COMAP Pathfinder – Season 2 results - II. Updated constraints on the CO(1–0) power spectrum

Astronomy

&

Astrophysics

A&A, 691, A336 (2024)

https://doi.org/10.1051/0004-6361/202451123

COMAP Pathfinder – Season 2 results

II. Updated constraints on the CO(1–0) power spectrum

N.-O. Stutzer

⋆

, J. G. S. Lunde

, P. C. Breysse

, D. T. Chung

, K. A. Cleary

, D. A. Dunne

H. K. Eriksen

, H. T. Ihle

, H. Padmanabhan

, D. Tolgay

, I. K. Wehus

, J. R. Bond

, S. E. Church

T. Gaier

, J. O. Gundersen

, A. I. Harris

, S. E. Harper

, R. Hobbs

, J. Kim

, J. W. Lamb

C. R. Lawrence

, N. Murray

, T. J. Pearson

, L. Philip

, A. C. S. Readhead

T. J. Rennie

, and D. P. Woody

(COMAP Collaboration)

Institute of Theoretical Astrophysics, University of Oslo, PO Box 1029 Blindern, 0315 Oslo, Norway

California Institute of Technology, 1200 E. California Blvd., Pasadena, CA 91125, USA

Canadian Institute for Theoretical Astrophysics, University of Toronto, 60 St. George Street, Toronto, ON M5S 3H8, Canada

Dunlap Institute for Astronomy and Astrophysics, University of Toronto, 50 St. George Street, Toronto, ON M5S 3H4, Canada

Department of Astronomy, Cornell University, Ithaca, NY 14853, USA

Center for Cosmology and Particle Physics, Department of Physics, New York University, 726 Broadway, New York, NY 10003,

USA

Department of Physics, Southern Methodist University, Dallas, TX 75275, USA

Departement de Physique Théorique, Universite de Genève, 24 Quai Ernest-Ansermet, 1211 Genève 4, Switzerland

Owens Valley Radio Observatory, California Institute of Technology, Big Pine, CA 93513, USA

Jodrell Bank Centre for Astrophysics, Alan Turing Building, Department of Physics and Astronomy, School of Natural Sciences,

The University of Manchester, Oxford Road, Manchester M13 9PL, UK

Department of Physics and Astronomy, University of British Columbia, Vancouver, BC Canada V6T 1Z1, Canada

Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA 91109, USA

Department of Physics, Korea Advanced Institute of Science and Technology (KAIST), 291 Daehak-ro, Yuseong-gu, Daejeon 34141,

Republic of Korea

Department of Physics, University of Toronto, 60 St. George Street, Toronto, ON M5S 1A7, Canada

David A. Dunlap Department of Astronomy, University of Toronto, 50 St. George Street, Toronto, ON, M5S 3H4, Canada

Brookhaven National Laboratory, Upton, NY 11973-5000, USA

Kavli Institute for Particle Astrophysics and Cosmology and Physics Department, Stanford University, Stanford, CA 94305, USA

Department of Physics, University of Miami, 1320 Campo Sano Avenue, Coral Gables, FL 33146, USA

Department of Astronomy, University of Maryland, College Park, MD 20742, USA

Received 14 June 2024 / Accepted 30 August 2024

ABSTRACT

We present updated constraints on the cosmological 3D power spectrum of carbon monoxide CO(1–0) emission in the redshift range

–

. The constraints are derived from the two first seasons of Carbon monOxide Mapping Array Project (COMAP) Pathfinder line

intensity mapping observations aiming to trace star formation during the epoch of galaxy assembly. These results improve on the pre-

vious Early Science results through both increased data volume and an improved data processing methodology. On the methodological

side, we now perform cross-correlations between groups of detectors (“feed groups”), as opposed to cross-correlations between single

feeds, and this new feed group pseudo power spectrum (FGPXS) is constructed to be more robust against systematic effects. In terms

of data volume, the effective mapping speed is significantly increased due to an improved observational strategy as well as a better data

selection methodology. The updated spherically and field-averaged FGPXS,

(

)

, is consistent with zero, at a probability-to-exceed

of around

34%

, with an excess of

in the most sensitive bin. Our power spectrum estimate is about an order of magnitude more

sensitive in our six deepest bins across

09 Mpc

−

73 Mpc

−

, compared to the feed-feed pseudo power spectrum (FPXS) of

COMAP ES. Each of these bins individually constrains the CO power spectrum to

(

)

2400

–

4900

μ

Mpc

95%

confi-

dence. To monitor potential contamination from residual systematic effects, we analyzed a set of 312 difference-map null tests and

found that these are consistent with the instrumental noise prediction. In sum, these results provide the strongest direct constraints on

the cosmological 3D CO(1–0) power spectrum published to date.

Key words.

methods: data analysis – methods: observational – galaxies: high-redshift – diffuse radiation – radio lines: galaxies

1. Introduction

By collecting the combined redshift-dependent line emission

from all sources, both diffusely emitting gas and all galaxies,

⋆

Corresponding author;

n.o.stutzer@astro.uio.no

bright and faint, line intensity mapping (LIM) aims to map the

Universe from large to small scales in three dimensions (see

Madau et al. 1997; Battye et al. 2004; Peterson et al. 2006; Loeb

& Wyithe 2008; Kovetz et al. 2017, 2019, and references therein

for details on LIM). Several emission lines of interest have been

proposed, among them

21 cm

, carbon monoxide (CO), ionized

A336, page 1 of 19

Open Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0),

which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This article is published in open access under the Subscribe to Open model. Subscribe to A&A to support open access publication.

Stutzer, N.-O., et al.: A&A, 691, A336 (2024)

carbon ([C

]),

, and

, each with different astrophysi-

cal and cosmological goals (Kovetz et al. 2017, 2019; Bernal &

Kovetz 2022).

