Simulations of galaxy formation based on the Lambda-Cold Dark Matter (ΛCDM) cosmological model predict that a large galaxy such as the Milky Way should have many dwarf satellite galaxies, perhaps thousands. However, only about 20 or 30 have been identified. Where are the rest? Are they really there? That question alludes to the “dwarf galaxy problem“.

Astrophysicists suspect that most satellite galaxies are much smaller than the galaxies they orbit. And, in addition, such dwarf galaxies may consist mainly of dark matter, with few visible stars, so they should be very difficult to detect, even if there are a lot of them. Since dwarf galaxies consisting mainly of dark matter are so difficult to find by visible light, there could be enough of them to reconcile the large number of dwarf galaxies that simulations predict to exist with the small number actually observed.

Surprisingly, recent research (Vegetti, et al) has been able to detect a very distant dwarf satellite galaxy by gravitational lensing effects – and from that it is possible to infer that a large number should exist.

The image shows an Einstein ring, which consists of a foreground galaxy (JVAS B1938+666) in the middle, and the distorted image of a more distant galaxy making up most of the ring. A detailed mathematical analysis of the image has confirmed that a minor irregularity in the ring is caused by the presence of a dwarf satellite galaxy of the lens galaxy.

For such a configuration to appear, the foreground galaxy must be almost exactly in front of the one behind it. Otherwise there will be only two (or, rarely, more) images of the more distant galaxy. In addition, the shape of the foreground galaxy has a small effect on the image. And, importantly, any satellite galaxies with sufficient mass (dark or otherwise) that are close enough to the lens galaxy may also affect the image.

In this case, the analysis yields a fairly precise estimate of the mass of the satellite galaxy: (1.9±0.1)×10^{8} M_{⊙}. That’s less than 1% of the mass of the lens galaxy, which is estimated at 2.46×10^{10} M_{⊙}, and less than 0.02% of the Milky Way’s mass. So the dwarf is actually pretty small. Since spectroscopy shows the redshift of the lens is z = 0.881, corresponding to a comoving distance of almost 10 billion light-years, this detection is rather impressive.

Still, that is only one dwarf satellite. So how does this help with the dwarf galaxy problem? The key detail that’s involved is the “mass function”, which describes the distribution of concentrations of cold dark matter close to the primary galaxy. The mass function counts the number of objects having a mass (including dark matter) in a certain small range within a fixed cubic unit of volume.

If N is the number density given by the mass function for a fixed range of masses around M, then simulations based on ΛCDM predict that N as a function of M is a power law of the form N = A×M^{1-α} + B, for constants A, B, and α. B = 0 can be assumed if the formula applies for large M. A is a scale factor that depends on mass units and the total mass of the system. In general, α > 1, and there is a simple proportionality for the derivative of this function with respect to M: (d/dM)N ∝ M^{-α}. (d/dM)N is just the slope of the mass function. (If α were < 1 the law would predict increasing numbers of objects as M increases, which doesn’t happen.) Since α > 1, the slope of the mass function is negative, meaning that as M increases there are fewer objects of a given mass M in a unit volume. (But the slope approaches 0 for large M.)

In order to determine the total number of satellite galaxies, it’s necessary to integrate over the whole mass range. Although many simulations based on ΛCDM indicate values of α close to 1.9 they are less consistent regarding the integrated totals, and in particular the percentage of total mass of the system represented by objects having masses in a given range.

Little is known of the bulk properties of dark matter, but in general it will not concentrate in objects too small to be held together gravitationally, nor (except for the central mass) in objects that are a large fraction of the central mass. So it is convenient to consider only objects that have masses between two values M_{min} and M_{max}.

Depending on how observations are done, there will be further limits on the masses of objects that can actually be detected with statistical significance (usually 95% confidence). If these are denoted by M_{low} and M_{high}, then M_{min} ≤ M_{low} < M_{high} ≤ M_{max}. Finally, the “mass fraction” f is defined to be the total mass of objects in the range M_{low} to M_{high} as a percentage of the total mass of the system. f can be derived from the mass function. Typically, f is only a few percent, with most of the mass being in the central object.

For example, the Milky Way’s brightest satellites, by far, are the Large and Small Magellanic Clouds (LMC and SMC), which are so obvious they’re visible to the naked eye. They have masses of about 10^{10} M_{⊙} and 7×10^{9} M_{⊙}, respectively. The masses are inferred from the velocities of stars within the galaxies, so they include dark matter. Many of the known satellites are so much less luminous than the LMC and SMC that they have only recently been discovered.

The mass of the Milky Way itself is in the range of 1 to 1.5×10^{12} M_{⊙}, so the two most prominent of its satellite galaxies together have only about 1.5% of its mass. Most of the other known satellites are a lot less massive than LMC and SMC. It appears that the situation is similar for other large galaxies.

Now comes the interesting part. If we knew the shape of the mass function in advance, we could predict the probabilities of observing (by a particular technique) a certain number of objects within a range of masses. If the technique uses gravitational lensing involving Einstein rings that would mean we could predict the number of objects of different masses that would be detected by being located close enough to the lens galaxy to affect the details of the ring. We could do this simultaneously for multiple rings.

But more interesting than that, if a certain number of objects having particular masses have been detected in Einstein rings, it’s possible using Bayesian inference to turn the analysis around and predict the probabilities that parameters of the mass function lie in specific ranges. In particular, it’s possible to predict the joint ranges (regions of parameter space) that can be taken by several parameters, with some desired level of probability. An earlier paper (Vegetti & Koopmans), whose lead author is the same as that of the present study of JVAS B1938+666, explained in detail how this analysis works.

