Overview of KAGRA : KAGRA science

The existence of massive stellar-mass BBH provides us a new insight about formation pathways of such heavy compact binaries GW150914PRL ; GW151226PRL . Since massive stars likely loose their masses by stellar winds driven by metal lines, dust, and pulsations of the stellar surface, heavy BHs with masses of $\gtrsim 20~{}M_{\odot}$ are not expected to be left as remnants at the end of their lifetime of massive stars with metallicity of the solar value LIGO_ApJ . In fact, the masses of BHs we have ever observed in EM waves (e.g., X-ray binaries) are significantly lower than those detected in GWs GWTC1 . This fact motivates us to explore the astrophysical origin of such massive stellar-mass BBH population in low-metallicity environments (below 50 % of solar metallicity or possibly less) LIGO_ApJ . So far, a number of authors have proposed formation channels; evolution of low-metallicity isolated binaries in the field Belczynski_2004 ; Dominik_2012 ; Belczynski_2016 ; Mapeli_2016 , dynamical processes in dense cluster systems (globular clusters, nuclear stellar cluster, or compact gaseous disks in active galactic nuclei) Portegies Zwart_2000 ; O'Leary_2009 ; BaeKimLee ; Rodriguez_2016a ; Antonini_2016 ; Stone_2017 ; Park_2017 , massive stars formed in extremely low-metallicity gas at high-redshift Universe (Population III stars, hereafter Pop III stars) Kinugawa_2014 ; Hartwig_2016 ; Inayoshi_2016 ; Inayoshi_2017 , and etc. As an alternative non-astrophysical possibility, a primordial BH population in the extremely early universe (e.g., originated by phase transitions, a temporary softening in the EOS, quantum fluctuations) have been attracting attention Carr_2016 ; Sasaki_2016 (Sasaki:2018dmp for a review).

In order to reveal the astrophysical origin of stellar-mass BBH detected by LIGO and Virgo, we need to explore the properties of coalescing BBH, depending on their formation pathway and environment where they form. In particular, the effective spin parameters of BBH $\chi_{\rm eff}$ (the dimensionless spin components aligned or anti-aligned with the orbital angular momentum) are expected to be useful to discriminate the evolution models Kushnir_2016 ; Rodriguez_2016b . In the field binary scenario, concerning binaries formed in a galactic plane, tidal torque exerting two stars (or one of them is a BH) in a close binary transport the orbital angular momentum into stellar spins, resulting in $\chi_{\rm eff}\gtrsim 0$. Since tidal synchronization occurs as quickly as $\dot{\Omega}\propto(R_{\star}/a)^{6}$, a binary with a short GW coalescence timescale (i.e., a small orbital separation) would be significantly spun to $\chi_{\rm eff}\sim 1$. Since BBH populations formed at high redshift take a long GW coalescence timescale, their BBH are hardly affected by tidal torque Hotokezaka_Piran_2017 . The underlying assumption is that the orbital angular momentum is not significantly changed via natal kicks that new-born BHs could receive during the core collapse of their progenitors Janka_2013 . Although a strong natal kick $\gtrsim 500~{}{\rm km~{}s}^{-1}$ is required to affect the effective spin parameter, observations of low-mass X-ray binaries show no evidence of such strong natal kicks of BHs (see, e.g., Mandel_2016 ).

On the other hand, in the dynamical formation scenario, concerning binaries formed in dense stellar environment, the distribution of effective spin would be isotropic at $-1\leq\chi_{\rm eff}\leq 1$, and negative values are allowed unlike the isolated binary scenario because the directions of stellar spins could be chosen randomly. As a robust result of the formation channel, the effective spin can have negative values at $-1\lesssim\chi_{\rm eff}\lesssim 0$. Moreover, in the dynamical capture model, a small fraction of the binaries/BBH could gain significantly high eccentricities ($e\gtrsim 0.1$), which will be imprinted in the GW waveform O'Leary_2009 .

For PBHs formed in the early Universe (radiation dominated era), the spin is suppressed to $\chi_{\rm eff}\lesssim 0.4$ Chiba:2017rvs or even smaller DeLuca:2019buf because they are formed from the collapses of density inhomogeneities right after the cosmological horizon entry. However, if the early matter phase exists after inflation, the spin can be large Harada:2017fjm . Because of the uncertainty in the early Universe scenario, it would be difficult to distinguish the PBH origin from the astrophysical ones.

In addition to the BBH population within a detection horizon of $z\lesssim 0.2$, BBH that coalesce at higher redshifts due to small initial separations are unresolved individually, but contribute to a GW background (GWB). The spectral shape of a GWB caused by compact binary mergers is characterized by a single-power law of $\Omega_{\rm GW}\propto f^{2/3}$ at lower frequencies ($f<f_{0}$), is flattened from it at $f\simeq f_{0}$, and has a cutoff at higher frequencies ($f>f_{0}$) Phinney_2001 . Importantly, this result hardly depends on the merger history of the BBH population qualitatively. Therefore, a GWB produced from low-redshift, relatively less massive BBH population is robustly given by $\Omega_{\rm GW}\propto f^{2/3}$ in the frequency range. This GWB signal could be detected by future observing run O5 with aLIGO and AdV LIGO_background , as shown in Fig. 2. In fact, the spectral flattening occurs outside the GWB sensitive frequency range ($f_{0}\gtrsim 100~{}{\rm Hz}$). On the other hand, a GWB caused by merger events of higher redshift, massive BBH population suggested by Pop III models are expected to be flattened at a lower frequency of $f_{0}\simeq 30~{}{\rm Hz}$ Inayoshi_2016 , where LIGO/Virgo/KAGRA are the most sensitive. A detection of the unique flattening at such low frequencies will indicate the existence of a high-chirp mass, high-redshift BBH population, which is consistent with the Pop III origin. The detailed study of the GWB would enable us to explore the properties of massive binary stars at higher redshift and the epoch of cosmic reionization.

LIGO-Virgo’s sensitivities during O1 and O2 have been proved to be efficient to detect stellar-mass BBH with a mass range between ${\cal O}$(1-100) (GWTC1, ). The typical detection frequency of these stellar-mass BBH are between 30-500 Hz. The bKAGRA’s sensitivity would be similar to aLIGO and AdV and we expect the observable sample for bKAGRA would be similar to those listed in GWTC-1 (GWTC1, ), i.e., stellar-mass BBH. The LIGO-Virgo-KAGRA (LVK) configuration would provide a better sky localization but without finding a host galaxy (via EM waves), it would be challenging to identify the location of a BBH (in a galactic disk or in a cluster, for example).

Toward the upgrades of bKAGRA, we expect the improved sensitivity toward the lower frequencies would be useful to increase the signal-to-noise ratio (SNR) for BBH populations. More observation means better number statistics in event rate estimation and distributions for underlying properties such as BH mass distribution. Furthermore, by observing earlier inspiral phase signals available at lower frequencies, it will be possible to achieve better accuracy in parameter estimation for individual masses and spin parameters. For a BBH population with small masses, say total mass below 10 $M_{\rm\odot}$, by improving the so-called ‘bucket’ sensitivity around a few hundred Hz would be most effective to increase the detectability as well as the precision of parameter estimation.

From the stellar-mass to supermassive populations, BHs are considered to have a broad mass spectrum. While a standard binary evolution scenario prefers a compact binary consisting of similar masses PostnovYungenson , some binaries may have mass ratio $1/q\equiv m_{\rm 1}/m_{\rm 2}$ much larger than 1, where $m_{\rm 1}$ is the primary mass and $m_{\rm 2}$ is the secondary mass of the binary. Dense stellar system is considered to be a factory to generate compact binaries through dynamical interactions, e.g. Benacquista . However, a toy model for the dynamic formation scenario for compact binaries still prefers BBH with $1/q<3$, e.g. BaeKimLee . Unequal-mass BBH with $1/q>$ a few are in particular interesting as the modulations in amplitude that is expected by post-Newtonian formalism in the inspiral phase become significant. Higher multiple modes (with the existence of spin(s)) are also more important for binaries with large mass ratios than equal-mass binaries Pekowsky_2013 . Therefore, detections of unequal-mass binaries require the overall improvement in the detector sensitivity corresponding to the detection frequency band in the ‘bucket.’ Indeed, the recent event, GW190412, has provided us the evidence to support the detection of the higher multiple models and proved that a broader frequency band play a crucial role for the detection LIGOScientific:2020stg .

In Table 1, we present the innermost stable circular orbit (ISCO) frequencies Bardeen:1972fi corresponding to stellar-mass BBH or BH-NS binaries with various masses. GW signals from binaries with the total mass of $\cal{O}$(10) M_⊙ or larger would be spanned between $f_{\rm low}$ of an interferometer and around the estimated $f_{\rm ISCO}$ assuming the duration of ringdown signals would be much shorter than inspiral signals. The actual detectability will depend on the sensitivity of the interferometer within the frequency range given the binary.

In Table 2 and 3, the measurement errors of the binary parameters for equal-mass BBH with $30\,M_{\odot}$ and $10\,M_{\odot}$ at $z=0.1$ are estimated with the Fisher information matrix. We assume the detector networks composed of A+, AdV+, and bKAGRA or KAGRA+ (LF, HF, $40\,{\rm kg}$, FDSQZ, or combined). The waveforms we use are the spin-aligned inspiral-merger-ringdown waveform (PhenomD) for BBH. There are significant improvements from 2G detectors to 2.5G detectors (bKAGRA, LF, HF, 40kg, FDSQZ, Combined) in the SNR and the errors. But this is not solely due to KAGRA’s contribution. The upgrade of bKAGRA to KAGRA+ among A+ and AdV+ modestly enhances the SNR and the sensitivities to the binary parameters in GW phase: the symmetric mass ratio $\eta$, the effective spin $\chi_{\rm eff}$, and the luminosity distance to a source $d_{\rm L}$. For the parameters in GW amplitude, the orbital inclination angle $\iota$ and the sky localization area $\Omega_{\rm S}$, the improvement of the errors is significant because the fourth detector is important to pin down the source direction and determine the other correlated parameters. The improvement factor depends on the configuration of the upgrade of KAGRA+. From the point of view to discriminate the formation scenarios, it would be better for KAGRA to improve the detector sensitivity at both low and middle frequencies. In addition, in order to increase the detectability of BBH coalescences at higher redshifts such as Pop III star binaries and primordial BH binaries, it is important to increase the horizon distance for BBH and analyze together with the data of a GWB measuring the spectral index and the spectral cutoff.

The direct detections of GW emission have revealed the existence of massive BHs with masses of $\gtrsim 20-50~{}M_{\odot}$, which are significantly heavier than those ever observed in X-ray binaries GW150914PRL ; LIGO_ApJ ; GWTC1 . On the other side, supermassive BHs with the order of $10^{6}-10^{9}~{}M_{\odot}$ almost ubiquitously exist at the centers of galaxies and are believed to be one of the most essential components of galaxies. However, the existence of a (binary) BH population between stellar-mass and supermassive regime, i.e., $100\lesssim M_{\rm BH}/M_{\odot}\lesssim 10^{5}$, has not been confirmed yet (though there are some candidates). The lack of such intermediate-mass binary BH (IMBBH) population is one of the most intriguing unsolved puzzles in astrophysics. Detection of GWs from IMBBH would be one of the best way to probe their existence and physical natures.

A plausible formation pathway of IMBHs is runaway collisions of massive stars in dense stellar systems (e.g., globular clusters and/or nuclear stellar clusters) Portegies_Zwart_2002 ; Freitag_2006 . In a dense young star cluster, massive stars sink down to the center due to mass segregation and begin to physically collide. During the collision processes, the most massive one gains more masses and grow to a very massive star (VMS) in a runaway fashion, and thus an IMBH with a mass of $M_{\rm IMBH}\simeq 100-10^{4}~{}M_{\odot}$ is left after gravitational collapse of the VMS. Even after IMBH formation, stellar-mass BHs (SBHs) are migrating to the central region and could form a binary system with the IMBH, which is so-called intermediate-mass ratio inspirals (IMRIs) Amaro-Seoane_2010 ; Fragione_2018 . GW emission from such IMRI systems can be detected not only by space-borne GW observatories such as LISA but also by ground-based detectors whose configurations are specifically designed to reach a better sensitivity at lower frequencies (e.g., A+ or KAGRA LF). For $M_{\rm IMBH}\sim 100-10^{3}~{}M_{\odot}$ and $M_{\rm SBH}\sim 10~{}M_{\odot}$, GWs produced from the IMRIs can be detected up to a distance of a few Gpc Shinkai_2017 . Since IMRIs systems are likely to have high eccentricities due to the formation process, the GW energy distribution is shifted to higher frequencies, increasing the SNR Amaro-Seoane_2018 . Although the IMRI coalescence rate is still highly uncertain, $R\simeq 1-30$ events yr^-1 would be expected from several theoretical studies Shinkai_2017 ; Fragione_2018 . In addition, Gurkan_2006 discussed the possibility that two IMBHs are formed in a massive dense cluster via runaway stellar collisions, and would form an IMBBH at a lower rate Amaro-Seoane_2010 .

