COMAP Pathfinder – Season 2 results - III. Implications for cosmic molecular gas content at z ~ 3

Astronomy

&

Astrophysics

A&A, 691, A337 (2024)

https://doi.org/10.1051/0004-6361/202451122

COMAP Pathfinder – Season 2 results

III. Implications for cosmic molecular gas content at

z

∼

D. T. Chung

⋆

, P. C. Breysse

, K. A. Cleary

, D. A. Dunne

, J. G. S. Lunde

, H. Padmanabhan

N.-O. Stutzer

, D. Tolgay

, J. R. Bond

, S. E. Church

, H. K. Eriksen

, T. Gaier

, J. O. Gundersen

S. E. Harper

, A. I. Harris

, R. Hobbs

, H. T. Ihle

, J. Kim

, J. W. Lamb

, C. R. Lawrence

N. Murray

, T. J. Pearson

, L. Philip

, A. C. S. Readhead

, T. J. Rennie

I. K. Wehus

, and D. P. Woody

(COMAP Collaboration)

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

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

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

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

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

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

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

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

Jodrell Bank Centre for Astrophysics, Department of Physics & Astronomy, The University of Manchester, Oxford Road,

Manchester M13 9PL, UK

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

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

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

Republic of Korea

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

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

Received 14 June 2024 / Accepted 23 September 2024

ABSTRACT

The Carbon monOxide Mapping Array Project (COMAP) Pathfinder survey continues to demonstrate the feasibility of line-intensity

mapping using high-redshift carbon monoxide (CO) line emission traced at cosmological scales. The latest COMAP Pathfinder power

spectrum analysis is based on observations through the end of Season 2, covering the first three years of Pathfinder operations. We use

our latest constraints on the CO(1–0) line-intensity power spectrum at

∼

to update corresponding constraints on the cosmological

clustering of CO line emission and thus the cosmic molecular gas content at a key epoch of galaxy assembly. We first mirror the

COMAP Early Science interpretation, considering how Season 2 results translate to limits on the shot noise power of CO fluctuations

and the bias of CO emission as a tracer of the underlying dark matter distribution. The COMAP Season 2 results place the most

stringent limits on the CO tracer bias to date, at

⟨

T b

⟩

μ

K, which translates to a molecular gas density upper limit of

⊙

Mpc

−

∼

given additional model assumptions. These limits narrow the model space significantly compared to previous

CO line-intensity mapping results while maintaining consistency with small-volume interferometric surveys of resolved line candidates.

The results also express a weak preference for CO emission models used to guide fiducial forecasts from COMAP Early Science,

including our data-driven priors. We also consider directly constraining a model of the halo–CO connection, and show qualitative hints

of capturing the total contribution of faint CO emitters through the improved sensitivity of COMAP data. With continued observations

and matching improvements in analysis, the COMAP Pathfinder remains on track for a detection of cosmological clustering of CO

emission.

Key words.

galaxies: high-redshift – diffuse radiation – radio lines: galaxies

1. Introduction

Line-intensity mapping (LIM) surveys map the large-scale struc-

ture of the Universe in large cosmological volumes, but not

⋆

Corresponding author;

dongwooc@cornell.edu

through discrete resolved tracer sources. Rather, LIM surveys

achieve this through unresolved emission in specific spectral

lines, including lines associated with different phases of the

star-forming interstellar medium (ISM) such as carbon monox-

ide (CO) and the [C

] line from singly ionized carbon (see

A337, page 1 of 14

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Chung, D. T., et al.: A&A, 691, A337 (2024)

Kovetz et al. 2019 and Bernal & Kovetz 2022 for recent reviews).

