# Importance of transient resonances in extreme-mass-ratio inspirals

Extreme-mass-ratio inspirals (EMRIs for short) are a promising source for the planned space-borne gravitational-wave observatory LISA. To detect and analyse them we need accurate models for the signals, which are exquisitely intricate. In this paper, we investigated a feature, transient resonances, which have not previously included in our models. They are difficult to incorporate, but can have a big impact on the signal. Fortunately, we find that we can still detect the majority of EMRIs, even without including resonances. Phew!

### EMRIs and orbits

EMRIs are a beautiful gravitational wave source. They occur when a stellar-mass black hole slowly inspirals into a massive black hole (as found in the centre of galaxies). The massive black hole can be tens of thousands or millions of times more massive than the stellar-mass black hole (hence extreme mass ratio). This means that the inspiral is slow—we can potentially measure tens of thousands of orbits. This is both the blessing and the curse of EMRIs. The huge numbers of cycles means that we can closely follow the inspiral, and build a detailed map of the massive black hole’s spacetime. EMRIs will give us precision measurements of the properties of massive black holes. However, to do this, we need to be able to find the EMRI signals in the data, we need models which can match the signals over all these cycles. Analysing EMRIs is a huge challenge.

EMRI orbits are complicated. At any moment, the orbit can be described by three orbital frequencies: one for radial (in/out) motion $\Omega_r$, one for polar (north/south if we think of the spin of the massive black hole like the rotation of the Earth) motion $\Omega_\theta$ and one for axial (around in the east/west direction) motion. As gravitational waves are emitted, and the orbit shrinks, these frequencies evolve. The animation above, made by Steve Drasco, illustrates the evolution of an EMRI. Every so often, so can see the pattern freeze—the orbits stays in a constant shape (although this still rotates). This is a transient resonance. Two of the orbital frequencies become commensurate (so we might have 3 north/south cycles and 2 in/out cycles over the same period [bonus note])—this is the resonance. However, because the frequencies are still evolving, we don’t stay locked like this is forever—which is why the resonance is transient. To calculate an EMRI, you need to know how the orbital frequencies evolve.

The evolution of an EMRI is slow—the time taken to inspiral is much longer than the time taken to complete one orbit. Therefore, we can usually split the problem of calculating the trajectory of an EMRI into two parts. On short timescales, we can consider orbits as having fixed frequencies. On long timescale, we can calculate the evolution by averaging over many orbits. You might see the problem with this—around resonances, this averaging breaks down. Whereas normally averaging over many orbits means averaging over a complicated trajectory that hits pretty much all possible points in the orbital range, on resonance, you just average over the same bit again and again. On resonance, terms which usually average to zero can become important. Éanna Flanagan and Tanja Hinderer first pointed out that around resonances the usual scheme (referred to as the adiabatic approximation) doesn’t work.

A non-resonant EMRI orbit in three dimensions (left) and two dimensions (right), ignoring the rotation in the axial direction. A non-resonant orbit will eventually fill the $r$$\theta$ plane. Credit: Rob Cole

For comparison, a resonant EMRI orbit. A 2:3 resonance traces the same parts of the $r$$\theta$ plane over and over. Credit: Rob Cole

Around a resonance, the evolution will be enhanced or decreased a little relative to the standard adiabatic evolution. We get a kick. This is only small, but because we observe EMRIs for so many orbits, a small difference can grow to become a significant difference later on. Does this mean that we won’t be able to detect EMRIs with our standard models? This was a concern, so back at the end of PhD I began to investigate [bonus note]. The first step is to understand the size of the kick.

A jump in the orbital energy across a 2:3 resonance. The plot shows the difference between the approximate adiabatic evolution and the instantaneous evolution including the resonance. The thickness of the blue line is from oscillations on the orbital timescale which is too short to resolve here. The dotted red line shows the fitted size of the jump. Time is measured in terms of the resonance time $\tau_\mathrm{res}$ which is defined below. Figure 4 of Berry et al. (2016).

