[PDF] Mauritius2017 - Testing strong-field gravity with pulsar timing & gravitational waves

Testing strong-ﬁeld gravity with
pulsar timing & gravitational waves
Lijing Shao
Max Planck Institute for Gravitational Physics (Albert Einstein Institute)
FUNDAMENTAL PHYSICS WITH THE SKA
MAY 1-5, MAURITIUS

Pulsars in the SKA era
• Pulsar search: e.g., simulations show that ~10,000 normal pulsars
(incl. ~1,800 millisecond pulsars) could be discovered with
• SKA1-LOW surveying |b| > 5°
• SKA1-MID surveying |b| < 10°
• Pulsar timing: SKA1-MID improves the precision by ~10, while the full
SKA improves probably by ~100 Shao et al., PoS (AASKA14) 042
Keane et al., PoS (AASKA14) 040
SKA-LOW SKA-MID
© SKAO © SKAO© Antoniadis
Pulsar-BH binaries?
see Michael Kavic’s & Fupeng Zhang’s talks

An example for better precision: Lense-Thirring effect4
Fig. 2. Improving sensitivity, ∆ωLT (red curves), for measuring the Lense-Thirring eﬀect based
on mock data simulations. The blue shaded area indicates the expected magnitude of the Lense-
Kehl, Wex, Kramer, Liu, arXiv:1605.00408

Gravitational-wave Astrophysics
FIG. 1. GW151226 observed by the LIGO Hanford (left column) and Livingston (right column) detectors, where tim
December 26, 2015 at 03:38:53.648 UTC. First row: Strain data from the two detectors, where the data are filtered w
bandpass filter to suppress large fluctuations outside this range and band-reject filters to remove strong instrumental sp
Also shown (black) is the best-match template from a nonprecessing spin waveform model reconstructed using a Bayes
with the same filtering applied. As a result, modulations in the waveform are present due to this conditioning and not d
effects. The thickness of the line indicates the 90% credible region. See Fig. 5 for a reconstruction of the best-match
filtering applied. Second row: The accumulated peak signal-to-noise ratio (SNRp) as a function of time when integrating
the best-match template, corresponding to a gravitational-wave frequency of 30 Hz, up to its merger time. The total ac
corresponds to the peak in the next row. Third row: Signal-to-noise ratio (SNR) time series produced by time shiftin
PRL 116, 241103 (2016) P H Y S I C A L R E V I E W L E T T E R S
properties of space-time in the strong-field, high-velocity
regime and confirm predictions of general relativity for the
nonlinear dynamics of highly disturbed black holes.
II. OBSERVATION
On September 14, 2015 at 09:50:45 UTC, the LIGO
Hanford, WA, and Livingston, LA, observatories detected
the coincident signal GW150914 shown in Fig. 1. The initial
detection was made by low-latency searches for generic
gravitational-wave transients [41] and was reported within
three minutes of data acquisition [43]. Subsequently,
matched-filter analyses that use relativistic models of com-
pact binary waveforms [44] recovered GW150914 as the
most significant event from each detector for the observa-
tions reported here. Occurring within the 10-ms intersite
FIG. 1. The gravitational-wave event GW150914 observed by the LIGO Hanford (H1, left column panels) and Livingston (L1, right
column panels) detectors. Times are shown relative to September 14, 2015 at 09:50:45 UTC. For visualization, all time series are filtered
with a 35–350 Hz bandpass filter to suppress large fluctuations outside the detectors’ most sensitive frequency band, and band-reject
filters to remove the strong instrumental spectral lines seen in the Fig. 3 spectra. Top row, left: H1 strain. Top row, right: L1 strain.
GW150914 arrived first at L1 and 6.9þ0.5
−0.4 ms later at H1; for a visual comparison, the H1 data are also shown, shifted in time by this
amount and inverted (to account for the detectors’ relative orientations). Second row: Gravitational-wave strain projected onto each
detector in the 35–350 Hz band. Solid lines show a numerical relativity waveform for a system with parameters consistent with those
recovered from GW150914 [37,38] confirmed to 99.9% by an independent calculation based on [15]. Shaded areas show 90% credible
regions for two independent waveform reconstructions. One (dark gray) models the signal using binary black hole template waveforms
[39]. The other (light gray) does not use an astrophysical model, but instead calculates the strain signal as a linear combination of
sine-Gaussian wavelets [40,41]. These reconstructions have a 94% overlap, as shown in [39]. Third row: Residuals after subtracting the
PRL 116, 061102 (2016) P H Y S I C A L R E V I E W L E T T E R S week ending
12 FEBRUARY 2016
LIGO & Virgo, PRL 116:061102 (2016) LIGO & Virgo, PRL 116:241103 (2016)
see Frans Pretorius’ talk
(14, 8) M(36, 29) M

