Four Readings on Gravitational Lensing and Imaging Dark Matter

Four readings on gravitational lensing and imaging dark matter:

1. Canada-France-Hawaii-Telescope (CFHT) story on "seeing" dark matter using gravitational lensing, by Emma Grout, last accessed 31 October 2017. http://www.cfhtlens.org/public/what-gravitational-lensing

2. "The Dark Side of the Universe," part of Chapter 5 of "Who Cares About Particle Physics?", by Pauline Gagnon (Oxford University Press, Oxford, 2016).

3. "Mapping the Dark Matter with Weak Gravitational Lensing," N. Kaiser and G. Squires, Ap. J 404, 441 (1993). Try to get through page 1 and then digest Figure 5. For the mathematically adept, you can try to work through Section 2.

4. BackRe(action) Blog on Bullet Cluster and Interpretation as DM "smoking gun", by Sabine Hossenfelder, 3 January 2017 http://backreaction.blogspot.ca/

What is Gravitational Lensing?

By Emma Grout http://www.cfhtlens.org/public/what-gravitational-lensing

Cosmology is the branch of astronomy which asks the biggest questions of all – what is the Universe made of? How did it form? How old is it? What will happen to our Universe in the distant future? How and why do the biggest structures in the Universe come about?

Humanity has been asking questions like this for millennia, but it is only in the past century that modern telescopes have been powerful enough to start providing meaningful answers. Our understanding of the Universe today can be summarised in one simple pie chart:

Image: physicsforme.wordpress.com

This chart shows the total mass-energy content of the Universe. Mass-energy equivalence means that we can equate the two; mass is just a measure of the internal energy content of an object. All the 'regular' matter in the Universe – the stuff that makes up galaxies, planets, stars, nebulae, dust, rocks and gas – is known as baryonic matter, and only makes up 4% of the mass-energy content of the Universe. The other two pieces of the pie are dark matter and dark energy, and together they make up almost all of the known Universe. Dark energy and dark matter are so named because cosmologists don't know what they actually are – we can't see them directly and can only infer their existence by the effect they have on the regular matter that we can see. It might seem strange to claim that most of the Universe is invisible and unknown to us, but the evidence for these mysterious entities is compelling. To learn more about dark matter and dark energy, follow the links on the left hand side.

When astronomers refer to lensing, they are talking about an effect called gravitational lensing. Normal lenses such as the ones in a magnifying glass or a pair of spectacles work by bending light rays that pass through them in a process known as refraction, in order to focus the light somewhere (such as in your eye).

Gravitational lensing works in an analogous way and is an effect of Einstein's theory of general relativity – simply put, mass bends light. The gravitational field of a massive object will extend far into space, and cause light rays passing close to that object (and thus through its gravitational field) to be bent and refocused somewhere else. The more massive the object, the stronger its gravitational field and hence the greater the bending of light rays - just like using denser materials to make optical lenses results in a greater amount of refraction.

Image: NASA/ESA Gravitational lensing happens on all scales – the gravitational field of galaxies and clusters of galaxies can lens light, but so can smaller objects such as stars and planets. Even the mass of our own bodies will lens light passing near us a tiny bit, although the effect is too small to ever measure.

So what are the effects of lensing? The kind of lensing that cosmologists are interested in is apparent only on the largest scales – by looking at galaxies and clusters of galaxies. When astronomers take a telescope image of a part of the night sky, we can see many galaxies on that image. However, in between the Earth and those galaxies is a mysterious entity called dark matter. Dark matter is invisible, but it does have mass, making up around 85% of the mass of the Universe. This means that light rays coming towards us from distant galaxies will pass through the gravitational field of dark matter and hence will be bent by the lensing effect.

Dark matter is found wherever 'normal' matter, such as the stuff that makes up galaxies, is found. For example, a large galaxy cluster will contain a very great amount of dark matter, which exists within and around the galaxies that make up that cluster. Light coming from more distant galaxies that passes close to a cluster may be distorted – lensed – by its mass. It is the dark matter in the cluster that does almost all of the lensing as it outweighs regular matter by a factor of six or so. The effects can be very strong and very strange; the images of the distant, lensed galaxies are stretched and pulled into arcs as the light passes close to the foreground cluster. This can be seen in the image below of the famous Abell 2218 cluster. The real galaxies are not this shape – they are usually elliptical or spiral shaped – they just appear this way because of lensing.

