Probing observational bounds on scalar-tensor theories
from standard sirens
Rocco D’Agostino
1,2,*
and Rafael C. Nunes
3,†
1
Dipartimento di Fisica, Universit`a di Napoli Federico II, Via Cinthia, I-80126, Napoli, Italy
2
Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Napoli, Via Cinthia 9, I-80126 Napoli, Italy
3
Divisão de Astrofísica, Instituto Nacional de Pesquisas Espaciais,
Avenida dos Astronautas 1758, São Jos´e dos Campos, 12227-010, SP, Brazil
(Received 16 July 2019; published 22 August 2019)
Standard sirens are the gravitational wave (GW) analog of the astronomical standard candles and can
provide powerful information about the dynamics of the Universe. In this work, we simulate a catalog with
1000 standard siren events from binary neutron star mergers, within the sensitivity predicted for the third
generation of the ground GW detector called the Einstein Telescope (ET). After correctly modifying the
propagation of GWs as input to generate the catalog, we apply our mock dataset on scalar-tensor theories
where the speed of GW propagation is equal to the speed of light. As a first application, we find new
observational bounds on the running of the Planck mass, when considering appropriate values within the
stability condition of the theory, and we discuss some consequences on the amplitude of the running of the
Planck mass. In the second part, we combine our simulated standard sirens catalog with other geomet ric
cosmological tests (supernovae Ia and cosmic chronometer measurements) to constrain the Hu-Sawicki
fðRÞ gravity model. We thus find new and non-null deviations from the standard ΛCDM model, showing
that in the future fðRÞgravity can be tested up to 95% confidence level. The results obtained here show that
the statistical accuracy achievable by future ground-based GW observations, mainly with the ET detector
(and planned detectors with a similar sensitivity), can provide strong observational bounds on modified
gravity theories.
DOI: 10.1103/PhysRevD.100.044041
I. INTRODUCTION
After 20 years of research, the nature of the physical
mechanism responsible for accelerating the Universe at
late times is still an open question, and a large variety of
cosmological models have been and are continually pro-
posed in the literature to explain such observations (see
[1–3] for review). This is essentially due to the difficulty of
discriminating among different scenarios that respond to
the observations in the same way, leading to a theoretical
degeneracy.
The observation of new astrophysical sources, through a
direct manifestation of gravitational effects, can provide
rich physical information about the nature of gravity, which
should play a key role to probe new (or rule out) additional
gravitational degree(s) of freedom, or exotic forms of
energy such as dark energy. The gravitational waves
(GWs) issued by binary systems, such as binary black
hole (BBH) and/or binary neutron star (BNS) detected by
LIGO/VIRGO, certainly open a new window to investigate
fundamental physics in this direction. At present, catalogs
of GWs from ten BBH mergers a nd one BNS merger
are available [4]. The latter, the GW170817 event [5],
observed at z ≃ 0.009, has imposed str ong constraints on
modified gravity and dar k energy models [6–11].Also,
GW170817 was the first standard siren (the gravitational-
wave analog of an astronomical standard candle) event
to be cataloged, once its electromagnetic counterpart
(GRB170817) was measured. These observations were
also used to measure H
0
at 12% accuracy, assuming a
fiducial ΛCDM cosmology [12]. We refer the reader to
[13–15] for proposals t o use the standard siren to m easure
H
0
with more accuracy.
Given the central importance of GW astronomy, beyond
the present performance of the LIGO and Virgo interfer-
ometers, plans for construction of several GW observatory
interferometers (on Earth and in space) are currently in
preparation, such like LIGO Voyager [16], Cosmic Explore
[16], Einstein Telescope (ET) [17,18], LISA [19], DECIGO
[20], and TianQin [21], among others, to observe GWs in
the most diverse frequencies bands and different types of
GW sources. In this paper, we are particularly interested to
use the sensitivity predicted for the ET [17,18], which is a
third-generation ground-based detector and it is envisaged
to be several times more sensitive in amplitude than the
*
rdagostino@na.infn.it
†
rafadcnunes@gmail.com
PHYSICAL REVIEW D 100, 044041 (2019)
2470-0010=2019=100(4)=044041(13) 044041-1 © 2019 American Physical Society
advanced ground-based detectors in operation, covering the
frequency band range from 1 to 10
4
Hz. Also, the ET is
expected to have a signal-to-noise ratio (SNR) for BBH and
BNS mergers several times larger than the current mea-
sures, as well as to observe hundreds or thousands of these
events throughout the whole operational time. Several
works have been done using the ET sensitivity to simulate
a GW standard siren in order to investigate diverse aspects
in cosmology [22–44].