### Resonance kicks

If there were no gravitational waves, the orbit would not evolve, it would be fixed. The orbit could then be described by a set of constants of motion. The most commonly used when describing orbits about black holes are the energy, angular momentum and Carter constant. For the purposes of this blog, we’ll not worry too much about what these constants are, we’ll just consider some constant $I$.

The resonance kick is a change in this constant $\Delta I$. What should this depend on? There are three ingredients. First, the rate of change of this constant $F$ on the resonant orbit. Second, the time spent on resonance $\tau_\mathrm{res}$. The bigger these are, the bigger the size of the jump. Therefore,

$|\Delta I| \propto F \tau_\mathrm{res}$.

However, the jump could be positive or negative. This depends upon the relative phase of the radial and polar motion [bonus note]—for example, do they both reach their maximum point at the same time, or does one lag behind the other? We’ll call this relative phase $q$. By varying $q$ we explore we can get our resonant trajectory to go through any possible point in space. Therefore, averaging over $q$ should get us back to the adiabatic approximation: the average value of $\Delta I$ must be zero. To complete our picture for the jump, we need a periodic function of the phase,

$\Delta I = F \tau_\mathrm{res} f(q)$,

with $\langle f(q) \rangle_q = 0$. Now, we know the pieces, we can try to figure out what the pieces are.

The rate of change $F$ is proportional the mass ratio $\eta \ll 1$: the smaller the stellar-mass black hole is relative to the massive one, the smaller $F$ is. The exact details depend upon gravitational self-force calculations, which we’ll skip over, as they’re pretty hard, but they are the same for all orbits (resonant or not).

We can think of the resonance timescale either as the time for the orbital frequencies to drift apart or the time for the orbit to start filling the space again (so that it’s safe to average). The two pictures yield the same answer—there’s a fuller explanation in Section III A of the paper. To define the resonance timescale, it is useful to define the frequency $\Omega = n_r \Omega_r - n_\theta \Omega_\theta$, which is zero exactly on resonance. If this is evolving at rate $\dot{\Omega}$, then the resonance timescale is

$\displaystyle \tau_\mathrm{res} = \left[\frac{2\pi}{\dot{\Omega}}\right]^{1/2}$.

This bridges the two timescales that usually define EMRIs: the short orbital timescale $T$ and the long evolution timescale $\tau_\mathrm{ev}$:

$T \sim \eta^{1/2} \tau_\mathrm{res} \sim \eta \tau_\mathrm{ev}$.

To find the form of for $f(q)$, we need to do some quite involved maths (given in Appendix B of the paper) [bonus note]. This works by treating the evolution far from resonance as depending upon two independent times (effectively defining $T$ and $\tau_\mathrm{ev}$), and then matching the evolution close to resonance using an expansion in terms of a different time (something like $\tau_\mathrm{res}$). The solution shows that the jump depends sensitively upon the phase $q$ at resonance, which makes them extremely difficult to calculate.

We numerically evaluated the size of kicks for different orbits and resonances. We found a number of trends. First, higher-order resonances (those with larger $n_r$ and $n_\theta$) have smaller jumps than lower-order ones. This makes sense, as higher-order resonances come closer to covering all the points in the space, and so are more like averaging over the entire space. Second, jumps are larger for higher eccentricity orbits. This also makes sense, as you can’t have resonances for circular (zero eccentricity orbits) as there’s no radial frequency, so the size of the jumps must tend to zero. We’ll see that these two points are important when it comes to observational consequences of transient resonances.

### Astrophysical EMRIs

Now we’ve figured out the impact of passing through a transient resonance, let’s look at what this means for detecting EMRIs. The jump can mean that the evolution post-resonance can soon become out of phase with that pre-resonance. We can’t match both parts with the same adiabatic template. This could significantly hamper our prospects for detection, as we’re limited to the bits of signal we can pick up between resonances.

We created an astrophysical population of simulated EMRIs. We used numerical simulations to estimate a plausible population of massive black holes and distribution of stellar-mass black holes insprialling into them. We then used adiabatic models to see how many LISA (or eLISA as it was called at the time) could potentially detect. We found there were ~510 EMRIs detectable (with a signal-to-noise ratio of 15 or above) for a two-year mission.