As part of a range of emerging interferometric and single-dish

LIM surveys from radio to sub-millimeter wavelengths, the CO

Mapping Array Project (COMAP; Cleary et al. 2022) is building

a dedicated centimeter-wave LIM program to map the cosmic

clustering of emission in the CO(1–0) and CO(2–1) lines from

the epochs of galaxy assembly (

∼

, just before so-called “cos-

mic noon”) and reionization (

∼

, “cosmic dawn”). COMAP

science encompasses both the astrophysics of the assembly of

molecular gas at these key epochs of galaxy evolution, and ulti-

mately the cosmological implications of observed high-redshift

large-scale structure traced by CO emission.

The first phase of COMAP is the COMAP Pathfinder, a 26–

34 GHz spectrometer comprising a single-polarization 19-feed

array of coherent receivers on a single 10-meter dish at the

Owens Valley Radio Observatory (Lamb et al. 2022). The focus

of the Pathfinder survey is on CO(1–0) emission from

∼

or a lookback time of

∼

Gyr. Around this “cosmic half-

past eleven”, we survey galaxies assembling towards the “cosmic

noon” of peak cosmic star formation activity (Somerville &

Davé 2015; Förster Schreiber & Wuyts 2020). Following the

Early Science analysis of Foss et al. (2022) and Ihle et al. (2022)

based on the first season of observations (Season 1), the Sea-

son 2 data analysis by Lunde et al. (2024) and Stutzer et al.

(2024) encompasses three years of observations and improved

data cleaning methods for almost an order-of-magnitude increase

in power spectrum sensitivity.

With such progress continuing to demonstrate the feasibil-

ity of CO LIM survey operations and low-level data analysis, we

present here the corresponding update on our understanding of

CO(1–0) emission at

∼

. To better understand high-redshift

CO(1–0) emission is to better understand the molecular gas con-

tent of the high-redshift universe. Previous studies (as reviewed

by, e.g., Carilli & Walter 2013) have established the strong

correlation between CO line luminosity and dust-obscured star

formation activity across a diversity of low- and high-redshift

galaxies, as should be expected for a tracer of cold molecular gas

that fuels such star formation activity. Using a LIM approach to

surveying CO line emission across large cosmological volumes

efficiently maps the luminosity density of this emission across

such volumes – integrated across faint and bright galaxies alike –

and thus sheds light on the cosmic molecular gas content and

thus cosmic star formation history in ways that strongly comple-

ment conventional surveys cataloguing individual line emitters

or star-forming galaxies (Pavesi et al. 2018; Decarli et al. 2020;

Lenki

c et al. 2020; Garratt et al. 2021). The Pathfinder survey at

“cosmic half-past eleven” is only the first step towards extending

our understanding of the history of cosmic gas and star formation

towards “cosmic dawn”, but a key step nonetheless in testing our

ability to use LIM statistics to derive astrophysical constraints of

noteworthy constraining power.

We carry out a high-level analysis of the power spectrum

constraints of Stutzer et al. (2024) to answer the following

questions:

– “How much does the increased data volume improve con-

straints on the clustering and shot noise power of cosmolog-

ical CO(1–0) emission at

∼

?”

– “Can COMAP Season 2 data better constrain the empir-

ical connection between CO emission and the underlying

structures of dark matter?”

We structure the paper as follows. In Sect. 2 we outline our

methodology for interpretation, including but no longer limited

to methods previously used in Chung et al. (2022). We dis-

cuss the results of our analysis in Sect. 3, and implications for

understanding CO emission and interpreting past and future CO

LIM surveys in Sect. 4. We end with our primary conclusions

and future outlook in Sect. 5.

We assume a

CDM cosmology with parameters

Ω

286

Ω

714

Ω

047

100

km s

−

Mpc

−

with

, and

, to maintain consistency

with previous COMAP simulations (starting with Li et al. 2016).

Distances carry an implicit

−

dependence throughout, which

propagates through masses (all based on virial halo masses, pro-

portional to

−

) and volume densities (

∝

). Logarithms are

base-10 unless stated otherwise.

