Title: Forecasting the constraints on optical selection bias and projection effects of galaxy cluster lensing with multiwavelength data
Authors: Conghao Zhou, Hao-Yi Wu, Andrés N. Salcedo, Sebastian Grandis, Tesla Jeltema, Alexie Leauthaud, Matteo Costanzi, Tomomi Sunayama, David H. Weinberg Tianyu Zhang, Eduardo Rozo, Chun-Hao To, Sebastian Bocquet, Tamas Varga, and Matthew Kwiecien
First Author’s Institution: Physics Department, University of California, Santa Cruz and Santa Cruz Institute for Particle Physics
Status: Published in Physical Review D [open access]
Background
While I personally love galaxies in their own right, it turns out they are more than just cool blobs of stars, gas, and dust. Galaxies—more specifically, galaxy clusters—are an important probe of the Universe’s large scale structure. Galaxy clusters are groups of galaxies held together by gravity. The galaxies in clusters sit in a massive dark matter halo which we otherwise wouldn’t be able to see. These dark matter halos link up to form the Cosmic Web. Cosmologists care about the large-scale distribution of dark matter because it evolved from the small-scale density fluctuations in the early Universe, giving us a peek at the structure of the Universe just after the Big Bang.
Since dark matter halos are, well, dark, it’s difficult to directly measure the amount of mass they hold. We sometimes measure halo masses using gravitational lensing, or the bending of light from a background object as it passes through the curved spacetime caused by a massive object (like a galaxy cluster or a supermassive black hole). It’s also possible to measure mass using the amount of hot gas in the halo (see the discussion of the Sunyaev-Zeldovich effect below) or by proxy using the number of galaxies in the halo (also called the “richness”). More massive halos will have more gas, more galaxies, and cause more lensing. There’s a catch, though—these methods (in particular richness and lensing) can be biased by objects in the line of sight.
Astronomers are limited in our observations. It’s relatively easy to measure the distance between two points on the sky. But this distance, also called a projected distance, is only part of the puzzle. One of the two points might be in front of the other, but since we can only see them on the celestial sphere, it can be tricky to measure the distance between the two in this third dimension (also called “in the line of sight”). This trickiness can lead us to over-count the number of galaxies in a cluster if, for example, there are a bunch of galaxies behind the cluster (see Figure 1). We would therefore overestimate the mass of the halo. The same thing is true for lensing—if there is another cluster behind the cluster we care about, it might contribute to the lensing signal, leading to an overestimate of the halo mass.
A demonstration of how projection effects. Left is a circle with four small circles inside which appear close to each other. Right is an alternate view of the same system in 3d, where it becomes clear that one of the circles is behind the other three.
Figure 1. A demonstration of the bias in richness measurements from projection effects. On the left is an image of four objects which are close to each other in projected radius. This represents the appearance of four galaxies on the sky, which might be counted as part of one cluster. On the right, we can see that one of the galaxies is much further from the others in 3D space. In this view, it is clear that only three galaxies are part of the same cluster. This projection effect can lead us to overestimate the number of galaxies in a cluster (a.k.a. the “richness” of a cluster).
The authors of today’s Physical Review paper attempt to measure the bias in measurements of the gravitational lensing signal resulting from line-of-sight effects. They argue that if we can measure this bias, we can apply a correction to our measurements to obtain the true distribution of dark matter halos of different masses. They also argue that using observations in multiple wavelengths can help constrain halo masses to better accuracy than using optical data alone.
A brief aside about the Sunyaev-Zeldovich Effect
An important element of this paper is the measurement of something called the “Sunyaev-Zeldovich effect.” Let’s take a moment to explore this important tracer of galaxy clusters.