We then calculated how much the signal-to-noise ratio would be reduced by passing through transient resonances. The plot below shows the distribution of signal-to-noise ratio for the original population, ignoring resonances, and then after factoring in the reduction. There are now ~490 detectable EMRIs, a loss of 4%. We can still detect the majority of EMRIs!

Distribution of signal-to-noise ratios for EMRIs. In blue (solid outline), we have the results ignoring transient resonances. In orange (dashed outline), we have the distribution including the reduction due to resonance jumps. Events falling below 15 are deemed to be undetectable. Figure 10 of Berry et al. (2016).

We were worried about the impact of transient resonances, we know that jumps can cause them to become undetectable, so why aren’t we seeing a bit effect in our population? The answer lies is in the trends we saw earlier. Jumps are large for low order resonances with high eccentricities. These were the ones first highlighted, as they are obviously the most important. However, low-order resonances are only encountered really close to the massive black hole. This means late in the inspiral, after we have already accumulated lots of signal-to-noise ratio. Losing a little bit of signal right at the end doesn’t hurt detectability too much. On top of this, gravitational wave emission efficiently damps down eccentricity. Orbits typically have low eccentricities by the time they hit low-order resonances, meaning that the jumps are actually quite small. Although small jumps lead to some mismatch, we can still use our signal templates without jumps. Therefore, resonances don’t hamper us (too much) in finding EMRIs!

This may seem like a happy ending, but it is not the end of the story. While we can detect EMRIs, we still need to be able to accurately infer their source properties. Features not included in our signal templates (like jumps), could bias our results. For example, it might be that we can better match a jump by using a template for a different black hole mass or spin. However, if we include jumps, these extra features could give us extra precision in our measurements. The question of what jumps could mean for parameter estimation remains to be answered.

arXiv: 1608.08951 [gr-qc]
Journal: Physical Review D; 94(12):124042(24); 2016
Conference proceedings: 1702.05481 [gr-qc] (only 2 pages—ideal for emergency journal club presentations)
Favourite jumpers: Woolly, Mario, Kangaroos

### Bonus notes

When discussing resonances, and their impact on orbital evolution, we’ll only care about $\Omega_r$$\Omega_\theta$ resonances. Resonances with $\Omega_\phi$ are not important because the spacetime is axisymmetric. The equations are exactly identical for all values of the the axial angle $\phi$, so it doesn’t matter where you are (or if you keep cycling over the same spot) for the evolution of the EMRI.

This, however, doesn’t mean that $\Omega_\phi$ resonances aren’t interesting. They can lead to small kicks to the binary, because you are preferentially emitting gravitational waves in one direction. For EMRIs this are negligibly small, but for more equal mass systems, they could have some interesting consequences as pointed out by Maarten van de Meent.

#### Extra time

I’m grateful to the Cambridge Philosophical Society for giving me some extra funding to work on resonances. If you’re a Cambridge PhD student, make sure to become a member so you can take advantage of the opportunities they offer.

#### Calculating jumps

The theory of how to evolve through a transient resonance was developed by Kevorkian and coauthors. I spent a long time studying these calculations before working up the courage to attempt them myself. There are a few technical details which need to be adapted for the case of EMRIs. I finally figured everything out while in Warsaw Airport, coming back from a conference. It was the most I had ever felt like a real physicist.

Transient resonances remind me of Spirographs. Thanks Frinkiac

# Perks and perils of a PhD

Pond in the gardens of Nijō Castle, Kyoto. A good spot for pondering. The castle has whistling floorboards, to warn you if an assassin is sneaking up on you. Modern buildings don’t do enough to warn you of assassins.

This blog has been neglected recently as I have been busy travelling, with conferences and meetings (with a little holiday in between) in Japan, Korea and Germany. I am now back in Birmingham where we have a veritable army of summer students. They are all enthusiastic, and seem to be doing well at both their projects and joining in lunchtime conversations. One asked whether it was a good idea to do a PhD? Travelling to interesting places in one of the perks of being an academic, but does it compensate all the hard work? Here are my thoughts on doing a PhD now mine is safely done but still fresh in my memory.

