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the experimental frontier, tireless efforts are 21. J. H. Taylor, R. A. Hulse, L. A. Fowler, G. E. Gullahorn, atnf.csiro.au/research/cmbr/index.html); Cosmic Anisot- underway to detect CDM directly in under- J. M. Rankin, Astrophys. J. 206, L53 (1976). ropy Polarization MAPper (CAPMAP, http://wwwphy. 22. J. H. Taylor, L. A. Fowler, J. M. Weisberg, Nature 277, princeton.edu/cosmology/); Cosmic Anisotropy Tele-

ECTION ground low-background experiments; to pro- 437 (1979). scope (CAT, www.mrao.cam.ac.uk/telescopes/cat/

S duce CDM in particle accelerators; to make 23. M. Begelman, Science 300, 1898 (2003). index.html); Cosmic Background Imager (CBI, www. measurements to determine the nature of the 24. L. Stella, M. Vietri, Astrophys. J. 492, L59 (1998). astro.caltech.edu/Ètjp/CBI/); CosmicMicrowave Polar- 25. I. Ciufolini, E. C. Pavlis, Nature 431, 958 (2004). ization at SmallScales (COMPASS, http://cmb.physics. dark energy (its sound speed, equation of 26. G. Gamow, Phys. Rev. 70, 572 (1946). wisc.edu/compass.html); Degree Angular Scale Inter- state, and evolution); and to test inflation and 27. R. A. Alpher, H. Bethe, G. Gamow, Phys. Rev. 73, 803 ferometer (DASI, http://astro.uchicago.edu/dasi/);

PECIAL determine what happened in the earliest mo- (1948). Princeton I, Q, and U Experiment (PIQUE, http://

