<p>A discussion of the analysis of LISA data is given by <a href="https://academic.oup.com/mnras/article/508/1/803/6372929" rel="nofollow noreferrer">Boileau et al. (2021)</a>. Specifically, they provide an in depth analysis of exactly the point you are interested in - to what extent is LISA able to &quot;resolve&quot; the gravitational waves from what are essentially monochromatic sources of gravitational wave radiation all over the Galaxy.</p>
<p>The majority of these sources will be binary systems involving white dwarfs - double white dwarfs mostly but also white dwarfs in few hour periods with M-dwarfs (so-called cataclysmic variables and related objects). There will also be some neutron star binaries but these should be much rarer. White dwarf binaries are common because (i) white dwarfs are common, (ii) their low mass progenitor stellar binaries are common, (iii) they can be brought closer together during mass transfer in the red giant phases and become short-period compact binaries that emit lots of gravitational waves at (twice their) orbital frequencies that fall inside the LISA sensitivity window.</p>
<p>Binary systems involving a pair of normal stars will not commonly be seen because these will have orbital periods of at least six hours (otherwise they would merge) and hence <span class="math-container">$f_{\rm GW} &lt; 10^{-4}$</span> Hz - starting to fall outside of LISA's sensitivity window. The populations with periods of 6 hours to days are sufficiently rare that combined with LISA's poor sensitivity at these frequencies, they won't make much of a contribution to LISA detections.</p>
<p>Boileau et al. make a very detailed model of the white dwarf binary population of the Milky Way and simulate the signals that will be received by LISA. The plot below is interesting. It shows the population of double white dwarf binary objects that will be detected above some SNR threshold (the solid line is the sensitivity threshold of LISA for its nominal 4-year mission) and also, of those, which will be &quot;resolvable&quot; (the blue points), in terms of presenting an isolated peak at a frequency separated from any other by more than a frequency that is the reciprocal of the mission lifetime (<span class="math-container">$\sim 10^{-8}$</span> Hz).</p>
<p>According to Boileau et al. <strong>only 1 in 1000</strong> of the <span class="math-container">$\sim 35$</span> million binary objects producing gravitational waves at <span class="math-container">$f &gt; 10^{-5}$</span> Hz will be resolvable. i.e. About 35,000 objects.</p>
<p>The resolvable objects clearly separate out into two groups at lower and higher frequencies. The low frequency objects (orbital periods of 1-3 hours) are relatively low mass systems. These generally have low gravitational wave &quot;luminosities&quot;, but the ones that become resolvable are those rare examples that are close to the Earth. In contrast, the high frequency resolvable objects, with orbital periods of 10 seconds to minutes, tend to be higher mass objects that are powerful gravitational wave emitters. These are intrinsically rare, so often separated from any other signal, and can be seen over a wide range of distances and amplitudes.</p>
<p>In between, at orbital periods of minutes to hours is the region that contains the bulk of the binary white dwarf population, that is <em>detectable</em>, but because there are so many examples, they all merge into a complete mish-mash of unresolved signals.</p>
<p><a href="https://i.sstatic.net/EwRHu.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/EwRHu.jpg" alt="Detection statistics for double white dwarfs" /></a></p>

CC BY-SA 4.0
2023-11-15T03:42:34.557Z
Oh that is *so darn cool!* (pardon my unscientific outburst, but this is a *really interesting* result!) Thanks!
uhoh
CC BY-SA 4.0
2023-11-14T20:39:46.313Z
2023-11-14T20:39:46.313Z
ProfRob
<p>LISA's data will be very different from LIGO's. It will typically "see" many sources at the same time. Most prominently:</p>

<ul>
<li><strong>Mergers of pairs of supermassive blackholes</strong>. These will be very much like a scaled version of the events that LIGO sees (probably more variation in the mass-ratio and ecccentricity though). They will be much louder (SNR>1000) than any event seen by LIGO though. They will (potentially) enter LISA's sensitivity band months before merger. Event rates are uncertain with estimates ranging from a few to hundreds per year. Depending on where this rate is in nature, we may get many overlapping signals.</li>
<li><strong>Wide binaries</strong> These are binaries of compact objects (mostly white dwarfs) that are far from merger. They evolve very slowly, and will therefore mostly show up as continuous almost monochromatic signals. There will be thousands of such signals "visible" at any one time.</li>
<li>There has been some suggestion that LISA may be able to see the earlier stages of <strong>GW150914 like events</strong> months to years before merger. The current design limitations make this somewhat unlikely though.</li>
<li><strong>Extreme mass ratio inspirals</strong> These are mergers of stellar mass compact object with a supermassive black hole. Due to the very small mass-ratio their evolution is very small, meaning that they will stat in the LISA sensitivity band years before merger. Their event rate is even more uncertain than that of supermassive BH mergers with estimates ranging from 1 to 1000s per year. In all likelihood these signal will overlap.</li>
</ul>

