Quantifying scalar field dynamics with DESI 2024 Y1 BAO measurements
Kim V. Berghaus ,
1
Joshua A. Kable ,
2
and Vivian Miranda
2
1
Walter Burke Institute for Theoretical Physics,
California Institute of Technology
, California 91125, USA
2
C.N. Yang Institute for Theoretical Physics,
Stony Brook University
, New York 11794, USA
(Received 27 August 2024; accepted 23 September 2024; published 15 November 2024)
Quintessence scalar fields are a natural candidate for evolving dark energy. Unlike the phenomenological
w
0
w
a
parameterization of the dark energy equation of state, they cannot accommodate the phantom regime
of dark energy
w
ð
z
Þ
<
−
1
, or crossings into the phantom regime. Recent baryon acoustic oscillation (BAO)
measurements by the Dark Energy Spectroscopic Instrument (DESI) indicate a preference for evolving
dark energy over a cosmological constant, ranging from
2
.
6
σ
−
3
.
9
σ
when fitting to
w
0
w
a
, and combining
the DESI BAO measurements with other cosmological probes. In this work, we directly fit three simple
scalar field models to the DESI BAO data, combined with cosmic microwave background anisotropy
measurements and supernova datasets. We find the best fit model to include a 2
–
4% kinetic scalar field
energy
Ω
scf
;
k
, for a canonical scalar field with a quadratic or linear potential. However, only the DESY-Y5
supernova dataset combination shows a preference for quintessence over
Λ
cold dark matter (CDM) at the
95% confidence level. Fitting to the supernova datasets Pantheon, Pantheon+, DES-Y5, and Union3, we
show that the mild tension (
n
σ
<
3
.
4
) under
Λ
CDM emerges from a BAO preference for smaller values of
fractional mass-energy density
Ω
m
<
0
.
29
, while all supernova datasets, except for Pantheon, prefer larger
values,
Ω
m
>
0
.
3
. The tension under
Λ
CDM remains noticeable (
n
σ
<
2
.
8
), when replacing two of the
DESI BAO redshift bins with effective redshifts
z
eff
¼
0
.
51
, and
z
eff
¼
0
.
706
with comparable BOSS DR
12 BAO measurements at
z
eff
¼
0
.
51
, and
z
eff
¼
0
.
61
. Canonical scalar fields as dark energy are successful
in mitigating that tension.
DOI:
10.1103/PhysRevD.110.103524
I. INTRODUCTION
Recent measurements of baryon acoustic oscillations
(BAO) in Galaxy, quasar, and Lyman-
α
forest tracers from
the first year of observations from the Dark Energy
Spectroscopic Instrument (DESI) find hints toward an
evolving equation of state of dark energy
[1]
. Spanning
a redshift range from
0
.
1
<z<
4
.
16
, the DESI BAO
measurements provide the first measurements of the trans-
verse comoving distance for redshifts larger than
z>
2
.
3
.
DESI BAO data alone are consistent with the concordance
Λ
cold dark matter (CDM) model of cosmology in which
dark energy is described by a cosmological constant such
that the equation of state of dark energy is
w
¼
−
1
.
However, when combined with supernova data, a mild
tension emerges under
Λ
CDM.
Allowing for a time-varying dark energy equation of
state, parametrized by
w
ð
z
Þ¼
w
0
þð
1
−
1
=
ð
1
þ
z
ÞÞ
w
a
,
combinations of DESI with cosmic microwave background
(CMB) or type Ia supernovae measurements find a pref-
erence for
w
0
>
−
1
, and
w
a
<
0
, ranging from
2
.
6
σ
−
3
.
9
σ
,
depending on the data combination and choice of super-
nova dataset
[1]
. This preference is consistent with results
obtained previously by both the Union3
[2]
and the Dark
Energy Survey (DES) Y5
[3]
supernova compilations when
combined with
Planck
CMB primary anisotropy
[4]
and
BAO data from BOSS
[5]
and eBOSS
[6]
, which suggests
this preference is not likely to be associated with a
systematic in solely the DESI data. With the DESI year
1 data release
[1,7,8]
providing the first influx of precision
data from Stage IV
[9]
dark energy experiments such as
DESI, Euclid
[10]
, and the Vera Rubin Observatory
[11]
[previously referred to as Large Synoptic Survey Telescope
(LSST)], this first potential hint of dynamical dark energy is
intriguing.