Testing Gravity with GWs
see Frans Pretorius’ talk
parameters is allowed to vary freely (in addition to masses,
spins, etc.), while the remaining ones are fixed to their GR
value, that is zero, and a multiple-parameter analysis in
which all of the parameters in one of the three sets
enumerated above are allowed to vary simultaneously.
The rationale behind our choices of single- and multiple-
parameter analyses comes from the following consider-
ations. In most known alternative theories of gravity
[13,14,88], the corrections to GR extend to all PN orders
even if, in most cases, they have been computed only at
leading PN order. Considering that GW150914 is an
inspiral-merger-ringdown signal sweeping through the
detector between 20 and 300 Hz, we expect to see signal
deviations from GR at all PN orders. The single-parameter
analysis corresponds to minimally extended models that
can capture deviations from GR that occur predominantly,
but not only, at a specific PN order. Nevertheless, should a
deviation be measurably present at multiple PN orders, we
expect the single-parameter analyses to also capture these.
In the multiple-parameter analysis, the correlations among
the parameters are very significant. In other words, a shift in
one of the testing parameters can always be compensated
for by a change of the opposite sign in another parameter
and still return the same overall GW phase. Thus, it is not
FIG. 6. 90% upper bounds on the fractional variations of the
known PN coefficients with respect to their GR values. The
orange squares are the 90% upper bounds obtained from
the single-parameter analysis of GW150914. As a comparison,
the blue triangles show the 90% upper bounds extrapolated
exclusively from the measured orbital-period derivative _Porb of
the double pulsar J0737-3039 [12,87], here, too, allowing for
possible GR violations at different powers of frequency, one at a
time. The GW phase deduced from an almost constant _Porb
PRL 116, 221101 (2016) P H Y S I C A L R E V I E W L E T T E R S week ending
3 JUNE 2016
spread as small as 0.07 for deviations in the 1.5PN
parameter φ3, which encapsulates the leading-order effects
of the dynamical self-interaction of spacetime geometry
(the “tail” effect) [148–151], as well as spin-orbit inter-
action [67,152,153].
In Fig. 8, we show the 90% credible upper bounds on the
magnitude of the fractional deviations in PN coefficients,
jδˆφij, which are affected by both the offsets and widths of
the posterior density functions for the δˆφi. We show bounds
thos
In
cont
regi
sign
som
A
form
GW
post
low
the m
used
wav
wea
info
imp
obse
the
grav
How
pola
align
expa
addi
gene
V
T
rate
prec
long
dete
FIG. 8. The 90% credible upper bounds on deviations in the PN
coefficients, from GW150914 and GW151226. Also shown are
joint upper bounds from the two detections; the main contributor
is GW151226, which had many more inspiral cycles in band than
GW150914. At 1PN order and higher, the joint bounds are
slightly looser than the ones from GW151226 alone; this is due to
the large offsets in the posteriors for GW150914.
LIGO & Virgo, PRL 116:221101 (2016) LIGO & Virgo, PRX 6:041015 (2016)
also see, Yunes et al., PRD 94:084002 (2016)