The Sunyaev-Zeldovich effect takes advantage of the fact that the entire Universe is backlit by the cosmic microwave background, or CMB. The CMB is the light that escaped when the Universe cooled down enough that protons and electrons combined to create neutral hydrogen for the first time. Recall that dark matter halos contain more than just galaxies. In particular, they contain a large reservoir of hot hydrogen gas that cycles into and out of galaxies. When light from the CMB reaches a galaxy cluster, it scatters off the electrons in the gas halo of the cluster and is boosted to higher energies in a process known as “inverse Compton scattering.” This leads to a dark spot in microwave maps of the sky where the CMB light has been boosted out of the wavelength range of the telescope. We call this the Sunyaev-Zeldovich effect.
The Sunyaev-Zeldovich effect is important for this paper because it provides a way of positively identifying clusters in another wavelength regime. Since the Sunyaev-Zeldovich effect traces the gas in galaxy clusters, and since the resolution of millimeter telescopes like the South Pole Telescope is worse than optical telescopes used to observe the galaxies inside of clusters, the Sunyaev-Zeldovich effect creates one point source per galaxy cluster in the line of sight. This means we know precisely where each cluster is.
Cluster library: the Monte Carlo Approach
The authors of today’s paper want to be able to infer the true lensing signal distribution from observations of richness and lensing signal. That means they need to be able to quantify the bias in lensing signal measurements and correct for it. They take a two-step approach: first, they construct a library of possible observed properties from a known halo mass distribution function using Monte Carlo methods. Then, they compare mock observations of simulated galaxy clusters to this library to infer the true lensing signal distribution.
In general, Monte Carlo methods involve taking a random sample from some known distribution. Imagine that you want to know the number of cards of a particular suit in a card deck. You could count all the cards, but that would take a long time. Instead, you can take a random sample of the card deck (say, a random 12 cards) and count the number of cards of each suit. As long as your sample is truly random, you should get the right fraction of, say, spades. This method works better as you take larger and larger samples.
Previous work gives an expected relationship between a cluster’s halo mass and its observable properties, including its richness and Sunyaev-Zeldovich effect. The authors generate a set of fake galaxy clusters and observables by drawing randomly from these relationships to mimic the scatter in real clusters. For example, more massive clusters tend to be both richer and have a stronger lensing signal, but any random cluster of a given halo mass might have slightly more or fewer galaxies than another cluster with the same mass. They predict the observed properties of the population of clusters for random values of richness, Sunyaev-Zeldovich effect, and true lensing signal, producing a library of possible observed lensing signals that they can use for comparison to real data.
Mock galaxy clusters from simulations
They test their model on a mock galaxy cluster catalog generated from cosmological simulations. The authors pull a set of dark matter halos from a simulation called Abacus Cosmos. Since this simulation only contains dark matter and not stars, they use the halo mass to assign a number of galaxies to each cluster while accounting for some scatter in the number of galaxies per halo. They then simulate the appearance of the galaxies in each halo using the galaxy cluster Red Sequence, a known relationship between the color and brightness of galaxies in any one cluster.
To get the observed quantities for each halo, the authors then pick a line of sight and measure the number of galaxies inside a cylinder along that line of sight to simulate the projection effects involved in real observations. They calculate the lensing signal along a line of sight by finding the overlap between dark matter and the galaxies they generated and calculating the expected distortion from the gravitational effect of the dark matter on the light from the galaxies. Finally, they use a previously-measured scaling relationship between halo mass and the Sunyaev-Zeldovich effect to calculate the intrinsic Sunyaev-Zeldovich effect for each halo. They convert this to an observed signal using the sensitivity limits of the South Pole Telescope.
Comparing “observations” to the model
For each method, the authors separate the clusters into two bins based on richness and whether they would be detected with the Sunyaev-Zeldovich effect. They measure the bias by comparing the true lensing signal from the simulation to the observed lensing signal. They explore the impact of richness and Sunyaev-Zeldovich effect on the bias. They then test their ability to recover the true lensing signal by iteratively comparing the observed lensing signal to the lensing signals in the library they constructed using Monte Carlo methods.