From a theoretical perspective, the small value of the
cosmological constant
Λ
, in the
Λ
CDM model of cosmol-
ogy suffers fine-tuning
[12]
; that is quantum corrections
of
Λ
are larger than the observed value. The dynamical
evolution of dark energy holds a promise to alleviate that
tuning
[13
–
18]
, motivating dynamical dark energy with a
redshift-dependent equation of state. Swampland conjec-
tures
[19
–
22]
, a set of conjectured criteria for theories in
the quantum gravity landscape, also favor dynamical dark
energy over a cosmological constant as a fundamental
description of dark energy.
Scalar fields are a fundamental description of evolving
dark energy that connects the property of a dominant
energy density with negative pressure to a particle descrip-
tion. Quintessence scalar fields with canonical kinetic terms
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=
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=
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=
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are theoretically well-motivated and exhibit thawing or
tracking behavior
[23
–
27]
. In this paper, we focus on scalar
fields that exhibit thawing behavior; that is, the Hubble
friction severely overdamps the scalar field at early times
such that the field is effectively frozen with an equation of
state approximately equal to
−
1
. At smaller redshifts,
Hubble friction becomes less efficient, and the scalar field
’
s
kinetic energy grows, leading to an equation of state that
increases over time such that
w
ð
0
Þ
>
−
1
. Simple renorma-
lizable power-law potentials exhibit this thawing dynamic.
On the other hand, exponential scalar field potentials,
for example, have tracking behavior, i.e., the dark energy
density tracks the dominant energy density of the Universe
[28]
, and the equation of state is larger than
−
1
at early
times
w
ð
z
Þ
>
−
1
, and approaches
w
ð
0
Þ
≈−
1
today. A third
model category, phantom scalar fields with noncanonical
kinetic terms, exhibit an evolving equation of state that is
less than
−
1
. This behavior requires a negative kinetic
energy term that can be thought of as a scalar field rolling
up its potential. Phantom models have theoretical pathol-
ogies
[29
–
32]
, one of which is a violation of the null energy
condition, that leads to the future death of the Universe in a
big rip
[31]
.
In light of the DESI BAO measurements
’
preference for
w
0
>
−
1
, we fit the BAO measurements to canonical scalar
fields with thawing behavior. This well-motivated descrip-
tion of dynamical dark energy provides a hypothesis for the
redshift dependence of the dark energy equation of state,
which is parametrized by one parameter beyond
Λ
CDM,
the value of
w
ð
0
Þ
, which directly maps onto the kinetic
energy of the scalar field today,
Ω
scf
;
k
. Unlike the
w
0
w
a
parametrization used in the DESI analysis, thawing scalar
field models asymptotically approach an equation of state
of
−
1
at large redshift, such that the phantom regime,
w
ð
z
Þ
<
−
1
is not part of the possible dynamics. By doing
so, they impose an arguably well-motivated theory prior,
which cuts out the dark energy phantom regime and
phantom crossings. In our analysis, we fit the simplest
canonical scalar fields with a linear or quadratic potential
to the DESI BAO data, directly addressing the points raised
in
[33]
, by quantifying the evidence for evolving dark
energy in a physical model beyond
w
0
w
a
.
Notably, taking as an example the best fit to
w
0
w
a
with
DESI BAO data
þ
CMB
þ
Union
3
,
w
0
¼
−
0
.
64
0
.
11;
w
a
¼
−
1
.
27
þ
0
.
40
−
0
.
34
, indicates
w
0
>
−
1
at
3
.
5
σ
, which points
toward a possible preference of thawing scalar field
behavior. However, the best fit crosses over into the
phantom regime,
w
ð
z
Þ
<
−
1
, for redshifts exceeding
z
¼
0
.
4
, which is not possible for canonical scalar field
dark energy. We analyze the preference of DESI BAO data
in combination with CMB and supernova datasets for
thawing scalar fields over
Λ
CDM, which we quantify by
evaluating the preference for scalar field dynamics, e.g.,
nonzero kinetic scalar field energy or nonzero dark energy
radiation
[34,35]
. Quintessence has already been invoked
as an explanation of the DESI results
[36,37]
. Here, we
perform a complete fit for concrete scalar field models to
the DESI BAO data, whereas previous works have pro-
jected select results of the
w
0
w
a
fit performed by the DESI
Collaboration onto quintessence cosmologies.
We focus on canonical scalar fields with a quadratic
or linear potential, as well as a scenario in which the
dynamical component is comprised of dark radiation
sourced by the scalar field, dark energy radiation
[34,35]
.
This paper is organized as follows. In Sec.
II
, we briefly
review the quantities measured by BAO and supernova
light curve measurements and discuss how they character-
ize the expansion history of our Universe and, conse-
quently, the equation of state of dark energy. In Sec.
