The dynamics of precessing binary black holes (BBHs) in the post-Newtonian regime has a strong timescale hierarchy: the orbital timescale is very short compared to the spin-precession timescale which, in turn, is much shorter than the radiation-reaction timescale on which the orbit is shrinking due to gravitational-wave emission. We exploit this timescale hierarchy to develop a multiscale analysis of BBH dynamics elaborating on the analysis of Kesden et al. [Phys. Rev. Lett. 114, 081103 (2015)]. We solve the spin-precession equations analytically on the precession time and then implement a quasiadiabatic approach to evolve these solutions on the longer radiation-reaction time. This procedure leads to an innovative "precession-averaged" post-Newtonian approach to studying precessing BBHs. We use our new solutions to classify BBH spin precession into three distinct morphologies, then investigate phase transitions between these morphologies as BBHs inspiral. These precession-averaged post-Newtonian inspirals can be efficiently calculated from arbitrarily large separations, thus making progress towards bridging the gap between astrophysics and numerical relativity.

Weak gravitational lensing by an intervening large-scale structure induces a distinct signature in the cosmic microwave background ͑CMB͒ that can be used to reconstruct the weak-lensing displacement map. Estimators for individual Fourier modes of this map can be combined to produce an estimator for the lensing-potential power spectrum. The naive estimator for this quantity will be biased upwards by the uncertainty associated with reconstructing individual modes; we present an iterative scheme for removing this bias. The variance and covariance of the lensing-potential power spectrum estimator are calculated and evaluated numerically in a ⌳CDM universe for Planck and future polarization-sensitive CMB experiments.

We study the expected spin misalignments of merging binary black holes formed in isolation by combining state-of-the-art population-synthesis models with efficient post-Newtonian evolutions, thus tracking sources from stellar formation to gravitational-wave detection. We present extensive predictions of the properties of sources detectable by both current and future interferometers. We account for the fact that detectors are more sensitive to spinning black-hole binaries with suitable spin orientations and find that this significantly impacts the population of sources detectable by LIGO, while this is not the case for third-generation detectors. We find that three formation pathways, differentiated by the order of core collapse and common-envelope phases, dominate the observed population, and that their relative importance critically depends on the recoils imparted to black holes at birth. Our models suggest that measurements of the "effective-spin" parameter χ eff will allow for powerful constraints. For instance, we find that the role of spin magnitudes and spin directions in χ eff can be largely disentangled, and that the symmetry of the effective-spin distribution is a robust indicator of the binary's formation history. Our predictions for individual spin directions and their precessional morphologies confirm and extend early toy models, while exploring substantially more realistic and broader sets of initial conditions. Our main conclusion is that specific subpopulations of black-hole binaries will exhibit distinctive precessional dynamics: these classes include (but are not limited to) sources where stellar tidal interactions act on sufficiently short timescales, and massive binaries produced in pulsational pair-instability supernovae. Measurements of black-hole spin orientations have enormous potential to constrain specific evolutionary processes in the lives of massive binary stars.

