in the red histogram in Fig. 4) is at distance $j=-3$. The
negative value means that at its closest Sheet B1 is in the cell that is
centred 15 Mpc from us in the direction $l=235^{\circ}$, $b=-46^{\circ}$. This
axis is tilted $40^{\circ}$ from the de Vaucouleurs pole. The larger angle and
the significant distance to Sheet B1 make it a candidate for a different flat
sheet in which galaxies also tend to be unusually common in a broad thin
pattern.

We must pause to consider the effect of distortions of distances by proper
motions. Sheet A1 passes through our position, so distorted distances derived
from redshifts do not tend to move objects into or out of the disk cell that
contains our position. Sheet B1 is 15 Mpc away, and in the direction where the
sheet is closest random peculiar motions $\sim 300$ km s-1 would produce
distance errors $\sim 4$ Mpc. If uncorrelated they would tend to move objects
out of the 5 Mpc thick disk cell, yet the two peaks in the histograms in
Figure 4 are similarly narrow. But in the disk cell at distance $h\sim-15$ Mpc
the effect is largest only toward the center of the cell, which is much
broader than 15 Mpc. In more distant samples to be discussed the disk
thicknesses are larger and less likely to be influenced by peculiar motions. I
conclude that peculiar motions are not likely to have affected this search for
flat patterns.

Figure 5: Counts of X-ray clusters of galaxies at $z<0.021$ measured along the
direction of the cell that contains Sheet A1, plotted in black, and Sheet B1,
plotted in red. The directions of the axes are set by the 2MRS galaxy sample.
The dotted histogram shows the means, as in Fig. 4.

###  4.2 Reproducibility Checks

The evidence of thin and flat sheet-like patterns is curious enough to call
for a check of reproducibility. We have the observations that clusters are
concentrated toward the plane of the Local Supercluster (Einasto et al. 1983;
Tully 1986, 1987). An addition to this evidence is obtained from the cluster
positions and redshifts compiled in the NASA MCXC Meta-Catalog of X-Ray
Detected Clusters of Galaxies. There are not large numbers of clusters in this
catalog, just 51 at $z<0.021$, so I do not apply a bound on the X-ray flux
densities.

Figure 5 shows cluster counts in the 5 Mpc thick disk cells as functions of
distance along the two orientations in equation (11), orientations derived
from the galaxy distribution. The colour choice is the same as as in Figure
(4). The peak counts are reasonably distinct, similar, and at the same
distances $h$ as in the galaxy counts in Figure 4. It might be objected that
this adds nothing new because cluster positions are known to be tightly
correlated with galaxies. But the two thin and extended sheets are not
intuitively expected in the standard $\Lambda$CDM theory, so it is important
to have good evidence that the sheets are not statistical accidents or
systematic errors in the measurement of a spatially homogeneous random process
without true sheets. Figure 5 is an important addition to the case that thin,
flat, and broad sheets of excess counts are real features of cosmic structure.

Figure 6: Counts of PSCz galaxies at $z<0.021$ along the directions of Sheets
A1 (the black histogram) and B1 (the red histogram).

The check in Figure 6 uses the PSCz redshift catalog that Saunders,
Sutherland, Maddox, et al. (2000) drew from point-like sources in the IRAS
(infrared astronomical satellite) sky survey at wavelengths from 12 to
$100\mu$. The data used here were downloaded from the NASA HEASARC IRASPSCZ
catalog, class GALAXY. The 2332 galaxies used to obtain Figure 6 have
$60\,\mu$ flux densities bounded by

$f>0.6(z_{\rm max}/z)^{2}\hbox{ Jy},\ z_{\rm max}=0.021.$ (12)

This is a smaller sample than the 8540 2MRS galaxies, and the number density
gradients indicated by the dotted line in Figure 6 are larger than in Figure
4, but the peak counts in the directions and distances $h$ established from
the 2MRS sample are clearly seen. At greater redshifts the PSCz galaxies
reveal no evidence of Sheets A or B, maybe because the samples are not large
enough.