III
,we
discuss the three scalar field cosmologies we fit to:
quadratic (SCF QUAD), linear (SCF LIN), and dark energy
radiation (SCF DER). We derive how the scalar field
dynamics map onto the usual cosmological parametriza-
tions of the dark energy equation of state. In Sec.
IV
,
we describe our methodology and dataset combinations,
which include CMB measurements, two combinations of
BAO measurements, as well as four supernova datasets.
In Sec.
V
, we present our results, the marginalized
posteriors for the three scalar field cosmologies, as well
as the mean values of the preferred scalar field dynamics
and their associated uncertainties,
Ω
scf
;
k
or
Ω
der
. We find
that evidence for evolving dark energy meets the 95% con-
fidence level for a simple canonical scalar field with a
quadratic or linear potential only in combination with the
DES-Y5 supernova dataset. Constraining the sum of
neutrino masses under the three scalar field cosmologies,
we find that the bounds become stronger, but remain
comparable to those derived under
Λ
CDM. We also
quantify the mild tension between BAO and supernova
measurements under
Λ
CDM, and the reduction of that
tension under the scalar field cosmologies, as well as
w
0
w
a
.
In Sec.
VI
, we summarize our main findings and conclude
with canonical scalar fields being promising candidates for
explaining the DESI BAO measurements.
In Appendix VI, we show results for various additional
w
0
w
a
parametrizations with priors chosen that mimic
thawing, tracking, and phantom scalar field behavior,
finding that thawing behavior is preferred over tracking
and phantom dynamics. We also perform a principle
component analysis (PC), quantifying the difference two
of the DESI redshift bins make in the mild tension of BAO
and supernova measurements under
Λ
CDM.
II. BACKGROUND
The baryon acoustic oscillation pattern, formed from
pressure waves in the baryon-photon fluid prior to the
decoupling of the photons from the baryons around
z
≃
1100
, is imprinted in both the distributions of the
CMB photons and the matter in the Universe. We refer to
the measurements of the baryon acoustic oscillation pattern
BERGHAUS, KABLE, and MIRANDA
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observed in visible matter as the BAO. Both the CMB and
BAO measurements are sensitive to the size of the sound
horizon at the time of decoupling,
r
d
,
1
and the expansion of
the Universe. The sound horizon is determined by
r
d
¼
Z
∞
z
d
c
s
ð
z
Þ
H
ð
z
Þ
dz;
ð
1
Þ
where
c
s
is the speed of sound prior to recombination,
determined by the ratio of baryons to photons,
c
s
ð
z
Þ¼
c
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3
1
þ
3
ρ
B
ð
z
Þ
4
ρ
γ
ð
z
Þ
r
:
ð
2
Þ
More specifically, the BAO measures a combination of
the transverse comoving distance and the sound horizon,
D
M
ð
z
Þ
=r
d
, as well as the equivalent distance variable and
the sound horizon,
D
H
ð
z
Þ
=r
d
. The transverse comoving
distance and the equivalent distance variable are respec-
tively defined as
D
M
ð
z
Þ¼
c
H
0
ffiffiffiffiffiffiffi
Ω
K
p
sinh
ffiffiffiffiffiffiffi
Ω
K
p
Z
z
0
dz
0
H
ð
z
Þ
=H
0
;
ð
3
Þ
and
D
H
ð
z
Þ¼
c
H
ð
z
Þ
;
ð
4
Þ
H
ð
z
Þ¼
H
0
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Ω
m
ð
1
þ
z
Þ
3
þ
Ω
3
ð
1
þ
w
ð
z
ÞÞ
DE
þ
Ω
K
ð
1
þ
z
Þ
2
q
ð
5
Þ
where
Ω
K
accounts for nonzero curvature, and we have
assumed a general dark energy,
Ω
DE
with equation of state
parameter that in general may vary with redshift,
w
ð
z
Þ
.
For conceptual clarity, in Eq.
(5)
we have neglected the
negligible amount of radiation around at redshifts mea-
suredbyBAO,aswellastheeffectsthatmassive
neutrinos have on the transfer of energy densities between
radiation and matter.
Combining measurements of the CMB and the BAO is a
powerful probe of cosmology because both measure the
same sound horizon scale at different points in the
evolution of the Universe. The CMB anisotropy measures
the distribution set at
z
≃
1100
, while DESI BAO, for
example, covers the range
0
.
4
≤
z
≤
4
.
16
. This has the
effect of breaking degeneracies between parameters and
thus providing tighter parameter constraints.