At the forefront of CO LIM is the CO Mapping Array Project

(COMAP), currently in its Pathfinder phase, which aims to mea-

sure the large-scale CO(1–0) line emission at redshifts of

∼

–

, tracing the star-forming galaxies around the epoch of

galaxy assembly (Cleary et al. 2022). The COMAP Pathfinder

instrument is a focal plane array of 19 detectors (which we refer

to as “feeds”), each with independent receiver electronics, fielded

on a 10.4 m Leighton telescope at the Owens Valley Radio

Observatory. It observes in a frequency range of

–

34 GHz

and

is sensitive to

115

27 GHz

CO(1–0) rotational line emission at

redshifts of

∼

–

. Based on the first year of observations

(“Season 1”), COMAP obtained the first direct limits on the 3D

CO(1–0) clustering power spectrum, already ruling out several

models from the literature. These results were published in a

series of eight Early Science (ES) papers, along with a preview

of our ongoing continuum survey of the Galaxy, a look at the

prospects for CO LIM at the epoch of reionization, and a cross-

correlation of ES data with an overlapping galaxy survey (Cleary

et al. 2022; Lamb et al. 2022; Foss et al. 2022; Ihle et al. 2022;

Chung et al. 2022; Rennie et al. 2022; Breysse et al. 2022; Dunne

et al. 2024).

In this paper, the second in a series of three, we update our

power spectrum results based on observations taken in our first

and second seasons (S2), following Ihle et al. (2022). We build

on the filtered and calibrated low-level COMAP data products

described in detail by Lunde et al. (2024). Implications for astro-

physical constraints and modeling are explored by Chung et al.

(2024a).

As is discussed by Lunde et al. (2024), the current exper-

imental design is overall very similar to ES, but takes into

account a few important lessons learned. For example, COMAP

Season 2 uses only constant elevation scans (CES), not Lis-

sajous scans, because one of the main conclusions of Ihle et al.

(2022) was that changes in elevation within a scan result in

significant residual systematic effects from changes in the atmo-

spheric and or ground pickup signals. We also avoid elevations

that are strongly contaminated by ground radiation received

in the sidelobes. In addition, the instrument drive speed was

decreased around May 2022 in order to reduce the stress on

the telescope (Lunde et al. 2024), and the effective instrumen-

tal properties therefore changed notably about halfway through

the second season. We denote periods before and after the speed

change the “fast-” and “slow-moving azimuth scans” respec-

tively (these are equivalent to the naming convention “Season

1+Season 2a” and “Season 2b” used by Lunde et al. (2024),

where “a” and “b” denote the period before and after the drive

changes).

For consistency with previous COMAP publications, we

adopt the same

CDM cosmological model as Chung et al.

(2022) and Li et al. (2016) when converting distances in our

map cubes from angular and spectral frequency units into phys-

ical units. Explicitly we set

Ω

286

Ω

714

Ω

047

100

km s

−

Mpc

−

, and

, which is roughly consistent with WMAP (Hinshaw et al.

2013). Unless otherwise stated all distances and distance-derived

quantities in megaparsecs carry an implicit

−

This paper is structured as follows, the power spectrum

methodology and updated null test framework are presented in

Sect. 2 and 3 respectively. In Sect. 4, we present the power

spectrum transfer function used to account for signal loss from

low-level filtering and instrumental effects. Sections 5 and 6

show the power spectrum results and the outcome of our null

tests. Our conclusions are presented in Sect. 7.

2. Power spectrum methodology

The power spectrum fully characterizes the information con-

tained in a Gaussian random field and so is one of the most

powerful statistics for cosmological density fields. While the

non-linear physics of galactic emissions to which COMAP is

sensitive is not fully Gaussian, the power spectrum is still a

useful statistic, and complementary to other summary statistics

such as the Voxel Intensity Distribution (VID); (Breysse et al.

2017; Ihle et al. 2019) or the Deconvolved Distribution Estimator

(DDE); (Breysse et al. 2023; Chung et al. 2023).

The COMAP Pathfinder uses three-dimensional maps of the

CO(1–0) emission to constrain models of star formation during

the epoch of galaxy assembly. While the maps already represent

the compression of hundreds of terabytes of raw time-ordered

data (TOD) into only a few gigabytes, it is possible to encode

and compress much of the relevant astrophysical and cosmo-

logical information contained within the maps even more by

using summary statistics like the power spectrum. As such the

power spectra are easier and more computationally efficient to

work with when constraining astrophysical and cosmological

information of the mapped emission field.

In the COMAP ES paper series, Ihle et al. (2022) devised a

novel cross-power spectrum methodology, the feed-feed pseudo-

cross-power spectrum (FPXS), constructed to be robust against

systematic errors. This work largely builds on the methodology

developed by Ihle et al. (2022) and lessons learned since the

ES data processing to improve the power spectrum constraints

of COMAP even further. In the following we summarize the

FPXS methodology used and outline what has changed from the

methodology developed by Ihle et al. (2022).

2.1. The feed group pseudo-cross-power spectrum

We begin by defining the general concepts of an auto- and cross-

power spectrum. The auto-power spectrum can simply be defined

as the variance of Fourier modes of a map. It can be written as

(

)

vox

⟨|F{

⟩

vox

⟨|

(

)

⟩

(1)

where

vox

is the volume of a voxel (i.e., three-dimensional

pixel) in units

Mpc

vox

is the number of voxels and

(

)

are

the Fourier coefficients of the map

at wavenumber

. The

units of

(

)

and

are, respectively, the same as the map’s

and

Mpc

−

. For the Fourier transform

}

of the map

we use the same convention previously used in ES (Ihle

et al. 2022; Harris et al. 2020). We can safely use the regular

Fourier basis in the case of COMAP, instead of the more gen-

eral spherical harmonics, as the fields are only

∼

◦

in diameter

and the flat-sky approximation is sufficient. Since our maps are

three-dimensional, so is the power spectrum derived from those

maps.

The auto-power spectrum will pick up all components that

contribute to the variance in the map: signal, noise, and system-

atic effects. It can thus be decomposed into

(

)

(

)

noise

(

)

syst

(

)

(2)

A336, page 2 of 19

Stutzer, N.-O., et al.: A&A, 691, A336 (2024)

Table 1.

Feed groups used in the feed group pseudo-cross-power

spectrum.

DCM1 (feed group)

Feeds

1, 4, 5, 12, 13

6, 14, 15, 16

2, 7, 18, 19

3, 8, 9, 10, 11

Notes.

“Feed groups” and their associated first down-conversion

(DCM1) electronics.

showing contributions from CO signal

, noise and systematic

effects, respectively. To obtain an unbiased estimate of the sig-

nal power spectrum, the systematic effects and noise properties

of the map have to be understood and modeled.