These three samples at $z<0.021$ were obtained in different ways: 2MRS
galaxies drawn from the 2MASS sky survey at $2\mu$ wavelength, PSCz galaxies
identified as point-like sources in the IRAS sky survey at wavelengths $\sim
60\mu$, and clusters identified by detections of X-ray emission by
intracluster plasma. The consistent signatures of flat large-scale patterns at
redshifts $z<0.021$ makes a good case for their reality.

###  4.3 Probes of conditions in Sheets A and B

Are Sheets A1 and B1 useful approximations to physically real flat thin
patterns? If so do the galaxies in sheets differ in natures and distributions
from objects outside the sheets, apart from the larger mean number densities?
Preliminary considerations are discussed here.

Figure 7: Ratios of the counts of 2MRS galaxies in disk cells at $z<0.021$ to
the mean count in that cell, separately for three morphological types, plotted
in black for Sheet A1 and red for Sheet B1.

The distributions of the most luminous of the elliptical and spiral galaxies
in the 2MRS catalog relative to the plane of the Local Supercluster are
compared in Peebles (2022). Figure 7 supplements this by comparing the
distributions of counts in cells of the far more numerous 2MRS galaxies within
the absolute magnitude limit in equation (10) at $z_{\rm max}=0.021$. The
galaxies are labeled spiral, S0, or elliptical according to the prescription
in Peebles (2022) for sorting the Huchra classification into these three
classes. At the cutoffs in redshift and luminosity there are 932 ellipticals,
2509 S0s, and 4105 spirals. The other 994 2MRS galaxies in the sample limited
only by distance and luminosity, and used in Figures 3 and 4, have
undetermined or enigmatic Huchra classifications.

The effect of the different numbers of the three galaxy types is reduced in
Figure 7 by plotting ratios of counts in a disk cell to the count in that cell
at fixed distance and averaged over $10^{6}$ random orientations of the
cylinder axis (as discussed in Sec. 2). Sheet A1 is particularly interesting
because it is a good candidate for the Extended Local Supercluster, but within
fluctuations the ratios of counts to the means as functions of morphology and
distance $h$ along the axis are similar for Sheets A1 and B1 (plotted in black
and red). By this measure the two sheets are equally interesting. In both
sheets the elliptical and S0 galaxies are more strongly concentrated to the
sheets than are the spirals. The difference is much more modest than for the
most luminous spirals and ellipticals (Peebles 2022), and it agrees with the
known tendency of the positions of ellipticals of more common luminosities to
be more strongly correlated than the positions of spirals (Davis and Geller
1976). The slight and expected greater concentrations of ellipticals and S0s
to the two sheets suggest that there is nothing special about the galaxies in
sheets, though a more detailed examination would be interesting.

Figure 8: Maps of positions of 2MRS galaxies at $z<0.021$, plotted as black
triangles, and clusters, plotted as red squares, viewed normal to the thin
disks. The cells in the top row are aligned with Sheet A1, and aligned with
Sheet B1 in the bottom row. The two centre maps show the distributions of
objects in the disks with the largest counts. The maps on each side show
positions in the disks on each side, at midplane distances $h_{j}$ from the
origin.

A related issue is whether the nature of the space distribution of galaxies in
sheets is unusual apart from the greater mean number density. Figure 8 shows
examples of the distributions of objects across the face of a disk cell 5 Mpc
thick with radius 74.25 Mpc. The objects are selected by the redshift and
luminosity bounds for Figure 4. The distance of an object from the centre of
each map is the perpendicular distance $r\sin(\psi)$ from the axis, which is
the normal of Sheet A1 or B1, where $r$ is the redshift distance and $\psi$ is
the angular distance of the object from the fiducial normal. The angular
position $\mu$ is the azimuthal position of the object around the fiducial
axis, with $\mu=0$ when the object is in line with the fiducial pole and the
North galactic pole. The horizontal and vertical axes in each map are
$x=r\sin(\psi)\cos(\mu)$ and $y=r\sin(\psi)\sin(\mu)$. Positions are projected
on the face of the disk cell, clusters marked by filled red squares, 2MRS
galaxies by triangles.

