Eur. Phys. J. C (2019) 79 :178 Page 3 of 11 178
where β is a parameter. This class of STT is indistinguishable
from pure GR in the weak field regime, while exhibiting non-
perturbative effects for strong fields. We will adopt a natural
non-negative scalar potential with self-interaction
V (ϕ) = 2m
2
ϕ
ϕ
2
+ λϕ
4
, (3)
where m
ϕ
is the mass of the scalar field ϕ and λ ≥ 0isthe
self-interaction parameter with dimensions length
−2
.
The field equations that follow from (1)are
R
μν
−
1
2
g
μν
R = 8π GT
μν
+ 2∇
μ
ϕ∇
ν
ϕ
−g
μν
g
αβ
∇
α
ϕ∇
β
ϕ −
1
2
V (ϕ)g
μν
, (4)
∇
μ
∇
μ
ϕ =−4π Gα(ϕ)T +
1
4
dV(ϕ)
dϕ
, (5)
where ∇
μ
is the covariant derivative with respect to g
μν
,
the coupling function α(ϕ) is defined via the conformal
factor α(ϕ) =
d ln A(ϕ)
dϕ
, and T is the trace of the energy-
momentum tensor. Using the field equations and the con-
tracted Bianchi identities we derive the conservation law of
the energy-momentum tensor in the Einstein frame:
∇
μ
T
μ
ν
= α(ϕ)T ∇
ν
ϕ. (6)
The Einstein frame energy-momentum tensor T
μν
and the
Jordan frame one
∗
T
μν
are related in the following way T
μν
=
A
2
(ϕ)
∗
T
μν
. We will consider stationary and axisymmetric
perfect fluid and scalar field configurations, for which the
transformation of the energy density ρ, the pressure p and
the 4-velocity u
μ
between the Einstein and the Jordan frame
are given as follows:
ρ = A
4
(ϕ)
∗
ρ, p = A
4
(ϕ)
∗
p, u
μ
= A
−1
(ϕ)
∗
u
μ
. (7)
Our metric ansatz is the standard one for stationary and
axisymmetric spacetime in the slow rotation approximation
[63]:
ds
2
=−e
2(r )
dt
2
+ e
2(r)
dr
2
+r
2
(dθ
2
+ sin
2
θdϑ
2
) − 2ω(r,θ)r
2
sin
2
θdϑdt. (8)
Since the rotational corrections to the metric functions
(except for ω(r,θ)), the scalar field, the fluid energy den-
sity and the pressure are of order O(
2
), where = u
ϑ
/u
t
is the angular velocity, this approximation allows us to cal-
culate the moment of inertia of the star I , while the rest
parameters, such as the mass and the radius, will coincide
with the static case. The asymptotic behavior of ω at large
distances from the star is ω → 2 J /r
3
, where J is the angular
momentum of the stars and I = J/. It turns out, though,
that it is numerically much more convenient to calculate the
moment of inertia via an integral throughout the volume of
the star that can be found in [30].
To obtain the results in the present paper, we solve numer-
ically the dimensionally reduced ODE system derived from
Eqs. (4) and (5) with the metric ansatz (8) by containing at
most terms linear in and supplementing it with the equa-
tion for hydrostatic equilibrium and an EOS. Further details
on the mathematical formulation of the problem one can find
in [30,64,65].
In the next section, where we present our numerical
results, we shall use dimensionless parameters m
ϕ
→ m
ϕ
R
0
and λ → λR
2
0
, where R
0
= 1.47664 km is one half of the
solar gravitational radius.
3 Numerical results
3.1 Preliminaries
As discussed earlier, STT with self-interacting massive scalar
field with conformal factor A(ϕ) = e
βϕ
2
/2
has three free
parameters – (β, m
ϕ
,λ) and for different sets of values, it
has varying degrees of development of spontaneous scalar-
ization. Namely, decreasing the value of β<0 increases
the deviations from GR, while the increase of either m
ϕ
or λ
suppresses them [30]. The constraints on the parameter space
are derived after confronting against the observations of the
gradual orbit contraction of binary pulsars due to the grav-
itational wave emission and the strongest constraints come
from [28,29]. The present data on the rate of orbital decay
matches very well the GR predictions, which suggests non
or negligible scalar gravitational radiation, and thus non or
very weak scalarization effect. As result, for the massless
STT, (β, m
ϕ
= 0,λ = 0), the observationally allowed val-
ues of β>−4.5 are such that possibility only for small
deviations from GR is left, since spontaneous scalarization
occurs roughly for β<−4.35 [13,15] and β<−3.9[19]
for the static and the rapidly rotating cases correspondingly.
If, however, one considers STT with massive scalar field,
(β, m
ϕ
,λ = 0), the parameter space that is in agreement
with the same binary observations is substantially expanded.
The reason is that the mass of the scalar field suppresses
the emission of scalar radiation, which reconciles already
discarded values of β. The lower boundary of the scalar field
mass can be estimated using the distance between the two
companions r
b
as follows: a negligible scalar gravitational
radiation implies that the Compton wavelength should be
much smaller than the orbital separation λ
ϕ
r
b
, which
for the observed binaries r
b
∼ 10
9
m translates into m
ϕ
10
−16
eV. The upper limit is calculated by the condition that
the mass of the scalar filed should be such that it does not
suppress the spontaneous scalarization in the stars, i.e. the
characteristic length of the star should be smaller than the
Compton wavelength, which leads to m
ϕ
10
−9
eV. Thus,
we will work with the following range of values for m
ϕ
:
10
−16
eV m
ϕ
10
−9
eV (9)
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