Similarly to the auto-power spectrum, we can define a cross-

power spectrum between two maps to be the covariance of

Fourier coefficients of the two. It can be written as

i j

(

)

vox

{

∗

(

)

(

)

}

(3)

where

and

represent Fourier coefficients of two different

maps

and

. The cross-power spectrum reduces to the auto-

power spectrum if the two maps

and

are chosen to be identical.

As opposed to the auto-power spectrum, a cross-power spec-

trum will only be sensitive to correlated common modes between

the two maps. Independent noise and independent systematic

effects will therefore be canceled out and we can decompose the

cross-spectrum as

(

)

(

)

i j

common

(

)

(4)

where

i j

common

(

)

represents the cross-spectrum contribution

from some common systematic effect between the two maps.

We can now see a powerful property of the cross-spectrum: if

we can choose two maps with independent noise properties and

statistically unique systematic effects, giving

i j

common

(

)

the cross-power spectrum will yield an unbiased estimator for

the signal power spectrum

(

)

This is the main property that the FPXS developed by Ihle

et al. (2022) is built to exploit. Because the COMAP Pathfinder

measures the sky with 19 feeds, each with its own receiver

signal chain, the maps from different detectors will have inde-

pendent noise properties. Additionally, several systematic effects

are believed to be unique to each feed, or specific group of

feeds. Therefore, a cross-spectrum between two detector maps

will not be biased by the noise contribution of the detectors or

feed-specific systematic contamination.

In this work, rather than cross-correlating individual feeds,

we instead cross-correlate groups of feeds. In particular, we

group feeds by their shared first down-conversion (DCM1) local

We note that technically

(

)

in this notation would include con-

tributions from both cosmic CO and all other astrophysical components

with non-trivial frequency structure that are not subtracted out by the

low-level data analysis steps, e.g., potential interloper line emission.

However, for CO(1–0) at

–

there are very few, if any, inter-

loper lines, apart from a

∼

10%

contamination from CO(2–1) at

–

(Breysse et al. 2022; Chung et al. 2024b) that could be picked up, and

we therefore use a CO-only notation.

oscillator (Lamb et al. 2022). Table 1 shows the feeds that are

grouped together in a given “feed group”.

The reason for this change is that some of the systematic

effects uncovered with the improved sensitivity of the current

data volume are correlated with the DCM1 feed groups as

has been shown by Lunde et al. (2024). Applying the origi-

nal FPXS, involving cross-correlation of feeds from the same

feed group, would not have been effective in canceling such sys-

tematic effects since they are common-mode for a given feed

group.

Instead, by grouping all feeds in a given feed group together,

detectors from the same feed group are never cross-correlated

when computing the average feed group pseudo-cross-power

spectra (FGPXS). This effectively cancels the systematic effects

that are common to each feed group, while retaining the CO

signal. Additionally, grouping together detectors in this way pro-

duces effective detector maps that have more sky overlap. Thus,

when cross-correlating these maps we obtain better constraints

on large-scale power spectrum modes and less mode-mixing

due to a larger cross-map footprint. The result from a lower

degree of mode mixing is also a lower amount of large-scale

systematic effects that can leak into the small- and intermediate-

scale power. This is especially important, as we know from

Lunde et al. (2024) that our most dominant systematic effects

are large-scale modes in the maps.

After splitting the data into feeds or feed groups, we split

the data additionally into halves, each with independent sys-

tematic effect contributions, such as high or low elevation, as

was done by Ihle et al. (2022). This further eliminates unwanted

contributions to the cross-spectrum.

However, even though the FGPXS is slightly more robust to

systematic effects, this comes at the price of a slight decrease in

sensitivity. The expected loss in sensitivity when using FGPXS

as opposed to FPXS should in theory follow the upper limit

found by Ihle et al. (2022):

split

feed

(

)

≥

−

split

−

feed

(

)

(5)

split

−

feed

(

)

(6)

where the uncertainty of a cross-spectrum,

split

feed

(

)

, is given

by the number of cross-correlated data splits and feeds,

split

and

feed

, compared to the optimal sensitivity,

(

)

, one can

obtain when using all available data in an auto-power spectrum.

To incorporate the effects of both cross-correlating feeds and an

additional cross-variable on the total sensitivity we have gener-

alized Eq. (14) of Ihle et al. (2022) to obtain Eq. (5). Because

we use the same elevation split as Ihle et al. (2022), splitting the

data into high or low elevation, the

split

dependency of Eq. (5)

reduces to the

√

loss in sensitivity in Eq. (6).

To give some intuition on Eq. (5), we show a grid of pos-

sible feed group and elevation split combinations in Fig. 1.

Equation (6) can be obtained by using a grid like the one seen

in Fig. 1 from the ratio between the total number of split combi-

nations (i.e. the optimal auto-spectrum sensitivity

) and the

number of all cross-combinations that do not constitute auto-

combinations between feeds or elevations (respectively dark and

light gray shading). From this we should expect there to be a loss

in sensitivity in the FGPXS compared to FPXS of

∼

12%

– that

A336, page 3 of 19

Stutzer, N.-O., et al.: A&A, 691, A336 (2024)

FG1

FG2

FG3

FG4

FG1

FG2

FG3

FG4

FG1

FG2

FG3

FG4

FG1

FG2

FG3

FG4

High elevation

Low elevation

High elevation

Fig. 1.

Example grid of possible feed group (FG1–4) and elevation

(high and low) split combinations. Combinations with dark and light

gray shading, respectively, represent auto-feed group and auto-elevation

combinations that are not used in the final averaged FGPXS. The cross-

combinations containing examples of a 2D cross-spectrum contribute

to the final average FGPXS as neither identical feed groups, nor eleva-

tions, are crossed.

is by, respectively, inserting values for the number

feed

of feed

groups,

feed

, and all individual feeds,

feed

, in Eq. (6)

and comparing the results

Nevertheless, we conservatively cluster feeds into the afore-

mentioned feed groups to avoid systematic effect contamination,

at the price of a minor loss in sensitivity. Apart from the rea-

sons stated above, there is in principle no difference between

the FPXS and FGPXS algorithmically; for instance, it would be

trivial to group the feeds in a different configuration if that were

found to be advantageous in the future for some reason. We can

thus describe the two methods using the same algorithmic rep-

resentation shown in the following. We can write the main steps

of the FGPXS algorithmically as follows:

1. Split the data into two halves

and

. As was done by

Ihle et al. (2022), we chose elevation as the main cross-

correlation variable to eliminate potential sidelobe pickup

from the ground.

