# Testing the $\mathbf{\Lambda}$CDM Cosmological Model with Forthcoming
Measurements of the Cosmic Microwave Background with SPT-3G

K. Prabhu  Department of Physics & Astronomy, University of California, One
Shields Avenue, Davis, CA 95616, USA  S. Raghunathan Center for AstroPhysical
Surveys, National Center for Supercomputing Applications, Urbana, IL, 61801,
USA  M. Millea Department of Physics, University of California, Berkeley, CA,
94720, USA  G. P. Lynch  Department of Physics & Astronomy, University of
California, One Shields Avenue, Davis, CA 95616, USA  P. A. R. Ade  School of
Physics and Astronomy, Cardiff University, Cardiff CF24 3YB, United Kingdom
E. Anderes  Department of Statistics, University of California, One Shields
Avenue, Davis, CA 95616, USA  A. J. Anderson Fermi National Accelerator
Laboratory, MS209, P.O. Box 500, Batavia, IL, 60510, USA  Kavli Institute for
Cosmological Physics, University of Chicago, 5640 South Ellis Avenue, Chicago,
IL, 60637, USA  Department of Astronomy and Astrophysics, University of
Chicago, 5640 South Ellis Avenue, Chicago, IL, 60637, USA  B. Ansarinejad
School of Physics, University of Melbourne, Parkville, VIC 3010, Australia  M.
Archipley Department of Astronomy, University of Illinois Urbana-Champaign,
1002 West Green Street, Urbana, IL, 61801, USA  Center for AstroPhysical
Surveys, National Center for Supercomputing Applications, Urbana, IL, 61801,
USA  L. Balkenhol Institut d’Astrophysique de Paris, UMR 7095, CNRS & Sorbonne
Université, 98 bis boulevard Arago, 75014 Paris, France  K. Benabed  Institut
d’Astrophysique de Paris, UMR 7095, CNRS & Sorbonne Université, 98 bis
boulevard Arago, 75014 Paris, France  A. N. Bender High-Energy Physics
Division, Argonne National Laboratory, 9700 South Cass Avenue., Lemont, IL,
60439, USA  Kavli Institute for Cosmological Physics, University of Chicago,
5640 South Ellis Avenue, Chicago, IL, 60637, USA  Department of Astronomy and
Astrophysics, University of Chicago, 5640 South Ellis Avenue, Chicago, IL,
60637, USA  B. A. Benson Fermi National Accelerator Laboratory, MS209, P.O.
Box 500, Batavia, IL, 60510, USA  Kavli Institute for Cosmological Physics,
University of Chicago, 5640 South Ellis Avenue, Chicago, IL, 60637, USA
Department of Astronomy and Astrophysics, University of Chicago, 5640 South
Ellis Avenue, Chicago, IL, 60637, USA  F. Bianchini Kavli Institute for
Particle Astrophysics and Cosmology, Stanford University, 452 Lomita Mall,
Stanford, CA, 94305, USA  Department of Physics, Stanford University, 382 Via
Pueblo Mall, Stanford, CA, 94305, USA  SLAC National Accelerator Laboratory,
2575 Sand Hill Road, Menlo Park, CA, 94025, USA  L. E. Bleem High-Energy
Physics Division, Argonne National Laboratory, 9700 South Cass Avenue.,
Lemont, IL, 60439, USA  Kavli Institute for Cosmological Physics, University
of Chicago, 5640 South Ellis Avenue, Chicago, IL, 60637, USA  F. R. Bouchet
Institut d’Astrophysique de Paris, UMR 7095, CNRS & Sorbonne Université, 98
bis boulevard Arago, 75014 Paris, France  L. Bryant  Enrico Fermi Institute,
University of Chicago, 5640 South Ellis Avenue, Chicago, IL, 60637, USA  E.
Camphuis  Institut d’Astrophysique de Paris, UMR 7095, CNRS & Sorbonne
Université, 98 bis boulevard Arago, 75014 Paris, France  J. E. Carlstrom
Kavli Institute for Cosmological Physics, University of Chicago, 5640 South
Ellis Avenue, Chicago, IL, 60637, USA  Enrico Fermi Institute, University of
Chicago, 5640 South Ellis Avenue, Chicago, IL, 60637, USA  Department of
Physics, University of Chicago, 5640 South Ellis Avenue, Chicago, IL, 60637,
USA  High-Energy Physics Division, Argonne National Laboratory, 9700 South
Cass Avenue., Lemont, IL, 60439, USA  Department of Astronomy and
Astrophysics, University of Chicago, 5640 South Ellis Avenue, Chicago, IL,
60637, USA  T. W. Cecil High-Energy Physics Division, Argonne National
Laboratory, 9700 South Cass Avenue., Lemont, IL, 60439, USA  C. L. Chang
High-Energy Physics Division, Argonne National Laboratory, 9700 South Cass
Avenue., Lemont, IL, 60439, USA  Kavli Institute for Cosmological Physics,
University of Chicago, 5640 South Ellis Avenue, Chicago, IL, 60637, USA
Department of Astronomy and Astrophysics, University of Chicago, 5640 South
Ellis Avenue, Chicago, IL, 60637, USA  P. Chaubal  School of Physics,
University of Melbourne, Parkville, VIC 3010, Australia  P. M. Chichura
Department of Physics, University of Chicago, 5640 South Ellis Avenue,
Chicago, IL, 60637, USA  Kavli Institute for Cosmological Physics, University
of Chicago, 5640 South Ellis Avenue, Chicago, IL, 60637, USA  T.-L. Chou
Department of Physics, University of Chicago, 5640 South Ellis Avenue,
Chicago, IL, 60637, USA  Kavli Institute for Cosmological Physics, University
of Chicago, 5640 South Ellis Avenue, Chicago, IL, 60637, USA  A. Coerver
Department of Physics, University of California, Berkeley, CA, 94720, USA  T.
M. Crawford Kavli Institute for Cosmological Physics, University of Chicago,
5640 South Ellis Avenue, Chicago, IL, 60637, USA  Department of Astronomy and
Astrophysics, University of Chicago, 5640 South Ellis Avenue, Chicago, IL,
60637, USA  A. Cukierman  Kavli Institute for Particle Astrophysics and
Cosmology, Stanford University, 452 Lomita Mall, Stanford, CA, 94305, USA
SLAC National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, CA,
94025, USA  Department of Physics, Stanford University, 382 Via Pueblo Mall,
Stanford, CA, 94305, USA  C. Daley Department of Astronomy, University of
Illinois Urbana-Champaign, 1002 West Green Street, Urbana, IL, 61801, USA  T.
de Haan  High Energy Accelerator Research Organization (KEK), Tsukuba, Ibaraki
305-0801, Japan  K. R. Dibert  Department of Astronomy and Astrophysics,
University of Chicago, 5640 South Ellis Avenue, Chicago, IL, 60637, USA  Kavli
Institute for Cosmological Physics, University of Chicago, 5640 South Ellis
Avenue, Chicago, IL, 60637, USA  M. A. Dobbs  Department of Physics and McGill
Space Institute, McGill University, 3600 Rue University, Montreal, Quebec H3A
2T8, Canada  Canadian Institute for Advanced Research, CIFAR Program in
Gravity and the Extreme Universe, Toronto, ON, M5G 1Z8, Canada  A. Doussot
Institut d’Astrophysique de Paris, UMR 7095, CNRS & Sorbonne Université, 98
bis boulevard Arago, 75014 Paris, France  D. Dutcher Joseph Henry Laboratories
of Physics, Jadwin Hall, Princeton University, Princeton, NJ 08544, USA  W.
Everett  Department of Astrophysical and Planetary Sciences, University of
Colorado, Boulder, CO, 80309, USA  C. Feng  Department of Physics, University
of Illinois Urbana-Champaign, 1110 West Green Street, Urbana, IL, 61801, USA
K. R. Ferguson Department of Physics and Astronomy, University of California,
Los Angeles, CA, 90095, USA  K. Fichman  Department of Physics, University of
Chicago, 5640 South Ellis Avenue, Chicago, IL, 60637, USA  Kavli Institute for
Cosmological Physics, University of Chicago, 5640 South Ellis Avenue, Chicago,
IL, 60637, USA  A. Foster Joseph Henry Laboratories of Physics, Jadwin Hall,
Princeton University, Princeton, NJ 08544, USA  S. Galli  Institut
d’Astrophysique de Paris, UMR 7095, CNRS & Sorbonne Université, 98 bis
boulevard Arago, 75014 Paris, France  A. E. Gambrel  Kavli Institute for
Cosmological Physics, University of Chicago, 5640 South Ellis Avenue, Chicago,
IL, 60637, USA  R. W. Gardner  Enrico Fermi Institute, University of Chicago,
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& Astronomy, University of California, One Shields Avenue, Davis, CA 95616,
USA  N. Goeckner-Wald  Department of Physics, Stanford University, 382 Via
Pueblo Mall, Stanford, CA, 94305, USA  Kavli Institute for Particle
Astrophysics and Cosmology, Stanford University, 452 Lomita Mall, Stanford,
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d’Astrophysique de Paris, UMR 7095, CNRS & Sorbonne Université, 98 bis
boulevard Arago, 75014 Paris, France  S. Guns  Department of Physics,
University of California, Berkeley, CA, 94720, USA  N. W. Halverson  CASA,
Department of Astrophysical and Planetary Sciences, University of Colorado,
Boulder, CO, 80309, USA  Department of Physics, University of Colorado,
Boulder, CO, 80309, USA  E. Hivon  Institut d’Astrophysique de Paris, UMR
7095, CNRS & Sorbonne Université, 98 bis boulevard Arago, 75014 Paris, France
G. P. Holder Department of Physics, University of Illinois Urbana-Champaign,
1110 West Green Street, Urbana, IL, 61801, USA  W. L. Holzapfel  Department of
Physics, University of California, Berkeley, CA, 94720, USA  J. C. Hood  Kavli
Institute for Cosmological Physics, University of Chicago, 5640 South Ellis
Avenue, Chicago, IL, 60637, USA  A. Hryciuk  Department of Physics, University
of Chicago, 5640 South Ellis Avenue, Chicago, IL, 60637, USA  Kavli Institute
for Cosmological Physics, University of Chicago, 5640 South Ellis Avenue,
Chicago, IL, 60637, USA  N. Huang  Department of Physics, University of
California, Berkeley, CA, 94720, USA  F. Kéruzoré  High-Energy Physics
Division, Argonne National Laboratory, 9700 South Cass Avenue., Lemont, IL,
60439, USA  L. Knox  Department of Physics & Astronomy, University of
California, One Shields Avenue, Davis, CA 95616, USA  M. Korman  Department of
Physics, Case Western Reserve University, Cleveland, OH, 44106, USA  K.
Kornoelje  Department of Astronomy and Astrophysics, University of Chicago,
5640 South Ellis Avenue, Chicago, IL, 60637, USA  Kavli Institute for
Cosmological Physics, University of Chicago, 5640 South Ellis Avenue, Chicago,
IL, 60637, USA  C.-L. Kuo  Kavli Institute for Particle Astrophysics and
Cosmology, Stanford University, 452 Lomita Mall, Stanford, CA, 94305, USA
Department of Physics, Stanford University, 382 Via Pueblo Mall, Stanford, CA,
94305, USA  SLAC National Accelerator Laboratory, 2575 Sand Hill Road, Menlo
Park, CA, 94025, USA  A. T. Lee  Department of Physics, University of
California, Berkeley, CA, 94720, USA  Physics Division, Lawrence Berkeley
National Laboratory, Berkeley, CA, 94720, USA  K. Levy  School of Physics,
University of Melbourne, Parkville, VIC 3010, Australia  A. E. Lowitz  Kavli
Institute for Cosmological Physics, University of Chicago, 5640 South Ellis
Avenue, Chicago, IL, 60637, USA  C. Lu  Department of Physics, University of
Illinois Urbana-Champaign, 1110 West Green Street, Urbana, IL, 61801, USA  A.
Maniyar  Kavli Institute for Particle Astrophysics and Cosmology, Stanford
University, 452 Lomita Mall, Stanford, CA, 94305, USA  Department of Physics,
Stanford University, 382 Via Pueblo Mall, Stanford, CA, 94305, USA  SLAC
National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, CA, 94025,
USA  F. Menanteau  Department of Astronomy, University of Illinois Urbana-
Champaign, 1002 West Green Street, Urbana, IL, 61801, USA  Center for
AstroPhysical Surveys, National Center for Supercomputing Applications,
Urbana, IL, 61801, USA  J. Montgomery  Department of Physics and McGill Space
Institute, McGill University, 3600 Rue University, Montreal, Quebec H3A 2T8,
Canada  Y. Nakato  Department of Physics, Stanford University, 382 Via Pueblo
Mall, Stanford, CA, 94305, USA  T. Natoli  Kavli Institute for Cosmological
Physics, University of Chicago, 5640 South Ellis Avenue, Chicago, IL, 60637,
USA  G. I. Noble Dunlap Institute for Astronomy & Astrophysics, University of
Toronto, 50 St. George Street, Toronto, ON, M5S 3H4, Canada  David A. Dunlap
Department of Astronomy & Astrophysics, University of Toronto, 50 St. George
Street, Toronto, ON, M5S 3H4, Canada  V. Novosad  Materials Sciences Division,
Argonne National Laboratory, 9700 South Cass Avenue, Lemont, IL, 60439, USA
Y. Omori  Department of Astronomy and Astrophysics, University of Chicago,
5640 South Ellis Avenue, Chicago, IL, 60637, USA  Kavli Institute for
Cosmological Physics, University of Chicago, 5640 South Ellis Avenue, Chicago,
IL, 60637, USA  S. Padin  Kavli Institute for Cosmological Physics, University
of Chicago, 5640 South Ellis Avenue, Chicago, IL, 60637, USA  California
Institute of Technology, 1200 East California Boulevard., Pasadena, CA, 91125,
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South Cass Avenue., Lemont, IL, 60439, USA  Kavli Institute for Cosmological
Physics, University of Chicago, 5640 South Ellis Avenue, Chicago, IL, 60637,
USA  Department of Physics, University of Chicago, 5640 South Ellis Avenue,
Chicago, IL, 60637, USA  P. Paschos  Enrico Fermi Institute, University of
Chicago, 5640 South Ellis Avenue, Chicago, IL, 60637, USA  K. A. Phadke
Department of Astronomy, University of Illinois Urbana-Champaign, 1002 West
Green Street, Urbana, IL, 61801, USA  Center for AstroPhysical Surveys,
National Center for Supercomputing Applications, Urbana, IL, 61801, USA  W.
Quan  Department of Physics, University of Chicago, 5640 South Ellis Avenue,
Chicago, IL, 60637, USA  Kavli Institute for Cosmological Physics, University
of Chicago, 5640 South Ellis Avenue, Chicago, IL, 60637, USA  M. Rahimi
School of Physics, University of Melbourne, Parkville, VIC 3010, Australia  A.
Rahlin Fermi National Accelerator Laboratory, MS209, P.O. Box 500, Batavia,
IL, 60510, USA  Kavli Institute for Cosmological Physics, University of
Chicago, 5640 South Ellis Avenue, Chicago, IL, 60637, USA  C. L. Reichardt
School of Physics, University of Melbourne, Parkville, VIC 3010, Australia  M.
Rouble  Department of Physics and McGill Space Institute, McGill University,
3600 Rue University, Montreal, Quebec H3A 2T8, Canada  J. E. Ruhl  Department
of Physics, Case Western Reserve University, Cleveland, OH, 44106, USA  E.
Schiappucci  School of Physics, University of Melbourne, Parkville, VIC 3010,
Australia  G. Smecher  Three-Speed Logic, Inc., Victoria, B.C., V8S 3Z5,
Canada  J. A. Sobrin Fermi National Accelerator Laboratory, MS209, P.O. Box
500, Batavia, IL, 60510, USA  Kavli Institute for Cosmological Physics,
University of Chicago, 5640 South Ellis Avenue, Chicago, IL, 60637, USA  A. A.
Stark  Harvard-Smithsonian Center for Astrophysics, 60 Garden Street,
Cambridge, MA, 02138, USA  J. Stephen  Enrico Fermi Institute, University of
Chicago, 5640 South Ellis Avenue, Chicago, IL, 60637, USA  A. Suzuki  Physics
Division, Lawrence Berkeley National Laboratory, Berkeley, CA, 94720, USA  C.
Tandoi  Department of Astronomy, University of Illinois Urbana-Champaign, 1002
West Green Street, Urbana, IL, 61801, USA  K. L. Thompson  Kavli Institute for
Particle Astrophysics and Cosmology, Stanford University, 452 Lomita Mall,
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Physics & Astronomy, University of California, One Shields Avenue, Davis, CA
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for Supercomputing Applications, Urbana, IL, 61801, USA  C. Tucker  School of
Physics and Astronomy, Cardiff University, Cardiff CF24 3YB, United Kingdom
C. Umilta Department of Physics, University of Illinois Urbana-Champaign, 1110
West Green Street, Urbana, IL, 61801, USA  A. Vitrier  Institut
d’Astrophysique de Paris, UMR 7095, CNRS & Sorbonne Université, 98 bis
boulevard Arago, 75014 Paris, France  J. D. Vieira  Department of Astronomy,
University of Illinois Urbana-Champaign, 1002 West Green Street, Urbana, IL,
61801, USA  Department of Physics, University of Illinois Urbana-Champaign,
1110 West Green Street, Urbana, IL, 61801, USA  Center for AstroPhysical
Surveys, National Center for Supercomputing Applications, Urbana, IL, 61801,
USA  Y. Wan  Department of Astronomy, University of Illinois Urbana-Champaign,
1002 West Green Street, Urbana, IL, 61801, USA  Center for AstroPhysical
Surveys, National Center for Supercomputing Applications, Urbana, IL, 61801,
USA  G. Wang  High-Energy Physics Division, Argonne National Laboratory, 9700
South Cass Avenue., Lemont, IL, 60439, USA  N. Whitehorn Department of Physics
and Astronomy, Michigan State University, East Lansing, MI 48824, USA  W. L.
K. Wu Kavli Institute for Particle Astrophysics and Cosmology, Stanford
University, 452 Lomita Mall, Stanford, CA, 94305, USA  SLAC National
Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, CA, 94025, USA  V.
Yefremenko  High-Energy Physics Division, Argonne National Laboratory, 9700
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Accelerator Laboratory, MS209, P.O. Box 500, Batavia, IL, 60510, USA  Kavli
Institute for Cosmological Physics, University of Chicago, 5640 South Ellis
Avenue, Chicago, IL, 60637, USA  J. A. Zebrowski  Kavli Institute for
Cosmological Physics, University of Chicago, 5640 South Ellis Avenue, Chicago,
IL, 60637, USA  Department of Astronomy and Astrophysics, University of
Chicago, 5640 South Ellis Avenue, Chicago, IL, 60637, USA  Fermi National
Accelerator Laboratory, MS209, P.O. Box 500, Batavia, IL, 60510, USA K. Prabhu
<EMAIL_ADDRESS>S. Raghunathan<EMAIL_ADDRESS>
###### Abstract