In this work, we generate a simulated catalog with 1000
standard siren events from BNS mergers, from the ET
power spectral density noise, in order to evaluate forecast-
ing observational constraints on scalar-tensor theories
where the speed of GW propagation is equal to the speed
of light. First, assuming a well-known parametric model for
the running of the Planck mass and assuming appropriate
stability conditions on the theory, we find new observa-
tional bounds on the amplitude of the running of the Planck
mass and we discuss its possible implications. In the second
part, we apply our simulated standard siren data on fðRÞ
gravity given by the Hu-Sawicki model in order to find new
observational limits on such a model. In both analyses,
we find that the parameters that characterize deviations
from general relativity (GR) may be non-null, within some
statistical borders.
The manuscript is organized as follows. In Sec. II, we set
our theoretical framework to show how the GW propaga-
tion is modified from scalar-tensor theories. In Sec. III,we
describe our methodology to generate standard siren mock
catalogs. In Secs. IV and V, we present our main results.
Finally, in Sec. VI, we outline our final considerations and
future perspectives.
Throughout the text, we use units such that c ¼ ℏ ¼ 1,
and M
P
¼ 1=
ffiffiffiffiffiffiffiffiffi
8πG
p
is the Planck mass. Moreover, we
adopt the flat Friedmann-Lemaître-Robertson-Walker
(FLRW) metric ds
2
¼ −dt
2
þ aðtÞ
2
δ
ij
dx
i
dx
j
, where a is
the scale factor, normalized to unity today. As usual
notation, we denote by a subscript “0” physical quantities
evaluated at the present time and by the prime and dot
symbols the derivatives with respect to the conformal time
(τ) and cosmic time (t), respectively, related by dt ¼ adτ.
Lastly, we express the Hubble constant (H
0
) results in units
of km=s=Mpc.
II. MODIFIED GRAVITATIONAL WAVE
PROPAGATION IN SCALAR-TENSOR GRAVITY
The Horndeski theories of gravity [45–47] (see [48,49]
for a review) are the most general Lorentz invariant scalar-
tensor theories with second-order equations of motion and
where all matter is universally coupled to gravity. They
include, as a subset, the archetypal modifications of gravity
such as metric and Palatini fðRÞ gravity, Brans-Dicke
theories, and Galileons, among others. The Horndeski
action reads
S ¼
Z
d
4
x
ffiffiffiffiffiffi
−g
p
X
5
i¼2
M
2
P
L
i
þ L
m
; ð1Þ
where g is the determinant of the metric tensor and
L
2
¼ G
2
ðϕ;XÞ; ð2Þ
L
3
¼ −G
3
ðϕ;XÞ□ϕ; ð3Þ
L
4
¼ −G
4
ðϕ;XÞR þ G
4;X
½ð□ϕÞ
2
− ϕ
;μν
ϕ
;μν
; ð4Þ
L
5
¼ −G
5
ðϕ;XÞG
μν
ϕ
;μν
−
1
6
G
5;X
½ð□ϕÞ
3
ð5Þ
þ 2ϕ
;μν
ϕ
;μσ
ϕ
;ν
;σ
− 3ϕ
;μν
ϕ
;μν
□ϕ: ð6Þ
Here, G
i
(i runs over 2, 3, 4, 5) are functions of a scalar
field ϕ and the kinetic term X ≡ −1=2∇
ν
ϕ∇
ν
ϕ, and
G
i;X
≡∂G
i
=∂X.ForG
2
¼ Λ, G
4
¼ M
2
P
=2 and G
3
¼
G
5
¼ 0, we recover GR with a cosmological constant.
For a general discussion on the model varieties for different
G
i
choices after GW170817, see [49].