2. Methods

The primary target of the COMAP Pathfinder is the power spec-

trum of spatial-spectral emission after subtraction of continuum

emission and systematic effects. Any residual fluctuations should

predominantly arise from clustered populations of CO-emitting

high-redshift galaxies, meaning that we interpret any constraints

on the residual emission as constraints on these CO emitters. In

the simplest possible model, the power spectrum as a function of

comoving wavenumber

consists of the matter power spectrum

(

)

scaled by some amplitude

clust

, plus a scale-independent

shot noise amplitude

shot

(

)

clust

(

)

shot

(1)

The matter power spectrum describes the distribution of mat-

ter density contrast across comoving space, and evolves with

redshift as large-scale structure forms and grows. The spatial-

spectral fluctuations in CO brightness temperature across cos-

mological scales trace the clustering of the underlying matter

fluctuations with some bias, which informs the clustering ampli-

tude

clust

. In combination with halo models of CO emission

that postulate average CO luminosities per halo of collapsed dark

matter, constraining

clust

(or related quantities) and

shot

allows

us to understand not just the global cosmic abundance of CO,

but also the relative contribution of different sizes of halos and

thus of galaxies. Estimation of the CO line power spectrum

(

)

is thus a key target of COMAP low-level analyses.

The goal of this section is to outline methods for the kind

of analyses suitable for the current level of sensitivity achieved

by the COMAP dataset. First, Sect. 2.1 reviews the COMAP

Season 2 power spectrum results in relation to previous work.

Then, Sect. 2.2 reviews a simple two-parameter analysis as car-

ried out by Chung et al. (2022) for COMAP Early Science,

constraining the clustering and shot noise amplitudes and only

indirectly using halo models to support physical interpretation.

Finally, Sect. 2.3 outlines a five-parameter analysis to directly

constrain a halo model of CO emission, as carried out by Chung

et al. (2022) to derive priors for COMAP Early Science but

incorporating COMAP data for the first time.

2.1. Foundational data: COMAP Season 2 power spectrum

constraints

The present work made use of the results of Stutzer et al. (2024),

which derived updated power spectrum constraints based on

COMAP Pathfinder survey data collected across 17500 hours

over three fields of 2–3 deg

each, between its commissioning

in May 2019 and the end of the second observing season in

November 2023. We also made use of the prior work of the CO

Power Spectrum Survey (COPSS; Keating et al. 2016), which

performed a pilot CO LIM survey targeting largely the same

A337, page 2 of 14

Chung, D. T., et al.: A&A, 691, A337 (2024)

−

(Mpc

−

)

k P

(

) (

Mpc

)

Models:

Padmanabhan2018,

duty

= 1

Pullen+2013, Model B

Keating+2020 clustering UL

Keating+2020

shot

estimate

Li+2016–Keating+2020

Chung+2022, UM+COLDz+COPSS

Li+2016

Padmanabhan2018,

duty

= 0.1

Yang+22 empirical fit

LIM observations:

(CO(1–0),

∼

2.8

, 95% UL /

CI)

COPSS

COMAP Season 1

COMAP Season 2 (this work)

Fig. 1.

COMAP Season 2 95% upper limits (given

(

)

) on the

∼

CO(1–0) power spectrum, with analogous limits from COPSS (Keating

et al. 2016) and COMAP Season 1 (Chung et al. 2022). Some

-bins in COPSS and COMAP Season 2 data show marginal excesses, influencing

analyses in this work; we thus show

intervals for these bins unlike in Stutzer et al. (2024). We also show predictions based on Chung et al.

(2022), Padmanabhan (2018), Pullen et al. (2013), Li et al. (2016), and Yang et al. (2022), plus a variation on the Li et al. (2016) model from Keating

et al. (2020), and the Keating et al. (2020) re-analysis of COPSS constraining clustering (triangles) and shot-noise amplitudes (dashed line).

observing frequencies, but with an interferometric dataset prob-

ing smaller scales. The COMAP observations are subject to the

effects of instrument and pipeline response, such as filtering,

pixelization, and beam smoothing. However, the results as con-

sidered in this work correct for these effects by applying the

inverse of the estimated power spectrum transfer function per

bin. We expect mode mixing in COMAP data is still at the level

of Ihle et al. (2022) at most, that is to say less than 20% for the

comoving wavenumber range of

≳

Mpc

−

that we consider.

