# Significant Figures and the Age of the Universe

(Note: This post originally contained a remarkably stupid error in an example. For some idiotic reason, I calculated as if a liter was a cubic meter. Which, duh, it isn’t. so I was off by a factor of 1000. Pathetic, I know. Thanks to the multiple readers who pointed it out!)

The other day, I got a question via email that involves significant figures. Sigfigs are really important in things that apply math to real-world measurements. But they’re poorly understood at best by most people. I’ve written about them before, but not in a while, and this question does have a somewhat different spin on it.

Here’s the email that I got:

Do you have strong credentials in math and/or science? I am looking for someone to give an expert opinion on what seems like a simple question that requires only a short answer.

Could the matter of significant figures be relevant to an estimate changing from 20 to less than 15? What if it were 20 billion and 13.7 billion?

If the context matters, in the 80s the age of the universe was given as probably 20 billion years, maybe more. After a number of changes it is now considered to be 13.7 billion years. I believe the change was due to distinct new discoveries, but I’ve been told it was simply a matter of increasing accuracy and I need to learn about significant figures. From what I know (or think I know?) of significant figures, they don’t really come into play in this case.

The subject of significant digits is near and dear to my heart. My father was a physicist who worked as an electrical engineer producing power circuitry for military and satellite applications. I’ve talked about him before: most of the math and science that I learned before college, I learned from him. One of his pet peeves was people screwing around with numbers in ways that made no sense. One of the most common ones of that involves significant digits. He used to get really angry at people who did things with calculators, and just read off all of the digits.

He used to get really upset when people did things like, say, measure a plate with a 6 inch diameter, and say that it had an are] of 28.27433375 square inches. That’s ridiculous! If you measured a plate’s diameter to within 1/16th of an inch, you can’t use that measurement to compute its area down to less than one billionth of a square inch!

Before we really look at how to answer the question that set this off, let’s start with a quick review of what significant figures are and why they matter.

When we’re doing science, a lot of what we’re doing involves working with measurements. Whether it’s cosmologists trying to measure the age of the universe, chemists trying to measure the energy produced by a reaction, or engineers trying to measure the strength of a metal rod, science involves measurements.

Measurements are limited by the accuracy of the way we take the measurement. In the real world, there’s no such thing as a perfect measurement: all measurements are approximations. Whatever method we chose for taking a measurement of something, the measurement is accurate only to within some margin.

If I measure a plate with a ruler, I’m limited by factors like how well I can align the ruler with the edge of the plate, by what units are marked on the ruler, and by how precisely the units are marked on the ruler.

Once I’ve taken a measurement and I want to use it for a calculation, the accuracy of anything I calculate is limited by the accuracy of the measurements: the accuracy of our measurements necessarily limits the accuracy of anything we can compute from those measurements.

For a trivial example: if I want to know the total mass of the water in a tank, I can start by saying that the mass of a liter of water is one kilogram. To figure out the mass of the total volume of water in the tank, I need to know its volume. Assuming that the tank edges are all perfect right angles, and that it’s uniform depth, I can measure the depth of the water, and the length and breadth of the tank, and use those to compute the volume.

Let’s say that the tank is 512 centimeters long, and 203 centimeters wide. I measure the depth – but that’s difficult, because the water moves. I come up with it being roughly 1 meter deep – so 100 centimeters.

The volume of the tank can be computed from those figures: 5.12 times 2.03 times 1.00, or 10,393.6 liters.

Can I really conclude that the volume of the tank is 10,393.6 liters? No. Because my measurement of the depth wasn’t accurate enough. It could easily have been anything from, say, 95 centimeters to 105 centimeters, so the actual volume could range between around 9900 liters and 11000 liters. From the accuracy of my measurements, claiming that I know the volume down to a milliliter is ridiculous, when my measurement of the depth was only accurate within a range of +/- 5 centimeters!