2. For parts

and

respectively make maps of each feed group

. We denote these by, for example,

for a map of part

with feed group

3. For each combination of feed groups

and

, and data splits

and

, compute cross-power spectra.

4. Compute a noise-weighted average FGPXS of all the result-

ing

feedgroup

(

feedgroup

−

(with

feedgroup

when using

feed groups and

feedgroup

if computing the ES FPXS)

individual FGPXS that do not involve the same detector or

We note that this number in practice tends to be a little larger because

we exclude auto-combinations between feed(-groups) and these contain

the largest fraction of the optimal total auto-spectrum sensitivity due to

better overlapping cross-sky maps.

elevation:

(

)

,

(

)

(

)

,

(

)

(7)

with a corresponding uncertainty of

(

)

,

(

)

(8)

This is what we refer to as the mean feed group pseudo-

cross-power spectrum (FGPXS) or feed-feed pseudo-cross-

power spectrum (FPXS), if (respectively) feed groups or

feeds are used as effective detectors.

In Fig. 1, we illustrate a grid of possible feed group and elevation

combinations used for an average FGPXS. Those shaded dark

and light gray represent auto-feed and auto-elevation combina-

tions (respectively). The combinations that cross neither feed

groups nor elevations, indicated with examples of 2D FGPXS

combinations, are used in the final average FGPXS in Eq. (7).

Due to the non-uniform coverage of our sky fields, as well as

a non-trivial survey footprint (see Lunde et al. 2024, for exam-

ples of maps), the maps are weighted prior to computing their

Fourier coefficients. We use the same weighting scheme as Ihle

et al. (2022), given for a cross-power spectrum by

∝

(9)

where

represents the uncertainty estimate in each voxel of a

feed group and elevation split map,

. These weights are then

applied to the map,

, before power spectrum estimation

with the Fourier coefficients

(

)

}

in Eq. (3). Regions

outside the map footprints are assigned zero weights. The power

spectra of these weighted maps are commonly referred to as

pseudo power spectra (Hivon et al. 2002). The pseudo power

spectra are a biased power spectrum estimator because differ-

ent Fourier modes become coupled via the applied weights (see

Hivon et al. 2002; Leung et al. 2022, for details on mode mix-

ing). It should be noted that we use

(

)

to denote pseudo spectra

in the later results sections, but we use the notation

(

)

(without

the tilde) in the methods sections as most of the methodology is

equivalently written for unbiased and pseudo spectra. A detailed

discussion of the COMAP-specific mode mixing can be found in

Fig. 1 and Appendix D of Ihle et al. (2022), which shows that

the effect is

≤

20%

over our k-range. Reversing the mode-mixing

will thus be left as a future exercise and is beyond the scope of

this work.

2.2. The binned power spectrum estimator

As COMAP produces line intensity maps spanning three-

dimensional redshift-space volumes, the resulting power spectra

also span three-dimensional Fourier-space volumes. It can, how-

ever, be easier to work with a power spectrum spherically

averaged down to one dimension. For the spherically averaged

power spectrum to contain all relevant information in the full

three-dimensional power spectrum, the emission field is tech-

nically required to remain statistically isotropic on large scales

and stationary across the mapped redshift range. This is not

A336, page 4 of 19

Stutzer, N.-O., et al.: A&A, 691, A336 (2024)

strictly known to be true for the CO emission field. Cosmic

star-formation, especially dust-obscured star-formation history

traced in the infrared, is poorly constrained in our targeted red-

shift range of

–

(see Madau & Dickinson 2014, for a

review of cosmic star-formation history). As a consequence, the

extent to which the mapped CO emission field is stationary is

largely unknown. The spherically averaged power spectrum of a

dynamic field will not be sensitive to changes in the CO emis-

sion across cosmic time, but it will measure the time-averaged

properties of the targeted CO structures. However, the distinc-

tion is moot for the current COMAP signal-to-noise ratio (S/N),

as no clear CO excess is observed in the power spectra. Thus we

present the spherically averaged, 1D power spectrum as our main

science product.

Additionally, in practice, the angular and the redshift axes

are observed in fundamentally different ways, and the low-level

filtering applied to the data (Lunde et al. 2024) as well as red-

shift space distortions and line-broadening (Chung et al. 2021)

can affect the signal and sensitivity differently along each axis.

Therefore, the angular and line-of-sight dimensions are conve-

nient to separate, and we bin the 3D power spectrum

(

)

, with

(

)

, into both a cylindrical and spherically averaged

power spectrum. The former of these conserves the structures

perpendicular and parallel to the line-of-sight by only merging

the two angular axes

(

⊥

∥

)

(10)

Meanwhile, the latter will average the 3D power spectrum into

1D bins of the form

(11)

The binned cylindrically averaged power spectrum estimator

will then become

(

)

≈

vox

modes

∈

{

∗

(

)

(

)

}

(12)

where the number of Fourier modes in bin

is given as

modes

The equation is completely analogous when binning to the spher-

ically averaged power spectrum. We henceforth refer to the

cylindrically and spherically averaged power spectrum estima-

tors as “2D” and “1D” due to the number of axes needed to

display them, but note that they still represent averages of a 3D

density field.

The bin edges are chosen to cover the scales to which

COMAP is most sensitive and correspond to those used in the

COMAP ES power spectra (Ihle et al. 2022), but due to our large

increase in sensitivity and better understanding of the origin of

correlations on large angular scales we conservatively excise all

perpendicular scales

⊥

≲

1 Mpc

−

for this publication. On

these scales, we are dominated by sub-optimal cross-map over-

lap that results in poor constraining power of the large-scale

structure as well as the possibility of large-scale residual sys-

tematic effect leakage through mode mixing into the small-scale

power spectrum modes (see Lunde et al. 2024, for examples of

our map-domain systematic effects). In the future, we aim to

recover the large scales at

⊥

≲

1 Mpc

−

. Lastly, we also mask

the bins corresponding to the highest

⊥

and

∥

used by Ihle

et al. (2022) to prevent issues with aliasing near the Nyquist fre-

quency of the two respective dimensions:

Nyquist

⊥

≈

22 Mpc

−

and

Nyquist

∥

≈

74 Mpc

−

2.3. Uncertainty estimation from randomized null maps

In order to compute the mean FGPXS and its errors, as is shown

in Eqs. (7) and (8), we need the power spectrum uncertainties

for each feed group and elevation cross-combination,

(

)

This can be done via two basic approaches: simulations and

data-driven methods. Here, we first detail some problems with

a simulation-based approach used previously in COMAP ES

(Ihle et al. 2022) and subsequently argue for why a data-driven

approach was chosen in this work.

