Investigating the effect of in-plane spin directions for Precessing BBH systems.

Since the advent of the Advanced LIGO TheLIGOScientific:2014jea and Virgo TheVirgo:2014hva gravitational-wave detectors in 2015, up to October 2019 there have been a total of 50 gravitational wave signals detected which include multiple binary-black hole mergers, binary neutron star mergers and possible neutron-star-black-hole merger LIGOScientific:2018mvr ; Abbott:2020niy ; Nitz_2020 ; Zackay:2019btq ; Venumadhav:2019lyq ; Abbott:2020khf ; LIGOScientific:2020stg ; Abbott:2020uma ; Abbott:2020tfl . The BBH observations have begun to reveal the astrophysical rate of black-hole mergers, and the astrophysical distribution of black-hole masses and spins LIGOScientific:2018jsj ; Abbott:2020mjq ; Abbott:2020gyp . To measure the binary’s properties the detector data are compared against a set of theoretical model waveforms. The accuracy of the measured parameters depends not only on the details of the source, the signal-to-noise ratio (SNR) of the signal, and parameter degeneracies, but also on the accuracy of the waveform models. Two families of waveform models, IMRPhenom Khan:2015jqa ; London:2017bcn ; Hannam:2013oca ; Khan:2018fmp ; Pratten:2020fqn ; Pratten:2020ceb and SEOBNR Bohe:2016gbl ; Cotesta:2018fcv ; Pan:2013rra ; Babak:2016tgq ; Taracchini:2013rva , were used to calculate the reported parameters during the first two observation runs. Both rely on several physical approximations, as discussed in, for example, Ref. Ramos-Buades:2020noq . In this paper we test the validity and impact of a subset of those approximations.

Binary-black-hole inspiral is the result of orbital energy and angular momentum loss through gravitational radiation. If the radiation from a binary is decomposed into spin-weighted spherical harmonics, ${}^{-2}Y_{\ell m}(\theta,\phi)$, the signal is dominated by the “quadrupole” contribution in the $\ell=2$ harmonics. Gravitational waves also carry linear momentum, and for nonspinning or aligned-spin binaries (where the black-hole spins $\mathbf{S}_{i}$ are parallel to the orbital angular momentum, $\mathbf{L}$, so that $\mathbf{L}\times\mathbf{S}_{i}=0$), the resultant recoil of the center-of-mass within the orbital plane is manifest in the signal through interplay between different multipoles; see, for example, Ref. Herrmann:2007ac . Current aligned-spin binary waveform models that include higher multipoles capture all of these physical effects, with varying degrees of accuracy London:2017bcn ; Garcia-Quiros:2020qpx ; Cotesta:2018fcv . In generic binaries, where the spins are mis-aligned with the orbital angular momentum, the orbital plane and spins precess during the inspiral. Generic binaries also radiate linear momentum perpendicular to the orbital plane. This effect, which shows up in the GW signal through an asymmetry between the $+m$ and $-m$ multipoles, is not present in current precessing SEOBNR Ossokine:2020kjp and IMRPhenom Khan:2019kot models. Although these models include the spin directions while computing the precession dynamics used to generate the precessing waveform, the effect of varying spin directions on the full waveforms remains un-modelled. Our goal is to make a first estimate of the effect of these omissions on GW source parameter measurements.

We begin by describing in more detail the phenomenology of BBH systems, and the construction of generic-binary waveform models.

A BBH system undergoing non-eccentric inspiral can be characterised by eight parameters, the individual masses $(m_{i})$, and the components of the two spin vectors $(\mathbf{S}_{i})$, specified at some fiducial point during the inspiral, for example a chosen orbital frequency. The GW signal is also parameterised by the binary’s sky-position $(\alpha,\delta)$, inclination $(\iota)$, coalescence phase ($\phi_{c}$), distance $(d_{L})$, polarisation $(\psi)$ and time of arrival $(t_{c})$ at the detector. As noted above, the complex GW strain can be decomposed into spin-weighed spherical harmonics as,

where $(\theta,\phi)$ give the position of the observer on a sphere centred on the centre-of-mass of the binary.

