# physics

## “Do you believe in the Big Bang?”

I originally wrote this as a “non-scientist's introduction” for my PhD thesis in astrophysics, but didn't end up including it.

The big bang is like biological evolution: the non-technical world makes a big fuss about it, but scientists simply shake their heads and move on. Our understanding of biology and astronomy is so inextricably linked with these theories that there would practically be nothing left if we suddenly decided to reject them out of hand. Indeed belief not the most useful word here, or in science generally. To be sure, science requires at least a working belief that we live in a physical universe governed by mathematical laws, but if a scientific theory were backed up by so little data that it required belief to be accepted, then it’s unlikely to be very useful. There is, of course, an intermediate ground for theories in development or proposed before technology exists to test them. No one would suggest discarding these theories outright, but they aren’t said to be accepted either. Rather, scientists say about string theory and supersymmetry, for instance, that the jury is still out.

In this essay, I will discuss the modern field of cosmology and the evidence which puts the big bang squarely in the realm of fact. First, though, an important point about terminology. Many scientists and science popularizers play fast and loose with what the term big bang actually refers to. Does it refer to the predicted singularity 13.8 billion years ago? Or both the singularity and its aftermath? Just its immediate aftermath, or the resulting expansion continuing into the present day? Or does it refer to the theory which predicts that event? I suspect this ambiguity arises in part out of efforts to wow audiences at public lectures, and in part because the definition of the big bang is really beside the point in the technical literature. Experimentalists focus on building instruments and studying distant galaxies and theorists work to better understand the predictions of quantum field theory and general relativity. Editors don’t make a big deal over the precise vocabulary used in paper introductions. Because of all this ambiguity, and because of the baggage the term big bang carries among the public, I prefer not to use it at all. Let’s just discuss what we know.

I could just describe our modern picture of the Galaxy, the universe and cosmology, but I fear it would be all to easy to simply dismiss as simply a philosophy like any other. So I’ll take a more historical approach and describe the main breakthroughs over the past century that have made cosmology the precision science that it is today.

Our story begins in the early 1920s, by which time a number of poorly understood spiral nebulae, fuzzy spiral blobs, had been observed in the night sky. There were essentially two possibilities. On the one hand, perhaps they were clouds of gas in the far reaches of our own galaxy; on the other, perhaps they were galaxies like our own except farther away than any object ever seen before. The implications were enormous. Is our galaxy all there is, or is it part of a space even more vast with countless others?

Things came to a head in the famous Great Debate between astronomers Harlow Shapley and Heber Curtis at the Smithsonian in 1920. The debate itself was mostly a stunt, scientific disputes aren’t settled by debate but by better data. At the time, there existed measurements supporting both points of view, but none of the data were extremely convincing. The dispute wasn’t settled until Edwin Hubble, using the new 100 inch Palomar telescope on Mount Wilson outside of Los Angeles, made the first measurements of the distances to these spiral nebulae, and demonstrated they were way outside our galaxy. Hubble sought and monitored a type of pulsating stars in these nebulae called Cepheid variables, which are known to pulsate with a period proportional to their intrinsic brightness. Then by comparing their apparent brightness, more distant objects generally appearing dimmer, their distances could be obtained. Many of the spiral nebulae were millions of light-years away, while our galaxy is (and was) known to be only about 50,000 light-years from side to side. The debate was settled, the universe was much larger than we had known.

For his next trick, Hubble turned his attention to even more distant galaxies, and the results were just as shocking. With only a few exceptions, all appeared to be flying away from us at staggering velocities, hundreds or thousands of kilometers per second. Fast enough to travel around the earth in less than a minute. Moreover more distant galaxies were flying away from us faster…linearly faster. Exactly what you’d expect for a uniform expansion of space itself. It was as if a rubber sheet with dots on it was being stretched out; no matter which dot you’re sitting on, all others are moving away from you at a rate proportional to their distance. This relationship is known as Hubble’s law, and the coefficient of proportionality, in units of velocity per distance, is knowns as the Hubble constant. Modern measurements place this value at roughly 70 km/s per Mpc, where 1 Mpc = 1,000,000 parsecs.

How can the universe itself be expanding? What does that even mean? A decade earlier, Einstein had applied the equations of General Relativity to the universe as a whole and found that static solutions are impossible, which suggested to him that his equations were not correct. He found that a single constant added to the central equation of the theory was sufficient to ensure a static solution, a cosmological constant. After hearing of Hubble’s results, though, Einstein famously abandoned that addition, calling it his greatest mistake. It wasn’t really a mistake per se, given that there wasn’t any data at the time to test his prediction, more a missed opportunity. He could, after all, have gambled and predicted the expansion of the Universe, and perhaps won a second Nobel prize for his trouble.

Despite this early progress, cosmology largely languished as a research backwater for decades, and for a time fell out of all memory, until the most unlikely creature of all, an engineer, made a breakthrough. In the course of developing ultra sensitive radio receivers at Bell Labs, Anro Penzias and Robert Wilson developed a horn antenna which seemed to pick up a very faint noise which they couldn’t explain.

Radio astronomers often quantify the power of a radio signal by the equivalent temperature of a thermal source emitting the same amount of power. Let us build some intuition here. Any object at a non-zero temperature emits electromagnetic radiation. The hotter it is, the higher frequency of the radiation. Objects at room temperature emit predominantly in the infrared band, while objects at a few thousand degrees, like the sun (and like tungsten filaments in incandescent bulbs), emit in the optical band. At the other end of the spectrum, only very cold objects emit predominantly in radio waves. Note of course that we are only talking about electromagnetic waves emitted by the random jostling around of atoms. By running currents through a wire, substantially more powerful radio waves may be produced which are not random at all, and are useful for transmitting Lady Gaga lyrics, among other things.

