Check me out on github.com/dgerosa/spinprecession.

This page contains data supporting some of my papers on black-hole binary spin precession. Most of these results are obtained with the public precession code. In particular, we provide animated versions of some of the figures from:

*Up-down instability of binary black holes in numerical relativity*. Vijay Varma, Matthew Mould, Davide Gerosa, Mark A. Scheel, Lawrence E. Kidder, and Harald P. Pfeiffer. arXiv:2012.07147.*Endpoint of the up-down instability in precessing binary black holes*. Matthew Mould, Davide Gerosa. Phys.Rev.D 101 (2020) 124037. arXiv:2003.02281.*Wide precession: binary black-hole spins repeatedly oscillating from full alignment to full anti-alignment*. Davide Gerosa, Alicia Lima, Emanuele Berti, Ulrich Sperhake, Michael Kesden, Richard O'Shaughnessy. Class.Quant.Grav. 36 (2019) no.10, 105003. arXiv:1811.05979.*Surprises from the spins: astrophysics and relativity with detections of spinning black-hole mergers*. Davide Gerosa. Journal of Physics: Conf. Series 957 (2018) 012014. arXiv:1711.10038.*Precessional instability in binary black holes with aligned spins*. Davide Gerosa, Michael Kesden, Richard O'Shaughnessy, Antoine Klein, Emanuele Berti, Ulrich Sperhake, Daniele Trifirò. Phys.Rev.Lett. 115 (2015) arXiv:1506.09116.*A multi-timescale analysis of phase transitions in precessing black-hole binaries*. Michael Kesden, Davide Gerosa, Ulrich Sperhake, Emanuele Berti, Richard O'Shaughnessy. Phys.Rev.D 92 (2015) 064016. arXiv:1506.03492.*Effective potentials and morphological transitions for binary black-hole spin precession*. Davide Gerosa, Michael Kesden, Richard O'Shaughnessy, Emanuele Berti, Ulrich Sperhake. Phys.Rev.Lett. 114 (2015) 081103. arXiv:1411.0674.

Figures published in the papers are snapshots (at fixed binary separation) of these animations. References to other figures and equations in the captions correspond to those in the relevant paper. Here is a Youtube playlist containing all animations.

We also provide older animated gif from:

*Resonant-plane locking and spin alignment in stellar-mass black-hole binaries: a diagnostic of compact-binary formation*. Davide Gerosa, Michael Kesden, Emanuele Berti, Richard O’Shaughnessy, Ulrich Sperhake. Phys.Rev.D 87 (2013) 104028. arXiv:1302.4442.*Spin alignment effects in black hole binaries*. Davide Gerosa. Caltech Undergraduate Research Journal (CURJ) Vol.15 No.1 (2014).

### Credit

You are more than welcome to use these results in your presentations! We kindly ask you to cite the relevant papers. If you want to cite the animations specifically, it's .

### Animations from 2012.07147

**Fig. 1 from 2012.07147:**

The up-down instability is demonstrated in NR. We consider binaries with mass ratios q = 0.9 and spin magnitudes χ1 = χ2 = 0.8, while the component spin vectors are initially perturbed from a perfectly aligned-spin configuration by an angle θpert = 10◦. The purple (orange) arrows represent the spin χ1 (χ2) of the heavier (lighter) BH near merger. The colored curves trace the evolution of χ1,2(t) as the binary precesses. Colors darken linearly in time. The spins of the up-up (top-left), down-down (top-right), and down-up (bottom-left) configurations precess stably about z, while those of the up-down (bottom-right) configuration are unstable and become largely misaligned. Smaller modulations occur on the shorter orbital timescale.

### Animations from 2003.02281

**Fig. 2 from 2003.02281:**

Numerical evolution of the normalized spins \hat Si = Si/Si of a BH binary with mass ratio q = 0.5 and dimensionless spins χ1 = χ2 = 1. The blue (red) curve traces the path of the spin vector S1 (S2) of the heavier (lighter) BH over the evolution. The integration is performed from a binary separation r = 1000M to 10M; the colors of the curves darken with decreasing separation. The binary is initialized with misalignments of 1◦ in the BH spins from the up-down configuration. The vertical z-axis is initially aligned to the total angular momentum, the x-axis is constructed such that the orbital angular momentum lies in the x-z plane, and the y-axis completes the orthogonal frame. The black dots show the location of the spins for r > rud+ ≃ 34M, before the onset of instability. The arrows show the orientation of the spins at the final separation r = 10M. The binary is approaching the endpoint listed in Eq. (2).

