Accuracy in Measuring the Neutron Star Mass in Gravitational Wave Parameter Estimation for Black Hole-Neutron Star Binaries

Recently, two gravitational wave (GW) signals, named as GW150914 and GW151226, have been detected by the two LIGO detectors. Although both signals were identified as originating from merging black hole (BH) binaries, GWs from systems containing neutron stars (NSs) are also expected to be detected in the near future by the Advanced detector network. In this work, we assess the accuracy in measuring the NS mass ($M_{ns}$) for the GWs from BH-NS binaries adopting the Advanced LIGO sensitivity with a signal-to-noise ratio of 10. By using the Fisher matrix method, we calculate the measurement errors ($\sigma$) in $M_{ns}$ assuming the NS mass of $1 \leq M_{ns}/M_{\odot} \leq 2$ and low mass BHs with the range of $4 \leq M_{bh}/M_{\odot} \leq 10$. We used the TaylorF2 waveform model where the spins are aligned with the orbital angular momentum, but here we only consider the BH spins. We find that the fractional errors ($\sigma/M_{ns} \times 100$) are in the range of $10\% - 50\%$ in our mass region for a given dimensionless BH spin as $\chi_{bh} = 0$. The errors tend to increase as the BH spin increases, and this tendency is stronger for higher NS masses (or higher total masses). In particular, for the highest mass NSs ($M_{ns}=2~M_{\odot}$), the errors $\sigma$ can be larger than the true value of $M_{ns}$ if the dimensionless BH spin exceeds $\sim 0.6$.Comment: 5 pages, 2 figures, submitted to JKP

Testing the validity of the phenomenological gravitational waveform models for nonspinning binary black hole searches at low masses

The phenomenological gravitational waveform models, which we refer to as PhenomA, PhenomB and PhenomC, generate full inspiral-merger-ringdown waveforms of coalescing binary back holes (BBHs). These models are defined in the Fourier domain, thus can be used for fast matched filtering in the gravitational wave search. PhenomA has been developed for nonspinning BBH waveforms, while PhenomB and PhenomC were designed to model the waveforms of BBH systems with nonprecessing (aligned) spins, but can also be used for nonspinning systems. In this work, we study the validity of the phenomenological models for nonspinning BBH searches at low masses, $m_{1,2}\geq 4 Msun$ and $m_1+m_2\equiv M \leq 30 Msun$, with Advanced LIGO. As our complete signal waveform model, we adopt EOBNRv2 that is a time-domain inspiral-merger-ringdown waveform model. To investigate the search efficiency of the phenomenological template models, we calculate fitting factors by exploring overlap surfaces. We find that only PhenomC is valid to obtain the fitting factors better than 0.97 in the mass range of $M<15 Msun$. Above $15 Msun$, PhenomA is most efficient in symmetric mass region, PhenomB is most efficient in highly asymmetric mass region, and PhenomC is most efficient in the intermediate region. Specifically, we propose an effective phenomenological template family that can be constructed by employing the phenomenological models in four subregions individually. We find that fitting factors of the effective templates are better than 0.97 in our entire mass region and mostly greater than 0.99.Comment: 13 pages, 4 figures, matched to the published version in CQ

Real-space Hamiltonian method for low-dimensional semiconductor heterostructures

We present a new method for calculating electronic states in low-dimensional semiconductor heterostructures, which is based on the real-space Hamiltonian in the envelope function approximation. The numerical implementation of the method is extremely simple; all subband energy levels and envelope functions are directly obtained by a single evaluation of the heterostructure Hamiltonian matrix. We test the method in the 6- and 8-band k \cdot p models as well as in a simple parabolic one-band model and demonstrate its great accuracy. The method can be straightforwardly generalized to a general n-band k \cdot p model. We describe three different approaches within the method which make it possible to investigate the origin and removal of the spurious or unphysical solutions, which has long been an important issue in the community.Comment: 44 pages, 15 figure

Note on maximal estimates of generalized Schr\"odinger equation

In this study we extend the recent works on the pointwise convergence for the solutions of Schr\"odinger equations based on Du, Guth, and Li and Du and Zhang to generalized Schr\"odinger equations. We establish the associated maximal estimates for a general class of phase functions, which give the pointwise convergence for $f \in H^s(\mathbb R^d)$ whenever $s > \frac{d}{2(d+1)}$.Comment: 30 page

Symmetric bilinear form on a Lie algebra

Let $\frak g$ be the finite dimensional simple Lie algebra associated to an indecomposable and symmetrizable generalized Cartan matrix $C=(a_{ij})_{n\times n}$ of finite type and let $\frak d$ be a finite dimensional Lie algebra related to a quantum group $D_{q,p^{-1}}(\frak g)$ obtained by Hodges, Levasseur and Toro \cite{HoLeT} by deforming the quantum group $U_q(\frak g)$. Here we see that $\frak d$ is a generalization of $\frak g$ and give a $\frak d$-invariant symmetric bilinear form on $\frak d$

Application of the effective Fisher matrix to the frequency domain inspiral waveforms