In COMAP ES, Ihle et al. (2022) chose a simple simu-

lation approach where the power spectrum uncertainties were

computed from an ensemble of simple white noise maps,

noise

∼N

)

, drawn from a zero-mean Gaussian distribu-

tion

with the voxel uncertainties

. These were then propa-

gated to the power spectrum level. The main advantage of this

approach is its computational efficiency. However, it can only

reflect the white noise level within the map, while residual cor-

related noise and the effect of the pipeline filters on the noise will

not be contained in the uncertainty from these simple white noise

maps (see, for instance, the power spectral density (PSD) of TOD

in Fig. 9 of Lunde et al. 2024, for an illustration of the noise

properties of the filtered data). The simplified simulations proved

an adequate method given the sensitivity of our ES data. With

the increased sensitivity achieved at the end of S2, obtaining

suitable power spectrum errors,

, through simulations would

require the sampling of noise from the TOD domain (ideally

with additional ground-up modeling of all contributing system-

atic effects), propagating it all the way through the low-level

pipeline (Lunde et al. 2024) up to the power spectrum. However,

this would be computationally expensive because the low-level

pipeline filters would have to be re-run for each ensemble, and

require significant additional data modeling.

Given the drawbacks of both the white noise and a poten-

tial TOD-level simulation-based approach, a data-driven method

was instead chosen for this work as it represents a relatively

computationally inexpensive method of estimating the power

spectrum uncertainties that automatically reflects all the prop-

erties of the data. In particular, we draw from the simple idea

that we can cancel the signal and systematic effects in a sub-

traction between data-half maps while leaving the correct noise

properties. In our case, we estimate

by what we refer to as

an ensemble of randomized null difference (RND) maps.

The first step in the RND calculation is to divide the set of

all scans in the data into two randomized halves,

and

, from

which we subsequently make maps

RND

and

RND

. This is done

for all random split realizations

. Both

RND

and

RND

should

contain the same signal, and due to the randomization of the

splits also the same systematic effects. Hence we can cancel both

the signal and systematic effects by computing the difference

between the two maps;

∆

RND

−

RND

(13)

The difference maps

∆

RND

now optimally capture the white

and correlated noise properties and biases (from low-level filters,

the instrumental beam, etc.) of the real maps, but are without any

of the signal or systematic effects. As such they reflect the true

properties of the data to a high degree.

Finally, to obtain the uncertainty of the power spectrum

we need to compute the FGPXS of each difference map

∆

RND

From the resulting ensemble of such feed group cross-spectra,

RND

∆

(

)

, we can compute the uncertainties

RND

(

)

by taking

A336, page 5 of 19

Stutzer, N.-O., et al.: A&A, 691, A336 (2024)

the standard deviation over the ensemble. These can then be used

when co-adding together feed group spectra to obtain the final

mean FGPXS as is explained in Eqs. (7) and (8).

3. Power spectrum null tests

With the increased effective COMAP data volume and the result-

ing increased sensitivity comes the need for more effective null

tests to ensure the data quality of our final power spectra.

As we explain in this section, the null tests devised in this

work are based on difference maps in a similar way to the

RND method used for uncertainty estimation described earlier

in Sect. 2.3, except we are now splitting the maps on meaningful

parameters instead of randomly. The goal then becomes finding

null variables (e.g. high or low humidity or left or right moving

scans; see Table C.2 for a list of all chosen variables) that cor-

relate to systematic effects in one of the null variable halves by

which we split the data.

We can write the difference map of some null variable

∆

null

−

(14)

where the maps

and

represent the maps of the two

halves of the data respectively. If the chosen null variable cor-

relates to a systematic effect, the difference map

∆

null

will

contain the systematic effect but cancel the signal. The differ-

ence maps can then be used to perform a null test, with the

null hypothesis being that the null maps are consistent with

the general noise properties of the maps. The associated voxel

uncertainty of the null map is then given by

null

∆

(15)

for uncertainties

and

of the maps

and

respectively.

For each of the null variables

we then take the difference

between the two maps as is described by Eqs. (14) and (15).

As we use a cross-elevation FGPXS we must compute a differ-

ence map for both high and low elevation. The data are therefore

split into four parts, two elevation ranges and two null variables

halves, where we subtract across the latter in the map domain and

cross-correlate the resulting null maps across the former using

the FGPXS method described earlier. With the set of resulting

null test FGPXS

∆

we can perform a

-test, with a null

hypothesis that the difference maps are consistent with noise,

by first computing

null







∆

−

∆













∆







(16)

with the expectation value of the null FGPXS

∆

under the

null hypothesis. Here

∆

is the uncertainty of the null FGPXS

∆

in bin

for null variable

that is estimated using the RND

method described earlier in Sect. 2.3.

Thereafter we can compute the probability-to-exceed (PTE)

that quantifies the probability of obtaining a value

null

higher. The PTE is defined as

PTE(

)

−

CDF(

)

(17)

where for a given probability distribution function

(

)

of the

null

values the corresponding cumulative distribution function

is denoted as

CDF(

)

In our case,

(

)

does not follow the usual analytical

distribution because the noise properties of the FGPXS are not

completely known analytically (see Watts et al. 2020; Nadarajah

& Pogány 2016; Gaunt 2019, for some examples of how cross-

spectrum noise properties can look). We thus compute the PTEs

numerically by using an ensemble of RND maps equivalent to

those we already use for estimating uncertainties as these will

perfectly reflect the noise properties and biases in the data, as

well as obey the null hypothesis. For each separate processing

run – over fields, fast- and slow-moving azimuth data – we com-

pute 244 RND maps, of which we use 61 for power spectrum

uncertainty estimation and the remaining 183 for measuring the

numerical

-distribution.

4. Transfer functions

As is described by Foss et al. (2022) and Ihle et al. (2022) the

COMAP maps are not unbiased as the low-level filtering of the

data, the binning of the data into voxels, and the finite resolution

of the telescope beam will attenuate the signal in the maps. In

this section, we explain how we de-bias our power spectrum esti-

mates using transfer functions for each of the main effects that

result in signal loss. The beam and voxel window smoothing of

the signal is corrected using analytically computed transfer func-

tions, while the low-level filtering attenuation is quantified using

simulations. Details on how each transfer function is estimated

are discussed by Lunde et al. (2024).