The double pulsar and more…
identical for the two pulsars; this equality
therefore does not implicitly make assumptions
about the validity of any particular theory of
gravity (see below). The same applies for the
orbital decay parameter ˙Pb. In contrast, the PK
parameters g (the gravitational redshift and time
dilation parameter) and s and r (the Shapiro-
delay parameters) are asymmetric in the masses,
and their values and interpretations differ for A
and B. In practical terms, the relatively low
timing precision for B does not require the
inclusion of g, s, r, or ˙Pb in the timing model.
We can, however, independently measure ˙wwB,
obtaining a value of 16.96- T 0.05- yearj1,
obtain the best possible accuracy for this param-
eter. We used the whole TOA data set for B in
order to measure B’s spin parameters P and ˙P,
given in Table 1. These parameters were then kept
fixed for a separate analysis of the concentrated
5-day GBT observing sessions at 820 MHz. On
the time scale of the long-term profile evolution
of B, each 5-day session represents a single-
epoch experiment and hence requires only a
single set of profile templates. The value of xB
obtained from a fit of this parameter only to
the two 5-day sessions is presented in Table 1.
Because of the possible presence of unmod-
eled intrinsic pulsar timing noise and because
ally somewhat smaller than quoted, this prac-
tice facilitates comparison with previous tests
of GR by pulsar observation. The timing model
also includes timing offsets between the data
sets for the different instruments represented by
the entries in table S1. The final weighted root
mean square post-fit residual is 54.2 ms. In
addition to the spin and astrometric parameters,
the Keplerian parameters of A’s orbit, and five
PK parameters, we also quote a tentative de-
tection of a timing annual parallax that is con-
sistent with the dispersion-derived distance.
Further details are given in (16).
Tests of general relativity. Previous obser-
vations of PSR J0737-3039A/B (8, 9) resulted
in the measurement of R and four PK param-
eters: w˙ , g, r, and s. Relative to these earlier
results, the measurement precision for these
parameters from PSR J0737-3039A/B has in-
creased by up to two orders of magnitude. Also,
we have now measured the orbital decay ˙Pb. Its
value, measured at the 1.4% level after only 2.5
years of timing, corresponds to a shrinkage of
the pulsars’ separation at a rate of 7 mm per day.
Therefore, we have measured five PK parame-
ters for the system in total. Together with the
mass ratio R, we have six different relationships
that connect the two unknown masses for A and
B with the observations. Solving for the two
masses using R and one PK parameter, we can
then use each further PK parameter to compare
its observed value with that predicted by GR for
the given two masses, providing four indepen-
dent tests of GR. Equivalently, one can display
these tests elegantly in a ‘‘mass-mass’’ diagram
(Fig. 1). Measurement of the PK parameters
gives curves on this diagram that are, in general,
different for different theories of gravity but
should intersect in a single point (i.e., at a pair
of mass values) if the theory is valid (12).
As shown in Fig. 1, we find that all mea-
sured constraints are consistent with GR. The
most precisely measured PK parameter current-
ly available is the precession of the longitude of
periastron, w˙ . We can combine this with the
theory-independent mass ratio R to derive the
masses given by the intersection region of their
curves: mA 0 1.3381 T 0.0007 MR and mB 0
1.2489 T 0.0007 MR , where MR is the mass of
the Sun (20). Table 2 lists the resulting four
independent tests that are currently available.
All of them rely on comparison of our mea-
sured values of s, r, g, and ˙Pb with predicted
values based on the masses defined by the
Fig. 1. Graphical summary of tests of GR parameters. Constraints on the masses of the two stars (A and
B) in the PSR J0737-3039A/B binary system are shown; the inset is an expanded view of the region of
principal interest. Shaded regions are forbidden by the individual mass functions of A and B because sin
i must be e1. Other constraining parameters are shown as pairs of lines, where the separation of the
lines indicates the measurement uncertainty. For the diagonal pair of lines labeled as R, representing
the mass ratio derived from the measured semimajor axes of the A and B orbits, the measurement
precision is so good that the line separation becomes apparent only in the inset. The other constraints
shown are based on the measured PK parameters interpreted within the framework of general relativity.
The PK parameter ˙ww describes the relativistic precession of the orbit, g combines gravitational redshift
and time dilation, and ˙PPb represents the measured decrease in orbital period due to the emission of
gravitational waves. The two PK parameters s and r reflect the observed Shapiro delay, describing a
onOctober24,2011www.sciencemag.orgDownloadedfrom
the literature (Will 1993; Damour & Esposito-Farèse 1992, 1993,
1996a,b, 1998). In strong-field conditions, notably within and near
a neutron star, the coupling constants α0 and β0 are modified by
self-gravity effects, and become body-dependent quantities, αA and
βA (A being a label for the body), which can be computed by
numerical integration of the field equations. One needs to assume
a specific EOS for nuclear matter in such integrations, and we will
use the moderate one of Damour & Esposito-Farèse (1996b) in the
Figure 7. Solar system and binary pulsar 1σ constraints on the matter–
scalar coupling constants α0 and β0. Note that a logarithmic scale is used
for the vertical axis |α0|, i.e. that GR (α0 = β0 = 0) is sent at an infinite
distance down this axis. LLR stands for Lunar Laser Ranging, Cassini for
the measurement of a Shapiro time-delay variation in the Solar system, and
SEP for tests of the strong equivalence principle using a set of neutron star–
WD low-eccentricity binaries (see text). The allowed region is shaded, and
it includes GR. PSR J1738+0333 is the most constraining binary pulsar,
although the Cassini bound is still better for a finite range of quadratic
Figu
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Kramer et al., Science 314:97 (2006)
Freire et al., MNRAS 423:3328 (2012)
see Gemma Janssen’s talk

“How to differentiate their powers quantitatively?”
A concrete example
Shao, Sennett, Buonanno, Kramer, Wex, arXiv:1704.07561

