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Considering an ansatz for $1+3+n$ dimensional static spacetime with uniform $n$-branes stacked in three-dimensional rotationally symmetric space, we found different classes of vacuum solutions of Einstein field equations. We analyzed these solutions through the Newtonian limit, Kretschmann scalar, and conserved energy from Landau--Lifshitz stress-energy pseudotensor. This revealed unusual physical properties for a class of solutions with nontrivial extension of the Schwarzschild spacetime with extra dimensions. Positive conserved energy is associated with repulsive gravity and physical singularity on the horizon, while negative conserved energy implies gravitational attraction and naked singularity. This sets the Schwarzschild solution in a broader context and provides new possible testing grounds for general relativity., Comment: 11 pages, 1 figure, 1 table

We study cosmological perturbations in a universe with only one matter component described by a triplet of fields. Configuration of these fields is the same as for body coordinates of a solid, and they enter the matter Lagrangian only through the kinetic term. We restrict ourselves only to cases with constant pressure to energy density ratio $w$. Superhorizon perturbations have no constant modes with scalar vector and tensor perturbations decaying or growing at different rates, and in cases with pressure to energy density ratio $w>(19-8\sqrt{7})/3\dot{=}-0.722$ perturbations propagate with superluminal sound speed. Regarding our universe, these results illustrate possible challenges with comparing the observational data to models similar to solid inflation, if the inflation is followed by a period during which the studied model is a sufficiently good approximation., Comment: 19 pages, 3 figures

The detection of the hyper-bright gamma-ray burst (GRB) 221009A enables us to explore the nature of GRB emission and the origin of very-high-energy (VHE) gamma-rays. We analyze the ${\it Fermi}$-LAT data and investigate GeV-TeV emission in the framework of the external reverse shock model. We show that early $\sim1-10$ GeV emission can be explained by the external inverse-Compton mechanism via upscattering MeV gamma-rays by electrons accelerated at the reverse shock, in addition to the synchrotron self-Compton component. The predicted early optical flux could have been brighter than the naked-eye GRB 080319B. We also show that proton synchrotron emission from accelerated ultra-high-energy cosmic rays (UHECRs) is detectable, and could potentially explain $\gtrsim \rm TeV$ photons detected by LHAASO or UHECR acceleration can be constrained. Our model suggests that the detection of $\mathcal{O}(10\rm~TeV)$ photons with energy up to $\sim18$ TeV is possible for reasonable models of the extragalactic background light without invoking new physics, and predicts anti-correlations between MeV photons and TeV photons, which can be tested with the LHAASO data., Comment: 9 pages, 4 figures, accepted for publication in ApJL. Results and figures are updated, and the main conclusions remain unchanged

The joint detection of gravitational waves and the gamma-ray counterpart of a binary neutron star merger event, GW170817, unambiguously validates the connection between short gamma-ray bursts and compact binary object (CBO) mergers. We focus on a special scenario where short gamma-ray bursts produced by CBO mergers are embedded in disks of active galactic nuclei (AGN), and we investigate the $\gamma$-ray emission produced in the internal dissipation region via synchrotron, synchrotron self-Compton and external inverse-Compton (EIC) processes. In this scenario, isotropic thermal photons from the AGN disks contribute to the EIC component. We show that a low-density cavity can be formed in the migration traps, leading to the embedded mergers producing successful GRB jets. We find that the EIC component would dominate the GeV emission for typical CBO mergers with an isotropic-equivalent luminosity of $L_{j,\rm iso}=10^{48.5}~\rm erg~s^{-1}$ which are located close to the central supermassive black hole. Considering a long-lasting jet of duration $T_{\rm dur}\sim10^2-10^3$ s, we find that the future CTA will be able to detect its $25-100$ GeV emission out to a redshift $z=1.0$. {In the optimistic case, it is possible to detect the on-axis extended emission simultaneously with GWs within one decade using MAGIC, H.E.S.S., VERITAS, CTA, and LHAASO-WCDA. Early diagnosis of prompt emissions with Fermi-GBM and HAWC can provide valuable directional information for the follow-up observations., Comment: 11 pages, 5 figures. Accepted for publication in The Astrophysical Journal

Fast-rotating pulsars and magnetars have been suggested as the central engines of super-luminous supernovae (SLSNe) and fast radio bursts, and this scenario naturally predicts non-thermal synchrotron emission from their nascent pulsar wind nebulae (PWNe). We report results of high-frequency radio observations with ALMA and NOEMA for three SLSNe (SN 2015bn, SN 2016ard, and SN 2017egm), and present a detailed theoretical model to calculate non-thermal emission from PWNe with an age of about 1-3 yr. We find that the ALMA data disfavors a PWN model motivated by the Crab nebula for SN 2015bn and SN 2017egm, and argue that this tension can be resolved if the nebular magnetization is very high or very low. Such models can be tested by future MeV-GeV gamma-ray telescopes such as AMEGO., Comment: 7 pages, 4 figures, 2 tables

For a perfect fluid, the quantity defined through mixed components of the stress-energy tensor $\widetilde{w}=(T_{i}^{\phantom{i}i}/3)/(-T_{0}^{\phantom{0}0})$ is independent on the choice of coordinates only for two values of the pressure to energy density ratio $w=p/\rho$: for radiation with $w=1/3$, and for dark energy with $w=-1$. With other choices of $w$, the quantity $\widetilde{w}$ is coordinate dependent, and $\widetilde{w}=w$ only in the local rest frame of the fluid. We show that the same is true also for solid matter with shear stress Lam\'e coefficient set to zero in a flat Friedmann--Lema\^itre--Robertson--Walker universe with perturbed metric as well as stress-energy tensor. We call the two different solids with coordinate independent $\widetilde{w}$ radiation-like solid and dark energy-like solid, and we restrict ourselves to these two special cases. By analysing second order perturbations we discover two one parametric sets of such solid matter models containing special cases of radiation and dark energy as perfect fluids. We also study equations for perturbations up to the second order for both sets of models., Comment: 20 pages

As a powerful source of gravitational waves (GW), a supermassive black hole (SMBH) merger may be accompanied by a relativistic jet that leads to detectable electromagnetic (EM) emission. We model the propagation of post-merger jets inside a pre-merger circumnuclear environment formed by disk winds, and calculate multi-wavelength EM spectra from the forward shock region. We show that the non-thermal EM signals from SMBH mergers are detectable up to the detection horizon of future GW facilities such as the Laser Interferometer Space Antenna (LISA). Calculations based on our model predict slowly fading transients with time delays from days to months after the coalescence, leading to implications for EM follow-up observations after the GW detection., Comment: 6 pages, 3 figures, accepted for publication in ApJL. Schematic diagrams added