**Fig. 7 from 2003.02281**:

The response of the up-down instability to different initial perturbations. Each panel shows a set of 1000 orbit-averaged evolutions. Binaries are initialized at r = 1000M with misalignments extracted from truncated Gaussians centered on the up-down configuration with widths δθ = 1◦, 5◦, 10◦, 20◦, increasing progressively from the left to the right panel. Blue (orange) histograms show the corresponding values of the total spin S at r = 1000M (r = 10M). In this example we fix q = 0.8 and χ1 = χ2 = 0.9. Vertical dashed lines at S = |S1 ± S2| mark the asymptotic locations of up-down binaries before and after the instability.

**Fig. 13 from 2003.02281:**

Joint distribution of cos θ1 and cos θ2 for binary BHs with initially aligned spins. Sources are evolved numerically from a separation r = 1000M to r = 10M. Mass ratios q are sampled according to the power law distribution p(q) ∝ q^6.7 [51]; dimensionless spins χi are sampled uniformly. We set q, χ1, χ2 ∈ [0.1, 1]. The populations, each containing 10^4 binaries, of up-up (blue), down-down (orange) and down-up (green) binaries remain in their initial distributions whereas the up-down (red) population does not, thus highlighting the precessional instability. By the end of the evolutions the up-down binaries split into two sub-populations: those which remain stable (bottom right corner) and those which do not (central region). The trend observed in the unstable subpopulation matches the prediction cos θ1 = cos θ2 of Sec. IV A.

### Animations from 1811.05979

**Fig. 3 from 1811.05979:**

Evolution of θ1 (top), θ2 (middle) and ∆Φ (bottom) for the wide binaries. Left (right) panels show a BH binary where the primary (secondary) spin undergoes wide precession at r = 100M. Binaries are characterized by q = 0.95, χ1 = 1, χ2 = 0.4 (left) and q = 0.95, χ1 = 0.4, χ2 = 1 (left). Light (dark) curves correspond to large (small) separations. Wide precession is marked with a dashed line. Binaries evolve along those curves on t_pre, while the curves themselves evolve on t_rad.

### Animations from 1711.10038

**Fig. 3 from 1711.10038**:

Precession-averaged PN evolutions of BH binaries. We evolve 500 systems with χ1=χ2=1 and q=0.8 with isotropic spin directions at large separations. The evolution is shown backwards, from r = 10Mto r = 10^6M, in the plane defined by the two spin orientations cos(θi)=Si·L. Each binary is colored according to its spin morphology at r=10M: blue for binaries librating about ∆Φ=0, green for binaries circulating in ∆Φ ∈ [0, π] and red for binaries librating about ∆Φ=π. Binaries with a given morphology at small separation (detection) originate from precise regions in the (θ1,θ2) plane at large separation (formation).

### Animations from 1506.09116

**Fig. 1 from 1506.09116**:

Effective-potential loops ξ±(S) for binary BHs with mass ratio q=0.9, dimensionless spins χ1=1, χ2=0.14, and total angular momentum J=|L+S1-S2|, corresponding to the up-down configuration. For binary separations r>rud+~337M, the up-down configuration at Smin marked by a red circle is also a minimum (marked by the lower triangle). At intermediate separations rud+>r>rud−~17M misaligned binaries with the same value of the conserved ξ exist along the dashed red line. Perturbations δJ, δξ will cause S to oscillate between the points S± where this line intersects the loop, making the up-down configuration unstable. For r<rud−, the up-down configuration is again a stable extremum, now a maximum (marked by the upper triangle).

**Fig. 2 from 1506.09116**:

The angles cos(θi) for spin-orbit resonances [extrema of ξ±(S)] for BHs with q=0.95, χ1=0.3, and

χ2=1. The solid (dashed) curves indicate the ∆Φ=0 (π) family. The up-down configuration (bottom right corner) belongs to the ∆Φ=0 family for r>rud+~ 2149M, to the ∆Φ=π family for r<rud-~13M, and is unstable for intermediate values rud−<r<rud+.

**Fig. 3 from 1506.09116**:

Precession-averaged radiation reaction dJ/dL as a function of J and ξ for binaries with q=0.8, χ1=χ2=1. Spin-orbit resonances including the up-up, down-down, and down-up configurations are extrema of ξ±(S) and constitute the boundary of the allowed region. All four aligned configurations are maxima where dJ/dL = 1, but the unstable up-down configuration (shown in the inset) is a cusp.