Fig. 1 shows these results alongside the range of expectations

for the

∼

CO(1–0) emission power spectrum from empirical

modeling in the decade leading up to this dataset (Pullen et al.

2013; Li et al. 2016; Padmanabhan 2018; Keating et al. 2020;

Chung et al. 2022; Yang et al. 2022). These models either pos-

tulate a connection between dark matter halo properties and CO

luminosity via intermediate galaxy properties such as the star

formation rate (SFR), or directly model the halo–CO connection

constrained by observed CO luminosity functions and CO LIM

measurements.

Of the models shown in Fig. 1, only the models of

Padmanabhan (2018) and Chung et al. (2022) fall into the latter

category. Keating et al. (2020) also provide empirical estimates

for the clustering and shot noise amplitudes, but this is sim-

ply based on decomposing the COPSS measurement of Keating

et al. (2016) into clustering and shot noise components, rather

than a detailed halo model. In a different context Keating et al.

(2020) do provide a halo model, which we term the Li et al.

(2016)–Keating et al. (2020) model, varying the Li et al. (2016)

model by using the same halo–SFR connection from Behroozi

et al. (2013a,b) but replacing the SFR–CO connection (via

infrared luminosity) derived from a compilation of local and

high-redshift galaxies (Carilli & Walter 2013) with one based

on a local sample observed by Kamenetzky et al. (2016).

Even before any detailed analyses, compared to COMAP

Season 1 we clearly see an increasing rejection of Model B

of Pullen et al. (2013) and of the Padmanabhan (2018) model

with CO emission duty cycle

duty

. We refer the reader to the

Early Science work of Chung et al. (2022) for the implications

of excluding these models. As with COMAP Early Science, we

exclude these models in the clustering regime, rather than the

shot-noise dominated scales surveyed by COPSS. However, the

COMAP Season 2 sensitivity is sufficient to exclude these mod-

els clearly in

individual

-bins of width

∆

(log

Mpc

)

155

rather than having to rely on a co-added measurement across all

as was necessary in Early Science. For reference, we show in

Appendix A the original data points behind these upper limits, in

a way that more closely resembles Fig. 4 of Stutzer et al. (2024).

Note also a weak tension against the previous positive

COPSS measurement in overlapping

-ranges. The original

COPSS analysis of Keating et al. (2016) measured the CO

A337, page 3 of 14

Chung, D. T., et al.: A&A, 691, A337 (2024)

power spectrum at

Mpc

−

to be

(

)

−

μ

Mpc

, for a best estimate of

(

Mpc

−

)

μ

Mpc

. This is co-added across the entire

-range

spanned by COPSS, with the highest sensitivity achieved around

–

Mpc

−

. By contrast, in a single

-bin spanning

–

Mpc

−

, the present COMAP data places a 95% upper

limit of

μ

Mpc

, lying below the COPSS co-added

best estimate. However, the COPSS result is itself only a tenta-

tive one at

≈

significance, and so there is no statistically

significant discrepancy. COMAP data are also entirely consis-

tent with the estimate of

shot

−

μ

Mpc

from

the later re-analysis of COPSS data by Keating et al. (2020),

which marginalized over the possible contribution to

(

)

from

clustering. In fact our power spectrum results show a marginal

excess at

≈

Mpc

−

that, while well below the upper limit

implied by the direct COPSS re-analysis of Keating et al. (2020),

does tentatively indicate a preference for models similar to the Li

et al. (2016)–Keating et al. (2020) model. The remainder of

this work establishes this preference more quantitatively, and

consider other implications of these results.