Alternatively, formation of VMSs in extremely low-metallicity environments (Pop III stars) can initiate IMBHs with masses of $\gtrsim 100~{}M_{\odot}$ Hirano_2014 . Since a molecular cloud forming massive Pop III stars would be very unstable against its self-gravity, the cloud would be likely to fragment into massive clumps with $\gtrsim 10~{}M_{\odot}$ and leave a very massive binary system that will collapse into IMBBH with $\sim 100+10~{}M_{\odot}$ Stacy_Bromm_2013 ; Susa_2014 ; Inayoshi_Haiman_2014 . Assuming a Salpeter-like initial stellar mass function and the merger delay-time distribution to be $dN/dt\propto t^{-1}$, the merger event rate for such IMBBH is inferred as $\sim 0.1-1~{}{\rm Gpc}^{-3}~{}{\rm yr}^{-1}$, and thus $R\sim$ a few events yr^-1 Kinugawa_2014 .

The possible target source that is detectable with ground-based detectors is IMRIs. See the next section 3.2.

The existence of IMBHs in globular clusters is suggested both by observational evidence and theoretical prediction Miller:2003sc . An IMBH in a globular cluster may capture a stellar-mass compact object surrounding it through some process (e.g. two-body relaxation), and may form an intermediate mass-ratio binaries (IMRB) (where “intermediate” mass-ratio means the range between the comparable mass ratio ($\sim 1$) and extreme-mass ratio ($\lesssim 10^{-4}$)).

The captured object in an IMBR orbits the IMBH many times during the inspiral phase before it plunges into the BH. Therefore, GW signals from IMRBs contain information on the geometry around the IMBHs, which can be used to test the general relativity in the strong field, for example, the no-hair theorem of BHs and the tidal coupling between the central BH and the orbit of the captured object Brown:2006pj . Also, if the observation of IMRBs are accumulated, it may give a constraint on the event rate and population of IMBHs.

The GW waveform from the inspiral phase of an IMRB can be estimated in the stationary phase approximation by

where ${\cal M}$, $D_{L}$ and $\Psi(f)$ are the redshifted chirp mass of the binary, the luminosity distance to the source, and the GW phase, respectively Berti:2004bd ; Isoyama:2018rjb . The black (solid, dash, dot) lines in Fig.3 show the strain amplitude of GWs, $2\sqrt{f}|\tilde{h}|$, from IMBRs for three different cases, in which a BH with $10M_{\odot}$ spirals into heavier BHs with masses of $10^{2}\,M_{\odot}$, $10^{3}\,M_{\odot}$, and $10^{4}\,M_{\odot}$ (these masses include the redshift factors) and the same spin parameter of $0.7$ at the luminosity distance of 100 Mpc.

The inspiral phase ceases around certain cutoff frequency, which corresponds to the frequency just before the plunge phase. For circular orbit cases, the cutoff is estimated by the frequency for an ISCO Bardeen:1972fi . The ISCO frequency, $f_{{\rm ISCO}}$ mainly depends on the mass and spin of the central BH. If the mass decreases or the spin increases, the ISCO frequency becomes higher and the overlap with the sensitive band of ground-based detectors becomes larger.

After the inspiral phase, the orbit of the captured object changes to the plunge phase through the transition phase. The shift of the frequency during the transition is roughly estimated by $\Delta f/f_{{\rm ISCO}}\sim\eta^{2/5}$, where $\eta$ is the mass ratio Ori:2000zn . The plunge of the object to the central IMBH will induce ringdown GWs, whose frequency of the dominant mode is given by $f_{{\rm RD}}\sim 32.3\,\mbox{Hz}\times(m_{2}/10^{3}M_{\odot})^{-1}g(\chi_{2})$, where $m_{2}$ and $\chi_{2}$ are the redshifted mass and spin of the IMBH and $g(\chi_{2})$ shows the spin dependence (see Ref.Berti:2005ys for details). Since the contribution from the transition phase and the ringdown phase is smaller than that from the inspiral phase, here we do not consider them (An analysis including the transition and ringdown is shown in Huerta:2010un ).

Since IMRBs with the redshifted mass of $10^{2}$-$10^{4}M_{\odot}$ are expected to be GW sources in $0.4-40$ Hz band Miller:2008fi , some of them may be possible targets for the low frequency band of ground-based detectors. To see the dependence of the detectability on the sensitivity in lower band, here we consider the detectable luminosity distance $D_{L}^{{\rm detect}}$ for the design sensitivity of the bKAGRA and KAGRA+ in the same way in Isoyama:2018rjb .

The left panel in Fig.4 shows the dependence of $D_{L}^{{\rm detect}}$ on the redshifted mass of the central IMBH ($m_{2}$) with fixed spin parameter ($\chi_{2}=0.7$). The right panel shows the dependence of $D_{L}^{{\rm detect}}$ on the spin parameter with fixed mass ($m_{2}=10^{3}M_{\odot}$). In both figures, we fix the mass ($m_{1}=10M_{\odot}$) and spin parameter ($\chi_{1}=0$) of the captured object, and the threshold of ${\rm{SNR}}=8$. For example, the values of $D_{L}^{{\rm detect}}$ with $(m_{1},m_{2})=(10M_{\odot},1000M_{\odot})$, $(\chi_{1},\chi_{2})=(0,0.7)$ and SNR$=8$ are given as 62, 126, 16, 59, 42, 67 Mpc for bKAGRA, LF, HF, 40kg, FDSQZ, and Combined, respectively. From the figures, we find that the detectability of IMBRs with $\sim 300M_{\odot}$ can be improved significantly by the LF, while it is difficult to detect IMBRs by the HF. To detect an event, the detection volume is a crucial quantity because it is roughly proportional to the event rate. Using the detector network composed of two A+, AdV+, and bKAGRA or KAGRA+ (LF, HF, 40kg, FDSQZ, and Combined), the improvement factors of the detection volume are LF 8.29, 40kg 0.85, FDSQZ 0.30, HF 0.02, Combined 1.27 for $10^{3}\,M_{\odot}$ - $10\,M_{\odot}$ BBH with spin 0.7.

Formation scenarios of BNS are similar to those of stellar-mass BBH. One is based on the standard binary evolution in a galactic disk (e.g., Tauris:2017 ; Chruslinska:2018 ) and the other involves dynamical interactions in dense stellar environments such as globular clusters (e.g., Pooley:2003 ). By precision measurements of the NS masses and orbital parameters such as eccentricity by GW observations, it would be possible to shed lights on the origin of the BNS formation as pointed out many previous works on the BNS population including Tauris:2017 . Although the expected kick involved in the BNS formation is small Tauris:2017 , the role of the natal kick from a supernova explosion can be relatively more significant for the BNS formation than that of BBH formation. The evidence of the natal kick is observed in the known BNS in our Galaxy, such as the Hulse-Taylor binary pulsar where the eccentricity is 0.67 WeisbergNiceTaylor . Even though a binary’s orbit could be significantly eccentric at the time of formation, the BNS orbit becomes circularized quite efficiently Peters and typical BNS within the detection frequency band of KAGRA is expected to have an eccentricity below $10^{-4}$. Similar to BBH, if there are eccentric NS binaries within the KAGRA frequency band, they are more likely to be formed through stellar interactions in dense stellar environment BaeKimLee ; Rodriguez_2016a . If eccentric binaries exist, measuring eccentricity would be one useful probe to distinguish the field population (formed by a standard binary evolution scenario) and the cluster population (formed by stellar interactions). We note that the measurement of eccentricity requires a much better sensitivity toward the lower frequency band below 20 Hz, therefore it is more plausible that a third generation detector may be better suited for searching eccentric BNSs.

Another difference between NS and BH populations is the mass range. While BH mass spectrum spans over nine orders of magnitudes, the NS mass range is narrow. For example, the NS mass range should depend on the formation mechanism but is estimated, assuming the Gaussian distribution, to be $1.33\pm 0.09\,M_{\odot}$ for double NSs, $1.54\pm 0.23\,M_{\odot}$ for recycled NSs, and $1.49\pm 0.19\,M_{\odot}$ for slow pulsars, which are likely to be NSs right after their birth ozelFreire:2016 . These are based on the known NS-pulsar binaries in our Galaxy before 2016 and do not include a possibly heavy NS recently discovered by GWs Abbott:2020khf , but even if included, the mass range of NSs is much narrower than that of BHs, which is advantageous of establishing a template bank. In addition, spin distribution would be something to be compared between NS and BH populations. NS spin distribution for those in BNS is not well constrained and we expect GW observations would shed light on the NS spin distribution by parameter estimation.

As of early 2019, there are fifteen BNS known in the Galactic disk (e.g., see Table 1 in (PML2019, )), where all systems consisting of at least one active radio pulsar ATNFPulsarCatalogue . Eight of them are expected to merge within a Hubble time. Measurements of binary’s orbital decay and other post-Keplerian parameters clearly showed the effects of gravitational radiation in these NS-pulsar binaries.

GW170817 is the first BNS discovered by GW observation. It is also the first extragalactic BNS. The binary is identified as BNS based on the mass estimation, where both $m_{1}$ and $m_{2}$ estimates are consistent with expected NS masses Abbott:2018wiz . The detection of GW170817 also makes it possible to constrain the BNS merger rate. Considering the first and second observing runs, the merger rate solely based on GW observation is estimated to be 110 - 3840$\,{\rm Gpc}^{-3}\,{\rm yr}^{-1}$ at 90% confidence interval GWTC1 . GW observation can observe extragalactic BNS, which are not accessible with current radio telescopes. The GW and radio observation of BNS would be complimentary to reveal underlying properties of the BNS.

In addition to the BNS population, GW observation would be able to provide rich information about neutron star interior (see Secs. 4.2 and 4.3) or the formation of young NSs (see also Secs. 6.3 and 7.1) if late-inspiral, merger, or even ringdown phases are observed by future detectors.

In terms of an observation, improving sensitivities toward lower and higher frequencies have significant implications for understanding astrophysics of BNS. A typical chirp mass of a BNS is $\approx 1.13\,M_{\odot}$ assuming $m_{1}=m_{2}=1.3\,M_{\odot}$. This implies the frequencies of merger and ringdown phases would be around 2 kHz where the current generation of GW detectors are not sensitive. However, the merger and ringdown phases of BNS coalescence would provide crucial hints on the remnant of the merger. On the other hand, the lower cutoff frequency of a detector is crucial to measure the early inspiral phase.

Effects of improved sensitivity of a detector can be considered in two folds: (a) At higher frequencies, better sensitivity would allows us to put stronger constrains for the NS EOS than what aLIGO and AdV could do for GW170817 Abbott:2018exr , not to mention improving accuracy for mass and spin parameters at higher post-Newetonian (PN) orders. Therefore, better sensitivities in high frequencies will be useful to have the observed sample of BNS via GWs as complete as possible in the mass-spin parameter space. (b) Improved sensitivity toward the lower frequencies below 20 Hz allows us observing early-inspiral signals. This would be crucial to constrain the orbital eccentricity and determine the origin of the binary formation.