Recently, the GW170817 event together with the
electromagnetic counterpart showed that the speed of
GW, c
T
, is very close to the speed of light for z<0.01,
i.e., jc
T
=c − 1j < 10
−15
[5]. In the context of Horndeski
gravity, in order to explain these constraints, the only
option is to consider G
4;X
≈ 0 and G
5
≈ const in the action
above. It is important to note that this restriction applies
only to the local Universe (≲40 Mpc). Thus, in principle,
nothing prevents one from considering the presence of
these terms at redshifts larger than z ¼ 0.01. In fact, only
future measurements at high z can confirm whether c
T
¼ c.
Here, we assume c
T
¼ c, without loss of generality in the
analysis we are going to develop. Under this condition, the
GW propagation obeys the equation of motion [50]
h
00
ij
þð2 þ α
M
ÞHh
0
ij
þ k
2
h
ij
¼ 0; ð7Þ
where h
ij
is the metric tensor perturbation and H ≡ a
0
=a
is the Hubble rate in conformal time. Moreover, α
M
is the
running of the Planck mass, which enters as a friction term
responsible for modifying the amplitude of GWs acting as a
damping term:
α
M
¼
1
HM
2
dM
2
dt
; ð8Þ
where M
is the effective Planck mass:
M
2
¼ 2ðG
4
− 2XG
4X
þ XG
5ϕ
−
_
ϕHXG
5X
Þ; ð9Þ
and H ≡
_
a=a is the Hubble parameter. Following the
methodology presented in [51] (see also [52,53]), we
ROCCO D’AGOSTINO and RAFAEL C. NUNES PHYS. REV. D 100, 044041 (2019)
044041-2
can write a generalized GW amplitude propagation for
scalar-tensor theories as
h ¼ e
−D
h
GR
; ð10Þ
where
D ¼
1
2
Z
τ
α
M
Hdτ
0
: ð11Þ
Note that due to the condition c
T
¼ c, that is, G
4;X
≈ 0 and
G
5
≈ const, we do not have phase corrections in Eq. (10).
As the GW amplitude is inversely proportional to the
distance, one can interpret the amplitude modification in
Eq. (10) as a correction to the luminosity distance, defining
an effective luminosity distance, or equivalently, an effec-
tive amplitude correction as [52,54,55]
d
GW
L
ðzÞ¼d
EM
L
ðzÞexp
1
2
Z
z
0
dz
0
1 þ z
0
α
M
ðz
0
Þ
; ð12Þ
where d
EM
L
is the standard electromagnetic luminosity
distance as a function of the redshift
1
:
d
EM
L
ðzÞ¼ð1 þ zÞ
Z
z
0
dz
0
Hðz
0
Þ
: ð13Þ
This generalization has been recently investigated in some
contexts of modified gravity (see, e.g., [51–63]).
It is usual to choose phenomenologically motivated
functional forms for the functions α
i
(see, e.g., [64–67]).
Typically, their evolution is tied to the scale factor aðtÞ or to
the dark energy density Ω
de
ðaÞ raised to some power n.On
the other hand, an important point within Horndeski gravity
is the stability conditions of the theory. Appropriate values
of the free parameters functions must be considered in order
to have a stable theory throughout the evolution of the
Universe (see [66] and reference therein). Following [66],
we adopt the parametrization α
M
¼ α
M0
a
n
, so that the
stability conditions can be summarized as follows:
(1) n>
5
2
: stable for α
M0
< 0;
(2) 0 <n<1 þ
3Ω
m0
2
: stable for α
M0
> 0.
Here, Ω
m0
is the present dimensionless matter density.
Under these considerations, we can note from Eq. (12)
that the changes in the GW amplitude propagation will be
sensitive to the sign of α
M0
. Possible corrections with
α
M0
> 0 or α
M0
< 0 will induce d
GW
L
>d
EM
L
and d
GW
L
<d
EM
L
,
respectively. We quantify these effects in Fig. 1. We note
that variations on α
M0
> 0 ð<0Þ can produce changes
up to 10% (5%), respectively, on the effective GW
amplitude, for reasonable values of the running of the
Planck mass today.
III. METHODOLOGY AND GW STANDARD
SIREN DATASET
In order to move on, we need to define the GW signal
h
GR
. In modeling the gravitational wave form, given a
GW strain signal hðtÞ¼AðtÞcos½ΦðtÞ, we can obtain its
Fourier transform
˜
hðfÞ using the stationary phase approxi-
mation for the orbital phase of an inspiraling binary system.