2.2. Two-parameter analysis: Constraining CO tracer bias

and shot noise

The most direct way to analyze the COMAP Season 2 con-

straints is to decompose the CO power spectrum into clustering

and shot noise terms as in Eq. (1), with a fixed cosmological

model and no assumptions around detailed astrophysical mod-

eling. The COMAP data then constrain the possible range of

values for

clust

and

shot

, which we may then compare to model

predictions for these amplitudes for the clustering and shot noise

contributions to the power spectrum.

However, physical interpretation requires some amount of

guidance from models. For a halo model of CO emission where

halos of virial mass

emit with CO luminosity

(

)

, we

might know the halo mass function

describing the dif-

ferential number density of halos of mass

, and the bias

(

)

with which the halo number density contrast traces the

continuous dark matter density contrast. Then the cosmologi-

cal fluctuations in CO(1–0) line temperature trace the underlying

dark matter fluctuations with a linear scaling of

⟨

T b

⟩

∝

(

)

(

)

(2)

This should be understood as a mean line temperature–bias

product, with appropriate normalization factors applied to con-

vert luminosity density to brightness temperature. We may also

ascribe a dimensionless bias

to CO emission contrast by

dividing out the average line temperature or luminosity density:

(

)

(

)

(

)

(

)

(

)

(3)

Furthermore, any halo model of

(

)

predicts the shot

noise, proportional to the second bias- and abundance-weighted

moment of the

(

)

function rather than the first moment:

shot

∝

(

)

(

)

(4)

The quantity

shot

directly describes the shot noise amplitude,

but the same is not true of

⟨

T b

⟩

in relation to the clus-

tering amplitude. In real comoving space we would expect

clust

⟨

T b

⟩

, but redshift-space distortions (RSD) enhance the

clustering term as large-scale structure coherently attracts galax-

ies (Kaiser 1987; Hamilton 1998). In the linear regime of small

, and given that

Ω

(

)

≈

at COMAP redshifts,

clust

≈

⟨

T b

⟩

(5)

Based on prior modeling efforts, we consider

to be a

fairly conservative lower bound on CO tracer bias, as outlined

by Chung et al. (2022). This bound on

in turn allows us to

bound

⟨

⟩

⟨

T b

⟩

based on an upper bound on

clust

We considered two variants of a two-parameter analysis of

the COMAP data, the same carried out in Chung et al. (2022).

1. The first variant is a model-agnostic evaluation of the likeli-

hood of different values of

clust

and

shot

given the

(

)

data

points available from the COPSS results of Keating et al.

(2016) and/or from COMAP data through Season 2. We only

invoke a conservative limit of

to obtain an upper bound

⟨

⟩

from our constraint on

clust

2. The second variant assumes that given values for

⟨

T b

⟩

and

shot

, we can expect specific values for

and for an

effective line width

eff

describing the suppression of the

high-

CO power spectrum from line broadening (Chung

et al. 2021). Exploration of an empirically constrained model

space informs fitting functions for

and

eff

given only

⟨

T b

⟩

and

shot

, as provided in Appendix B of Chung et al. (2022),

which then enter into calculation of the redshift-space

(

)

accounting for RSD and line broadening. We can directly

compare this

(

)

to our

(

)

data to evaluate the likelihood

of different values of

⟨

T b

⟩

and

shot

. We refer to this variant

as the “

- and

eff

-informed” analysis, versus the first “

- and

eff

-agnostic” version.

We may then compare likely and unlikely regions of this two-

parameter space to model predictions.

2.3. Five-parameter analysis: Directly constraining the

halo–CO connection

Neither variant of our two-parameter analysis truly directly con-

strains the physical picture of CO emission, only a clustering

term and a shot noise term. Given a fixed set of power spec-

trum measurements, the two-parameter analysis broadly projects

likelihood contours favouring either high clustering and low shot

noise, or low clustering and high shot noise. Yet physical models

should impose a strong prior such that clustering and shot noise

co-vary, given that the shot noise also tracks with luminosity

density, albeit at a higher order – cf. Eq. (4).

Directly modeling and constraining

(

)

thus has its uses.