BH-NS binaries are expected to exist and their evolution would be similar to those of stellar-mass BBH and BNS. As mentioned above, as there are several parameters necessary to be measured in order to discriminate the formation scenarios, we estimate with the Fisher information matrix the measurement errors of the binary parameters for a $10\,M_{\odot}$ BH-NS binary is at $z=0.06$ and a BNS at $z=0.03$. The waveform we use is the spin-aligned inspiral waveform up to 3.5PN in phase. The maximum frequency for the inspiral part is set to the ISCO frequency. We assume the detector networks composed of A+, AdV+, and bKAGRA or KAGRA+ (LF, HF, $40\,{\rm kg}$, FDSQZ, or combined). The measurement errors of the binary parameters are shown in Table 4 and 5. There are significant improvements from 2G detectors to 2.5G detectors (bKAGRA, LF, HF, 40kg, FDSQZ, Combined). But this is not solely due to KAGRA’s contribution. The upgrade of bKAGRA to KAGRA+ among A+ and AdV+ modestly enhances the SNR and the sensitivities to the binary parameters in GW phase: the symmetric mass ratio $\eta$, the effective spin $\chi_{\rm eff}$, and the luminosity distance to a source $d_{\rm L}$. For the parameters in GW amplitude, the orbital inclination angle $\iota$ and the sky localization area $\Omega_{\rm S}$, the improvement of the errors is significant because the fourth detector is important to pin down the source direction and determine the other correlated parameters. The improvement factor depends on the configuration of the upgrade of KAGRA+. From the point of view to discriminate the formation scenarios, it would be better for KAGRA to improve the detector sensitivity at both low and middle frequencies.

BNS mergers provide us rich information to study nuclear astrophysics. Since NSs consist of ultra-dense matter Lattimer:2015nhk . In the BNS system, a NS is deformed by the tidal field generated by the companion star in the late-inspiral stage Flanagan:2007ix ; Hinderer:2007mb ; Damour:2012yf . The tidal deformability originating from the matter effect encodes the information of NS EOS. Therefore, the measurement of the tidal deformabilities from GWs of the BNS mergers provides the information about nuclear physics Wade:2014vqa ; Hinderer:2009ca ; Hotokezaka:2011dh . In this section, we investigate the ability to measure the tidal deformability of NSs with current and future detector networks including bKAGRA and KAGRA+ in addition to the A+ and AdV+.

During the late stages of the BNS inspiral, at the leading order, the induced quadrupole moment tensor $Q_{ij}$ is proportional to the external tidal field tensor ${\cal E}_{ij}$ as $Q_{ij}=-\lambda{\cal E}_{ij}$. The information about the NS EOS can be quantified by the tidal deformability parameter $\lambda=(2/3)k_{2}R^{5}/G$, where $k_{2}$ is the second Love number and $R$ is the stellar radius Flanagan:2007ix ; Hinderer:2007mb . The leading-order tidal contribution to the GW phase evolution appears through the binary tidal deformability Wade:2014vqa

which is a mass-weighted linear combination of the both component tidal parameters, where $\Lambda_{1,2}$ is the dimensionless tidal deformability parameter $\Lambda=G\lambda[c^{2}/(Gm)]^{5}$. It is first imprinted at relative 5PN order.

The first detection of a GW signal from a BNS system, GW170817 GW170817PRL , provides an opportunity to extract information about NSs via measuring the tidal deformability. The LIGO-Virgo collaboration has placed an upper bound on the tidal deformability as $\tilde{\Lambda}\leq 800$ when restricting the magnitude of the component spins GW170817PRL using the restricted TF2 model Buonanno:2009zt ; Blanchet:2013haa (this limit is later corrected to be $\tilde{\Lambda}\leq 900$ in Ref. Abbott:2018wiz ). In Ref. Abbott:2018wiz ; GWTC1 , the updated analysis by the LIGO-Virgo collaboration by using a numerical-relativity (NR) calibrated waveform model, NRTidal Dietrich:2017aum have been reported. By using EOS-insensitive relations among several properties of NSs Yagi:2016bkt , the constraints on $\tilde{\Lambda}$ can be improved Abbott:2018exr (but see also Kastaun:2019bxo ). As found in Refs. Abbott:2018wiz ; LIGOScientific:2019eut , the stiffer EOSs (large radii) yielding larger tidal deformabilities such as MS1 and MS1b Mueller:1996pm are disfavored by the data of GW170817. Independent analyses have also been done by assuming the common EOS for both NSs De:2018uhw ; Capano:2019eae . In Ref. Narikawa:2019xng , the authors indicate that there is a difference in estimates of $\tilde{\Lambda}$ for GW170817 between NR calibrated waveform models (the KyotoTidal and NRTidalv2 models). Here, the KyotoTidal model is one of NR calibrated waveform models of inspiraling BNS Kiuchi:2017pte ; Kawaguchi:2018gvj and NRTidalv2 model is an upgrade of the NRTidal model Dietrich:2019kaq . Several other studies have also derived constraints on the NS EOS via measuring tidal deformability from GW170817 Margalit:2017dij ; Bauswein:2017vtn ; Ruiz:2017due ; Annala:2017llu ; Zhou:2017pha ; Fattoyev:2017jql ; Paschalidis:2017qmb ; Nandi:2017rhy ; Most:2018hfd ; Raithel:2018ncd ; Landry:2018prl . A review of these and other results are available in Baiotti:2019sew .

In order to investigate the expected error in the measurement $\tilde{\Lambda}$, we use the Fisher matrix analysis with respect to the source parameters $\{{\rm ln}d_{L},t_{c},\phi_{c},{\cal M},\eta,\tilde{\Lambda}\}$, where $d_{L}$ is the luminosity distance, $t_{c}$ and $\phi_{c}$ are the time and phase at coalescence, ${\cal M}$ is the chirp mass, and $\eta$ is the symmetric mass ratio. We use the restricted TF2_PNTidal model, which employ the 3.5PN-order formula for the phase and only the Newtonian-order evolution for the amplitude as the point-particle part Buonanno:2009zt ; Blanchet:2013haa and the 1PN-order (relative 5+1PN-order) tidal-part phase formula (see e.g., Damour:2012yf ). We take the frequency range from 23 Hz to the ISCO frequency. Here, we consider non-spinning binary for simplicity.

In Table 6, we show the fractional errors in the tidal deformability $\Delta\tilde{\Lambda}/\tilde{\Lambda}$ in the cases of some NS EOS models, using the inspiral PN waveform, the TF2_PNTidal model. We assume future detector networks including bKAGRA and KAGRA+ in addition to two A+ and AdV+. We assume 1.35$M_{\odot}$-1.35$M_{\odot}$ BNS located at a distance of 100 Mpc and study three different NS EOS models Takami:2014tva : APR4 Akmal:1998cf ($\tilde{\Lambda}$=321.7), SLy Douchin:2001sv ($\tilde{\Lambda}$=390.2), and high-$\tilde{\Lambda}$ ($\tilde{\Lambda}$=1000). In one example, for the SLy, the fractional error on the tidal deformability are 0.069, 0.070, 0.061, 0.068, 0.067, and 0.062 for bKAGRA, LF, HF, 40kg, FDSQZ, and Combined, respectively. We find that the measurement precision of the tidal deformability can be slightly improved for the HF, while cannot be done for the LF. We also present the fractional errors in the tidal deformability for the single detector case in Table 7 to see how much KAGRA can contribute to the global networks.

In Ref. Narikawa:2018yzt , the authors have found a discrepancy in the tidal deformability of GW170817 between Hanford and Livingston detectors of aLIGO. While the two distributions look consistent with each other and also consistent with what we would expect from noise realization, the Livingston data are not very useful to determine the tidal deformability for GW170817. Their results suggest that the measurement of the tidal deformability by the third detector such as AdV or bKAGRA might be helpful in order to improve the constraint on the tidal deformability. In Ref. Narikawa:2019xng , their results indicate that the systematic error for the NR calibrated waveform models will be significant to measure $\tilde{\Lambda}$ from upcoming GW detections. It is needed to improve current waveform models for the BNS mergers.

Investigating the evolution of the object that forms after the merger of a BNS system is more difficult, but also, possibly, rewarding (see bai17 ; Duez2019 ; Baiotti:2019sew for recent reviews). In addition to the connection between such a merged object and ejecta powering the kilonova (which will be treated in detail in Sec. 11.1), strong interest in observations of the post-merger phase comes also from the fact that these would at some point yield information about the EOS of matter at densities higher (several times the nuclear density) than typical densities in inspiralling stars and also at much higher temperatures (up to $\sim 50$ MeV).

Interestingly, the post-merger phase may be luminous in GW emission, perhaps more so than the preceding merger phase, but, since post-merger GW frequencies are higher (from 1 to several kHz [116, 118]), the SNR for current and projected detectors is smaller than the pre-merger phase. However, when analyzed in different ways, such GW emission gives us a novel opportunity to prove for a hypermassive NS (HMNS) in the immediate post-merger phase (see below in this subsection). Also gravitational collapse to a rotating BH, should it occur, opens an additional window to emission from a BH-torus system at relatively lower frequencies (van19b, ).

Theoretical estimates from numerical simulations of emission from the post-merger HMNS are less reliable than those of the inspiral, because such simulations are intrinsically more difficult to carry out accurately. This is due to the presence of strong shocks, turbulence, large magnetic fields, various physical instabilities, neutrino cooling, viscosity and other microphysical effects. Currently there exist no reliable determinations of the phase of post-merger gravitational radiation, but only of its spectrum (see, e.g., Takami:2014zpa ; Maione2017 ; Paschalidis_and_Stergioulas_2017 ; Breschi2019 ).

Measuring even just the main frequencies of the post-merger waveform from the putative HMNS may, however, give precious hints on the supranuclear EOS. It has been found, in fact, that the frequencies of the main peaks of the post-merger power spectrum strongly correlate with properties (radius at a fiducial mass, compactness, tidal deformability, etc.) of a zero-temperature spherical equilibrium star (see, among others, Bauswein2011 ; Takami:2014zpa ; Maione2017 ; Paschalidis_and_Stergioulas_2017 and, for applications, Bose2017 ; Chatziioannou2017 ; Torres-Rivas2019 ; Breschi2019 ). For example, finding the tidal deformability of the post-merger object to be very different from that estimated from the inspiral may hint at phase transitions in the high-density matter (see Baiotti:2019sew for a review). A similar hint would be given by an abrupt change in the post-merger main frequency Weih2020 ; Bauswein2020 . The complicated morphology of these post-merger signals makes constructing accurate templates challenging and thus matched-filtering less efficient Breschi2019 .

In addition to predictions on the GW spectrum, simulations also show, in most cases, delayed gravitational collapses to a rotating BH with dimensionless spin parameter $a/M\simeq 0.76-0.84$ (bai08, ), surrounded by a disc. Strong magnetic fields (rez11, ) and baryon-rich disk winds are also predicted, though with inferior accuracy, and they are prescient to kilonova light curves (kiu15, ).

However, simulations cannot accurately predict the time of delay in gravitational collapse of the putative HMNS to a BH, representative for the lifetime of the HMNS initially supported against collapse by differential rotation. The lifetime of the HMNS may be measured in EM-GW observations. Furthermore, because of constraints in computational resources, simulations of post-merger BH-disk or torus systems are limited to tens of ms (e.g. bai17, ; rad18, ). Yet, rotating BHs provide a window to powerful emission over a secular Kelvin-Helmholtz time-scale $\tau_{\rm KH}=E_{J}/L_{H}$, where $E_{J}=2Mc^{2}\sin^{2}(\lambda/4)$   $\left(\sin\lambda=a/M\right)$ is the spin-energy in angular momentum $J$ of a Kerr BH of mass $M$ (ker63, ), $c$ is the velocity of light and $L_{H}$ is the total BH luminosity most of which may be irradiating surrounding matter (van99, ). For gamma-ray bursts, canonical estimates show $\tau_{\rm KH}$ to be tens of seconds (van19b, ).

EM-GW observations promise to fill the missing links in our picture of BNS mergers, namely the lifetime of the HMNS in delayed gravitational collapse to a BH (e.g. dep19a, ; kli19, ). If finite, continuing emission may extend over the lifetime of BH spin by $\tau_{\rm KH}$ above. For GW170817, if the HMNS experienced delayed collapse, the merger sequence

offers a unique window to post-merger EM-GW emission powered by the energy reservoir $E_{J}$ of the remnant compact object. Crucially, gravitational collapse $b$ to a BH in the process (3) can increase $E_{J}$ significantly above canonical bounds on the same at formation $a$ of the progenitor HMNS (hae09, ) in the immediate aftermath of the merger.

In the following subsection, we will focus only on calorimetric studies of the post-merger object, leaving the other topics to the above-mentioned reviews bai17 ; Duez2019 ; Baiotti:2019sew .

Post-merger EM-calorimetry on GRB170817A and the kilonova AT 2017gfo con17 ; sav17 ; moo18a ; moo18b show a combined post-merger energy in EM radiation limited to ${\cal E}_{\rm EM}\simeq 0.5\%M_{\odot}c^{2}$. This output is well-below the scale of total energy output in GW170817, and is insufficient to break the degeneracy between a HMNS or BH remnant.