Ideally, I might want to know a strong estimate on the bounds of the accuracy of a computation based on measurements. I can compute that if I know the measurement error bounds on each error measurement, and I can track them through the computation and come up with a good estimate of the bounds – that’s basically what I did up above, to conclude that the volume of the tank was between 9,900 and 11,000 liters. The problem with that is that we often don’t really know the precise error bounds – so even our estimate of error is an imprecise figure! And even if we did know precise error bounds, the computation becomes much more difficult when you want to track error bounds through it. (And that’s not even considering the fact that our error bounds are only another measured estimate with its own error bounds!)

Significant figures are a simple statistical tool that we can use to determine a reasonable way of estimating how much accuracy we have in our measurements, and how much accuracy we can have at the end of a computation. It’s not perfect, but most of the time, it’s good enough, and it’s really easy.

The basic concept of significant figures is simple. You count how many digits of accuracy each measurement has. The result of the computation over the measurements is accurate to the smallest number of digits of any of the measurements used in the computation.

In the water tank example, we had three significant figures of accuracy on the length and width of the tank. But we only had one significant figure on the accuracy of the depth. So we can only have one significant figure in the accuracy of the volume. So we conclude that we can say it was around 10 liters, and we can’t really say anything more precise than that. The exact value likely falls somewhere within a bell curve centered around 10 liters.

Returning to the original question: can significant figures change an estimate of the age of the universe from 20 to 13.7?

Intuitively, it might seem like it shouldn’t: sigfigs are really an extension of the idea of rounding, and 13.7 rounded to one sigfig should round down to 10, not up to 20.

I can’t say anything about the specifics of the computations that produced the estimates of 20 and 13.7 billion years. I don’t know the specific measurements or computations that were involved in that estimate.

What I can do is just work through a simple exercise in computations with significant figures to see whether it’s possible that changing the number of significant digits in a measurement could produce a change from 20 to 13.7.

So, we’re looking at two different computations that are estimating the same quantity. The first, 20, has just one significant figure. The second, 13.7 has three significant digits. What that means is that for the original computation, one of the quantities was known only to one significant figure. We can’t say whether all of the elements of the computation were limited to one sigfig, but we know at least one of them was.

So if the change from 20 to 13.7 was caused by significant digits, it means that by increasing the precision of just one element of the computation, we could produce a large change in the computed value. Let’s make it simpler, and see if we can see what’s going on by just adding one significant digit to one measurement.

Again, to keep things simple, let’s imagine that we’re doing a really simple calculation. We’ll use just two measurements $x$ and $y$, and the value that we want to compute is just their product, $x \times y$.

Initially, we’ll say that we measured the value of $x$ to be 8.2 – that’s a measurement with two significant figures. We measure $y$ to be 2 – just one significant figure. The product $x\times y = 8.2 \times 2 = 16.4$. Then we need to reduce that product to just one significant figure, which gives us 20.

After a few years pass, and our ability to measure $y$ gets much better: now we can measure it to two significant figures, with a new value of 1.7. Our new measurement is completely compatible with the old one – 1.7 reduced to 1 significant figure is 2.

Now we’ve got equal precision on both of the measurements – they’re now both 2 significant figures. So we can compute a new, better estimate by multiplying them together, and reducing the solution to 2 significant figures.

We multiply 8.2 by 1.7, giving us around 13.94. Reduced to 2 significant figures, that’s 14.

Adding one significant digit to just one of our measurements changed our estimate of the figure from 20 to 14.

Returning to the intuition: It seems like 14 vs 20 is a very big difference: it’s a 30 percent change from 20 to 14! Our intuition is that it’s too big a difference to be explained just by a tiny one-digit change in the precision of our measurements!

There’s two phenomena going on here that make it look so strange.