For a coalescing binary system with component masses m
1
and m
2
,wehave
˜
hðfÞ¼QAf
−7=6
e
iΦðfÞ
; ð14Þ
where A is the GW inspiral amplitude computed perturba-
tively within the so-called post-Newtonian (PN) formalism
up until three PN corrections,
A ¼
ffiffiffiffiffi
5
96
r
M
5=6
c
π
2=3
d
GW
L
X
6
i¼0
A
i
ðπfÞ
i=3
; ð15Þ
FIG. 1. Correct io ns on the effective luminosity distance [see Eq . (12)], as a functio n of the redsh ift, for d ifferent values of the
running of the Planck mass today. The theoretical curves correspond to the case α
M0
> 0 and n ¼ 3 (left panel) and α
M0
> 0 and
n ¼ 1 (right p anel).
1
The redshift is defined as z ¼ a
−1
− 1.
PROBING OBSERVATIONAL BOUNDS ON SCALAR-TENSOR … PHYS. REV. D 100, 044041 (2019)
044041-3
where d
GW
L
is the modified luminosity distance as in
Eq. (12), and the coefficients A
i
are given in the
Appendix A. The function Q is expressed by
Q
2
¼ F
2
þ
ð1 þ cos
2
ðιÞÞ
2
þ 2F
2
×
cos
2
ðιÞ; ð16Þ
where ι is the inclination angle of the binary orbital angular
momentum with respect to the line of sight and F
2
þ
and F
2
×
are the two antenna pattern functions. In Eq. (14), the
function ΦðfÞ is the inspiral phase of the binary system:
ΦðfÞ¼2πft
c
− ϕ
c
−
π
4
þ
3
128ηv
5
1 þ
X
7
i¼2
α
i
v
i
; ð17Þ
where the coefficients α
i
are the corrections up to the
3.5 PN corrections. In Appendix A, we also provide the
expressions for these coefficients. In the above equation,
we have defined v ≡ðπMfÞ
1=3
, M ≡ m
1
þ m
2
, η ≡ m
1
m
2
=
ðm
1
þ m
2
Þ
2
, and M
c
≡ ð1 þ zÞMη
3=5
to be the inspiral
reduced frequency, total mass, symmetric mass ratio, and
the redshifted chirp mass, respectively. The quantities t
c
and ϕ
c
are the time and phase of coalescence, respectively.
After having defined the modified GW signal for
compact binaries, in what follows we summarize the
already known methodology used to estimate d
L
ðzÞ mea-
sures from GW standard sirens. We refer to [68,69] for
pioneer works in this regard.
For a high enough SNR and a given waveform model
hðf; θ
i
Þ, with free parameters θ
i
, we can use the Fisher
matrix analysis to provide upper bounds for the free
parameters of the models by means of the Cramer-Rao
bound [70,71]. We refer the reader to [72–77] for a
discussion on the Fisher analysis to estimate parameters
in binary systems for a given GW signal. Once the
waveform model is defined, the root-mean-squared error
on any parameter is determined by
Δθ
i
¼
ffiffiffiffiffiffi
Σ
ii
p
; ð18Þ
where Σ
ij
is the covariance matrix, i.e., the inverse of the
Fisher matrix, Σ
ij
¼ Γ
−1
ij
. The Fisher matrix is given by
Γ
ij
¼
∂
˜
h
∂θ
i
∂
˜
h
∂θ
j
: ð19Þ
The inner product between two waveform models is
defined as
ð
˜
h
1
j
˜
h
2
Þ ≡ 2
Z
f
upper
f
low
˜
h
1
˜
h
2
þ
˜
h
1
˜
h
2
S
n
ðfÞ
df; ð20Þ
where the “star” stands for complex conjugation and S
n
ðfÞ
is the detector spectral noise density. With this definition of
the inner product, the SNR is defined as
SNR
2
≡ 4Re
Z
f
upper
f
low
jhðfÞj
2
S
n
df: ð21Þ
In what follows, we consider the ET detector power
spectral density noise. The ET is a third-generation ground-
based detector of GWs and it is envisaged to be 10 times
more sensitive in amplitude than the advanced ground-
based detectors in operation nowadays, covering the
frequency range 1–10
4
Hz. Unlike the current detectors,
from the ET conceptual design study, the expected rates of
BNS detections per year are of the order of 10
3
–10
7
[23].