Image: NASA/ESA

This strange shape distortion comes from the fact that galaxies are large objects, and the light rays leaving one side of the galaxy (e.g. the left hand side from our point of view) will pass through a different part of space than the light rays leaving the other side (e.g. the right hand side). The light rays will therefore pass through different parts of the dark matter's gravitational field and will be bent in slightly different ways. The net effect of this is a distortion to the shape of the galaxy image, which can in some cases be very severe. Another interesting effect that can occur due to lensing is the formation of multiple images of the same galaxy. This occurs because light rays from a distant galaxy that would otherwise diverge may be focused together by lensing. From the point of view of an observer on the Earth, it looks as if two very similar light rays have travelled along straight lines from different parts of the sky. You can see this in the orange lines in the schematic above - we can see more than one image of the same galaxy in different places. Lensing can also act like a magnifying glass, allowing us to see images of galaxies that would otherwise be too faint to see.

An example of multiple images is shown below in an image from the Hubble Space Telescope. There are 3 images of the same galaxy, and 5 images of a type of galaxy called a quasar. The images are not the same shape or size because each image will have passed through a different region of space on its journey to us, and hence will have been distorted differently. A technique known as spectroscopy is used to determine which images came from the same galaxy.

Image: NASA/ESA, K Sharon (Tel Aviv University), E. Ofek (Caltech)

Weak Lensing

If the lensing effect is strong enough to be seen by the human eye on an astronomical image, like in Abell 2218, we call this strong lensing. Strong lensing only happens when a massive cluster of galaxies lies between us and some other galaxies - it is the further-away galaxies that have their shapes changed by lensing. In this case, it is easy to see and measure the effects of lensing. However, there are not that many clusters in the sky that are so big that they cause such a large lensing effect - most of the time, we don’t see galaxies stretched into arcs or multiply-imaged. So these instances of strong lensing are very useful - and pretty - but rare.

However, the fact that there is some dark matter in between us and every distant galaxy we see means that ALL galaxies are lensed - even if it is only slightly. In fact, most galaxies are lensed such that their shapes are altered by only 1%, an effect we call weak gravitational lensing. We can never see this shape modification with our own eyes on an image because it is too small - but if we have some way to measure this, it could tell us a lot about how dark matter behaves across the whole sky (and not just in massive clusters) as it is a ubiquitous effect. But if we can’t see the effect, how do we measure it? How do we know how strong the lensing effect is on a particular galaxy?

It turns out that we don't need to know how much an individual galaxy image has been lensed – we can instead work out the average lensing effect on a set of galaxies. To do so, cosmologists have to make a couple of assumptions: firstly, that all galaxies are roughly elliptical in overall shape, and secondly that they are orientated randomly on the sky, as shown in the left hand side of the figure below. In the presence of a lensing effect, we would expect that the galaxies in a patch of sky would appear to align themselves together slightly on the sky, as lensing stretches all their images in the same direction. In this way, any deviation from a random distribution of galaxy shape orientations is a direct measure of the lensing signal in that patch of sky. Weak lensing can thus be used to measure the gravitational lensing signal on any part of the sky.

Image: E Grocutt, IfA, Edinburgh

Why is lensing useful?

Gravitational lensing is useful to cosmologists because it is directly sensitive to the amount and distribution of dark matter. This is because the amount of light bending is sensitive only to the strength of the gravitational field it passes through*, which is mostly generated by the mass of the dark matter in the Universe. This means that to measure the amount of lensing on a patch of sky, we don't need to know anything about what kind of galaxies we are observing, how they form and behave or what colour light they emit. This makes gravitational lensing a very clean and reliable cosmological probe as it relies on few assumptions or approximations.

Lensing can therefore help astronomers work out exactly how much dark matter there is in the Universe as a whole (the fraction of the pie chart at the top of the page that dark matter takes up), and also how it is distributed. An example dark matter map constructed from CFHTLenS data is shown below.