Another important cosmological probe is the measure-
ment of the luminosity distances,
d
L
ð
z
Þ
, of type Ia super-
nova as a function of redshift. These measurements
are sensitive to the expansion history of the Universe by
measuring the flux of light,
F
, from an object with known
luminosity
L
. In particular,
F
¼
L
4
π
D
M
ð
z
Þ
2
ð
1
þ
z
Þ
2
¼
L
4
π
d
L
ð
z
Þ
2
:
ð
6
Þ
The two factors of (
1
þ
z
) arise because the expansion of
the Universe causes light to redshift, losing energy, as well
as changes to the rate of reception of photons. In the latter
case, two photons emitted at times separated by
Δ
t
e
, will be
observed to be separated in time by
Δ
t
o
¼ð
1
þ
z
Þ
Δ
t
e
.
Hence, the luminosity distance is
[38]
d
L
ð
z
Þ¼
c
ð
1
þ
z
Þ
H
0
ffiffiffiffiffiffiffi
Ω
K
p
sinh
ffiffiffiffiffiffiffi
Ω
K
p
Z
z
0
dz
0
H
ð
z
Þ
=H
0
:
ð
7
Þ
Including CMB anisotropy measurements breaks the
degeneracy between
H
0
and
r
d
. They also constrain the
baryon density, as well as
Ω
m
.
The flat
Λ
CDM model sets
Ω
K
¼
0
, which implies
Ω
DE
¼
Ω
Λ
¼
1
−
Ω
m
, as well as
w
¼
−
1
for the dark
energy equation of state. In the
w
0
w
a
parametrization the
equation of state is given by
w
ð
z
Þ¼
w
0
þð
1
−
1
1
þ
z
Þ
w
a
,
which allows the dark energy density to evolve in redshift.
In this work, we go beyond phenomenological parametri-
zations of the equation of state of dark energy and fit BAO
and supernova directly to scalar field models.
III. SCALAR FIELD MODELS
A. Quintessence
We consider a canonical scalar field
φ
to comprise the
dark energy density whose evolution is governed by its
scalar field potential
V
ð
φ
Þ
and the Hubble expansion rate
of the Universe,
H
¼
̇
a=a
, where
a
is the scale factor, and
̇
a
≡
da
dt
. We take the entire energy density of dark energy to
be comprised of the scalar field such that there is no
additional cosmological constant. We also assume that
there is no curvature,
Ω
K
¼
0
.
The scalar field
’
s homogeneous equation of motion is
̈
φ
ð
t
Þþ
3
H
ð
t
Þ
̇
φ
ð
t
Þþ
dV
d
φ
¼
0
:
ð
8
Þ
The Hubble expansion under this hypothesis is given by
H
ð
z
Þ¼
H
0
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Ω
m
ð
1
þ
z
Þ
3
þ
V
ð
φ
ð
z
ÞÞþ
̇
φ
ð
z
Þ
2
=
2
ρ
c
s
;
ð
9
Þ
1
The BAO are sensitive to the size of the sound horizon when
the baryons decouple from the photons,
z
d
≃
1060
. Because there
were many orders of magnitude more photons than baryons, a
relatively small amount of photons was sufficient for the baryons
to stay coupled, delaying when the baryon acoustic oscillation
pattern was set in the distribution of baryonic matter. This is
referred to as the drag epoch.
QUANTIFYING SCALAR FIELD DYNAMICS WITH DESI 2024
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PHYS. REV. D
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where
Ω
m
þ
V
ð
φ
ð
0
ÞÞþ
̇
φ
ð
0
Þ
2
=
2
ρ
c
¼
1
:
ð
10
Þ
The critical density of the Universe today is defined as
ρ
c
≡
3
M
2
pl
H
2
0
. For two choices of scalar field potential,
we quantify the preference for nonzero kinetic energy that
the DESI BAO data exhibit by themselves and in combi-
nation with other datasets. We define
Ω
scf
;
k
≡
̇
φ
ð
z
¼
0
Þ
2
=
2
ρ
c
;
ð
11
Þ
as the fraction of kinetic scalar field energy today over the
critical density of the Universe today. The kinetic scalar
energy at early times is fixed by attractor initial conditions
that effectively freeze the field due to large Hubble friction
such that
̇
φ
ð
z
→
∞
Þ
→
0
for the linear and quadratic
potentials we consider. Thus,
Ω
scf
;
k
is a derived parameter,
which is fully determined by the slope of the potential
V
.