The first is that significant figures are an absolute error measurement. If I’m measuring something in inches, the difference between 15 and 20 inches is the same size error as the difference between 90 and 95 inches. If a measurement error changed a value from 90 to 84, we wouldn’t give it a second thought; but because it reduced 20 to 14, that seems worse, even though the absolute magnitude of the difference considered in the units that we’re measuring is exactly the same.

The second (and far more important one) is that a measurement of just one significant digit is a very imprecise measurement, and so any estimate that you produce from it is a very imprecise estimate. It seems like a big difference, and it is – but that’s to be expected when you try to compute a value from a very rough measurement. Off by one digit in the least significant position is usually not a big deal. But if there’s only one significant digit, then you’ve got very little precision: it’s saying that you can barely measure it. So of course adding precision is going to have a significant impact: you’re adding a lot of extra information in your increase in precision!

# Big Science News! Inflation, Gravity, and Gravitational Waves

So, big announcement yesterday. Lots of people have asked if I could try to explain it! People have been asking since yesterday morning, but folks, I’ve got a job! I’ve been writing when I have time while running slow tests in another window, so it’s taken more than a day to get to it.

The announcement is really, really fascinating. A group has been able to observe gravity wave fluctuations in the cosmic microwave background. This is a huge deal! For example, Sean Carroll (no relation) wrote:

other than finding life on other planets or directly detecting dark matter, I can’t think of any other plausible near-term astrophysical discovery more important than this one for improving our understanding of the universe.

Why is this such a big deal?

This is not an easy thing to explain, but I’ll do my best.

We believe that the universe started with the big bang – all of the matter and energy, all of the space in the universe, expanding outwards from a point. There’s all sorts of amazing evidence for the big bang – not least the cosmic microwave background.

But the big-bang theory has some problems. In particular, why is everything the same everywhere?

That sounds like a strange question. Why wouldn’t it be the same everywhere?

Here’s why: because for changes to occur at the same time in different places, we expect there to be some causal connection between those places. If there is no plausible causal connection, then there’s a problem: how could things happen at the same time, in the same way?

That causal connection is a problem. To explain why, first I need to explain the idea of the observable universe.

Right now, there is some part of the universe that we can observe – because light from it has reached us. There’s also some part of the universe that we can’t observe, because light from it hasn’t reached us yet. Every day, every moment, the observable universe gets larger – not because the universe is expanding (it is, but we’re not talking about the size of the universe, but rather of the part of the universe that we can observe). It’s literally getting larger, because there are parts of the universe that are so far away from us, that the first light they emitted after the universe started didn’t reach us until right now. That threshold, of the stuff that couldn’t possible have gotten here yet, is constantly expanding, getting farther and farther away.

There are parts of the universe that are so far away, that the light from them couldn’t reach us until now. But when we look at that light, and use it to see what’s there, it looks exactly like what we see around us.

The problem is, it shouldn’t. If you just take the big bang, and you don’t have a period of inflation, what you would expect is a highly non-uniform universe with a very high spatial curvurature. Places very far away shouldn’t be exactly the same as here, because there is no mechanism which can make them evolve in exactly the same way that they did here! As energy levels from the big bang decrease, local fluctuations should have produced very different outcomes. They shouldn’t have ended up the same as here – because there’s many different ways things could have turned out, and they can’t be causally connected, because there’s no way that information could have gotten from there to here in time for it to have any effect.

Light is the fastest thing in the universe – but light from these places just got here. That means that until now, there couldn’t possibly be any connection between here and there. How could all of the fundamental properties of space – its curvature, the density of matter and energy – be exactly the same as here, if there was never any possible causal contact between us?

The answer to that is an idea called inflation. At some time in the earliest part of the universe – during a tiny fraction of the first second – the universe itself expanded at a speed faster than light. (Note: this doesn’t mean that stuff moved faster than light – it means that space itself expanded, creating space between things so that the distance between them expanded faster than light. Subtle distinction, but important!) So the matter and energy all got “stretched” out, at the same time, in the same way, giving the universe the basic shape and structure that it has now.

