by Max Tegmark
Cross-checking is a bad thing in ice hockey but a good thing in science, where it can reveal hidden mistakes. Boomerang let us cosmologists make two cross-checks on the cosmic-matter budget:
1. We measured the dark-energy fraction in two different ways (with supernovae Ia and with cosmic microwave–background peaks) and the answers agreed.
2. We measured the ordinary-matter fraction in two different ways (with Big Bang nucleosynthesis and with cosmic microwave–background peaks) and the answers disagreed, so at least one of the two methods was messed up.
The Bump Is Back
A year later, I’m in a swanky press conference room in Washington DC, glued to my seat, feeling like Santa Claus is about to arrive three times over. First up was John Carlstrom, reporting results from a microwave telescope called DASI at the South Pole. After the usual blah blah about technical details I already knew—boom!—the most amazing power-spectrum plot I’d ever seen! With as many as three peaks clearly visible. Then came Santa 2: John Ruhl from the Boomerang team. Blah blah—boom! Another amazing power spectrum with three peaks, in beautiful agreement with the independent DASI measurements. And the once-so-anemic second peak was bigger this time, after they’d improved the modeling of their telescope. Finally, Santa 3: Paul Richards reported measurements from a balloon experiment called MAXIMA, agreeing well with the others’ data. I was simply amazed. After all these years of dreaming about these elusive clues encoded in the microwave-background sky, here they were! It had felt so hubristic to assume that we humans knew what our Universe was doing just a few hundred thousand years after our Big Bang, yet we’d been right. That night I quickly reran my model-fitting software with the new microwave-background data, and now that the second peak was higher, my code predicted about 5% atoms, in beautiful agreement with Big Bang nucleosynthesis. The atom cross-check had gone from failure to success, and order was restored in the cosmos. And the order remained: by now, WMAP, Planck and other experiments have measured the power-spectrum curve way more accurately, as you can see in Figure 4.2, showing that these early experiments really got it right.
Precision Galaxy Clustering
By 2003, the cosmic microwave–background radiation had become arguably cosmology’s greatest success story ever. It became widely perceived as a panacea that could solve all our problems and let us measure all the key numbers in our cosmological model. But this perception was incorrect. Suppose that you measure my weight to be two hundred pounds. This clearly isn’t enough information for you to determine my height and my width, since my weight depends on both: I could be either tall and slim or short and chubby. We have analogous problems when trying to measure key numbers about our Universe. For example, the characteristic microwave background–spot sizes that correspond to the horizontal locations of the cosmic microwave–background peaks in Figure 4.2 depend on both the curvature of space (which magnifies/demagnifies the spots) and on the dark-energy density (which changes the expansion rate of our Universe and therefore the distance to the plasma surface with its spots, also causing them to look larger or smaller). So although many journalists claimed that experiments such as Boomerang and WMAP had shown space to be flat, they hadn’t: our Universe could be either flat with about 70% dark energy or curved with a different amount of dark energy. There are other pairs of cosmological parameters that are similarly hard for the microwave background to tease apart, for example the amplitude of clumping in our early Universe and the time when the first stars formed, both of which affect the Figure 4.2 power spectrum in similar ways (in this case changing the peak heights). As we’re taught in high-school algebra, we need more than one equation to determine two unknown quantities. In cosmology we want to determine about seven numbers, and the microwave background alone simply doesn’t contain enough information to allow this. So we need additional information from other cosmological measurements. Such as 3-D galaxy maps.
Galaxy Redshift Surveys
When we make a 3-D map of where the galaxies are in our Universe, we first analyze 2-D photos of the sky to find galaxies, then make additional measurements to figure out how far away they are. The most ambitious 3-D mapping project to date is called the Sloan Digital Sky Survey, and I had the great fortune to get to join it when I was a postdoc in Princeton even though a small army of people had already spent almost a decade organizing the project, building the telescope hardware, and making things work. It’s made the 2-D sky map shown in Figure 4.4 by spending over a decade imaging over a third of the sky with a custom-built 2.5 meter telescope in New Mexico. Jim Gunn, a Princeton professor who reminded me of a friendly bearded wizard, used his magical powers to build this amazing digital camera for the telescope, then the largest camera ever made for astronomical purposes.
Figure 4.4: The Sloan Digital Sky Survey contains an astonishing amount of information. The left panel, where the sphere represents the whole sky, contains almost a terapixel, a million megapixels. The successive zooms focus behind the Big Dipper on the so-called Whirlpool galaxy, but the same level of detail is available everywhere you zoom. (Image credit: Mike Blanton and David Hogg/SDSS Collaboration)
Click here to see a larger image.