Image: CFHTLenS Collaboration

Lensing has also been used to help verify the existence of dark matter itself. The image below is known as the Bullet Cluster, and it has been observed in both optical (visible) light and in X-ray. The majority of the light coming from the Bullet cluster comes from hot X-ray emitting gas, and has been overlaid onto the visible-light image in pink. Superimposed in blue is the location of the dark matter in the cluster, determined from measuring the lensing signal from the visible-light images of the galaxies. The offset between the pink X-ray gas and the blue dark matter regions tells us that what we are observing is actually the aftermath of a collision between two galaxy clusters. During the collision, the baryonic X-ray gas particles (the 'normal' matter) will interact with each other through both gravity and electrostatic forces, slowing and shocking one another. The dark matter particles, however, only interact through gravity and can pass through each other unimpeded by electrostatic interactions. This means that the X-ray gas lags behind the dark matter as the two clusters escape the collision, causing the observed offset - most of the visible matter is now in the centre of the image, but lensing tells us that most of the mass lies further out.

Image: Composite Credit: X-ray: NASA/CXC/CfA/ M.Markevitch et al.; Lensing Map: NASA/STScI; ESO WFI; Magellan/U.Arizona/ D.Clowe et al. Optical: NASA/STScI; Magellan/U.Arizona/D.Clowe et al.

Some scientists believe that since the only observed effects of dark matter are gravitational, then perhaps our understanding of gravity is incomplete. It is possible that we are not observing a new type of matter, but that the laws of gravity as we understand them are wrong. As a result, many different modified gravity theories have arisen to explain the dark matter phenomenon. The Bullet cluster provides strong evidence for the existence of dark matter, as this offset between the light and mass is exactly what scientists expect to see if dark matter is real, and it is hard to explain under many theories of modified gravity.

If we know something about the distances to the galaxies we look at with our telescopes, lensing can also tell us about the nature of dark energy because the amount of dark energy affects how galaxies and clusters form and develop. Measuring their distribution with distance through gravitational lensing can help us constrain the amount of dark energy in the Universe to a higher degree of precision. The light from distant galaxies began travelling towards us many millions (or even billions) of years ago, providing a window into the early Universe. This means that it is also possible to work out if the amount of dark energy changes over time by observing galaxy structures at different distances from us. Thus, gravitational lensing is a clean probe of the Universe and has much to tell us about its two most mysterious components - dark matter and dark energy.

*In fact, this is one way in which gravitational lensing differs from optical lensing, as gravitational lensing is independent of the wavelength (colour) of the light. All light rays are bent the same amount by gravity. Optical lenses cause light of different colours to bend by varying amounts in a process called diffraction, resulting in the splitting of light into rainbows. There is no such analogous effect with gravitational lensing.

Author: Emma Grocutt

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BackRe(action), a blog by Sabine Hossenfelder

The Bullet Cluster as Evidence against Dark Matter

Once upon a time, at the far end of the universe, two galaxy clusters collided. Their head-on encounter tore apart the galaxies and left behind two reconfigured heaps of stars and gas, separating again and moving apart from each other, destiny unknown.

Four billion years later, a curious group of water-based humanoid life-forms tries to make sense of the galaxies’ collision. They point their telescope at the clusters’ relics and admire its odd shape. They call it the “Bullet Cluster.”

In the below image of the Bullet Cluster you see three types of data overlaid. First, there are the stars and galaxies in the optical regime. (Can you spot the two foreground objects?) Then there are the regions colored red which show the distribution of hot gas, inferred from X-ray measurements. And the blue- colored regions show the space-time curvature, inferred from gravitational lensing which deforms the shape of galaxies behind the cluster.

The Bullet Cluster. [Img Src: APOD. Credits: NASA]

The Bullet Cluster comes to play an important role in the humanoids’ understanding of the universe. Already a generation earlier, they had noticed that their explanation for the gravitational pull of matter did not match observations. The outer stars of many galaxies, they saw, moved faster than expected, meaning that the gravitational pull was stronger than what their theories could account for. Galaxies which combined in clusters, too, were moving too fast, indicating more pull than expected. The humanoids concluded that their theory, according to which gravity was due to space-time curvature, had to be modified.

Some of them, however, argued it wasn’t gravity they had gotten wrong. They thought there was instead an additional type of unseen, “dark matter,” that was interacting so weakly it wouldn’t have any consequences besides the additional gravitational pull. They even tried to catch the elusive particles, but without success. Experiment after experiment reported null results. Decades passed. And yet, they claimed, the dark matter particles might just be even more weakly interacting. They built larger experiments to catch them.

