Tidal deformation of a slowly rotating material body: Interior metric and Love numbers

The metric outside a compact body deformed by a quadrupolar tidal field is universal up to its Love numbers, constants which encode the tidal response's dependence on the body's internal structure. For a non-rotating body, the deformed external geometry is characterized by the familiar gravitational Love numbers $K_2^{\text{el}}$ and $K_2^{\text{mag}}$. For a slowly rotating body, these must be supplemented by rotational-tidal Love numbers, which measure the response to couplings between the body's spin and the external tidal field. By integrating the interior field equations, I find that the response of a barotropic perfect fluid to spin-coupled tidal perturbations is described by two rotational-tidal Love numbers, which I calculate explicitly for polytropes. Two other rotational-tidal Love numbers identified in prior work are found to have a fixed, universal value for all barotropes. Equipped with the complete interior solution, I calculate the amplitude of the time-varying internal currents induced by the gravitomagnetic part of the tidal field. For a typical neutron star in an equal-mass binary system, the size of the equatorial velocity perturbation is on the order of kilometers per second.Comment: 19 pages, 7 figures; updated figures and corrected typos; matches the published versio

Relativistic theory of surficial Love numbers

A relativistic theory of surficial Love numbers, which characterize the surface deformation of a body subjected to tidal forces, was initiated by Damour and Nagar. We revisit this effort in order to extend it, clarify some of its aspects, and simplify its computational implementation. First, we refine the definition of surficial Love numbers proposed by Damour and Nagar, and formulate it directly in terms of the deformed curvature of the body's surface, a meaningful geometrical quantity. Second, we develop a unified theory of surficial Love numbers that applies equally well to material bodies and black holes. Third, we derive a compactness-dependent relation between the surficial and (electric-type) gravitational Love numbers of a perfect-fluid body, and show that it reduces to the familiar Newtonian relation when the compactness is small. And fourth, we simplify the tasks associated with the practical computation of the surficial and gravitational Love numbers for a material body.Comment: 12 pages, 2 figure

Tidal deformation of a slowly rotating material body. External metric

We construct the external metric of a slowly rotating, tidally deformed material body in general relativity. The tidal forces acting on the body are assumed to be weak and to vary slowly with time, and the metric is obtained as a perturbation of a background metric that describes the external geometry of an isolated, slowly rotating body. The tidal environment is generic and characterized by two symmetric-tracefree tidal moments E_{ab} and B_{ab}, and the body is characterized by its mass M, its radius R, and a dimensionless angular-momentum vector \chi^a << 1. The perturbation accounts for all couplings between \chi^a and the tidal moments. The body's gravitational response to the applied tidal field is measured in part by the familiar gravitational Love numbers K^{el}_2 and K^{mag}_2, but we find that the coupling between the body's rotation and the tidal environment requires the introduction of four new quantities, which we designate as rotational-tidal Love numbers. All these Love numbers are gauge invariant in the usual sense of perturbation theory, and all vanish when the body is a black hole.Comment: 17 pages, 0 figures, matches the published versio

Gravitomagnetic response of an irrotational body to an applied tidal field

The deformation of a nonrotating body resulting from the application of a tidal field is measured by two sets of Love numbers associated with the gravitoelectric and gravitomagnetic pieces of the tidal field, respectively. The gravitomagnetic Love numbers were previously computed for fluid bodies, under the assumption that the fluid is in a strict hydrostatic equilibrium that requires the complete absence of internal motions. A more realistic configuration, however, is an irrotational state that establishes, in the course of time, internal motions driven by the gravitomagnetic interaction. We recompute the gravitomagnetic Love numbers for this irrotational state, and show that they are dramatically different from those associated with the strict hydrostatic equilibrium: While the Love numbers are positive in the case of strict hydrostatic equilibrium, they are negative in the irrotational state. Our computations are carried out in the context of perturbation theory in full general relativity, and in a post-Newtonian approximation that reproduces the behavior of the Love numbers when the body's compactness is small.Comment: 14 pages, 4 figure

Constraints on the moment of inertia of PSR J0737-3039A from GW170817

Continued observation of PSR J0737-3039, the double pulsar, is expected to yield a precise determination of its primary component's moment of inertia in the next few years. Since the moment of inertia depends sensitively on the neutron star's internal structure, such a measurement will constrain the equation of state of ultra-dense matter, which is believed to be universal. Independent equation-of-state constraints have already been established by the gravitational-wave measurement of the neutron-star tidal deformability in GW170817. Here, using well-known universal relations among neutron star observables, we translate the reported 90%-credible bounds on tidal deformability into a direct constraint, $I_{\star} = 1.15^{+0.38}_{-0.24} \times 10^{45} \text{ g cm}^2$, on the moment of inertia of PSR J0737-3039A. Should a future astrophysical measurement of $I_{\star}$ disagree with this prediction, it could indicate a breakdown in the universality of the neutron-star equation of state.Comment: 8 pages, 4 figures; matches the published versio

Non-parametric inference of the neutron star equation of state from gravitational wave observations

