Inverse-Chirp Imprint of Gravitational Wave Signals in Scalar Tensor Theory

The scalar tensor theory contains a coupling function connecting the quantities in the Jordan and Einstein frames, which is constrained to guarantee a transformation rule between frames. We simulate the supernovae core collapse with different choices of coupling functions defined over the viable region of the parameter space and find that a generic inverse-chirp feature of the gravitational waves in the scalar tensor scenario.


I. INTRODUCTION
The detection of gravitational waves (GWs) by kilometric-size laser-interferometer systems such as the Laser Interferometric Gravitational wave Observatory (LIGO) in US, Virgo in Italy and the KAmioka GRAvitational Wave Detector (KAGRA) in Japan, will initiate the test of gravitational theories. Particularly, the gravity effects in the strong-field regime can be verified through the observations, where the underlying gravity theories may deviate from General Relativity (GR). In practice, several alternative theories of gravity have been proposed. Among them, the scalar tensor (ST) theory is the most natural extension to GR, in which gravity can be mediated by a scalar field in addition to the metric one.
This additional field introduces the spontaneous scalarization phenomenon, which is a nonperturbative deviation from GR, within the gravitation field of neutron stars [1]. In a recent work [2], it has been shown that one can observe the presence of such phenomenon in the supernova core collapse scenario as well.
Furthermore, it has been proved by Damour and Esposito-Farèse [1,3] that under the assumption that there exists a transformation rule between Jordan and Einstein frames, a two-parameter family of the ST theory is sufficient to parametrize the most general post-Newtonian deviations from GR, and to include nonperturbative strong-field effects. Many works have also been done along this direction [2,[4][5][6][7][8]. However, several problems related to the assumption have been discussed in the literature [9][10][11][12][13][14][15]. The transformation between two frames includes a Weyl transformation of metric and a redefinition of the scalar field, dφ/dϕ. It has been demonstrated in [14,15] that there is a criterion concerning about the scalar field redefinition, which leaves a constrain on the parameter space of the coupling function α(ϕ, α 0 , β 0 ) defined in the Einstein frame [15]. In light of such criterion/constrain, it is prospective to further constrain the ST theory with the signals of gravitational waves (GWs). To capture the features of those signals, we consider the stellar core collapse systems in the massive ST scenario.
In this work, we first express how the aforementioned criterion manifests itself as a constraint on the parameter space. We then numerically simulate the supernovae core collapse in the viable region of the parameter space by using the code introduced in [2] to study the profile of the genuine strong-field effects.
The paper is organized as follows. In Sec. II, we briefly introduce the theoretical frame-work of the ST theory and discuss the constraint of the scalar field. The numerical simulations of the supernovae core collapse are represented in Sec. III. Our conclusions are given in Sec. VI.

II. SCALAR TENSOR THEORY
The ST theory can be formulated in both Jordan and Einstein frames, which are conformally related. In the Jordan frame, the action takes the form where F (φ) and ω(φ) are the regular coupling functions of the scalar field φ, and S m corresponds to the action of ordinary matter. It has been revealed since the original Brans-Dicke paper [16,17] that another formulation of the theory is possible. Through a Weyl transfor- where g ⋆ µν is the transformed metric and A(φ) is the coupling function defined by F = A −2 , and a redefinition of the scalar field [15] dϕ dφ one can recast the theory into the so-called Einstein frame, in which the action is given by where V (ϕ) := A 4 U(φ)/4. As a result, the field equations are the usual Einstein equations with the scalar field as a source together with an equation of motion of the scalar field, where and T ⋆ := g ⋆µν T ⋆ µν with the stress energy tensor In the mathematical viewpoint, having the scalar field ϕ in the Einstein frame to be viable in the Jordan frame, one should be able to represent ϕ as a function of φ. Subsequently, the existence of φ(ϕ) indicates that [15] Hence, the solution to the scalar equation in the Einstein frame must satisfy (8). Otherwise, it is not a solution to the scalar equation in the Jordan frame.
In this paper, we adopt the conformal factor A(ϕ) as discussed in [1,4,5,18], given by where ϕ 0 is the asymptotic value of ϕ at spatial infinity. The constants α 0 and β 0 are defined as The coupling function in (9) leads to and By substituting (11) into (5b), one can get the equation of motion where we define the square of the effective mass for ϕ as In [1,18], Damour and Esposito-Farse described the dramatic deviation from GR for some specific values of the coupling constants, dubbed as "spontaneous scalarization." To trigger this sudden behavior of the scalar field, β 0 should be smaller than a specific value, which is −4.35 for the static neutron stars, while it would increase a bit but still negative [19] for the rotating ones. In this work, we set α 0 > 0 and β 0 < 0.