Based on the black-hole (BH) spin configurations, coalescing BBH systems with spins can be considered to be either:

• •

  Aligned-Spin: The BH spins are parallel or anti-parallel to $\mathbf{L}$, so $\mathbf{L}\times\mathbf{S}_{i}=0$, where $i=1,2$ for each BH. From the symmetries of the system, the BHs orbit in a fixed plane, i.e., the direction of the orbital angular momentum $\mathbf{\hat{L}}$ remains fixed. In the frame where $\mathbf{\hat{L}}\parallel\hat{z}$, symmetry also implies that

  $$h_{\ell,m}=(-1)^{\ell}h_{\ell,-m}^{*},$$

  (2)

  and that any linear momentum emission is perpendicular to $\mathbf{L}$; although the orientation of the orbital plane remains fixed, the center-of-mass can recoil within this plane.

• •

  Precessing: One or both BHs have non-zero spin components perpendicular to $\mathbf{\hat{L}}$. We denote the parallel components by $\mathbf{S}_{i}^{\parallel}$ and the perpendicular components by $\mathbf{S}_{i}^{\perp}$. The presence of $\mathbf{S}_{i}^{\perp}$ causes the orbital plane to precess over the course of the coalescence. This leads to modulations of the amplitude and phase of the waveform. Emission of linear momentum is now also possible perpendicular to the orbital plane, which breaks the symmetry of Eq. (2) between the $\pm m$ multipoles.

As was shown in previous studies Schmidt:2012rh ; Boyle:2011gg ; OShaughnessy:2011pmr , a precessing waveform can be decomposed into the waveform as observed in a co-precessing frame, $h^{\mathrm{CP}}$, and a time- or frequency-dependent rotation that describes the precessional dynamics. The rotation can be expressed in terms of three Euler angles, $(\alpha,\beta,\gamma)$, and the $\ell=2$ modes of the precessing-binary waveform $h^{\mathrm{P}}$ constructed as

where $d^{2}_{mm^{\prime}}$ denote the $\ell=2$ Wigner-d matrices. In the current precessing models (IMRPhenom and SEOBNR), the co-precessing-frame waveform is based on an underlying non-precessing-binary model (with some modifications), and this procedure preserves its orbital-plane symmetry, Eq. (2). These models therefore do not include the $\pm m$ mode asymmetry of full precessing-binary waveforms.

The magnitude and direction of the out-of-plane angular momentum loss $\dot{p}_{\parallel}$ (and therefore the level of mode asymmetry) is related to the angles between the in-plane spins $\mathbf{S}_{i}^{\perp}$ and the separation vector between the two black holes $\mathbf{\hat{n}}$, as most easily seen in the PN treatment in Sec. III.E of Ref. PhysRevD.52.821 . During one orbit the spin directions change little, so $\dot{p}_{\parallel}$ oscillates approximately on the orbital timescale. In the “twisted-up” models described above, this effect is not present, and an overall rotation of the spin(s) in the orbital plane introduces only an offset in the precession angle $\alpha$, which is degenerate with the azimuthal angle, $\phi$, since it enters the spin-weighted spherical harmonics as $e^{im\phi}$. The model waveforms are therefore degenerate with respect to a constant rotation of the in-plane spins, while true waveforms include an additional effect that varies sinusoidally with respect to this spin rotation.

Out of plane recoil in the context of mode asymmetries has been discussed in NR simulations in Brugmann:2007zj , and further illustration of the effect in GW signals is shown in Ramos-Buades:2020noq . Earlier studies on in-plane effects on waveforms and/or mode-asymmetries for precessing systems include OShaughnessy:2012iol ; Pekowsky:2013ska ; Boyle:2014ioa .

In this study, we investigate the effects of varying the in-plane spin direction for single-spin precessing NR waveforms for a given combination of mass-ratio and spin. We also consider the special case of the “super-kick” configuration Campanelli:2007cga ; Gonzalez:2007hi ; Brugmann:2007zj : these are equal-spin configurations where the spins lie entirely in the orbital plane, and $\mathbf{S}_{1}=-\mathbf{S}_{2}$. Due to the symmetry of this configuration, the orbital plane does not precess, but does bob up and down due to linear momentum loss, making this an especially clean system for the study of mode asymmetry. We choose these configurations to estimate the importance of mode-asymmetric content on parameter measurements. Using the waveform with in-plane spin initially aligned to the position vector as a proxy template, we compute matches (see Sec. III.1) against systems with different in-plane spin directions. Using a relationship between the match value and SNR at which two signals are distinguishable, we provide an estimate of the SNR at which mode asymmetries will impact parameter measurements. We also use a selection of waveforms with the same spin direction as the proxy template but with differing in-plane spin magnitude to estimate the relative strength of the effect of varying spin direction versus varying spin magnitude.

