Our standard model of the Universe requires only 6 numbers to describe everything, and cosmologists have been measuring these numbers with increasing accuracy for decades. Two of the six correspond to how much stuff there is in the Universe: the “baryon” (normal stuff) density, and the dark matter density.
Month: May 2020
WIMP #31: How many zeros?
The quantum mechanical calculation of the “vacuum energy”, differs from the observed cosmological constant (or dark energy), by about 10 to the power of 120. I.e. 10 followed by about 120 “0”s. I will leave you all to type that out for yourself and see what it looks like…
WIMP #30: Bottom up!
In the usual model of the Universe, because the dark matter is “cold” (it moves relatively slowly), the structure in the Universe forms “from the bottom, up”. I.e. the smaller structures form first, followed by the larger structures.
Model independence: Cinderella and the search for modified gravity
Have you ever thought about how inefficient the prince in Cinderella was?
Consider his plight: He has a shoe and he needs to find the correct owner in a country of millions of people. So, he tries the shoe on each person in turn until he finds the person who fits the shoe. This could take him a long time!
So, what could he do instead? He could measure the length of the shoe and send out a royal proclamation asking that people whose feet are roughly the correct size come to the palace to try the shoe on.
With any luck, he is now trying the shoe on a much smaller number of people, so he is much more likely to find his Cinderella during his lifetime.
We can be in a similar situation sometimes in physics: we have lots of possible theories we can come up with, and testing them all against the data one-by-one can be very laborious and inefficient. This is one of the reasons for using something called “model-independent approaches”.
This is a topic that is close to my heart, and which links two recent papers (HERE) and (HERE), so I thought this was a golden opportunity to say a little about it.
(Note: I am an author on both of these papers. I hope the reader can forgive my self-indulgence in this blog post. The first of these papers also has other results concerning a currently topical problem in cosmology; I want to focus here on the theme that links these two papers, so I will discuss this extra result another time).
Model independent approaches: What are they?
Typically these are ways to try to write down a more general description of a system, rather than just how the system looks according to a particular theory. This general description allows the system to behave in extra or new ways (perhaps ways that our currently favoured theory doesn’t let it behave in).
These approaches can take many forms. Sometimes they are about removing an assumption that is used to derive the equations in our current theory, necessitating a set of more complex equations.
Sometimes these approaches take a value (e.g. the strength of gravity) that never changes in our current theory, and allow it to change with time or location.
For the scientists developing them, these approaches often amount to writing down a set of “master” equations, that are more general than the equations in any particular theory. By making different assumptions or choices, one can then find the (simpler) equations for specific models within these master equations.
We can then go further and compare the master equations to the data, so we can see which parts are actually needed to describe the data we have.
If this is done well, the clever bit is that the master equations might even contain theories within them that we haven’t even thought of yet. So by testing the master equations, we can not just rule out a couple of alternative theories, but rule out many theories at once, including ones that no-one has written down yet!
Why do them? Why not do them?
These approaches have some clear downsides. They are not always easy to create, and if you find some unexpected behaviour using one of these approaches, then it isn’t always clear which theory or classes of theory might give that behaviour.
I like them because I think they are more efficient and more definitive. The efficiency is related to the Cinderella example: we can rule out many theories in one go, with a lot less effort than ruling them all out individually. The definitiveness comes from the fact that if you see nothing, this is a much stronger result than just ruling out a particular theory, because any theory that would give this kind of behaviour is now ruled out. Even ones we haven’t thought of yet.
Similarly, these approaches mean that we don’t have to come up with a good/clever alternative to our existing theory in order to test the existing theory. This approach is often called a “null test”.
A null test can be described in several ways, and is basically a test of behaviour that isn’t allowed by our current theory, such as comparing two things that our current theory says should be the same. I.e. it is a test that should return “null” (nothing interesting) if our current theory is correct.
For example, the idea that a potato and a feather are equally accelerated by the Earth’s gravity (in the absence of air resistance) is a common theme to our theories of gravity. Testing this idea would be a null test: we don’t expect to see a difference, but we might, and if we don’t then it allows us to be more sure of the assumptions and ideas underlying our theories.
First paper: Generalised dark matter
In the standard picture of the Universe, the dark matter is “cold dark matter”. Translated, this basically means “the simplest, most boring kind of matter that we could add to the Universe”.
In particular, dark matter in the standard picture moves relatively slowly, just like the “normal” matter that makes up the Earth (and each of us). It also has zero pressure, just like a bunch of rocks: these don’t push on the walls of their container the way something like air does.
This “simplest possible” picture of dark matter is appropriate if dark matter is “WIMP” particles (see HERE), which have long been thought to be the most likely kind of dark matter. However, despite extensive searches, these have still not been found.
The first of these two new papers uses a model independent approach for dark matter, which allows the dark matter to have extra properties. In particular it is allowed to move a bit faster and have a small pressure. These properties are also allowed to change with time.
In total, the dark matter was allowed to have extra features in 26 different ways (in contrast only 6 numbers are needed to describe the entirety of the rest of the Universe!), but the data showed no evidence for any of these extra features to be turned on. See figure 2 for an example.
This is a powerful “null test” of our current model of dark matter: the equations allowed for 26 ways in which the data could choose something different to our simple picture of dark matter, and the data rejected all of them.
Second paper: Modified gravity
Since we only have gravitational evidence for dark matter and dark energy (i.e. we have only inferred their existence from how other things move, we haven’t found either of them in a lab), it is natural to consider whether instead of these things existing, we have just got the laws of gravity wrong.
Quite a few alternative gravity theories have been proposed, putting us into a similar position to the prince in the Cinderella story: testing them all one-by-one is potentially very time consuming. Plus, it is a good idea to get a handle on what the data says about possible differences from our standard picture in general. Luckily, model-independent approaches for alternative gravity theories have a venerable history.
There is a famous approach developed during the 20th century for testing alternative gravity theories in the Solar System, called “PPN: Parameterised Post Newtonian.” This constructs a set of quantities, and ways to measure them, that describe the different ways that gravity could be different from Einstein’s gravity in our Solar System.
Over the last 15 years, cosmologists have attempted to do the same thing for testing gravity outside our Solar System. Several successful and well developed frameworks exist, and these have been used to see how much deviation from Einstein’s gravity is currently allowed by the data.
Unfortunately, gravitational calculations for the whole Universe are hard, so we do them in different ways on large scales and small scales (see HERE). The model independent frameworks are designed using the large scale approach, so they only apply on the biggest scales in the Universe. This means that we won’t be able to use all of the data from planned surveys, because these surveys mix together large, intermediate and small scales.
This new paper points out that gravity in the large and small scale limits is similar in certain ways, and they overlap (i.e. on intermediate scales we can use the equations from either limit). This means that existing large scale frameworks can be extended relatively easily to work on all scales.
Importantly, this means that we will be able to use all of the data from future surveys (such as the Euclid satellite, HERE) to test for alternative gravity theories in a model independent way.
This could be the most definitive test yet for Einstein’s theory and, if it passes this test, we will have comprehensively tested it in ways that Einstein could never have imagined when he first wrote his theory down.
Figure credits
1) Stock clipart image from HERE.
2) Taken from the first paper, HERE.
WIMP #29: Are we special?
One of the building blocks of most modern theories of the Universe is the “Copernican principle”, which says that there is nothing special about the Earth’s or its location in the Universe.
Some theories try to remove dark energy by relaxing this requirement a little, and allowing the Earth to be in a “different from average” environment, such that dark energy phenomena arise from our misinterpretation of the observations.
Dark Matters 