We derive an effective potential for binary black hole (BBH) spin precession at second post-Newtonian order. This effective potential allows us to solve the orbit-averaged spin-precession equations analytically for arbitrary mass ratios and spins. These solutions are quasiperiodic functions of time: after a fixed period, the BBH spins return to their initial relative orientations and jointly precess about the total angular momentum by a fixed angle. Using these solutions, we classify BBH spin precession into three distinct morphologies between which BBHs can transition during their inspiral. We also derive a precessionaveraged evolution equation for the total angular momentum that can be integrated on the radiation-reaction time and identify a new class of spin-orbit resonances that can tilt the direction of the total angular momentum during the inspiral. Our new results will help efforts to model and interpret gravitational waves from generic BBH mergers and predict the distributions of final spins and gravitational recoils. Introduction.-The classic two-body problem was a major engine of historical progress in physics and astronomy. This problem can be solved analytically in Newtonian gravity; its solutions are the well-known Keplerian orbits. The analogs to Newtonian point masses in general relativity are binary black holes (BBHs). Astrophysical BBHs have spins S i [1] in addition to their masses m i [the masses determine the total mass M ≡ m 1 þ m 2 , mass ratio q ≡ m 2 =m 1 ≤ 1 and symmetric mass ratio η ≡ m 1 m 2 =M 2 ¼ q=ð1 þ qÞ 2 ]. Full solutions to the two-body problem in general relativity must, therefore, include spin evolution in addition to orbital motion. Einstein's equations must be solved numerically [2][3][4] when the binary separation r is comparable to the gravitational radius r g ≡ GM=c 2 , but post-Newtonian (PN) approximations may be used when r ≫ r g . BBH evolution in the PN limit occurs on three distinct time scales: the orbital time t orb ∼ ðr 3 =GMÞ 1=2 on which the binary separation r evolves, the precession time t pre ∼ c 2 r 5=2 = ½ηðGMÞ 3=2 ∼ ðt orb =ηÞðr=r g Þ on which the spin directions change, and the radiation-reaction time t RR ∼ E= jdE GW =dtj ∼ c 5 r 4 =½ηðGMÞ 3 ∼ ðt orb =ηÞðr=r g Þ 5=2 on which the energy E ¼ −GηM 2 =ð2rÞ and orbital angular momentum L ¼ ηðrGM 3 Þ 1=2 decrease.The hierarchy t orb ≪ t pre ≪ t RR implies that when considering evolution on one time scale, quantities evolving on a shorter (longer) time scale can be averaged (held constant). This has been used to derive orbit-averaged spin-precession equations _ S i ¼Ω i × S i [5][6][7][8], where the precession frequenciesΩ i depend on the orbital angular momentum L and spins S i but not on the instantaneous

Inflationary gravitational waves (GW) contribute to the curl component in the polarization of the cosmic microwave background (CMB). Cosmic shear -gravitational lensing of the CMB-converts a fraction of the dominant gradient polarization to the curl component. Higher-order correlations can be used to map the cosmic shear and subtract this contribution to the curl. Arcminute resolution will be required to pursue GW amplitudes smaller than those accessible by the Planck surveyor mission. The blurring by lensing of small-scale CMB power leads with this reconstruction technique to a minimum detectable GW amplitude corresponding to an inflation energy near 10 15 GeV. Observation of acoustic oscillations in the temperature anisotropies of the cosmic microwave background (CMB) [1] strongly suggests an inflationary origin for primordial perturbations [2]. It has been argued that a new "smokinggun" signature for inflation would be the detection of a stochastic background in inflationary gravitational waves (IGWs) [3]. These IGWs produce a distinct signature in the CMB in the form of a contribution to the curl, or magneticlike, component of the polarization [4]. Since there is no scalar, or density-perturbation, contribution to these curl modes, curl polarization was considered to be a direct probe of IGWs.There is, however, another source of a curl component. Cosmic shear (CS) -weak gravitational lensing of the CMB due to large-scale structure along the line of sightresults in a fractional conversion of the gradient mode from density perturbations to the curl component [5]. The amplitude of the IGW background varies quadratically with the energy scale E infl of inflation, and so the prospects for detection also depend on this energy scale. In the absence of CS, the smallest detectable IGW background scales simply with the sensitivity of the CMB experiment -as the instrumental sensitivity is improved, smaller values of E infl become accessible [3,6]. More realistically, however, the CS-induced curl introduces a noise from which IGWs must be distinguished. If the IGW amplitude (or E infl ) is sufficiently large, the CS-induced curl will not be a problem. However, as E infl is reduced, the IGW signal becomes smaller and will get lost in the CS-induced noise. This confusion leads to a minimum detectable IGW amplitude [7].In addition to producing a curl component, CS also introduces distinct higher-order correlations in the CMB temperature pattern. Roughly speaking, lensing can stretch the image of the CMB on a small patch of sky and thus lead to something akin to anisotropic correlations on that patch of sky, even though the CMB pattern at the surface of last scatter had isotropic correlations. By mapping these effects, the CS can be mapped as a function of position on the sky [8]. The observed CMB polarization can then be corrected for these lensing deflections to reconstruct the intrinsic CMB polarization at the surface of last scatter (in which the only curl component would be that due to IGWs). In this Letter we evaluate...