Eclipses of continuous gravitational waves as a probe of stellar structure

Understanding how stars work is a fundamental problem in astrophysics. We can’t open up a star to investigate its inner workings, which makes it difficult to test our models. Over the years, we have developed several ways to sneak a peek into what must be happening inside stars, such as by measuring solar neutrinos, or using asteroseismology to measure how sounds travels through a star. In this paper, we propose a new way to examine the hearts of stars using gravitational waves.

Gravitational waves interact very weakly with stuff. Whereas light gets blocked by material (meaning that we can’t see deeper than a star’s photosphere), gravitational waves will happily travel through pretty much anything. This property means that gravitational waves are hard to detect, but it also means that there’ll happily pass through an entire star. While the material that makes up a star will not affect the passing of a gravitational wave, its gravity will. The mass of a star can lead to gravitational lensing and a slight deflecting, magnification and delaying of a passing gravitational wave. If we can measure this lensing, we can reconstruct the mass of star, and potentially map out its internal structure.

Eclipsing gravitational wave sources

Two types of eclipse: the eclipse of a distant gravitational wave (GW) source by the Sun, and gravitational waves from an accreting millisecond pulsar (MSP) eclipsed by its companion. Either scenario could enable us to see gravitational waves passing through a star. Figure 2 of Marchant et al. (2020).

We proposed looking at gravitational waves for eclipsing sources—where a gravitational wave source is behind a star. As the alignment of the Earth (and our detectors), the star and the source changes, the gravitational wave will travel through different parts of the star, and we will see a different amount of lensing, allowing us to measure the mass of the star at different radii. This sounds neat, but how often will we be lucky enough to see an eclipsing source?

To date, we have only seen gravitational waves from compact binary coalescences (the inspiral and merger of two black holes or neutron stars). These are not a good source for eclipses. The chances that they travel through a star is small (as space is pretty empty) [bonus note]. Furthermore, we might not even be able to work out that this happened. The signal is relatively short, so we can’t compare the signal before and during an eclipse. Another type of gravitational wave signal would be much better: a continuous gravitational wave signal.

How common are eclipsing gravitational wave sources?

Probability of observing at least one eclipsing source amongst a number of observed sources. Compact binary coalescences (CBCs, shown in purple) are the most rare, continuous gravitational waves (CGWs) eclipsed by the Sun (red) or by a companion (red) are more common. Here we assume companions are stars about a tenth the mass of the neutron star. The number of neutron stars with binary companions is estimated using the COSMIC population synthesis code. Results are shown for eclipses where the gravitational waves get within distance b of the centre of the star. Figure 1 of Marchant et al. (2020).

Continuous gravitational waves are produced by rotating neutron stars. They are pretty much perfect for searching for eclipses. As you might guess from their name, continuous gravitational waves are always there. They happily hum away, sticking to pretty much the same note (they’d get pretty annoying to listen to). Therefore, we can measure them before, during and after an eclipse, and identify any changes due to the gravitational lensing. Furthermore, we’d expect that many neutron stars would be in close binaries, and therefore would be eclipsed by their partner. This would happen each time they orbit, potentially giving us lots of juicy information on these stars. All we need to do is measure the continuous gravitational wave…

The effect of the gravitational lensing by a star is small. We performed detailed calculations for our Sun (using MESA), and found that for the effects to be measurable you would need an extremely loud signal. A signal-to-noise ratio would need to be hundreds during the eclipse for measurement precision to be good enough to notice the imprint of lensing. To map out how things changed as the eclipse progressed, you’d need signal-to-noise ratios many times higher than this. As an eclipse by the Sun is only a small fraction of the time, we’re going to need some really loud signals (at least signal-to-noise ratios of 2500) to see these effects. We will need the next generation of gravitational wave detectors.

