## Is the universe a doughnut?

http://www.cosmosmagazine.com/node/1566

Ιs the universe a doughnut?

What doesThe Simpsonshave to say about cosmology? Find out in this extract fromWhat’s Science Ever Done for Us?– an unauthorised and amusing look at science in the cult cartoon series. Someday spacecraft will be powerful enough perhaps to journey at extraordinary speeds, spanning the vast interstellar voids. Our technology might develop until we become a vast, powerful intergalactic society, capable of resolving the deepest quandaries ever known. Only then could we definitely answer what is perhaps the ultimate question: “Is the universe shaped like a doughnut?” This last question pertains to an idea attributed to Homer and mentioned by guest star Stephen Hawking in an episode ofThe Simpson’s. In the episode, Lisa Simpson joins Springfield’s chapter of the brainy organisation Mensa, which assumes mayoral duties and vows to remake Springfield into a perfect society. The prospect of experiencing a blossoming utopia attracts the attention of the British cosmologist Hawking, who – in his first animated appearance on the show – decides to visit and see it for himself. “We were looking for someone much smarter than all the Mensa members, and so we naturally thought of him,” says the cartoon’s executive producer Al Jean, explaining his decision to invite Hawking on the show. “He seemed pretty interested in coming on right away.” (Hawking is in fact a known Simpson’s fan and has a Lisa keyring dangling from his computer and a Homer clock on the wall in his University of Cambridge office.) In the episode, Hawking arrives just in time to see the ensuing mayhem, and escapes with Lisa in tow, using a flying device attached to his wheelchair. Later, he is seen telling Homer over a beer, “Your theory of a doughnut-shaped universe is intriguing… I may have to steal it.”

IN MATHEMATICS, A DOUGHNUT SHAPEis known as a torus, the three-dimensional generalisation of a ring. A ring lies in a single plane; so topologically speaking there is one closed path around it that lies just outside it (a loop around the ring). Because a torus has one more dimension, you can travel along closed paths around it in two perpendicular directions. If you imagine a doughnut on a plate, one of these is a larger loop around the periphery, parallel to the plate, and the other is a smaller loop through the hole, toward and away from the plate. The generalization of a torus, any closed curve spun in a circle around an axis, is called atoroid. Curiously, there are genuine scientific theories that the universe is toroidal. Modern cosmology is mathematically modelled through solutions to Einstein’s general theory of relativity. Recall that general relativity explains gravity through a mechanism in which matter curves the fabric of space and time. It is expressed in terms of an equation that relates the geometry of a region to its distribution of mass and energy. For example, an enormous star warps space-time much more, and therefore bends the paths of objects in its neighbourhood by a greater amount, than does a tiny satellite. Soon after general relativity was published, a number of theorists, including Einstein himself, delved for solutions that could describe the universe in general, not just the stars and other objects within it. The researchers discovered a plethora of diverse geometries and behaviours, each a distinct way of characterising the cosmos. Some of these models imagined space as resembling an unbounded plain or endless flat landscapes, only uniform in three directions, not just two. Two parallel straight lines, in such a spatial vista, would just keep going in the same direction indefinitely, like outback railroad tracks. Physicists call these flat cosmologies. Other solutions possess spaces that curve in a saddle shape, technically known as hyperbolic geometries with negative curvature. This curvature couldn’t be seen directly, unless you could somehow step out of three-dimensional space itself, but rather would make itself known through the behaviour of parallel lines and triangles. In a flat geometry (called Euclidean), if you draw a straight line and a point not on it, you can construct just one single line through that point parallel to the first line. For a saddle-shaped geometry, in contrast, there are an infinite number of parallel lines fanning out from that point, like the tracks out of a major city’s terminal train station. Moreover, while triangles in flat space have angles that add up to 180 degrees, in saddle-space the angles add up to less than 180 degrees. Yet another possibility, called positive curvature, resembles the spherical surface of an orange. Like the saddle-shape, its form could be seen only indirectly, through altered laws of geometry.