Dark matter was a convenient invention. It could be distributed in just the right amounts wherever necessary and that way the data of every galaxy and galaxy cluster could be custom-fit. But while dark matter worked well to fit the data, it failed to explain how regular the modification of the gravitational pull seemed to be. On the other hand, a modification of gravity was difficult to work with, especially for handling the dynamics of the early universe, which was much easier to explain with particle dark matter.

To move on, the curious scientists had to tell apart their two hypotheses: Modified gravity or particle dark matter? They needed an observation able to rule out one of these ideas, a smoking gun signal – the Bullet Cluster.

The theory of particle dark matter had become known as the “concordance model” (also: ΛCDM). It heavily relied on computer simulations which were optimized so as to match the observed structures in the universe. From these simulations, the scientists could tell the frequency by which galaxy clusters should collide and the typical relative speed at which that should happen.

From the X-ray observations, the scientists inferred that the collision speed of the galaxies in the Bullet Cluster must have taken place at approximately 3000 km/s. But such high collision speeds almost never occurred in the computer simulations based on particle dark matter. The scientists estimated the probability for a Bullet-Cluster-like collision to be about one in ten billion, and concluded: that we see such a collision is incompatible with the concordance model. And that’s how the Bullet Cluster became strong evidence in favor of modified gravity.

However, a few years later some inventive humanoids had optimized the dark-matter based computer simulations and arrived at a more optimistic estimate of a probability of 4.6×10-4 for seeing something like the Bullet-Cluster. Briefly later they revised the probability again to 6.4×10−6.

Either way, the Bullet Cluster remained a stunningly unlikely event to happen in the theory of particle dark matter. It was, in contrast, easy to accommodate in theories of modified gravity, in which collisions with high relative velocity occur much more frequently.

It might sound like a story from a parallel universe – but it’s true. The Bullet Cluster isn’t the incontrovertible evidence for particle dark matter that you have been told it is. It’s possible to explain the Bullet Cluster with models of modified gravity. And it’s difficult to explain it with particle dark matter.

How come we so rarely read about the difficulties the Bullet Cluster poses for particle dark matter? It’s because the pop sci media doesn’t like anything better than a simple explanation that comes with an image that has “scientific consensus” written all over it. Isn’t it obvious the visible stuff is separated from the center of the gravitational pull?

But modifying gravity works by introducing additional fields that are coupled to gravity. There’s no reason that, in a dynamical system, these fields have to be focused at the same place where the normal matter is. Indeed, one would expect that modified gravity too should have a path dependence that leads to such a delocalization as is observed in this, and other, cluster collisions. And never mind that when they pointed at the image of the Bullet Cluster nobody told you how rarely such an event occurs in models with particle dark matter.

No, the real challenge for modified gravity isn’t the Bullet Cluster. The real challenge is to get the early universe right, to explain the particle abundances and the temperature fluctuations in the cosmic microwave background. The Bullet Cluster is merely a red-blue herring that circulates on social media as a shut-up argument. It’s a simple explanation. But simple explanations are almost always wrong.

Posted by Sabine Hossenfelder at 11:47 PM Labels: Astrophysics, Papers, Physics Email ThisBlogThis!Share to TwitterShare to FacebookShare to Pinterest

104 comments:

TheBigHenry said...

There, again, is that “scientific consensus” ruse, which conflates "consensus of opinion" with science.

12:35 AM, January 04, 2017

Cyberax said...

I don't get it. How does modified gravity explain displacement of visible and gravitating matter in the case of Bullet Cluster?

The improbability of conditions that are needed to set up the collision is another question.

1:02 AM, January 04, 2017

Q said...

The real challenge is to get the early universe right, to explain the particle abundances and the temperature fluctuations in the cosmic microwave background.

I'd love to hear more about how different the early universe could be under MOND compared to LCDM. Is it possible we have something as basic as the age of the universe wrong?

1:20 AM, January 04, 2017

Jeff said...

Perhaps Diogenes would have been surprised to find that you weren't a man. But he would have appreciated you anyway. As do I.

1:29 AM, January 04, 2017

Sabine Hossenfelder said... Cyberax,

The idea of modifying gravity is that there's no particle dark matter. I mean, strictly speaking it could be both, but if you have particle dark matter anyway, you don't need modified gravity, so it's kinda pointless to combine them. So, you modify gravity instead of adding particle dark matter.