When performing a power spectrum analysis of our maps, as

is described in Sect. 2, we obtain an estimate of the signal that

is biased by several different effects. To see how the signal is

biased we can write the FGPXS signal estimator as

(

)

(

)

(

⊥

)

(

⊥

)

(

∥

)

(18)

where the transfer function

(

)

is the product of the filter trans-

fer function

(

)

, the beam smoothing transfer function

(

⊥

)

as well as the pixel and spectral channel windows,

(

⊥

)

and

(

∥

)

. The transfer function can be written in this multiplicative

form in the Fourier domain because the low-level filtering and

the smoothing of small-scale structures due to the instrumental

beam and voxel window of the map grid can all (approximately)

be expressed as a convolution in map domain. In Fig. 2 we

show the full transfer function product

(

)

, while the individ-

ual transfer function elements are shown in detail in Sect. 6. of

Lunde et al. (2024).

Using our transfer function estimate we can de-bias the

FGPXS by deconvolution;

(

)

(19)

with the uncertainties of the signal estimator being affected in a

similar manner,

(

)

(20)

becoming large whenever the transfer function

(

)

becomes

small.

A336, page 6 of 19

Stutzer, N.-O., et al.: A&A, 691, A336 (2024)

0.1

0.2

0.4

0.8

[Mpc

]

0.025

0.05

0.1

0.2

0.4

[Mpc

]

0.0

0.1

0.2

0.3

0.4

(

)

34.0

17.0

8.5

4.3

Angular scale [arcmin]

Fig. 2.

Full power spectrum transfer function used to account for the sig-

nal losses due to the low-level filtering pipeline, the instrumental beam

smoothing, and the voxel window of the maps (see Lunde et al. 2024,

for details on each individual transfer function). Thin green contours

indicate the bin edges of the (1D) spherically averaged FGPXS.

5. Power spectrum results

In this section, we present the main power spectrum results of

this paper. The raw data going into the power spectra are filtered,

calibrated, and binned into maps after a set of data selection steps

that remove scans that are likely contaminated by systematic

effects. This is described in detail by Lunde et al. (2024).

As one of the main lessons learned from COMAP ES was

to employ only CES scans, and no longer use a Lissajous scan-

ning strategy, the data presented here consist only of CES data

(Foss et al. 2022; Ihle et al. 2022). Specifically, we include all

data obtained up to November 2023, both the ES (Season 1) CES

data as well as all data gathered in S2. The data volume obtained

in S2 is, as has been explained by Lunde et al. (2024), effectively

around eight times larger than the Season 1 CES data after data

selection. In addition, the ES analysis of Season 1 data excluded

several detectors that were either offline or excluded due to clear

signs of systematic excess in reduced

-tests or in visual inspec-

tions of feed cross-spectra. We are able to include these in the

S2 analysis because all feeds were functioning during S2 and

the map-domain PCA described in Lunde et al. (2024) strongly

suppresses detector-specific systematic effects.

We note that in the ES analysis Ihle et al. (2022) removed

feed-feed cross-spectra both through a reduced

-test, and man-

ual inspection of misbehaving feed combinations. Due to better

low-level data processing, we were able to remove all data-driven

cuts in the power spectrum domain, with the increased set of

null tests since ES working as an additional safeguard against

systematic effects (see Sect. 6 for discussion of null test results).

Lastly, in Appendix D, we show a simple end-to-end sig-

nal injection test as a qualitative test of our pipeline’s ability

to recover a known signal’s amplitude within the estimated

experimental errors and power spectrum transfer function.

5.1. The cylindrically averaged power spectrum result

In Fig. 3, we show the cylindrically averaged (2D) mean FGPXS

for all three fields separately, as well as in combination. The

figure also shows the sensitivity per

(

⊥

∥

)

-bin as well as the

FGPXS in units of the sensitivity.

When looking at the 2D FGPXS in Fig. 3 we note that the

noise blows up on small scales, particularly so in the angular

direction, due to the COMAP transfer function seen in Fig. 2

(Lunde et al. 2024). However, we see no obvious patterns in the

2D FGPXS that would indicate a systematic effect. In fact, the

spectra resemble white noise.

As was mentioned earlier, in Sect. 2.2, we want to avoid

issues with poorly constrained large-scale modes, strong mode

mixing, and possible residual large-scale systematic effects. We

mitigate these issues by excluding 2D bins at

⊥

1 Mpc

−

An example of spurious fluctuations induced by poor overlap

can be seen in the COMAP ES cylindrically averaged FPXS of

Field 1 (see Ihle et al. 2022, noting that Field 1 is especially

susceptible to poor detector overlap due to its position at decli-

nation zero) as correlated structures along constant

⊥

at scales

⊥

1 Mpc

−

. These correlations have since been understood

to originate from sub-optimal detector overlap, and are pushed

to larger scales due to a larger sky overlap when computing

cross-spectra between feed groups instead of individual feeds.

In interim estimates we found the average of the maximum cor-

relations between a bin and all the others to be around

15%

scales

⊥

≥

1 Mpc

−

, while the correlations at

⊥

≤

1 Mpc

−

are somewhere in the

–

70%

regime. Improved modeling of

these correlations will be the aim of future work.

5.2. The spherically averaged power spectrum result

As interpreting the 2D cylindrically averaged FGPXS can be

somewhat unintuitive we can bin the spectra into 1D by perform-

ing a full spherical averaging. This is done as is described in

Sect. 2.2, in which the 1D bin-edges are indicated as thin green

contours in Fig. 3. When doing so we obtain the spherically

averaged FGPXS for the three fields, as well as the combination

thereof, as is seen in Fig. 4.

As was discussed in Sect. 5.1, we excluded scales

⊥

1 Mpc

−

from the power spectrum analysis to avoid issues

with poor cross-map overlap, mode mixing and large correlations

between large scale bins. Therefore, Fig. 4 only shows FGPXS

data points on scales

1 Mpc

−

. Similar to the discussion in

Sect. 5.1, we estimate the average of the maximum correlation

between a 1D bin and all the others, on scales

1 Mpc

−

to be

≲

10%

after excluding the large

⊥

scales and performing

the spherical averaging. Given this

≲

10%

level, we shall assume

for Season 2 analyses downstream that the spherically averaged

1D FPGXS bins are approximately uncorrelated. As with the 2D

FGPXS discussed in Sect. 5.1, we intend to improve the exact

modeling of these correlations in future work.