The recent detection of TeV photons from two gamma-ray bursts (GRBs), GRB 190114C and GRB 180720B, has opened a new window for multi-messenger and multi-wavelength astrophysics of high-energy transients. We study the origin of very-high-energy (VHE) $\gamma$-rays from the short GRB 160821B, for which the MAGIC Collaboration reported a $\sim 3 \sigma$ statistical significance. Short GRBs are often accompanied by extended and plateau emission, which is attributed to internal dissipation resulting from activities of a long-lasting central engine, and Murase et al. (2018) recently suggested the external inverse-Compton (EIC) scenario for VHE counterparts of short GRBs and neutron star mergers. Applying this scenario to GRB 160821B, we show that the EIC flux can reach $\sim 10^{-12}\rm~erg~cm^{-2}~s^{-1}$ within a time period of $\sim 10^3 - 10^4\rm~s$, which is consistent with the MAGIC observations. EIC $\gamma$-rays expected during the extended and plateau emission will be detectable with greater significance by future detectors such as the Cherenkov Telescope Array (CTA). The resulting light curve has a distinguishable feature, where the VHE emission is predicted to reach the peak around the end of the seed emission., Comment: 6 pages, 2 figures, accepted for publication in ApJL

The detection of TeV gamma-ray bursts (GRBs) brought new opportunities for studying the physics of particle acceleration at relativistic shocks. The \hess telescopes recently observed very-high-energy (VHE) emission from a nearby low-luminosity GRB, GRB 190829A. Follow-up observations with, e.g., Swift-XRT, revealed unusual flare activities at $\sim 10^3\rm~s$, which can be caused by a long-lasting central engine. We show that the VHE emission during the H.E.S.S. observation time is naturally produced in the external inverse-Compton (EIC) scenario, where seed photons supplied by the flares or other late-time dissipation are upscattered to VHE energies by the non-thermal electrons accelerated at the external forward shock. Our calculations show that the EIC flare nearly coincides with the late-prompt flare, but extends $\sim3-4$ times longer than the duration of the late-prompt flare. The preferred kinetic energy and initial Lorentz factor used in our model are $\sim 10^{52}\rm~erg$ and $\sim 20$, respectively. Understanding the mechanisms of the VHE emission from low-luminosity GRBs will help us constrain the properties of the outflow and the central engine activities, as well as the particle acceleration mechanism., Comment: 12 pages, 6 figures, accepted for publication in ApJ. HESS data are added, figures are updated, appendix is extended, main conclusions unchanged

We present a generic theoretical model for the structuring of a relativistic jet propagating through the ejecta of a binary neutron star merger event, introducing the effects of the neutron conversion-diffusion, which provides a baryon flux propagating transversely from the ejecta towards the jet axis. This results naturally in an increased baryon load structure of the outer jet with the approximate isotropic energy distribution $E_{iso}(\theta) \propto \theta^{-4}$, which is compatible with the first gravitational wave and short gamma-ray burst event GW170817/GRB 170817A observed at an off-axis angle of the jet., Comment: 17 pages, 10 figures, Accepted for publication in MNRAS

Supermassive black hole (SMBH) coalescences are ubiquitous in the history of the Universe and often exhibit strong accretion activities and powerful jets. These SMBH mergers are also promising candidates for future gravitational wave detectors such as Laser Space Inteferometric Antenna (LISA). In this work, we consider neutrino counterpart emission originating from the jet-induced shocks. The physical picture is that relativistic jets launched after the merger will push forward inside the premerger disk wind material, and then they subsequently get collimated, leading to the formation of internal shocks, collimation shocks, forward shocks and reverse shocks. Cosmic rays can be accelerated in these sites and neutrinos are expected via the photomeson production process. We formulate the jet structures and relevant interactions therein, and then evaluate neutrino emission from each shock site. We find that month-to-year high-energy neutrino emission from the postmerger jet after the gravitational wave event is detectable by IceCube-Gen2 within approximately five to ten years of operation in optimistic cases where the cosmic-ray loading is sufficiently high and a mildly super-Eddington accretion is achieved. We also estimate the contribution of SMBH mergers to the diffuse neutrino intensity, and find that a significant fraction of the observed very high-energy ($E_\nu\gtrsim1$ PeV) IceCube neutrinos could originate from them in the optimistic cases. In the future, such neutrino counterparts together with gravitational wave observations can be used in a multimessenger approach to elucidate in greater detail the evolution and the physical mechanism of SMBH mergers., Comment: 17 pages, 9 figures, 1 table. Accepted for publication in PRD

We investigate hadronic and leptonic scenarios for the GeV--TeV gamma-ray emission from jets of the microquasar SS 433. The emission region of the TeV photons coincides with the X-ray knots, where electrons are efficiently accelerated. On the other hand, the optical high-density filaments are also located close to the X-ray knots, which may support a hadronic scenario. We calculate multi-wavelength photon spectra of the extended jet region by solving the transport equations for the electrons and protons. We find that both hadronic and leptonic models can account for the observational data, including the latest {\it Fermi} LAT result. The hadronic scenarios predict higher-energy photons than the leptonic scenarios, and future observations such as with the Cherenkov Telescope Array (CTA), the Large High-Altitude Air Shower Observatory (LHAASO), and the Southern Wide-field Gamma-ray Observatory (SWGO) may distinguish between these scenarios and unravel the emission mechanism of GeV--TeV gamma-rays. Based on our hadronic scenario, the analogy between microquasars and radio galaxies implies that the X-ray knot region of the radio-galaxy jets may accelerate heavy nuclei up to ultrahigh energies., Comment: 9 pages, 2 figures, 2 tables, accepted for publication in ApJ, discussion added, conclusions unchanged