TO UNDERSTAND THESE DIFFERENCES, slice an orange in half widthwise and cut the top half into quarters. Pick up one of the slices and look at the skin, and you’ll notice that it is bounded by two edges that start out seemingly straight and parallel (where the widthwise slice was made) but end up meeting at the top. They are like any two lines of longitude on Earth, appearing parallel near the equator, but converging at the North Pole. This demonstration shows that no two lines on a positively curved surface are truly parallel. What about lines of latitude, or the equivalent produced by slicing an onion in repeated widthwise segments? They appear parallel enough, never meeting. Strangely enough, on the spherical surface of Earth, they aren’t true lines because they do not comprise the shortest distance between two points, technically known as geodesics. If you want to experience this yourself, purchase a ticket on a non-stop flight from Vancouver to Paris, both approximately the same latitude. Chances are, the flight will veer north, then south, rather than maintaining close latitude, because minimizing distance requires taking an orange-slice path – a geodesic – rather than an onion-slice path. It is these geodesics that must always meet somewhere, as seen in flight pattern maps. Because geometry, in general relativity, influences dynamics, the shape of the cosmos bears greatly on its destiny. The vast majority of astronomers believe that the universe, regardless of its shape, started off as an ultra-dense point of extremely compact, perhaps infinitesimal, size, called the Big Bang, and expanded to its enormous present-day size. The exact manner of this expansion, and where it will ultimately lead, is partly determined by what geometry the universe possesses. If spatial geometry were the only determinant, then by knowing if the universe has negative, zero, or positive curvature you could predict if it will expand forever (in the case of negative or zero curvature) or someday reverse its expansion and contract back down to a point (in the case of positive curvature). Geometry, however, cannot be the only influence on the dynamics of the universe. Another factor is an antigravity term, the cosmological constant, which was first suggested by Einstein. This term has come into prominence in recent years with the discovery by Adam Riess, Saul Perlmutter, Brian Schmidt, and their co-workers in various research teams that the universe is not just expanding, but is also currently speeding up in its expansion. This cosmic acceleration cannot be explained through geometry, but requires an additional outward boost, represented by the cosmological constant and known as dark energy. Models with a cosmological constant can have zero, negative, or positive curvatures, with the specific geometry affecting how and when the influence of the dark energy dominates the dynamics.

YOU MIGHT WONDER WHYin this discussion we have mentioned flat shapes, saddle shapes, and orange shapes, but not yet doughnut shapes. It turns out that there has been traditionally much greater interest in hyperplanes (generalizations of infinite, flat surfaces), hyperboloids (generalizations of saddle shapes), and hyperspheres (generalizations of orange shapes) than in toroidal, doughnut-shaped cosmologies. Why are orange-like shapes, for instance, more favoured in the literature than doughnuts? Actually, the bias in favour of hyperplanes, hyperboloids, and hyperspheres has more to do with their mathematical simplicity than anything else. They represent the most basic isotropic (appearing the same in all directions) three-dimensional surfaces, possessing the simplest topologies. Topology is different from geometry in that it concerns itself with how surfaces are connected, not their specific shapes and sizes. For example, topologically speaking, solid footballs, baseballs, basketballs, and even books about sports are all equivalent because they don’t have holes through them, and you could theoretically transform one into another (assuming they were elastic enough) without cutting. Doughnuts, coffee cups with handles, tires, and hollow frames each have single holes, and therefore share common topologies distinct from continuous objects. Even if they are stretched, the holes are still there. A flat two-dimensional planar geometry – a square, let’s say – can be transformed into a cylinder by identifying the far left side with the far right side, essentially gluing the two sides together. If an object travels far enough to the left, it ends up on the right. Something moving continuously to the left or right would experience the same region again and again in periodic fashion, like the animation loops common to cartoons from the 1960s and 1970s. Used to save time and effort, animation loops occur when the characters pass by the same background scenes again and again. For example, when Fred and Barney from The Flintstones drove down a road, they seemed to encounter the same array of rocks and trees over and over again. If you could explore a cylindrical universe, surviving somehow for tens of billions of years while travelling in what appeared to be a straight line, you’d have the same repetitive experience. Although you’d imagine that you’re ploughing directly ahead, you’d eventually circumnavigate space and pass the same array of galaxies once more in a topological déjà vu.

SPACE COULD BEeven more interconnected than that. Take a vertical cylinder and connect its upper and lower circles; what you get then is a torus. Now there are two perpendicular ways you can loop around the space: left-right and up-down. It’s a bit like the 1980s arcade mainstay, the game of Pac-Man and its variants. When the colourful moving blobs exit the maze through any border portal, they miraculously pop up on the other side. Show them the back door and they gleefully return through the front, begging for more quarters. An even more intricate arrangement links the extremes of all three spatial dimensions into a kind of “über-doughnut.” Imagine space as a colossal cube; these connections would equate to the left and right, top and bottom, and front and back faces. Such a layout, a generalization of the torus with a three-dimensional instead of a two-dimensional surface, would be hard to visualize. Paradoxically, it merges a straightforward “flat” geometry (in the sense that parallel straight lines remain straight and parallel) with a mindbogglingly complex topology. Picture living in a house in which an ascending stairway in your attic leads to your basement, your front window has a scenic view of your rear kitchen, and your next-door neighbours are yourself. If the pipes under your living room happen to leak, the water would trickle down through all the lower levels, return through the upper floors, and ruin your living room furniture. Because there’d be nothing coming in from the outside world, everything in your residence would need to be recycled. You’d never be able to leave, just make the rounds through its doors and rooms again and again. Such would be life in a toroidal abode—not recommended for the claustrophobic. Could the entire universe have such a topology? The most reliable current data on the shape and configuration of space stem from missions to measure the cosmic microwave background (CMB), the cooled-down relic radiation from the Big Bang. The universe began its life very small, very hot, and very mixed up. Particles of matter and energy were bound together in a sizzling gumbo. Then, approximately 380,000 years after the initial burst, the stew cooled enough for complete atoms (mostly hydrogen) to coagulate, leaving the leftover photons (particles of light) as a kind of broth. At the point of separation, known as recombination, the matter was in some places a bit lumpier than in others, making the energy broth slightly uneven in temperature. These minute temperature differences have persisted throughout the ages, while the expansion of the universe has cooled down the energy broth significantly. From thousands of degrees Kelvin (above absolute zero) it’s been reduced to a mere 2.73. Now it’s a frigid backdrop of radio waves distributed throughout the universe.