The impact of richness and SZE on lensing bias
In Figure 2 of this Astrobite (Figure 3 of the paper), the authors find that the bias is lower for clusters where they detect the Sunyaev-Zeldovich effect than for those without. The most biased clusters are those without Sunyaev-Zeldovich effect detection. This is because it is easier to detect the Sunyaev-Zeldovich effect in more massive galaxy clusters, so those without a detection tend to be lower mass. Low-mass clusters suffer the most from projection effects. (Consider a cluster with three background galaxies. If the cluster already has 80 galaxies, the impact of including three background galaxies is smaller than for a cluster with 25 galaxies.) Similarly, the lensing impact of a more massive cluster will dominate background lensed sources.
Figure 3 from Zhou et al. 2024. The amount of bias in gravitational lensing measurements increases as you move out from the cluster center. It is larger in richer galaxies without hot gas detections.
Figure 3 from Zhou et al. 2024. The amount of bias in gravitational lensing measurements increases as you move out from the cluster center. It is larger in richer galaxies without hot gas detections.
Can we recover the true lensing signal with their model?
If the observed lensing signal were a perfect match to a particular entry in the library, it would mean the parameters associated with that entry (i.e., the richness, Sunyaev-Zeldovich effect, and true lensing signal) are the same as the parameters for that simulated galaxy cluster. The authors’ iterative approach means that they can assign a likelihood to each parameter value based on exactly how similar the lensing signals are. For example, it may be very likely that a particular cluster contains 45 galaxies, slightly less likely that it contains 40 or 50 galaxies, and extremely unlikely that it contains 15 or 75 galaxies.
Their analysis also takes into account prior assumptions about galaxy clusters based on our understanding of cluster physics. For example, if there is a strong lensing signal for a particular cluster, it is likely that the cluster is more massive. That should be correlated with higher richness and a stronger Sunyaev-Zeldovich effect. The authors combine their prior knowledge with the observations to create a probability distribution for the true parameters of a given cluster. The process of assigning a probability distribution for the true parameters given an observation and a model is also known as Bayesian statistics.
In Figure 3 of this Astrobite (which is Figure 5 of the paper), the authors show that their methods can recover the unbiased lensing relationship for galaxies with and without Sunyaev-Zeldovich detections and in high- and low-richness clusters. By dividing the lensing signal from observations with the true lensing signal, the authors can quantify the projection effect bias as a function of the observed parameters. They can then correct for the bias in real observations, where we don’t know the true lensing signal distribution but do know the observed richness, lensing signal, and Sunyaev-Zeldovich effect.
Figure 5 from Zhou et al. (2024). Their model correctly recovers the unbiased lensing signal given biased lensing observations.
Figure 5 from Zhou et al. (2024). Their model correctly recovers the unbiased lensing signal given biased lensing observations.
Improving the model and applications to real data
The authors currently use a dark matter simulation, Dark Emulator, to calculate the lensing signal for each cluster. However, clusters are made of more than just dark matter, and the gravitational impact of gas and stars can be important for lensing on small scales. The authors plan to incorporate baryons, or non-dark atomic matter, into their lensing calculations in the future. They also want to account for errors that come from misidentifying the center of a halo when we observe it with optical telescopes.
Once the authors have implemented these effects into their simulations, we might be able to apply their corrections to real data to get a better measurement of the number of clusters at different halo masses. This halo mass distribution will help cosmologists get a better handle on the evolution of large-scale structure since the Big Bang, granting us a more in-depth view of the earliest evolution of the Universe.
Astrobite edited by Alexandra Masegian
Featured image credit: X-ray: NASA/CXC/CfA/M.Markevitch, Optical and lensing map: NASA/STScI, Magellan/U.Arizona/D.Clowe, Lensing map: ESO WFI
Author
I am a third year graduate student at the University of Illinois Urbana-Champaign. I study the connection between supermassive black hole transients and their host galaxies. I am also an avid knitter and reader, and I am passionate about opening up STEM opportunities for people of all backgrounds.
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