High-energy neutrinos are a promising tool for identifying astrophysical sources of high and ultra-high energy cosmic rays (UHECR). Prospects of detecting neutrinos at high energies ($\gtrsim$TeV) from blazars have been boosted after the recent association of IceCube-170922A and TXS 0506+056. We investigate the high-energy neutrino, IceCube-190331A, a high-energy starting event (HESE) with a high likelihood of being astrophysical in origin. We initiated a Swift/XRT and UVOT tiling mosaic of the neutrino localisation, and followed up with ATCA radio observations, compiling a multiwavelength SED for the most likely source of origin. NuSTAR observations of the neutrino location and a nearby X-ray source were also performed. We find two promising counterpart in the 90% confidence localisation region and identify the brightest as the most likely counterpart. However, no Fermi/LAT $\gamma$-ray source and no prompt Swift/BAT source is consistent with the neutrino event. At this point it is unclear whether any of the counterparts produced IceCube-190331A. We note that the Helix Nebula is also consistent with the position of the neutrino event, and we calculate that associated particle acceleration processes cannot produce the required energies to generate a high-energy HESE neutrino., Comment: 10 pages, 8 figures. Accepted for publication by MNRAS

The Universe is filled with a diffuse background of MeV gamma-rays and PeV neutrinos, whose origins are unknown. Here, we propose a scenario that can account for both backgrounds simultaneously. Low-luminosity active galactic nuclei have hot accretion flows where thermal electrons naturally emit soft gamma rays via Comptonization of their synchrotron photons. Protons there can be accelerated via turbulence or reconnection, producing high-energy neutrinos via hadronic interactions. We demonstrate that our model can reproduce the gamma-ray and neutrino data. Combined with a contribution by hot coronae in luminous active galactic nuclei, these accretion flows can explain the keV -- MeV photon and TeV -- PeV neutrino backgrounds. This scenario can account for the MeV background without non-thermal electrons, suggesting a higher transition energy from the thermal to non-thermal Universe than expected. Our model is consistent with X-ray data of nearby objects, and testable by future MeV gamma-ray and high-energy neutrino detectors., Comment: Published in Nature Communications. Conclusions unchanged. Calculation methods and presentations are improved. Discussions are extended

Some short gamma-ray bursts (SGRBs) show a longer lasting emission phase, called extended emission (EE) lasting $\sim 10^{2-3}\,\rm s$, as well as a plateau emission (PE) lasting $\sim10^{4-5}\,\rm s$. While a long-lasting activity of the central engines is a promising explanation for powering both emissions, their physical origin and their emission mechanisms are still uncertain. In this work, we study the properties of the EEs and their connection with the PEs. First, we constrain the minimal Lorentz factor $\Gamma$ of the outflows powering EEs, using compactness arguments and find that the outflows should be relativistic, $\Gamma\gtrsim10$. We propose a consistent scenario for the PEs, where the outflow eventually catches up with the jet responsible for the prompt emission, injecting energy into the forward shock formed by the prior jet, which naturally results in a PE. We also derive the radiation efficiency of EEs and the Lorentz factor of the outflow within our scenario for 10 well-observed SGRBs accompanied by both EE and PE. The efficiency has an average value of $\sim3\,\%$ but shows a broad distribution ranging from $\sim0.01$ to $\sim100\%$. The Lorentz factor is $\sim20-30$, consistent with the compactness arguments. These results suggest that EEs are produced by a slower outflow via more inefficient emission than the faster outflow which causes the prompt emission with a high radiation efficiency., Comment: 10 pages, 4 figures, 3 tables, accepted for publication in MNRAS

We investigate prolonged engine activities of short gamma-ray bursts (SGRBs), such as extended and/or plateau emissions, as high-energy gamma-ray counterparts to gravitational waves (GWs). Binary neutron-star mergers lead to relativistic jets and merger ejecta with $r$-process nucleosynthesis, which are observed as SGRBs and kilonovae/macronovae, respectively. Long-term relativistic jets may be launched by the merger remnant as hinted in X-ray light curves of some SGRBs. The prolonged jets may dissipate their kinetic energy within the radius of the cocoon formed by the jet-ejecta interaction. Then the cocoon supplies seed photons to non-thermal electrons accelerated at the dissipation region, causing high-energy gamma-ray production through the inverse Compton scattering process. We numerically calculate high-energy gamma-ray spectra in such a system using a one-zone and steady-state approximation, and show that GeV--TeV gamma-rays are produced with a duration of $10^2-10^5$ seconds. They can be detected by {\it Fermi}/LAT or CTA as gamma-ray counterparts to GWs., Comment: 8pages, 3 figures, accepted for publication in ApJL

The cores of active galactic nuclei (AGNs) have been suggested as the sources of IceCube neutrinos, and recent numerical simulations have indicated that hot AGN coronae of Seyfert galaxies and radiatively inefficient accretion flows (RIAFs) of low-luminosity AGNs (LLAGNs) may be promising sites of ion acceleration. We present detailed studies on detection prospects of high-energy multi-messenger emissions from RIAFs in nearby LLAGNs. We construct a model of RIAFs that can reproduce the observational features of the current X-ray observations of nearby LLAGNs. We then calculate the high-energy particle emissions from nearby individual LLAGNs, including MeV gamma rays from thermal electrons, TeV--PeV neutrinos produced by non-thermal protons, and sub-GeV to sub-TeV gamma rays from proton-induced electromagnetic cascades. We find that, although these are beyond the reach of current facilities, proposed future experiments such as e-ASTROGAM and IceCube-Gen2 should be able to detect the MeV gamma rays and the neutrinos, respectively, or else they can place meaningful constraints on the parameter space of the model. On the other hand, the detection of high-energy gamma rays due to the electromagnetic cascades will be challenging with the current and near-future experiments, such as Fermi and Cherenkov Telescope Array. In an accompanying paper, we demonstrate that LLAGNs can be a source of the diffuse soft gamma-ray and TeV--PeV neutrino backgrounds, whereas in the present paper, we focus on the prospects for multi-messenger tests which can be applied to reveal the nature of the high-energy neutrinos and photons from LLAGNs., Comment: 15 pages, 6 figures, 3 tables, Typos in Figure 3 are corrected, conclusions are unchanged

Fast radio bursts (FRBs) are short duration ($\sim$millisecond) radio transients with cosmological origin. The simple sharp features of the FRB signal have been utilized to probe two fundamental laws of physics, namey, testing Einstein's weak equivalence principle and constraining the rest mass of the photon. Recently, \cite{hessels18} found that after correcting for dispersive delay, some of the bursts in FRB 121102 have complex time-frequency structures that include sub-pulses with a time-frequency downward drifting property. Using the delay time between sub-pulses in FRB 121102, here we show that the parameterized post-Newtonian parameter $\gamma$ is the same for photons with different energies to the level of $\left|\gamma_1-\gamma_2\right|<2.5\times 10^{-16}$, which is 1000 times better than previous constraints from FRBs using similar methods. We also obtain a stringent constraint on the photon mass, $m_{\gamma} < 5.1\times10^{-48}$ g, which is 10 times smaller than previous best limits on the photon mass derived through the velocity dispersion method., Comment: 5 pages, 1 table, 1 figure