We forecast constraints on cosmological parameters enabled by three surveys
conducted with SPT-3G, the third-generation camera on the South Pole
Telescope. The surveys cover separate regions of 1500, 2650, and 6000 ${\rm
deg}^{2}$ to different depths, in total observing 25% of the sky. These
regions will be measured to white noise levels of roughly $2.5$, $9$, and $12\
\mu\mathrm{K}{\rm-arcmin}$, respectively, in CMB temperature units at 150 GHz
by the end of 2024. The survey also includes measurements at 95 and 220 GHz,
which have noise levels a factor of $\sim$1.2 and 3.5 times higher than 150
GHz, respectively, with each band having a polarization noise level
$\sim\sqrt{2}$ times higher than the temperature noise. We use a novel
approach to obtain the covariance matrices for jointly and optimally estimated
gravitational lensing potential bandpowers and unlensed CMB temperature and
polarization bandpowers. We demonstrate the ability to test the $\Lambda{\rm
CDM}$ model via the consistency of cosmological parameters constrained
independently from SPT-3G and Planck data, and consider the improvement in
constraints on $\Lambda{\rm CDM}$ extension parameters from a joint analysis
of SPT-3G and Planck data. The $\Lambda{\rm CDM}$ cosmological parameters are
typically constrained with uncertainties up to $\sim$2 times smaller with
SPT-3G data, compared to Planck, with the two data sets measuring
significantly different angular scales and polarization levels, providing
additional tests of the standard cosmological model.

##  1 Introduction

Observations of the cosmic microwave background (CMB) temperature and
polarization anisotropies have played a crucial role in the field of
cosmology, particularly in the establishment of a six-parameter standard
cosmological model, $\Lambda{\rm CDM}$. This model which provides an excellent
fit to CMB data from Planck (Planck Collaboration et al., 2020), the Atacama
Cosmology Telescope (ACT) (Aiola et al., 2020; Choi et al., 2020;
Madhavacheril et al., 2023) and the South Pole Telescope (SPT) (Simard et al.,
2018; Dutcher et al., 2021; Balkenhol et al., 2021; Pan et al., 2023), as well
as a great variety of other astrophysical datasets (Alam et al., 2021; DES
Collaboration, 2024; Abbott et al., 2022; Kuijken et al., 2019), has been
enormously successful in providing a consistent framework for understanding
the universe.

These successes are perhaps surprising, since critical ingredients to the
model (dark matter, dark energy, and the generator of primordial fluctuations)
appear disconnected from the physics we know of through laboratory
experiments. The desire for clues that could deepen our understanding of these
ingredients motivates us to continue our testing of the model. One
particularly compelling line of testing emerges from the fact that the
standard cosmological model, conditioned on data from the Planck satellite,
makes extremely precise predictions for the CMB temperature, polarization, and
lensing power spectra on angular scales not well measured by Planck.

Additional motivation for testing $\Lambda{\rm CDM}$ (beyond our ignorance
about dark matter, dark energy, and the generator of primordial fluctuations)
comes from cosmological tensions and anomalies as these might be evidence for
beyond-$\Lambda{\rm CDM}$ physics. A prominent issue is the Hubble tension – a
notable discrepancy between classical distance ladder determinations (Murakami
et al., 2023; Riess et al., 2022) and $\Lambda{\rm CDM}$-dependent
determinations (Planck Collaboration et al., 2020) of the current expansion
rate of the universe. An additional, though statistically weaker tension, is
in the values of $\sigma_{8}$, the root mean square of the density field
smoothed over $8h^{-1}$ Mpc radius spheres, derived from weak lensing
observations in optical galaxy surveys and those inferred from CMB
data111Although this tension is partially resolved in the analysis of Planck
PR4 maps (Tristram et al., 2024) (Asgari et al., 2021; Joudaki et al., 2020;
Abbott et al., 2022). There are also peculiar patterns in the CMB temperature
anisotropies on large angular scales, which, by some calculations, are
extremely unlikely in a $\Lambda{\rm CDM}$ universe (Copi et al., 2010; Givans
& Kamionkowski, 2023).

Altogether these discrepancies offer some evidence that $\Lambda{\rm CDM}$ is
not the final word in cosmology, encouraging us to search for
beyond-$\Lambda{\rm CDM}$ signals in our surveys. Indeed, there are
alternative models that reduce the $\sigma_{8}$ or $H_{0}$ tensions and make
different predictions for the power spectra than is the case for $\Lambda{\rm
CDM}$. In this context, it becomes pertinent to investigate the extent to
which upcoming observational data can distinguish between the standard
$\Lambda{\rm CDM}$ model and some of these alternatives.

SPT-3G, the third-generation camera on the SPT, has the potential to test the
$\Lambda{\rm CDM}$ model by precisely mapping the primary and lensing
anisotropies of the CMB. The full SPT-3G data set will comprise three surveys,
which, altogether, cover 10000 ${\rm deg}^{2}$, or $\sim$25% of the sky. The
SPT-3G Main survey (1500 ${\rm deg}^{2}$), observed over five austral winters,
is approaching the target depths of the next generation CMB-S4 Deep and Wide
survey of 70% of the sky planned to begin next decade (Abazajian et al.,
2022). Next-deepest is the SPT-3G Summer survey, a completed collection of
three fields observed during four austral summers totaling 2650 ${\rm
deg}^{2}$. The shallowest is the 6000 ${\rm deg}^{2}{}$ SPT-3G Wide survey,
which, after one observing season, will still be deeper than surveys from any
other existing high-resolution CMB experiment of comparable survey size.

Besides mapping the primary CMB and lensing anisotropies, SPT-3G will provide
a powerful set of data for studying a broad range of topics in cosmology and
astrophysics. These include the delensing of lensing-induced B-modes (Ade et
al., 2021); production of mass-limited catalogs of galaxy clusters out to high
redshifts using the thermal Sunyaev-Zel’dovich (tSZ) signature (Bleem et al.,
2015; Huang et al., 2020; Bleem et al., 2020, 2023) constraining cosmology
with these catalogs (Bocquet et al., 2024); robust measurements of the
kinematic Sunyaev-Zel’dovich (kSZ) signal for velocity reconstruction and
constraining the epoch of reionization (Schiappucci et al., 2023; Raghunathan
& Omori, 2023; Raghunathan et al., 2024); production of mm-wave point source
catalogues (Everett et al., 2020); and the detection and monitoring of mm-wave
galactic, extragalactic, and solar system transient, variable, and moving
objects (Whitehorn et al., 2016; Guns et al., 2021; Chichura et al., 2022;
Hood et al., 2023; Tandoi et al., 2024).

Here we demonstrate the capacity of the SPT-3G surveys to test $\Lambda{\rm
CDM}$ in several ways, focusing exclusively on the constraining power of CMB
and lensing power spectrum measurements. We forecast how well we will measure
the temperature and $E$-mode polarization power spectra, their cross power
spectrum, and the power spectrum of the lensing potential.

We also propagate the power spectrum uncertainties forward to constraints on
cosmological parameters. From SPT-3G data alone we can obtain constraints on
the parameters of the $\Lambda{\rm CDM}$ model that are comparable to, and in
some cases, better than those from Planck. When compared to Planck, a greater
fraction of the weight of these constraints comes from polarization (see Galli
et al., 2014, for example), from smaller angular scales, and from lensing.
Therefore, comparison of these parameter estimates with those from Planck will
be an excellent test of the $\Lambda{\rm CDM}$ model.

The outcome of such a test could provide evidence for physics beyond
$\Lambda{\rm CDM}$. Looking ahead to this scenario, we also forecast
constraints on single- and double-parameter extensions of $\Lambda{\rm CDM}$
from the combination of Planck and SPT-3G data and report how these
constraints improve upon those from Planck alone. The test could equally well
reveal consistency with $\Lambda{\rm CDM}$, an outcome that should not be
dismissed given the model’s track record. If this is the case, joint
Planck+SPT-3G $\Lambda{\rm CDM}$ parameter constraints will be of interest,
and we provide forecasts for the combined constraints on $\Lambda{\rm CDM}$ as
well.

In addition to our comparison with Planck, we compare with forecasts we make
for the high-resolution survey of the Simons Observatory (SO, Simons
Observatory Collaboration 2019). SO is expected to survey roughly 40% of the
sky (much more sky is available to survey from mid-latitude sites such as the
SO site in the Atacama Plateau in Chile compared to the South Pole) to noise
levels similar to or slightly better than the SPT-3G Summer fields, leading to
constraints on $\Lambda{\rm CDM}$ parameters that are comparable to what we
expect from the combined SPT-3G surveys.

For an ongoing analysis of the first two full years of data from the SPT-3G
Main survey, one of our analysis approaches is a Bayesian one, which is in
principle unbiased and optimal (information lossless) despite the presence of
gravitational lensing, and naturally results in joint constraints on the
lensing power spectrum and unlensed CMB power spectra. To speed up estimation
of the unlensed and lensing power spectra, from this likelihood, we employ the
Marginal Unbiased Score Expansion (MUSE) technique (Millea & Seljak, 2021). We
refer to this whole inference pipeline from maps to the bandpowers and their
covariance matrix as MUSE. Since MUSE can be used to obtain a covariance
matrix for all the relevant spectra, it is quite convenient for us to
propagate these uncertainties to a parameter error covariance matrix, and
compare the results with more traditional forecasting techniques.

This paper is structured as follows: In Sec. 2, we describe the specifications
of the SPT-3G instrument, the survey regions and our assumptions about the
noise and foregrounds. In Sec. 3 we describe the methodology used to obtain
these forecasts. We report the forecasts of the parameter constraints for
$\Lambda{\rm CDM}$ with SPT-3G alone and for extensions with SPT-3G combined
with Planck in Sec. 4. In Sec. 5, we discuss prospects for SPT-3G to detect
departures from $\Lambda{\rm CDM}$ predictions and test alternate models.
Finally, we summarize our findings in Sec. 6.