When looking at Fig. 4 we note that Fields 2 and 3 have the

highest sensitivity, while Field 1 has around

50%

larger errors

than the two other fields. This is because, of the three COMAP

fields, Field 1 is most affected by the low-level data cuts (see

Lunde et al. 2024), and due to its location at zero declination, is

particularly susceptible to poor detector overlap. In addition, by

rejecting poorly overlapping detector combinations, which are

also the least sensitive, we prevent the resulting strong mode

mixing and potential consequent leakage of systematic effects

into smaller scales at the cost of a relatively minimal loss in

A336, page 7 of 19

Stutzer, N.-O., et al.: A&A, 691, A336 (2024)

0.025

0.05

0.1

0.2

0.4

[Mpc

]

34.0

17.0

8.5

4.3

Angular scale [arcmin]

Field 1

34.0

17.0

8.5

4.3

Angular scale [arcmin]

34.0

17.0

8.5

4.3

Angular scale [arcmin]

0.025

0.05

0.1

0.2

0.4

[Mpc

]

Field 2

0.025

0.05

0.1

0.2

0.4

[Mpc

]

Field 3

0.1

0.2

0.4

0.8

[Mpc

]

0.025

0.05

0.1

0.2

0.4

[Mpc

]

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

(

)

[10

(Mpc)

]

0.1

0.2

0.4

0.8

[Mpc

]

Combined

(

)

[ K

(Mpc)

]

0.1

0.2

0.4

0.8

[Mpc

]

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

(

)

Fig. 3.

Cylindrically averaged (2D) feed group pseudo-cross-spectra. Columns show, from left to right, the full spectra, the corresponding

uncertainty, and the ratio between the two. Rows show, from top to bottom, Fields 1, 2, 3, and all three combined. The approximate angular scale,

assuming the central COMAP redshift at

, corresponding to each

⊥

is shown as a twin axis on the upper row of plots. Thin green contours

indicate the bin edges of the (1D) spherically averaged FGPXS.

A336, page 8 of 19

Stutzer, N.-O., et al.: A&A, 691, A336 (2024)

3.5

0.0

3.5

7.0

(

) [10

Mpc

]

Combined

Field 1

Field 2

Field 3

0.1

0.2

0.4

0.8

[Mpc

]

Fig. 4.

Spherically averaged FGPXS with

uncertainties for Fields 1,

2 and 3 (orange, red and blue respectively) as well as the combination

of the three (black) in units

μ

Mpc

(

upper panel

) and in units of the

power spectrum uncertainties (

lower panel

). The data points have

been slightly offset from their true

-position to increase readability

(see Table B.1 for an overview over bin centers, FGPXS values, and

uncertainties).

sensitivity. As a consequence, we keep

of the feed group

cross-spectra of Field 1 in the analysis.

As we can see from Fig. 4, the cross-spectra are largely con-

sistent with zero to within

in most bins. However, perhaps

the most notable feature is the high power in the second and

most sensitive

-bin, at

12 Mpc

−

18 Mpc

−

, which

is respectively at around

and

significance above zero

for Fields 1 and 2. Meanwhile, for Field 3 the same bin is con-

sistent with zero power. When combining the three fields, the

co-added data point in the second

-bin has a value that is

away from zero. For each of the spherically averaged FGPXS

of the three fields, we compute their

PTE to check their

constancy with zero power. In doing so, we obtain PTEs of,

respectively,

, and

for Fields 1–3. The field-

combined spherically averaged FGPXS results in a

34%

PTE.

As for the null tests, the PTE is estimated from the numeri-

cal RND

ensemble. While the combined 1D

∼

power

12 Mpc

−

18 Mpc

−

bin is interesting, we do not

consider it a statistically significant excess given the estimated

PTEs. Thus, we shall have to wait for future analyses, and more

data, to answer definitively whether this excess is simply noise

or not.

Although we do not consider the field-combined

point

statistically significant, the agreement between two of the three

fields is interesting to note. As Fields 1 and 2 are quite different

in terms of their path across the sky as it is seen from the tele-

scope (Field 1 being a zero declination field while Field 2 is at

a declination of

◦

; see Foss et al. (2022) for details on the

fields) they also are expected to have some independent system-

atic effects. However, Fields 2 and 3 are more alike and would

be expected to share certain systematic effects.

5.3. Comparison to COMAP Early Science and COPSS

Having computed the spherically and field-averaged FGPXS

from the data we can compare it to the previous COMAP release

as well as the CO Power Spectrum Survey (COPSS), the only

other comparable CO(1–0) LIM survey in the literature with

published data (Keating et al. 2016; Kovetz et al. 2017, 2019;

Bernal & Kovetz 2022). This is illustrated in Fig. 5 where

the field-averaged FGPXS is plotted together with the COMAP

ES constant-elevation-scan FPXS of Ihle et al. (2022) and the

individual COPSS data points from Keating et al. (2016).

The first thing we notice when considering Fig. 5 is the dra-

matic reduction in the uncertainty of the current measurement

compared to that from our ES phase (Ihle et al. 2022). Compared

to the Ihle et al. (2022) FPXS the current level of sensitivity has

increased by a factor

∼

–

across our six most sensitive bins

09 Mpc

−

73 Mpc

−

. This illustrates the significant

increase in the effective data volume by around a factor of eight.

Even though the low-level data selection procedure detailed by

Lunde et al. (2024) is somewhat more strict than the one in

COMAP ES (Foss et al. 2022; Ihle et al. 2022), this is more

than compensated by the lack of data cuts in the power spectrum

domain, resulting in a significant increase in sensitivity overall.

In other words, we have demonstrated that uncertainties in the

power spectra integrate down in accordance with expectations

for noise-dominated data.

The two highest

-bins have somewhat larger errors in the

current result compared to the COMAP ES spectrum. This is due

to a combination of the analytical beam transfer function now

applied and a stricter 2D

-space mask. The beam transfer func-

tion now applied is somewhat more strict than the numerically

computed one of Ihle et al. (2022) on scales closer to the Nyquist

limit in the angular direction. Additionally, to avoid problems

with aliasing we have masked the outer-most bins in both

∥

and

⊥

. As a result, the outer-most 1D

-bins contain a lower number

of samples than they would have for the same 1D bins of Ihle

et al. (2022).