The centre maps in both rows show positions on the face of the disk cell that
contains the largest count of galaxies and, presumably, the thin flat sheet.
The maps to the left and right show positions of objects in the two
immediately adjacent disks centered 5 Mpc away. The considerably larger counts
in the two central maps are evident, and expected from the prominent peaks in
the two histograms in Figure 4. The evidence the figure adds is that the
greater numbers of objects in the central disk cells are not in orderly or
otherwise unusual-looking arrangements. On and off the peak we see the
familiar clumpy distributions of objects on scales ranging up to the distance
across each of these maps. The simple difference is that number densities in
the central disks are larger than in the neighboring disks. The situation
looks similar in and around Sheets A1 and B1.

Figure 9: Variation of the galaxy count in a disk cell as the cell is tilted
from the direction of maximum count.

Might the counts in a cell with the special orientations shown in Figure 3 be
exceptionally large because the cell contains a single particularly rich
concentrations of galaxies? That is not my impression of the distributions of
galaxies and clusters in the central maps in Figure 8. And a single
concentration would not produce the well-defined direction shown in Figure 3.
If the count in a thin disk cell were particularly large because the cell
contains a straight filament of objects then there would be large counts in
disk cells in other orientations that also contain the filament. This is not
suggested by the concentrations of points in the two directions in Figure 3 .
A filament of galaxies that is curved in two dimensions but flat in the third
would serve as what I term a thin flat sheet.

An elaborated version of these thoughts starts from the consideration that the
disk cells are much thinner than they are broad. The cell thickness $\Delta h$
divided by the distance to the edge of the disk cell,
$r_{\perp}=\sqrt{2}(N+1/2)\Delta h$, defines the characteristic angle,

$\Theta=[\sqrt{2}(N+1/2)]^{-1}\sim 4^{\circ}.$ (13)

If the sheet of galaxies is scattered across the face of the disk, in the
clumpy fashion suggested by the maps in Figure 8, then a tilt
$\theta\sim\Theta$ from the orientation at maximum count in a cell, while the
distance $h_{j}$ from the origin is fixed, moves the edge of the disk cell
outside the original cell, meaning an appreciable part of the tilted cell is
outside the nominal thin sheet. The expected size of the effect is difficult
to estimate because we do not know how thick the sheet is, and the effect of
the tilt depends on the abundance of objects just outside the cell before it
was tilted. But a perceptible decrease in the count at tilt $\theta\sim\Theta$
is expected of an interesting approximation to extra galaxies in a flat sheet.

The data for the two scatter plots in Figure 9 are the 8540 2MRS galaxies that
were used for Figures 3 and 4. The upper plot shows counts in cells centered
on our position, $h_{0}=0$, as a function of the tilt $\theta$ from the normal
of Sheet A1 (in eq. 11) in random directions. The lower plot shows counts in
cells centered at distance $h=-15$ Mpc as a function of the tilt from Sheet B1
in random directions.

In this experiment the cell at $j=0$ is tilted while the centre of the cell is
fixed. The cell at $j=-3$ is both tilted and shifted. Perhaps the scatter in
counts at given tilt is greater around Sheet B1 in part because of this shift
as well as tilt. But in both cases the counts at $\theta\sim\Theta\sim
4^{\circ}$ are about 80 percent of the counts at $\theta\sim 0$, roughly what
one might expect of a thin sheet containing an unusually large count spread
reasonably well across the diameter of the disk cell. By this measure Sheets
A1 and B1 have the properties of thin flat regions in which the density is on
average larger than usual across the width of the region sampled by the cell.

##  5 More Distant Sheets

The smaller number densities at greater distances in the catalogs used here
are compensated by doubling lengths in the second redshift bin, meaning
redshifts $z=0.021$ to 0.042, 10 Mpc thick disk cells with radius
$r_{\perp}=148.49$ Mpc. The data are kept disjoint by the use of a straight
hollow cylinder, or tube, of thin disk cells with the inner radius given by
equation (6) (in the form of a machinist’s washer). That eliminates objects in
the smaller redshift bin, as wanted. It reduces the use of data, but that is
compensated by the advantage of disjoint samples scanned in an isotropic way.
These dimensions are doubled again in the third redshift bin $0.042<z<0.085$.