We are currently thinking about the next generation of gravitational wave detectors [bonus note]. The leading ideas are successors to LIGO and Virgo: detectors which cover a large range of frequencies to detect many different types of source. These will be expensive (billions of dollars, euros or pounds), and need international collaboration to finance. However, I also like the idea of smaller detectors designed to do one thing really well. Potentially these could be financed by a single national lab. I think eclipsing continuous waves are the perfect source for this—instead of needing a detector sensitive over a wide frequency range, we just need to be sensitive over a really narrow range. We will be able to detect continuous waves before we are able to see the impact of eclipses. Therefore, we’ll know exactly what frequency to tune for. We’ll also know exactly when we need to observe. I think it would be really awesome to have a tunable narrowband detector, which could measure the eclipse of one source, and then be tuned for the next one, and the next. By combining many observations, we could really build up a detailed picture of the Sun. I think this would be an exciting experiment—instrumentalists, put your thinking hats on!

Let’s reach for(the centres of) the stars.

arXiv: 1912.04268 [astro-ph.SR]
Journal: Physical Review D; 101(2):024039(15); 2020
Data release: Eclipses of continuous gravitational waves as a probe of stellar structure
CIERA story: Using gravitational waves to see inside stars
Why does the sun really shine? The Sun is a miasma of incandescent plasma

Bonus notes

Silver lining

Since signals from compact binary coalescences are so unlikely to be eclipsed by a star, we don’t have to worry that our measurements of the source property are being messed up by this type of gravitational lensing distorting the signal. Which is nice.

Prospects with LISA

If you were wondering if we could see these types of eclipses with the space-based gravitational wave observatory LISA, the answer is sadly no. LISA observes lower frequency gravitational waves. Lower frequency means longer wavelength, so long in fact that the wavelength is larger than the size of the Sun! Since the size of the Sun is so small compared to the gravitational wave, it doesn’t leave a same imprint: the wave effectively skips over the gravitational potential.

Second star to the right and straight on ’til morning—Astrophysics white papers

What will be the next big thing in astronomy? One of the hard things about research is that you often don’t know what you will discover before you embark on an investigation. An idea might work out, or it might not, or along the way you might discover something unexpected which is far more interesting. As you might imagine, this can make laying definite plans difficult…

However, it is important to have plans for research. While you might not be sure of the outcome, it is necessary to weigh the risks and rewards associated with the probable results before you invest your time and taxpayers’ money!

To help with planning and prioritising, researchers in astrophysics often pull together white papers [bonus note]. These are sketches of ideas for future research, arguing why you think they might be interesting. These can then be discussed within the community to help shape the direction of the field. If other scientists find the paper convincing, you can build support which helps push for funding. If there are gaps in the logic, others can point these out to ave you heading the wrong way. This type of consensus building is especially important for large experiments or missions—you don’t want to spend a billion dollars on something unless you’re really sure it is a good idea and lots of people agree.

I have been involved with a few white papers recently. Here are some key ideas for where research should go.

Ground-based gravitational-wave detectors: The next generation

We’ve done some awesome things with Advanced LIGO and Advanced Virgo. In just a couple of years we have revolutionized our understanding of binary black holes. That’s not bad. However, our current gravitational-wave observatories are limited in what they can detect. What amazing things could we achieve with a new generation of detectors?

It can take decades to develop new instruments, therefore it’s important to start thinking about them early. Obviously, what we would most like is an observatory which can detect everything, but that’s not feasible. In this white paper, we pick the questions we most want answered, and see what the requirements for a new detector would be. A design which satisfies these specifications would therefore be a solid choice for future investment.

Binary black holes are the perfect source for ground-based detectors. What do we most want to know about them?

  1. How many mergers are there, and how does the merger rate change over the history of the Universe? We want to know how binary black holes are made. The merger rate encodes lots of information about how to make binaries, and comparing how this evolves compared with the rate at which the Universe forms stars, will give us a deeper understanding of how black holes are made.
  2. What are the properties (masses and spins) of black holes? The merger rate tells us some things about how black holes form, but other properties like the masses, spins and orbital eccentricity complete the picture. We want to make precise measurements for individual systems, and also understand the population.
  3. Where do supermassive black holes come from? We know that stars can collapse to produce stellar-mass black holes. We also know that the centres of galaxies contain massive black holes. Where do these massive black holes come from? Do they grow from our smaller black holes, or do they form in a different way? Looking for intermediate-mass black holes in the gap in-between will tells us whether there is a missing link in the evolution of black holes.
Detection horizon as a function of binary mass for Advanced LIGO, A+, Cosmic Explorer and the Einstein Telescope

The detection horizon (the distance to which sources can be detected) for Advanced LIGO (aLIGO), its upgrade A+, and the proposed Cosmic Explorer (CE) and Einstein Telescope (ET). The horizon is plotted for binaries with equal-mass, nonspinning components. Adapted from Hall & Evans (2019).