### Animations from 1506.03492

**Fig. 3 from 1506.03492**:

Analytical solutions given by Eq. (20) for the evolution of the angles θ1 (top panel), θ2 (middle panel), and ∆Φ (bottom panel) during a precession cycle. The evolution of three binaries with ξ = 0.25 (blue), 0.3 (green) and 0.35 (red) is shown for q=0.8,χ1 =1,χ2 =0.8 and evolved on the radiation time using Eq.(38). In particular, J=1.29M^2 for all binaries at r=20M. The evolution of θ1 and θ2 is monotonic during each half of a precession cycle and is bounded by the dotted lines for which cos φ = ∓1 [these curves can be found by substituting ξ±(S) for ξ in Eq. (20)]. Three classes of solutions are possible and define the binary morphology: ∆Φ can oscillate about 0 (ξ = 0.25), circulate (ξ = 0.3) or oscillate about π (ξ = 0.35).

**Fig. 5 from 1506.03492**:

The (J,ξ) parameter space for BBHs with different minimum allowed total angular momentum J_min. BBH spin morphology is shown with different colors, as indicated in the legend. The extrema ξ_min(J ) and ξ_max(J ) of the effective potentials constitute the edges of the allowed regions and are marked by solid blue (red) curves for ∆Φ = 0 (π). Dashed lines mark the boundaries between the different morphologies. The parameters q, χ1, χ2 and r are chosen as in Fig.4, whose panels can be thought of as vertical (constant J) “sections” of this figure (where we suppress the S dependence). The lowest allowed value of ξ occurs at J = |L − S1 − S2| in all three panels at all separations. Three phases are present for each vertical section with J > |L − S1 − S2|. This condition may either cover the entire parameter space or leave room for additional regions where vertical sections include two different phases in which ∆Φ oscillates about π and a circulating phase in between or only a single phase where the spins librate about ∆Φ = π.

**Fig. 6 from 1506.03492**:

Time-dependent solutions for the total spin-magnitude S (top panel) and the orbital-angular-momentum phase ΦL (bottom panel). We set q = 0.7, χ1 = 0.7, χ2 = 0.9; binaries are evolved such that J = 1.48M^2 r=30M. We integrate Eq. (26) for three values of ξ corresponding to the three different spin morphologies at r=30M: ∆Φ oscillates about 0 (ξ = 0.17, blue), circulates (ξ = 0.25, green), and oscillates about π (ξ = 0.34, red). Initial conditions have been chosen such that S=S− and ΦL=0 at t=0. The oscillations in S induce small wiggles in ΦL on top of a mostly linear drift. Spin-orbit resonances (horizontal dashed lines, top panel) correspond to configurations for which S is constant and can be interpreted as zero-amplitude limits of generic oscillatory solutions. The projections of the effective potentials, i.e. parametric curves [τ(ξ)/2, S+(ξ)] and [τ(ξ),S−(ξ)], are shown with dotted lines.

**Fig. 8 from 1506.03492**:

Precession-averaged BBH inspirals as described in Sec.IIIC (purple/darker) compared to numerical integration of the orbit-averaged PN equations [35,36] (orange/lighter). Marginalized distributions of the spin angles θ1, θ2, and |∆Φ| (rows) are shown at several separations along the inspirals. The three initial spin distributions are isotropic (top panels), one aligned BH (middle panels), and Gaussian spikes (bottom panels) as described in Sec.IIIC. The two approaches are in good agreement except for minor deviations in the distribution of ∆Φ at r∼10M. We take q=0.7, χ1=0.8 and χ2=0.4 for all BBHs.

**Fig. 10 from 1506.03492**:

Precessional solutions ∆Φ(S) of Eqs. (20) as J and L evolve during inspirals according to Eq.(38). These solutions are colored according to the separation r/M as shown in the color bar on the right (orange/lighter for large separations and black/darker for small separations). Binaries in the left (right) panel transition from the circulating morphology to the morphology in which ∆Φ librates about 0 (π) at the transition radius r_tr ≃ 152M (18.9M).

**Fig. 13 from 1506.03492**:

The fraction f of isotropic BBHs for which ∆Φ circulates (green, middle region), oscillates about 0 (blue, bottom region), or oscillates about π (red, top region) as functions of the mass ratio q. Dashed lines separate the different morphologies. Each panel corresponds to a different value of χ1 (columns) and χ2 (rows). The fraction of BBHs in librating morphologies increases as the mass asymmetry decreases (q→1). For nearly equal masses (q>0.9), asymmetry in the spin magnitudes increases the fraction of binaries in the circulating morphology as can be seen by comparing panels on and off of the diagonal.