<p>Consequently, LISA will be observing thousands of sources at any one time, which will be a significant challenge for the data analysis.</p>

CC BY-SA 4.0
2019-08-30T10:13:45.497Z
`+1` Beautiful! Thank you for the concise yet thorough answer. "... thousands of sources at any one time." We can assume then that they'll be using more than 1024 samples in their FFTs ;-)
uhoh
CC BY-SA 4.0
2019-08-30T10:05:38.490Z
2019-08-30T10:05:38.490Z
TimRias
<p>Our galaxy hosts probably more than <span class="math-container">$10^6$</span> binary stars with periods ranging from one day to about 100 days. Each is thought to have at least six planets orbiting it. Each planet has a different orbital period and causes phase modulation with the corresponding frequency. The first planet causes at least two spectral lines (sidebands). Each additional planet doubles this number, as can be checked with FFT. With 6 planets, <span class="math-container">$2^6=64$</span> spectral lines surround the carrier frequency <span class="math-container">$f_{GW}$</span>.</p>
<p>Roughly calculated, ~<span class="math-container">$10^8$</span> spectral lines fill the frequency range between 0.2 <span class="math-container">$\mu$</span>Hz and 12 <span class="math-container">$\mu$</span>Hz. You would have to accumulate data for several hundred years to achieve the necessary frequency resolution. Second problem: How do you find a related group of spectral lines that fits one of the many binary systems. Neither the number of planets nor their orbital periods nor the frequency drift is known.</p>

CC BY-SA 4.0
2023-11-14T20:50:39.440Z
LISA is not sensitive to $\mu$Hz gravitational waves. The frequency resolution of LISA wil be about $10^{-8}$ Hz (depending on mission lifetime). The typical source it will detect will have frequency of $\sim 10^{-3}$ Hz. Short period planets cannot exist around binary white dwarfs. They will have cleared out anything within a few au during their prior evolution. The orbital periods of any planets will be longer than a few years. Any sidebands this will produce are separated by $<10^{-8}$ Hz and unresolved in any case.
ProfRob
CC BY-SA 4.0
2023-11-14T20:53:13.417Z
You are correct though that the vast majority of detectable binaries will be unresolved. However, there are various parts of the amplitude-frequency domain that are sparsely populated - either because the objects are intrinsically rare or because the detectable examples have to be nearby (and are therefore rare). These objects are resolvable.
ProfRob
CC BY-SA 4.0
2023-11-15T16:03:15.423Z
The observation period $T_{obs}$ determines the frequency resolution $\Delta f$ (see https://en.wikipedia.org/wiki/Küpfmüller's_uncertainty_principle). With an observation period of $T_{obs}>1$ year, the necessary frequency resolution $\Delta f\approx 10^{-8}$ can be achieved. Useful resolution *cannot* be achieved with only 1024 samples.
9herbert9
CC BY-SA 4.0
2023-11-15T18:17:24.420Z
You are repeating back to me what I said? But where does the 1024 samples number come from?
ProfRob
CC BY-SA 4.0
2023-11-15T21:42:59.367Z
As far as I know @uhoh mentioned this value, I repeated it. How useful 1024 samples are depends on the sampling frequency. In my analysis I prefer $2^{16} samples or more.
9herbert9
CC BY-SA 4.0
2023-11-05T20:21:00.133Z
2023-11-05T20:21:00.133Z
9herbert9
<p><a href="https://astronomy.stackexchange.com/a/32783/7982">@RobJeffries' clear answer</a> to <em>Which things “LIGO can see things that LISA can't”, and vice-versa?</em> explains that the <a href="https://en.wikipedia.org/wiki/Laser_Interferometer_Space_Antenna" rel="nofollow noreferrer">Laser Interferometer Space Antenna</a> or LISA will only be sensitive to gravitational waves with frequencies below about 1 Hz and so will be sensitive to much slower rotating objects than those that LIGO records.</p>

<blockquote>
  <p>Lower frequency GWs can only be seen with LISA. This would include stellar binary systems with orbital periods longer than about 10 seconds, merging supermassive black holes and maybe GWs from the big bang</p>
</blockquote>

<p>This can might include "event-like" signals, for example (possibly) a merger of two supermassive black holes that are so large in diameter that even during the merger their orbital frequency is below 1 Hz, but it can also include signals from the rotation of bodies not yet in contact but are slowly losing angular momentum though continuous GW radiation. While these would be much weaker, they would provide a very stable and very narrow peak in a spectrogram with an extremely slow drift, so you could accumulate data for years in order to bring them up out of the noise.</p>