Gravitational radiation provides a radically new opportunity to probe a transient source which, “if detected, promises to reveal the physical nature of the trigger” cut02 . GW-calorimetry builds on earlier applications of indirect calorimetry to, notably, PSR1913+105 hul75 and pulsar wind nebulae wei78 . Power-excess methods GW170817:GRB ; abb19a have thus far proven ineffective, however, by a threshold of ${\cal E}_{\rm gw}=6.5\,M_{\odot}c^{2}$ sun19 , given a total mass $M<3M_{\odot}$ of the BNS progenitor and hence the mass of its remnant compact object.

Independent EM and GW observations corroborate the process (3) with a lifetime $t_{b}-t_{a}$ of the HMNS of less than about one second (van19b, ; gil19, ). In EM, the lifetime of the HMNS is inferred from an initially blue component in the kilonova AT 2017gfo  sma17 ; pia17 ; rad18 ; poo18 ; gil19 ; luc19 , whereas the same is inferred from the time-of-onset $t_{s}$ of post-merger GW-emission in time-frequency spectrograms produced by butterfly filtering, developed originally to identify broadband Kolmogorov spectra of light curves of long GRBs in the BeppoSAX catalogue van14 ; van17 . Application to the LIGO snippet of 2048 s of H1-L1 O2 data covering GW170817 serendipitously shows observational evidence of $\sim 5\,$s post-merger gravitational radiation van19a . Subsequent signal injection experiments indicate an output ${\cal E}_{\rm gw}=(3.5\pm 1)\%M_{\odot}c^{2}$ in this extended emission (van19b, ). Its time-of-onset $t_{s}<1$ s post-merger falls in the 1.7 s gap between GW170817 and GRB170817A, satisfying causality at birth of a central engine of GRB170817A.

${\cal E}_{\rm gw}$ exceeds the maximal spin-energy of a (HM)NS hae09 , yet it readily derives from $E_{J}$ of a BH following gravitational collapse thereof at $b$ in the process (3). Moreover, aforementioned ${\cal E}_{\rm EM}$ is quantitatively consistent with model predictions of ultra-relativistic baryon-poor jets and baryon-rich disk winds from a BH-torus system, the size of which derives from the frequency of its gravitational-wave emission van19b .

At current detector sensitivities, witnessing the formation and evolution of the progenitor HMNS in the immediate aftermath of a merger is extremely challenging. Null-results on GW170817 (GW170817:PMS, ) are entirely consistent with the energetically moderate and relatively high-frequency ($>1$kHz) spectra expected from numerical simulations (bai17, ).

Once the BH forms, it has no memory of its progenitor except for total mass and angular momentum. This suggests pursuing GW-calorimetry in (3) to catastrophic events more generally, including mergers of a NS with a BH companion and nearby core-collapse supernovae (CCSNe), notably the progenitors of type Ib/c of long gamma-ray bursts (GRBs) van19c . While the latter is rare, the fraction of failed GRB-supernovae that are nevertheless luminous in gravitational radiation may exceed the local GRB rate and hence the rate of mergers involving a NS (e.g. van14, ). Blind all-sky butterfly searches may thus produce candidate signals with and without merger precursors from events within distances on par with GW170817, that may be followed up by time-slide correlation analysis and optical-radio signals from existing transient surveys of the local Universe. It also appears opportune to consider directed searches for events in neighboring galaxies (e.g. heo16, ), notably M51 ($D\simeq 7$Mpc) and M82 ($D\simeq 3.5$Mpc) each offering about one radio-loud CCSN per decade. By their close proximity, such appears of interest also independent of any association with progenitors of GRBs (and13, ; aas14, ).

For a planned KAGRA upgrade, the above suggests optimizing the broadband window of 50-1000 Hz to pursue GW-calorimetry on extreme transient sources using modern heterogeneous computing, in multi-messenger approaches involving existing and planned high-energy missions such as THESUES ama18a ; stra18a . Jointly with LIGO and Virgo, KAGRA is expected to give us a window to modeled and unmodeled transient events out to tens of Mpc.

There are several classes of continuous GWs (CWs). One major population of CW sources is systems involving a spinning NS that has an asymmetry with respect to its rotation axis Riles2017 . Potential candidates include pulsars, NSs in supernova remnants, and NSs in binary systems. In particular, an accreting NS in a low-mass X-ray binary is a promising target. In this system, the central NS accretes materials from an orbiting late-type companion star. Finite quadrupole moment of an accreting NS can be possibly induced by various processes such as the laterally asymmetric distribution of accreted material, elastic strain in the crust as well as magnetic deformation Ushomirsky2000 .

Sco X–1 is the prime target for directed searches of CWs from X-ray binaries because it is the brightest persistent X-ray binary. The source is relatively nearby ($\sim 2.8$ kpc) and its X-ray luminosity suggests that the source is accreting near the Eddington limit for a NS. If the torque-balance (i.e. balance between accretion-induced spin-up torque and the total spin-down torque due to gravitational and EM radiation) can be maintained in the binary system, Sco X–1 is a promising target for CW signals Bildsten .

Theoretical understanding of torque-balance and spin wandering effects of X-ray binaries are poorly understood Bildsten ; Mukherjee2018 . By combining CWs from X-ray binaries and multi-wavelength observations, we may track the evolution of spin torques and orbit, shedding light on disk dynamics and maximum rotation frequency of a NS in a binary system Riles2017 .

No CW signal from X-ray binaries has been observed so far. During LIGO S6, upper limits have been estimated for Sco X–1 and XTE J1751–305 while the O1 observation of Sco X–1 yields a null detection as well Meadors2017 ; Abbott2017a ; Abbott2017b . In O2, by using a hidden Markov model (HMM) to track spin wandering, a more sensitive search for Sco X–1 has been performed Abbott2019scox1 . No evidence of CW can be found in the frequency range of 60 - 650 Hz. An upper limit (95% confidence) of $h\sim 3.5\times 10^{-25}$ is placed at 194.6 Hz, which is the tightest constraint has been placed on this system so far Abbott2019scox1 .

CW signals from accreting X-ray binaries are extremely small comparing to all known compact binary mergers. Assuming a torque-balance, the GW amplitude can be linked with the X-ray flux presumably associated with the accretion rate Bildsten :

Sco X–1 is the brightest persistent X-ray binary ($F_{X}\sim 4\times 10^{-7}$ erg cm^-2 s^-1) and the expected GW signals are of the order of $h\sim 10^{-25}$ or smaller. To search for such a weak signal, we need to integrate the data over a long period of time. One major challenge for Sco X–1 is that the spin period is unknown. We therefore require to search for a broad range of frequencies. Moreover, because of the spin wandering effect due to changing accretion rates onto the NS Mukherjee2018 , it is difficult to integrate the signals with a long coherence time (hence better sensitivity). If we could discover the pulsation of Sco X–1 via X-ray observations in the future, it will narrow the parameter space enhancing our chance to make the first detection of CWs from an X-ray binary.

Apart from Sco X–1, there are other luminous X-ray binaries such as GX 5–1 and GX 349+2. However, their X-ray flux is at least an order of magnitude lower than that of Sco X–1, making them even more difficult to be detected with current GW detectors. Although Sco X–1 is still the only promising persistent X-ray binary for CW signals, we may detect CWs from a very luminous outburst of a NS X-ray binary in the future. For example, the X-ray outbursts from Cen X–4 are even brighter than Sco X–1 although they only lasted for about a month Kaluzienski1980 .

In order to optimise the search for CWs from X-ray binaries and Sco X–1 in particular, the best frequencies should be from a few tens to a few hundreds Hz. A stable (with high duty cycle) long-term (months to years) observing run is required to integrate the data for the weak signals of the order of $h\sim 10^{-25}$ or smaller. Unless we know the spin frequency of the NS from EM observations, we have to search for a large frequency range implying a high demand of computation time. In any case, better computing algorithms have to be developed along with implementation of GPU computation in order to increase the detection efficiency Messenger2015 ; Meadors2018 . If we define the SNR ratio averaged over relevant frequencies, we can quantitatively compare performances of various configurations of possible KAGRA upgrades. We define the ratio $r_{\alpha/\beta}\equiv r_{\alpha}/r_{\beta}$ for the configurations $\alpha$ and $\beta$ where

and a similar equation holds for the $\beta$ configuration. In the case of an accreting X-ray binary, for a typical frequency range between $f_{\rm low}=30$ Hz and $f_{\rm high}=200$ Hz, the SNR ratios of KAGRA+ to bKAGRA are LF 0.04, 40kg 1.62, FDSQZ 1.38, HF 0.89, Combined 2.25.

Recently, a novel methodology of searching CWs by deep learning has been proposed Dreissigacker2019 . Since the CW is expected to be weak, it is necessary to integrate the data with very long time span to result in SNR above the detection threshold. This poses a computational challenge to the traditional coherent matched-filtering search. First, it is difficult to apply such method on a data span longer than weeks. Moreover, as we do not know the spin period and period derivatives precisely, we have to perform blind search in large parameter space at the same time which makes the problem more computationally demanding. On the other hand, once a deep learning network has been trained, prediction on any inputs can be executed very fast which is very favorable for searching CWs. Although the sensitivity of the first proof-of-principle deep learning CW search is far from being optimal Dreissigacker2019 , it does demonstrate an excellent ability in generalization. Further investigation on this methodology (e.g. exploring which network architecture is optimal for CW search) can be fruitful.

GWs that last more than $\sim$ 30 minutes, when we search for them using ground-based GW detectors, are susceptible to Doppler modulation due to the Earth spin and orbital motion. Those long-lasting GWs are called “continuous” GWs, or CWs.

Spinning stars with non-axisymmetric deformations may emit CWs. Pulsars may have such non-axisymmetric deformations and are good candidates for GW observatories. It is estimated that there are roughly $\sim$ 160,000 normal pulsars and $\sim$ 40,000 millisecond pulsars in our galaxy. As of writing this paper, the ATNF pulsar catalogue ATNFPulsarCatalogue includes $\sim$ 2800 pulsars among which $\sim$ 480 pulsars ($\sim$ 17 %) spin at faster than 10 Hz. While $\sim$ 10 % of pulsars are in binaries, the fraction increases to more than 57 % when limited to pulsars with $f_{\rm spin}\geq 10$Hz. The Square Kilometer Array (SKA) is expected to find much more pulsars in near future Smits_2011 .

A rapidly spinning NS may emit GWs (roughly) at its spin frequency $f_{\rm spin}$, 4/3, and/or twice of it, depending on emission mechanisms. If NS emits at $\sim f_{\rm spin}$, it may mean the NS is “wobbling” (freely precessing). Detection of “wobbling mode” CW gives information on interaction between crust and fluid core of the NS. If a NS has a non-axisymmetric mass quadrupole deformation, it may emit CWs at $2f_{\rm spin}$. CW frequency can be different from twice the spin frequency estimated from an EM observation, if a component producing EM radiations does not completely couple with that producing GWs. We sometimes call the $2f_{\rm spin}$ mode “mountain mode”. “Mountains” on a star may be due to, e.g., fossil deformations developed during NS formation supported by crustal strain and/or strong magnetic field within the NS, or thermal gradient (in case of an accreting NS in a binary). If both GWs of the wobbling mode and the mountain mode are detected from a single NS, one may be able to determine its mass OnoEdaItoh . R-mode may be unstable within a young NS and it may emit CWs at $\sim 4f_{\rm spin}/3$. Detection of “r-mode” CW gives us information on evolution history of the star, as damping time-scale depends on interior temperature. See, e.g., Paschalidis_and_Stergioulas_2017 ; Glampedakis_and_Gualtieri_2018 ; Sieniawska_and_Bejger_2019 , for recent reviews on physics of possible mountain mode, r-mode, and wobbling mode CWs.

Due to the lack of the space, we focus on isolated NSs that emit mountain mode CWs in this section. LVC has been searching for mountain mode CWs LVC_CW_O1CW_APJ_2017 ; LVC_CW_O2CW_APJ_2019 . Readers may be referred to LVC_CW_O2CW_APJ_2019 for a wobbling mode CW search (or more generally, $m=1$ mode CWs) and LVC_CW_SNR_APJ_2019 for a r-mode search. Mountain mode CWs typically last for more than observation time $T_{0}$ ($\sim$ year). As such, the detectable CW amplitude $h$ at frequency $f_{\rm gw}$ scales as

where $C$ depends on a search method and a pre-defined threshold for detection. The LSC Bayesian time-domain search for known pulsars LVC_CW_O1CW_APJ_2017 adopts $C\simeq 10.8$, where “known” means their timing solutions as well as their positions on the sky are known to sufficient accuracies and there is little, if any, need to search over parameter space in the search.