However, we can expect only a small fraction (∼10
−3
)of
them accompanied with the observation of a short γ-ray
burst. If we assume that the detection rate is in the middle
range around Oð10
5
Þ, we can expect to see Oð10
2
Þ events
with short γ-ray bursts per year.
Thus, let us consider in our simulations a mock GW
standard siren dataset composed by 1000 BNS merger
events. Assuming that the errors on d
L
are uncorrelated
with errors on the remaining GW parameters, we have
σ
2
d
L
¼
∂
˜
hðfÞ
∂d
L
;
∂
˜
hðfÞ
∂d
L
−1
: ð22Þ
Since
˜
hðfÞ ∝ ðd
GW
L
Þ
−1
, then σ
d
L
¼ d
L
=SNR. However,
when we estimate the practical uncertainty of the mea-
surements of d
L
, we should take the orbital inclination into
account. The maximal effect of the inclination on the SNR
is a factor of 2 (between ι ¼ 0° and ι ¼ 90°). Therefore, we
add this factor to the instrumental error for a conservative
estimation. Thus, the estimate of the instrumental error is
given by σ
d
L
¼ 2d
L
=SNR. On the other hand, GWs are
lensed in the same way as the electromagnetic waves,
resulting into a weak lensing effect error, which we model
as σ
lens
d
L
¼ 0.05 zd
L
ðzÞ [23,78]. In our study, we do not
consider possible errors induced from the peculiar velocity
due to the clustering of galaxies. Since we are interested in
simulating events at high z mainly, we can neglect such
contributions, which are significant only for z ≪ 1. In fact,
at high z, the dominant source of uncertainty is the one due
to weak lensing. Therefore, the total uncertainty σ
d
L
on the
luminosity distance measurements associated to each event
is obtained by combining the instrumental and weak
lensing uncertainties as
σ
d
L
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðσ
ins
d
L
Þ
2
þðσ
lens
d
L
Þ
2
q
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2d
L
ðzÞ
SNR
2
þð0.05zd
L
ðzÞÞ
2
s
: ð23Þ
The redshift distribution of the BNS sources is taken to be
of the form
ROCCO D’AGOSTINO and RAFAEL C. NUNES PHYS. REV. D 100, 044041 (2019)
044041-4
PðzÞ ∝
4πd
2
C
ðzÞrðzÞ
HðzÞð1 þ zÞ
; ð24Þ
where d
C
ðzÞ is the comoving distance and rðzÞ describes
the time evolution of the burst rate:
rðzÞ¼
8
>
>
<
>
>
:
1 þ 2z; z ≤ 1;
ð15 − 3zÞ=4; 1 <z<5;
0;z≥ 5.
ð25Þ
The distribution of the neutron star masses is chosen to
be randomly sampled from uniform distributions within
½1 − 2 M
⊙
, also under the condition m
1
≳ m
2
and
η < 0.25. In this case, in the mock data generation we
take χ
1
¼ χ
2
¼ 0, where χ
1
and χ
2
are the associated spin
magnitudes on each mass component. Then, we simulate
BNS mergers up to z ¼ 2, which represents the maximum
distance at which these events can be observed from the
power spectral density noise from the ET [23]. Also, we
checked that, beyond z ¼ 2, the SNR presents low values,
which also can limit the use of the Fisher information for
mock data. Now, in order to realistically generate a mock
catalog using modified gravity, we shall consider nonzero
values for the parameters α
M0
and n, which are compatible
with the current cosmological observation as well as with
the stability criteria of the theory. Lastly, when generating
our mock GW standard siren dataset, we only consider
BNS mergers with SNR > 8.
In order to estimate the observational constraints on
the free parameters of the models, we apply the Markov
chain Monte Carlo (MCMC) method through the
Metropolis-Hastings algorithm [79], where the likelihood
function for the GW standard siren mock dataset is built
in the form
L
GW
∝ exp
−
1
2
X
1000
i¼1
d
obs
L
ðz
i
Þ − d
th
L
ðz
i
Þ
σ
d
L
;i
: ð26Þ
Here, d
obs
L
ðz
i
Þ are the 1000 simulated BNS merger events
with their associated uncertainties σ
d
L
;i
, while d
th
L
ðz
i
Þ is the
theoretical prediction on each ith event.