Then the question is what's the difference? Well, if you add particle dark matter you add quantum fields to the standard model of particle physics. If you modify gravity otoh, you add classical fields to general relativity. The main difference is (besides the one being quantized and the other not) the way that the additional fields couple. For what is relevant here, however, is only that there is a priori no reason for the focus of the additional fields in modified gravity to be located where the 'normal' dark matter is.

I say 'a priori' because to figure out where it is you have to solve the dynamical equations. Which I haven't done. On that matter I can merely tell you that people working on modified gravity claim they can fit the Bullet Cluster without too many problems. I haven't looked into this too deeply and can't say much about this. Hence, all I am saying here is that at least for what the theoretical structure is concerned the focus of gravity can be offset from the normal matter distribution in modified gravity too. Best,

3:49 AM, January 04, 2017

Phillip Helbig said...

"I'd love to hear more about how different the early universe could be under MOND compared to LCDM. Is it possible we have something as basic as the age of the universe wrong?"

Doing early-universe calculations with MOND is a hard problem. However, even most MOND supporters concede that dark matter (not necessarily particle dark matter; it could be primordial black holes, though many mass ranges have been ruled out) works better in this context. (The asymmetry comes about because most "conventional" astrophysicists don't admit that MOND works better in the "MOND regime".)

Age of the universe wrong? No. First, there is non-cosmological evidence for the same age. Second, it doesn't look like MOND would predict something different here.

4:24 AM, January 04, 2017

Guy Roberts said...

Can you explain to a layman, given that projects like Gaia are discovering lots of unknown stars and not-shiny objects, isn't it possible that we have under estimated the amount of ordinary matter in the galaxy ? Perhaps we have not looked carefully enough yet.

4:39 AM, January 04, 2017

Able Lawrence said...

How would Verlindt's Entropic "Emergent" be expected to perform under the circumstances. He himself does not provide a solution for dynamic situations like bullet cluster and early universe. At least the idea provides theoretical motivation and derives both Einsteins field equations and MOND from first principles.

4:50 AM, January 04, 2017

Maurice said...

Who are the "people working on MG" you refer to? The MONDians can fit the Bullet Cluster because to fit clusters they always needed to invoke dark matter anyway.

5:14 AM, January 04, 2017

akidbelle said...

Hi Sabine,

very instructive post to me.

Out of the subject I'd really like to know the scientific consensus about pop sci media.

6:26 AM, January 04, 2017

Fernando Almatax said...

Why additional fields in modified gravity? Why not just a modified dynamics for the same fields?

6:32 AM, January 04, 2017

Sabine Hossenfelder said...

Fernando,

Because it doesn't work, or at least I don't know a good example.

8:25 AM, January 04, 2017

Mandar said...

Can modified gravity explain the gravitational lensing ? If not then it would still point towards the dark matter, however unlikely the collision might have been. Is it not ?

Sean Carroll indulged in a nice FB hangout recently where he argued in favour of dark matter against rhe modified gravity. He presented a wide variety of arguments, for instance the one involving CMB oscillations. The video is available on his blog.

8:25 AM, January 04, 2017

Sabine Hossenfelder said...

Maurice,

I really don't understand why people keep talking about MOND. MOND is a non-relativistic limit. You don't expect it to work in general. It's an approximation. Complaining that MOND doesn't work is beating a dead horse. If you want to know if modified gravity is a promising explanation, you really have to look at a relativistic theory. As to who says what, you find the relevant references here, see section 8.3. Best,

8:32 AM, January 04, 2017

Sabine Hossenfelder said...

Guy,

That's an old idea which isn't entirely dead but it's extremely disfavored. The reason is that if there were sufficiently many such objects to make up all the dark matter (we know how much it has to be in total) these would cause frequent gravitational lensing which hasn't been seen. The situation is somewhat murky however for objects that are significantly lighter than typical solar masses because in this case the lensing wouldn't be strong enough. This stuff is called 'macro dark matter' and I wrote about this here. The problem in this case is to understand how they would form and what they would be made of. Best,

8:36 AM, January 04, 2017

Sabine Hossenfelder said...

Able,

Last time I looked Verlinde didn't derive the field equations. Besides this, I've run into some trouble trying to reproduce what's in the paper and I'm hence somewhat confused about what he does or doesn't show. Sorry for being vague, I presently really don't know a good answer. Best,

8:39 AM, January 04, 2017

Sabine Hossenfelder said...