A mountain mode CW depends on strain amplitude $h$, GW frequency $f_{\rm gw}$, its higher order time derivatives, direction to the source, inclination angle between the line of sight and the spin angular momentum, GW initial phase, and polarization angle. As a persistent source, a single detector can detect and locate a CW source. Since the detector beam pattern changes during the observation time, one can test general relativity (GR) by searching for CWs having nontensorial polarizations LVC_CW_NonTensorialCW_PRL_2018 . GW frequency and its higher time derivatives together with $h$ may tell us how the source loses its energy and spin angular momentum.

The amplitude $h$ of the dominant mountain mode CW depends on $\ell=m=2$ mass quadrupole deformation $Q_{22}$, distance to the source $r$, and the GW frequency as

$Q_{22}$ is related with the stellar ellipticity $\epsilon$ in the literature as $\epsilon{\cal I}_{3}=\sqrt{8\pi/15}Q_{22}$ where ${\cal I}_{3}$ is the moment of the inertia of the star with respect to its spin axis.

Roughly speaking, $Q_{22,{\rm max}}\propto\mu\sigma R^{6}/M$ where $\mu$ is the shear modulus, $\sigma$ is the breaking strain of the crust, $R$ is the stellar radius, and $M$ is its mass. As it depends on the stellar radius and mass, maximum possible $Q_{22,{\rm max}}$ depends on EOS Johnson-McDaniel_2013a ; Johnson-McDaniel_2013b . For a normal NS Pitkin_2011 ,

If we find some “NS” has $Q_{22}$ much larger than $Q_{22,{\rm max}}$, it means that that particular star may not be “a normal NS” (or we do not understand “the normal NS” good enough to predict $Q_{22,{\rm max}}$).

Real NSs may have $Q_{22}$ much smaller than $Q_{22,{\rm max}}$. The LIGO-Virgo collaboration reported upper limits on $Q_{22}$ for more than 200 pulsars, many of which already surpassed the theoretical upper limit significantly. For those pulsars, even if their GWs are detected, we may not be able to obtain information on the NS EOS from measurement of $Q_{22}$.

Figures of merit for isolated pulsar search may be the $h$ value given by Eq. (7) for possible $Q_{22}$ values, and the so-called spin down upper limit. Many pulsars show spin-downs which indicate the rate of the loss of the stellar rotational energy. Assuming a part of the rotational energy is radiated as GWs, we can estimate the maximum possible GW amplitude, called the spin down upper limit on GW amplitude.

where this equation assumes that $100\eta\%$ of energy is radiated as GWs.

Figure 5 shows the detector sensitivity curves for CW search (assuming 1 year integration, coherent search) as well as the figures of merit for the pulsars that the LIGO-Virgo collaboration has searched for using O2 data LVC_CW_O2CW_APJ_2019 . It is clear that KAGRA+ HF is more useful for CW search.

Figure 6 shows the upper limits on $Q_{22}$ that LIGO and Virgo obtained in their O2 search (stars), as well as possible upper limits that are expected by using various configurations of KAGRA+ for the same pulsars. The horizontal thick line indicates the theoretical maximum for $Q_{22}$ for a normal NS. The LIGO-Virgo collaboration has already beaten $Q_{22,{\rm max}}$ at higher frequencies than $\sim 100$Hz. KAGRA+ LF does not have any power to explore normal NS CWs.

It is possible to conduct unknown pulsar search (e.g., all-sky search, wide-band frequency search). In this case, a possible figure of merit is reach of a detector, which is shown for various detector configurations in Fig. 7, assuming coherent 1 year integration (although it is impossible in practice to conduct a fully coherent unknown pulsar search due to computational resource limitation). This plot again shows that we may be able to have more chance of detection if we put more emphasis on higher frequencies.

We can quantitatively compare performances of various configurations of possible KAGRA upgrades by introducing the gain in the SNR averaged over relevant frequencies. Namely, we define the ratio $r_{\alpha/\beta}\equiv r_{\alpha}/r_{\beta}$ for the configurations $\alpha$ and $\beta$ where $r_{\alpha}$ is defined in Eq. (5) and a similar equation holds for the $\beta$ configuration. In the case of a continuous wave search, we may be interested in the sensitivity at higher frequency, say, $f_{\rm low}=500$ Hz and $f_{\rm high}=1$ kHz where the possible constraints on $Q_{22}$ would be the tightest. For $\alpha=$“KAGRA+” and $\beta=$“bKAGRA”, $r_{\alpha/\beta}$’s then are 1.04 (“40kg”), 3.27 (“FDSQZ”), 5.89 (“HF”), and 4.53 (“Combined”).

Giant flares in soft gamma-ray repeaters (SGR) which are a class of strongly magnetized NSs, so-called magnetars, were observed: March 5, 1979 (SGR 0526-66); August 27, 1998 (SGR 1900+14); December 27, 2004 (SGR 1806-20). Event rate is quite rare, but huge amount of energy is known to be radiated in EM X and gamma bands. For example, in the last SGR 1806-20 hyperflare, the peak luminosity was $10^{47}$ erg/s and total energy released in a spike within 1 second was estimated as $10^{46}$ ergs 2005ApJ…628L..53I . Quasi-periodic oscillations were also observed in the burst tail. Their frequencies are 20-2000 Hz, and are identified by seismic modes excited in magnetar flare. The (quasi)-periodic oscillations are separately discussed in Sec. 6.3. The property and mechanism of the flare are not certain at present, owing to limited number of the events. Especially, the energy carried by GWs is unclear. Theoretical estimate of GW energy associated with flares ranges from $\sim 10^{40}$ erg 2001MNRAS.327..639I ; 2011PhRvD..83j4014C to $\sim 10^{49}$ erg 2011MNRAS.418..659L ; 2012PhRvD..85b4030Z . Dipole field-strength derived by stellar spin and its time-derivative is normally used for the energy deposited before a flare, but much stronger fields might be hidden inside. Thus, the problem will be solved only by actual observations due to many uncertain factors. Actually, GW energy at the SGR 1806-20 hyperflare was limited by LIGO. At that time, one interferometer at Hanford was operated. Upper limit on the GW amplitude is $5\times 10^{-22}$ Hz^-1/2 and GW energy is less than $8\times 10^{46}$ erg Abbott:2007zzb . It is very important to further proceed the similar argument to each flare observation.

Another interesting astrophysical event is a pulsar glitch. Sudden change in spin angular frequency is $\Delta\Omega/\Omega\sim 10^{-6}$ in a typical giant glitch and rotation energy changes by $I\Omega^{2}/2\times\Delta\Omega/\Omega\sim 10^{41}$ ergs as for a maximum value. The change is likely to be much smaller, since the rearrangement of stellar structure is partial in the transition.

These burst events will be probably first detected in EM signs, and GW signals should be carefully searched at the burst epoch. There is no reliable waveform, so that GW signals are identified by combinations of multiple detectors. The burst is a dynamical event in a timescale of millisecond, and therefore a sharp peak at kHz range is expected. Coincidence among different GW/EM detectors is crucial. The observational strategy of GW analysis may depend on the EM information. When QPO frequencies are identified in EM data, we should search for GW counterparts in the same frequency range. GW signals in LIGO O2 data2019ApJ…874..16 were searched for magnetar bursts , which are less energetic events. Two software packages, X-Pipeline and STAMP were used: Clusters of bright pixels above a threshold in each time-frequency map are searched in the X-Pipeline, and directional excess power due to arrival time-delay between detectors in the STAMP.

As for the future extension of bKAGRA, improvements in a higher frequency band are favored for these events. If we fix the frequency of the signal to 1 kHz, SNR ratios of KAGRA+ to bKAGRA at 1 kHz are 40kg 1.03, FDSQZ 3.98, HF 7.47, Combined 5.01. However, it is not clear at present to predict the detection level of meaningful signals by improvements. We will catch them or upper limits by chance, when bursts will happen. The stable operation is desirable.

In order to observationally extract the properties of astronomical objects, asteroseismology is a very powerful technique, where one could get the information about the objects via their specific frequencies. This is similar to seismology in the Earth and helioseismology in the Sun. For this purpose, GWs must be the most suitable astronomical information, owing to their high permeability. In fact, several modes in GWs are expected to be radiated from the compact objects KS99 , and each mode depends on different aspect of internal physics. The GWs have complex frequencies, because the GWs carry out the oscillation energy, where the real and imaginary parts correspond to the oscillation frequency and damping rate, respectively. So, by identifying the observed frequencies (and damping times) of GWs to the corresponding specific modes, one would get the physics behind the phenomena. In general, the GWs are classified into two families with their parity. The axial parity (toroidal) oscillations are incompressible oscillations, which do not involve the density variation, while the polar parity (spheroidal) oscillations involve the density variation and stellar deformation. Thus, from observational point of view, the polar type oscillations are more important, although those are coupled with the axial type oscillations on the non-spherical background.

Some of the GWs from NSs are classified in the same way as the oscillations of usual (Newtonian) stars. That is, the fundamental ($f$) and the pressure ($p_{i}$) modes are excited as acoustic oscillations of NSs, while the gravity ($g_{i}$) modes are excited by the buoyancy force due to the existence of the density discontinuity or composition gradients inside the star. The Rossby ($r$) modes are also excited by the Coriolis force in the rotating stars, although they are axial parity oscillations. In addition to these modes associated with the fluid motions, the oscillations related to the spacetime curvature are also excited, i.e., the so-called GW ($w_{i}$) modes, which can be discussed only in the relativistic framework. The typical frequencies of $f$, $p_{1}$, $g_{1}$, and $w_{1}$ modes for standard NSs are around a few kHz, higher than $4-7$ kHz, smaller than a hundred Hz, and around 10 kHz, respectively KS99 . With these specific modes from the NSs, it has been discussed to extract the NS information via the direct detection of GWs.

One of the good examples of the GW asteroseismology may be the results shown by Andersson and Kokkotas AK96 ; AK98 , where they found the universal relations for the frequencies of the $f$ and $w_{1}$ modes as a function of the average density of star and the compactness, respectively, almost independently of the EOS for NS matter. This is because the frequency of $f$ mode, which is an acoustic wave, should depend on the crossing time inside the star with the sound velocity, while the frequency of $w$ mode is strongly associated with the strength of the gravitational field. So, if one would simultaneously detect two modes in GWs from the NS, one could get the average density and compactness of the source object, which tells us the NS mass and radius. On the other hand, $g$ mode GWs may tell us the existence of the density discontinuity due to the phase transition F87 ; STM01 ; MPBGF03 ; SYMT11 , or they may become more important in the newborn NSs or protoneutron stars (PNSs) RG92 ; FGP07 ; PAH16 . The nonaxisymmetric $r$ mode oscillations may also be interesting due to their instability against the gravitational radiations A98 ; FM98 . That is, for highly rotating case, the GW amplitude can exponentially grow up due to the instability, which may become a detectable signal. Eventually, GW amplitude would be suppressed via nonlinear coupling AFMSTW03 . We note that even the $f$ mode oscillations may become unstable in rapidly rotating relativistic NSs GGKZ11 . In any case, the detection of GWs from the isolated (cold) NS may be quite difficult with the current GW detectors, because most of their frequencies are more than around kHz.

On the other hand, the detection of GWs from the PNSs would be more likely if they are formed within our galaxy. This is because, since the PNSs are initially less massive and larger radius, i.e., their average density is smaller than the cold NSs, one can expect the frequency of $f$ mode GW would be smaller than the typical one for cold NSs. In practice, the $f$ mode frequency of PNS is around a few hundred Hz in the early phase after core-bounce ST16 . However, the studies of the asteroseismology in PNSs are very few, unlike cold NSs. This is because of the difficulty for providing the background models. That is, the cold NSs can be constructed with the zero-temperature EOS, i.e., the relation between the energy density and pressure, while one has to consider the distribution of the electron fraction and entropy per baryon together with the pressure distribution as a function of density to construct the PNS models. But, such distributions can be determined only after the numerical simulation of core-collapse supernovae. Even so, the study of asteroseismology in PNSs is becoming possible with the numerical data of simulation.