If you carefully look through the survey’s sky images, like those in Figure 4.5, you’ll find lots of stars, galaxies and other objects—more than half a billion of them, in fact. This multiplicity means that if you tell a grad student to find all the objects at a rate of one per second, working eight hours a day without breaks for weekends or holidays, you’ll be waiting for fifty years—and get the award for worst thesis advisor ever. This object finding is a surprisingly tricky problem even for a computer: it needs to be able to distinguish between galaxies (which look fuzzy and spread-out), stars (which would look pointlike were it not for atmospheric blurring), comets, satellites and various instrumental artifacts. Worse yet: sometimes objects overlap, as when a nearby star is annoyingly located in front of a distant galaxy. After a large group of people had struggled with this problem for years, it was solved through a heroic programming effort by Robert Lupton, a chipper Englishman who used “Robert Lupton the Good” as his email name and was always barefoot (Figure 4.5).
The next step is to figure out how far away each galaxy is. In Chapter 3 we saw how Edwin Hubble’s law v = Hd means that our Universe is expanding, so the greater the distance d to a faraway galaxy, the higher the velocity v with which it recedes from us. Now that Hubble’s law has been firmly established, we can use it backwards, as a method to measure distances: by measuring how fast a galaxy recedes using the redshifting of its spectral lines, we learn its distance. Basically, measuring redshifts and velocities is easy in astronomy while measuring distances is hard, so Hubble’s law can save you work: once you’ve measured the Hubble parameter H using some nearby galaxies, you just need to measure the velocities v of distant galaxies from their redshifted spectra, then divide by H to get a good estimate of their distance.
Figure 4.5: A small fraction of the Sloan Digital Sky Survey map has been used to decorate an entire wall in the Princeton University Astronomy Department, which Robert Lupton can be seen scrutinizing with my kids. After Robert’s software identifies all objects in the map, the distances to the most interesting galaxies are measured, producing a 3-D map (left) with us at the center and every dot representing a galaxy. You can see the “Sloan Great Wall” about a third of the way from the top in the image.
Click here to see a larger image.
From the catalog of objects churned out by Robert Lupton’s software, the most interesting million or so were selected to have their spectra measured. The twenty-four galaxy spectra that Edwin Hubble had used to discover our cosmic expansion took weeks to collect at the time. In contrast, the Sloan Digital Sky Survey could mass-produce spectra at a rate of 640 per hour, all measured at the same time. The trick was to position 640 optical fibers at the places in the focal plane of the telescope where Robert’s catalog said that the galaxy images would be, and then have these
fibers lead the galaxy light to a spectrograph that split it into 640 separate rainbows imaged by a digital camera. Another software package, this one spearheaded by David Schlegel and colleagues, analyzed these rainbows to figure out the distance to each galaxy (from the redshifting of its spectral lines) and other galaxy properties.
On the leftmost side of Figure 4.5, I’ve plotted a 3D slice of our Universe, with each point representing a galaxy; when I feel I need to get away from it all for a while, I like to fly around among the galaxies with a 3-D cosmological-flight simulator I have. Doing this reveals something I find quite beautiful: we’re part of something grander. Not only is our planet a part of a solar system and our Solar System part of a galaxy, but our Galaxy is part of a cosmic web of galaxy groups, clusters, superclusters and gigantic filamentary structures. While poring over this map and noticing what’s now become known as the “Sloan Great Wall” (Figure 4.5, left), I was so flabbergasted by its size that I first suspected a bug in my code. But some of my collaborators discovered it independently and it’s definitely real: being 1.4 billion light-years long, it’s the largest known structure in our Universe. These large-scale clustering patterns are a cosmological treasure trove, encoding the sort of valuable information that the cosmic microwave background is missing.
From Derision Cosmology to Precision Cosmology
These patterns in the galaxy distribution are really the same patterns that we saw manifest themselves in cosmic microwave–background maps—only billions of years later, amplified by gravity. In a region of space where gas was once 0.001% denser than its surroundings, causing a spot in the WMAP map (see Figure 3.4), there might today be a cluster of one hundred galaxies. In this sense, we can think of the cosmic microwave–background fluctuations as the cosmic DNA, the blueprints for what our Universe will grow to become. By comparing the slight past clustering seen in the cosmic microwave background to the strong current clustering seen in a 3-D galaxy map, we can measure the detailed nature of the stuff whose gravity caused the clustering to grow between then and now.
Just as the microwave-background clustering is characterized by a power-spectrum curve (see Figure 4.2), so is the galaxy clustering. However, measuring this curve really accurately turns out to be really hard: the Sloan Digital Sky Survey galaxy–power spectrum measurement shown in Figure 4.6 took me six years (six!) to finish, despite lots of help from colleagues, and ended up being my most exhausting project ever. Time and again, I’d think, Thank goodness I’m finally almost done, since I just can’t take this any longer! just to discover a major new problem with the analysis.