What can we do to answer these questions?

  1. Increase sensitivity! Advanced LIGO and Advanced Virgo can detect a 30 M_\odot + 30 M_\odot binary out to a redshift of about z \approx 1. The planned detector upgrade A+ will see them out to redshift z \approx 2. That’s pretty impressive, it means we’re covering 10 billion years of history. However, the peak in the Universe’s star formation happens at around z \approx 2, so we’d really like to see beyond this in order to measure how the merger rate evolves. Ideally we would see all the way back to cosmic dawn at z \approx 20 when the Universe was only 200 million years old and the first stars light up.
  2. Increase our frequency range! Our current detectors are limited in the range of frequencies they can detect. Pushing to lower frequencies helps us to detect heavier systems. If we want to detect intermediate-mass black holes of 100 M_\odot we need this low frequency sensitivity. At the moment, Advanced LIGO could get down to about 10~\mathrm{Hz}. The plot below shows the signal from a 100 M_\odot + 100 M_\odot binary at z = 10. The signal is completely undetectable at 10~\mathrm{Hz}.

    Gravitational wave signal from a binary of two 100 solar mass black holes at a redshift of 10

    The gravitational wave signal from the final stages of inspiral, merger and ringdown of a two 100 solar mass black holes at a redshift of 10. The signal chirps up in frequency. The colour coding shows parts of the signal above different frequencies. Part of Figure 2 of the Binary Black Holes White Paper.

  3. Increase sensitivity and frequency range! Increasing sensitivity means that we will have higher signal-to-noise ratio detections. For these loudest sources, we will be able to make more precise measurements of the source properties. We will also have more detections overall, as we can survey a larger volume of the Universe. Increasing the frequency range means we can observe a longer stretch of the signal (for the systems we currently see). This means it is easier to measure spin precession and orbital eccentricity. We also get to measure a wider range of masses. Putting the improved sensitivity and frequency range together means that we’ll get better measurements of individual systems and a more complete picture of the population.

How much do we need to improve our observatories to achieve our goals? To quantify this, lets consider the boost in sensitivity relative to A+, which I’ll call \beta_\mathrm{A+}. If the questions can be answered with \beta_\mathrm{A+} = 1, then we don’t need anything beyond the currently planned A+. If we need a slightly larger \beta_\mathrm{A+}, we should start investigating extra ways to improve the A+ design. If we need much larger \beta_\mathrm{A+}, we need to think about new facilities.

The plot below shows the boost necessary to detect a binary (with equal-mass nonspinning components) out to a given redshift. With a boost of \beta_\mathrm{A+} = 10 (blue line) we can survey black holes around 10 M_\odot30 M_\odot across cosmic time.

Boost to detect a binary of a given mass at a given redshift

The boost factor (relative to A+) \beta_\mathrm{A+} needed to detect a binary with a total mass M out to redshift z. The binaries are assumed to have equal-mass, nonspinning components. The colour scale saturates at \log_{10} \beta_\mathrm{A+} = 4.5. The blue curve highlights the reach at a boost factor of \beta_\mathrm{A+} = 10. The solid and dashed white lines indicate the maximum reach of Cosmic Explorer and the Einstein Telescope, respectively. Part of Figure 1 of the Binary Black Holes White Paper.

The plot above shows that to see intermediate-mass black holes, we do need to completely overhaul the low-frequency sensitivity. What do we need to detect a 100 M_\odot + 100 M_\odot binary at z = 10? If we parameterize the noise spectrum (power spectral density) of our detector as S_n(f) = S_{10}(f/10~\mathrm{Hz})^\alpha with a lower cut-off frequency of f_\mathrm{min}, we can investigate the various possibilities. The plot below shows the possible combinations of parameters which meet of requirements.