##  2 Experiment Specifications

Figure 1:  Footprints of the three SPT-3G surveys: SPT-3G Main (green), SPT-3G
Summer (yellow), and SPT-3G Wide (red). Also shown are the footprints of other
surveys: BICEP3 (pink dash-dotted), DES (black dash-dotted), and SO/LSST (blue
dash-dotted). The background is the map of the galactic dust produced by the
Planck satellite (Planck Collaboration et al., 2016a).

In this section, we briefly describe the specifications of the SPT-3G
experiment along with the details of the three survey regions and a
description of the noise characteristics.

###  2.1 Survey Specifications

Table 1: Sky fraction and temperature white noise levels in the three bands for different SPT surveys considered in this work. The polarization white noise levels are expected to be $\sim$$\sqrt{2}$ times higher.  Survey | Area | RA | Dec. | Years observed | Noise level ($\Delta_{T}$) [$\mu\mathrm{K}$-arcmin]
---|---|---|---|---|---
[${\rm deg}^{2}$] | center [deg] | center [deg] | 95 GHz | 150 GHz | 220 GHz | Coadded
Completed: |  |
SPT-3G Main | 1500 | 0 | -57.5 | 2019-2023 | 3.0 | 2.5 | 8.9 | 1.9
SPT-3G Summer-A | 1210 | 75.0 | -42.0 | 9.4 | 8.7 | 29.5 | 6.2
SPT-3G Summer-B | 570 | 25 | -36 | 10.0 | 9.7 | 30.3 | 6.8
SPT-3G Summer-C | 860 | 187.5 | -38 | 10.0 | 9.2 | 27.8 | 6.6
Ongoing+Future: |  |
SPT-3G Main | 1500 | 0 | -57.5 | 2019-2023 | 2.5 | 2.1 | 7.6 | 1.6
\+ 2025-2026
SPT-3G Wide | 6000 | Multiple | 2024 | 14 | 12 | 42 | 8.8
Table 2: Atmospheric noise specifications for the three SPT bands. Specification | Observation band
---|---
90 GHz | 150 GHz | 220 GHz
SPT-3G Main:
$\ell_{\text{knee},T}$ | 1200 | 2200 | 2300
$\alpha_{T}$ | -3 | -4 | -4
$\ell_{\text{knee},P}$ | 300
$\alpha_{P}$ | -1
SPT-3G Summer & SPT-3G Wide:
$\ell_{\text{knee},T}$ | 1600 | 2600 | 2600
$\alpha_{T}$ | -4.5 | -4 | -3.9
$\ell_{\text{knee},P}$ | 300 | 490 | 500
$\alpha_{P}$ | -2.2 | -2 | -2.5
Figure 2: Expected $TT$ and $EE$ noise power spectra for the three SPT-3G
surveys: Beam-deconvolved noise curves in the three bands for SPT-3G Main (5
years), Summer (4 years) and Wide (1 year) surveys are shown as colored
curves. The Minimum Variance (MV) combination of the LC residuals is the thick
black solid curve in all the panels. The gray curves represent the CMB power
spectra. Top panels are for temperature and the bottom panels are for
polarization. The LC residuals are dominated by foregrounds for temperature
for SPT-3G Main while for polarization they are roughly equal to the inverse
variance weighted noise spectra from the three bands. Figure 3: $S/N$ per
multipole on the primary and lensing CMB power spectra for the SPT-3G Main
(green) and the Ext-10k (orange) SPT-3G surveys, with values for Planck
(black) plotted for comparison. The constraining power of the full Ext-10k
survey will be larger than Planck at $\ell\gtrsim 1800$ in $TT$, $\ell\gtrsim
800$ in $TE$, $\ell\gtrsim 450$ in $EE$ and effectively for all the modes in
$\phi\phi$. The non-shaded area corresponds to the range of multipoles used in
the forecasts.

SPT is a 10-meter diameter telescope specifically engineered for low-noise and
high-resolution measurements of the millimeter-wave sky. The telescope is
currently equipped with SPT-3G, the third CMB camera installed on SPT. SPT-3G
is a significant upgrade compared to previous cameras, incorporating a
polarization sensitive tri-chroic focal plane with almost 16,000 detectors,
operating in three frequency bands, 95, 150, and 220 GHz. For more details
about the instrument, we refer the readers to Benson et al. (2014); Bender et
al. (2018); Anderson et al. (2018); Sobrin et al. (2018, 2022).

One of the primary science goals of SPT-3G is to create a template of the
gravitational-lensing-induced B modes in the patch of sky observed by the
BICEP-Keck (BK) family of experiments. This template can be subtracted from
the BK data, reducing the primary source of variance in the BK B-mode analysis
and potentially improving the sensitivity to primordial gravitational waves by
a factor of three. To that end, since 2019, SPT-3G has been used primarily
(eight months per year) to observe the 1500 ${\rm deg}^{2}$ SPT-3G Main
survey, which overlaps completely with the BICEP3 and BICEP Array sky coverage
(see Fig. 1). During the austral summer season (roughly at the beginning of
December), when the Sun is above the horizon and gets sufficiently close to
the SPT-3G Main survey region, it is picked up by the telescope far sidelobes
. Therefore, until sunset, we switch to observing three fields that are safe
from sun contamination. These comprise the 2650 ${\rm deg}^{2}$ SPT-3G Summer
survey (Fig. 1). We refer to these three fields as Summer-A, Summer-B, and
Summer-C.

Even with the added sky coverage of the Summer survey, the constraints on
$\Lambda{\rm CDM}$ parameters from SPT-3G data are limited by sample variance.
For this reason, we have paused our standard observing strategy for one year
to undertake a new survey of all the sky viewable from the South Pole at
instrumentally feasible observing elevations and with minimal Galactic
foreground contamination. We have empirically determined that, during the
winter, we are able to observe with SPT-3G between elevations of roughly
$20^{\circ}$ and $80^{\circ}$ (limited by detector linearity at the low end
and refrigerator performance on the high end). The intersection of these
elevation limits (which, at the South Pole, correspond directly to declination
limits) and the Planck GAL080 (Planck Collaboration et al., 2016b) Galactic
mask defines our new SPT-3G Wide survey, which totals 6000 ${\rm deg}^{2}$ and
is shown in Fig. 1. Observations of the SPT-3G Wide survey commenced in
December of 2023 and will continue through the austral winter of 2024. We will
refer to the combination of the SPT-3G Main and the SPT-3G Summer surveys as
the Ext-4k survey. Similarly, we will refer to the combination of SPT-3G Main,
SPT-3G Summer, and SPT-3G Wide surveys as the Ext-10k survey. In Table 1, we
summarize the areas and depths for the SPT-3G Main, Summer, and Wide surveys.

The survey footprints for SPT-3G and other surveys are shown in Fig. 1. As is
evident from the figure, all the three SPT-3G surveys have an excellent
overlap with SO, which will enable a variety of cross-checks to be performed
between the two surveys. We list the noise levels and the field centers for
the SPT-3G surveys in Table 1. In this work, we produce forecasts both for
current noise levels (2019-2023) and for the ones expected from future SPT-3G
observations (2024-2026). Besides instrumental noise, our data is also
affected by atmospheric noise, which increases towards large angular scales.
We model the temperature noise power spectrum as

$N_{\ell}=\Delta_{T}^{2}\left[1+\left(\frac{\ell}{\ell_{\text{knee},T}}\right)^{\alpha_{T}}\right]\,,$
(1)

where $\Delta_{T}$ represents the white noise level at a given band in
$\mu\mathrm{K}{\rm-arcmin}$, $\ell_{\text{knee},T}$ and $\alpha_{T}$ are used
to parameterize the atmospheric noise. For the SPT-3G Main survey, we adopt
the values of $\ell_{\text{knee}}$ and $\alpha$ in Table 2 from previous
analyses, but we find that the results of our forecasting are insensitive to
the exact values of $\ell_{\text{knee}}$ and $\alpha$. For the SPT-3G Summer
survey, during which the atmospheric opacity and precipitable water vapor
levels are higher, we fit real noise data to the model in Eq. 1, and the
parameters in the Table 2 are the result of those fits. We also adopt the
SPT-3G Summer survey parameters for the SPT-3G Wide survey (which is likely
pessimistic for the higher-elevation Wide fields). We extrapolate the white
noise levels for deeper integration times. We also fit the noise model
parameters for the polarization maps in a similar way. We note that the white
noise levels in the polarization maps are roughly $\sqrt{2}\times$ higher than
in the temperature maps. In Table 2, we list the $\ell_{\text{knee}}$ and
$\alpha$ values adopted for different bands for both temperature and
polarization. Finally, we deconvolve the experimental beam window function
$B_{\ell}$ from the noise spectra as
$N^{\prime}_{\ell}\equiv\dfrac{N_{\ell}}{B_{\ell}^{2}}$ where the beam window
function at each frequency is estimated as detailed in Dutcher et al. (2021).
We show the model noise spectra in different bands (90 GHz in green, 150 GHz
in yellow, and 220 GHz in red) for all the SPT-3G surveys in Fig. 2. These
spectra assume integration times of 5 years for the SPT-3G Main survey, 4
years for the SPT-3G Summersurvey, and 1 year for the SPT-3G Wide survey.

##  3 Forecasting methodology

In this section, we outline the two methods used for calculating the bandpower
covariance matrix from which a Fisher matrix for cosmological parameters
(Tegmark et al., 1997) can be derived. The first method relies on an analytic
approximation (See § 3.2) for the covariance matrix. This approach has been
widely adopted in various forecasting studies, and is quite simple and
flexible. Our second method employs MUSE (see § 3.3), which is one of the
analysis pipelines developed for SPT-3G data. MUSE is more accurate as it
readily incorporates non-ideal factors such as covariance between multipoles,
covariance between different types of power spectra, and impacts of delensing.
Although previous works (for example, Hotinli et al., 2022; Trendafilova,
2023), suggest that these factors may not significantly alter our forecasts,
adopting both these methods improves the robustness and reliability of our
forecasts. We also describe the method we use to combine Planck data with
SPT-3G. We begin with a brief description of how we optimally combine data
from multiple frequency bands.

###  3.1 Linear Combination of Frequency Bands

We combine data from multiple frequency bands listed in Table 1 using a scale-
dependent linear combination (LC) technique (Cardoso et al., 2008; Planck
Collaboration et al., 2014). In harmonic space denoted by subscript $\ell$,
this corresponds to

$S_{\ell m}=\sum_{i=1}^{N_{\mathrm{bands}}}w_{\ell}^{i}M_{\ell m}^{i}\,,$ (2)

where $S$ is the desired sky signal (in our case the CMB), $\mathbf{w}_{\ell}$
is the weight, $\mathbf{M}_{\ell m}$ is the spherical harmonic transform of
the input map, and the subscript $i$ runs over all the frequency bands. The
multipole-dependent weights $\mathbf{w}_{\ell}$ are tuned in order to minimize
the overall variance from noise and foregrounds. They are derived as

$\mathbf{w}_{\ell}=\frac{\mathbf{C}_{\mathrm{\ell}}^{-1}\mathbf{A_{s}}}{\mathbf{A_{s}}^{\dagger}\mathbf{C}_{\mathrm{\ell}}^{-1}\mathbf{A_{s}}}\,.$
(3)

Here $\mathbf{C}_{\mathrm{\ell}}$ is a $N_{\mathrm{bands}}\times
N_{\mathrm{bands}}$ matrix containing the covariance of noise and foregrounds
across different frequencies at a given multipole $\ell$; $\mathbf{A_{s}}=[1\
1\ ..\ 1]$ is the frequency response of the CMB in different bands and is a
$N_{\mathrm{bands}}\times 1$ vector. We model the foreground contribution
using results from Reichardt et al. (2021) and the noise modeling is described
in § 2.1.

In Fig. 2, we show the LC residuals for all three SPT-3G surveys as the thick
black solid curve. The top and bottom panels in the figure are for CMB
temperature ($TT$) and polarization ($EE$) power spectra, denoted as gray
curves. In the absence of foregrounds, the LC residuals simply correspond to
the inverse variance combination of noise from different frequency bands,
which is almost true for $EE$ (bottom panels).

In Fig. 3, we present signal-to-noise ($S/N$) per multipole for
$TT/EE/TE/\phi\phi$ spectra for SPT-3G Main (green) and Ext-10k (orange)
surveys along with Planck (black). The power spectrum errors were calculated
using the analytical formula (see Eq.5) (Knox, 1995; Jungman et al., 1996;
Zaldarriaga et al., 1997). Planck has a higher $S/N$ on large scales because
of the larger sky coverage. However, the $S/N$s of SPT-3G Main and Ext-10k
surveys are much higher than Planck on small scales. In particular, the $S/N$s
for the Ext-10k survey will surpass Planck beyond $\ell=$ 1800, 800, and 450
for $TT$, $TE$, and $EE$ respectively. For $\phi\phi$, the $S/N$ are better
than Planck on all scales. As a result, we can expect significant improvement
on cosmological parameters that control structure formation and that are
sensitive to the damping tail from SPT-3G compared to Planck as we show later
in this work.