The non-vanishing property of (8) together with the definition (6) implies that the parameter α can never be zero, which results in that there is a critical value for ϕ by (11), This shows that the solution space of the scalar field in the Einstein frame is divided into two disconnected branches by the value of ϕ c with ϕ = ϕ c to be a no-crossing line for the value of the scalar field in the Einstein frame.
The metric in the Jordan frame is partially determined by ϕ, which is a field in the Einstein frame. Technically speaking, once one ensures that the signals from the simulation are reversible to the Jordan frame, the measurable amplitude of the scalar signal can be expressed as where and which are the breathing and longitudinal modes of GWs, respectively [20]. The amplitude of the longitudinal mode h L is proportional (up to a sign) to h B , and the coefficient depends on the Compton length of the scalar field, ω comp := m eff / , hence to the effective mass term of ϕ. As a result, h s is clearly proportional to ϕ. We shall show this amplitude has an inversechirp evolution along with ϕ, which is the key profile of the additional modes of GW in the massive ST theory. Practically, the GW signals from the stellar collapse in the theory are monochromatic soon after its emission, which are likely detectable with the ground-based GW detectors [2].

III. NUMERICAL SIMULATIONS
We follow the method used in [2]. The supernovae core collapse is considered with the spherically symmetric metric of the form where all metric functions depend only on the coordinates r and t. We take matter as a perfect fluid, whose energy-momentum tensor in the Jordan frame can be expressed in the spherical coordinate as where ρ, p and u µ are the energy density, pressure and 4-velocity of matters, respectively.
The enthalpy is defined by where ǫ is the internal energy. The quantities are connected to those in the Einstein frame, noted with the asterisk, via the relations The field equations are solved numerically by utilizing the modification of the code introduced in [2], which is developed from GR1D [21]. In the simulation, the hybrid equation of state (EOS) is used to account for the stiffening of the nuclear and to model the response of the shocked material by the forms of p = p c + p th and ǫ = ǫ c + ǫ th with the thermal effects, where the cold parts of pressure and internal energy are given as where ρ nuc = 2 × 10 14 g/cm 3 and K 1 = 4.9345 × 10 14 [cgs] with K 2 and E 3 naturally followed by the continuity. An additional relation to close up Eqs. (22) and (23) is given by Thus, we have three parameters (Γ 1 , Γ 2 , Γ th ), set to be (1.3, 2.5, 1.35), to specify the EOS.
As a first study of the influence of the constraint of (8), we will present the results for a specific progenitor of the supernova core collapse, which is coded as WH20 in [23] and pave the density of the atmosphere being 2 g/cm 3 outside the progenitor. Note that our methodology can be generalized to all systems.
We consider the action where the coupling function A(ϕ) is given in (9). The effective mass can be obtained from (14) If GWs are to be detectable inside the LIGO sensitivity window, the mass should be bounded above by 10 −13 eV since the low-frequency modes of GWs with ω < ω comp will damp out instead of radiating outward to infinity [22]. In addition, the mass less than 10 −15 eV would not be able to generate the strong scalarization and satisfy binary pulsar constraints [23,24] at the same time. Hence, we fix the mass to be 10 −14 eV hereafter.
In principle, one can define the coupling functions by fixing α 0 and β 0 in (11) with ϕ 0 = 0 [2,15] to carry out the simulation. To illustrate our results, we consider the set containing pairs (α 0 , β 0 ) with a constant ratio of −k, namely It is clear that S k is defined by a certain critical value. One can view the solutions of the scalar field for (α 0 , β 0 ) within S k as an one-parameter family curve by choosing β 0 as the parameter for the later analysis. In Fig. 1, we show that the amplitudes of the GW signals at the distance of 5 × 10 9 cm away from the stellar core are too small by comparing with the critical value k = 0.05, where ϕ = k = 0.05 is a horizontal line far above the signals on the plot. One can further notice that the signals for the cases with β 0 = −2 and −4 are obviously different from the others since they do not or barely possess the second twist before reaching the peak around 0.3s. This will be explained next.