S 28. H. A. Bethe, Phys. Rev. 55, 103 (1939). wwwphy.princeton.edu/cosmology/); Polarization ments of our universe. 29. S. Perlmutter et al., Astrophys. J. 517, 565 (1999). Observations of Large Angular Regions (POLAR, 30. A. G. Riess et al., Astron. J. 116, 1009 (1998). http://cmb.physics.wisc.edu/polar/); Saskatoon/ References and Notes 31. R. A. Alpher, R. C. Herman, Phys. Rev. 75, 1089 (1949). Toco/Mobile Anisotropy Telescope (SK/TOCO/MAT, 1. A. Einstein, Ann. der Phys. 18, 639 (1905). 32. J. C. Mather, D. J. Fixsen, R. A. Shafer, C. Mosier, D. T. http://wwwphy.princeton.edu/cosmology/); Python 2. A. S. Eddington, Address to the Mathematical and Wilkinson, Astrophys. J. 512, 511 (1999). (http://astro.uchicago.edu/cara/research/cmbr/python. Physical Science Section (Section A) of the Royal Astro- 33. J. R. Bond, G. Efstathiou, Astrophys. J. 285, L45 (1984). html); Tenerife (www.iac.es/project/cmb/rad/index. nomical Society in Cardiff, and Nature 106, 14 (1920). 34. For examples of balloon-borne CMB experiments, see html); Very Small Array (VSA, www.mrao.cam.ac.uk/ 3. H. A. Bethe, Phys. Rev. 55, 103 (1939). the following Web sites: Far Infra-Red Survey (FIRS, telescopes/vsa/index.html); Jodrell Bank; Viper; Owens 4.W.A.Fowler,G.R.Burbidge,E.M.Burbidge, http://physics.princeton.edu/Ècmb/firs.html); Bal- Valley Radio Observatory (OVRO, www.astro.caltech. Astrophys. J. 122, 271 (1955). loon Observations Of Millimetric Extragalactic Radia- edu/Ètjp/OVRO-CMB/intrinsic.html). 5. A. Einstein, Ann. der Phys. 17, 132 (1905). tion and Geophysics (BOOMERanG, http://cmb. 36. The Legacy Archive for Microwave Background Data 6. A. Einstein, Ann. der Phys. 19, 289 (1906). phys.cwru.edu/boomerang/); Millimeter Anisotropy Analysis (LAMBDA) data center, available at http:// 7. A. Einstein, Phys. Z. 18, 121 (1917). eXperiment Imaging Array (MAXIMA, http://cosmology. lambda.gsfc.nasa.gov, has links to data and further 8. A. Einstein, Ann. der Phys. 17, 549 (1905). berkeley.edu/group/cmb/index.html); QMAP (http:// information about the CMB experiments. 9. A. Einstein, Ann. der Phys. 17, 891 (1905). physics.princeton.edu/Ècmb/qmap/qmap.html); 37. C. L. Bennett et al., Astrophys. J. 148, 1 (2003). 10. A. Einstein, Ann. der Phys. 49, 769 (1916). Medium Scale Anisotropy Measurement and TopHat 38. W. L. Freedman et al., Astrophys. J. 553, 47 (2001). 11. Letter from Isaac Newton to Robert Hooke, 5 (MSAM/TopHat, http://topweb.gsfc.nasa.gov/); 39. W. J. Percival et al., Mon. Not. R. Astron. Soc. 337, February 1675. Archeops (www.archeops.org/index_english.html); 1068 (2002). 12. E. B. Fomalont, R. A. Sramek, Phys. Rev. Lett. 36, Balloon-borne Anisotropy Measurement (BAM, http:// 40. M. Tegmark et al., Phys. Rev. D69, 103501 (2004). 1475 (1976). cmbr.physics.ubc.ca/bam/experimental.html); ARGO 41. P. J. E. Peebles, B. Ratra, Rev. Mod. Phys. 75, 559 13. I. I. Shapiro, Phys. Rev. Lett. 13, 789 (1964). (http://oberon.roma1.infn.it/argo/); Background Emis- (2003). 14. B. Bertotti, L. Iess, P. Tortora, Nature 425, 374 (2003). sion Anisotropy Scanning Telescope (BEAST/ACE, 42. P. Steinhardt, Philos. Trans. Math. Phys. Eng. Sci. 15. F. Zwicky, Phys. Rev. 51, 290 (1937). www.deepspace.ucsb.edu/research/Sphome.htm). 361, 2497 (2003). 16. F. Zwicky, Phys. Rev. 51, 679 (1937). 35. For examples of ground-based CMB experiments, see 43. Connecting Quarks with the Cosmos: Eleven Science 17. V. C. Rubin, N. Thonnard, W. K. Ford Jr., Astrophys. J. the following Web sites: Arcminute Cosmology Questions for the New Century (National Academies 225, L107 (1978). Bolometer Array Receiver (ACBAR, http://cosmology. Press, Washington, DC, 2003). 