IV. CONSTRAINTS ON THE RUNNING
OF THE PLANCK MASS
In this section, we present and discuss our results
regarding the future observational constraints that GW
standard sirens can impose on a possible time variation
of the Planck mass within the ET sensitivity. The running
of the Planck mass is an important physical quantity,
which essentially is present in any and all modified gravity
models. To generate a simulated d
L
ðz
i
Þ catalog using
modified gravity, we assume realistic values for the pair
(α
M0
, n), on each triplet [z
i
, d
L
ðz
i
Þ; σ
d
L
ðz
i
Þ
] evaluated at
each point i, as follows.
We first note that the parameter n is statistically
degenerate. This fact is already well known and expected
to happen. In the literature, it is usual to assume n ¼ 1,but
here we follow the stability conditions discussed in Sec. II
and weakly generate random values for n, within the range
of stability of the theory: (i) for the case α
M0
> 0,we
randomly sampled from uniform distributions: α
M0
∈
½0; 0.5 and n ∈ ½0; 1.40; (ii) for the case α
M0
< 0,we
randomly sampled α
M0
∈ ½−0.5; 0 and n ∈ ½2.5; 3.5.We
found that different prior ranges on n change the simulated
catalogs very weakly. Only very different prior ranges on
α
M0
can significantly change the pair [d
L
ðz
i
Þ; σ
d
L
ðz
i
Þ
]. The
range assumed on α
M0
is fully compatible with current
constraints [80–90]. We used as input values H
0
¼
67.4 km=s=Mpc and Ω
m0
¼ 0.31 for the Hubble constant
and matter density parameter, respectively, in agreement
with the most recent Planck cosmic microwave background
(CMB) data [91]. Hence, these values are reasonable for
our purpose to generate GW standard siren mock data.
In the realization of the MCMC analysis, the sampling
has been done assuming the following uniform priors for
the cosmological parameters: H
0
∈ ½55; 90, Ω
m0
∈ ½0; 1,
and α
M0
∈ ½−1; 0, α
M0
∈ ½0; 1 for each case. Due to the
large statistical degeneracy on n, as commented above,
we fixed n ¼ 3 and n ¼ 1 for the cases of α
M0
> 0 and
α
M0
< 0, respectively. Table I summarizes the constraints at
the 68% and 95% confidence levels (C.L.). In Fig. 2,we
show the parametric space and the one-dimensional mar-
ginalized distribution for the parameters Ω
m0
, H
0
and α
M0
,
in both α
M0
> 0 and α
M0
< 0 cases. We note from both
analyses that the parameter α
M0
is non-null at the 68% C.L.
Assuming the stability condition where α
M0
is negative,
we find the new lower limit α
M0
> −0.2 at the 95% C.L.
TABLE I. Summary of the MCMC results for the cases α
M0
> 0 and α
M0
< 0. The upper and lower values next to
the mean value of each parameter denote the 68% and 95% C.L. errors, respectively.
Stability conditions H
0
Ω
m0
α
M0
α
M0
< 067.466
þ0.036ð0.143Þ
−0.067ð0.179Þ
0.328
þ0.015ð0.028Þ
−0.014ð0.028Þ
−0.100
þ0.051ð0.092Þ
−0.043ð0.085Þ
α
M0
> 067.390
þ0.047ð0.098Þ
−0.050ð0.095Þ
0.297
þ0.029ð0.083Þ
−0.044ð0.072Þ
0.199
þ0.069ð0.178Þ
−0.097ð0.167Þ
PROBING OBSERVATIONAL BOUNDS ON SCALAR-TENSOR … PHYS. REV. D 100, 044041 (2019)
044041-5
![FIG. 1. Corrections on the effective luminosity distance [see Eq. (12)], as a function of the redshift, for different values of the running of the Planck mass today. The theoretical curves correspond to the case αM0 > 0 and n ¼ 3 (left panel) and αM0 > 0 and n ¼ 1 (right panel).](/figures/fig-1-corrections-on-the-effective-luminosity-distance-see-yr6kjtwa.webp)