Figure 10: Evidence of continuations of Sheets A1 and B1 in the second
redshift bin, $0.021<z<0.042$. Black points show directions where the galaxy
count in the central cell, $j=0$, exceeds 220, and the red points show where
the count at $j=-1$ or $-2$ exceeds 220. A direction below the curved line is
the same as a direction above the line and in the opposite direction.

###  5.1 Sheets A and B at Redshifts 0.021 to 0.042

Figure 10 shows evidence of detection of the continuations of Sheets A1 and B1
from the first redshift bin to the second, $0.021<z<0.042$. A threshold count
of 220 galaxies in a hollow disk cell makes the continuations visible, but
cell counts in cells in other orientations also exceed the threshold. The
confusion is reduced by using the prediction that a continuation of Sheet A1
would contain the origin, $h\sim 0$, and a continuation of Sheet B1 would be
at $h\sim-15$ Mpc at its closest. Accordingly, the black dots in the figure
show orientations in which the galaxy counts exceed 220 in cells at the
origin, $-5<h<5$ Mpc, and the red points show where counts are this large in
one of the two cells that span the range of distances from the origin
$-25<h<-5$ Mpc. The continuations of Sheets A1 and B1 at close to the
predicted directions are labeled in Figure 10. The angular positions and
distances from the origin at close to maximum cell counts of the two labeled
clumps of points are

$\displaystyle\hbox{ Sheet
A${}_{2}$:}\,l=63^{\circ},\,b=16^{\circ},\,-5<h<5\hbox{ Mpc};$
$\displaystyle\hbox{ Sheet
B${}_{2}$:}\,l=62^{\circ},\,b=60^{\circ},\,-25<h<-5\hbox{ Mpc}.$ (14)

The subscripts indicate that these are based on the data in the second
redshift bin.

The red points near A in Figure 10 represent large counts in cells with
normals near Sheet A2 and further than 5 Mpc from the origin. The interesting
concentration of disk cells with large galaxy counts and normals near $l\sim
100^{\circ}$, $b\sim 0$ seems to continue to smaller longitudes and lower
latitudes to $l\sim 300^{\circ}$, $b\sim-40^{\circ}$. I have not attempted to
understand what this might mean.

Figure 11: Counts of galaxies in the upper panel and clusters in the lower at
$0.021<z<0.042$ along the directions of Sheets A2 (black) and B2 (red) in
equation (14). Dotted histograms are means.

Figure 11 shows counts of galaxies and clusters of galaxies in the rows of
cells in the two directions in equation (14). In the black histogram the disk
cell centred on the origin contains Sheet A2, and in the red histogram the
cell at $j=-2$ contains Sheet B2. As before the dotted histogram is the cell
count averaged over direction as a function of distance $h$ of the disk cell
from the origin.

The fluctuations of counts around the means are large, but the two peak counts
seem significant. The peak of Sheet A2 is in the distance interval $-5<h<5$
Mpc, consistent with the interval $-2.5<h<2.5$ Mpc for Sheet A1. The normal is
tilted $8^{\circ}$ from Sheet A1. Sheet B2 peaks in the interval $-25<h<15$
Mpc, consistent with the interval $-17.5<h<-12.5$ Mpc in the first redshift
bin. Its normal is $15^{\circ}$ from Sheet B1.

The lower panel in Figure 11 shows counts of clusters along the orientations
of the galaxy counts in the upper panel. It might be taken as somewhat
encouraging that the two largest counts are at the expected distances from the
origin, $h\sim 0$ and $h\sim-15$ Mpc. But there are other peak counts that are
not much smaller, so this is a modest addition to the evidence of extensions
of Sheets A and B to the second redshift bin.

The case for detection of the extensions of Sheets A and B in the 2MRS galaxy
distribution rests on the similar directions labeled A and B in Figure 10,
along with the similar measures of the distances $h$ of the peak counts in
Figure 11. This passes the test of predictions from the data in the first
redshift bin. It is important evidence of reality of the close to flat sheet-
like patterns.