Figure 4.6: The clumping of matter in our Universe is described by the power-spectrum curve shown here. The fact that the curve equals 10% at 1,000 million light-years crudely speaking means that if you measure the amount of mass in a sphere of that radius, then the answer you’ll get will vary by 10%, depending on where in space you put the sphere. In contrast to when I started my career, highly accurate measurements now exist, and they agree with theoretical prediction. I find it particularly remarkable that the five different measurements of this curve agree with each other even though the data, the people and the methods involved are totally different.
Click here to see a larger image.
Why was it so hard? Well, it would be easy if we knew the exact position of every galaxy in our Universe and had an infinitely powerful computer with which to analyze them. In practice, we can’t see many of the galaxies because of various complications, and some of the ones we do see have a different distance and luminosity than we think. If we ignore these complications, we get an incorrect power spectrum that translates into incorrect conclusions about our Universe.
The first 3-D galaxy maps were so small that it wasn’t worth putting lots of time into analyzing them. My colleague Michael Vogeley gave me a nice plot summarizing all the measurements up to 1996 or so, and when I asked him why he hadn’t put error bars on them to indicate the measurement uncertainty, he said, “Because I don’t believe them.” He had good reason for his skepticism: some teams claimed ten times more power than others, so they couldn’t all be right.
Groups around the world gradually made bigger 3-D maps and shared them online. I felt that when so many people were putting so much hard work into making these maps, they deserved a really careful analysis. So I teamed up with my friend Andrew Hamilton to go the extra mile, measuring galaxy power spectra with the same sorts of information-theory methods we’d developed for cosmic microwave–background analysis.
Andrew is an incurably cheerful Brit with a mischievous, bright smile, and one of my favorite collaborators. I once showed up late at a restaurant where I was supposed to meet Andrew as well as my friends Wayne Hu and David Hogg, who had recently shaved his head. When I asked a waitress if she’d seen a trio who looked like Robert Redford, Bruce Lee and Kojak, she thought for a moment, then smiled and said: “I can see Robert Redford.…” We first analyzed progressively larger 3-D maps with obscure names such as IRAS, PSCz, UZC and 2dF, with about 5,000, 15,000, 20,000 and 100,000 galaxies, respectively. He lived in Colorado, and we had endless conversations about the mathematical intricacies of power-spectrum measurement by email, by phone, and while hiking in the Alps and the Rocky Mountains.
The Sloan Digital Sky Survey map was the largest and cleanest survey of all, based on all-digital imaging and meticulous quality control, so I felt that it also deserved the most painstaking analysis. Because the results would only be as good as the weakest link, I spent years working on many of the dirty-laundry issues that people considered the most boring. Professor Jill Knapp, one of the driving forces in the project and also Jim Gunn’s wife, would organize weekly meetings in Princeton where she’d spoil us all with irresistible food while we tried to identify all the skeletons in the analysis closet and figure out what to do about them. For example, how many galaxies we’d map in a particular direction depended on how bad the weather was while it was imaged, how much Galactic dust was in the way, and on the fraction of the visible galaxies that could be covered by optical fibers. Frankly, this stuff truly was boring, so I’ll spare you the details, but I nonetheless got huge amounts of help from many people, particularly Professor Michael Strauss and his then grad student Mike Blanton. In parallel, there was the seemingly never-ending cycle of computing terabytes of number tables called matrices during multiweek computer runs, looking at messed-up result plots, debugging my code, and trying again.
After six years of this, I finally submitted two papers with results in 2003, both with over sixty coauthors. I’ve never in my entire life felt as relieved to finish something, except perhaps for this book. The first paper concerned the galaxy–power spectrum measurement in Figure 4.6, and the second dealt with a measurement of cosmological parameters from combining this with the microwave-background power spectrum. I’ve listed some of the highlights in Table 4.1; here I’ve updated the numbers to the most recent measurements by others, but the values haven’t changed significantly even though the uncertainties have decreased. I still had in fresh memory the wild debates from my grad-student days about whether our Universe was 10 billion or 20 billion years old, and now we were arguing over whether it was 13.7 or 13.8 billion! Precision cosmology had finally arrived, and I felt excited and honored to have gotten to play a small part in this.
At a personal level, this outcome was quite lucky for me: they evaluated me for tenure at MIT in the fall of 2004, and I’d been told that to get it, I needed “a home run, or at least a couple of doubles.” Just as musicians have their top-ten sales charts, we scientists have our citation lists: every time someone cites your paper, it counts as a feather in your hat. The citation business can be rather random and silly, prone to bandwagon effects, since lazy authors tend to copy citations from others without actually reading the papers they cite, but promotion committees nonetheless care as much about citation rates as baseball coaches care about batting averages. And now, just when I could really use some luck, these two
papers suddenly became my most-cited ones ever, one even grabbing the spot as the most-cited physics paper of 2004—that distinction didn’t last long, but long enough for the tenure decision. My dumb luck continued with the magazine Science deciding that the number-one “Breakthrough of the Year: 2003” was that cosmology had finally become believable, mentioning both the WMAP results and our Sloan Digital Sky Survey analysis.