For all results, we use only the $(\ell=2,m=\pm 2)$ modes of the waveforms in the co-precessing frame. Higher modes are much weaker than the $(\ell=2,|m|=2)$ multipoles, but far stronger than the asymmetry contribution to the dominant modes, and we choose to consider only the dominant modes in order to more easily isolate effects due to the mode asymmetry.

The paper is organised as follows. Sec. II provides details of the simulations generated for this study, Sec. III.1 and Sec. III.2 discuss the computation of precessing matches and the connection between the match and detectable SNR respectively. The specific results presented are motivated in Sec. IV with the actual results in Sec. V. The conclusions we draw from this work, and some of its limitations and potential future extensions, are discussed in Sec. VI.

For this study, a set of 12 new NR simulations were performed with the BAM code Husa_2008 ; PhysRevD.77.024027 . Configurations are defined by the mass ratio, $q=m_{2}/m_{1}$, where we choose the convention $m_{2}>m_{1}$, and the spin vectors specified at the start of the simulation, $\mathbf{S}_{i}$. In the unequal-mass simulations, only the larger black hole is spinning, so that $\mathbf{S}_{1}=0$. (We could also assign spin to the secondary black hole, but placing spin on one black hole is sufficient to produce the asymmetry effects that we wish to study.) For these simulations we can completely specify the spin direction at the beginning of the simulation by two angles, i) the angle between the spin vector and angular momentum vector, which we call $\theta_{SL}$, and ii) the angle between the separation vector ($\vec{n}$) and the projection of spin onto the orbital plane ($\mathbf{S}_{i}^{\perp}$), which we call $\phi_{\mathrm{Sn}}$. The codes available at the beginning of this study for initial data generation did not allow for user specified $(\theta_{SL},\phi_{\mathrm{Sn}})$ values, and so an iterative method was developed for obtaining the required initial parameters for single-spin precessing systems. The initial data generation method is described in detail Appendix. A. Sec. II.1 gives the details of the simulations with the parameters of all the simulations described in Tab: 1.

This section provides the details of the match computation procedure employed for computing matches between the various precessing waveforms and the connection between the match and the SNR at which the template and signal can be distinguished from each other. This is the primary method we use to interpret the results in Sec. V.

In subsequent sections we study the SNR ($\rho_{c}$) at which configurations with differing spin directions or spin magnitudes are distinguishable, as defined by Eq. (9). In this section we make some general comments on the accuracy of $\rho_{c}$ for our simulations, some general properties of the waveforms with respect to changes in the in-plane spin direction, $\phi_{\mathrm{Sn}}$, and examine how $\rho_{c}$ varies with respect to different binary orientations. This motivates the way we will present our results for the remainder of this paper.

Let us first discuss accuracy. As reported in Sec. II, the matches between the 80- and 96-point runs (for q2a07p0 and q2a07p90) are $\sim$0.9995 - 0.99995, which translates to $\rho_{c}$ between 110 and 350. This suggests that we can identify two waveforms as indistinguishable up to SNRs of at least 110. We also computed the matches between the $\phi_{\mathrm{Sn}}=0,\pi/2$ systems using the corresponding 80- and 96-point waveforms over a range of $(\theta,\phi)$ values, and found that the relative error between them is $\mathcal{O}(0.05\%)$. These numbers suggest that although we should be cautious when interpreting very large values of $\rho_{c}$, we expect the qualitative behaviour of the matches to remain unchanged with more accurate simulations.