Multi-messenger astrophysics, a long-anticipated extension to traditional and multiwavelength astronomy, has recently emerged as a distinct discipline providing unique and valuable insights into the properties and processes of the physical universe. These insights arise from the inherently complementary information carried by photons, gravitational waves, neutrinos, and cosmic rays about individual cosmic sources and source populations. Realizing the observation of astrophysical sources via non-photonic messengers has presented enormous challenges, as evidenced by the fiscal and physical scales of the multi-messenger observatories. However, the scientific payoff has already been substantial, with even greater rewards promised in the years ahead. In this review we survey the current status of multi-messenger astrophysics, highlighting some exciting recent results, and addressing the major follow-on questions they have raised. Key recent achievements include the measurement of the spectrum of ultra-high energy cosmic rays out to the highest observable energies; discovery of the diffuse high energy neutrino background; the first direct detections of gravitational waves and the use of gravitational waves to characterize merging black holes and neutron stars in strong-field gravity; and the identification of the first joint electromagnetic + gravitational wave and electromagnetic + high-energy neutrino multi-messenger sources. We then review the rationales for the next generation of multi-messenger observatories, and outline a vision of the most likely future directions for this exciting and rapidly advancing field., Comment: Revised version of MS originally submitted on March 31, 2019; appeared in Nature Reviews Physics, 1:585 (2019); https://www.nature.com/articles/s42254-019-0101-z

Motivated by the recently reported evidence of an association between a high-energy neutrino and a gamma-ray flare from the blazar TXS 0506+056, we calculate the expected high-energy neutrino signal from past, individual flares, from twelve blazars, selected in declinations favourable for detection with IceCube. To keep the number of free parameters to a minimum, we mainly focus on BL Lac objects and assume the synchrotron self-Compton mechanism produces the bulk of the high-energy emission. We consider a broad range of the allowed parameter space for the efficiency of proton acceleration, the proton content of BL Lac jets, and the presence of external photon fields. To model the expected neutrino fluence we use simultaneous multi-wavelength observations. We find that in the absence of external photon fields and with jet proton luminosity normalised to match the observed production rate of ultra-high-energy cosmic rays, individual flaring sources produce a modest neutrino flux in IceCube, $\lesssim10^{-3}$ muon neutrinos with energy exceeding 100 TeV, stacking ten years of flare periods selected in the >800 MeV Fermi energy range from each source. Under optimistic assumptions about the jet proton luminosity and in the presence of external photon fields, we find that the two most powerful sources in our sample, AO 0235+164, and OJ 287, would produce, in total, $\approx 3$ muon neutrinos during ten years of Fermi flaring periods, in future neutrino detectors with total instrumented volume $\sim$ten times larger than IceCube,or otherwise, constrain the proton luminosity of blazar jets., Comment: 22 pages, 10 figures, submitted to MNRAS

We study effect of superhorizon tensor perturbations on scalar perturbations, so called effect of clustering fossils, in cosmological model in which inflation is driven by both solid matter and scalar field. The effect deforms primordial scalar power spectrum causing asymmetry in it, which leaves imprint on CMB anisotropies and the cosmic structure. Parameter space of this combined model allows for enhancement of logarithmic scale dependence of this deformation, as opposed to simpler models in which such scale dependence is suppressed by slow-roll parameters., Comment: 22 pages plus 3 appendices, 3 figures

We investigate the blazar contribution to the cumulative neutrino intensity assuming a generic relationship between neutrino and gamma-ray luminosities, $L_{\nu}\propto (L_{\rm ph})^{\gamma_{\rm lw}}$. Using the gamma-ray luminosity functions for blazars including flat spectrum radio quasars (FSRQs) and BL Lac objects, as well as the $Fermi$-LAT detection efficiency, we estimate contributions from {blazars resolved by $Fermi$-LAT as well as the unresolved counterpart.} Combining the existing upper limits from stacking analyses, the cumulative neutrino flux from all blazars ({including $Fermi$-LAT resolved and unresolved ones}) are constrained { in the range $0\lesssim\gamma_{\rm lw}\lesssim2.5$}. We also evaluate the effects of the redshift evolution and the effective local number densities for each class of FSRQs, BL Lacs, and all blazars, by which we place another type of constraints on the blazar contribution using the non-detection of high-energy neutrino multiplets. We demonstrate that these two upper limits are complementary, and that the joint consideration of the stacking and multiplet analyses { not only supports the argument that blazars are disfavored as the dominant sources of the 100-TeV neutrino background, but it extends this argument by including also $Fermi$-LAT-unresolved blazars as well, for a more generic luminosity correlation $L_{\nu}\propto (L_{\rm ph})^{\gamma_{\rm lw}}$}., Comment: Accepted for publication in The Astrophysical Journal. 8 pages, 4 figures

Mysteries about the origin of high-energy cosmic neutrinos have deepened by the recent IceCube measurement of a large diffuse flux in the 10-100 TeV range. Based on the standard disk-corona picture of active galactic nuclei (AGN), we present a phenomenological model enabling us to systematically calculate the spectral sequence of multimessenger emission from the AGN coronae. We show that protons in the coronal plasma can be stochastically accelerated up to PeV energies by plasma turbulence, and find that the model explains the large diffuse flux of medium-energy neutrinos if the cosmic rays carry only a few percent of the thermal energy. We find that the Bethe-Heitler process plays a crucial role in connecting these neutrinos and cascaded MeV gamma rays, and point out that the gamma-ray flux can even be enhanced by the reacceleration of secondary pairs. Critical tests of the model are given by its prediction that a significant fraction of the MeV gamma-ray background correlates with about 10 TeV neutrinos, and nearby Seyfert galaxies including NGC 1068 are promising targets for IceCube, KM3Net, IceCube-Gen2, and future MeV gamma-ray telescopes., Comment: 7+6 pages, 4+6 figures, 0+2 tables, version to appear in PRL (submitted on Apr/08/2019), with clarifications and extended descriptions (including discussion on NGC 1068). Conclusions unchanged