18. R. V. Pound, G. A. Rebka Jr., Phys. Rev. Lett. 4, 337 (1960). berkeley.edu/group/swlh/acbar/); Advanced Cosmic 44. I thank R. Weiss, E. L. Wright, and G. Hinshaw for 19. R. F. C. Vessot et al., Phys. Rev. Lett. 45, 2081 (1980). Microwave Explorer (ACME/HACME, www.deepspace. helpful comments that led to an improved paper. I am 20. A pulsar is a rapidly rotating neutron star that emits ucsb.edu/research/Sphome.htm); Antarctic Plateau also grateful to B. Griswold for expert graphics as- highly directional radiation. Pulses are observed as Anisotropy CHasing Experiment (APACHE, www. sistance. Support for this work was provided by NASA. the rotation periodically sweeps the beam across the bo.iasf.cnr.it/Èvalenziano/APACHE/apache.htm); sky toward the observer, similar to a lighthouse. Australia Telescope Compact Array (ATCA, www. 10.1126/science.1106444

REVIEW Inflationary Cosmology: Exploring the Universe from the Smallest to the Largest Scales

Alan H. Guth* and David I. Kaiser*

Understanding the behavior of the universe at large depends critically on insights blend of concepts from particle physics and about the smallest units of matter and their fundamental interactions. Inflationary gravitation. The last few years have been a cosmology is a highly successful framework for exploring these interconnections remarkably exciting time for cosmology, with between particle physics and gravitation. Inflation makes several predictions about new observations of unprecedented accuracy the present state of the universe—such as its overall shape, large-scale smoothness, yielding many surprises. Einstein_s legacy is and smaller scale structure—which are being tested to unprecedented accuracy by a flourishing in the early 21st century. new generation of astronomical measurements. The agreement between these Inflation was invented a quarter of a century predictions and the latest observations is extremely promising. Meanwhile, ago, and has become a central ingredient of physicists are busy trying to understand inflation’s ultimate implications for the current cosmological research. Describing dra- nature of matter, energy, and spacetime. matic events in the earliest history of our universe, inflationary models generically pre- The scientific community is celebrating the In the span of just a few months during 1905 dict that our universe today should have several International Year of Physics in 2005, Einstein introduced key notions that would distinct features—features that are currently honoring the centennial of Albert Einstein_s dramatically change our understanding of being tested by the new generation of high- most important year of scientific innovation. matter and energy as well as the nature of precision astronomical measurements. Even as space and time. The centennial of these inflation passes more and more stringent seminal developments offers an enticing empirical tests, theorists continue to explore Center for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics, Massachusetts opportunity to take stock of how scientists broader features and implications, such as what Institute of Technology, Cambridge, MA 02139, USA. think about these issues today. We focus in might have come before an inflationary epoch, *To whom correspondence should be addressed. particular on recent developments in the field how inflation might have ended within our E-mail: [email protected] (A.H.G.); [email protected] of inflationary cosmology, which draws on a observable universe, and how inflation might