Up to now, two different approaches are proposed for providing the PNS models, i.e, one is that the PNS surface is defined with a specific density around $10^{10-12}$ g/cm³ ST16 ; SKTK17 ; MRBV18 ; SKTK19 ; Sotani19 , and the other is that the oscillations inside the shock radius are considered TCPF18 ; TCPOF19 , where the different boundary conditions should be adopted according to the approaches. In contrast to the cold NSs, matter widely exists outside the PNSs, which makes difficult to treat the boundary of the background models. Anyway, even with either approach, it seems that the mode frequencies of GWs are expected to range in wide frequencies and that such frequencies would change with time. If one could identify the frequencies of GWs with a specific mode, it must be very helpful for understanding the physics of PNS cooling via understanding of the PNS properties. In practice, the time evolution of the identified frequencies is considered as a result of the changing of the PNS mass and radius, i.e., the mass increases due to the accretion and the radius decreases due to the both of a relativistic effect and neutrino cooling, with time. So, in the same way as for the cold NSs, if one could simultaneously identify the frequencies with the $f$ and $w_{1}$ modes GWs from PNSs, one would get the information of the average density and compactness at each time, which enables us to know the evolution of mass and radius of PNSs SKTK17 . That is, unlike the cold NSs, in principle one can expect to make a severe constraint on the EOS even with one event of GW detection. It should be emphasized that the frequencies of $w_{1}$ mode GWs from the PNSs becomes typically around a few kHz, because the compactness is smaller than that of cold NSs.

We make a comment about the detectability of these modes in GWs. The GW amplitudes have been discussed in Ref. AK96 ; AK98 for isolated cold NSs, assuming the radiation energy of GWs with each mode. Since the radiation energy with each mode can not be estimated as far as one examines the GW asteroseismology with linear perturbation analysis, one should eventually estimate such radiation energy with numerical simulation to see the detectability. The situation in PNSs born after the core-collapse supernova is the same. Even so, adopting the formula shown in Ref. AK96 ; AK98 , the effective gravitational amplitude of $w_{1}$ mode from the PNSs may be estimated with its energy, $E_{w_{1}}$, at each time step as

where $f_{w_{1}}$ and $D$ denote the frequency of considering $w_{1}$ mode and the distance to the source object, respectively SKTK17 . The total radiation energy of the $w_{1}$ mode in the GWs from PNS, $E_{w_{1}}^{T}$, may be estimated as $E_{w_{1}}^{T}\approx E_{w_{1}}T_{w_{1}}/\tau_{w_{1}}$, where $\tau_{w_{1}}$ denotes the damping time of $w_{1}$ mode with the frequency of $f_{w_{1}}$ at each time step. In this estimation, we assume that $w_{1}$ mode GW emission would last for the duration of $T_{w_{1}}$, as the frequency of $f_{w_{1}}$ evolves, where we simply adopt $T_{w_{1}}=250$ ms. Then, the effective amplitude is plotted as in Fig. 8, where the circles, squares, diamond, triangles, and upside-down triangles correspond to the expectations with $E_{w_{1}}^{T}/M_{\odot}=10^{-4}$, $10^{-5}$, $10^{-6}$, $10^{-7}$, and $10^{-8}$, respectively.

To detect the GW signal due to the NS oscillations, improvements in a higher frequency must be more suitable, although the amount energy of GWs are unknown at present. If we fix the frequency of the signal to 1 kHz as in the previous section 6.2, the SNR ratios of KAGRA+ to bKAGRA at 1 kHz are 40kg 1.03, FDSQZ 3.98, HF 7.47, Combined 5.01. The improvements even in a middle frequency band may be suitable for detecting the $f$-mode GWs from the PNS, but it must be acceptable only in the early phase after the core-bounce, because the frequency increases with time and eventually becomes $\sim$ kHz.

Accumulating observational evidence from multi-wavelength EM observations such as of ejecta morphologies and spatial distributions of heavy elements have all pointed toward CCSNe being generally aspherical (e.g., casA ; tanaka17 and references therein). However, these signals are secondary for probing the multidimensionality of the supernova engine because they are only able to provide images of optically thin regions far away from the central core. The GWs from CCSNe are the primary observables, which imprint a live information of the central engine.

From a theoretical point of view, understanding the origin of the explosion multidimensionality is indispensable for clarifying the yet uncertain CCSN mechanisms. After a half-century of continuing efforts, theory and neutrino radiation-hydrodynamics simulations are now converging to a point that multi-dimensional (multi-D) hydrodynamics instabilities play a pivotal role in the neutrino mechanism (bethe ), the most favoured scenario to trigger explosions. In fact, self-consistent simulations in three spatial dimensions (3D) now report shock revival of the stalled bounce shock into explosion by the multi-D neutrino mechanism (see bernhard16 ; janka16 ; burrows13 ; Kotake12 for recent reviews).

The most distinct GW emission process commonly seen in recent self-consistent three-dimensional (3D) models is associated with the excitation of core/PNS oscillatory modes (e.g., radice18 ; andresen18 ; andresen17 ; kuroda16 ). The dominant modes are the quadrupolar ($\ell=2$) deformations of the PNS excited by the non-spherical mass accretion onto it (see also Sec. 6.3). The characteristic GW frequency increases almost monotonically with time due to an accumulating accretion onto the PNS, which ranges from $\sim 500$ to 1000Hz (see astone18 for the dedicated scheme to detect the ramp-up GW signature from the noises).

In addition to the prime GW signature, the Standing Accretion Shock Instability (SASI) (e.g., thierry15 ) has been also considered as a key to generate GW emission in the postbounce supernova core. It has become apparent radice18 ; andresen18 ; kuroda16 that the predominant GW emission from CCSNe is produced in the same region behind the postshock region where both the SASI and neutrino-driven convection vigorously develop (e.g., Kotake13 for a review). The GW emission from this region–from just above the neutrinospheres at densities of roughly 10¹¹ g/cm³ and inward–is determined by the effects of the super-nuclear EOS, as well as the impact of hydrodynamic instabilities present in the post-shock flow (e.g., andresen17 ). The typical frequency of the SASI-induced GWs is in the range of $\sim$100 – 250 Hz, which is in the best sensitivity range of the currently running interferometers.

Rapid rotation in the iron core leads to significant rotational flattening of the collapsing and bouncing core, which produces a time-dependent quadrupole (or higher) GW emission (e.g., ott_rev for a review). For the bounce signals having a strong and characteristic signature, the iron core must rotate rapidly enough³³3Although rapid rotation and the strong magnetic fields in the core is attracting great attention as the key to solve the dynamics of collapsars and magnetars, one should keep in mind that recent stellar evolution calculations predict that such an extreme condition can be realized only in a special case woos06 ($\lesssim$ 1% of massive star population).. The GW frequency associated with the rapidly rotating collapse and bounce is in the range of $\sim 600-1000$ Hz.

Another distinguished feature of GWs from a CCSN is a circular polarization. The importance of detecting circular GW polarization from CCSNe was first discussed by Hayama16 in the context of rapidly rotating core-collapse. More recently Hayama18 presented analysis of the GW polarization using results from 3D general-relativistic simulations of a non-rotating $15M_{\odot}$ star kuroda16 . The amplitude of the GW polarization was shown to become larger for the 3D general-relativistic model that exhibits strong SASI activity. The non-axsymmetric flows associated with spiral SASI give rise to the circular polarization. If detected, it will provide us a new probe into the SASI activity in the preexplosion supernova core. Note that the spiral SASI develops mainly outside of the PNS, so that the circular polarization from the spiral SASI has different features in the spectrogram comparing to the stochastic, weak circular polarization from the PNS oscillations that have no preferential direction to develop Hayama18 . By estimating the SNR of the GW polarization, they pointed out the possibility that the detection horizon of the circular polarization could extend by more than a factor of several times farther comparing to that of the GW amplitude. In order to detect the GW circular polarization with a high SNR, the joint GW observation with KAGRA in addition to two LIGOs and Virgo is indispensable.

Each phase of a CCSN has a range of characteristic GW signatures and the GW polarization that can provide diagnostic constraints on the evolution and physical parameters of a CCSN and on the explosion hydrodynamics. The GW signatures described in this section commonly appear in the frequency range of $100$ - $1000$ Hz in recent simulations of the CCSN. Among them, the PNS oscillatory modes are currently recognized as the model-independent GW signature. The characteristic frequencies of the fundamental ($f/g$) modes are predominantly dependent on the PNS mass ($M$) and the radius ($R$), however, in a non-trivial manner sotani17 . In order to break the degeneracy, the detection of other eigen-modes ($p,w$ modes) of the PNS oscillations is mandatory. In fact, the GW asteroseismology of the PNS has just started forne19 ; morozova18 ; sotani17 (see also Sec. 6.3). The outcomes of which should reveal the requirement of KAGRA (and the next-generation detectors) to detect the whole of the eigenmodes (presumably extending up to several kHz) for the future CCSN event. This could be done in the next five years.

For the detectability of GWs from PNSs, see Sec. 6.3. The GWs due to SASI for a Galactic event may be detected with the SNR of the GW signals predicted in the most recent 3D CCSN models in the range of $\sim$ 4 – 10 for the aLIGO andresen17 . With the third-generation detectors online (e.g., Cosmic Explorer), these signals would be never missed for the CCSN events throughout the Milky way. A current estimate of SNR for the GWs from rapidly rotating collapse and bounce based on predictions from a set of 3D models using a coherent network analysis shows that these GW signals could be detectable up to about $\sim 20$ kpc for the most rapidly rotating model (hayama15 , see also powell19 ; gossan16 ). If the spectra of the supernovae are assumed to be flat in the frequency range of 100 Hz - 1 kHz, we can quantitatively compare performances of various configurations of possible KAGRA upgrades by defining the SNR ratio $r_{\alpha/\beta}\equiv r_{\alpha}/r_{\beta}$ for the configurations $\alpha$ and $\beta$ where $r_{\alpha}$ is defined in Eq. (5) and a similar equation holds for the $\beta$ configuration. For the frequency range between $f_{\rm low}=100$ Hz and $f_{\rm high}=1$ kHz, the SNR ratios of KAGRA+ to bKAGRA averaged over relevant frequencies are 40kg 1.51, FDSQZ 2.35, HF 3.43, Combined 3.55. However, since the spectral evolution of the CCSN GWs has not been completely clarified yet and it does depend upon the details of CCSN modeling (such as on the initial rotation rates and the progenitor masses), it is too early to make a quantitative discussion about the best-suited sensitivity curves for KAGRA+. Because recent CCSN simulations indicate that the important GW frequency to unveil the physics of CCSNe will be $\geq 100$Hz, all the candidate upgrades except for LF should help enhance the chance of detection.

Inflation, an accelerated expansion phase of the very early Universe, was proposed in the 1980s to solve problems in the standard model of the Big Bang theory, such as the horizon, flatness, and monopole problems inflation1 ; inflation2 ; inflation3 ; inflation4 . A key feature of inflation is that the rapid expansion stretches quantum fluctuations in a scalar field to classical scales and generates primordial density fluctuations scalar1 ; scalar2 ; scalar3 ; scalar4 ; scalar5 , which become the seeds of galaxies. The nearly scale invariant spectrum of scalar perturbations, predicted by the inflationary theory, is strongly supported by cosmological observations of the cosmic microwave background and galaxy surveys. The same mechanism also applies to quantum fluctuations in space-time, which produce a stochastic background of GWs over a wide range of frequencies with a scale-invariant power spectrum tensor1 ; tensor2 ; tensor3 . The detection of inflationary GWs is a key to test inflation and distinguish a number of models since the amplitude and spectral index strongly depend on the model.

The amplitude of the GWs is typically far below the sensitivity of the ground-based detectors. The dimensionless parameter, $\Omega_{\rm GW}$, which describes the energy density of GWs per logarithmic frequency interval, is roughly given by $\Omega_{\rm GW}\simeq 10^{-15}(H_{\rm inf}/10^{14}{\rm GeV})^{2}$ for a scale invariant primordial spectrum $\Omega_{\rm GW}\propto f^{0}$, where $H_{\rm inf}$ is the Hubble expansion rate during inflation. Inflation predicts a slightly red-tilted spectrum, which further reduces the amplitude at high frequency. For example, the model predicts $\Omega_{\rm GW}\sim 10^{-16}$ at $100$Hz for chaotic inflation and $\Omega_{\rm GW}\sim 10^{-17}$ for $R^{2}$ inflation, while the cross correlation between LIGO, Virgo, and KAGRA with their final design sensitivities is expected to reach a stochastic background of $\Omega_{\rm GW}\sim 10^{-9}$. For details on the shape of the spectrum, see Kuroyanagi:2008ye ; Kuroyanagi:2014qaa .