While dark matter halos are not themselves the direct source of

CO emission or indeed any baryonic physics, a halo model of CO

emission still serves as a simple way to physically ground inter-

pretation of our CO measurements and introduce priors based on

other empirical constraints on the galaxy–halo connection.

2.3.1. Parameterization and derivation of “UM+COLDz”

posterior

To model

(

)

, we used the same parameterization and data-

driven procedure as in Chung et al. (2022). One of the datasets

driving this procedure is provided by the CO Luminosity Den-

sity at High-

(COLDz) survey (Pavesi et al. 2018; Riechers

et al. 2019), which identified line candidates at

∼

through

an untargeted interferometric search. In Chung et al. (2022) we

A337, page 4 of 14

Chung, D. T., et al.: A&A, 691, A337 (2024)

also introduced somewhat informative priors based loosely on

the work of Behroozi et al. (2019), which devised the U

VERSE

ACHINE

(UM) framework for an empirical model of

the galaxy–halo connection by connecting halo accretion histo-

ries to a minimal model of stellar mass growth. We thus once

again combined these “UM” priors with COLDz data and a basic

(

)

parameterization, just as in Chung et al. (2022).

We assumed a double power law for the linear average

(

)

. In observer units,

′

(

)

K km s

−

(

)

(

)

(6)

For CO(1–0),

(

)

⊙

−

′

(

)

K km s

−

(7)

We also modeled stochasticity albeit in a highly simplistic

fashion, assuming some level of log-normal scatter

(in units

of dex) about the average relation. We inherit this practice from

the common use of log-normal distributions to model intrinsic

scatter in, e.g., the halo–SFR connection (e.g.: Behroozi et al.

2013a,b) and the halo–CO connection as modeled for previous

early COMAP forecasts (Li et al. 2016).

The somewhat informative “UM” priors for the five free

parameters of

(

)

are as follows:

−

(8)

(9)

log

(10)

log (

⊙

)

(11)

For log-normal scatter, we assumed an initial prior of

(dex), taking cues from Li et al. (2016) for the central value

and slightly broadening the prior. We review the derivation of

priors for all of these parameters in somewhat more detail in

Appendix B.

We then narrowed these priors further by matching the lumi-

nosity function constraints of the COLDz survey. The matching

procedure is similar to that used in Chung et al. (2022). How-

ever, that procedure used a snapshot from the Bolshoi–Planck

simulation, used by Behroozi et al. (2019) but slightly discrepant

against our fiducial cosmology. Here, we used our own peak-

patch mocks (Stein et al. 2019) with virial masses matched to

the halo mass function of Tinker et al. (2008). We extracted

halos from

∈

45)

(or

∈

(5720

5844

19)

Mpc)

from these peak-patch mocks, since we are specifically trying

to match a luminosity function constraint at

∼

. We thus

obtained 161 independent realizations of a halo catalogue from

1140

124

Mpc

Gpc

box, comparable to the

Bolshoi–Planck snapshot with a box size of

(250

678)

Mpc (or

a volume of

Gpc

). A Markov chain Monte Carlo (MCMC)

procedure identifies the posterior (narrowed prior) based on a

likelihood calculation in addition to the mildly informative priors

outlined above. At each step:

– We used the sampled

(

)

parameters to convert a random

peak-patch realization of halo masses into CO luminosities.

– We then fit a Schechter function to the resulting CO lumi-

nosity function of that randomly chosen peak-patch box.

– We calculated the log-likelihood by comparing the Schechter

function fit against the COLDz posterior for the Schechter

function parameters via a kernel density estimator.

The MCMC used 250 walkers for 4242 steps, and we discarded

the first 1000 steps as a burn-in phase. The result is an informed

distribution, which we call the “UM+COLDz” posterior, for the

five parameters

{

,σ

}

of our

(

)

model.