We calculate spectra of escaping cosmic rays (CRs) accelerated at shocks produced by expanding Galactic superbubbles powered by multiple supernovae producing a continuous energy outflow in star-forming galaxies. We solve the generalized Kompaneets equations adapted to expansion in various external density profiles, including exponential and power-law shapes, and take into account that escaping CRs are dominated by those around their maximum energies. We find that the escaping CR spectrum largely depends on the specific density profiles and power source properties, and the results are compared to and constrained by the observed CR spectrum. As a generic demonstration, we apply the scheme to a superbubble occurring in the centre of the Milky Way, and find that under specific parameter sets the CRs produced in our model can explain the observed CR flux and spectrum around the second knee at $10^{17}$ eV., Comment: 11 pages, 3 figures, accepted by MNRAS

Shocks arising in galaxy mergers could accelerate cosmic-ray (CR) ions to TeV-PeV energies. While propagating in the intergalactic medium, these CRs can produce high-energy neutrinos, electron-positron pairs and gamma-rays. In the presence of intergalactic magnetic fields, the secondary pairs will radiate observable emissions through synchrotron radiation and inverse Compton scattering. In this paper, we demonstrate that these emissions can explain the radio and X-ray fluxes of merging galaxies such as NGC 660 and NGC 3256. Using our model in combination with the observations, we can constrain the gas mass, shock velocity, magnetic field and the CR spectral index $s$ of these systems. For NGC 660 a single-zone model with a spectral index $2.1\lesssim s\lesssim2.2$ is able to reproduce simultaneously the radio and X-ray observations, while a simple one-zone scenario with $s\sim2$ can describe the radio and a large fraction of X-ray observations of NGC 3256. Our work provides a useful approach for studying the dynamics and physical parameters of galaxy mergers, which can play an important part in future multi-messenger studies of similar and related extragalactic sources., Comment: 11 pages, 6 figures. Accepted by ApJ

The ANITA collaboration have reported observation of two anomalous events that appear to be $\varepsilon_{\rm cr} \approx 0.6$ EeV cosmic ray showers emerging from the Earth with exit angles of $27^\circ$ and $35^\circ$, respectively. While EeV-scale upgoing showers have been anticipated as a result of astrophysical tau neutrinos converting to tau leptons during Earth passage, the observed exit angles are much steeper than expected in Standard Model (SM) scenarios. Indeed, under conservative extrapolations of the SM interactions, there is no particle that can propagate through the Earth with probability $p > 10^{-6}$ at these energies and exit angles. We explore here whether "beyond the Standard Model" (BSM) particles are required to explain the ANITA events, if correctly interpreted, and conclude that they are. Seeking confirmation or refutation of the physical phenomenon of sub-EeV Earth-emergent cosmic rays in data from other facilities, we find support for the reality of the ANITA events, and three candidate analog events, among the Extremely High Energy Northern Track neutrinos of the IceCube Neutrino Observatory. Properties of the implied BSM particle are anticipated, at least in part, by those predicted for the "stau" slepton ($\tilde{\tau}_R$) in some supersymmetric models of the fundamental interactions, wherein the stau manifests as the next-to-lowest mass supersymmetric partner particle., Comment: PRD submitted. 15 pages, 3 figures

The detection of gravitational waves and electromagnetic counterparts from a binary neutron star (BNS) merger confirmed that it is accompanied by the launch of fast merger ejecta. Analogous to supernova remnants, forward shocks formed by the interaction of the ejecta with interstellar material will produce high-energy cosmic rays. We investigate the possibility that Galactic neutron star merger remnants (NSMRs) significantly contribute to the observed cosmic rays in the energy range between the knee and the ankle. Using typical parameters obtained by modeling of GW170817, we find that NSMRs can accelerate iron nuclei up to $\sim500$ PeV. We calculate the cosmic-ray spectrum and composition observed on Earth, and show that the Galactic NSMR scenario can account for the experimental cosmic-ray data in the 20 -- 1000 PeV range. Our model can naturally explain the hardening feature around 20 PeV for total cosmic-ray spectrum, which has been observed by the Telescope Array Low Energy extension and the IceTop air shower array., Comment: 8 pages, 1 figure, 1 table, accepted for publication in ApJ

The observations of a macronova/kilonova accompanied by gravitational waves from a binary neutron star merger (GW170817) confirmed that neutron star coalescences produce copious ejecta. The coincident gamma-ray detection implies the existence of a relativistic jet in this system. During the jet's propagation within the ejecta, high-energy photons and neutrinos can be produced. The photons are absorbed by the ejecta, while the neutrinos escape and can be detected. Here, we estimate such trans-ejecta neutrino emission, and discuss how neutrino observations could be used to differentiate between gamma-ray burst emission scenarios. We find that neutrinos from the internal shocks inside the ejecta may be detectable by IceCube within a few years of operation, and will likely be detected with IceCube-Gen2. The neutrino signals coincident with gravitational waves would enable us to reveal the physical quantities of the choked jets even without electromagnetic signals., Comment: 12 pages, 5 figures, 2 tables, accepted for publication in PRD

The merger of dark matter halos and the gaseous structures embedded in them, such as proto-galaxies, galaxies, and groups and clusters of galaxies, results in strong shocks that are capable of accelerating cosmic rays (CRs) to $\sim10~\rm PeV$. These shocks will produce high-energy neutrinos and $\gamma$-rays through inelastic $pp$ collisions with ambient gaseous environments. In this work, we study the contributions of these halo mergers to the diffuse neutrino flux measured in IceCube and to the non-blazar portion of the extragalactic $\gamma$-ray background measured by $Fermi$. In order to calculate them, we formulate the redshift dependence of the shock velocity, galactic radius, halo gas content and galactic/intergalactic magnetic fields over the dark matter halo distribution up to a redshift $z=10$. We find that high-redshift mergers contribute a significant amount of the cosmic-ray energy luminosity density, and the resulting neutrino spectra could explain a large part of the observed diffuse neutrino flux above 0.1 PeV up to several PeV. We also show that our model can somewhat alleviate tensions with the extragalactic $\gamma$-ray background. First, since a larger fraction of the CR energy luminosity density comes from high redshifts, the accompanying $\gamma$-rays are more strongly suppressed through $\gamma\gamma$ annihilations with the cosmic microwave background (CMB) and the extragalactic background light (EBL). Second, mildly radiative-cooled shocks may lead to a harder CR spectrum with spectral indices of $1.5\lesssim s\lesssim2.0$. Our study suggests that halo mergers, a fraction of which may also induce starbursts in the merged galaxies, can be promising neutrino emitters without violating the existing $Fermi$ $\gamma$-ray constraints on the non-blazar component of the extragalactic $\gamma$-ray background., Comment: 13 pages, 7 figures, ApJ accepted