The equation of state of the scalar field dark energy as a
function of redshift is determined by
w
ð
z
Þ
≡
p
ð
z
Þ
ρ
ð
z
Þ
¼
̇
φ
ð
z
Þ
2
2
−
V
ð
φ
ð
z
ÞÞ
̇
φ
ð
z
Þ
2
2
þ
V
ð
φ
ð
z
ÞÞ
:
ð
12
Þ
Analytical expressions for
w
ð
z
Þ
, valid in the regime in
which
w
ð
z
Þ
is near
−
1
, can be found in
[39,40]
.In
that regime in which
̇
φ
2
=
2
≪
V
, the equation of state
simplifies to
w
ð
z
Þ
≈−
1
þ
2
Δ
w
ð
z
Þ¼
−
1
þ
̇
φ
ð
z
Þ
2
V
ð
φ
ð
z
ÞÞ
:
ð
13
Þ
Comparing this with the
w
0
w
a
parametrization
w
ð
z
Þ¼
w
0
þ
1
−
1
1
þ
z
w
a
;
ð
14
Þ
one can identify
w
0
þ
1
≈
̇
φ
ð
0
Þ
2
V
ð
φ
ð
0
ÞÞ
;
ð
15
Þ
which indicates that the preference for a positive deviation
from
w
0
¼
−
1
found in the DESI BAO data may map onto
a preference for nonzero kinetic scalar field energy today at
redshift
z
¼
0
. Translating the scalar field description in a
flat Universe onto the
w
0
w
a
parametrization, one finds
w
0
≈
2
Ω
scf
;
k
1
−
Ω
m
−
1
;
and
ð
16
Þ
w
a
¼
1
þ
w
0
;
ð
17
Þ
where we have approximated
V
ð
φ
ð
0
ÞÞ
=
ρ
c
≈
1
−
Ω
m
. Many
scalar field potentials have been considered historically
in the context of quintessence dark energy (see, for
instance,
[23,41
–
44]
). In this work, we focus on two
simple scenarios that describe the local shape of the
potential. For this purpose, we consider a quadratic scalar
field potential,
V
ð
φ
Þ¼
1
2
m
2
φ
2
;
ð
SCF QUAD
Þð
18
Þ
and a linear potential scalar field potential,
V
ð
φ
Þ¼
−
C
φ
;
ð
SCF LIN
Þð
19
Þ
which we label as
“
SCF QUAD,
”
and
“
SCF LIN
”
hence-
forth. Both choices of potential exhibit thawing behavior,
implying that the kinetic energy of the scalar field is largest
today, and the dark energy equation of state asymptotes
to
−
1
at large redshift as shown in Fig.
1
. The scalar field
dynamics cannot accommodate the best fit for
w
0
and
w
a
,
FIG. 1. Comparison of the dark energy equation of state
w
ð
z
Þ
as
a function of redshift for the three scalar field cosmologies
described in Sec.
III
for the same value
w
ð
0
Þ¼
−
0
.
914
, as well as
the phenomenological
w
0
w
a
thawing parametrization, described
in Appendix VI, that mimics thawing scalar field dynamics. The
curves correspond to values of
Ω
scf
;
k
¼
0
.
03
,
Ω
der
¼
0
.
09
, and
w
thawing
¼
−
0
.
914
, for fixed
Ω
m
¼
0
.
3
for all curves. The dashed
vertical lines indicate the effective redshifts
z
eff
of the DESI BAO
measurements.
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found with the DESI BAO dataset in combination with
supernova data due to the preference for a crossing into the
phantom regime
w
ð
z
Þ
<
−
1
which is not possible in a
standard canonical scalar field model. We vary the linear
slope
C
∈
½
2
.
2
×
10
−
8
;
2
.
4
×
10
−
7
of the scalar potential in
units of
½
M
pl
Mpc
−
2
. The steepest slope in this prior allows
for up to
Ω
scf
;
k
≈
0
.
2
, and the shallowest slope allows for
values as low as
Ω
scf
;
k
¼
0
.
002
. We do not expand to
smaller priors due to numerical instabilities in our modified
CLASS
implementation. For the quadratic potential we vary
m
∈
½
1
×
10
−
5
;
1
×
10
−
3
in units of
½
Mpc
−
1
. The largest
mass allows for up to
Ω
scf
;
k
≈
0
.
6
, and the smallest value of
the prior corresponds to
Ω
scf
;
k
≈
0
.
0001
(
w
0
¼
−
0
.
9997
),
effectively asymptoting to
Λ
CDM, which corresponds to
Ω
scf
;
k
¼
0
(
w
0
¼
−
1
).
The masses that feature evolving dark energy with
O
ð
1
Þ
dynamics are
m
∼
H
0
≈
10
−
33
eV.