As introduced in [25], the amplitudes of the signals at the wave zone of their propagations with different Compton lengths of the scalar field have an approximate universal relation. If we ignore the source term of −4πGα 0 T ⋆ in (13), the Compton length would be a function of β 0 , so that the signals within the same window S k have a homologous form. This universality will be adulterated due to the presence of −4πGα 0 T ⋆ , which may be measured by the absolute value of the ratio between the coefficients of the zeroth and first order terms in O(ϕ) on the right hand side of (13), given by where γ = m 2 /T ⋆ . Consequently, we have that δ ≈ −k = −0.05 for S 0.05 as the linear term dominates, and δ → 0 − otherwise. The behaviors of δ as a function of β 0 with several values of γ (γ = 1) are plotted on the left (right) panel of Fig. 2. For the signals in Fig. 1, our simulation on the right panel of Fig. 2 shows that there are deviations of 0.79 − 1.31% from the first order dominating limitation when β 0 = −6, −8 and −10, whereas they are about twice even three times as much as the cases with β 0 = −2 and −4. As seen in Fig. 2, one can graphically notice that the homologous form has been distorted for the later two cases.
For S 1×10 −5 shown in Fig. 3, the amplitudes of the GW signals at the same extraction distance are comparable to the critical value ϕ = k = 1 × 10 −5 , and hence the constraint is more stringent in this case. Any solution crossing the dashed line ϕ = k will be ruled out.
In these cases in Fig. 3, δ is confined in 0.20 − 0.26% with respective to −k = −1 × 10 −5 , which is small enough so that the shapes of the signals do not deviate much. From Figs. 1  Fig. 1, but with ϕ c = k = 1 × 10 −5 , where the scalar field labeled by β 0 = −40 is ruled out by the argument in the context. and 3, one can observe that as β 0 decreases, the peaks of the curves increase accordingly until they touch the non-crossing line. We then define the corresponding parameter as the critical value of β 0 , denoted by β c . Therefore, there exists a value of β c such that the peak of ϕ reaches the value of ϕ c . Consequently, any case with |β 0 | > |β c | will be forbidden due to its crossing with the line of ϕ = ϕ c .
In Fig. 4, we select seven values of 1 × 10 −5 , 8 × 10 −6 , 7 × 10 −6 , 5 × 10 −6 , 2.5 × 10 −6 , 1 × 10 −6 , and 5 × 10 −7 for k as the constraints on β 0 . The parameter space is split into two pieces bounded by the solid line, in which the parameters in the left (yellow) region are ruled out. This is to say, the Jordan and Einstein frames do not correspond to each other in the shaded area in Fig. 4. We note that the shade area will change for the different initial data/progenitors.
Moreover, within the same parameter set, the amplitude of the scalar field with a larger |β 0 | or α 0 is bigger. The contribution of the scalar field in the Einstein frame to the gravitational waveform in the Jordan frame comes through Eq. (5.6) of [3] 2α 0 ϕη µν , which implies that the scalar field ϕ affects more as α 0 increases. However, to some extend, the solution will touch the no-crossing line and this very α 0 puts the upper limit for the detectability of the contribution of the scalar field in GWs.
We have simulated the supernovae core collapse and found the generic feature of scalar GWs in the ST scenario with V = (1/2)mϕ 2 in the Einstein frame, which has an inversechirp behavior. We have shown that the ST theory should be defined in the viable region for the parameter space in order to have the signal to be recognized as GWs in the Jordan frame.
In particular, to ensure that one can transform fields from the Jordan frame to Einstein fone, and vice versa, there is a constraint on the parameter space in the ST theory. For the supernovae core collapse, we have illustrated the upper bound on −β 0 for each S k in (27).
Using the bounds for the different sets of S k , we have obtained the viable region for the parameters in the particular ST theory. In this area of the parameter space, we have carried out the numerical simulations with several pairs of (α 0 , β 0 ).
Even though the intrinsic amplitude of the scalar field is insensitive to (α 0 , β 0 ) [2], the measurable scalar signal with the amplitude h s in terms of ϕ is closely related to the Weyl transformation associated with the viable region of these parameters. Furthermore, the constraint of the parameter would affect the contribution of the scalar mode in the gravitational waveform of the tensor mode, which has been shown in (29). Therefore, we have demonstrated that the inverse-chirp profile of the GW signals is generic so that it can be used as a probe to test the ST theory.