###  3.2 Analytic method

Given a set of power spectra of the lensed CMB and the lensing potential,
$X,Y\in[TT,EE,TE,\phi\phi]$, the Fisher matrix is given by,

$\mathcal{F}_{ij}=\sum_{X,Y}\sum_{\ell}\dfrac{\partial
C_{\ell}^{X}}{\partial\theta_{i}}\cdot\left(\Sigma_{\ell}^{-1}\right)_{XY}\cdot\dfrac{\partial
C_{\ell}^{Y}}{\partial\theta_{j}},$ (4)

where $\theta$s correspond to the cosmological parameters being constrained.
The partial derivatives of the spectra with respect to the parameters
$\theta$, denoted as $\partial C_{\ell}/\partial\theta_{j}$, are obtained
using the finite difference method. $\Sigma_{\ell}$ is the covariance of the
power spectra, given by Eq.(5)

$\displaystyle\Sigma_{\ell}=\dfrac{2}{\left(2\ell+1\right)f_{\rm
sky}}\begin{pmatrix}(\tilde{C}_{\ell}^{TT})^{2}&(C_{\ell}^{TE})^{2}&\tilde{C}_{\ell}^{TT}C_{\ell}^{TE}&(C_{\ell}^{T\phi})^{2}\\\
(C_{\ell}^{TE})^{2}&(\tilde{C}_{\ell}^{EE})^{2}&\tilde{C}_{\ell}^{EE}C_{\ell}^{TE}&(C_{\ell}^{E\phi})^{2}\\\
\tilde{C}_{\ell}^{TT}C_{\ell}^{TE}&\tilde{C}_{\ell}^{EE}C_{\ell}^{TE}&\frac{1}{2}\left[(C_{\ell}^{TE})^{2}+\tilde{C}_{\ell}^{TT}\tilde{C}_{\ell}^{EE}\right]&C_{\ell}^{T\phi}C_{\ell}^{E\phi}\\\
(C_{\ell}^{T\phi})^{2}&(C_{\ell}^{E\phi})^{2}&C_{\ell}^{T\phi}C_{\ell}^{E\phi}&(\tilde{C}_{\ell}^{\phi\phi})^{2}\end{pmatrix},$
(5)

where $f_{\textrm{sky}}$ is the fraction of the sky for the survey under
consideration and

$\tilde{C}_{\ell}^{X}=C_{\ell}^{X}+N_{\ell}^{X},$ (6)

$X=\\{TT,EE,\phi\phi\\}$. $N_{\ell}^{TT}$ and $N_{\ell}^{EE}$ are the residual
noise and foreground power after combining data from all the bands using the
LC method. $N_{\ell}^{\phi\phi}$ is the lensing reconstruction noise derived
using Quadratic Lensing Estimator (Hu & Okamoto, 2002). In Table 3 we list the
fiducial values of the cosmological parameters at which the Fisher matrix is
evaluated and associated priors.

Table 3: Fiducial values of the parameters and priors used in this work. All
the applied priors are Gaussian with the widths given below in the table.

Parameter | Fiducial | Prior
---|---|---
$\Lambda{\rm CDM}$ parameters:
Amplitude of scalar | 3.044 | -
fluctuations ${\rm ln}(10^{10}A_{s})$
Dark matter density $\Omega_{c}h^{2}$ | 0.1200
Baryon density $\Omega_{b}h^{2}$ | 0.02237
Scalar spectral index $n_{s}$ | 0.9649
Hubble parameter $h$ | 0.6732
Reionization optical depth $\tau_{\rm re}$ | 0.0544 | 0.007
Extensions:
Sum of neutrino masses $\sum m_{\nu}$ | 0.06 eV | -
Number of relativistic species $N_{\mathrm{eff}}$ | 3.044
Running of the spectral index $n_{\rm run}$ | 0
Spatial curvature $\Omega_{k}$ | 0
Equation of state of dark energy $w_{0}$ | -1
Helium abundance $Y_{P}$${}^{\dagger}$${}^{\dagger}$footnotemark:  | 0.245

††footnotetext: Set from BBN (Big Bang nucleosynthesis) consistency when
$Y_{P}$ is not included in the extension.

We calculate the Fisher matrix for each of the SPT-3G surveys given in Table 1
and combine them to obtain the final SPT-3G Fisher matrices as

$\displaystyle\mathcal{F}_{\rm Summer}$ $\displaystyle=\mathcal{F}_{\rm
Summer-A}+\mathcal{F}_{\rm Summer-B}+\mathcal{F}_{\rm Summer-C}$
$\displaystyle\mathcal{F}_{\rm Ext-4k}$ $\displaystyle=\mathcal{F}_{\rm
Main}+\mathcal{F}_{\rm Summer}$ $\displaystyle\mathcal{F}_{\rm Ext-10k}$
$\displaystyle=\mathcal{F}_{\rm Main}+\mathcal{F}_{\rm
Summer}+\mathcal{F}_{\rm Wide}\,.$ (7)

The parameter covariance matrices are then obtained by inverting the Fisher
matrix. We fix the $\ell$ ranges for all the SPT-3G surveys to the following
for the Fisher matrix calculation: $300\leq\ell\leq 3500$ for $TT/TE$;
$300\leq\ell\leq 4000$ for $EE$; and $30\leq L\leq 3500$ for $\phi\phi$. For
lensing reconstruction, we use $300\leq\ell\leq 3500$ for T and
$300\leq\ell\leq 4000$ for P. We ignore $\ell>3500$ in $TT/TE$ because of the
difficulties in modeling the extragalactic foreground signals in the
temperature data. We have checked the correlations between the SPT-3G Main and
the contiguous SPT-3G Summer surveys and we have found that they are
negligible. This is due to the fact that we do not include the large-scale
modes $\ell\leq 300$. Since the extragalactic foregrounds are largely
unpolarized (Datta et al., 2019; Gupta et al., 2019), we can set a higher
$\ell_{\rm max}$ for $EE$. We find that extending $\ell_{\rm max}^{EE}>4000$
results in marginal improvements, indicating those modes are noise dominated,
as can be inferred from Fig. 3.

###  3.3 MUSE

MUSE (Millea & Seljak, 2021) is a general algorithm for approximate
marginalization over arbitrary latent parameters which yields Gaussianized,
asymptotically unbiased and near-optimal constraints on parameters of
interest. In our application of MUSE, the parameters of interest are the
unlensed $TT$, $TE$, $EE$, and $\phi\phi$ bandpowers, which are weighted
averages of the respective power spectra across defined multipole bins. These
bandpowers depend on the latent parameters, which are the pixel values of the
lensing map and the unlensed CMB field maps.

As previously demonstrated by Millea et al. (2021) using SPTpol data, the
standard Quadratic Lensing Estimator is sub-optimal for lensing reconstruction
at noise levels below $\Delta_{T}\sim 5\mu\mathrm{K}{\rm-arcmin}$ (Hirata &
Seljak, 2003a, b) and hence, using MUSE has an advantage in terms of the
lensing reconstruction. Another advantage of MUSE is that it performs optimal
delensing on the lensed CMB maps and produces estimates of the unlensed power
spectra, lensing potential power spectra and their joint covariance. This
covariance naturally includes lensing-induced correlations between different
bandpowers ($\ell-\ell^{\prime}$); and also between delensed CMB $TT/EE/TE$
and lensing $\phi\phi$ spectra (Peloton et al., 2017; Trendafilova, 2023).

Despite the above advantages, we do not expect a significant improvement in
the constraints from using MUSE over QE methods. This is because the lensing
bandpower errors are dominated by sample variance over the angular scales for
which MUSE reduces $\phi$ map noise, and the lensing-induced correlations are
not important for the current noise levels. Moreover, the optimality of MUSE
is non-negligibly better only for the SPT-3G Main survey, and not for the
others given their higher noise levels (see Table 1).

The optimality of the MUSE estimates is useful for other purposes like
delensing the lensing-induced B-modes and cross correlations, at least for the
SPT-3G Main survey with its low noise levels (see Table 1). We demonstrate
this in Fig. 4 which shows the lensing reconstruction noise
$N_{L}^{(0,\phi\phi)}$ for different surveys. For the SPT-3G Main survey shown
in green, the lensing reconstruction noise goes down by roughly $\times 2.5\
(\times 1.9)$ for polarization-only (including temperature information)
lensing estimators when we replace QE (solid) with MUSE (dash-dotted). As
mentioned above, the difference between QE and MUSE is negligible for other
surveys and hence, not shown.

Figure 4: Left panel: Lensing reconstruction noise $N_{L}^{(0,\phi\phi)}$
power spectra for the SPT-3G surveys along with Planck, SO and ACT. For the
SPT-3G Main survey, we show two curves: solid green for QE and dash-dotted
green for MUSE. For SPT-3G Main survey noise levels, lensing noise from MUSE
is roughly $\times 2$ better than QE at large scales. For other surveys, the
difference between MUSE and QE is negligible and hence, not shown. Right
panel: Bandpower errors for the lensing power spectrum (gray) from different
surveys. While the lensing reconstruction noise $N_{L}^{(0,\phi\phi)}$ is
lower on large scales for MUSE, the bandpower errors are not significantly
different as they are dominated by the sample variance.

In MUSE, we express the lensing and unlensed CMB power spectra as scaled
versions of the fiducial power spectra, which, for our purposes, are based on
the Planck 2015 spectra (Planck Collaboration et al., 2016b). This is given by

$C_{\ell}=A_{\ell}C_{\ell}^{\text{fid}}\,,$ (8)

where $C_{\ell}^{\text{fid}}$ represents the fiducial power spectrum from the
Planck 2015 data, and $A_{\ell}$ is the scaling factor applied to this
fiducial spectrum. Here, $C_{\ell}$ is used as a general notation that
encompasses all types of power spectra, including unlensed $TT/TE/EE$ and
$\phi\phi$.

In our application of MUSE, we do not estimate the power at each individual
multipole but instead bandpower parameters, $A_{b}$, that each scale a
fiducial power spectrum across a band of width $\Delta\ell=100$. As a result,
MUSE produces a covariance matrix of these scaling coefficients
$\Sigma_{A_{b}}\equiv\left<A_{b}A_{b}^{*}\right>$. The $A_{b}$ can be
understood as the binned version of $A_{\ell}$.

As described below, this covariance is subsequently utilized to construct the
cosmological parameter-Fisher matrix, from which we derive the constraints on
parameters. We provide more details about MUSE in the appendices.
Specifically, we describe the derivation of the MUSE covariance matrix in
Appendix A, the simulations used for estimating the covariance matrix in
Appendix A.1, and the lensing reconstruction noise in Appendix A.2.

The standard Fisher formula described in the previous section Eq.(4) can be
adapted to incorporate the MUSE covariance matrix $\Sigma_{A_{b}}$. Let the
covariance matrix of the power spectra be represented by $\Sigma_{C_{\ell}}$
and the covariance matrix of the amplitudes be represented by
$\Sigma_{A_{\ell}}$, then the Fisher matrix can be written as,

$\displaystyle\mathcal{F}_{ij}$
$\displaystyle=\sum_{XY}\sum_{\ell\ell^{\prime}}\dfrac{\partial
C_{\ell}^{X}}{\partial\theta_{i}}\left(\Sigma_{C_{\ell}}^{-1}\right)_{XY}^{\ell\ell^{\prime}}\dfrac{\partial
C_{\ell^{\prime}}^{Y}}{\partial\theta_{j}}$ (9)
$\displaystyle=\frac{1}{C^{\text{fid}}_{\ell}}\frac{\partial
C_{\ell}}{\partial\theta_{i}}\cdot\left(\Sigma_{A_{\ell}}^{-1}\right)\cdot\frac{1}{C^{\text{fid}}_{\ell}}\frac{\partial
C_{\ell}}{\partial\theta_{j}}$ (10) $\displaystyle\approx\dfrac{\partial
A_{b}}{\partial\theta_{i}}\cdot\left(\Sigma_{A_{b}}^{-1}\right)\cdot\dfrac{\partial
A_{b}}{\partial\theta_{j}}.$ (11)

For ease of notation, we drop the summation symbols and transition to using
matrix products in the second line. In the last line we assume that the Fisher
matrix is approximately equivalent to its binned version. We obtain the
derivatives of $A_{b}$ with respect to the cosmological parameters via a
minimum-variance binning of $\partial C_{\ell}/\partial\theta$ given by

$\dfrac{\partial A_{b}}{\partial\theta}=\dfrac{1}{\sum_{\ell\in
b}\dfrac{1}{\sigma_{A_{\ell}}^{2}}}\sum_{\ell\in
b}\dfrac{1}{\sigma_{A_{\ell}}^{2}}\dfrac{\partial
C_{\ell}/\partial\theta}{C_{\ell}^{\text{fid}}}\,,$ (12)

where we use an analytic ansatz for the weighting function
$\sigma_{A_{\ell}}^{2}$. For $X\in[TT,EE,\phi\phi]$ this is given by

$\sigma_{A_{\ell}^{X}}^{2}=\frac{\sigma^{2}_{C_{\ell}^{X}}}{(C_{\ell}^{X,\text{fid}})^{2}}\equiv\frac{2}{2\ell+1}\frac{(C_{\ell}^{X}+N_{\ell}^{X})^{2}}{(C_{\ell}^{X,\text{fid}})^{2}}$
(13)

and for $TE$ it is

$\sigma_{A_{\ell}^{TE}}^{2}=\frac{1}{2\ell+1}\frac{(C_{\ell}^{TE})^{2}+(C_{\ell}^{TT}+N_{\ell}^{TT})(C_{\ell}^{EE}+N_{\ell}^{EE})}{(C_{\ell}^{TE,\text{fid}})^{2}}.$
(14)

Figure 5: Ratio of the $\Lambda{\rm CDM}$ parameter constraints derived from
MUSE to those derived from analytical covariance matrix for the SPT-3G Main
survey (dark green circle) and the SPT-3G Wide survey (orange triangle). We
note that they agree with each other within 10%.

In Fig. 5, we show the ratio of parameter constraints obtained from MUSE and
the analytic covariance for SPT-3G Main (green) and the SPT-3G Wide (orange)
surveys. We find that these methods are consistent with each other within a
10% margin, which we deem sufficient to satisfy our validation criteria.

There are a number of reasons to expect these errors to differ to some degree,
which we already brought up when describing the advantages of MUSE. Another
reason is that the analytic results are not binned, but we found that for both
the Wide and Main surveys, binning to $\Delta\ell=100$ only leads to at most
2% degradation in constraints. The largest difference is the 8% tighter
constraint on $\Omega_{b}h^{2}$ from the analytic forecast for the SPT-3G Wide
survey. This is because the analytic estimate does not include the
$\ell$-by-$\ell$ lensing-induced correlations among different CMB multipoles,
and that can lead to overly-optimistic constraints. By artificially removing
off-diagonal elements from the MUSE covariance matrix222Specifically we did
this by inverting it, removing the off-diagonals, and inverting back, we were
able to largely reproduce this reduced analytic error for both
$\Omega_{b}h^{2}$ and $n_{s}$.

###  3.4 Combining with Planck data

Comparison of the SPT-only $\Lambda{\rm CDM}$ constraints with those from
Planck, as we will see, will enable a powerful test of the $\Lambda{\rm CDM}$
model. If these tests are consistent with $\Lambda{\rm CDM}$ then we will be
interested in using the combined SPT-3G+Planck data to estimate $\Lambda{\rm
CDM}$ parameters, as Planck provides complementary information on large
angular scales. Consistent or not, we will also be interested in using the
combined data to constrain $\Lambda{\rm CDM}$ extensions. Hence, we report the
expected constraints from the combined datasets under the assumption of
$\Lambda{\rm CDM}$ as well as under the assumption of various extensions.