B. Dark energy radiation
Dark energy radiation is a novel description of dark
energy, which proposes that the dynamical component of
dark energy is dominated by a thermal bath of relativistic
particles sourced by thermal friction from a slowly rolling
scalar field
[34,35]
. Such a model is motivated by consid-
ering couplings of the dark energy scalar field to other light
particles
[45
–
47]
. For example, an axionlike field coupling
to non-Abelian gauge fields leads to the phenomenology of
dark energy radiation
[34,35,47,48]
. The homogeneous
equations governing the coupled evolution of the scalar
field and the dark radiation are
̈
φ
ð
t
Þþð
3
H
ð
t
Þþ
Υ
Þ
̇
φ
ð
t
Þþ
dV
d
φ
¼
0
;
̇
ρ
der
ð
t
Þþ
4
H
ð
t
Þ
ρ
der
ð
t
Þ¼
Υ
̇
φ
ð
t
Þ
2
:
ð
SCF DER
Þð
20
Þ
We follow a toy model approach of dark energy radiation
that treats the thermal friction coefficient
Υ
as a constant.
We also choose our priors such that
Υ
≫
H
. In this regime,
the kinetic energy of the scalar field is highly suppressed,
̇
φ
2
=
2
≪
ω
der
, and there is a degeneracy between the choice
of scalar field potential, and friction coefficient, e.g., a
steeper slope with larger friction
Υ
leads to the same
phenomenology as a shallower slope with smaller friction
coefficient. The Hubble parameter is given by
H
ð
z
Þ¼
H
0
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Ω
m
ð
1
þ
z
Þ
3
þ
V
ð
φ
ð
z
ÞÞþ
ρ
der
ð
z
Þ
ρ
c
s
;
ð
21
Þ
where
Ω
m
þ
V
ð
φ
ð
0
ÞÞþ
ρ
der
ð
0
Þ
ρ
c
¼
1
:
ð
22
Þ
One extra parameter beyond
Λ
CDM, the amount of dark
energy radiation today, defined as
Ω
der
≡
ρ
der
ð
z
¼
0
Þ
ρ
c
;
ð
23
Þ
parametrizes this model. The behavior of dark energy is
comprised of the sum of the scalar field and the dark energy
radiation
ρ
der
. The equation of state is given by
w
ð
z
Þ
≡
p
φ
ð
z
Þþ
p
der
ð
z
Þ
ρ
φ
ð
z
Þþ
ρ
der
ð
z
Þ
;
ð
24
Þ
which due to
̇
φ
2
=
2
≪
ρ
der
, simplifies to
w
ð
z
Þ
≈−
1
þ
2
Δ
w
ð
z
Þ¼
−
1
þ
1
3
ρ
der
ð
z
Þ
V
ð
φ
ð
z
ÞÞ
:
ð
25
Þ
A deviation in the
w
0
parameter from
−
1
indicates a
nonzero component of dark energy radiation
w
0
≈
1
3
2
Ω
der
1
−
Ω
m
−
1
:
ð
26
Þ
Compared to quintessence scalar field models in which
kinetic energy comprises the dynamical component
described in Sec.
III A
, the equation of state of dark energy
in dark energy radiation asymptotes slower to
−
1
at larger
redshifts. This behavior is illustrated in Fig.
1
. The
dynamical evolution of dark energy radiation is insensitive
to the choice of scalar field potential, due to the additional
thermal friction severely overdamping the scalar field.
Consequently, we explore the phenomenology of dark
energy radiation on the example of a single choice for
the scalar field, which we take to be linear, as given by
Eq.
(19)
. Due to the degeneracy between slope and friction,
we fix the linear slope to be
C
¼
1
×
10
−
6
M
pl
Mpc
−
2
, and
vary the friction coefficient
Υ
in units of [Mpc
−
1
]as
log
10
Υ
∈
½
−
1
;
1
. The lower prior limit of
Υ
allows for
Ω
der
≈
0
.
1
, and the upper limit corresponds to the asymp-
totic limit of a frozen scalar field with
Ω
der
≈
0
. We refer to
this model as SCF DER.
IV. DATA AND METHODOLOGY
We compare the scalar field dark energy cosmologies
outlined in Sec.
III
(SCF QUAD, SCF LIN, SCF DER) as
well as the phenomenological
w
0
w
a
model to a standard
flat
Λ
CDM Universe. In Appendix VI, we also examine
more phenomenological dark energy parametrizations. In
all cases, we vary the usual six
Λ
CDM base parameters: the
amplitude
A
s
of the primordial curvature power spectrum at
k
¼
0
.