Recent results from the Pierre Auger Collaboration have shown that the composition of ultrahigh-energy cosmic rays (UHECRs) becomes gradually heavier with increasing energy. Although gamma-ray bursts (GRBs) have been promising sources of UHECRs, it is still unclear whether they can account for the Auger results because of their unknown nuclear composition of ejected UHECRs. In this work, we revisit the possibility that low-luminosity GRBs (LL GRBs) act as the sources of UHECR nuclei, and give new predictions based on the intrajet nuclear composition models considering progenitor dependencies. We find that the nuclear component in the jet can be divided into two groups according to the mass fraction of silicon nuclei, Si-free and Si-rich. Motivated by the connection between LL GRBs and transrelativistic supernovae, we also consider the hypernova ejecta composition. Then, we discuss the survivability of UHECR nuclei in the jet base and internal shocks of the jets, and show that it is easier for nuclei to survive for typical LL GRBs. Finally, we numerically propagate UHECR nuclei ejected from LL GRBs with different composition models and compare the resulting spectra and composition to Auger data. Our results show that both the Si-rich progenitor and hypernova ejecta models match the Auger data well, while the Si-free progenitor models have more difficulty in fitting the spectrum. We argue that our model is consistent with the newly reported cross correlation between the UHECRs and starburst galaxies, since both LL GRBs and hypernovae are expected to be tracers of the star-formation activity. LL GRBs have also been suggested as the dominant origin of IceCube neutrinos in the PeV range, and the LL GRB origin of UHECRs can be critically tested by near-future multimessenger observations., Comment: 17 pages, 16 figures, minor corrections, added the discussion about the possible correlation with starburst galaxies, version accepted by PRD

Solid inflation is a cosmological model where inflation is driven by fields which enter the Lagrangian in the same way as body coordinates of a solid matter enter the equation of state, spontaneously breaking spatial translational and rotational symmetry. We construct a simple generalization of this model by adding a scalar field with standard kinetic term to the action. In our model the scalar power spectrum and the tensor-to-scalar ratio do not differ from the ones predicted by the solid inflation qualitatively, if the scalar field does not dominate the solid matter. The same applies also for the size of the scalar bispectrum measured by the non-linearity parameter, although our model allows it to have different shapes. The tensor bispectra predicted by the two models do not differ from each other in the leading order of the slow-roll approximation. In the case when contribution of the solid matter to the stress-energy tensor is much smaller than the contribution from the scalar field, the tensor-to-scalar ratio and the non-linearity parameter are amplified by factors $\epsilon^{-1}$ and $\epsilon^{-2}$ respectively., Comment: 23 pages, 3 figures

We discuss the electromagnetic radiation from newborn binary black holes (BBHs). As a consequence of the evolution of massive stellar binaries, a binary consisting of a primary black hole (BH) and a secondary Wolf-Rayet star is expected as a BBH progenitor system. We investigate optical transients from the birth of BBHs powered by the Bondi-Hoyle-Lyttleton accretion onto the primary BH, which occur $\sim1-10$ Gyr earlier than gravitational wave signals at the BH-BH merger. When the secondary massive star collapses into a BH, it may eject a fraction of its outer material and may form a disk around the primary BH and induces a powerful disk wind. These primary-induced winds can lead to optical transients with a kinetic energy of $\sim10^{47}-3\times10^{48}$ erg, an ejecta velocity of $10^8-10^9\rm~cm~s^{-1}$, a duration of a few days, and an absolute magnitude ranging from about $-11$ to $-14$. The light curves and late-time spectra of these transients are distinctive from those of ordinary supernovae, and detection of this type of transient is possible by future optical transient surveys if the event rate of this transient is comparable to the merger rate of BBHs. This paper focuses on the emissions from disk-driven transients induced by the primary BH, different from the first paper that focuses on wind-driven transients from the tidally-locked secondary massive star., Comment: 9 pages, 3 figures, accepted for publication in ApJ, arXiv:1702.07337v2 was split into two papers according to referee's suggestion. This is Paper II. See also arXiv:1702.07337v3 for Paper I

The recent detection of gravitational waves and electromagnetic counterparts from the double neutron star merger event GW+EM170817, supports the standard paradigm of short gamma-ray bursts (SGRBs) and kilonovae/macronovae. It is important to reveal the nature of the compact remnant left after the merger, either a black hole or neutron star, and their physical link to the origin of the long-lasting emission observed in SGRBs. The diversity of the merger remnants may also lead to different kinds of transients that can be detected in future. Here we study the high-energy emission from the long-lasting central engine left after the coalescence, under certain assumptions. In particular, we consider the X-ray emission from a remnant disk and the non-thermal nebular emission from disk-driven outflows or pulsar winds. We demonstrate that late-time X-ray and high-frequency radio emission can provide useful constraints on properties of the hidden compact remnants and their connections to long-lasting SGRB emission, and we discuss the detectability of nearby merger events through late-time observations at $\sim30-100$ d after the coalescence. We also investigate the GeV-TeV gamma-ray emission that occurs in the presence of long-lasting central engines, and show the importance of external inverse-Compton radiation due to up-scattering of X-ray photons by relativistic electrons in the jet. We also search for high-energy gamma-rays from GW170817 in the Fermi-LAT data, and report upper limits on such long-lasting emission. Finally, we consider the implications of GW+EM170817 and discuss the constraints placed by X-ray and high-frequency radio observations., Comment: 16 pages, 12 figures, accepted for publication in ApJ, minor changes