2.3.2. Derivation of posteriors incorporating CO LIM data

To derive posteriors based on CO LIM power spectrum mea-

surements, we reran the same MCMC procedure used to derive

the “UM+COLDz” distribution with additional contributions to

the likelihood from any discrepancy with the CO LIM results. In

other words, we introduced additive log-likelihood terms,

∆

(log

)

∝−

[

model

(

)

−

data

(

)]

[

data

(

)]

(12)

evaluated against each dataset

data

(

)

with error

[

data

(

)]

for

the model

model

(

)

drawn at each MCMC step

Using our fiducial cosmology and the halo mass function

of Tinker et al. (2008), we numerically evaluated closed-form

expressions describing the CO power spectrum at

∼

We evaluated the above log-likelihood terms against the pre-

dicted

model

(

)

without imposing positivity priors, which would

be redundant with the always positive predictions of our

(

)

model.

We considered three (combinations of) datasets:

– The “UM+COLDz+COPSS” posterior derives from consid-

ering only the addition of COPSS data points as shown

in Keating et al. (2016).

– The “UM+COLDz+COPSS+COMAP S1” posterior derives

from considering both COPSS and COMAP Early Science

(

)

constraints from Season 1 data.

– The “UM+COLDz+COPSS+COMAP S2” posterior derives

from considering constraints from both COPSS and the

present work on COMAP Pathfinder data through Season 2.

While the MCMC procedure itself evaluates posteriors for

{

,σ

}

, we can use the resulting sampling of parame-

ter space to obtain posterior distributions for derived quantities

such as

(

)

⟨

⟩

, and

shot

. In doing so, we can look for

how (if at all) the “UM+COLDz+COPSS+COMAP S2” poste-

rior distinguishes itself from posteriors based on only previous

data.

3. Results

Having outlined the datasets and methods used in the analy-

ses, we now review the results in relation to previous models

and results. We consider outcomes of the two-paramater analy-

sis identifying overall amplitudes for clustering and shot noise

power in Sect. 3.1, followed by outcomes of the five-parameter

analysis fitting for the

(

)

relation in Sect. 3.2.

3.1. Two-parameter analysis

We summarize the results of the two-parameter analysis of the

COMAP results in Table 1, and show in Fig. 2 the probability

distributions when considering only COMAP data up to Season

This approximates the likelihood as Gaussian and independent

between

-bins, which we consider to be a reasonable approximation

at least for COMAP Season 2 data. In obtaining the

(

)

result, Stutzer

et al. (2024) found that on average, any single

-bin correlated with any

other

-bin at a level of

≲

10%

A337, page 5 of 14

Chung, D. T., et al.: A&A, 691, A337 (2024)

Table 1.

Results from two-parameter analyses of CO power spectrum measurements.

- and

eff

-agnostic:

- and

eff

-informed:

- and

eff

-agnostic:

- and

eff

-informed:

clust

shot

⟨

T b

⟩

shot

⟨

⟩

⟨

⟩

Data

(

μ

)

(

μ

Mpc

)

(

μ

)

(

μ

Mpc

)

(

μ

(

⊙

Mpc

−

)

(

μ

(

⊙

Mpc

−

)

COPSS

630

−

345

−

COMAP S1

COMAP S1+COPSS

−

COMAP S2

COMAP S2+COPSS

Notes.

We show constraints obtained for clustering amplitude (

clust

⟨

T b

⟩

) and shot noise power (

shot

), assuming any deviation from zero

describes CO(1–0) emission at

∼

. For comparison, we also show results from using only COPSS data or COMAP data through Season 1;

we indicate in bold type the results from using COMAP data through Season 2 (without COPSS data). We quote 68% intervals for

shot

in the

“COPSS” and “COMAP S1+COPSS” analyses; otherwise we quote 95% upper limits.