884 11 FEBRUARY 2005 VOL 307 SCIENCE www.sciencemag.org E INSTEIN’S L EGACY S PECIAL arise in the context of our latest understanding to fundamental particle physics continues to tions or directions—Einstein’s gravitational of the structure of space, time, and matter. evolve. equations give a particularly simple result. The Particle theory has been changing rapidly, expansion of the universe may be described by and these theoretical developments have pro- Inflationary Basics introducing a time-dependent ‘‘scale factor,’’ S vided just as important a spur to inflationary According to inflationary cosmology (1–3), a(t), with the separation between any two ob- ECTION cosmology as have the new observations. Dur- the universe expanded exponentially quickly jects in the universe being proportional to a(t). ing the 1960s and 70s, particle physicists dis- for a fraction of a second very early in its Einstein’s equations prescribe how this scale covered that if they neglected gravity, they history—growing from a patch as small as factor will evolve over time, t. The rate of ac- could construct highly successful descriptions 10j26 m, one hundred billion times smaller celeration is proportional to the density of mass- of three out of the four basic forces in the than a proton, to macroscopic scales on the energy in the universe, r, plus three times its universe: electromagnetism and the strong and order of a meter, all within about 10j35 s— pressure, p: d2a/dt2 0 j4pG (r þ 3p)a/3, weak nuclear forces. The Bstandard model of before slowing down to the more stately rate where G is Newton’s gravitational constant particle physics,[ describing these three forces, of expansion that has characterized the uni- (and we use units for which the speed of light was formulated within the framework of quan- verse’s behavior ever since. The driving force c 0 1). The minus sign is important: Ordinary tum field theory, the physicist_s quantum- behind this dramatic growth, strangely enough, matter under ordinary circumstances has both mechanical description of subatomic matter. was gravity. [For technical introductions to in- positive mass-energy density and positive (or Inflationary cosmology was likewise first formu- flationary cosmology, see (4–6); a more popu- zero) pressure, so that (r þ 3p) 9 0. In this lated in terms of quantum field theory. Now, lar description may be found in (7).] Although case, gravity acts as we would expect it to: however, despite (or perhaps because of) the this might sound like hopeless (or, depending All of the matter in the universe tends to attract spectacular experimental success of the stan- all of the other matter, causing the expansion dard model, the major thrust of particle phys- of the universe as a whole to slow down. ics research is aimed at moving beyond it. The crucial idea behind inflation is that For all its successes, the standard model matter can behave rather differently from this says nothing at all about the fourth force: grav- familiar pattern. Ideas from particle physics ity. For more than 50 years physicists have suggest that the universe is permeated by sought ways to incorporate gravity within a scalar fields, such as the Higgs field of the quantum-mechanical framework, initially with standard model of particle physics, or its more no success. But for the past 25 or more years, exotic generalizations. (A scalar field takes an ever-growing group of theoretical physicists exactly one value at every point in space and has been pursuing superstring theory as the time. For example, one could measure the bright hope for solving this problem. To accom- Fig. 1. In simple inflationary models, the uni- temperature at every position in a room and plish this task, however, string theorists have verse at early times is dominated by the repeat the measurements over time, and repre- been forced to introduce many novel departures potential energy density of a scalar field, f. sent the measurements by a scalar field, T,of f from conventional ideas about fundamental Red arrows show the classical motion of . temperature. Electric and magnetic fields are When f is near region (a), the energy density forces and the nature of the universe. For one ; vector fields, which carry three distinct com- will remain nearly constant, r rf, even as the thing, string theory stipulates that the basic universe expands. Moreover, cosmic expansion ponents at every point in space and time: the units of matter are not pointlike particles (as acts like a frictional drag, slowing the motion fieldinthex direction, in the y direction, and treated by quantum field theory), but rather of f: Even near regions (b) and (d), f behaves in the z direction. Scalar fields are introduced one-dimensional extended objects, or strings. more like a marble moving in a bowl of molas- in particle physics to describe certain kinds of Moreover, in order to be mathematically self- ses, slowly creeping down the side of its particles, just as photons are described in quan- potential, rather than like a marble sliding down consistent, string theories require the existence the inside of a polished bowl. During this period tum field theories in terms of electromagnetic of several additional spatial dimensions. Where- of ‘‘slow roll,’’ r remains nearly constant. Only fields.) These scalar fields can exist in a spe- as our observable universe seems to contain after f has slid most of the way down its cial state, having a high energy density that one timelike dimension and three spatial potential will it begin to oscillate around its cannot be rapidly lowered, such as the arrange- dimensions—height, width, and depth—string minimum, as in region (c), ending inflation. ment labeled (a) in Fig. 1. Such a state is called theory postulates that our universe actually a ‘‘false vacuum.’’ Particle physicists use the contains at least six additional spatial dimen- on one’s inclinations, interesting) speculation, word ‘‘vacuum’’ to denote the state of lowest sions, each at right angles to the others and in fact inflationary cosmology leads to several energy. ‘‘False vacua’’ are only metastable, yet somehow hidden from view. quantitative predictions about the present be- not the true states of lowest possible energy. For measurements at low energies, string havior of our universe—predictions that are In the early universe, a scalar field in such theory should behave effectively like a quan- being tested to unprecedented accuracy by a a false vacuum state can dominate all the con- tum field theory, reproducing the successes new generation of observational techniques. tributions to the total mass-energy density, r. of the standard model of particle physics. So far the agreement has been excellent. During this period, r remains nearly constant, Yet the interface between cosmology and How could gravity drive the universal even as the volume of the universe expands ; 0 string theory has been a lively frontier. For repulsion during inflation? The key to this rapidly: r rf constant. This is quite different example, some theorists have been con- rapid expansion is that in Einstein’s general from the density of ordinary matter, which structing inflationary models for our uni- relativity (physicists’ reigning description of decreases when the volume of its container verse that make use of the extra dimensions gravity), the gravitational field couples both increases. Moreover, the first law of thermo- that string theory introduces. Others have to mass-energy (where mass and energy are dynamics, in the context of general relativity, 0 2 ; been studying the string theory underpin- interchangeable thanks to Einstein’s E mc ) implies that if r rf while the universe ex- nings for inflationary models, exploring such and to pressure, rather than to mass alone. In pands, then the equation of state for this spe- ;j topics as the nature of vacuum states and the the simplest scenario, in which at least the cial state of matter must be p rf, a negative 2 2 0 question of their uniqueness. As we will see, observable portion of our universe can be pressure. This yields d a/dt 8pGrf a/3: inflation continues to occupy a central place approximated as being homogeneous and Rather than slowing down, the cosmic expan- in cosmological research, even as its relation isotropic—that is, having no preferred loca- sion rate will grow rapidly, driven by the neg-