The standard approach to obtain this combination is to add the Fisher matrices
from SPT-3G and Planck similar to Eq.(7) as

$\displaystyle\mathcal{F}_{{\rm SPT}+{\it Planck}}$
$\displaystyle=\mathcal{F}_{\rm\widehat{SPT}}+\mathcal{F}_{\widehat{{\it
Planck}}},$ (15)

where $\mathcal{F}_{\rm\widehat{SPT}}$ is the modified SPT-3G Fisher matrix
obtained by taking the inverse variance combination of SPT-3G and Planck noise
in the field under consideration; and $\mathcal{F}_{\widehat{{\it Planck}}}$
is the modified Planck Fisher matrix after scaling to remove the overlapping
sky region between SPT-3G and Planck. For example, if $f_{\mathrm{sky}}=1$ for
$\mathcal{F}_{{\it Planck}}$, then for $\mathcal{F}_{\widehat{{\it Planck}}}$,
we use $f_{\mathrm{sky}}=1-f_{\mathrm{sky}}^{\rm SPT}$.

However, obtaining Eq.(15) is non-trivial as it requires us to know the sky
fraction used by Planck, which varies for different bands and power spectra
(Planck Collaboration et al., 2020). Instead, we follow a different approach
which involves two steps: First, we derive the Planck Fisher matrix directly
from the publicly available Planck PR3 chains (Planck Collaboration et al.,
2020); second, when combining the above Planck Fisher matrix with SPT-3G, we
modify the SPT-3G Fisher matrix by removing the large-scale modes from it, to
avoid double counting information in the overlapping region.

Figure 6: Ratio of power spectrum errors for Planck to those of SPT-3G for the
same sky fraction. $\ell_{min}$, the multipole value at which we switch from
Planck to SPT-3G in combined constraints (shown as the vertical dash-dotted
lines), is chosen to be where this ratio roughly crosses $2$. The blue, green,
and orange colors correspond to $TT,TE,EE$ respectively. For $\phi\phi$ this
ratio is greater than $2$ for all the multipoles used in the forecast, and
therefore it is not shown. The darker curves correspond to the SPT-3G Main
survey whereas the lighter curves represent the Ext-10k survey. The choice of
$\ell_{\rm min}$ does not change significantly for different SPT-3G surveys.

We use BASE_PLIKHM_TTTEEE_lowL_lowE_lensing333Planck chains are downloaded
from https://wiki.cosmos.esa.int/planck-legacy-
archive/index.php/Cosmological_Parameters chains for the first step unless
specified otherwise. These chains, which contain samples from the probability
distribution of cosmological parameters, are used to construct a parameter
covariance matrix. By inverting this covariance matrix, we obtain the Planck
Fisher matrix. For the second step, we restrict the information we use from
SPT power spectra to multipoles above some $\ell_{\rm min}$ or $L_{\rm min}$
threshold, set by the approximate multipole at which the $S/N$ for Planck is
$\times 2$ lower than that for SPT in the same patch. We find that the choice
of $\ell_{min}$ does not change significantly for different SPT-3G surveys. As
can be seen from Fig. 6, which shows the ratio of Planck $TT$, $TE$, and $EE$
power spectrum uncertainties to those projected for SPT-3G, this threshold is
roughly 2000, 1000, and 750 for $TT$, $TE$, and $EE$ respectively, and we
adopt these values for the joint forecasting. For lensing power spectra, we
set $L_{\rm min}=30$ (the same as our fiducial setting) because Planck’s
lensing $S/N$ is much lower than SPT on all scales, as can be seen from Fig.
3. However, we find our forecasts to be insensitive to the choice of $L_{\rm
min}$. For example, with $L_{\rm min}=400$ for SPT, our constraints only
weaken by $\lesssim 5\%$.

We validate this approach of using $\ell_{\rm min}$ cuts rather than the
$f_{\mathrm{sky}}$ scaling in Eq.(15) by creating a toy model for a Planck-
like Fisher matrix with $f_{\mathrm{sky}}=0.5$. For this toy model, we use the
following multipole ranges: $\ell\in[2,2000],\ [30,2000],\ [30,2000]$
respectively for $TT$, $TE$, and $EE$ and $L\in[30,400]$ for $\phi\phi$. Next,
we compare the Fisher forecasts from Eq.(15) and the $\ell_{\rm min}$-cut
approach discussed above. We find the results from the $\ell_{\rm min}$ cut
approach to be $\lesssim 8\%$ worse than the traditional method. Given that
the two methods agree well and given that our $\ell_{\rm min}$-cut-based
forecasts are slightly on the conservative side, we choose to use this
approach for the rest of this work.

##  4 Results

Figure 7: Figure of merit (FoM) for $\Lambda{\rm CDM}$ compared to Planck: We
present the FoM for SPT-3G Main (green), Ext-4k (yellow), and Ext-10k (red).
The transparent yellow circle represents the improvement in Ext-4k, if we were
to continue the SPT-3G Main survey during 2024 as opposed to conducting the
SPT-3G Wide survey. The evident increase in FoM with the new strategy
emphasizes its potential efficacy over continuing the SPT-3G Main survey for
cosmological parameter constraints. For reference, we also show the FoM
expected for the two SO configurations (Baseline in blue and Goal in orange)
under the assumption that the observations will begin in early 2025.  Figure
8: Constraints on the standard $\Lambda{\rm CDM}$ parameters expected from
combinations of the three SPT-3G surveys (SPT-3G Main in green, Ext-4k in
yellow, and Ext-10k in red) compared to Planck (grey shaded region). We expect
significant improvements on all the parameters except $n_{s}$ and $\tau_{\rm
re}$ which depend on the largest scales that ground-based surveys cannot
easily measure.

In this section, we begin by presenting our forecasts on cosmological
parameters using all three SPT-3G surveys, assuming the 6-parameter
$\Lambda{\rm CDM}$ model. Subsequently, we project the extent to which
combined SPT-3G and Planck data can constrain both single- and double-
parameter extensions to the $\Lambda{\rm CDM}$ model. We also qualitatively
investigate how well SPT-3G data can differentiate between $\Lambda{\rm CDM}$
and alternative models that have been proposed in the literature to address
current cosmological tensions (for example, Cyr-Racine et al., 2022;
Schöneberg et al., 2022; Meiers et al., 2023; Hughes et al., 2023; Khalife et
al., 2023; Schöneberg et al., 2022). The models we consider are fully
consistent with current CMB and lensing power spectra measurements but make
different predictions, from those of $\Lambda{\rm CDM}$, for signals that can
be measured well by SPT-3G.

###  4.1 Forecasts of constraints on $\Lambda{\rm CDM}$ parameters

We start first with forecasts of constraints on the six parameters of the
$\Lambda{\rm CDM}$ model, namely baryon density $\Omega_{b}h^{2}$, cold dark
matter density $\Omega_{c}h^{2}$, Hubble constant $H_{0}$, optical depth to
reionization $\tau_{\rm re}$, amplitude (defined at the pivot scale
$k_{0}=0.05$ Mpc-1) and spectral index of scalar fluctuations, $A_{s}$ and
$n_{s}$. For these constraints we fix the neutrino mass to the minimal value
of $0.06$ eV. We also include a prior on $\sigma(\tau_{\rm re})=0.007$ from
Planck (Planck Collaboration et al., 2020). We use the MUSE approach as
described in § 3.3 to derive these constraints, and remind the reader that
there is agreement between MUSE and the traditional analytic approach to
better than 10%, as shown in Fig. 5.

As we will see below, SPT-3G can produce constraints on $\Lambda{\rm CDM}$
parameters that are tighter than those we have from Planck. Since these
constraints are obtained from largely different signals than those from
Planck, they provide a robust consistency test for the $\Lambda{\rm CDM}$
model. As mentioned earlier in § 3.4, we also present $\Lambda{\rm CDM}$
forecasts by combining SPT-3G and Planck.

In Fig. 7, we present the projected Figure of Merit (FoM), which is inversely
proportional to the volume of allowed 6-dimensional $\Lambda{\rm CDM}$
parameter space. It is calculated as

${\rm FoM}=\sqrt{1/{\rm det}(\mathcal{F}^{-1})}$ (16)

for different SPT-3G surveys (SPT-3G Main in green, Ext-4k in yellow, and
Ext-10k in orange-red) along with SO (blue). The FoM is inversely proportional
to the 6-dimensional volume of the 68% confidence region.444In our Gaussian
approximation, this proportionality holds independent of whether it is 68%,
95% or any other such threshold for defining the confidence region We plot it
divided by the FoM for Planck.

As is evident from the figure, SPT-3G surveys significantly increase the FoM
relative to that from Planck. From just the first two years (2019-2020) of
SPT-3G Main survey data, the FoM increases by $\times 2.5$ over Planck, and
that jumps to $\times 13$ with the addition of the first two years of SPT-3G
Summer data. Both of these are further improved by roughly $\times 1.7$ with
the inclusion of SPT-3G Main and SPT-3G Summer survey data through 2023.

The figure also demonstrates the benefit of expanding sky coverage rather than
continuing to go deeper on the SPT-3G Main survey patch. One can see this from
the minimal improvement to Ext-4k ($\times 1.1$) that we anticipate if we
continue with the SPT-3G Main survey patch observations for the 2024 winter,
indicated by the semi-transparent yellow circle. On the other hand, switching
to the SPT-3G Wide observations in the 2024 season improves the constraining
power by $\times 5$ compared to Ext-4k and by $\times 130$ compared to Planck.

Were we to observe the SPT-3G Wide patch for one more year we would not notice
a significant increase in the FoM. Instead of doing this we plan to switch
back to the SPT-3G Main survey during the 2025 and 2026 winter seasons.
Further noise reduction in the SPT-3G Main survey will not increase the FoM
much, as can be seen in Fig. 7, but will significantly improve the
measurements of the secondary CMB anisotropies including lensing, which will
further improve the subtraction of lensed B-modes and hence, the search for
B-modes from primordial gravitational waves (Ade et al., 2021).

Table 4: Forecasted $1\sigma$ errors on $\Lambda{\rm CDM}$ parameters for various survey regions using SPT data. The numbers in the parentheses correspond to the constraints obtained when SPT-3G is combined with Planck. Note that when SPT-3G is combined with Planck, we remove information below $\ell_{\rm min}=2000,1000,750,30$ for $TT,TE,EE$ and $\phi\phi$ power spectra respectively from SPT-3G, to avoid double counting information. Parameter | SPT-3G Main | SPT-3G Summer | SPT-3G Wide | Ext-4k | Ext-10k
---|---|---|---|---|---
$A_{s}$ [$10^{-11}$] | 2.79 (2.47) | 2.66 (2.50) | 2.54 (2.42) | 2.53 (2.38) | 2.35 (2.29)
$H_{0}[{\rm km/s/Mpc}]$ [$10^{-1}$] | 3.28 (3.16) | 3.91 (3.47) | 3.49 (3.26) | 3.03 (2.95) | 2.78 (2.78)
$n_{s}$ [$10^{-3}$] | 7.23 (3.14) | 6.12 (3.16) | 5.10 (3.03) | 4.76 (3.00) | 3.66 (2.83)
$\Omega_{b}h^{2}$ [$10^{-5}$] | 12.9 (7.87) | 14.9 (8.68) | 11.9 (7.66) | 9.5 (6.7) | 7.37 (5.67)
$\Omega_{c}h^{2}$ [$10^{-4}$] | 8.86 (7.71) | 9.84 (8.34) | 8.92 (7.97) | 8.07 (7.39) | 7.36 (7.10)
$\tau_{\rm re}$ [$10^{-3}$] | 6.85 (6.64) | 6.78 (6.65) | 6.59 (6.47) | 6.63 (6.42) | 6.28 (6.19)

In Table 4 we list our forecasts of $\Lambda{\rm CDM}$ parameter errors for
different SPT-3G datasets. We also present them in Fig. 8, where the bars show
the ratios of the values of $\sigma(\theta)$ from Planck to those from SPT.

Let us begin by noting two things relevant for interpretation of our $A_{s}$
and $\tau_{\rm re}$ results. First, the calibration of SPT-3G maps relies on
data from Planck (Dutcher et al., 2021), and, consequently, SPT-3G constraints
on the normalization parameter $A_{s}$ are not entirely independent of Planck.
Second, for SPT-only constraints in this work, we include a Planck-like prior
on the optical depth, $\sigma(\tau_{\rm re})=0.007$, since $\tau_{\rm re}$ is
primarily constrained by the large-scale $\ell\lesssim 10$ reionization bump
in $EE/TE$, not accessible by SPT. All the $\tau_{\rm re}$ constraints are
dominated by this prior. These two points are not unrelated since the
combination $A_{s}\ e^{-2\tau_{\rm re}}$ is very tightly constrained by CMB
data at $\ell>20$, so improvements in $\tau_{\rm re}$ determination translate
into improvements in $A_{s}$ determination, and vice versa. The small
improvements that we do see in $\tau_{\rm re}$ are due to the dependence of
the amplitude of gravitational lensing on $A_{s}$, an amplitude unaffected by
$\tau_{\rm re}$.

Focusing on the other four parameters, we find that the SPT-only constraints
are generally comparable to, or better than, those obtained from Planck alone,
for each of our three SPT-3G survey combinations. For the case of Ext-10k, the
improvements in $H_{0}$ and $\Omega_{b}h^{2}$ are nearly a factor of 2, and
for $\Omega_{c}h^{2}$ they are just more than a factor of 1.5. For $n_{s}$,
the relative errors from SPT-3G Main survey and Ext-4k survey are larger
compared to those from Planck due to Planck’s significantly broader multipole
range. Nonetheless, the Ext-10k survey still manages to achieve slightly
tighter constraints on $n_{s}$ than what we have from Planck.

####  4.1.1 Comparison with SO

In Fig. 7, we also show the expected constraining power from the Simons
Observatory for two configurations: SO-Baseline (blue dash-dotted) and SO-Goal
(orange dash-dotted), based on the noise levels in Simons Observatory
Collaboration (2019). We use the original SO noise model
code555https://github.com/simonsobs/so_noise_models to get the noise curves
but use the same multipole ranges described in § 3.2 for forecasting. As a
check, we compare the constraints that we get from our forecasts for SO
against what is reported in Simons Observatory Collaboration (2019) but note
that we do not include Planck unlike SO’s original forecasts.