〈

T b

〉

∼

clust

30.4

(95% UL)

shot

3720

Mpc

(95% UL)

COMAP S2 analysis

clust

∼〈

T b

〉

(

)

shot

(

Mpc

)

Li+2016–Keating+2020

Chung+2022, UM+COLDz+COPSS

Li+2016

Padmanabhan2018,

duty

= 0.1

Yang+22 empirical fit

〈

T b

〉

22.7

(95% UL)

shot

4774

Mpc

(95% UL)

COMAP S2 analysis

〈

T b

〉

(

)

shot

(

Mpc

)

Li+2016–Keating+2020

Chung+2022, UM+COLDz+COPSS

Li+2016

Padmanabhan2018,

duty

= 0.1

Yang+22 empirical fit

Fig. 2.

Likelihood contours and marginalized probability distributions

for the clustering and shot-noise amplitudes of the CO power spec-

trum, conditioned on COMAP Season 2 data, in

- and

eff

-agnostic

(upper) and -informed (lower) analyses. Black solid lines plotted with

the 1D marginalized distributions indicate the 95% upper limits for each

parameter. The solid and dashed 2D contours are meant to encompass

39% and 86% of the probability mass (delineated at

∆

{

}

relative

to the minimum

, corresponding to

and

for 2D Gaussians). We

show the clustering and shot noise amplitudes for a subset of the models

plotted in Fig. 1. Models shown in Fig. 1 but not shown here have values

clust

⟨

T b

⟩

well beyond the 2

regions shown.

2 (“COMAP S2” in Table 1). We find a factor of 5 improve-

ment in our ability to constrain

shot

from above with COMAP

data alone up to Season 2 compared to COMAP Early Science

alone, and a factor of 2 improvement in upper limits for the clus-

tering amplitude. In fact, framing sensitivity to clustering purely

in terms of the upper limit achieved downplays our gain. Where

the COMAP Early Science analysis effectively gave a maximum

a posteriori (MAP) estimate of zero for

clust

and

⟨

T b

⟩

, Fig. 2

shows that the likelihood distributions peak at positive values of

these parameters under COMAP Season 2 constraints.

We also show in Fig. 2 model predictions for

clust

and

shot

, or for

⟨

T b

⟩

and

shot

. As expected, all models not shown

to be excluded by the COMAP Season 2 data at 95% confi-

dence in Fig. 1 are consistent to within

of the MAP estimate

from the COMAP Season 2 likelihood analysis, including the

COMAP Early Science fiducial model from Chung et al. (2022).

That said, the most favoured model (within

of the MAP esti-

mate) is the Li et al. (2016)–Keating et al. (2020) model used to

explain the results of the mm-wave Intensity Mapping Experi-

ment (mmIME; Keating et al. 2020). This finding is consistent

between the

- and

eff

-agnostic and -informed analyses.

The resulting constraints on

⟨

⟩

given

are also con-

sistent between these analyses to within a few percent. Going

forward we quote

⟨

T b

⟩

μ

K or

⟨

⟩

μ

K, consistent

with both of our “COMAP S2” standalone analyses as well

as both of the “COMAP S2+COPSS” joint analyses as we

show in Table 1. As in Chung et al. (2022) we can convert

any estimate of

⟨

⟩

into an estimate for cosmic molecular gas

density:

⟨

⟩

(

)

(13)

We show the resulting bounds on

in Table 1 alongside the

original bounds on

⟨

⟩

, given

⊙

(

K km s

−

)

−

and the Hubble parameter

(

)

at the central COMAP redshift

∼

. Although some works have advocated for values of

(Bolatto et al. 2013; Scoville et al. 2016) higher by as much

as a factor of two – and in Sect. C we discuss in more detail

what motivates different values of

– our chosen value fol-

lows the one most commonly used by previous CO line search

and line-intensity mapping analyses (e.g.: Riechers et al. 2019;

Decarli et al. 2020; Lenki

c et al. 2020; Keating et al. 2020), with

this value originally identified in three

∼

galaxies (Daddi

et al. 2010). Our top-line result of

⟨

⟩

μ

K corresponds to a

bound of

⊙

Mpc

−

A337, page 6 of 14