Inflation is the process that created the uniform universe. This process, which happened to the entire universe, had tiny but uniform fluctuations because of the basic quantum structure of the universe. Those fluctuations were the same everywhere – because when they happened, they were causally connected! Inflation expanded space, but those fluctuations provided the basic structure on which the stuff we observe in the universe developed. Since that basic underlying structure is the same everywhere, everything built on top it is the same as well.

We’ve seen lots of evidence for inflation, but it hasn’t quite been a universally accepted idea.

The next piece of the puzzle is gravity. Gravity at least appears to be very strange. All of the other forces in our universe behave in a consistent way. In fact, we’ve been able to show that they’re ultimately different aspects of the same underlying phenomena. All of the other forces can be described quantum mechanically, and they operate through exchange particles that transmit force/energy – for example, electromagnetic forces are transmitted by photons. But not gravity: we have no working quantum theory for how gravity works! We strongly suspect that it must, but we don’t know how, and up to now, we never found any actual proof that it does behave quantumly. But if it did, and if inflation happened, that means that those quantum fluctations during expansion, the things that provided the basic lattice on which matter and energy hang, should have created an echo in gravity!

Unfortunately, we can’t see gravity. The combination of inflation and quantum mechanics means that there should be gravitational fluctuations in the universe – waves in the basic field of gravity. We’ve predicted those waves for a long time. But we haven’t been able to actually test that prediction, because we didn’t have a way to see gravitational waves.

So now, I can finally get to this new result.

They believe that they found gravity waves in the cosmic microwave background. They used a brilliant scheme to observe them: if we look at the cosmic microwave background – not at any specific celestial object, but just at the background – gravitational waves would created a very subtle tensor polarization effect. So they created a system that could observe polarization. Then they removed all of the kinds of polarization that could be explained by anything other than gravitational waves. What they were left with was a very clear wave pattern in the polarization – exactly what was predicted by quantum inflation! You can see one of their images of this wave pattern at the top of this post.

If these new observations are confirmed, that means that we have new evidence for two things:

1. Inflation happened. These gravity waves are an expected residue of inflation. They’re exactly what we would have expected if inflation happened, and we don’t have any other explanation that’s compatible with them.
2. Gravity is quantum! If gravity wasn’t quantum, then expansion would have completely smoothed out the gravitational effects, and we wouldn’t see gravitational waves. Since we do see waves, it’s strong evidence that gravity really does have a quantum aspect. We still don’t know how it works, but now we have some really compelling evidence that it must!

# The Elegance of Uncertainty

I was recently reading yet another botched explanation of Heisenberg’s uncertainty principle, and it ticked me off. It wasn’t a particularly interesting one, so I’m not going disassemble it in detail. What it did was the usual crackpot quantum dance: Heisenberg said that quantum means observers affect the universe, therefore our thoughts can control the universe. Blah blah blah.

It’s not worth getting into the cranky details. But it inspired me to actually take some time and try to explain what uncertainty really means. Heisenberg’s uncertainty principle is fascinating. It’s an extremely simple concept, and yet when you realize what it means, it’s the most mind-blowingly strange thing that you’ve ever heard.

One of the beautiful things about it is that you can take the math of uncertainty and reduce it to one simple equation. It says that given any object or particle, the following equation is always true:

$sigma_x sigma_p ge hbar$

Where:

• $sigma_x$ is a measurement of the amount of uncertainty
about the position of the particle;
• $sigma_p$ is the uncertainty about the momentum of the particle; and
• $hbar$ is a fundamental constant, called the reduced Plank’s constant, which is roughly $1.05457173 times 10^{-34}frac{m^2 kg}{s}$.