Comparing our forecasts for SO to Table 3 of Simons Observatory Collaboration
(2019), our forecasts are slightly better for $H_{0}$ and $\Omega_{c}h^{2}$
but they agree within 10%. For $n_{s}$ and $\Omega_{b}h^{2}$, we obtain 15-35%
worse constraints, which is presumably because we do not add the large-scale
information from Planck. On the other hand, we obtain roughly 35% (20%) better
constraints on $A_{s}$ ($\tau_{\rm re}$). When we increase the prior
uncertainty on optical depth to $\sigma(\tau_{\rm re})=0.009$ to match SO’s
forecasting code (see caption of Table 3 in Simons Observatory Collaboration
2019), our results on $A_{s}$ and $\tau_{\rm re}$ match the numbers quoted in
Simons Observatory Collaboration (2019). Switching to SO-Goal configuration
reduces the errors on all parameters by 5% except for $\Omega_{c}h^{2}$ where
the constraining power improves by 15%.

We also note from Fig. 7, the constraining power of the SPT-3G Ext-10k survey
for $\Lambda{\rm CDM}$ parameters will be roughly similar to the first two
years of SO-Baseline. Such similar levels of constraining power between SPT
and SO along with significant sky overlap (see Fig. 1) will allow for
comparisons of the two datasets at both the power spectrum and map level (such
as the ACT-Planck and SPT-Planck comparisons shown in Louis et al. 2014 and
Hou et al. 2018) which will provide useful opportunities for identifying
and/or limiting systematic errors in individual datasets.

####  4.1.2 Real-world effects

In this section, we examine how our forecasts change as we vary our baseline
modeling assumptions, and introduce additional non-ideal aspects into our
model of the data on which the forecasts are based. We modify our noise curves
to account for filtering effects and to include a higher level of complexity
in the noise due to the correlated atmospheric signal on large scales in the
temperature data. We also investigate the degradation in constraints that
could come from galactic foregrounds. We also check the impact of including
uncertainties in the calibration of our temperature and polarization maps. As
detailed below, we find the degradation of parameter errors relative to the
baseline case, from all of these effects, is $\lesssim 10\%$. Finally, we
build further confidence in our forecasts by comparing them to constraints
from an ongoing MUSE analysis of the two-year SPT-3G Main data.

##### Atmospheric noise

The first thing we examine is the impact of the filtering applied to the
detector time-ordered data (TOD) before it is processed into maps. To avoid
excess noise in the scan direction in the resulting maps, primarily as a
result of the large-scale atmospheric noise, we apply filters to remove the
long-timescale information from the detector TOD. While this improves the
quality of the maps, it results in a reduction of data volume that increases
the final power spectrum errors by a factor equivalent to raising map noise
levels by roughly 10%. We test the impact of this by increasing the white
noise levels in Table 1 by 10%. We found that this results in changes to the
parameter constraints at the $\leq 1\%$ level. Secondly, since SPT-3G uses
multichroic detectors that can simultaneously observe in multiple frequencies,
the large-scale atmospheric noise is expected to be correlated between bands.
From the 2019-20 dataset, we estimate the correlation at the largest scales to
be $\sim 90\%$ between the 150 and 220 GHz bands, and around $70\%$ between
the 90 and 150 GHz bands as well as between the 90 and 220 GHz bands. This
degree of correlation diminishes as we move to small scales. To take this into
account, we set the cross-frequency noise to be correlated across bands and
rerun the LC step to get the residuals. This correlated noise, in fact,
reduces the large-scale LC residuals slightly. When we propagate these to
parameter constraints, we find changes to the parameter constraints at the
$\leq 1\%$ level, indicating that our forecasts are robust to these details
about the noise spectra.

##### Galactic foregrounds and mask

Given that the SPT-3G Wide survey extends close to the plane of the galaxy
(see Fig. 1), it is crucial to understand the impact of galactic foregrounds
on the parameter constraints. To assess this, we make use of the PySM
simulations (Thorne et al., 2017) of galactic foregrounds and the galactic
foreground masks (GAL070, GAL080, GAL090) produced by the Planck collaboration
(Planck Collaboration et al., 2016b). The names of the above masks correspond
to the sky retained after masking the galactic plane. For instance, GAL090
masks 10% of the region with a high level of foregrounds close to the galactic
plane while GAL070 removes 30% of the sky and is the most conservative amongst
the three.

We use these masks and calculate the level of galactic dust and synchrotron
signals in PySM simulations in the following regions: Case (A) GAL080-GAL070;
Case (B) GAL090-GAL070; and Case (C) GAL090-GAL080. The three cases correspond
to the differences between the respective masks; for instance, GAL080-GAL070
represents the sky area included in mask GAL080 but excluded in GAL070. As
expected, we find the galactic foreground in Case (C) to be the highest
followed by Cases (B) and (A).

We perform a conservative test by assuming that the galactic foreground levels
across the entire SPT-3G Wide survey match one of these three cases. With this
assumption, we derive the LC residuals and the subsequent impact on the
parameter constraints. Even for Case (C), where the galactic foregrounds are
the highest, we find the constraints degrade by only $8\%$. This is not
surprising given that the galactic foregrounds are mostly important on the
largest scales and the constraining power of SPT-3G is predominately driven by
the high-$\ell$ region.

In the preceding sections, we mentioned that the observing strategy to
minimize galactic foreground contamination in the SPT-3G Wide involves
observing within the confines of the Planck GAL080 mask, which corresponds to
roughly $6000\,{\rm deg}^{2}$ of sky coverage. In the scenario where we might
need to adopt a more conservative Planck GAL070 mask, we found that this would
lead to a modest degradation in parameter constraints, at 4-5%. While we do
not address potential biases due to mismodeling of the galactic foregrounds,
we remain confident that any such biases would be minimal, given that the
levels of galactic foregrounds are significantly lower than the CMB power
spectra levels at the multipoles relevant to this analysis.

##### Temperature and polarization calibration

Next, we assess the impact of calibration uncertainty on the temperature and
polarization maps by including two absolute calibration parameters, $T_{\rm
cal}$ and $P_{\rm cal}$ (as defined in Dutcher et al., 2021), in our forecasts
and marginalizing over them. In the case where we do not apply a prior, we
find a significant $\times 2$ hit on some of the parameters, which is not
surprising since parameters like $A_{s}$ are fully degenerate with the
calibration parameters. By comparing SPT-3G Main survey maps from 2019-2020 to
Planck maps in that same region, we are able to constrain $T_{\rm cal}$ and
$P_{\rm cal}$ to roughly 0.2% and 1%, respectively (where the errors are a
combination of statistical uncertainty in the comparison and uncertainties in
the Planck calibration). If we adopt these as priors, the resulting
degradation in the parameter constraints are $\leq 5\%$ for all parameters. If
we loosen the $T_{\rm cal}$ prior to 1%, the degradation increases to $\sim
20\%$. If we assume either parameter is estimated perfectly and allow for
uncertainty in only one calibration parameter, we find the resulting
degradation to be negligible, even without any prior, consistent with the
results from Galli et al. (2021).

Finally, we replace the covariance matrix for the 2-year SPT-3G Main survey
from this work with the one from an ongoing MUSE analysis performed on the
real data from the same survey. The latter takes into account certain effects
that are present in the real data but not taken into account in our forecasts.
These include anisotropic noise, anisotropic filtering, and masking. With this
covariance swap, we find that the degradation in the parameter constraints is
$\leq 10\%$.

###  4.2 Single-parameter extensions to $\Lambda{\rm CDM}$

Figure 9: Ratio of 1$\sigma$ constraints on single-parameter extensions from
Planck to those forecasted from the combination of Ext-10k and Planck. The
gray shaded region corresponds to the current best constraints from Planck
alone. The actual errors are expressed in the text boxes.  Figure 10:
Degradation in the $\Lambda{\rm CDM}$ constraints due to single-parameter
extensions compared to the original constraints reported in Fig. 8 for the
Ext-10k+Planck dataset. The horizontal axis shows the parameter that is varied
in our extensions. Except for $H_{0}$, which is affected by a geometric
parameter degeneracy, the degradation in constraints for other $\Lambda{\rm
CDM}$ parameters is quite mild.

Now we turn our attention towards extensions to the $\Lambda{\rm CDM}$ model.
For extensions, we always combine SPT-3G with Planck. We consider 7 different
cases by varying the following parameters one at a time along with the 6
$\Lambda{\rm CDM}$ parameters: dark-energy equation of state ($w_{0}$), Helium
mass fraction ($Y_{P}$), running of the spectral index ($n_{\rm run}\equiv{\rm
d}n_{s}/{\rm d}\ln k$), sum of neutrino masses ($\sum m_{\nu}$), effective
number of neutrino species ($N_{\mathrm{eff}}$) and the spatial curvature
parameter ($\Omega_{k}$). The forecasted constraints for Ext-10k are reported
in Fig. 9. As before, we show the improvement in constraints compared to
Planck using the bars. The actual parameter constraints are given in the text
boxes. The constraints from other SPT-3G surveys are presented in Table 5 of
Appendix B.

We find a significant ($\times 2-\times 3$) reduction of the errors in many of
these parameters, namely $\sum m_{\nu}$, $w_{0}$, $Y_{P}$, and
$N_{\mathrm{eff}}$. For other parameters, namely $\Omega_{k}$, and $n_{\rm
run}$, the improvement is $\times 1.4-\times 1.7$ compared to Planck. Some of
these improvements are driven by the inclusion of lensing ($\Omega_{k},w_{0}$)
while others come from higher-redshift information in the damping tail of the
CMB, whose extraction benefits from wide sky coverage ($n_{\rm run}$, $Y_{P}$,
and $N_{\mathrm{eff}}$).

We also investigate the degradation in $\Lambda{\rm CDM}$ constraints due to
degeneracies with the extension parameters. We present these error
degradations in Fig. 10 for Ext-10k+Planck; the full list can be found in
Table 6 of Appendix B. As can be inferred from the figure, the constraints on
the Hubble constant $H_{0}$ degrade significantly when we free $\theta\in[\sum
m_{\nu},N_{\mathrm{eff}},\Omega_{k},w_{0}]$. Note that these degradations
largely disappear with the addition of external datasets that constrain the
redshift-distance relationship down to $z\lesssim 0.1$, namely supernova
brightness measurements. This is the reason why these extensions are not
viable solutions to the $H_{0}$ tension (Knox & Millea, 2020). The degradation
is less severe ($\sim\times 1.6$) for $n_{s},\Omega_{b}h^{2}$ when we vary
$\theta\in[N_{\mathrm{eff}},Y_{P}]$. For the rest, the constraints are robust
and the degradation is $\lesssim\times 1.3$.

In expanding the parameter space from $\Lambda{\rm CDM}$ to $\Lambda{\rm
CDM}$+$w_{0}$, we notice an unexpected tightening in the constraints for
parameters $\Omega_{b}h^{2}$, $\Omega_{c}h^{2}$, and $n_{s}$. We suspect that
this arises from our use of a Gaussian approximation for a posterior that is
significantly non-Gaussian. The non-Gaussianity is visually evident in the one
and two-dimensional marginal posteriors derived from the appropriate Planck
chains. With the addition of an $H_{0}$ prior, the range of acceptable $w$
values tightens significantly, reducing the degree of non-Gaussianity, and
this anomalous behavior of our forecasts no longer appears.

###  4.3 Double-parameter extensions to $\Lambda{\rm CDM}$

Figure 11: Marginalized two-dimensional parameter contours for the three pairs
of double-parameter extensions to $\Lambda{\rm CDM}$ considered in this work.
The red contours represent the joint constraints from the Ext-10k+Planck, and
the black contours show the constraints from Planck alone. The inverse of the
allowed volume of the two-dimensional parameter space (FoM) are given in the
legend. Compared to the constraints from Planck alone, we find reductions in
individual parameter errors that approach or exceed a factor of two except for
$n_{\rm run}$, where we obtain $\times 1.3$ reduction.

We also explore two-parameter extensions similar to those considered in Planck
Collaboration et al. (2020). These are (a) $N_{\mathrm{eff}}$ and $Y_{P}$, (b)
$N_{\mathrm{eff}}$ and $n_{\rm run}$, and (c) $N_{\mathrm{eff}}$ and $\sum
m_{\nu}$. In Fig. 11, we present the marginalized two-dimensional contours for
the above extensions from both Planck (black) and Ext-10k+Planck (orange). The
FoM for the two-parameter extensions, calculated similar to Fig. 7, is given
in the legend. In most cases, we find an improvement of greater than a factor
of 2 relative to Planck alone.

In case (a), we vary the effective number of neutrino species
$N_{\mathrm{eff}}$ and the fraction of baryonic mass in helium $Y_{P}$
simultaneously. Such constraints are useful for constraining scenarios in
which the value of $N_{\rm eff}$ at big bang nucleosynthesis differs from that
at recombination (Millea et al., 2015; Cyburt et al., 2016; Particle Data
Group et al., 2020). Since both these parameters affect the damping tail of
the CMB, they are partially degenerate and, as expected, the constraints on
both parameters degrade by more than $\times 2$ compared to the single-
parameter extensions in Fig. 9.

In case (b), we vary $N_{\mathrm{eff}}$ and $n_{\rm run}$ which are also
degenerate to some level leading to $\times 1.5$ weaker constraints compared
to Fig. 9.

In case (c), we vary $N_{\mathrm{eff}}$ and the sum of neutrino masses $\sum
m_{\nu}$ since one can expect some level of correlation between the two
(Planck Collaboration et al., 2020). However, this correlation depends on the
datasets being considered. In our case, $\sum m_{\nu}$ is primarily
constrained by lensing while $N_{\mathrm{eff}}$ constraints are dominated by
the small-scale $TT$, $EE$, and $TE$ power spectra. Subsequently, the
degradation in the constraints on these two parameters is negligible compared
to what we obtain for the single-parameter extensions in Fig. 9.