The coincident detection of a gravitational-wave (GW) event GW170817 with electromagnetic (EM) signals (e.g., a short gamma-ray burst SGRB 170817A or a macronova) from a binary neutron star merger within the nearby galaxy NGC 4933 provides a new, multimessenger test of the weak equivalence principle (WEP), extending the WEP test with GWs and photons. Assuming that the arrival time delay between the GW signals from GW170817 and the photons from SGRB 170817A or the macronova is mainly attributed to the gravitational potential of the Milky Way, we demonstrate that the strict upper limits on the deviation from the WEP are $\Delta \gamma<1.4\times10^{-3}$ for GW170817/macronova and $\Delta \gamma <5.9\times10^{-8}$ for GW170817/SGRB 170817A. A much more severe constraint on the WEP accuracy can be achieved ($\sim0.9\times10^{-10}$) for GW170817/SGRB 170817A when we consider the gravitational potential of the Virgo Cluster, rather than the Milky Way's gravity. This provides the tightest limit to date on the WEP through the relative differential variations of the $\gamma$ parameter for two different species of particles. Compared with other multimessenger (photons and neutrinos) results, our limit is 7 orders of magnitude tighter than that placed by the neutrinos and photons from supernova 1987A, and is almost as good as or is an improvement of 6 orders of magnitude over the limits obtained by the low-significance neutrinos correlated with GRBs and a blazar flare., Comment: 10 pages, no figures. Accepted by JCAP on Nov 8, 2017

We investigate current and future prospects for coincident detection of high-energy neutrinos and gravitational waves (GWs). Short gamma-ray bursts (SGRBs) are believed to originate from mergers of compact star binaries involving neutron stars. We estimate high-energy neutrino fluences from prompt emission, extended emission, X-ray flares, and plateau emission, and show that neutrino signals associated with the extended emission are the most promising. Assuming that the cosmic-ray loading factor is $\sim10$ and the Lorentz factor distribution is lognormal, we calculate the probability of neutrino detection from extended emission by current and future neutrino detectors, and find that the quasi-simultaneous detection of high-energy neutrinos, gamma rays, and GWs is possible with future instruments or even with current instruments for nearby SGRBs having extended emission. We also discuss stacking analyses that will also be useful with future experiments such as IceCube-Gen2., Comment: 8 pages, 2 figures, 3 tables, accepted for publication in ApJL

We compute the CMB angular power spectrum in the presence of the radiation-like solid with the same pressure to energy density ratio as for the radiation but with nonzero shear modulus. The effect can be observable not only for large-angle anisotropies as shown by Balek and \v{S}kovran [2015] but also for very small-angle ones., Comment: 7 pages, 1 figure, conference: Week of Doctoral Students 2016, Prague

There are indications that $\gamma$-ray dark objects such as supernovae (SNe) with choked jets, and the cores of active galactic nuclei may contribute to the diffuse flux of astrophysical neutrinos measured by the IceCube observatory. In particular, stripped-envelope SNe have received much attention since they are capable of producing relativistic jets and could explain the diversity in observations of collapsar explosions (e.g., gamma-ray bursts (GRBs), low-luminosity GRBs, and Type Ibc SNe). We use an unbinned maximum likelihood method to search for spatial and temporal coincidences between Type Ibc core-collapse SNe, which may harbor a choked jet, and muon neutrinos from a sample of IceCube up-going track-like events measured from May 2011-May 2012. In this stacking analysis, we find no significant deviation from a background-only hypothesis using one year of data, and are able to place upper limits on the total amount of isotropic equivalent energy that choked jet core-collapse SNe deposit in cosmic rays ${\mathcal E}_{\rm cr}$ and the fraction of core-collapse SNe which have a jet pointed towards Earth $f_{\rm jet}$. This analysis can be extended with yet to be made public IceCube data, and the increased amount of optically detected core-collapse SNe discovered by wide field-of-view surveys such as the Palomar Transient Factory and All-Sky Automated Survey for Supernovae. The choked jet SNe/high-energy cosmic neutrino connection can be more tightly constrained in the near future., Comment: 21 pages, 7 figures, 1 table, accepted for publication in JCAP

High-energy neutrinos are expected to originate from different stages in a gamma-ray burst (GRB) event. In this work we revisit the dissipative photospheric scenario, in which the GRB prompt emission is produced around the photospheric radius. Meanwhile, possible dissipation mechanisms (e.g., internal shocks or magnetic reconnection) could accelerate cosmic-rays (CRs) to ultra-high energies and then produce neutrinos via hadronuclear and photohadronic processes, which are referred to as prompt neutrinos. In this paper, we obtain the prompt neutrino spectrum of a single GRB within a self-consistent analytical framework, in which the jet-cocoon structure and possible collimation effects are included. We investigate a possible neutrino signal from the cocoon, which has been ignored in the previous studies. We show that if a GRB event happens at a distance of the order of Mpc, there is a great chance to observe the neutrino emission from the cocoon by IceCube, which is even more promising than jet neutrinos since the opening angle of the cocoon is much larger. We also figure out the diffuse neutrino flux of GRB cocoons and find that it could be comparable with that of the jets. Our results are consistent with the latest result reported by the IceCube collaboration that no significant correlation between neutrino events and observed GRBs is seen in the new data (Aartsen et al. 2017)., Comment: 6 pages, 6 figures, accepted for publication in ApJ

Violations of Lorentz invariance can lead to an energy-dependent vacuum dispersion of light, which results in arrival-time differences of photons arising with different energies from a given transient source. In this work, direction-dependent dispersion constraints are obtained on nonbirefringent Lorentz-violating effects, using the observed spectral lags of the gamma-ray burst GRB 160625B. This burst has unusually large high-energy photon statistics, so we can obtain constraints from the true spectral time lags of bunches of high-energy photons rather than from the rough time lag of a single highest-energy photon. Also, GRB 160625B is the only burst to date having a well-defined transition from positive lags to negative lags, which provides a unique opportunity to distinguish Lorentz-violating effects from any source-intrinsic time lag in the emission of photons of different energy bands. Our results place comparatively robust two-sided constraints on a variety of isotropic and anisotropic coefficients for Lorentz violation, including first bounds on Lorentz-violating effects from operators of mass dimension ten in the photon sector., Comment: 6 pages, 3 figures, 1 table. Accepted for publication in ApJ