Mandar,

Yes, the 2nd acoustic peak is the somewhat more enlightened argument. Yes, for all I know modified gravity can reproduce the lensing. As I mentioned above, for a quick summary and literature references, see here, section 8.3.

8:41 AM, January 04, 2017

John Fredsted said...

My personal inclination towards the concepts of "dark matter" and "dark energy" is that they are "parametrizations of our ignorance", rather than something refering to some genuine physical substances. But without having these fields as areas of expertise, I might of course be completely misguided, as well as appearing somewhat arrogant. In both cases, I apologize. But as long as all experiments, ever more sensitive, searching for dark matter particles turn up empty handed, I have at least some empirical backup for this point of view.

I wonder if we are missing something in connection with angular momentum, including spin: Einsteins theory of general relativity, GR, is based on (pseudo-)Riemannian spacetime where curvature is generally present, but torsion is non-present. The teleparallel equivalent (?) of GR is based on Weitzenböck spacetime where torsion is generally present, but curvature is non-present. Although locally equivalent, as they have the same Euler-Lagrange equations, as far as I recall, these theories are, I guess, topologically non-equivalent; or am I wrong?!

Perhaps more importantly, though, The Einstein-Cartan theory, which is still a viable theory, uses spacetimes in which both curvature and torsion are present, curvature coupling to matter, and torsion coupling to spin. But, as far as I recall from some articles by Trautman I read some years ago, the torsion field arising from spin does not propagate because it is algebraically, i.e., non- differentially, related to the spin density.

Coming now to my point: It has always puzzled me why of all the properties of particles - their mass, electric charge, isospin, color, and spin - it seems that only spin is not the source of some propagating field. This in connection with the rotation curves of galaxies, the origin of the "dark matter" concept, being readily related to angular momentum, makes me wonder if we are missing some coupling of angular momentum to some (propagating part of a) gravitational field. What it should be, though, I am completely ignorant.

8:45 AM, January 04, 2017

David Schroeder said... This is an eye-opening post, and I really enjoyed the playful way Bee introduces this controversial subject matter to us. If Bee hadn't researched the facts about the exceedingly low probability of explaining this cluster's dynamics from Lambda-CDM, I, (and many others), likely would never have become aware of this important detail. Despite battling a bad case of the flu, fogging the mind, this provides much fodder to chew on, and think about, in the coming days.

9:01 AM, January 04, 2017

Phillip Helbig said...

"Can you explain to a layman, given that projects like Gaia are discovering lots of unknown stars and not-shiny objects, isn't it possible that we have under estimated the amount of ordinary matter in the galaxy ? Perhaps we have not looked carefully enough yet."

There are various ways to estimate the mass of the Milky Way. Of course we can't just add up the currently visible matter and expect to account for all of the mass. The fact that this doesn't work is the motivation for dark matter in the galaxy. For various reasons, it is extremely unlikely that new objects found by Gaia would make up a significant amount of this dark matter.

9:12 AM, January 04, 2017

Phillip Helbig said...

But as long as all experiments, ever more sensitive, searching for dark matter particles turn up empty handed, I have at least some empirical backup for this point of view.

Would you have argued the same about the neutrino between postulation and detection? Or the Higgs?

Absence of evidence is not evidence of absence, unless you have searched the entire parameter space and found nothing. Of course we find something in the last place we look, because after that we stop looking.

There are several reasons to believe that the cosmological constant is "real": 1. It was introduced a hundred years ago, before it was known that the universe is expanding, much less accelerating; it's not a fudge factor nor even an additional parameter invented to fit the data. 2. The value derived from astronomical observations, together with other astronomical observations which suggest a flat universe, result in a value for the density parameter which is confirmed by several other independent observations. 3. It would be an incredible case of fine tuning if something which is fundamentally different than the cosmological constant just happened to behave exactly like it. If it walks like a duck, and quacks like a duck, then it is a duck. 4. There is not a shred of evidence that "dark energy" is anything but the cosmological constant, i.e. the equation of state is w = -1 and this and the value are both constant in time. There is a reason that it is known as the "concordance" model; any alternative explanation for one phenomenon (say, accelerated expansion) has to fit all other observations (say, cluster abundance as a function of redshift, CMB power spectrum, age of the universe) with the samey value.