For a simple example of Bayesian inference, consider statistics of heights of adult males in a particular country. Assuming a survey has been done of a large number of people, a probability distribution of heights could be produced. This would generally be a “normal distribution” with a certain mean and standard deviation (σ). From that one could predict that the probability an individual selected at random from the population would have a height that is within 2σ of the mean is about .95. If, for example, the mean is 178 cm and σ is 15 cm, then the height of the selected individual is between 163 and 193 cm with 95% confidence. On the other hand suppose the same standard deviation is assumed but the mean isn’t known. If a few individuals are chosen and measured, a prediction could be made about the range of values of the mean, with a certain confidence level. Clearly, the more individuals that can be sampled, the higher confidence level will be for predictions about the “true” mean.

All this works the same way for predicting possible values of α and f in the case of mass functions associated with Einstein ring galaxies. But with a sample of only 1, it’s not possible to place constraints that are very narrow on the possible values. Ideally, a sample of 100s or even 1000s could be obtained, and that would put very tight constraints on the values. Eventually, given much larger telescopes, such as the European Extremely Large Telescope, and enough time data will be available, but that might be 20 years in the future.

Fortunately, one other Einstein ring system has already been studied in this way – by the same principal author (Vegetti, et al). The lens galaxy is SDSS J0946+1006, at redshift z=0.222 (comoving distance 2.9 billion light-years). The mass of the lens galaxy is estimated at 2.45×10^{10} M_{⊙} – almost the same as JVAS B1938+666. However, the one detected substructure (dwarf galaxy) is estimated to be (3.51±0.15)×10^{9} M_{⊙}. That’s 18 times the mass of the substructure of JVAS B1938+666, and half the mass of the Milky Way’s SMC. This high value for the substructure, despite the relatively low mass of the lens galaxy, makes a large difference in estimates of the mass function parameters.

In applying Bayesian estimation, some probability distribution has to be assumed for the slope α. One possible assumption is a normal distribution with mean of 1.9, as suggested by simulations. If predictions are made using this distribution, the predictions will amount to a test of the ΛCDM model. If the predictions eventually prove incorrect, the model must be rejected – but this won’t be known for quite awhile. An alternative assumption is that α is uniformly distributed in the range of 1.0 to 3.0. This assumption would be used to predict the “real” value of α when models other than ΛCDM are admissible.

When JVAS B1938+666 is considered by itself the results are as follows. The mass fraction f is considered for the range of substructure size from 4.0×10^{6} M_{⊙} to 4.0×10^{9} M_{⊙} within the distance from the lens galaxy required to affect the ring. Assuming a normal distribution of α with mean 1.9, the range of f is 0.6% to 3.0%, at the 68% confidence level. With a uniformly distributed α between 1.0 and 3.0, f is 1.5% to 7.5% at the same confidence level.

By comparison, simulations yield f from 0.0% to 0.4%, in the same mass range and distance from the lens galaxy. Thus the results predicted from observation are actually somewhat higher than given by simulations.

For SDSS J0946+1006 the comparable predictions for f were 0.9% to 4.2% for normally distributed α and 1.06% to 5.82% for uniformly distributed α. If the two observations are combined, the range for the average value of f is 1.5% to 6.9%. The slope of the mass function is predicted to have a range from 0.7 to 1.7, somewhat smaller than given by simulations. So this indicates the presence of slightly larger substructures than simulations predict.

These estimates should be taken as very provisional, since they are based on only 2 actually detected substructures that are probably dwarf satellite galaxies, and a theoretically reasonable but scarcely tested method for estimating mass functions. However, since the method does not require dwarf galaxy candidates to have any particular abundance of stars, it could provide solid evidence for large numbers of dark dwarf galaxies once sufficient numbers of Einstein ring systems are identified and studied in future galaxy surveys.

It may seem surprising that a great deal could be learned about a difficult issue from just a couple of studies. However, there’s a possible reason that this research is so informative. In order for a small satellite galaxy to have a detectable effect on an Einstein ring, the satellite has to be in a rather small region relative to both the lens and the background source. Namely, it must be in a region that projects to a narrow annulus on the plane of the lens perpendicular to our line of sight. This area occupies only a rather small fraction of the total around the lens that may contain satellites. Consequently, in order to actually observe a satellite in that area, there should probably be a lot of them. Especially if this happens with a large percentage of Einstein rings that are eventually studied. It’s like trying to count flies in a room by observing only those that land on random but very small spots on the wall.

Vegetti, S., Lagattuta, D., McKean, J., Auger, M., Fassnacht, C., & Koopmans, L. (2012). Gravitational detection of a low-mass dark satellite galaxy at cosmological distance Nature, 481 (7381), 341-343 DOI: 10.1038/nature10669 |

**Further reading:**

Astronomers find a dark matter galaxy far, far away

Most Distant Dwarf Galaxy Detected

Astronomers find dark matter galaxy far, far away

Faint ‘satellite galaxy’ discovered

Cataract in a gravitational lens may be a tiny galaxy

How to spot a dark-matter galaxy

Einstein ring reveals dwarf dark-matter galaxy

Galaxy formation: Distant dwarfs

Gravitational detection of a low-mass dark satellite galaxy at cosmological distance

Detection of a dark substructure through gravitational imaging

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