Scalar-tensor gravity of Damour & Esposito-Farèse
In this paper we focus on the class of mono-scalar-tensor
theories that are defined by the following action in the
Einstein-frame [4, 5, 14, 15],
S =
c4
16πG∗
d4
x
c
√
−g∗ R∗ − 2gµν
∗ ∂µϕ∂νϕ − V(ϕ)
+S m ψm; A2
(ϕ)g∗
µν , (1)
where G∗ is the bare gravitational coupling constant, g∗
µν is
the Einstein metric with its determinant g∗, R∗ ≡ gµν
∗ R∗
µν is the
Ricci scalar, ψm collectively denotes the matter content, and
A(ϕ) is the (conformal) coupling function that depends on the
scalar field, ϕ. Henceforth, for simplicity, we assume that the
potential, V(ϕ), is a slowly varying function that changes on
scales much larger than typical length scales of the system that
we consider, thus, we set V(ϕ) = 0 in our calculation.
The field equations are derived with the least-action princi-
ple [7, 8] for g∗
µν and ϕ,
R∗
µν = 2∂µϕ∂νϕ +
8πG∗
c4
T∗
µν −
1
2
T∗
g∗
µν , (2)
∗ ϕ = −
4πG∗
α(ϕ)T , (3)
FIG. 1. Illustration of
in comparison to indiv
SLy4 and |α0| = 10−5
bottom) β0 = −4.5, −4
differ in β0 in steps of
upper limits on the eff
pulsars listed in Table
gap” at mA ∼ 1.7 M⊙.
S =
c4
16πG∗
d4
x
c
√
−g∗ R∗ − 2gµν
∗ ∂
+S m ψm; A2
(ϕ)g∗
µν ,
where G∗ is the bare gravitational coupl
the Einstein metric with its determinant g∗
Ricci scalar, ψm collectively denotes the
A(ϕ) is the (conformal) coupling function
scalar field, ϕ. Henceforth, for simplicity,
potential, V(ϕ), is a slowly varying funct
scales much larger than typical length scale
we consider, thus, we set V(ϕ) = 0 in our
The field equations are derived with the
ple [7, 8] for g∗
µν and ϕ,
R∗
µν = 2∂µϕ∂νϕ +
8πG∗
c4
T∗
µν −
g∗ ϕ = −
4πG∗
c4
α(ϕ)T∗ ,
e bare gravitational coupling constant, g∗
µν is
tric with its determinant g∗, R∗ ≡ gµν
∗ R∗
µν is the
m collectively denotes the matter content, and
formal) coupling function that depends on the
Henceforth, for simplicity, we assume that the
is a slowly varying function that changes on
ger than typical length scales of the system that
us, we set V(ϕ) = 0 in our calculation.
ations are derived with the least-action princi-
ν and ϕ,
= 2∂µϕ∂νϕ +
8πG∗
c4
T∗
µν −
1
2
T∗
g∗
µν , (2)
= −
4πG∗
c4
α(ϕ)T∗ , (3)
y-momentum tensor of matter fields, Tµν
∗ ≡
m/δg∗
µν, and the field-dependent coupling
n the scalar field and the trace of the energy-
or of matter fields, α(ϕ) ≡ ∂ ln A(ϕ)/∂ϕ.
amour and Esposito-Farèse [7, 15], we con-
ial form for ln A(ϕ) up to quadratic order, that
β0ϕ2
/2 , and denote α0 ≡ α(ϕ0) = β0ϕ0
ymptotic value of ϕ at infinity. This partic-
or theory (henceforth, DEF theory) is com-
FIG. 1. Illustration of spontaneous scalarization in the DE
in comparison to individual binary-pulsar limits, for a NS
SLy4 and |α0| = 10−5
. The blue curves correspond to (f
bottom) β0 = −4.5, −4.4, −4.3, and −4.2; the grey curves
differ in β0 in steps of 0.01. We indicate with triangles th
upper limits on the effective scalar coupling |αA| from the
pulsars listed in Table I. We can clearly see a “scalariza
gap” at mA ∼ 1.7 M⊙.
much smaller 1
. In Fig. 1 we show an example of spo
scalarization for a NS with the realistic EOS SLy4,
pare it to existing individual binary-pulsar constrain
In general, if two compact bodies in a binary have
scalar couplings, αA and αB, they produce gravitatio
lar radiation ∝ (∆α)2
, with ∆α ≡ αA − αB, which is
post-Newtonian (PN) order than the canonical quadr
diation in GR [15] 2
. In Ref. [16], Damour and
Farèse for the first time compared limits on the D
ity arising from Solar system and binary pulsar exp
with expected limits from ground-based GW dete
LIGO and Virgo. The analysis in Ref. [16] is base
with the energy-momentum tensor of matter fields, Tµν
∗ ≡
2c (−g∗)−1/2
δSm/δg∗
strength between the scalar field and the trace of the energy-
momentum tensor of matter fields, α(ϕ) ≡ ∂ ln A(ϕ)/∂ϕ.
Following Damour and Esposito-Farèse [7, 15], we con-
sider a polynomial form for ln A(ϕ) up to quadratic order, that
is A(ϕ) = exp β0ϕ2
with ϕ0 the asymptotic value of ϕ at infinity. This partic-
ular scalar-tensor theory (henceforth, DEF theory) is com-
pletely characterized by two parameters (α0, β0) and for sys-