Given the accuracy limits of our simulations, we are now in a position to study how the waveforms vary with respect to different initial directions of the in-plane spin. We begin by noting an approximate symmetry between systems with a $\phi_{\mathrm{Sn}}$ difference of $\pi$. An in-plane spin rotation of $\pi$ corresponds to flipping the direction of the out-of-plane recoil, and therefore we would expect that the signal from a system with a given value of $\phi_{\mathrm{Sn}}$ to be identical to that from a system with $\phi_{\mathrm{Sn}}+\pi$, if observed from the opposite side of the orbital plane, i.e., with $\theta\rightarrow\theta+\pi$. We have verified that the optimal match is indeed found when $\theta_{\mathrm{template}}\approx(\theta_{\mathrm{signal}}+\pi$). In Fig. 1 we plot the $\rho_{c}$ across signal $\theta$ for a range of signal $\phi$ for which $\mathcal{M}$ is computed with $(\theta_{\mathrm{template}},\phi_{\mathrm{template}})=(\theta_{\mathrm{signal}}+\pi,\phi_{\mathrm{signal}})$ and the match is optimized only over template $\psi$. The $\rho_{c}\geq 100$ for all $\phi_{\mathrm{Sn}}=\pi$ signals for the $q=1$ and $q=2$ systems, with the $\rho_{c}\geq 50$ for each $\theta_{\mathrm{signal}}$ for $q=4$ system. We do not observe an exact symmetry (i.e., a mismatch of zero), because in the $q=1$ and $q=4$ systems, we did not simply rotate the spin between each configuration, but instead calculated initial parameters individually for each value of $\phi_{\mathrm{Sn}}$, so these do not form a one-parameter family. Even with rotated spins within a series, as in the $q=2$ series, we have not changed the momenta; we would expect even lower mismatches if the out-of-plane momenta, in the initial data, had been reflected in the orbital plane between the $\phi_{\mathrm{Sn}}$ and $\phi_{\mathrm{Sn}}+\pi$ configurations.

When we rotate the initial in-plane spin direction $\phi_{\mathrm{Sn}}$, this will, by definition, rotate the initial orbital angular momentum about the total angular momentum, i.e., a change in the $\alpha$ Euler angle in Eq. (3). But we do not expect it to have a significant impact on the opening angle $\beta$ between the orbital and total angular momenta, at least during the inspiral. In Fig. 2 we plot the differences in the $\beta$ Euler angle for the $q=2$ and $q=4$ systems with differences in $\phi_{\mathrm{Sn}}$ of $\pi/2$ and $\pi$. We can see that $\Delta\beta\sim 0$ for $q=2$, but rises to a few $\sim\mathcal{O}(1^{\circ})$ for $q=4$; this is consistent with our expectation as the construction of the $q=2$ series configurations better represent the symmetry. We note that in comparing the $\phi_{\mathrm{Sn}}=0$ and $\phi_{\mathrm{Sn}}=\pi/2$ configurations, there is a clear difference in $\beta$ during the merger and ringdown.

We now wish to estimate how easily two configurations can be distinguished for different choices of binary orientation, $(\theta,\phi)$. In all of these comparisons, we average $\rho_{c}$ over four choices of waveform polarisation. As mentioned in Sec. III.1, for each signal system we have ($25\times 30\times 4=3000$) match values and the SNR for each. To average across different choices of signal polarisation, $\psi^{s}$, we follow Refs. Khan:2018fmp ; Harry:2017weg ; Buonanno:2002fy and average the match for each $(\theta^{s},\phi^{s})$ across the signal polarisation $\psi^{s}$ by weighting them with their SNR. This approximately accounts for the likelihood of the signal being detected. This SNR-averaged match is defined as,

where the sum is over all four signal polarisation values. So, for a given system, we have 750 values of the SNR averaged match.

Fig. 3 shows contour plots of the variation of $\rho_{c}$ across the signal $(\theta,\phi)$ for the q2a07p90 and q4a08p90 signals as seen by the q2a07p0 and q4a08p0 systems respectively, where the match at each $(\theta,\phi)$ point is maximised over template $(\theta,\phi,\psi)$ and then averaged over the signal $\psi$ values using Eq. (10) and $\rho_{c}$ is computed using Eq. (9). For these systems with $\phi_{\mathrm{Sn}}$ differences of $\pi/2$, the major contribution to the mismatch would be from a combination of their slightly different precession motion (as seen from Fig. 2) and mode-asymmetry behaviour. For the $q=2$ system, $20\lesssim\rho_{c}\lesssim 72$, whereas for the $q=4$ system, $11\lesssim\rho_{c}\lesssim 32$. Precession effects are more pronounced at edge-on than face-on inclination, so the lower $\rho_{c}$ for q2a07p90 at $\theta\sim\pi/2$ is expected. For q4a08p90, this behaviour seems to reverse (higher $\rho_{c}$ at edge-on compared to face-on). In Sec. V.3, we remove the mode-asymmetry from the waveforms and compute the matches between the symmetrized waveforms for the $\phi_{\mathrm{Sn}}=\pi/2$ signals as seen by $\phi_{\mathrm{Sn}}=0$ template. For the $q=2$ system, the band of low $\rho_{c}$ near edge-on inclination broadens for all $\phi$ and for the $q=4$ system, the peaks of high $\rho_{c}$ shift slightly away from near edge-on inclinations with an overall increase in $\rho_{c}$ for both cases. This implies that the behaviour of $\rho_{c}$ across the ($\theta,\phi$) space can be strongly affected by mode-asymmetric content for these systems.