##  5 Power spectrum predictions from $\Lambda{\rm CDM}$ and alternatives

Figure 12: Left panels: Current data from CMB experiments and some alternative
models with either higher $H_{0}$ or lower $\sigma_{8}$ (see text). Except for
lensing data, only the most constraining observations in a given multipole
range have been plotted for clarity. Each spectrum has the best-fit
$\Lambda{\rm CDM}$ spectrum given Planck subtracted as a reference and is
normalized in an $\ell$-dependent manner to reduce dynamic range, as described
in the $y$-axis label. Note that for the ADM model, neither the model
prediction nor the reference model includes non-linear corrections to the
lensing power spectrum, as existing non-linear correction procedures do not
account for the modified dark sector physics. All spectra are taken from
chains constrained using Planck and BAO data, but the specific BAO datasets
used differ between models. The HDGDM constraints also used uncalibrated
supernovae data as well as the SH0ES $H_{0}$ measurement. Right panels: The
same models as in the left panels, forecasted bandpower errors for Ext-10k,
and the 68% confidence region given Planck and the assumption of $\Lambda{\rm
CDM}$. The bandpower errors have been binned with $\Delta\ell=100$. Note that
the $\Lambda{\rm CDM}$ predictions conditioned on Planck are very tight, even
in regions where Planck data on their own are completely unconstraining.

In this section we discuss, in a qualitative manner, prospects for using
SPT-3G data to detect departures from $\Lambda{\rm CDM}$ predictions for CMB
power spectra conditioned on Planck data. Many alternative models that resolve
cosmological tensions, such as the $H_{0}$ and $\sigma_{8}$ tensions, predict
power spectra that differ from $\Lambda{\rm CDM}$ predictions at small angular
scales, and we use a selection of such models to guide our discussion. In Fig.
12, we present the $\Lambda{\rm CDM}$ predictions for angular power spectra
given Planck data, the predictions of alternative models that fit the Planck
data well while either having a high $H_{0}$ or a low $\sigma_{8}$, and the
forecasted Ext-10k error bars.

The selected alternative models are (a) a phenomenological model of non-
perturbatively modified recombination (ModRec) (Lynch et al. 2024, in
preparation); (b) an atomic dark matter (ADM) model arising from a hidden dark
sector charged under a U(1) gauge interaction (Hughes et al., 2023); and (c) a
model featuring an additional dark fluid with an equation of state parameter
that is a free function of redshift, hereafter “high dimensional generalized
dark matter” (HDGDM) (Meiers et al., 2023).

We present one set of spectra from each of these models in Fig. 12 as a
relative shift $\Delta C_{\ell}^{X}=C_{\ell}^{X}-C_{\ell,{\rm ref}}^{X}$
compared to the reference $\Lambda{\rm CDM}$ mean Planck $C_{\ell,{\rm
ref}}^{X}$, normalized as described in the figure caption. Here $X$
corresponds to one of the primary lensed CMB $TT/EE/TE$ or the lensing
$\phi\phi$ power spectra. Each of these spectra are drawn from posterior
distributions conditioned on Planck and BAO data, although the HDGDM spectra
also includes uncalibrated supernovae data in the fit. It should be noted that
the specific BAO datasets used in these constraints differ: please see the
appropriate references for details.

The left-hand panels of Fig. 12 show these spectra in relation to existing CMB
measurements from Planck, SPTpol (Bianchini et al., 2020), SPT-3G 2018
(Dutcher et al., 2021) and ACT DR6 (Qu et al., 2024). For angular scales that
have been probed by multiple experiments, only the most constraining dataset
has been plotted. This set of panels highlights that although these models
make predictions that differ from $\Lambda{\rm CDM}$, they are currently not
ruled out by the data. To the right, we show the 68% confidence region for
angular power spectrum predictions conditioned on Planck data and assuming
$\Lambda{\rm CDM}$, along with forecasted bandpower errors for Ext-10k in
black.

The Planck data, along with the assumption of $\Lambda{\rm CDM}$, make tight
predictions for the power spectrum residuals even at angular scales not probed
by Planck. This is evident from the righthand panels of Figure 12, where the
gray confidence interval extends to no more than about 2% for the $TT$, $TE$,
and $EE$ power spectra. This range represents the expected region that the
power spectra observed by SPT-3G will fall in if measurements continue to be
consistent with $\Lambda{\rm CDM}$. As is evident in these panels, the
alternative models presented here exhibit deviations which are larger than
this expected range, and larger than the expected precision of SPT-3G. We now
discuss each of these models in turn.

The potential role of changes to recombination in alleviating the $H_{0}$
tension has been explored in Chiang & Slosar (2018); Hart & Chluba (2020a);
Sekiguchi & Takahashi (2021); Lee et al. (2023); Cyr-Racine et al. (2022).
Physical models which have a modified recombination epoch include models with
varying fundamental constants (Ade et al., 2015; Hart & Chluba, 2018, 2020b;
Sekiguchi & Takahashi, 2021), primordial magnetic fields (Jedamzik & Pogosian,
2020; Galli et al., 2022; Rashkovetskyi et al., 2021), or energy injection
from decaying dark matter (Galli et al., 2013; Slatyer & Wu, 2017; Finkbeiner
et al., 2012). The ModRec model presented here is an extension of $\Lambda{\rm
CDM}$ that allows for a purely phenomenological modification of the ionization
fraction as a function of redshift through the epoch of hydrogen
recombination, in order to study the role of these changes independently of
the details of particular physical models. We note that the different modified
ionization fractions predicted by the above physical models can be well
approximated within the ModRec model using seven parameters characterizing
deviations from the standard recombination scenario.

This additional freedom allows for higher values of $H_{0}$ to provide good
fits to the Planck data. The ModRec model accommodates a high value of $H_{0}$
by altering the recombination process, shifting last scattering to higher
redshifts and increasing the width of the visibility function. This reduces
the comoving size of the sound horizon at recombination, so that the shorter
distance to last-scattering associated with a higher $H_{0}$ does not shift
the well-determined angular size of the sound horizon. It also impacts the
damping scale, an effect ameliorated by the change to the visibility function
width.

In Fig. 12 we show predictions within the ModRec model for a set of parameters
which fit Planck and eBOSS BAO data at least as well as a $\Lambda{\rm CDM}$
best-fit, while having $H_{0}=70.48$ km/s/Mpc. To maintain the fit with Planck
data, residuals with respect to the fiducial cosmology get shifted to higher
multipoles, which, while unprobed by Planck, are expected to be measured with
high SNR by SPT-3G. We see that SPT-3G has clear potential to rule out such a
model, particularly through measurements of the $TE$ spectrum. We note that
without including BAO data, much higher values of $H_{0}$ are achievable
within the ModRec model while staying consistent with Planck. In this case, if
SPT-3G measurements remain consistent with $\Lambda{\rm CDM}$, solutions to
the $H_{0}$ tension which modify recombination will be seriously challenged by
CMB data alone.

Another approach to resolving the $H_{0}$ tension is to leave the
recombination history unmodified and to instead introduce more freedom in the
dark sector. The HDGDM model is another phenomenological model that uses the
generalized dark matter (GDM) model of Hu (1998) and allows the equation of
state parameter for the GDM to be an arbitrary function of redshift. Meiers et
al. (2023) used such a model to approximate, and perturb around, a more
physical model, the “step-like” model of Aloni et al. (2022). These authors
introduce a massive particle which becomes non-relativistic before decaying,
in equilibrium, into a massless final state. The equation of state parameter
$w(z)$ for such a species begins at 1/3, drops as the particle becomes non-
relativistic, and returns to 1/3 in the final massless state. This new
component affects the expansion history in a non-trivial way, and alters the
time-dependent gravitational driving of the plasma. They found that a
transition at $z\sim 20000$ could accommodate a higher $H_{0}$. In Figure 12,
we present the best-fit spectrum of the “step-like restricted” model of Meiers
et al. (2023) constrained using the “$\mathcal{D}+$” data combination in that
work, which includes Planck BAO, uncalibrated supernovae data, as well a
Gaussian likelihood centered at the SH0ES measurement $H_{0}=73.04\pm 1.04$
km/s/Mpc (Riess et al., 2022). This model has $H_{0}=71.88$ km/s/Mpc,
illustrating how the HDGDM model can accommodate higher values of $H_{0}$.
However, the TT damping tail predictions from this model differ from the
$\Lambda{\rm CDM}$ prediction using the Planck mean parameters at a more than
2% for $\ell>3000$, which should be distinguishable with SPT-3G.

In addition to the $H_{0}$ tension, in recent years there has been increased
attention to the potential discrepancy between different measurements of the
amplitude of matter fluctuations on $8h^{-1}$ Mpc scales, $\sigma_{8}$.
Inferences from Planck, that depend on assuming the $\Lambda{\rm CDM}$ model,
give a value of $\sigma_{8}=0.811\pm 0.006$, whereas the KiDS-1000 survey,
measuring cosmic shear, gives a value of $\sigma_{8}=0.76^{+0.025}_{-0.020}$
(Planck Collaboration et al., 2020; Heymans et al., 2020). Although, DES sees
less of a $\sigma_{8}$ tension compared to KiDS (Dark Energy Survey and Kilo-
Degree Survey Collaboration, 2023). In the ADM model considered here, this
tension is addressed through the introduction of two dark fermions interacting
with a dark photon, where the dark fermions make up some fraction
$f_{\text{ADM}}$ of the dark matter. This results in dark sector dynamics
analogous to the dynamics of the baryon-photon plasma. The pressure support
from the dark photon at early times, prior to the formation of dark atoms,
suppresses the growth of structure. In general, this leads to effects in the
CMB power spectra observable by Planck but if the dark sector temperature is
cool enough, the epoch of significant dark photon pressure support is early
enough, and the Planck observables are relatively unaffected. More direct
inferences of $\sigma_{8}$ are impacted by smaller scales than those to which
Planck is sensitive; so, at the right dark photon temperature with a
suppressed $\sigma_{8}$, Planck observables are unaffected. In Fig. 12, we
present the best-fit model identified in Hughes et al. (2023), coming from a
region of parameter space consistent with Planck data, with
$\sigma_{8}=0.7869$, a low dark photon temperature, and high $f_{\text{ADM}}$.
For this fit, both CMB data from Planck were used, as were BAO measurements
(see Hughes et al. 2023 for details).

From this survey of alternatives to $\Lambda{\rm CDM}$ we conclude that
measurements of temperature, polarization, and lensing power spectra from
SPT-3G are capable of clarifying existing cosmological tensions.

##  6 Conclusions

We presented forecasts of constraints on cosmological parameters that can be
derived from the estimates of lensing, temperature and E-mode polarization
power spectra to come from completed and ongoing SPT-3G sky surveys. We made
forecasts for three combinations of the three SPT-3G surveys of varying sizes
and depths, with particular emphasis on the combination of all three of these
surveys, called Ext-10k, covering $\sim 10000\,{\rm deg}^{2}$ (25% of the
sky). We showed that SPT-3G will enable some powerful tests of the
$\Lambda{\rm CDM}$ model.

We showed that the constraining power of the full SPT-3G dataset will surpass
that of Planck for the power spectra at modes $\ell\gtrsim 1800$ in $TT$,
$\ell\gtrsim 800$ in $TE$, at $\ell\gtrsim 450$ for $EE$ and at $L\gtrsim 30$
in $\phi\phi$. We calculated the predictions of the $\Lambda{\rm CDM}$ model,
conditioned on Planck data, and found that they are highly precise in the
power spectra on these scales, setting us up for a powerful test of
$\Lambda{\rm CDM}$.

We also demonstrated that SPT-3G will facilitate testing of the $\Lambda{\rm
CDM}$ model at the level of cosmological parameters. By propagating the
uncertainties on power spectra to uncertainties on cosmological parameters,
our analysis suggests that data from SPT-3G will be able to achieve similar,
or in some cases, tighter constraints than those from Planck. Since these
constraints incorporate higher weights from polarization, from small angular
scales in temperature, and from lensing when compared to those from Planck, a
consistency check between the two will be a powerful test of $\Lambda{\rm
CDM}$. The constraints on the current expansion rate ($H_{0}$) and baryon
density ($\Omega_{b}h^{2}$) improve by roughly a factor of 2 when compared
with the constraints from Planck, while for the dark matter density
($\Omega_{c}h^{2}$), the improvement is a factor of 1.5. We find that the full
SPT-3G dataset is expected to achieve an improvement in the $\Lambda{\rm CDM}$
FoM of 130; i.e., SPT-3G will reduce the allowed six-dimensional parameter
volume by a factor of 130 compared to Planck.

This reduction in volume is comparable to what is expected from the first two
years of the SO-Baseline dataset. This similarity in constraining power, plus
the substantial amount of overlap in sky coverage, provides an opportunity for
significant map-level consistency tests. These can be used to detect, or
limit, sources of systematic error, increasing the robustness of the parameter
determinations from both SO and SPT-3G.

We also explored how well we can test $\Lambda{\rm CDM}$ by forecasting
constraints on single- and double- parameter extensions with the combination
of Planck+SPT-3G. We found that the combined data will result in considerable
improvements over Planck alone. For example, we found that the constraints on
single-parameter extensions improve by more than $\times 2$ for
$\theta\in[\sum m_{\nu},N_{\mathrm{eff}},w_{0},Y_{P}]$ and roughly $\times
1.5$ for $\theta\in[n_{\rm run},\Omega_{k}]$ compared to Planck-only
constraints. We also showed that including these extensions does not
significantly degrade the constraints on the $\Lambda{\rm CDM}$ model
parameters, except for $H_{0}$, which is highly degenerate with some of the
extensions like $w_{0}$.

In addition to showing the highly-precise predictions of the $\Lambda{\rm
CDM}$ model on angular scales to be measured by SPT-3G, we showed that there
are also alternative models that are consistent with current data, with
significantly different predictions on these same angular scales. We
considered three different models that could address existing cosmological
tensions. While all of these models are consistent with Planck data, they
exhibit deviations from the $\Lambda{\rm CDM}$ predictions—deviations that are
larger than the forecasted SPT-3G error bars. This set of alternative models
illuminates the potential discovery space of SPT-3G observations.

These SPT-3G tests of $\Lambda{\rm CDM}$ will begin soon and then improve in
precision over the next five years. Later this year, we expect to release the
first cosmological constraints from the first two years of the SPT-3G Main
survey observations, and soon afterwards, from the first two years of Ext-4k
survey. We forecast that this early dataset will offer constraining power on
the $\Lambda{\rm CDM}$ model parameters that is comparable to that achieved by
Planck, improving the $\Lambda{\rm CDM}$ FoM by a factor of 2 (SPT-3G Main)
and 13 (Ext-4k) relative to Planck. A significant portion of the SPT-3G Wide
survey has already been completed and we could expect Ext-10k results as early
as sometime in 2025.