05
Mpc
−
1
as
10
9
A
s
; the tilt
n
s
of this spectrum; the
angular size
θ
of the sound horizon, the physical density
Ω
b
h
2
of baryons; the physical density
Ω
c
h
2
of cold dark
matter; and the optical depth
τ
to reionization, where we have
chosen broad, uninformative priors. We adopt the standard
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PHYS. REV. D
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neutrino description with one massive (
m
ν
¼
0
.
06
eV) and
two massless neutrinos where we quantify the constraints on
the sum of the neutrino masses under our scalar field dark
energy cosmologies. In some select cases, we study varying
the sum of the neutrino masses.
We modified the Boltzmann code
CLASS
[49]
to include
the scalar field dark energy cosmologies described in
Sec.
III
. For all
Λ
CDM and all
w
0
w
a
models explored
in this work, we use the
CAMB
Boltzmann code
[50]
.To
determine the posterior distributions for the various model
parameters, we perform Markov Chain Monte Carlo
(MCMC) runs using the publicly available
COBAYA
code
[51]
. To assess the convergence of the MCMC chains, we
use a Gelman-Rubin converge criterion of
R
−
1
¼
0
.
02
.
To constrain cosmological models, we primarily use
observational data from CMB, BAO, and supernova. For
CMB, we use the
Planck
2018 multifrequency half mission
temperature (T) and polarization (E) auto and cross power
spectra, which are referred to as the TT, TE, and EE power
spectra
[4,52,53]
. For each of the TT, TE, and EE power
spectra, we make a multipole cut so that we only include
l
≤
1296
. We do this to avoid nonlinear lensing effects as
we do not have the modeling of nonlinear scales for the
scalar field dark energy models. In addition, we use
Planck
l
<
30
TT and EE data.
For BAO, we primarily use the DESI DR1 BAO results
[1]
,
2
which include measurements of the BAO signal in
galaxies and quasars
[7]
as well as in Lyman alpha
forests
[8]
. In their analysis, the DESI Collaboration noted
a preference for a
2
σ
deviation from
Λ
CDM predictions in
the luminous red galaxy data in the redshift range
0
.
4
≤
z
≤
0
.
6
. Nevertheless DESI found the preference
for
w
0
w
a
CDM over
Λ
CDM persisted, though at weaker
significance, when DESI data below
z
¼
0
.
6
were replaced
with SDSS data. Similar to the DESI analysis, in some
cases in this analysis, we replace select DESI measure-
ments with comparable redshift BOSS DR12 measure-
ments to assess the robustness of the DESI preference for
noncosmological constant dark energy models
[5]
.In
particular, we replace the
D
M
=r
d
and
D
H
=r
d
measurements
from DESI in redshift bins
0
.
4
≤
z
≤
0
.
6
(
z
eff
¼
0
.
51
)
and
0
.
6
≤
z
≤
0
.
8
(
z
eff
¼
0
.
706
) with BOSS DR12
measurements of
D
M
=r
d
and
Hr
d
at effective redshifts
z
eff
¼
0
.
51
and
z
eff
¼
0
.
61
. This case is referred to as
DESI
⋆
þ
BOSS
⋆
DR
12
.
For supernova, we use four different supernova catalogs
to determine how each of these catalogs affects the
constraints on cosmological models. In particular, we
use supernova from the Pantheon compilation
[38]
,
the successor analysis Pantheon
þ
[54]
, the Union3
analysis
[2]
, and the Dark Energy Survey Y5 analysis
[3]
.
Importantly, we only include one of these supernova
compilations at a time, and we note that these compilations
have supernovas in common, which suggests that these
results will be correlated with each other.
V. RESULTS AND DISCUSSION
We explore the ability of the CMB, BAO, and supernova
data outlined in Sec.
IV
to constrain the
Λ
CDM, and scalar
field dark energy models. In Fig.
2
, we show posteriors
for
Λ
CDM when fit to various combinations of CMB,
BAO, and supernova data. Fitting
Λ
CDM to BAO data in
combination with CMB and supernova unveils that every
supernova dataset, except for Pantheon, prefers larger
values for
Ω
m
as evidenced by the negative correlation
between
χ
2
SN
(the
χ
2
for the corresponding supernova data)
and
Ω
m
. On the other hand, BAO data favor the opposite
regime, indicated by the decreasing
χ
2
BAO
for smaller
Ω
m
.
While more pronounced when using the DESI BAO data-
set, the trend persists when replacing the two DESI
measurements in the redshift bins for
0
.
4
≤
z
≤
0
.
6
and
0
.
6
≤
z
≤
0
.
8
, which were claimed to drive the preference
for
w
0
w
a
in
[1]
, with BOSS DR12 measurements, as shown
in the right panel of Fig.