The background radiation detected by Penzias and Wilson was equivalent to that emitted by a very cold object, within a few degrees of absolute zero. 3 kelvin to be precise. Moreover, it was almost exactly the same brightness in any direction they looked away from our galaxy. Their discovery set off a flashbang in the cosmology community. Based on calculations of how atoms could have formed in the hot and dense early universe, Ralph Alpher and Robert Herman calculated that the ambient temperature of empty space today would have to be roughly 5 kelvin (See also this and this).

The exact temperature was somewhat uncertain, and later estimates ranged from a few kelvin to tens of kelvin, but Robert Dicke immediately recognized that the observed 3 kelvin radiation was cosmic in origin, corresponding exactly to the aftermath of that hot, dense early universe, what had been mocked as the the big bang by astronomers who opposed the theory. But much as it’s a losing battle to keep non-scientists from referring the Higgs boson as the God particle, the name big bang stuck.

Two additional discoveries made in the early 1990s confirmed these results and began the era of precision cosmology. Both came from a satellite-born experiment named the Cosmic Background Explorer (COBE) designed to better measure the cosmic microwave background (CMB) discovered by Penzias and Wilson. The first key result was a precise measurement of its spectrum, the amount of power emitted as a function of frequency. Purely thermal emitters have a very characteristic spectrum known as the blackbody curve or the Planck function. Radiation emitted by stars and galaxies often looks somewhat thermal, but atoms and dust inevitable emit or absorb some radiation at different frequencies, always resulting in an imperfect frequency spectrum. In fact, the only known source of such a perfect thermal spectrum is the hot, dense early universe when everything was a sort of primordial soup, or perhaps more precisely, a purée. All scientific opposition to the hot, dense early universe model, ie, the big bang, evaporated after the publication of this spectrum.

The second important discovery was the icing on the cake. On top of clear evidence of this exotic hot, dense, homogenous early universe, COBE saw hints of how the modern clumpy universe could have emerged. To the imprecise radio antenna of Penzias and Wilson, the cosmic microwave background appeared just as bright in every direction, but the more sensitive COBE satellite distinguished two interesting patterns. First, after subtracting the sky-averaged intensity corresponding to the 3 kelvin radiation, they observed a bi-polar pattern in the sky. That is, the sky was uniformly brighter in one direction and fainter in the other, just as you would expect due to the motion of our galaxy relative to other galaxies. A doppler shift. It’s as if our dot on the expanding rubber sheet is actually moving slowly across the surface while the balloon is expanding, so dots on one side don’t seem to be receding as quickly, while those on the other recede faster than they otherwise would. Then after subtracting this bi-polar pattern, we see an incredible, random-looking field of fluctuations. This anisotropy of the cosmic microwave background shows us the slight inhomogenieties in the nearly smooth early universe.

Over time, we predict, but have yet to directly observe, that the denser regions slowly drew in more matter and collapsed due to gravitational attraction, eventually forming stars and galaxies. More recent CMB surveys by the WMAP and Planck satellites have confirmed and extended these results with exquisite precision, and truly made the past two decades the golden age of cosmology.

Lastly, I would be remiss if I didn’t mention the most shocking discovery in cosmology over the past decades: the acceleration of the expansion of the universe. In 1997 and 1998, two groups attempting to reproduce, refine, and extend Hubble’s original measurements observed that galaxies a thousand times more distant that Hubble’s appeared only half as bright as they should given their distance. Recall that Hubble’s law relates the recession velocity of a galaxy to its distance, and thus, to its apparent brightness. An obvious possibility was obscuration by dust, but both groups went to great lengths to demonstrate this was not the case. Dust absorbs preferentially red light, drastically altering the spectrum, but the spectra of these galaxies appeared normal. They were just fainter than their other properties suggested they should be.

The consensus conclusion is that not only is the universe is expanding, but it is accelerating, due to exactly a term in Einstein’s field equation like the cosmological constant he artificially added. But this time with living proof. The acceleration of the universe is often described as a dark energy, often meant as a more general theory than a cosmological constant, but it remains a big question mark. Don’t be discouraged, though. Many of the best breakthroughs in physics have occurred after observations of the unexpected. How fortunate we are to witness one of them!

#physics

Posted by Abraham

## Why can't you fold paper more than 7 times?

Mythbusters was one of my favorite series growing up, and it was definitely one of my early inspirations to study experimental physics. But over the course of the show, Adam Savage and Jamie Hyneman did plenty of experiments that were basically unnecessary had they done a two line physics calculation. Yes, it was fun to see the guys make a football field-sized sheet of paper, and use a hydraulic press to fold it, but...well...you could just do the math instead. Standard printer paper is $t=0.1\text{mm}$ thick, and $d=25\text{cm}$ long. It gets twice as thick and half as long each time you fold it, so after $N$ folds it is $2^Nt$ thick and $d/2^N$ long. Roughly speaking, you can fold paper in half until it is as thick as it is long, i.e., $2^Nt = d/2^N$, which implies $N = \frac{1}{2}\log_2 d/t\approx 5.6$. So maybe if you really force it: 6 times. Because this result depends only logarithmically on the length and thickness, it is fairly insensitive to their values. For example, if the sheet of paper is as large as a football field, $d=100\text{m}$, 400 times bigger than standard printer paper, then $N=9.9$. So experiments are great, but sometimes a spherical cow is all you need.

#physics

Posted by Abraham