tems dominated by strong-field gravity, such as NSs, can give
rise to potentially observable, nonperturbative physical phe-
nomena [14, 23]. Weak-field Solar-system experiments, gen-
erally, only probe the α0-dimension or the combination β0α2
0
in the (α0, β0) parameter space (see Refs. [10, 27] and refer-
ences therein).
Using a perfect-fluid description of the energy-momentum
tensor for NSs in the Jordan frame, in 1993 Damour and
Esposito-Farèse derived the Tolman-Oppenheimer-Volkoff
(TOV) equations [14] for a NS in their scalar-tensor gravity
theory. Interestingly, they discovered a phase-transition phe-
nomenon when β0 −4, largely irrespective of the α0 value
(a nonzero α0 tends to smooth the phase transition [15]). The
phenomenon was named spontaneous scalarization. With a
suitable (α0, β0), the “effective scalar coupling” that a NS de-
scalarization for a
pare it to existing
In general, if tw
scalar couplings, α
lar radiation ∝ (∆α
post-Newtonian (P
diation in GR [15
Farèse for the firs
ity arising from S
with expected lim
LIGO and Virgo.
(by now excluded
the LIGO/Virgo e
PSR B1913+16 li
as a 1.4 M⊙-10 M
Farèse come to th
would generally b
on the parameters
as LIGO and Vir
1 For sufficiently nega
ing their maximum m
above a certain criti
the EOS [14, 15].
2 In this paper, gene
to the leading Newt
∗ ≡
2c (−g∗)−1/2
δSm/δg∗
0
ences therein).
pare it to existing indi
In general, if two co
scalar couplings, αA a
,
post-Newtonian (PN)
.
Farèse for the first tim
ity arising from Solar
with expected limits
LIGO and Virgo. Th
(by now excluded [28
the LIGO/Virgo expe
PSR B1913+16 like m
as a 1.4 M⊙-10 M⊙ N
Farèse come to the co
would generally be ex
on the parameters (α0
as LIGO and Virgo.
1 For sufficiently negative β
ing their maximum mass,
above a certain critical m
the EOS [14, 15].
c
∗ ≡
2c (−g∗)−1/2
δSm/δg∗
0
ences therein).
phenomenon was named spontaneous scalarization. With a
suitable (α0, β0), the “effective scalar coupling” that a NS de-
velops, αA ≡ ∂ ln mA/∂ϕ0 (the baryonic mass of NS is fixed
scalarization for a NS with the realistic
pare it to existing individual binary-puls
In general, if two compact bodies in a
scalar couplings, αA and αB, they produ
, with ∆α ≡ αA − α
post-Newtonian (PN) order than the cano
. In Ref. [16], D
Farèse for the first time compared limi
ity arising from Solar system and binar
with expected limits from ground-base
LIGO and Virgo. The analysis in Ref.
(by now excluded [28, 29]), medium an
the LIGO/Virgo experiment it assumes
PSR B1913+16 like masses (1.44 M⊙ a
as a 1.4 M⊙-10 M⊙ NS-BH merger. D
Farèse come to the conclusion that binar
would generally be expected to put more
on the parameters (α0, β0) than ground-
as LIGO and Virgo. Since then, sever
1 For sufficiently negative β0 ( −4.6), NSs do no
ing their maximum mass, i.e. spontaneous scalar
above a certain critical mass, which depends on
the EOS [14, 15].
2 In this paper, generally we denote with nPN
to the leading Newtonian dynamics (equations
gravitational dipolar radiation reaction is at 1.5
Two theory parameters: α0 & β0
2
ENA
ensor
the
(1)
∗
µν is
s the
and
n the
at the
es on
m that
inci-
FIG. 1. Illustration of spontaneous scalarization in the DEF gravity,
in comparison to individual binary-pulsar limits, for a NS with EOS
SLy4 and |α0| = 10−5
. The blue curves correspond to (from top to
bottom) β0 = −4.5, −4.4, −4.3, and −4.2; the grey curves in between
differ in β0 in steps of 0.01. We indicate with triangles the 90% CL
upper limits on the effective scalar coupling |αA| from the individual
dipolar emission (10) by giving the first star a charge close
to the maximum value allowed by the ST theory (1 $
max ), and an almost zero scalar charge to the second star
(2 % 0), the scalar field grows rapidly inside the second
star, which quickly develops a charge 2 % 1 when the
ferromagne
field [11,12
configuratio
over the ini
Quite rem
stars that ar
its magnitud
In fact, it h
zation is lik
e.g., in (at
nonscalariz
1:85M, m
FIG. 2 (color online). The scalar field ’G1=2 (color code) and
the NS surfaces (solid black line) at t ¼ f1:8; 3:1; 4:0; 5:3g ms for