Figure 12: Evidence of continuation of Sheet A to the redshift bin
$0.042<z<0.0845$. The points show directions where the galaxy count in a
hollow disk cell is $\geq 25$ at distances from the origin $-30<h<-10$ Mpc.

###  5.2 Sheet A at Redshifts 0.042 to 0.085

The counts in this third redshift bin are in a hollow cylinder or tube of disk
cells 20 Mpc thick with inner and outer radii 182 and 297 Mpc, where the lower
bound eliminates data in the second redshift bin. A survey of the sky in this
redshift range and allowing the full range of distances $h$ from the origin
reveals many directions where cell counts of galaxies reach 25, a relatively
large value for the sample. As for the second bin this means the survey for
Sheets A and B must be confined to a narrow range of distances $h$. Figure 12
shows directions where the count of galaxies in a cell is 25 or larger in the
disk cell at $j=-1$, which spans the distance range $-30<h<-10$ Mpc. The
choice $j=-1$ was empirical: it shows the most prominent concentration of
directions near Sheet A with large galaxy counts.

I have not found evidence of the presence of Sheet B in this redshift range
and in trial scans of the sky in tubes at distance $h$ in the range $-4\leq
j\leq 1$.

Figure 13: Evidence of continuation of Sheet A to $0.042<z<0.0845$ from the
distributions of galaxies and clusters. The points show directions where the
galaxy count is $\geq 25$ or the cluster count is $\geq 30$ at distances
$-30<h<-10$ Mpc.

Figure 13 shows an enlarged region of the sky near the direction of the
candidate for continuation of Sheet A. The points are directions where the
count of galaxies in a cell is 25 or larger, in the left-hand figure, and
where the count of clusters is 30 or larger, on the right, again with
$-30<h<-10$ Mpc. Counts of galaxies and clusters both are close to maximum in
the direction222As remarked in Sec. 2 it is meaningless to specify directions
at maximum count to better than one degree, but I yield to the temptation to
specify the galactic longitude to a half degree because this happens to bring
both counts of galaxies and clusters close to maximum.

$\hbox{ Sheet A${}_{3}$:}\,l=57.5^{\circ},\,b=15^{\circ},\,-30<h<-10\hbox{
Mpc}.$ (15)

Figure 14: Counts of galaxies (dashed line) and clusters (solid line) at
$0.042<z<0.085$ along the axis in equation (15).

Figure 14 shows counts of galaxies and clusters of galaxies as functions of
the distance $h$ of the disk cell from the origin along the direction in
equation (15). In this direction the disk cell at $h=-1$ contains Sheet A3.
The counts of clusters are plotted as the solid line, and counts of galaxies
as the dashed line. The dotted lines are the means, where the means for
galaxies are larger than the means for clusters at the left-hand side of the
figure.

At the redshift upper limit, $z_{\rm max}=0.085$, the absolute magnitude bound
in equation (10) selects only the most luminous galaxies, which likely are in
rich groups or clusters, though not necessarily the clusters selected by their
X-ray emission. It is not surprising therefore that clusters and these
luminous galaxies point to similar directions in Figure 13. But I emphasize
once again that this agreement is important because it is a check that the
small numbers of objects in the cells have not allowed statistical
fluctuations to point to a meaningless direction.

It is important also that the normal of Sheet A3 in equation (15) is tilted
just $5^{\circ}$ from the normal of Sheet A2 in the second redshift bin (eq.
[14]), $3^{\circ}$ from Sheet A1 in the first bin (eq. [11]), $17^{\circ}$
from the plane of the Nearby Universe (eq. 7), and $13^{\circ}$ from the
supergalactic pole. The sheet of the Nearby Universe with Sheet A1 and Sheet
A2 all pass close to the origin, at our position. The midplane of the disk
cell that contains Sheet A3 is 20 Mpc from the origin (Fig. 14). This means
the identification of the feature in Figure 13 as a continuation of Sheet A
requires the postulate of a bend or warp. The warp is modest compared to the
600 Mpc diameter of the disk cells, and it is in the spirit of acceptance of
the modest bends in directions required to identify continuations of the
sheets across disjoint redshift bins.