Noise curve requirements for intermediate-mass black hole detection

Requirements on the low-frequency noise power spectrum necessary to detect an optimally oriented intermediate-mass binary black hole system with two 100 solar mass components at a redshift of 10. Part of Figure 2 of the Binary Black Holes White Paper.

To build up information about the population of black holes, we need lots of detections. Uncertainties scale inversely with the square root of the number of detections, so you would expect few percent uncertainty after 1000 detections. If we want to see how the population evolves, we need these many per redshift bin! The plot below shows the number of detections per year of observing time for different boost factors. The rate starts to saturate once we detect all the binaries in the redshift range. This is as good as you’ll ever going to get.

Detections per redshift bin as a function of boost factor

Expected rate of binary black hole detections R_\mathrm{det} per redshift bin as a function of A+ boost factor \beta_\mathrm{A+} for three redshift bins. The merging binaries are assumed to be uniformly distributed with a constant merger rate roughly consistent with current observations: the solid line is about the current median, while the dashed and dotted lines are roughly the 90% bounds. Figure 3 of the Binary Black Holes White Paper.

Looking at the plots above, it is clear that A+ is not going to satisfy our requirements. We need something with a boost factor of \beta_\mathrm{A+} = 10: a next-generation observatory. Both the Cosmic Explorer and Einstein Telescope designs do satisfy our goals.

Yes!

Data is pleased. Credit: Paramount

Title: Deeper, wider, sharper: Next-generation ground-based gravitational-wave observations of binary black holes
arXiv:
1903.09220 [astro-ph.HE]
Contribution level: ☆☆☆☆☆ Leading author
Theme music: Daft Punk

Extreme mass ratio inspirals are awesome

We have seen gravitational waves from a stellar-mass black hole merging with another stellar-mass black hole, can we observe a stellar-mass black hole merging with a massive black hole? Yes, these are a perfect source for a space-based gravitational wave observatory. We call these systems extreme mass-ratio inspirals (or EMRIs, pronounced em-rees, for short) [bonus note].

Having such an extreme mass ratio, with one black hole much bigger than the other, gives EMRIs interesting properties. The number of orbits over the course of an inspiral scales with the mass ratio: the more extreme the mass ratio, the more orbits there are. Each of these gives us something to measure in the gravitational wave signal.

The intricate structure of an EMRI orbit

A short section of an orbit around a spinning black hole. While inspirals last for years, this would represent only a few hours around a black hole of mass M = 10^6 M_\odot. The position is measured in terms of the gravitational radius r_\mathrm{g} = GM/c^2. The innermost stable orbit for this black hole would be about r_\mathrm{g} = 2.3. Part of Figure 1 of the EMRI White Paper.

As EMRIs are so intricate, we can make exquisit measurements of the source properties. These will enable us to:

Event rates for EMRIs are currently uncertain: there could be just one per year or thousands. From the rate we can figure out the details of what is going in in the nuclei of galaxies, and what types of objects you find there.

With EMRIs you can unravel mysteries in astrophysics, fundamental physics and cosmology.

Have we sold you that EMRIs are awesome? Well then, what do we need to do to observe them? There is only one currently planned mission which can enable us to study EMRIs: LISA. To maximise the science from EMRIs, we have to support LISA.

Lisa Simpson dancing

As an aspiring scientist, Lisa Simpson is a strong supporter of the LISA mission. Credit: Fox

Title: The unique potential of extreme mass-ratio inspirals for gravitational-wave astronomy
arXiv:
1903.03686 [astro-ph.HE]
Contribution level: ☆☆☆☆☆ Leading author
Theme music: Muse

Bonus notes

White paper vs journal article

Since white papers are proposals for future research, they aren’t as rigorous as usual academic papers. They are really attempts to figure out a good question to ask, rather than being answers. White papers are not usually peer reviewed before publication—the point is that you want everybody to comment on them, rather than just one or two anonymous referees.

Whilst white papers aren’t quite the same class as journal articles, they do still contain some interesting ideas, so I thought they still merit a blog post.

Recycling

I have blogged about EMRIs before, so I won’t go into too much detail here. It was one of my former blog posts which inspired the LISA Science Team to get in touch to ask me to write the white paper.