<p>While LIGO's detected events have lasted of the order of 1 second or less, it sounds like it is possible that LISA could potentially record continuous GW radiation from a large number of pairs at the same time since the "lifetimes" might be years or millennia rather than seconds, provided that the sensitivity to these weaker signals is sufficient.</p>

<p><strong>Question:</strong> What is the expected nature of LISA's data; will it be more like a forest of fairly static or slowly moving peaks in frequency space, or a series of individual events? Presumably this has been modeled based on some predictions of the number of different kinds of pairs at various separations expected to be found nearby.</p>

<observational-astronomy><gravitational-waves><ligo><lisa>
Expected nature of LISA's data; will it be more like a forest of static peaks, or a series of individual events?
uhoh
# Expected nature of LISA's data; will it be more like a forest of static peaks, or a series of individual events?

(Asked by: username_0 on 2019-07-25)

[@RobJeffries' clear answer](https://astronomy.stackexchange.com/a/32783/7982) to *Which things “LIGO can see things that LISA can't”, and vice-versa?* explains that the [Laser Interferometer Space Antenna](https://en.wikipedia.org/wiki/Laser_Interferometer_Space_Antenna) or LISA will only be sensitive to gravitational waves with frequencies below about 1 Hz and so will be sensitive to much slower rotating objects than those that LIGO records.

> Lower frequency GWs can only be seen with LISA. This would include stellar binary systems with orbital periods longer than about 10 seconds, merging supermassive black holes and maybe GWs from the big bang

This can might include "event-like" signals, for example (possibly) a merger of two supermassive black holes that are so large in diameter that even during the merger their orbital frequency is below 1 Hz, but it can also include signals from the rotation of bodies not yet in contact but are slowly losing angular momentum though continuous GW radiation. While these would be much weaker, they would provide a very stable and very narrow peak in a spectrogram with an extremely slow drift, so you could accumulate data for years in order to bring them up out of the noise.

While LIGO's detected events have lasted of the order of 1 second or less, it sounds like it is possible that LISA could potentially record continuous GW radiation from a large number of pairs at the same time since the "lifetimes" might be years or millennia rather than seconds, provided that the sensitivity to these weaker signals is sufficient.

**Question:** What is the expected nature of LISA's data; will it be more like a forest of fairly static or slowly moving peaks in frequency space, or a series of individual events? Presumably this has been modeled based on some predictions of the number of different kinds of pairs at various separations expected to be found nearby.

**Tags:** observational-astronomy, gravitational-waves, ligo, lisa

---

## 3 answers

### Answer by username_1 on 2023-11-14. Score: 2 | Accepted answer

A discussion of the analysis of LISA data is given by [Boileau et al. (2021)](https://academic.oup.com/mnras/article/508/1/803/6372929). Specifically, they provide an in depth analysis of exactly the point you are interested in - to what extent is LISA able to "resolve" the gravitational waves from what are essentially monochromatic sources of gravitational wave radiation all over the Galaxy.

The majority of these sources will be binary systems involving white dwarfs - double white dwarfs mostly but also white dwarfs in few hour periods with M-dwarfs (so-called cataclysmic variables and related objects). There will also be some neutron star binaries but these should be much rarer. White dwarf binaries are common because (i) white dwarfs are common, (ii) their low mass progenitor stellar binaries are common, (iii) they can be brought closer together during mass transfer in the red giant phases and become short-period compact binaries that emit lots of gravitational waves at (twice their) orbital frequencies that fall inside the LISA sensitivity window.

Binary systems involving a pair of normal stars will not commonly be seen because these will have orbital periods of at least six hours (otherwise they would merge) and hence $f\_{\rm GW} < 10^{-4}$ Hz - starting to fall outside of LISA's sensitivity window. The populations with periods of 6 hours to days are sufficiently rare that combined with LISA's poor sensitivity at these frequencies, they won't make much of a contribution to LISA detections.

Boileau et al. make a very detailed model of the white dwarf binary population of the Milky Way and simulate the signals that will be received by LISA. The plot below is interesting. It shows the population of double white dwarf binary objects that will be detected above some SNR threshold (the solid line is the sensitivity threshold of LISA for its nominal 4-year mission) and also, of those, which will be "resolvable" (the blue points), in terms of presenting an isolated peak at a frequency separated from any other by more than a frequency that is the reciprocal of the mission lifetime ($\sim 10^{-8}$ Hz).

According to Boileau et al. **only 1 in 1000** of the $\sim 35$ million binary objects producing gravitational waves at $f > 10^{-5}$ Hz will be resolvable. i.e. About 35,000 objects.