2
. This indicates that the
combination of CMB, BAO, and supernova datasets are
in mild tension under
Λ
CDM, independently of the two
redshift bins in question.
We quantify this tension for
Λ
CDM as well as for
phenomenological
w
0
w
a
parametrizations in Appendix VI.
Overall, we find all of the models fit the supernova
similarly relative to the number of degrees of freedom
for each supernova catalog; however, we find for
Λ
CDM,
mild to moderate tension (an effective
n
σ
3
≈
2
–
3
) for
Pantheon
þ
, DES-Y5, and Union3 between the parameter
constraints resulting from MCMCs using these supernova
catalogs alone and an MCMC using only CMB and BAO
data. This disagreement is reduced when using evolving
dark energy models, which motivates exploring the con-
straints of our scalar field cosmologies.
The results for our scalar field dark energy cosmologies
are shown in Fig.
3
(SCF QUAD and SCF LIN) and Fig.
4
(SCF DER). Additionally, we provide select parameter
constraints for these cosmologies in Table
I
. The
χ
2
SN
row in
Fig.
3
shows that the hypothesis of thawing scalar fields
with a quadratic or linear potential is able to ease the
preference for larger
Ω
m
from the supernova datasets as
seen by the allowed parameter space with
χ
2
SN
<
0
and
2
Note that the effective redshifts in Table 1 of
[1]
are rounded
and introduce a significant truncation (
Δ
χ
2
∼
2
) error when used
directly instead of the official DESI likelihood, which was
released within the
COBAYA
package only after
[1]
.
3
Note this is not directly interpretable as a number of standard
deviations away from the mean. The effective
n
σ
values are
calculated using the Tensiometer code,
https://github.com/
mraveri/tensiometer
. This code uses a two-tailed statistical test,
which is outlined in Appendix G of
[55]
. For reference, a
probability to exceed of 0.045 corresponds approximately to
2
σ
. We use this effective
n
σ
throughout the remainder of this work.
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Ω
m
<
0
.
31
. This opening of parameter space helps accom-
modate the BAO preference for lower
Ω
m
, which persists
even in these scalar field dark energy models, partially
because the
χ
2
BAO
−
Ω
m
posterior is less steep (i.e., smaller
shifts in
χ
2
BAO
as
Ω
m
increases) when
Ω
m
<
0
.
31
.
Moreover, the
Ω
scf
;
k
rows, for both SCF QUAD and SCF
LIN cases, show weak statistical evidence for nonzero
kinetic scalar field energy. The mean values are largest for
the cases where the DES-Y5 and Union3 compilations are
used, which for both a quadratic and linear scalar field are
around 4% kinetic energy. For Pantheon and Pantheon+,
the preference for nonzero kinetic energy is weaker, around
2%. In all cases explored, except the DES-Y5 dataset, the
1D 95% confidence intervals for the scalar field kinetic
energy include the lower boundary allowed by our prior,
which indicates the
Λ
CDM limit of the model.
FIG. 2. Top panels: supernova and BAO
χ
2
posteriors for
Λ
CDM in combination with CMB Planck and various supernova datasets.
Two BAO dataset combinations are shown, DESI to the left, and DESI
⋆
þ
BOSS
⋆
DR
12
, where we have replaced the DESI BAO
z
eff
¼
0
.
51
, and
z
eff
¼
0
.
706
data points with comparable redshift BOSS DR12 measurements. The CMB Planck data have a reduced
l
<
1296
range to avoid nonlinear lensing effects. The opposing preference for
Ω
m
in
χ
2
SN
and
χ
2
BAO
indicates a mild tension of the
datasets under
Λ
CDM. The shown supernova
χ
2
SN
for Pantheon, Pantheon+, DES-Y5, and Union3 have, respectively, the constant offset
χ
2
SN
;
offset
¼ð
1034
.
8
;
1405
.
7
;
1648
.
4
;
31
.
5
Þ
. Bottom panels: probability distribution of the difference
ΔΩ
m
between
Ω
m
density derived
from CMB
þ
BAO versus SN-only MCMC chains. Replacing DESI bins at
z
eff
¼
0
.
51
, and
z
eff
¼
0
.
706
with BOSS DR12
measurement does not significantly reduce the tension on
Ω
m
values between CMB
þ
BAO and newer Type-Ia supernova
measurements that drive the detection of dynamical dark energy. Appendix VI shows that the CMB
þ
DESI combination is perfectly
compatible with the cosmological constant as long as type-Ia supernova constrains the cold dark matter density to be
Ω
m
≈
0
.
289
, which
is the range preferred by the Pantheon dataset.
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