That last constant deserves a bit of extra explanation. Plank’s constant describes the fundamental granularity of the universe. We perceive the world as being smooth. When we look at the distance between two objects, we can divide it in half, and in half again, and in half again. It seems like we should be able to do that forever. Mathematically we can, but physically we can’t! Eventually, we get to a point where where is no way to subdivide distance anymore. We hit the grain-size of the universe. The same goes for time: we can look at what happens in a second, or a millisecond, or a nanosecond. But eventually, it gets down to a point where you can’t divide time anymore! Planck’s constant essentially defines that smallest unit of time or space.

Back to that beautiful equation: what uncertainty says is that the product of the uncertainty about the position of a particle and the uncertainty about the momentum of a particle must be at least a certain minimum.

Here’s where people go wrong. They take that to mean that our ability to measure the position and momentum of a particle is uncertain – that the problem is in the process of measurement. But no: it’s talking about a fundamental uncertainty. This is what makes it an incredibly crazy idea. It’s not just talking about our inability to measure something: it’s talking about the fundamental true uncertainty of the particle in the universe because of the quantum structure of the universe.

Let’s talk about an example. Look out the window. See the sunlight? It’s produced by fusion in the sun. But fusion should be impossible. Without uncertainty, the sun could not exist. We could not exist.

Why should it be impossible for fusion to happen in the sun? Because it’s nowhere near dense or hot enough.

There are two forces that you need to consider in the process of nuclear fusion. There’s the electromagnetic force, and there’s the strong nuclear force.

The electromagnetic force, we’re all familiar with. Like charges repel, different charges attract. The nucleus of an atom has a positive charge – so nuclei repel each other.

The nuclear force we’re less familiar with. The protons in a nucleus repel each other – they’ve still got like charges! But there’s another force – the strong nuclear force – that holds the nucleus together. The strong nuclear force is incredibly strong at extremely short distances, but it diminishes much, much faster than electromagnetism. So if you can get a proton close enough to the nucleus of an atom for the strong force to outweigh the electromagnetic, then that proton will stick to the nucleus, and you’ve got fusion!

The problem with fusion is that it takes a lot of energy to get two hydrogen nuclei close enough to each other for that strong force to kick in. In fact, it turns out that hydrogen nuclei in the sun are nowhere close to energetic enough to overcome the electromagnetic repulsion – not by multiple orders of magnitude!

But this is where uncertainty comes in to play. The core of the sun is a dense soup of other hydrogen atoms. They can’t move around very much without the other atoms around them moving. That means that their momentum is very constrained – $sigma_p$ is very small, because there’s just not much possible variation in how fast it’s moving. But the product of $sigma_p$ and $sigma_x$ have to be greater than $hbar$, which means that $sigma_x$ needs to be pretty large to compensate for the certainty about the momentum.

If $sigma_x$ is large, that means that the particle’s position is not very constrained at all. It’s not just that we can’t tell exactly where it is, but it’s position is fundamentally fuzzy. It doesn’t have a precise position!

That uncertainty about the position allows a strange thing to happen. The fuzziness of position of a hydrogen nucleus is large enough that it overlaps with the the nucleus of another atom – and bang, they fuse.

This is an insane idea. A hydrogen nucleus doesn’t get pushed into a collision with another hydrogen nucleus. It randomly appears in a collided state, because it’s position wasn’t really fixed. The two nuclei that fused didn’t move: they simply didn’t have a precise position!

So where does this uncertainty come from? It’s part of the hard-to-comprehend world of quantum physics. Particles aren’t really particles. They’re waves. But they’re not really waves. They’re particles. They’re both, and they’re neither. They’re something in between, or they’re both at the same time. But they’re not the precise things that we think of. They’re inherently fuzzy probabilistic things. That’s the source uncertainty: at macroscopic scales, they behave as if they’re particles. But they aren’t really. So the properties that associate with particles just don’t work. An electron doesn’t have an exact position and velocity. It has a haze of probability space where it could be. The uncertainty equation describes that haze – the inherent uncertainty that’s caused by the real particle/wave duality of the things we call particles.