### The third degree

A PhD is not a simple continuation of your studies. One of the things that surprised me was how different research is from study (although they may share many of the same skills). At school and undergraduate you learn: you pay attention in class, you do assignments and projects, you revise and you take assessments. If you work hard at these, you pick up new knowledge and skills, and end up doing well. (Wooh!) In research, you have to solve problems, to figure out how to do things that have never been done before (which may require picking up new knowledge and skills). This can be extremely exciting: you could be the only person in the world to know how to do something, but since you are trying something new, it could also turn out not to work… You can work hard in a particular area for days, weeks or even years, and it all come to nothing.

Research projects at an undergraduate level are different from those at postgraduate. The former are usually designed to be safely solvable, and even if things don’t work out, you come to the end of your time and be given marks for effort. It’s much harder to put together a PhD dissertation without results, and a lack of progress (perhaps especially if through no fault of your own) can be especially demotivating.

When asked about doing a PhD, the current PhDs showed varying levels of enthusiasm. This is usually correlated with how things are going and how close they are to finishing. Maggie, who is always keen on encouraging people to learn about science, has put together a list of 5 reasons why you should do a PhD. I think these neatly sum up some of main motivations for doing a PhD.

### 1. Freedom and flexibility

Academia enjoys a lot of freedom. And I don’t just mean that you don’t have to wear a tie

You don’t have to work standard office hours, but can often schedule things around you. This can be especially handy if you have family commitments, you don’t function well in the morning, or just fancy an afternoon off. It can also have the downside of blurring the work/life divide. Working from home gives you the flexibility to work in your pyjamas, but it can also makes it easy to work evenings and weekends—perhaps the weekends are the best time to come in to the lab because there are fewer people trying to use the most shiny equipment. It can be difficult to maintain a healthy work/life balance, and it can also lead to ridiculous expectations that you can work all the time (or should feel guilty if you’re not). Of course, sometimes you have to visit the lab every two hours to look after your experiment, and there’s no flexibility at all.

As well of freedom in when you schedule work, there is also freedom in what you do. It’s difficult to predict where a PhD will go, but you can focus in on what you are interested in and what you enjoy. Your supervisor, future examiners and potential employers may disagree with you about what’s worthwhile researching, so you do have some constraints; however, as long as solve an interesting problem, it doesn’t matter as much as in industry if it’s a different one to the one you started with. Some of the best PhD projects I have seen (or been involved with) come about because the student came across a new technique they wanted to play with, read up on a different area out or just wanted to help answer someone else’s question. Procrastination can have some useful side-effects.

### 2. The title

Being a doctor is pretty cool. Not as cool as being the Doctor, but still, it can command some respect. However, that doesn’t mean you receive (or deserve) any special treatment. Contrary to popular opinion, your title doesn’t go on your passport. It does indicate that you are an expert in one particular area; however, this might be an obscure and unhelpful one.  If you are ever on a flight and the attendant asks “Is there a doctor of astrophysics on board?” you are probably sufficiently doomed that you might as well just stay seated and try to finish up your peanuts in whatever time remains.

In the end, it is having completed the difficult research and produced a quality thesis that is worth the respect, and not the extra letters with your name. If you are not interested in the the former, the latter will not give you the motivation to put in the time and effort to complete it.

### 3. To prove you’re smart

Leading on from the last point, a doctorate is a seal of academic quality. However, I really wouldn’t suggest doing a PhD because you need to validate your intelligence. You are intelligent whether or not you decide to go to graduate school, and one should never assume that someone is less smart because they lack a PhD—first, because you do not know what opportunities they may or may not have had in life, and second, intelligence is about more than academic achievements. If you’re a lazy writer, giving a character a PhD is an easy way to establish they are clever without having to think of way for them to show it. In real life, people will soon figure out how smart you are by interacting with you (if they are only interested in titles, find someone else).