**Fig. 14 from 1506.03492**:

Spin morphologies at evolving separation r_f as functions of the asymptotic values of the spin angles θi∞. The mass ratio q and spin magnitudes χi for each panel are indicated in the legends. Evolving BBHs along the four lines cosθi=±1 at r_f out to r/M → ∞ using our new precession-averaged approach yields the dashed curves separating the different final morphologies: ∆Φ oscillates about 0 (blue), oscillates about π (red), circulates without ever having experienced a phase transition (plain green), or circulates after having experienced a phase transition to libration and then a second phase transition back to circulation (hatched green). The morphology within each region defined by the dashed boundaries is determined by which of the conditions cosθi=±1 these boundaries satisfy, as described in Sec. IV C. The points show the locations of binaries in the cosθ1 − cosθ2 plane at r_f and are colored by their morphology at that separation [∆Φ oscillates about 0 (blue circles), oscillates about π (green squares), or circulates (red trianges)]. Because morphology depends on ∆Φ in addition to θ1 and θ2 at finite separation, the projection onto the cosθ1 − cosθ2 plane can lead points of different morphologies to occur at the same positions, particularly for comparable-mass binaries q≃1 where the θi’s oscillate with greater amplitude.

### Animations from 1411.0674

**Fig. 2 from 1411.0674**:

The three morphologies of BBH spin precession. The angular momenta J, L, and Si are all in the xz plane at S=S±. In all three panels, the BBHs have maximal spins and q=0.8. Binaries have J=0.85M^2 at r=10M (L=0.781M^2) as in Fig. 1. The left, middle, and right panels correspond to ξ=−0.025, 0.025, and 0.15, respectively. If the components of Si perpendicular to L are aligned with each other at both roots S, ΔΦ librates about 0°. If they are aligned at one root and antialigned at the other, ΔΦ circulates. If they are antialigned at both roots, ΔΦ librates about 180°.

### Older animated gif from 1302.4442

**tides-isotropic.gif ****Fig. 5 from 1302.4442 **:

Scatter plots of the PN inspiral of maximally spinning BH binaries with mass ratio q = 0.8 from an initial separation a_PNi just above 1000M to a final separation aPNf = 10M. The left panel shows this evolution in the (θ1, θ2) plane and the right panel shows the evolution in the (∆Φ, θ12) plane. Darker (red) and lighter (green) dots refer to the SMR and RMR scenarios, respectively. The initial distribution for these Monte Carlo simulations was constructed from an astrophysical model with efficient tides and isotropic kicks.

**tides-polar.gif ****Fig. 6 from 1302.4442 **: Scatter plots of the PN inspiral of maximally spinning BH binaries with mass ratio q = 0.8 from an initial separation a_PNi just above 1000M to a final separation aPNf = 10M. The left panel shows this evolution in the (θ1, θ2) plane and the right panel shows the evolution in the (∆Φ, θ12) plane. Darker (red) and lighter (green) dots refer to the SMR and RMR scenarios, respectively. The initial distribution for these Monte Carlo simulations was constructed from an astrophysical model with efficient tides and polar kicks.

**notides-isotropic.gif ****Fig. 7 from 1302.4442 **: Scatter plots of the PN inspiral of maximally spinning BH binaries with mass ratio q = 0.8 from an initial separation a_PNi just above 1000M to a final separation aPNf = 10M. The left panel shows this evolution in the (θ1, θ2) plane and the right panel shows the evolution in the (∆Φ, θ12) plane. Darker (red) and lighter (green) dots refer to the SMR and RMR scenarios, respectively. The initial distribution for these Monte Carlo simulations was constructed from an astrophysical model with inefficient tides and isotropic kicks.

**notides-polar.gif ****Fig. 8 from 1506.09116**: Scatter plots of the PN inspiral of maximally spinning BH binaries with mass ratio q = 0.8 from an initial separation a_PNi just above 1000M to a final separation aPNf = 10M. The left panel shows this evolution in the (θ1, θ2) plane and the right panel shows the evolution in the (∆Φ, θ12) plane. Darker (red) and lighter (green) dots refer to the SMR and RMR scenarios, respectively. The initial distribution for these Monte Carlo simulations was constructed from an astrophysical model with inefficient tides and polar kicks.

**CURJ Vol.15 No.1 (2014)**: animations available on github.