=ð4GÞ ¼ À4:5, and the binary of Fig. 1.
α
FIG. 3 (colo

=ð4GÞ ¼
BARAUSSE et al.
(Shao et al. 2017)
(Barausseetal.2013)
Nonperturbative “phase-transition” phenomena can happen when the
compactness of a NS (GmNS/RNS) [DEF 1993, PRL 70:2220] or of a BNS (GMBNS/rBNS)
[Barausse et al. 2013, PRD 87:081506] reaches a critical point
spontaneous scalarization dynamical scalarization

Testing dipole radiation with binary pulsars
We select five best NS-WD systems in
testing dipole radiation and explore 7-d
parameter space {α0, β0, ρc
(i)} (i=1,2,…,5)
with emcee for eleven EOSs
5
in the above equations with the Newtonian gravitational con-
stant GN = G∗(1 + α2
0), since |α0| ≪ 1 (e.g., from the Cassini
spacecraft [27, 52]).
We construct the logarithmic likelihood for the MCMC runs
as,
ln L ∝ −
1
2
N
i=1
⎡
⎢⎢⎢⎢⎢⎢⎢⎣
⎛
⎜⎜⎜⎜⎜⎜⎝
˙Pint
b − ˙Pth
b
σobs
˙Pb
⎞
⎟⎟⎟⎟⎟⎟⎠
2
+
⎛
⎜⎜⎜⎜⎝
mp/mc − q
σobs
q
⎞
⎟⎟⎟⎟⎠
2
⎤
⎥⎥⎥⎥⎥⎥⎥⎦ , (9)
where for PSRs J1909−3744 and J2222−0137 we replace the
second term in the squared brackets with mp − mobs
p /σobs
mp
2
.
In Eq. (9), the predicted orbital decay from the theory is ˙Pth
b ≡
˙P
dipole
b + ˙P
quad
b , and σobs
X is the observational uncertainty for
X ∈ ˙Pint
b , q, mp , as given in Table I.
For each EOS, we perform four separate MCMC runs:
(i) 1 pulsar: PSR J0348+0432 (J0348);
(ii) 1 pulsar: PSR J1738+0333 (J1738);
(iii) combining 2 pulsars: PSRs J0348+0432 and
J1738+0333 (2PSRs);
(iv) combining 5 pulsars: PSRs J0348+0432, J1012+5307,
J1738+0333, J1909−3744 and J2222−0137 (5PSRs).
We pick J0348 and J1738 due to their mass difference
(2.01 M⊙ and 1.46 M⊙ respectively), and their high timing
precision (see Table I), which leads to interesting differences
in the constraints on the DEF parameters, especially on β0.
FIG. 2. The marginalized 2-d distribution of (log10 |α0|, −β0) from
MCMC runs on the five pulsars listed in Table I, for the EOS SLy4.
The marginalized 1-d distributions and the extraction of upper limits
are illustrated in upper and right panels.
0.0054 0.2130(24) 1.293(25)
Then, for the decay of the binary’s orbital period, which
nters the likelihood function (see Eq. (9)), we use the dipolar
ontribution from the scalar field and the quadrupolar contri-
ution from the tensor field as given by the following, well
nown, formulae [7, 55],
˙P
dipole
b = −
2πG∗
c3
g(e)
2π
Pb
mpmc
mp + mc
(αA − α0)2
, (5)
˙P
quad
b = −
192πG5/3
∗
5c5
f(e)
2π
Pb
5/3
mpmc
mp + mc
1/3
, (6)
ith
g(e) ≡ 1 +
e2
2
1 − e2 −5/2
, (7)
f(e) ≡ 1 +
73
24
e2
+
37
96
e4
1 − e2 −7/2
. (8)
We find that higher order terms, as well as the subdominant
In a NS-WD binary with effective scalar couplings αA and α0, we
have dipole radiation [DEF 1992, CQG 9:2093; Will 1994, PRD 50:6058]
An example
(Shao et al. 2017)
Del Pozzo Vecchio 2016, MNRAS 462:L21