Getting a PhD isn’t just a case of being smart, it’s not a prize on a game show. As much as intelligence, a doctorate requires determination. Undergraduate is like a sprint, you can work really hard for a short stretch (around exams) and then collapse. The quickest people will come out on top, but you could still take it at a jog and make it to the end provided you don’t mind being second. A doctorate is more like a marathon, its not enough to be fast, you need to be able to keep going, to pace yourself and to pick yourself up if you trip. Both can be exhausting and painful, but it’s much more important to figure out if you can really go the distance before starting on the 26 miles.

Perhaps you are unsure if you’re up-to-scratch and want to try a PhD to see? Finding out that you can do it may be a huge confidence boost! However, academia can also batter your ego as you’ll be surrounded by other equally intelligent people. I guess you just need to be happy with who you are.

Finally, parents may like to show off the achievements of their children, and you may make your friends proud, but it’s not them who have to spend the time in the library. Making people you love proud is wonderful, but so is spending time with them. A PhD can consume huge amounts of time, energy and attention (especially while writing up). It should be something that you want to do, not something other people want you to do.

Academia does give you the chance to visit new places and work with people all over the world. I really enjoyed my summer travels and Maggie is currently observing at a telescope in Chile. Of course you don’t always have that luxury: sometimes conferences aren’t in interesting places or funding could be running short. My first two conferences were in Glasgow and Cardiff. Both are lovely cities to visit, but neither was a once-in-a-lifetime opportunity. If you are really keen on travel, then there are other careers that give you better excuses to travel. Or you could take a better paid job and just pay for yourself to go on holiday. Travel, like free coffee, is a perk, but it’s not enough to justify doing a PhD.

I visited the in-construction Kamioka Gravitational Wave Detector (KAGRA) in Japan. It is being built underground, in an old mine in the Hida Mountains. You can see part of the vacuum tubing for one of the laser-interferometer arms in the foreground, and where they’re going to suspend a mirror from the room above in the background. It’s amazing engineering and the views outside are impressive too! They’re on a tight schedule, aiming for a first run (albeit at terrible sensitivity) this year.

More importantly, a PhD gives you the opportunity to come across new ideas and ways of looking at a problem; to work with interesting, intelligent people from a range of backgrounds, and time to examine the world (or Universe) in detail in many different ways. That might all be from your cluttered desk, but it can be really exciting.

### 5. For knowledge

Over the course of a doctorate you will learn many things: the best seminars for free food, how to manage you supervisor and lots of transferable skills. However, the big thing is your thesis. Through your research, you will contribute something to the sum of human knowledge. It may be revolutionary, it’s more likely to be something that will go towards a bigger question (with help from lots of others), but it could also be the discovery that this particular thing doesn’t work. You research will push back the boundary of the unknown. You will become a world expert in your area: no-one will know your research as well as you do. If there is a topic that really interests you, if there is something that you want to know more about, then a PhD gives you the chance to explore this.

In my opinion, this is the only reason to do a PhD. There are other benefits and perks, but this should be your motivation. A PhD is not just a training course, but is another step towards understanding everything. I think that is amazing.

### The forbidden motivation

Having been through the list, you may think there is something missing. What about doing a PhD to get a job? There are few careers that require a PhD, and it may not serve any more advantage than a Masters. Doing a PhD probably won’t make you rich. It may make you more attractive to some employers, but maybe spending the same amount of time working your way up from a lower rung would be just as effective? Extremely few employers have any kind of hiring scheme for PhDs, so in many cases you would start at a similar level as someone with an undergraduate degree. Some areas, of course, have strong industrial links, so it’s easy to move across. In this case, doing a PhD can be a great option: you can even get to work with potential future employers during your study (and possibly get some extra funding). The usefulness of a PhD strongly depends on the area.

There is one domain where a PhD is the well-established first step. Academia. Many think of academia as the logical progression, but it is not. You are not guaranteed an academic job with a PhD. In the sciences, most PhDs will not continue in academia. According to a report from the Royal Society, only 3.5% of science PhDs in the UK end up with a permanent academic position, and only 0.45% become professors. Competition is extremely tough: the number of PhDs awarded is increasing rapidly, but the number of faculty positions is remaining constant. I do not think the situation is better in the arts. A PhD student should not expect to get a job in academia.