## Acknowledgments

SR acknowledges support by the Illinois Survey Science Fellowship from the
Center for AstroPhysical Surveys at the National Center for Supercomputing
Applications. Work at Argonne National Lab is supported by UChicago Argonne
LLC, Operator of Argonne National Laboratory (Argonne). Argonne, a U.S.
Department of Energy Office of Science Laboratory, is operated under contract
no. DE-AC02-06CH11357.

The South Pole Telescope program is supported by the National Science
Foundation (NSF) through award OPP-1852617. Partial support is also provided
by the Kavli Institute of Cosmological Physics at the University of Chicago.

This work made use of the following computing resources: Illinois Campus
Cluster, a computing resource that is operated by the Illinois Campus Cluster
Program (ICCP) in conjunction with the National Center for Supercomputing
Applications (NCSA) and which is supported by funds from the University of
Illinois at Urbana-Champaign; the computational and storage services
associated with the Hoffman2 Shared Cluster provided by UCLA Institute for
Digital Research and Education’s Research Technology Group; the computing
resources provided on Crossover, a high-performance computing cluster operated
by the Laboratory Computing Resource Center at Argonne National Laboratory;
and the National Energy Research Scientific Computing Center, which is
supported by the Office of Science of the U.S. Department of Energy under
Contract No. DE-AC02-05CH11231.

##  Appendix A Derivation of the MUSE covariance matrix

In the context of parameter inferences from CMB data, the observed maps ($x$),
do not depend just on cosmological parameters ($\theta$) but also on
unobserved latent variables ($z$) such as the unlensed CMB field and the
lensing potential field. In such a scenario, the likelihood
($\mathcal{L}(x\mid\theta)$) involves marginalizing the joint likelihood
$\mathcal{L}(x,z\mid\theta)$ over the latent space.

$\mathcal{L}(x\mid\theta)=\int\mathrm{d}^{N}z\mathcal{L}(x,z\mid\theta)=\int\mathrm{d}^{N}z\mathcal{L}(x\mid
z,\theta)\mathcal{L}(z\mid\theta),$ (A1)

where $\mathcal{L}(z\mid\theta)$ represents the likelihood of the unlensed CMB
field or the lensing potential $z$, given a set of cosmological parameters
$\theta$. For parameter inferences, we are often interested in the gradient of
this marginal likelihood, a quantity called the marginal score $s_{i}$.

$s_{i}(\theta,x)\equiv\frac{d}{d\theta_{i}}\log\mathcal{L}(x\mid\theta).$ (A2)

However, performing the integral over the maps of unlensed CMB and lensing
potential which contain an order of million pixels is a computationally
challenging task and analytic solutions only exist for the simplest cases.
MUSE gets around this problem by approximating the marginal score. The key
ingredient of the MUSE score is a quantity referred to as $s^{\mathrm{MAP}}$,
which is the gradient of the joint likelihood evaluated at the maximum a
posteriori (MAP) estimate of the unlensed CMB field and the lensing potential,
$\hat{z}_{\theta,x}\equiv\underset{z}{\operatorname{argmax}}\log\mathcal{L}(x,z\mid\theta)$

$s_{i}^{\mathrm{MAP}}(\theta,x)\equiv\frac{d}{d\theta_{i}}\log\mathcal{L}\left(x,\hat{z}_{\theta,x}\mid\theta\right)$
(A3)

and the MUSE score is defined as

$s_{i}^{\mathrm{MUSE}}(\theta,x)\equiv
s_{i}^{\mathrm{MAP}}(\theta,x)-\left\langle
s_{i}^{\mathrm{MAP}}\left(\theta,x^{\prime}\right)\right\rangle_{x^{\prime}\sim\mathcal{P}\left(x^{\prime}\mid\theta\right)},$
(A4)

where the second term is calculated by averaging over simulations $x^{\prime}$
drawn from a probability distribution generated at the specified value of
$\theta$. Once we have the MUSE score, we can define an estimator for the
bandpowers $\theta$ as the root of the equation
$s_{i}^{\mathrm{MUSE}}(\theta^{\mathrm{MUSE}},x)=0$

and the bandpower covariance can be constructed by two related matrices, J and
H defined as:

$\displaystyle J_{ij}=$ $\displaystyle\left\langle
s_{i}^{\mathrm{MAP}}\left(\theta^{*},x\right)s_{j}^{\mathrm{MAP}}\left(\theta^{*},x\right)\right\rangle_{x\sim\mathcal{P}\left(x\mid\theta^{*}\right)}$
(A5) $\displaystyle-\left\langle
s_{i}^{\mathrm{MAP}}\left(\theta^{*},x\right)\right\rangle\left\langle
s_{j}^{\mathrm{MAP}}\left(\theta^{*},x\right)\right\rangle_{x\sim\mathcal{P}\left(x\mid\theta^{*}\right)}$
$\displaystyle H_{ij}=$
$\displaystyle\left.\frac{d}{d\theta_{j}}\left[\left\langle
s_{i}^{\mathrm{MAP}}\left(\theta^{*},x\right)\right\rangle_{x\sim\mathcal{P}(x\mid\theta)}\right]\right|_{\theta=\theta^{*}}.$

$J$ is the covariance of the MAP gradients at the true value of bandpowers
$\theta^{*}$. We use simulations of the data in order to calculate this
quantity. $H$ can be thought of as the response matrix of the MAP gradients
with respect to small changes in parameters that generate the simulated data.
The bandpower covariance is constructed as follows:

$\Sigma_{ij}^{\mathrm{MUSE}}\equiv\left\langle\Delta\hat{\theta}_{i}^{\mathrm{MUSE}}\Delta\hat{\theta}_{j}^{\mathrm{MUSE}}\right\rangle=\left(H^{-1}JH^{-1\dagger}\right)_{ij}.$
(A6)

###  A.1 Simulations

We create simulations of the data, which consists of the lensed CMB along with
noise as described in Table 1. Utilizing the best-fit Planck ’15 spectrum, we
produce a suite of 1000 flat-sky realizations of unlensed $T,Q,$ and $U$ CMB
maps on a 512x512 pixel grid, spanning an area of $300$ ${\rm deg}^{2}$, which
are subsequently lensed by the lenseflow algorithm (Millea et al., 2019). We
do not simulate the entire observation patch to limit the computational cost,
thereby allowing us to run a large number of simulations. We do not expect
this to have a significant impact as lensing is a local operation and the
induced deflections are coherent only across a few degrees. Subsequently, we
scaled this covariance matrix using an appropriate factor that corresponds to
the ratio between our survey area and the simulated patch size. For the
purposes of this forecast, we use a simplified version of the data model,
which will be used in the actual analysis, whereby we ignore the effect of
masking and filtering as is generally done with traditional fisher forecasts.
We don’t anticipate these to have a significant impact on the parameter
constraints.

The data model used in this work can be written as:

$\displaystyle d=\mathbb{F}\cdot\mathbb{B}\cdot\mathbb{L}(\phi)f+n$
$\displaystyle
f\sim\mathbb{C}_{f}\left(\boldsymbol{C}_{\ell}^{\mathrm{TT}/\mathrm{TE}/\mathrm{EE}}\right)$
$\displaystyle\phi\sim\mathbb{C}_{\phi}\left(\boldsymbol{C}_{\ell}^{\phi\phi}\right)$
$\displaystyle n\sim\mathbb{C}_{n}(N_{\ell}),$

where, $d$ is the data; $f$ is a Gaussian realization of the unlensed CMB map;
$\phi$ is a Gaussian realization of the gravitational lensing potential;
$\mathbb{L}(\phi)$ is a linear lensing operator acting on $f$; $\mathbb{B}$ is
the instrumental beam function; $\mathbb{F}$ is a mid-pass Fourier filter that
cuts off $\ell$s below 300 and above 3500(4000) for $TT,TE(EE)$, and
$N_{\ell}$ is the white+$1/f$ noise power spectrum as defined in Table 1.

###  A.2 Effective lensing reconstruction map noise

The effective lensing reconstruction noise is defined such that for an
unbiased estimate of the lensing potential harmonic coefficients,
$\hat{\phi}_{LM}$, the per-mode variance of the residual to the true $\phi$
map is

$N_{L}^{\phi\phi}=\big{\langle}(\hat{\phi}_{LM}-\phi_{LM}^{\rm
true})^{2}\big{\rangle}.$ (A7)

For the quadratic estimate, for example, this is given by the sum of the
$N^{0}$, $N^{1}$, $N^{3/2}$, etc.. noise bias terms (Hu & Okamoto, 2002;
Madhavacheril et al., 2020). Note that if one has a biased estimate of the
lensing potential, $\bar{\phi}=A\phi+n$ with some unknown normalization bias,
$A$, and effective noise, $n$, the noise power can be written as

$N_{L}^{\phi\phi}=C_{L}^{\phi\phi}\left(\frac{1}{\rho_{L}}-1\right),$ (A8)

where,

$\rho_{L}=\frac{\langle\phi_{LM}^{\rm
true}\bar{\phi}_{LM}\rangle}{\sqrt{\langle\phi_{LM}^{\rm true}\phi_{LM}^{\rm
true}\rangle\langle\bar{\phi}_{LM}\bar{\phi}_{LM}\rangle}}$ (A9)

is the average correlation coefficient between the true and estimated lensing
potential. This allows computing the effective noise without ever explicitly
debiasing the lensing estimate to remove the bias, $A$.

The per-mode noise enters the error bars on the estimated lensing power
spectrum as,

$\big{\langle}(\Delta
C_{L}^{\phi\phi})^{2}\big{\rangle}=\frac{2(C_{L}^{\phi\phi}+N_{L}^{\phi\phi})^{2}}{(f_{\rm
sky}(2L+1))}.$ (A10)

Our MUSE forecasts do not compute an unbiased lensing map estimate,
$\hat{\phi}_{LM}$, nor do they explicitly compute $N_{L}^{\phi\phi}$ or use
Eq.(A10). Instead, MUSE directly computes the total posterior bandpower
covariance, including signal and noise contributions, using Eq.(A6). It is
useful however to explore the effective noise levels which are reached. This
can be conveniently performed by computing the averages in $\rho_{L}$ via
Monte Carlo methods, where the biased lensing map estimate, $\bar{\phi}$, is
the MAP estimate of $\phi$, which is a byproduct of the MUSE inference
procedure.

##  Appendix B Additional details about parameter constraints

In Table 5 we present the constraints on the single-parameter extensions to
$\Lambda{\rm CDM}$ from all the SPT-3G datasets. A comparison of the Ext-10k
dataset with Planck is presented in Fig. 9. In Table 6, we present the
degradation in cosmological parameters when we add one more parameter to the
$\Lambda{\rm CDM}$ model. This is the same as the information presented in
Fig. 10.

Figure 13: Marginalized 1-dimensional and 2-dimensional 68% contours for Hubble constant($H_{0}$), scalar spectral index ($n_{s}$), Cold dark matter density ($\Omega_{c}h^{2}$), and optical depth ($\tau_{\rm re}$) with(solid) and without (dashed) lensing for SPT-3G Main (dark green), Ext-10k (dark red) and Planck (black). The inclusion of the lensing spectra significantly helps to break the degeneracies especially for the SPT-3G Main survey. The 1 $\sigma$ errors are listed in the legend. Table 5: Forecasted $1\sigma$ errors on single-parameter extensions to the $\Lambda{\rm CDM}$ for various SPT-3G surveys in combination with Planck. Parameter | SPT-3G Main | SPT-3G Summer | SPT-3G Wide | Ext-4k | Ext-10k
---|---|---|---|---|---
$N_{\mathrm{eff}}$ [$10^{-2}$] | 10.1 | 10.9 | 9.23 | 8.27 | 6.75
$n_{\rm run}$ [$10^{-3}$] | 4.94 | 4.99 | 4.59 | 4.44 | 3.93
$\Omega_{k}$ [$10^{-3}$] | 2.46 | 2.49 | 2.38 | 2.35 | 2.26
$\sum m_{\nu}$ [eV] [$10^{-2}$] | 5.99 | 6.15 | 5.91 | 5.78 | 5.57
$w_{0}$ [$10^{-}2$] | 10.23 | 10.44 | 10.03 | 9.91 | 9.64
$Y_{P}$ [$10^{-3}$] | 6.48 | 6.68 | 5.75 | 5.34 | 4.43
Table 6: Degradation in constraints on standard cosmological parameters when the $\Lambda{\rm CDM}$ model is extended by single additional parameters. Note that $H_{0}$ is poorly constrained when $w_{0}$ is allowed to vary. Parameter |  $A_{s}$[$10^{-11}$] |  $H_{0}$ [$10^{-1}$] |  $n_{s}$ [$10^{-3}$] |  $\Omega_{b}h^{2}$[$10^{-5}$] |  $\Omega_{c}h^{2}$ [$10^{-4}$] |  $\tau_{\rm re}$[$10^{-3}$]
---|---|---|---|---|---|---
$\Lambda{\rm CDM}$ | 2.29 | 2.78 | 2.83 | 5.67 | 7.10 | 6.19
$N_{\mathrm{eff}}$ | 2.51 | 6.18 | 4.88 | 9.61 | 10.80 | 6.39
$n_{\rm run}\equiv{\rm d}n_{s}/{\rm d}{\rm ln}k$ | 2.43 | 2.83 | 2.94 | 6.90 | 7.14 | 6.35
$\Omega_{k}$ | 2.55 | 8.73 | 3.10 | 5.73 | 9.56 | 6.42
$\sum m_{\nu}$[eV] | 2.78 | 7.84 | 3.00 | 5.77 | 9.19 | 6.90
$w_{0}$ | 2.75 | 28.35 | 2.72 | 5.62 | 6.53 | 7.02
$Y_{P}$ | 2.52 | 3.31 | 4.44 | 9.34 | 7.49 | 6.67

In Fig. 13, we demonstrate the parameter constraints from Planck in black, and
SPT-3G Main in green and Ext-10k in orange. For SPT-3G, we show the
constraints that we get from $TT/EE/TE$ CMB-only as the dash-dotted and the
ones with the inclusion of lensing $\phi\phi$ as the solid curves. It is
evident from the figure that the inclusion of lensing breaks parameter
degeneracies, which helps in improving the constraints on many cosmological
parameters.

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