The Fisher matrix (FM) has been generally used to predict the accuracy of the gravitational wave parameter estimation. Although a limitation of the FM has been well known, it is still mainly used due to its very low computational cost compared to the Monte Carlo simulations. Recently, Rodriguez et al. [Phys. Rev. D 88, 084013 (2013)] performed Markov chain Monte Carlo (MCMC) simulations for nonspinning binary systems with total masses $M \leq 20 M_{\odot}$, they found systematic differences between the predictions from FM and MCMC for $M>10 M_{\odot}$. On the other hand, an effective Fisher matrix (eFM) was recently introduced by Cho et al. [Phys. Rev. D 87, 24004 (2013)]. The eFM is a semi-analytic approach to the standard FM, in which the partial derivative is taken by a quadratic fitting function to the local overlap surface. In this work, we apply the eFM method to several nonspinning binary systems and find that the error bounds in eFM are qualitatively in good agreement with the MCMC results of Rodriguez et al. in all mass regions. In particular, we provide concrete examples showing an importance of taking into account the template-dependent frequency cutoff of the inspiral waveforms.Comment: 13 pages, 5figures; final version accepted for publication in CQG; changed significantly from v

Validity of the Effective Fisher matrix for parameter estimation analysis: Comparing to the analytic Fisher matrix

The effective Fisher matrix method recently introduced by Cho et al. is a semi-analytic approach to the Fisher matrix, in which a local overlap surface is fitted by using a quadratic fitting function. Mathematically, the effective Fisher matrix should be consistent with the analytic one at the infinitesimal fitting scale. In this work, using the frequency-domain waveform (TaylorF2), we give brief comparison results between the effective and analytic Fisher matrices for several non-spinning binaries consisting of binary neutron stars with masses of (1.4, 1.4)M_sun, black hole-neutron star of (1.4, 10)M_sun, and binary black holes of (5, 5) and (10, 10)M_sun for a fixed signal to noise ratio (SNR=20) and show a good consistency between two methods. We also give a comparison result for an aligned-spin black hole-neutron star binary with a black hole spin of \chi=1, where we define new mass parameters (Mc, \eta^-1, \chi^7/2) to find good fitting functions to the overlap surface. The effective Fisher matrix can also be computed by using the time-domain waveforms which are generally more accurate than frequency-domain waveform. We show comparison results between the frequency-domain and time-domain waveforms (TaylorT4) for both the non-spinning aligned-spin binaries.Comment: 8 pages, 3 figure

Gravitational Wave Searches for Aligned-Spin Binary Neutron Stars Using Nonspinning Templates

We study gravitational wave searches for merging binary neutron stars (NSs). We use nonspinning template waveforms towards the signals emitted from aligned-spin NS-NS binaries, in which the spins of the NSs are aligned with the orbital angular momentum. We use the TaylorF2 waveform model, which can generate inspiral waveforms emitted from aligned-spin compact binaries. We employ the single effective spin parameter $\chi_{\rm eff}$ to represent the effect of two component spins ($\chi_1, \chi_2$) on the wave function. For a target system, we choose a binary consisting of the same component masses of $1.4 M_{\odot}$ and consider the spins up to $\chi_i= 0.4$, We investigate fitting factors of the nonspinning templates to evaluate their efficiency in gravitational wave searches for the aligned-spin NS-NS binaries. We find that the templates can achieve the fitting factors exceeding $0.97$ only for the signals in the range of $-0.2 \lesssim \chi_{\rm eff} \lesssim 0$. Therefore, we demonstrate the necessity of using aligned-spin templates not to lose the signals outside that range. We also show how much the recovered total mass can be biased from the true value depending on the spin of the signal.Comment: 4 pages, 2 figure

Semiclassical limits of Ore extensions and a Poisson generalized Weyl algebra

We observe \cite[Proposition 4.1]{LaLe} that Poisson polynomial extensions appear as semiclassical limits of a class of Ore extensions. As an application, a Poisson generalized Weyl algebra $A_1$ considered as a Poisson version of the quantum generalized Weyl algebra is constructed and its Poisson structures are studied. In particular, it is obtained a necessary and sufficient condition such that $A_1$ is Poisson simple and established that the Poisson endomorphisms of $A_1$ are Poisson analogues of the endomorphisms of the quantum generalized Weyl algebra.Comment: 10 page

On inhomogeneous Strichartz estimates for fractional Schr\"odinger equations and their applications

In this paper we obtain some new inhomogeneous Strichartz estimates for the fractional Schr\"odinger equation in the radial case. Then we apply them to the well-posedness theory for the equation $i\partial_{t}u+|\nabla|^{\alpha}u=V(x,t)u$, $1<\alpha<2$, with radial $\dot{H}^\gamma$ initial data below $L^2$ and radial potentials $V\in L_t^rL_x^w$ under the scaling-critical range $\alpha/r+n/w=\alpha$.Comment: To appear in Discrete Contin. Dyn. Syst., 22 pages, 2 figure