I conclude that the measures in the bin $0.042<z<0.085$ provide a serious
addition to the evidence of the presence of Sheet A.

Figure 15: Distributions of radial distances of the galaxies, black, and
clusters, red, that are in the disk cell $j=-1$ that contains Sheet A3. The
dotted histograms are counts averaged over directions.

I have not found interesting evidence of the continuation of Sheets A or B in
the 497 MCXC clusters in the redshift bin $0.085<z<0.17$, or the 209 XMM X-ray
cluster detections (Koulouridis, Clerc, Sadibekova, et al. 2021), and there
are few 2MRS galaxies at this distance. This makes it interesting to estimate
how far detection of Sheet A3 extends into the third redshift bin. The minimum
and maximum radial distances from the origin in the $j=-1$ cell are 182.2 and
298.5 Mpc. The solid histograms in Figure 15 (black for 2MRS galaxies, red for
MCXC clusters) show the distributions of radial distances, from redshifts, of
the objects that are in the disk cell $j=-1$ when the tube axis is in the
direction in equation (15). The dotted histograms show the counts of these
objects as a function of radial distance in cell $j=-1$ in the average over
statistically isotropic random directions. The fluctuations in these dotted
histograms are real, determined by the positions of the objects. We see that
at distances greater than 230 Mpc there are only marginally more objects in
the cell oriented along the continuation of Sheet A than would expected in the
absence of this feature. The counts above average at distances less than 230
are modest but systematically larger than random, as we see in Figure 14. I
conclude that Sheet A3 is reasonably well detected to about 230 Mpc distance,
or redshift $z\sim 0.05$.

##  6 Discussion

In the standard and well-tested $\Lambda$CDM theory the primeval mass
distribution is a stationary random Gaussian process with a close to scale-
invariant power spectrum without the patterns under discussion here. The
theory follows in a natural way from the cosmological inflation picture for
the evolution of the very early universe, and the theory fits precision
measurements of the anisotropy of the cosmic microwave background and a
considerable network of other critical and well-checked tests that
persuasively establish the theory as a useful approximation to reality (e.g.
Peebles, Page, and Partridge 2009). But dark matter and dark energy were put
in the theory as the simplest way to get agreement with the empirical evidence
we had decades ago (Peebles 1982, 1984), and although the Gaussian initial
conditions are a natural consequence of inflation they are not a prediction.
These considerations are meant to remind the reader of two points. First,
evidence of phenomena that would require adjusting the $\Lambda$CDM theory
must be treated with caution because the theory passes demanding tests. But
second, the search for evidence of phenomena that would require adjusting the
theory is essential because the theory is incomplete and might be improved by
following hints from empirical evidence.

###  6.1 The Evidence

The phenomena discussed here are the signatures of large-scale thin and flat
sheet-like patterns in cosmic structure. The method of detection of these
phenomena can equally well be applied to realizations of theoretical
predictions of the spatial distribution of galaxies or clusters of galaxies.
This offers the prospect of an interesting and maybe critical comparison of
theory and observation that might lead us to a still better theory.

I have treated with caution the evidence of phenomena that seems to require
adjusting the successful $\Lambda$CDM theory. To begin, at redshifts $z<0.021$
Sheets A and B are detected in catalogues obtained in the three quite
different ways reviewed in Section 4.2. This makes a good case that the
detection is not some exceedingly subtle systematic error in the data.

Even more compelling evidence of reality of Sheet A is the continuity of
detection of the patterns in data that are disjoint (or very close to it in
the Nearby Universe). The Nearby Universe and Local Supercluster are known to
be flat; it was the motivation for this study. The probe used to detect flat
patterns applied to this region at distances $\lower
2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}7$ Mpc gives the indication of its
flatness in Figure 2. By this measure the Nearby Universe is tilted
$5^{\circ}$ from the de Vaucouleurs supergalactic pole. Figures 3 to 6 show
evidence of the presence of Sheet A1 in the redshift bin $z<0.021$, the Local
Universe. The normal to this sheet is tilted $11^{\circ}$ from the
supergalactic pole and $15^{\circ}$ from the sheet of the Nearby Universe. The
Nearby Universe sheet and Sheet A1 both contain the origin, our position. We
can take the continuity in position and near continuity in direction to mean
the Nearby Universe is a moderately bent part of Sheet A1 in the Local
Universe. This is an important example of continuity of detection in different
ranges of length scales.