We see from Fig. 3 that there is a wide variation in the SNR at which these configurations would be distinguishable, depending on their orientation to the detector. Also, the exact way in which precessional motion and mode-asymmetry affect distinguishability over the $(\theta,\phi)$ space is hard to characterize. For the main results in this paper, we find it more instructive to use a measure that would give an idea of the variation of $\rho_{c}$ across all orientations. For a given signal and template configuration, we define the quantity $\Gamma(\rho)$ which gives the percentage of signals distinguishable at a given SNR by the template. This quantity can be defined formally as,

where $\mathcal{S}_{\text{id2}:\text{id1}}[\rho_{c}<\rho]$ is the set of all signals with distinguishability SNR ($\rho_{c}$) smaller than a given SNR value ($\rho$) and $\mathcal{S}_{\text{id2}:\text{id1}}$ is the set of all the available signals. In the following sections, this cumulative measure of the fraction of signal distinguishable at a given SNR or lower, will be used as our main tool to quantify the differences between binary configurations.

We now consider in detail the distinguishability of our NR configurations. To reiterate, for each system at a given mass-ratio, the $\phi_{\mathrm{Sn}}=0$ system is used as the proxy template waveform. For the results hence, compared to the proxy templates, the signals either have a different spin direction or the magnitude. Also, due to the approximate symmetry between the $\phi_{\mathrm{Sn}}\pm\pi$ systems, the results for the $\phi_{\mathrm{Sn}}=3\pi/2$ and $\phi_{\mathrm{Sn}}=\pi/2$ are very similar, and hence, we will only present results for the $\phi_{\mathrm{Sn}}=\pi/2$ systems.

We first summarise the main differences that we observe between the different configurations, by averaging the match and distinguishability SNR $\rho_{c}$ over all 3000 orientations and polarisations. The results are shown in Tab. 2. We see that variations in the in-plane spin direction can be distinguishable at an SNR of 60, and sometimes as low as $\sim$20. Similarly, waveforms from systems with different spin magnitudes can be distinguishable are SNRs of $\sim$60 for spin differences on the order of 0.1. These results encapsulate the two key conclusions we derive from this study: differences in the waveforms between configurations with different in-plane spin directions may be measurable with observations in the near future (the highest BBH SNR to date has been GW150914, with an SNR of $\sim$24 TheLIGOScientific:2016pea ), and the SNRs at which in-plane spin magnitudes could be measured are comparable to those at which the spin direction will also impact the results. This strongly suggests that waveform changes due to the in-plane spin direction (beyond an overall offset in the precession angle $\alpha$) need to be included in waveform models.

These remainder of this paper considers these results in more detail, and we also attempt to isolate the physical effects that lead to these waveform differences.

In Sec V.1 we compare the full NR waveforms (using all the $\ell=2$ multipoles). This allows us to identify the range of SNRs in which the configurations with different choices of $\phi_{\mathrm{Sn}}$ will be distinguishable, and to compare this with the effect of changing the in-plane spin magnitude.

We then attempt to isolate the causes of these differences. In Sec. V.2, we transform the waveforms into the co-precessing frame (where modes with $|m|<2\approx 0$) and study the matches between the waveform with symmetrized $(l,|m|)=(2,2)$ modes for $\phi_{\mathrm{Sn}}\pm\pi/2$ systems. This allows us to estimate the distinguishability of two waveforms when both precession and mode-asymmetry effects are muted, due primarily to small differences in the inspiral rate and merger-ringdown differences. These co-precessing-frame symmetrized modes are then transformed back to the inertial frame and Sec. V.3 presents the results of analysis with those waveforms. These results estimate the impact of neglecting mode-asymmetry on the distinguishability of precessing-binary waveforms.