Einstein's weak equivalence principle (WEP) states that any freely falling, uncharged test particle follows the same identical trajectory independent of its internal structure and composition. Since the polarization of a photon is considered to be part of its internal structure, we propose that polarized photons from astrophysical transients, such as gamma-ray bursts (GRBs) and fast radio bursts (FRBs), can be used to constrain the accuracy of the WEP through the Shapiro time delay effect. Assuming that the arrival time delays of photons with different polarizations are mainly attributed to the gravitational potential of the Laniakea supercluster of galaxies, we show that a strict upper limit on the differences of the parametrized post-Newtonian parameter $\gamma$ value for the polarized optical emission of GRB 120308A is $\Delta\gamma<1.2\times10^{-10}$, for the polarized gamma-ray emission of GRB 100826A is $\Delta\gamma<1.2\times10^{-10}$, and for the polarized radio emission of FRB 150807 is $\Delta\gamma<2.2\times10^{-16}$. These are the first direct verifications of the WEP for multiband photons with different polarizations. In particular, the result from FRB 150807 provides the most stringent limit to date on a deviation from the WEP, improving by one order of magnitude the previous best result based on Crab pulsar photons with different energies., Comment: 7 pages, 1 table. Updated to match the published version

We discuss likely sources of cosmic rays in the $10^{15}-10^{20}$ eV range and their possible very high energy neutrino and gamma-ray signatures which could serve to identify these sources and constrain their physics. Among these sources we discuss in particular low luminosity gamma-ray bursts, including choked and shock-breakout objects, tidal disruption events and white dwarf mergers. Among efforts aimed at simultaneous secondary multi-messenger detections we discuss the AMON program., Comment: 9 pages; Based on a talk given at the Origin of Cosmic Rays: Beyond the Standard Model 2016 conference in San Vito di Cadore, Dolomites; This is not a comprehensive review; it is weighted towards work the author has been more personally involved with. Fig 3 has been updated to reflect the published work

We propose a new type of electromagnetic transients associated with the birth of binary black holes (BBHs), which may lead to merger events accompanied by gravitational waves in $\sim0.1-1$ Gyr. We consider the newborn BBHs formed through the evolution of isolated massive stellar binaries. For a close massive binary, consisting of a primary black hole (BH) and a secondary Wolf-Rayet (WR) star that are orbiting around each other, the spin period of the WR star can be tidally synchronized to its orbital period. Then, the angular momentum of the outer material of the WR star is large enough to form an accretion disk around a newborn, secondary BH, following its core-collapse. This disk may produce an energetic outflow with a kinetic energy of $\sim10^{50}-10^{52}$ erg and an outflow velocity of $\sim10^{10}\rm~cm~s^{-1}$, resulting in an optical transient with an absolute magnitude from $\sim -14$ to $\sim-17$ with a duration of around a day. This type of transient also produces detectable radio signals $\sim1-10$ years after the birth of BBHs, via synchrotron emission from non-thermal electrons accelerated at external shocks. The predicted optical transients have a shorter duration than ordinary core-collapse supernovae. Dedicated optical transient surveys can detect them, and distinguish it from ordinary SNe using the different features of its light curve and late-time spectrum. In the paper I, we investigate disk-driven outflows from the secondary BH, whereas possible signals from the primary BH will be discussed in the paper II., Comment: v3: 8 pages, 3 figures, accepted for publication in ApJ. arXiv:1702.07337v2 was split into two papers according to referee's suggestion. This is Paper I. See also arXiv:1711.01015 for Paper II

We compute the CMB angular power spectrum in the presence of a radiation-like solid - elastic matter with the same pressure to energy density ratio as radiation but with nonzero shear modulus. For the values of shear modulus that are close enough to zero, so that the effect of the solid on large-scale anisotropies remains within cosmic variance, we find that there is an observable effect of the solid on small-scale anisotropies., Comment: 16 pages, 3 figures

Possible violations of Lorentz invariance (LIV) have been investigated for a long time using the observed spectral lags of gamma-ray bursts (GRBs). However, these generally have relied on using a single photon in the highest energy range. Furthermore, the search for LIV lags has been hindered by our ignorance concerning the intrinsic time lag in different energy bands. GRB 160625B, the only burst so far with a well-defined transition from $positive$ lags to $negative$ lags provides a unique opportunity to put new constraints on LIV. Using multi-photon energy bands we consider the contributions to the observed spectral lag from both the intrinsic time lag and the lag by LIV effects, and assuming the intrinsic time lag to have a positive dependence on the photon energy, we obtain robust limits on LIV by directly fitting the spectral lag data of GRB 160625B. Here we show that these robust limits on the quantum gravity energy scales are $E_{\rm QG,1}\geq0.5\times10^{16}$ GeV for the linear, and $E_{\rm QG,2}\geq1.4\times10^{7}$ GeV for the quadratic LIV effects, respectively. In addition, we give for the first time a reasonable formulation of the intrinsic energy-dependent time lag., Comment: 6 pages, 4 figures, 1 table. Accepted for publication in ApJ Letters

X-ray and $\gamma$-ray observations by the Swift satellite revealed that a fraction of tidal disruption events (TDEs) have relativistic jets. Jetted TDEs have been considered as potential sources of very high-energy cosmic-rays and neutrinos. In this work, using semi-analytical methods, we calculate neutrino spectra of X-ray bright TDEs with powerful jets and dark TDEs with possible choked jets, respectively. We estimate their neutrino fluxes and find that the non-detection would give us an upper limit on the cosmic-ray loading factor, $\xi_{\rm cr}\lesssim20$, for Sw J1644+57. We show that X-ray bright TDEs make a sub-dominant ($\lesssim5-10$%) contribution to IceCube's diffuse neutrino flux, and study possible contributions of X-ray dark TDEs given that particles are accelerated in choked jets or disk-winds. We discuss future prospects for multi-messenger searches of the brightest TDEs., Comment: v3: 9 pages, 7 figures, accepted in ApJ

We consider some general implications of bright gamma-ray counterparts to fast radio bursts (FRBs). We show that even if these manifest in only a fraction of FRBs, gamma-ray detections with current satellites (including Swift) can provide stringent constraints on cosmological FRB models. If the energy is drawn from the magnetic energy of a compact object such as a magnetized neutron star, the sources should be nearby and be very rare. If the intergalactic medium is responsible for the observed dispersion measure, the required gamma-ray energy is comparable to that of the early afterglow or extended emission of short gamma-ray bursts. While this can be reconciled with the rotation energy of compact objects, as expected in many merger scenarios, the prompt outflow that yields the gamma-rays is too dense for radio waves to escape. Highly relativistic winds launched in a precursor phase, and forming a wind bubble, may avoid the scattering and absorption limits and could yield FRB emission. Largely independent of source models, we show that detectable radio afterglow emission from gamma-ray bright FRBs can reasonably be anticipated. Gravitational wave searches can also be expected to provide useful tests., Comment: 8 pages, version to appear in ApJL, minor changes (references and discussions added)