Detection of Sheet A2 at $0.021<z<0.042$ is shown in Figure 10. It is tilted
$8^{\circ}$ from Sheet A1, $21^{\circ}$ from the Nearby Universe, and
$18^{\circ}$ from the supergalactic pole. The disk cell used here has a hole
in its center, to avoid the galaxies used in the first redshift bin, but the
centerplane of the cell passes through the origin (Fig. 11), along with the
Local Supercluster. The detection of this feature about where it would be
expected in position and orientation amounts to a confirmation of a
prediction. This is substantial evidence of the physical reality of Sheet A.

At the third redshift bin, $0.042<z<0.085$, the evidence that Sheet A3 is
detected is seen in Figures 12 and 13. Sheet A3 is tilted $5^{\circ}$ from
Sheet A2, $3^{\circ}$ from Sheet A1,$17^{\circ}$ from the plane of the Nearby
Universe, and $13^{\circ}$ from the supergalactic pole. The centerplane of the
cell that contains Sheet A3 is 20 Mpc from the origin. This is modest compared
to the dimensions of the disk cell, 20 Mpc thick by 600 Mpc diameter, but the
offset in distance from the pole is real. Maybe it signifies a bend or warp in
the sheet tracing back to the sheet of the Nearby Universe, or maybe something
more complicated. But the postulate of a warp is in the philosophy of
tolerating modest bends in the segments of redshift that detect Sheet A.

Sheet B1 in the first redshift bin looks similar to Sheet A1. This is seen in
the counts in disk cells as functions of distance from the cells containing
the sheets (Fig. 4), the distributions of galaxy morphologies (Fig. 7), the
maps (Fig. 8), and the sensitivity to tilts of the disk cell (Fig. 9). Sheet
B2 is in the interval of distance from the origin $-25<h<-15$ Mpc, consistent
with the interval $-17.5<h<-12.5$ Mpc in the first redshift bin. The normal of
Sheet B2 is $15^{\circ}$ from Sheet B1. This signature of a modestly bent
continuation of Sheet B in two disjoint intervals of redshift is significant
evidence of another thin close to flat sheet in cosmic structure. It might be
expected that the sheet of the Local Supercluster cannot be unique, but the
evidence of another specific sheet is important.

It is curious that the 51 clusters in the first redshift bin yield clear
signatures of Sheets A1 and B1 (Fig. 5), and the 351 clusters in the third bin
give a clear signature of Sheet A3 (Figs. 13 and 14), but the 139 clusters in
the second bin yield at best a modest suggestion of detection (in the lower
plot in Fig. 11). But there are reasonably clear signatures of Sheets A and B
in the distribution of galaxies in the second bin (Figs. 10 and 11).

To summarize, the near continuity in directions and distances of the normals
of the sheet-like patterns in three disjoint redshift bins, and in the close
to disjoint Nearby Universe, is serious evidence of reality of Sheet A, the
Extended Local Supercluster, which extends to redshift $z\sim 0.05$.

###  6.2 Consequences

The probe that establishes this case for reality of flat sheets could readily
be applied to spatial distributions of simulated galaxies, or of dark matter
halos characteristic of galaxies or clusters of galaxies. Of greatest interest
are positions established by numerical simulations of the growth of cosmic
structure from the initial conditions and physics of the present standard
$\Lambda$CDM theory. A likely first step would be a check for occurrence in
numerical simulations of the flat Nearby Universe, the part of the Local
Supercluster shown in Figure (2). Neuzil, Mansfield, and Kravtsov (2020)
remark on the “flat sheetlike shape” of the Local Supercluster, and find that
the fit of an ellipsoid to the distribution of galaxies closer than 8 Mpc has
ratio of largest to smallest axes $a/c\sim 6$. This is in the spirit of Figure
2, though not as flat as suggested by the figure. Neuzil et al. (2020) find
that their measure of flattening is “not exceedingly rare in the $\Lambda$CDM
model.” The pronounced concentration of galaxies in the central sheet 0.5 Mpc
thick and 15 Mpc across in the Nearby Universe intuitively seems more
demanding than the Neuzil el al. (2020) criterion, but a check certainly would
be of great interest.