The COPSS power spectrum estimate (Keating et al. 2016)

primarily covers scales smaller than COMAP, but the two exper-

iments overlap at

3 Mpc

−

≲

≲

0 Mpc

−

, where they are

largely consistent with each other. The only noteworthy disagree-

ment between COPSS and the field-combined FGPXS is a mild

∼

tension in terms of the combined error between the two

power spectrum estimates at

∼

6 Mpc

−

. As we can see from

Fig. 5 this point of mild tension coincides with one of the two

COPSS points in which they reported a

excess above zero.

Albeit with large uncertainties, we see that compared to

COPSS and COMAP ES the updated COMAP data points clus-

ter significantly closer to, and are consistent with, the two

brightest models that were not already excluded in ES (Chung

et al. 2022), namely the COMAP fiducial model

and the Li-

Keating model of Keating et al. (2020). For more discussion

of the consistency of COPSS with the current COMAP result,

including modeling implications, we refer the interested reader

to Chung et al. (2024a).

When comparing the power spectrum sensitivity of COMAP

to that of COPSS, we must take into account the smaller

-bins

in the COPSS analysis. Although the two experiments have a

certain region of overlap in

-space, the different bin sizes of

COPSS and COMAP result in a different intrinsic within-bin

variance. To mitigate this effect we can define the normalized

sensitivity

√

∆

, where

∆

is the volume of a spher-

ical

-shell defined by the bin

-space. Two bins with the

same value for

would have the same sensitivity,

, if they

A double power-law model relating halo masses in cosmological sim-

ulations to CO luminosities; see specifically “UM+COLDz+COPSS” in

Table 1 of Chung et al. (2022) for their fiducial model definition.

A336, page 9 of 19

Stutzer, N.-O., et al.: A&A, 691, A336 (2024)

(

) [10

Mpc

]

This work

COMAP ES

COPSS

COMAP fiducial

Li-Keating

0.1

0.2

0.4

0.8

[Mpc

]

0.1

0.2

0.4

0.8

[Mpc

]

(

) [10

Mpc

]

Fig. 5.

Detailed overview of our field-averaged FGPXS, as well as other datasets and some selected models from the literature.

Upper panel

spherically averaged FGPXS with

uncertainties for the field-combined data presented in this paper (black), the COMAP ES field-averaged

FPXS (blue; Ihle et al. 2022), and the COPSS power spectrum (orange; Keating et al. 2016).

Lower panel

: Corresponding power spectra divided

by their respective

uncertainty.

Inset

: zoom-in of the COMAP data points and two comparable models from the literature, namely the fiducial

second season COMAP model (Chung et al. 2022) and the Li-Keating model Li et al. (2016); Keating et al. (2020) model. None of the models

includes any line-broadening discussed by Chung et al. (2021). Our data points and those of COMAP ES have been slightly offset from their true

-position to increase readability (see Table B.1 for an overview over bin centers, FGPXS values and uncertainties).

were binned to a standardized bin size. In other words,

traces

the underlying continuous sensitivity of each experiment and

scale, and as a result, the normalized sensitivity across

-bins

and surveys becomes comparable. We illustrate the normalized

sensitivity,

, in Fig. 6 for all data points shown in Fig. 5.

The figure nicely illustrates the scales to which COMAP and

COPSS are most sensitive. We see that COMAP is most sensi-

tive on large scales at

1 Mpc

−

3 Mpc

−

, while COPSS

is most sensitive on small scales,

5 Mpc

−

0 Mpc

−

where the COMAP beam starts to dominate. However, the cur-

rent FGPXS result has a peak sensitivity increase of around a

factor of eight and ten compared to COMAP ES and COPSS

respectively (see Table A.1 for a detailed list of exact normal-

ized sensitivity improvements). This improvement in relative

sensitivity compared to COPSS and COMAP ES is expected

to increase further as the COMAP instrument gathers more

data, and illustrates our ability to remove systematic effects to

below the noise level and integrate the noise of the incoming

data. In fact, as COPSS to-date remains the only comparable

CO(1–0) LIM experiment with published data, COMAP cur-

rently provides the most sensitive CO(1–0) LIM constraints in

the field.

5.4. Upper limits on the power spectrum

Given the factors of, respectively, eight and ten times the sen-

sitivity of our power spectrum result compared to the COMAP

ES and COPSS data, it is interesting to consider the upper limits

(UL) at

95%

confidence on a non-zero CO(1–0) power spectrum

that can be derived from the data points. These are shown in

Fig. 7 for the spherically and field-averaged COMAP FGPXS,

the COMAP ES (Ihle et al. 2022) data points as well as the

COPSS (Keating et al. 2016) power spectrum estimate. As we

are at the level of sensitivity where it becomes more informative

to look at the ULs per

-bin we only consider the ULs derived per

-bin in this work. We show only bin-wise derived ULs from the

COMAP ES (Ihle et al. 2022) and COPSS (Keating et al. 2016)

data to facilitate a direct comparison to our result. All ULs are

computed under the assumption that the astrophysical CO signal

must be positive. For comparison, two of the closest models from

the literature are included in the plot; the COMAP fiducial model

from Chung et al. (2022) and the Li-Keating model – a version

of the Li et al. (2016) model from Keating et al. (2020). While

the

12 Mpc

−

18 Mpc

−

FGPXS bin has a

excess

above zero, we still present it as a

95%

upper limit in Fig. 7 as

we do not consider the excess statistically significant.

As in Fig. 6, the ULs we present in Fig. 7 reflect

regions in which each survey is most sensitive. The

95%

ULs of this work and those derived from COMAP ES (Ihle

et al. 2022) are most constraining in the six most sensitive

COMAP bins at

09 Mpc

−

73 Mpc

−

. Meanwhile the

COPSS Keating et al. (2016) ULs are at their lowest around

7 Mpc

−

6 Mpc

−

, beyond where the COMAP beam

and voxel window dominate and blow up the noise. Compared

to COMAP ES, we see a significant improvement in the current

ULs per

-bin. Specifically, each of our six most sensitive

bins can individually constrain

(

)

2400

–

4900

Mpc

95%

confidence. The maximum improvement between the

A336, page 10 of 19