Can NSs still be scalarized?
Scalarization depends strongly on NS EOS [Shibata et al. 2014, PRD 89:084005], and by
combining binary pulsars with different NS masses, we constrain the
scalarization parameter β0 tightly
While satisfying current constraints, NSs with suitable masses can still have
effective scalar couplings of O(10-2-10-1), for some EOSs
and
5307,
Rs).
erence
iming
rences
on β0.
les to
ng the
EOS,
0,000
espec-
4 runs
ile we
meters
-d dis-
e case
ribute
ectan-
ations,
onger
he ob-
0| and
d 90%
all 44
facts.
mit on
at the
r than
ly the
from
hould
n GR)
FIG. 2. The marginalized 2-d distribution of (log10 |α0|, −β0) from
MCMC runs on the five pulsars listed in Table I, for the EOS SLy4.
The marginalized 1-d distributions and the extraction of upper limits
are illustrated in upper and right panels.
FIG. 3. Marginalized upper limits on |α0| (upper) and −β0 (lower)
at 90% CL. These limits are obtained from PSRs J0348+0432
(J0348), J1738+0333 (J1738), a combination of them (2PSRs),
and a combination of PSRs J0348+0432, J1012+5307, J1738+0333,
J1909−3744 and J2222−0137 (5PSRs). The color coding for differ-
ent EOSs is kept consistent for all figures in this paper.
1.46 M⊙ for J1738, and 2.01 M⊙ for J0348. For EOSs that
favour spontaneous scalarization at around 1.4–1.5 M⊙, J1738
gives a better limit, while for EOSs that favour spontaneous
scalarization at around 2 M⊙, J0348 gives a better limit. This
trend is also consistent with Fig. 4 (to be introduced below).
Third, by combining two pulsars (2PSRs), NSs are limited to
scalarize at neither 1.4–1.5 M⊙ nor ∼ 2 M⊙. Therefore, almost
for all EOSs, β0 is well constrained. This result demonstrates
the power of properly using multiple pulsars with different NS
SLy4 5.2 × 10−5
4.23 1.71 1.1 × 10−3
1.4 × 1
WFF1 5.3 × 10−5
4.21 1.58 9.1 × 10−4
1.3 × 1
WFF2 5.5 × 10−5
4.24 1.68 1.2 × 10−3
1.4 × 1
FIG. 4. The effective scalar coupling |αA| that an isolated NS could
still develop after taking into account the 95% CL constraints from
the five pulsars (see Table II). The point of the maximum |αA| is
marked with a dot, and the values (and the corresponding masses)
are listed in Table II.
masses to constrain the DEF parameter space for any EOS.
Fourth, we obtain the most stringent constraints with five pul-
sars (5PSRs). This is especially true for β0, which is con-
strained at the level of ∼ −4.2 (68% CL) and ∼ −4.3 (90%
CL) for all EOSs. Finally, we list in Table II the marginalized
1-d limits for 5PSRs. We shall use them in the next section
when combining binary pulsars with laser-interferometer GW
observations.
Considering the results that we have obtained when com-
bining the five pulsars (5PSRs), one could wonder whether
tive scalar
the NS m
at 90% CL
the maxim
90% CLs,
(marked a
Figure
scalarizati
imally all
large as O
used, whi
its at 68%
Furthermo
scalarizat
Fig. 1 for
ing. The
are 1.46 M
1.76 M⊙ (
2.01 M⊙ (
the 11 EO
we have c
ever, some
acquire la
masses. A
EOSs AP
if the NS
NSs with
mA ≃ 1.92
strongly s
scalarizati
nary pulsa
itational d
we shall d
gaps also
(Shao et al. 2017) (Shao et al. 2017)