Note that a search for a stochastic GW background is performed by cross-correlating data from different detectors over a long time period. The sensitivity is determined by noise curves of the multiple detectors as well as the observation time. Since the two detectors of LIGO have advantages in both sensitivity and observation time, the sensitivity of the four-detector network $\Omega_{\rm GW}\sim 10^{-9}$ will be dominated by LIGO detectors, while KAGRA with the current design sensitivity would not contribute to increase the overall sensitivity. Currently LIGO-Virgo O2 data provide an upper limit of $\Omega_{\rm GW}<6.0\times 10^{-8}$ for a scale invariant spectrum LIGOScientific:2019vic .

Although the ground-based detectors are not sensitive enough to detect standard inflationary GWs, some models going beyond the standard picture predict a blue-tilted spectrum, e.g, string gas cosmology Brandenberger:2006xi , super-inflation models Piao:2004tq ; Baldi:2005gk , G-inflation Kobayashi:2010cm , non-commutative inflation Calcagni:2004as ; Calcagni:2013lya , particle production during inflation Cook:2011hg ; Mukohyama:2014gba , Hawking radiation during inflation Mohanty:2014kwa , a spectator scalar field with small sound speed Biagetti:2013kwa , massive gravity Fujita:2018ehq , and so on. The amplitude of the inflationary GW spectrum is constrained at low frequencies by the cosmic microwave background $\Omega_{\rm GW}\lesssim 10^{-15}$, but the amplitude can be large at the frequency of $\sim 100$ Hz if the spectrum is blue-tilted. There would be a chance for the ground-based detectors to detect inflationary GWs if the spectral index of the spectrum $n_{T}$ is larger than $\sim 0.5$.

Another possibility for the ground-based detectors is to probe a non-standard Hubble expansion history of the early Universe Peebles:1998qn ; Giovannini:1998bp ; Giovannini:1999bh ; Giovannini:1999qj ; Tashiro:2003qp ; Giovannini:2008tm . The information on the expansion rate, which is determined by the dominant component of the Universe, is imprinted in the spectrum of inflationary GWs. For a scale invariant primordial spectrum, the spectral dependence of the stochastic background becomes $\Omega_{\rm GW}\propto f^{2(3w-1)/(3w+1)}$, where $w$ is the EOS of the Universe. In the standard cosmological model, the matter-dominated ($w=0$) and radiation-dominated ($w=1/3$) phases yield $\Omega_{\rm GW}\propto f^{-2}$ and $f^{0}$, respectively. When one considers a kination dominated phase ($w=1$) soon after inflation, the spectral amplitude is enhanced at high frequency by the dependence $\Omega_{\rm GW}\propto f$ and peaks at the frequency corresponding to the size of the Universe at the end of inflation, which is typically higher than $\sim 100$ Hz. Although it is difficult to find a case where GWs are detectable by the ground-based experiments satisfying the Big-Bang nucleosynthesis (BBN) bound Figueroa:2018twl ; Ahmad:2019jbm , $\Omega_{\rm GW}\lesssim 10^{-5}$ for $f\gtrsim 10^{-10}$ Hz, the upper bound on the stochastic background is still helpful to explore early-Universe physics beyond the standard cosmology.

The SNR for a stochastic GW background is computed by taking cross-correlation between detector signals Allen:1997ad , and given by

where $H_{0}$ is the Hubble constant, $T_{\rm obs}$ is the observation time, and $S_{n,I}(f)$ is the noise spectral density with $I$ and $J$ labeling two different detectors. The overlap reduction function $\gamma_{IJ}(f)$ is given by the detector responses of the two polarization modes $F^{+}$ and $F^{\times}$ as

where ${\bf x}$ describes the position of the detector and $\hat{\bf\Omega}$ specifies the propagation direction of GWs. For a network of $N$ detectors, the total SNR is given by

As seen in the overlap reduction function, the SNR depends on the distance and relative orientation of the detectors. For KAGRA, correlating with AdV+ is better than with A+ because of the relative orientation. Assuming the flat spectrum of $\Omega_{\rm GW}(f)$, the sensitivities of the pair of KAGRA and AdV+ to $\Omega_{\rm GW}$ with ${\rm SNR}=5$ are $1.1\times 10^{-7}$ for bKAGRA, $3.4\times 10^{-8}$ for LF, $8.3\times 10^{-8}$ for 40kg, $1.1\times 10^{-7}$ for FDSQZ, $3.3\times 10^{-7}$ for HF, and $5.0\times 10^{-8}$ for Combined. The lower frequency upgrades give better sensitivities. However, the sensitivity between two A+ is $4.0\times 10^{-9}$, which is an order of magnitude better than those with KAGRA because of much closer separation of the detector pair. Therefore, KAGRA is not good for detection but will be able to play an important role for determining the spectral shape and measurig anisotropy, non-Gaussianity and polarizations, which are discussed in Sec. 8.3.

The thermal history of the Universe after BBN, which takes place at a temperature scale of order of 1 MeV, has been well established thanks to amazing progress made by cosmological observations. On the other hand, our knowledge about the Universe earlier than BBN is quite limited. If we extrapolate our understanding of particle physics, several phase transitions should occur in such early stages of the Universe. These include the QCD phase transition, the electroweak phase transition, the GUT phase transition and some phase transitions of new models beyond the Standard Model of particle physics. The nature of a phase transition is classified according to the evolution of the order parameter: first-order or second-order. In addition, the case where the order parameter changes smoothly and no phase transition occurs is referred to as a crossover. For a first-order phase transition, a latent heat is released, supercooling emerges and phases are mixed like boiling water. Non-uniform motion of vacuum bubbles during the first-order phase transition, GWs are produced. Therefore, one can survey new physics models involving a first-order phase transition by observing GWs Grojean:2006bp .

GWs produced by the first-order phase transition in the early Universe are stochastic: isotropic, stationary and unpolarized. They are characterized only by the frequency. There are three known sources for GWs from the first-order phase transition: collisions of vacuum bubble walls, sound waves in the plasma and plasma turbulence. In general, one has to sum these contributions. For typical non-runaway bubbles, the contribution from sound waves is dominant and the peak energy density ⁴⁴4The energy density of gravitational waves $\Omega_{\rm GW}$ and the power spectral density $S_{h}$ is related through $\Omega_{\rm GW}H_{0}^{2}=(2\pi^{2}/3)f^{3}S_{h}$. is roughly Caprini:2015zlo

and the peak frequency is

Here, $T_{*}$ is the transition temperature, $\alpha$ is the ratio of the released energy (latent heat) to the radiation energy density, $\beta$ is the inverse of the time duration of the phase transition, $H_{*}$ is the Hubble expansion parameter at $T_{*}$, $\kappa$ is the ratio of vacuum energy converted into fluid motion, and $v$ is the bubble wall velocity. If one selects a new physics model and chooses a model parameter set, one can compute the finite temperature effective potential, from which the above quantities are derived. Namely, the above quantities are functions of the underlying model parameters. Therefore, the number of free model parameters can be generally reduced by measuring the stochastic GW spectrum.

Among new physics models that involve phase transitions, models with an extended Higgs sector achieving the electroweak phase transition are of most importance and should be addressed. Although the Higgs boson has been discovered at the CERN Large Hadron Collider and the spontaneous breaking of the electroweak symmetry has been established, the dynamics of the electroweak symmetry is still unknown. The baryon asymmetry of the Universe may be accounted for by electroweak baryogenesis, which requires a strongly first-order electroweak phase transition. Since the transition temperature of the electroweak phase transition is of the order of 100 GeV, unfortunately the peak frequency of the predicted GWs spectrum falls in the range of $10^{-3}$ Hz to 0.1 Hz, regardless of the details of the extended Higgs models. Therefore, the detection of GWs from the electroweak phase transition and determination of the model parameters of extended Higgs models are challenging for KAGRA upgrade and need future space-based interferometers, such as LISA and DECIGO Kakizaki:2015wua ; Hashino:2018wee . Conversely, one can envisage and construct hypothetical models with the phase transition temperature of $10^{5}$ GeV to $10^{7}$ GeV so that the peak frequency matches with the KAGRA frequency band Dev:2016feu . Even in such models, an enormously large latent heat and/or an very long transition duration are necessary for GW signals to hit the sensitivity curves for KAGRA upgrade candidates. Another issue concerns uncertainties in computing the spectrum of generated stochastic GWs at a first-order phase transition. In particular, numerical simulations of the dynamics of nucleated bubbles are rather cumbersome. With the development of computational techniques, the result in Eq. (14) might be revised by one order of magnitude or more. Therefore, although it is difficult to predict the detectability of GWs from a phase transition, KAGRA will play an important role in constraining models of first-order phase transitions that take place at temperatures of around $10^{5}$ GeV to $10^{7}$ GeV.

GWs generated in the early Universe are typically in the form of a stochastic background, which is a continuous noise-like signal spreading in all directions, while numerous astrophysical GWs also become a stochastic background by overlapping one another. Even if a stochastic background is detected, it may be difficult to identify its origin among many cosmological and astrophysical candidates.

A standard data analysis provides information only on the amplitude and the tilt of the spectrum. In order to discriminate the origin of the GW background, we need to go beyond conventional data analysis and extract more information, such as the spectral shape Kuroyanagi:2018csn , anisotropy Allen:1996gp ; TheLIGOScientific:2016xzw , polarization Nishizawa:2009bf ; Abbott:2018utx , and non-Gaussianity (or so-called popcorn background) Drasco:2002yd ; Thrane:2013kb ; Martellini:2014xia . Anisotropy and non-Gaussianity could arise in a GW background formed by overlapping GW bursts. Typical examples are the ones originating from astrophysical sources and cosmic strings. Additional GW polarizations appear when the generation process is related to physics beyond general relativity (see Sec. 9.4). Therefore, they can be used as indicators of specific generation mechanisms.

The spectral shape of the stochastic background depends strongly on its origin and it can be generally used to distinguish a wide range of generation models. Many models predict a spectral shape spanning a broad range of frequencies. In this case, the ground-based detectors observe a part of the spectrum because of the limited sensitivity range of frequency, and multi-band observations are needed to determine the whole spectral shape. However, some models predict a peaky spectral shape; one good example testable by the ground-based detector network itself is a GW background generated by superradiant instabilities Brito:2017wnc . One way to obtain the spectral shape in data analysis is to use broken power-law templates instead of the currently used single power-law templates Kuroyanagi:2018csn ; Bose:2005fm . The uncertainties in the parameter determination decrease in proportional to the inverse of the SNR, and at least SNR $\gtrsim 10$ is necessary to obtain useful information on the entire spectral shape. If detection were made with higher SNR, the analysis could be extended to the whole spectral shape reconstruction with multiple frequency bins, which evaluates the spectral index in each bin (see Figueroa:2018xtu for the discussion for LISA, which could be applied to ground-based experiments).

In general, a higher SNR is necessary not only to determine the spectral shape but also to obtain the above mentioned additional properties of the stochastic background. Since inflationary GWs have a red-tilted spectrum in terms of strain amplitude, an improvement of sensitivity at lower frequency would be helpful to test more properties of the stochastic background. On the other hand, as previously mentioned, the sensitivity of the ground-based detectors is not enough to detect a standard inflationary GW background, and a blue-tilted spectrum is necessary to enhance the amplitude for detection. If $n_{T}>3$, where we define the spectral tilt as $\Omega_{\rm GW}\propto f^{n_{T}}$, an improvement of sensitivity at higher frequencies becomes important (see Fig. 1 of Kuroyanagi:2018csn ). Astrophysical GWs also have a blue-tilted spectrum $n_{T}=2/3$, but when the tilt is not so steep, sensitivity at mid-frequencies ($20-50$Hz) is more important.

Apart from sensitivity for the amplitude, multiple detectors facing different directions provide independent information about anisotropy and polarization, and thus the inclusion of KAGRA is essential.

There are currently no alternatives to Einstein’s GR that are physically viable. As such, one cannot test GR by comparing the data with waveforms in alternative theories. Instead, one can test for the consistency of the signal with GR. In doing so, one must ensure that any deviation from GR cannot be attributed to noise artifacts, but instead are intrinsic to the GW signal itself. However, this must be done without explicitly assuming an alternative model for gravity that governs the morphology of the GW signal.