THE CMB WAS FIRST DISCOVEREDin the mid-1960s by the Bell Labs researchers Arno Penzias and Robert Wilson. While completing a radio wave survey, their horn shaped antenna picked up a strange hiss. After they reported the result to the physicist Robert Dicke of Princeton, he calculated its temperature and found that it matched the predictions of the Big Bang theory. This discovery confirmed the existence of an ultra-hot beginning to the universe. It was not precise enough, however, to reveal the fine details of the primordial distribution of matter and energy. A far more detailed examination of the CMB came in the early 1990s, thanks to the Nobel prize–winning work of John Mather and George Smoot. Using NASA’s Cosmic Background Explorer (COBE) satellite, Mather and his team of researchers mapped out the precise frequency distribution of the microwave background radiation and established, beyond a shadow of a doubt, that it matched precisely what would be expected for a once-fiery universe cooled down over billions of years. Smoot and his group discovered a mosaic of minute temperature fluctuations (called anisotropies) throughout the sky, pointing to subtle early differences in the densities of various regions of the cosmos. These fluctuations showed how in the nascent universe slightly denser “seeds” existed that would attract more and more mass and eventually grow into the hierarchical structures (stars, galaxies, clusters of galaxies, and so forth), that we observe today. The quest to map out the ripples in the CMB with greater and greater precision has continued throughout the past two decades. Uniquely, these provide a wealth of accessible information about the state of the cosmos many billions of years ago. It’s like a rare cuneiform tablet that, with improving translations, provides richer and richer insights into ancient history each time it’s read. In 2001, the Wilkinson Microwave Anisotropy Probe (WMAP) was launched, offering an extraordinarily detailed mapping out of the CMB. From these data, astronomers have assembled an ultrarefined snapshot of the matter and energy distribution of the early cosmos. This information has furnished critical resolution of many long-standing cosmological riddles. For example, in the decades before WMAP there was considerable disagreement as to the age of the universe since the Big Bang. WMAP pinned down the value to be approximately 13.7 billion years—a fantastic achievement in the history of scientific measurement.

WHAT OF THE SHAPE OF SPACE?WMAP says much about that, too. Astronomers have gleaned the specific geometry of the universe by examining how the brightest patches in the CMB are stretched out or compressed in angle compared to what you would expect for pure flatness. While positive curvature would stretch these spots to 1.5 degrees and negative curvature would compress them to 0.5 degrees, zero curvature (flat) leaves them at 1 degree across. The third case appears to be true, so, based on that litmus test, space seems indeed to be flat. In 1993, the U.C. Berkeley researchers Daniel Stevens, Donald Scott, and Joseph Silk proposed a way of sifting through CMB data to assess the topology of space as well. In their paper “Microwave Background Anisotropy in a Toroidal Universe,” they showed how a universe with a multiply connected, torus-like topology would force the radiation into certain detectable wave patterns. Because such patterns seemed to be absent from the COBE data, the researchers did not find support for a toroidal cosmos. Later work by Neil Cornish of Case Western University, David Spergel of Princeton University, and Glenn Starkman of the University of Maryland extended this technique to consider a wider range of possible topologies. Such a method has been applied to the WMAP results, examining the possibility that it could have a complex topology— not a toroid perhaps, but rather a dodecahedron (a bit like a soccer ball, but with all sides equivalent in size and shape). Although preliminary data (analysed in 2003) seemed to rule out this model, more recent looks at the WMAP findings have revived the idea that if you venture far enough out into space you’ll return to your starting point. Hence Homer’s doughnut theory may have at least a sprinkling of truth: the universe could indeed have loops. If the universe is truly loopy, what does it loop around? The two-dimensional surface of a sphere curves along a third dimension. Hence, fruits have cores as well as skins. What, then, would lie at the core of a looped three-dimensional cosmos? Could there be a higher spatial dimension beyond the limits of observation?

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