The resolvable objects clearly separate out into two groups at lower and higher frequencies. The low frequency objects (orbital periods of 1-3 hours) are relatively low mass systems. These generally have low gravitational wave "luminosities", but the ones that become resolvable are those rare examples that are close to the Earth. In contrast, the high frequency resolvable objects, with orbital periods of 10 seconds to minutes, tend to be higher mass objects that are powerful gravitational wave emitters. These are intrinsically rare, so often separated from any other signal, and can be seen over a wide range of distances and amplitudes.

In between, at orbital periods of minutes to hours is the region that contains the bulk of the binary white dwarf population, that is *detectable*, but because there are so many examples, they all merge into a complete mish-mash of unresolved signals.

[![Detection statistics for double white dwarfs](https://i.sstatic.net/EwRHu.jpg)](https://i.sstatic.net/EwRHu.jpg)

> **Comments:**
> **username_0**: Oh that is *so darn cool!* (pardon my unscientific outburst, but this is a *really interesting* result!) Thanks!  

---

### Answer by username_2 on 2019-08-30. Score: 9

LISA's data will be very different from LIGO's. It will typically "see" many sources at the same time. Most prominently:

* **Mergers of pairs of supermassive blackholes**. These will be very much like a scaled version of the events that LIGO sees (probably more variation in the mass-ratio and ecccentricity though). They will be much louder (SNR>1000) than any event seen by LIGO though. They will (potentially) enter LISA's sensitivity band months before merger. Event rates are uncertain with estimates ranging from a few to hundreds per year. Depending on where this rate is in nature, we may get many overlapping signals.
* **Wide binaries** These are binaries of compact objects (mostly white dwarfs) that are far from merger. They evolve very slowly, and will therefore mostly show up as continuous almost monochromatic signals. There will be thousands of such signals "visible" at any one time.
* There has been some suggestion that LISA may be able to see the earlier stages of **GW150914 like events** months to years before merger. The current design limitations make this somewhat unlikely though.
* **Extreme mass ratio inspirals** These are mergers of stellar mass compact object with a supermassive black hole. Due to the very small mass-ratio their evolution is very small, meaning that they will stat in the LISA sensitivity band years before merger. Their event rate is even more uncertain than that of supermassive BH mergers with estimates ranging from 1 to 1000s per year. In all likelihood these signal will overlap.

Consequently, LISA will be observing thousands of sources at any one time, which will be a significant challenge for the data analysis.

> **Comments:**
> **username_0**: `+1` Beautiful! Thank you for the concise yet thorough answer. "... thousands of sources at any one time." We can assume then that they'll be using more than 1024 samples in their FFTs ;-)  

---

### Answer by username_3 on 2023-11-05. Score: 1

Our galaxy hosts probably more than $10^6$ binary stars with periods ranging from one day to about 100 days. Each is thought to have at least six planets orbiting it. Each planet has a different orbital period and causes phase modulation with the corresponding frequency. The first planet causes at least two spectral lines (sidebands). Each additional planet doubles this number, as can be checked with FFT. With 6 planets, $2^6=64$ spectral lines surround the carrier frequency $f\_{GW}$.

Roughly calculated, ~$10^8$ spectral lines fill the frequency range between 0.2 $\mu$Hz and 12 $\mu$Hz. You would have to accumulate data for several hundred years to achieve the necessary frequency resolution. Second problem: How do you find a related group of spectral lines that fits one of the many binary systems. Neither the number of planets nor their orbital periods nor the frequency drift is known.

> **Comments:**
> **username_1**: LISA is not sensitive to $\mu$Hz gravitational waves. The frequency resolution of LISA wil be about $10^{-8}$ Hz (depending on mission lifetime). The typical source it will detect will have frequency of $\sim 10^{-3}$ Hz. Short period planets cannot exist around binary white dwarfs. They will have cleared out anything within a few au during their prior evolution. The orbital periods of any planets will be longer than a few years. Any sidebands this will produce are separated by $<10^{-8}$ Hz and unresolved in any case.  
> **username_1**: You are correct though that the vast majority of detectable binaries will be unresolved. However, there are various parts of the amplitude-frequency domain that are sparsely populated - either because the objects are intrinsically rare or because the detectable examples have to be nearby (and are therefore rare). These objects are resolvable.  
> **username_3**: The observation period $T_{obs}$ determines the frequency resolution $\Delta f$ (see https://en.wikipedia.org/wiki/Küpfmüller's_uncertainty_principle). With an observation period of $T_{obs}>1$ year, the necessary frequency resolution $\Delta f\approx 10^{-8}$ can be achieved. Useful resolution *cannot* be achieved with only 1024 samples.  
> **username_1**: You are repeating back to me what I said? But where does the 1024 samples number come from?  
> **username_3**: As far as I know @uhoh mentioned this value, I repeated it. How useful 1024 samples are depends on the sampling frequency. In my analysis I prefer $2^{16} samples or more.  

---