Existing efforts largely focus on comparing different portions or aspects of the signal and require consistency amongst them. For example, one can verify the self-consistency of the post-Newtonian description of the inspiral phase, and from the phenomenological GR model of the merger-ringdown LiEtAl:2012a ; LiEtAl:2012b ; Agathos:2013upa ; Cornish:2011ys ; Sampson:2013jpa ; Meidam:2017dgf . In addition, one can also test the final remnant mass and spin, as determined from the low-frequency (inspiral) and high-frequency (post-inspiral) phases of the signal for mutually consistent Ghosh:2016qgn ; Ghosh:2017gfp . Furthermore, one can also compare the residual signal after subtracting the best-fitting GR waveform to the expected statistical properties of the noise. For the first set of BBH detections, all tests so far have not revealed any significant deviations from GR TheLIGOScientific:2016src ; TheLIGOScientific:2016pea ; Abbott:2018lct ; LIGOScientific:2019fpa . Instead, constraints on various possible beyond-GR effects have been placed. For example, the fractional deviations to the lowest post-Newtonian parameters have been constrained to tens of percent, whereas the higher post-Newtonian orders and the merger-ringdown parameters are constrained to around a hundred percent. Moreover, the possibility of dipolar radiation has been constrained to the percent level, and the mass of the graviton has been constrained to $m_{g}\leq 4.7\times 10^{-23}\,\mathrm{eV/c}^{2}$.

In the future, as the LVK network operates at their full strengths, we can anticipate several effects that will improve our ability to constrain general relativity. Firstly, the improved detection rate will allow us see many events for which the different aspects of their dynamics are covered by the network’s sensitive frequencies. Secondly, more detections also means that our constraints will be improved statistically. While we do not expect improvements of orders of magnitude, it will help to shrink the parameter space of possible deviations from GR. Thirdly, a full LVK network will greatly enhance tests that require multiple detectors, including tests for alternative polarisations (see Sec. 9.4). These tests are expected to greatly improve as KAGRA, as the fourth detector, improves its sensitivity. Finally, the prospect of observing novel source classes, including highly eccentric systems, highly precessing systems and even head-on collisions will open up a entirely new suite of tests.

In the era of KAGRA+, we expect more detections of BBH and BNS, and anticipate detections of BH-NS systems. The precise way in which KAGRA+’s sensitivity develops will govern the way it will contribute to tests of GR. For example, with the focus on low-frequency sensitivity, one improves the insight into the merger and ringdown of heavy BHs, where the gravitational fields are the strongest. In contrast, high frequency sensitivity allows one to track lower mass binary merger for a longer period of time, which improves the precision of the consistency checks. Additionally, by improving the sky coverage and increasing the multiple-detector live time, KAGRA+ will increase the number of detected events and thereby reducing the statistical errors due to detector noise by combining results from multiple events.

We can raise several motivations to study modified gravity: a) GR is not ultraviolet (UV) complete theory. It should be modified to describe physics near Planck scale. b) Cosmology based on GR requires dark components to explain the observed Universe, such as dark matter and dark energy. It is attractive if we can naturally explain the observed Universe by a well-motivated modification of gravity without introducing cosmological constant. c) To test GR, it is not enough to know the prediction based on GR when we use GWs as a probe. Since GW data is noisy, an appropriate projection of the data in the direction predicted by the possible modification of gravity is necessary to obtain a meaningful constraint.

As possible modifications of gravity, there are several categories. One is to add a new degree of freedom like a scalar field in scalar-tensor gravity. Such a kind of extra degrees of freedom is strongly constrained by existing observations, since they introduce a fifth force. There are various ways, however, to hide the fifth force. 1) Sufficiently small coupling to the ordinary matter. 1-a) Coupling through higher derivative terms, which becomes important only under strong field environment. 1-b) Some non-linear mechanism to reduce the coupling like Chameleon/Vainstein mechanism Khoury:2003aq ; Vainshtein:1972sx . 2) Giving a sufficiently large mass to the field that mediates the fifth force.

There is also a possibility to modify gravity without introducing a new degree of freedom. One example is so-called minimal extension of massive gravity. Phenomenologically, we can give small mass to graviton in this kind of model, but the graviton mass at the present epoch is already strongly constrained by the observation of GW170817.

We focus on compact binary inspirals. In this context, the most probable modification of the generation mechanism of GWs is the one caused by the change in orbital evolution. The orbital evolution is governed by two aspects. One is the conservative part of the gravitational interaction and the other is the dissipative effects of radiation. It is not so obvious which effect is more important even in the weak field regime. Hence, we need a little more careful investigation of each model.

Whichever effect dominates, the main purpose of this study is the same. What we want to clarify is whether or not there exists a new effective propagating degree of freedom which appears only when we consider extremely compact objects like BHs and NSs.

There might be various possibilities. From the observational point of view it is convenient to introduce the parametrized post-Einstein (PPE) framework YunesPretorius2009 . The waveform of quasi circular inspiral in Fourier domain will be given in the form of

in GR. The waveform is characterized by the functions $A(f)$ and $\Psi(f)$ and they are modified in the extended theories like

with $u:=\pi Mf$. Since the test of gravity using GW data is more sensitive to the modification in the phase $\Psi(f)$, we will focus on it.

The smallest non-vanishing value of $b_{i}$ is related to the leading post-Newtonian order. The $n$-th post Newtonian correction appears as $b=(2n-5)/3$. Here we list representative possible modifications of gravity with the post-Newtonian order of the leading correction in phase in Table 8. BH evaporation is negligible in the ordinary scenario, but can be accelerated if there are many massless hidden degrees of freedom. For the dipole radiation to be emitted, charge not proportional to the mass of each constituent object is required. One typical example is the scalar tensor theory. Although BHs are thought not to bear a scalar hair in many cases, they can have a scalar charge in some models like in Einstein-Gauss-Bonnet-dilaton theory Yagi:2011xp . If we naively consider that gravitons have non-vanishing mass, the dispersion relation is also modified to give 1PN order correction, although this is not the effect in the generation of gravitational waves. However, if we consider covariant theories of massive gravity, we need to consider a bigravity model Hassan:2011zd . As a modification that appears at the 1PN order, one may think of a model in which the mass of each constituent object depends on the Newton potential.

As a typical modification of GR, the scalar-tensor theory is often discussed. If the binary members have some scalar charges coupled to the extra degree of freedom, we have dipolar radiation. Although the overall magnitude also depends on the coupling strength, if we focus only on the frequency dependence, the flux due to dipolar radiation is larger at a low frequency by the factor $(v/c)^{-2}\sim(m\omega)^{-2/3}$ compared with that due to GW radiation. On the other hand, the extra scalar degree of freedom changes the conservative part of the force, but there is no such enhancement in the sense of post Newtonian order. Hence, at low frequency, where $v/c\ll 1$ the modification to the conservative gravitational interaction can be neglected.

For the modification that predicts negative post-Newtonian order, the deviation form GR is relatively larger at lower frequencies, compared with GR contributions. Therefore, it is not so easy for GW observations to compete with the constraint from the observations at lower frequencies, such as the pulsar timing.

There are some more detailed aspects that cannot be expressed by the PPE framework. One possibility is the spontaneous scalarization. When we consider BNS, each NS in isolation may not have a scalar charge. However, there is a region in the model parameter space in which instability occurs as the binary separation decreases and both NSs get charged. Another possibility is that the mediating scalar field is massive. This is quite likely to avoid the instability similar to the spontaneous scalarization to occur in the whole Universe after the temperature of the Universe becomes $10\,{\rm MeV}$ or lower. If the instability occurs, the success of the big-bang nucleosynthesis will be significantly damaged. This gives the constraint on the mass of the mediating field to be greater than $10^{-16}$ eV or so Ramazanoglu2016 , which is comparable to the constraint from pulsar timing (PSRJ0348+0432). On the other hand, ground-based detectors are sensitive to modification if the mass is smaller than $10^{-12}$ eV. If the mass is in the range between these two, the existence of the extra degree of freedom might be probed only by GWs.

Basically we check the deviation of the observed GW signal from GR templates. For this purpose, it is better to have a better theoretical template based on each candidate of modified gravity. In particular this demand is higher when we consider the models whose leading order modification appears at the positive post-Newtonian order. In such models, we have more chances to observe the largest deviation at the final stage of inspirals. In many cases, this task remains to be a challenge for the theorists.

As long as we consider BHs with the mass larger than a few times the solar mass and NSs, it seems difficult to use a high frequency band to give constraints on modified gravity. In the case of BHs, we do not have much signals at very high frequencies. In the case of binary NSs, high frequency signal can be expected but it might be difficult to disentangle the effects of modified gravity from the uncertainty of the complex physics of NSs.

In general, when we wish to give a tighter constraint on the parameters contained in the waveform template, broader observational sensitivity band is advantageous. A broadband observation is not necessarily realized by a single detector. It would work well if a wider range of the frequency band is covered by plural detectors as a whole. In this sense, it would be necessary to negotiate among the various plans of future upgrade of detectors.

In near future we will establish a network composed of LIGO, Virgo and KAGRA. On the point of view of detecting modification in the gravitational wave generation, the improvement of the constraints depends on the increase of statistics and finding of some golden events with appropriate binary parameters and a large signal-to-noise.

The properties of GW propagation are changed when a gravity theory is modified from GR. Particularly, such modifications are paid much attentions at cosmological distance because the extended theories of gravity may explain the accelerating expansion of the Universe. In other words, testing GW propagation can constrain the possible modification of gravity at cosmological distance and give us an implication about the physical mechanism of the cosmic accelerating expansion Lombriser:2015sxa . According to Saltas:2014dha , the modifications of GW propagation in the effective field theory is expressed in general as

and are characterized by four properties: propagation speed $c_{\rm T}$, amplitude damping rate $\nu$, graviton mass $\mu$, and a source term $\Gamma$. Each modification appears differently in a specific theory, but this framework allows us to test many classes of gravity theories at the same time. Among the modifications, the propagation speed is important to test the violation of the Lorentz symmetry and the quantum nature of spacetime (modified dispersion relation). The amplitude damping rate is equivalent to the variation of the gravitational constant for GWs and enables us to test the equivalence principle. The graviton mass is a characteristic quantity in massive gravity. If parity in gravity is violated, it modifies in general the propagation speeds and the amplitude damping rates in Eq. (18) differently for left-handed and right-handed polarizations with opposite signs as shown in Nishizawa:2018srh ; Zhao:2019xmm .

The coincidence detection of GW170817/GRB170817A GW170817PRL brought us the first opportunity to measure the speed of a GW from the arrival time difference between a GW and an EM radiation⁵⁵5Before GW170817/GRB170817A, the measurements of the speed of a GW have done with GWs from BBH based on the arrival time difference between detectors Blas:2016qmn ; Cornish:2017jml ., as was predicted in Nishizawa2014PRD , and constrained the deviation from the speed of light at the level of $10^{-15}$ GW170817:GRB . Based in this constraint on GW speed, a large class of theories as alternatives to dark energy have already been almost ruled out Baker:2017hug ; Creminelli:2017sry ; Sakstein:2017xjx ; Ezquiaga:2017ekz . The graviton mass $\mu$ has been tightly constrained from the observation of BBH mergers in the range of $\mu<5.0\times 10^{-23}\,{\rm{eV}}$ LIGOScientific:2019fpa , which is tighter than that obtained in the Solar system Bernus:2019rgl . On the other hand, the amplitude damping rate $\nu$ has also been measured from GW170817/GRB170817A for the first time in Arai:2017hxj , but the constraint is still too weak, $-75.3\leq\nu\leq 78.4$. However, the anomalous amplitude damping rate is one of the prominent signatures of modified gravity, which is likely to be accompanied by the gravity modification explaining the cosmic acceleration, and plays a crucial role when we test gravity at cosmological distance Nishizawa:2019rra .

In generic parity-violating gravity including recently proposed ghost-free theories with parity violation Crisostomi:2017ugk , not only the amplitude damping rate but also the propagation speed are modified, in contrast to Chern-Simons modified gravity in which the GW speed is the speed of light. From GW170817/GRB170817A, since a GW is required to propagate almost at the speed of light, the parity violation in the GW sector of the theory has been constrained almost to that in Chern-Simons gravity, allowing only the amplitude damping rate to differ from GR Nishizawa:2018srh .