We develop a non-parametric method for inferring the universal neutron star (NS) equation of state (EOS) from gravitational wave (GW) observations. Many different possible realizations of the EOS are generated with a Gaussian process conditioned on a set of nuclear-theoretic models. These synthetic EOSs are causal and thermodynamically stable by construction, span a broad region of the pressure-density plane, and can be selected to satisfy astrophysical constraints on the NS mass. Associating every synthetic EOS with a pair of component masses $M_{1,2}$ and calculating the corresponding tidal deformabilities $\Lambda_{1,2}$, we perform Monte Carlo integration over the GW likelihood for $M_{1,2}$ and $\Lambda_{1,2}$ to directly infer a posterior process for the NS EOS. We first demonstrate that the method can accurately recover an injected GW signal, and subsequently use it to analyze data from GW170817, finding a canonical deformability of $\Lambda_{1.4} = 160^{+448}_{-113}$ and $p(2\rho_{\mathrm{nuc}})=1.35^{+1.8}_{-1.2}\times 10^{34}~\mathrm{dyn}/\mathrm{cm}^2$ for the pressure at twice the nuclear saturation density at 90$\%$ confidence, in agreement with previous studies, when assuming a loose EOS prior. With a prior more tightly constrained to resemble the theoretical EOS models, we recover $\Lambda_{1.4} = 556^{+163}_{-172}$ and $p(2\rho_{\mathrm{nuc}})=4.73^{+1.4}_{-2.5}\times 10^{34}~\mathrm{dyn}/\mathrm{cm}^2$. We further infer the maximum NS mass supported by the EOS to be $M_\mathrm{max}=2.09^{+0.37}_{-0.16}$ ($2.04^{+0.22}_{-0.002}$) $M_\odot$ with the loose (tight) prior. The Bayes factor between the two priors is $B^{\mathcal{A}}_{\mathcal{I}} \simeq 1.12$, implying that neither is strongly preferred by the data and suggesting that constraints on the EOS from GW170817 alone may be relatively prior-dominated.Comment: 27 pages, 12 figures; references adde

Dynamical response to a stationary tidal field

We demonstrate that a slowly rotating compact body subjected to a stationary tidal field undergoes a dynamical response, in which the fluid variables and the interior metric vary on the time scale of the rotation period. This dynamical response requires the tidal field to have a gravitomagnetic component generated by external mass currents; the response to a gravitoelectric tidal field is stationary. We confirm that in a calculation carried out to first order in the body's rotation, the exterior geometry bears no trace of this internal dynamics; it remains stationary in spite of the time-dependent interior.Comment: 12 page

CELLO-3D: Estimating the Covariance of ICP in the Real World

The fusion of Iterative Closest Point (ICP) reg- istrations in existing state estimation frameworks relies on an accurate estimation of their uncertainty. In this paper, we study the estimation of this uncertainty in the form of a covariance. First, we scrutinize the limitations of existing closed-form covariance estimation algorithms over 3D datasets. Then, we set out to estimate the covariance of ICP registrations through a data-driven approach, with over 5 100 000 registrations on 1020 pairs from real 3D point clouds. We assess our solution upon a wide spectrum of environments, ranging from structured to unstructured and indoor to outdoor. The capacity of our algorithm to predict covariances is accurately assessed, as well as the usefulness of these estimations for uncertainty estimation over trajectories. The proposed method estimates covariances better than existing closed-form solutions, and makes predictions that are consistent with observed trajectories

Modeling and Solving Alternative Financial Solutions Seeking

In this paper we build a method to optimize Multi-Year Prospective Budgets. First we present a systemic model of Local Community Finances. Then, from two acceptable Multi-Year Prospective Budgets the method implements a Genetic Algorithm to generate a collection of admissible Multi-Year Prospective Budgets among which Decision-Makers can choose. The method is tested on simplified cases and on in operational situation and gives satisfactory results

Rotational-tidal phasing of the binary neutron star waveform

Tidal forces cause inspiralling binary neutron stars to deform, leaving a measurable imprint on the gravitational waves they emit. The induced stellar multipoles are an added source of gravitational radiation and modify the orbital dynamics, producing a slight acceleration of the coalescence which manifests as a phase shift in the waveform relative to point-particles. The dominant piece of this tidal phase comes from the mass quadrupoles, which contribute at fifth post-Newtonian order (5PN). Current quadrupoles and mass octupoles contribute at higher orders. For spinning neutron stars, additional multipole moments are induced by nonlinear couplings between spin and tides. We calculate these rotational-tidal deformations assuming the stars are rotating slowly and the tides are weak and quasi-stationary. The stellar multipole moments are read off from an asymptotically flat metric that encodes the difference between their tidal response and a black hole's. The multipoles are subsequently inserted into post-Newtonian formulas for the orbit and the gravitational radiation. We find that, at leading order, the rotational-tidal deformations make a 6.5PN contribution to the tidal phase. Their effect on the waveform is thus larger than that of the mass octupoles, and nearly as large as that of the current quadrupoles, in systems with non-negligible spin.Comment: 10 page