BNS inspirals with GW detectors
For future GW detectors on Earth, we
investigate the projected constraints on
(αA-αB) for nonspinning BNSs at DL = 200
Mpc, using the Fisher matrix with 3.5 PN
phasing, augmented with a dipole term
[Will 1994, PRD 50:6058; Buonanno et al. 2009, PRD 80:084043]
• SNR = 11/450/153 for aLIGO/CE/ET
If BNSs of suitable masses are seen:
• aLIGO might outperform current pulsar
limits if NS EOSs are described by
certain EOSs
• CE, ET can outperform current pulsar
limits with all EOSs considered here
FIG. 6. The sensitivities of aLIGO, CE, and ET to |∆α| (namely
the uncertainty, σ (|∆α|), obtained from the inverse Fisher matrix)
are depicted with dashed lines, as a function of mB, for a pattern-
averaged BNS inspiral signal with rest-frame component masses
(mA = 1.25 M⊙, mB). The starting frequencies of GW detectors are
labeled. Luminosity distance DL = 200 Mpc is assumed. The sen-
sitivity to |∆α| from GW detectors scales with SNR as ρ−1/2
. The
maximum available values of |∆α| for 11 EOSs, saturating the limits
from binary pulsars at 90% CL, are shown in solid lines. If a sensi-
to
co
th
th
ex
6
W
m
u
la
th
p
sa
th
as
R
FIG. 6. The sensitivities of aLIGO, CE, and ET to |∆α| (namely
the uncertainty, σ (|∆α|), obtained from the inverse Fisher matrix)
are depicted with dashed lines, as a function of mB, for a pattern-
averaged BNS inspiral signal with rest-frame component masses
(mA = 1.25 M⊙, mB). The starting frequencies of GW detectors are
labeled. Luminosity distance DL = 200 Mpc is assumed. The sen-
sitivity to |∆α| from GW detectors scales with SNR as ρ−1/2
. The
maximum available values of |∆α| for 11 EOSs, saturating the limits
from binary pulsars at 90% CL, are shown in solid lines. If a sensi-
tivity curve (dashed) is below a solid curve, the corresponding GW
detector has the potential to improve the limit from binary pulsars for
this particular EOS, with BNSs of suitable masses.
use the waveform parameters ln A, ln η, ln M, tc, Φc, (∆α)2
to construct the 6 × 6 Fisher matrix, Γab. The inverse of
the Fisher matrix is the correlation matrix for these parame-
ters, from where we can read their uncertainties and correla-
tions [17, 58, 68, 69].
In Fig. 6 we plot in dashed lines the uncertainties in |∆α|
obtained with three GW detectors (aLIGO, CE, and ET) for
an asymmetric BNS with rest-frame masses mA = 1.25 M⊙
and mB 1.25 M⊙, located at DL = 200 Mpc. For a BNS of
masses, for example, (1.25 M⊙, 1.63 M⊙) which are the most
probable masses for the newly discovered asymmetric double-
NS binary pulsar PSR J1913+1102 [73], we ﬁnd that aLIGO,
CE, and ET can detect its merger at 200 Mpc with ρ = 10.6,
We stress that those c
matrix analysis, and sh
using more sophistica
With the results abo
late the limits from aL
the spontaneous scala
pare them to existing
sars [16]. Shibata et
there exists a simple r
as spontaneous scalar
Ref. [19]), which in o
αA ≃
With this equation at
from ground-based GW
−4, into limits for |α0|
|α
Figure 8 gives the re
tions and the EOS AP
lead to less constraini
tors and binary pulsar
current Solar system a
constraining than wha
and ET, only inspiral
provide constraints th
FIG. 6. The sensitivities o
the uncertainty, σ (|∆α|), o
are depicted with dashed li
averaged BNS inspiral sig
(mA = 1.25 M⊙, mB). The s
labeled. Luminosity distanc
sitivity to |∆α| from GW d
maximum available values o
from binary pulsars at 90%
tivity curve (dashed) is belo
detector has the potential to
this particular EOS, with BN
use the waveform param
to construct the 6 × 6 F
the Fisher matrix is the c
ters, from where we can
tions [17, 58, 68, 69].
In Fig. 6 we plot in d
obtained with three GW
an asymmetric BNS wit
and mB 1.25 M⊙, locat
masses, for example, (1.
probable masses for the n
NS binary pulsar PSR J1
CE, and ET can detect it
450, and 153, respectivel
(Shao et al. 2017)
How about future pulsar limits?
SKA tells…

BNS late inspirals/merge with GW detectors
If dynamical scalarization
occurs early enough, it could
be detectable with aLIGO
We show that this might be the
case for certain choices of
BNS masses and NS EOS
12
FIG. 9. Scalar mass as a function of orbital angular frequency for equal-mass BNS systems with masses (1.3 M⊙, 1.3 M⊙), (1.5 M⊙, 1.5 M⊙),
1.7 M⊙, 1.7 M⊙), and (1.9 M⊙, 1.9 M⊙). We use the limits on (α0, β0) at 90% CLs, given in Table II, for each EOS. The corresponding GW
requency is given along the top axis, with fGW = Ω/π. Dashed vertical lines highlight the conservative detectability criterion for aLIGO that
fDS 50 Hz, derived in [34, 78].
The sharp feature for the WFF1 EOS in the 1.9 M⊙–1.9 M⊙
system occurs because of the relatively low mass at which
spontaneous scalarization occurs for this particular EOS. We
provide a more detailed analysis of this phenomenon in Ap-
pendix B. Similarly abrupt transitions occur for other EOSs in
spontaneously scalarized stars (i.e., those with appreciable ef-
fective scalar coupling even in isolation) are demarcated as
scalarizing below 1 Hz; as noted above, these systems would
be indistinguishable to GW detectors from those that dynam-
ically scalarize below 1 Hz.
Four equal-mass examples
(vertical dashed lines show 50 Hz)
(Shao et al. 2017)
Sampson et al. 2014, PRD 90:124091
Sennett Buonanno 2016, PRD 93:124004

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