We have investigated when changes in the in-plane spin direction of binary-black-hole systems, the effects of which are not included in current Phenom and EOB models, will be distinguishable in GW measurements. To do that, we use a set of NR simulations obtained from the BAM code (see Tab. 1). We quantify the distinguishability of systems with different choices of in-plane spin direction $\phi_{\mathrm{Sn}}$ by calculating matches between them. This approach allows us to estimate the SNR at which the signals will be distinguishable. Our study is restricted to a small number of configurations at mass ratios $q=1,2,4$, and large in-plane spin magnitudes of 0.7 and 0.8, with two configurations with moderate in-plane spin of 0.4. All of our calculations are performed on systems with total mass $100\,M_{\odot}$.

Changes in $\phi_{\mathrm{Sn}}$ have several effects on the binary dynamics and the waveform. One effect that we discuss in detail is the asymmetry between the $\pm m$ waveform modes. Another is small changes in the phasing of the binary, and in the merger and ringdown signal. By removing asymmetry and/or precession effects from our waveforms, we show that all of these effects contribute to the waveform variations between different choices of $\phi_{\mathrm{Sn}}$. When mode-asymmetries are muted, the distinguishability SNR $\rho_{c}$ for all the systems (different $\phi_{\mathrm{Sn}}$ and different $\chi_{p}$) show a marked increase across the ($\theta,\phi$) space (see Fig. 7). Disregarding mode asymmetries increases $\rho_{c}$ by factors of $\sim 1.5-1.9$ between systems of different $\phi_{\mathrm{Sn}}$, indicating that this is a significant feature of these waveforms.

Our main results are shown in Sec. V.1, and show that for large in-plane spins, variations in $\phi_{\mathrm{Sn}}$ will be distinguishable at moderate SNRs. More importantly, these effects will influence measurements at SNRs comparable to those at which in-plane spin magnitudes become measurable. For example, in the $q=2$ systems we considered, a change in spin magnitude of 0.3 will be distinguishable at a comparable SNR to a change in spin direction of $\pi/2$. This effect will be reduced for smaller spins, but so will our ability to measure the spin magnitude. Precession effects and in-plane spin magnitude, typically captured by the parameter $\chi_{p}$, have not yet been identified in individual observations Fairhurst:2019srr . Our results suggest that when they are, the absence of in-plane spin direction effects in the modelling could lead to significant parameter biases. We plan to study the impact on parameter estimation in future work.

There are a number of questions that require further work. We have limited ourselves to small number of configurations, and to one choice of total mass. We have also neglected the effect of $\ell>2$ modes, which also impact parameter estimation for systems with mass ratios of $q\geq 2$ Kalaghatgi:2019log . The impact of changes in $\phi_{\mathrm{Sn}}$, and the importance of mode asymmetries, also needs to be studied for systems with lower masses, where the inspiral contributes more power to the waveform, with mode-asymmetric effects being weaker but with a larger number of precessional cycles. However, in order to fully understand the importance of these physical effects, we require models that include them, which can then be used in parameter-estimation studies. This work has provided strong evidence that these effects must be taken into account in order to make unbiassed physical measurements from GW observations, and therefore already provide a strong motivation for such modelling. This has already been done for the surrogate models described in Refs. Blackman:2017pcm ; Varma:2019csw . Since these models are valid only for high-mass systems and a limited range of mass ratios, it would be advantageous to be extended to other classes of model.

We thank Frank Ohme and Sebastian Khan for useful discussions. We thank Edward Fauchon-Jones, Eleanor Hamilton, Charlie Hoy and Dave Yeeles for their help in performing the comparison cases NR simulations.

This work was supported by Science and Technology Facilities Council (STFC) grant ST/L000962/1, European Research Council Consolidator Grant 647839. We are grateful for computational resources provided by Cardiff University, and funded by an STFC grant supporting UK Involvement in the Operation of Advanced LIGO.

Numerical simulations were performed on the DiRAC@Durham facility managed by the Institute for Computational Cosmology on behalf of the STFC DiRAC HPC Facility (www.dirac.ac.uk). The equipment was funded by BEIS capital funding via STFC capital grants ST/P002293/1, ST/R002371/1 and ST/S002502/1, Durham University and STFC operations grant ST/R000832/1. DiRAC is part of the National e-Infrastructure.