I have not found discussions of whether the $\Lambda$CDM theory might be
challenged by the considerable extension of the plane of the Local
Supercluster found by Einasto et al. (1983), Tully (1986), and Shaver (1990).
The extension has been more precisely characterized here, and again a check
for the presence of close to flat patterns similar to Sheets A and B in
simulations of cosmic structure certainly would be interesting.

If the patterns of flat thin sheets prove to be real and not predicted by the
$\Lambda$CDM theory then the theory will remain a very useful approximation,
but it will have been established that we have an interesting hint to an even
better theory to be found.

If as seems intuitively likely the signatures of flat and extended sheets
presented here are real and are not predicted by the standard $\Lambda$CDM
cosmology then thoughts will turn to the passages of long nearly straight
cosmic strings that produced near-planar density concentrations (e.g.
Shellard, Brandenberger, Kaiser, and Turok 1987; Vachaspati and Vilenkin 1991;
da Cunha, Harnois-Deraps, Brandenberger, Amara, and Refregier 2018). Could
cosmic strings be straight enough to produce extended flat sheets and less
massive enough to agree with the constraint from data assembled by the
International Pulsar Timing Array (Falxa M., Babak S., Baker P. T. et al.,
2023)? If so would cosmic string perturbations to the galaxies and clusters in
sheets agree with the observations? But the immediately pressing question is
whether cosmic strings capable of producing the flat patterns discussed here
would have observable effects on the angular distribution of the CMB. One
would look for the distinctive line discontinuities Kaiser and Stebbins (1984)
and Gott (1985) predicted. The effect on the CMB might be detected by N-point
statistics (e.g. Planck Collaboration et al. 2014 and references therein), or
maybe by a probe sensitive to the expected nature of perturbations to the CMB
temperature caused by moving strings. Danos and Brandenberger (2010) discussed
a probe of temperature gradients. Another possibility is an adaption of the
filter used in this study.

If cosmic strings fail we will need a new idea. Situations of this kind have
led to many advances in the development of natural science.

## acknowledgements

I have been suggesting that the curious extent of the plane of the Local
Supercluster might be teaching us something of value for more than a quarter
century, beginning with the plane traced by clusters in Peebles (1993, Fig.
3.7c), then following Shaver’s (1990, Fig. 1; 1991, Fig. 5) elegant
demonstration of the alignment of radio galaxies the argument that we ought to
be thinking about this in Peebles (1999) and four later publications. My first
step to follow this advice is in Peebles (2022). In the present paper I have
taken the Extended Local Supercluster to be the prototype for the search for
thin flat patterns in cosmic structure. I remember with pleasure discussions
of this and more with Peter Shaver. I am grateful for information and advice
on this paper from Miguel Aragon, Robert Brandenberger, Roger Clowes, Maret
Einasto, Andrey Kravtsov, Biswajit Pandey, Daniel Pomarède, Joe Silk, Brent
Tully, and Yen-Ting Lin. I am grateful to Princeton University for giving me
the space to do this research. I have not received any financial support for
this research, and I am not complaining.

## Data Availability

This research made use of the data compiled by the High Energy Astrophysics
Science Archive Research Center (HEASARC), which is a service of the
Astrophysics Science Division at NASA/GSFC; the SIMBAD and VizieR catalogue
access tool, CDS, Strasbourg, France; the Two Micron All Sky Survey, which is
a joint project of the University of Massachusetts and the Infrared Processing
and Analysis Center/California Institute of Technology, funded by the National
Aeronautics and Space Administration and the National Science Foundation; the
NASA Astrophysics Data System Bibliographic Services; and the NASA/IPAC
Extragalactic Database. I have not generated any new data.

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