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ative pressure created by this special state of for this ratio, inflation predicts that W 0 1 Before that time, the ambient temperature matter. Under these circumstances, the scale within our observable universe to extremely of matter in the universe was so high that º Ht ECTION factor grows as a e , where the Hubble high accuracy. Until recently, uncertainties in would-be atoms were broken up by high- j1 S parameter, H K a da/dt, which measures the measurement of W allowed any value in energy photons as soon as they formed, so the universe’s rate of expansion, assumes the the wide range, 0.1 e W e 2, with many that the photons were effectively trapped, ; 1/2 ; constant value, H (8pGrf/3) . The uni- observations seeming to favor W 0.3. A new constantly colliding into electrically charged verse expands exponentially until the scalar generation of detectors, however, has dra- matter. Since stable atoms formed, however, the

PECIAL field rolls to near the bottom of the hill in the matically changed the situation. The latest CMB photons have been streaming freely. S potential energy diagram. observations, combining data from the Wilkin- Their temperature today is terrifically uniform: What supplies the energy for this gigantic son Microwave Anisotropy Probe (WMAP), the After adjusting for the Earth’s motion, CMB expansion? The answer, surprisingly, is that no SloanDigitalSkySurvey(SDSS),andobser- photons measured from any direction in the energy is needed (7). Physicists have known vations of type Ia supernovae, have measured sky have the same temperature to one part in 0 þ0.018 5 since the 1930s (8) that the gravitational field W 1.012j0.022 (10)—an amazing match 10 (12). carries negative potential en- between prediction and Without inflation, this large-scale smooth- ergy density. As vast quan- observation. ness appears quite puzzling. According to or- tities of matter are produced In fact, inflation offers dinary (noninflationary) big bang cosmology, during inflation, a vast a simple explanation for these photons should never have had a chance amount of negative poten- why the universe should to come to thermal equilibrium: The regions tial energy materializes in be so flat today. In the in the sky from which they were released the gravitational field that standard big bang cosmol- would have been about 100 times farther fills the ever-enlarging re- ogy (without inflation), apart than even light could have traveled gion of space. The total W 0 1isanunstable solu- between the time of the big bang and the energy remains constant, tion: If W were ever-so- time of the photons’ release (1, 4–6). Much and very small, and possi- slightly less than 1 at an like the flatness problem, inflation provides a bly exactly equal to zero. early time, then it would simple and generic reason for the observed There are now dozens rapidly slide toward 0. For homogeneity of the CMB: Today’s observable of models that lead to this example, if W were 0.9 at universe originated from a much smaller generic inflationary behav- 1 s after the big bang, it region than that in the noninflationary scenar- ior, featuring an equation of would be only 10j14 to- ios. This much-smaller patch could easily state, p ;jr, during the day. If W were 1.1 at t 0 have become smooth before inflation began. early universe (4, 9). This 1s,thenitwouldhave Inflation would then stretch this small homo- entire family of models, grown so quickly that the geneous region to encompass the entire moreover, leads to several universe would have re- observable universe. main predictions about to- collapsed just 45 s later. A third major prediction of inflationary day’s universe. First, our In inflationary models, on cosmology is that there should be tiny depar- observable universe should the other hand, any orig- tures from this strict large-scale smoothness be spatially flat. Einstein’s inal curvature of the ear- and that these ripples (or ‘‘perturbations’’) general relativity allows for ly universe would have should have a characteristic spectrum. Today all kinds of curved (or ‘‘non- been stretched out to near- these ripples can be seen directly as fluctua- Euclidean’’) spacetimes. flatness as the universe un- tions in the CMB. Although the ripples are Homogeneous and isotropic derwent its rapid expansion believed to be responsible for the grandest spacetimes fall into three (Fig. 3). Quantitatively, structures of the universe—galaxies, super- classes (Fig. 2), depending kWj 1k º 1/(aH)2,sothat clusters, and giant voids—in inflationary mod- on the value of the mass- Fig. 2. According to general rela- while H ; constant and els they arise from quantum fluctuations, 9 tivity, spacetime may be warped º Ht energy density, r.Ifr rc, a e during the infla- usually important only on atomic scales or K 2 or curved, depending on the den- where rc 3H /(8pG), then tionary epoch, W gets driv- smaller. The field f that drives inflation, like Einstein’s equations imply sity of mass-energy. Inflation pre- en rapidly to 1. all quantum fields, undergoes quantum fluctua- dicts that our observable universe that the spacetime will be should be spatially flat to very The second main pre- tions in accord with the Heisenberg uncertainty positively curved, or closed high accuracy. diction of inflation is that principle. During inflation these quantum (akin to the two-dimensional the presently observed uni- fluctuations are stretched proportionally to surface of a sphere); paral- verse should be remark- a(t), rapidly growing to macroscopic scales. lel lines will intersect, and the interior angles ably smooth and homogeneous on the largest The result: a set of nearly scale-invariant of a triangle will always add up to more than astronomical scales. This, too, has been mea- perturbations extending over a huge range of - G 180 .Ifr rc, the spacetime will be sured to extraordinary accuracy during the past wavelengths (13). Cosmologists parameter- negatively curved, or open (akin to the two- decades. Starting in the 1960s, Earth-bound, ize the spectrum of primordial perturbations

dimensional surface of a saddle); parallel balloon-borne, and now satellite detectors have by a spectral index, ns. A scale-invariant 0 lines will diverge and triangles will sum to measured the cosmic microwave background spectrum would have ns 1.00; inflationary - 0 0 less than 180 . Only if r rc will spacetime (CMB) radiation, a thermal bath of photons models generically predict ns 1 to within be spatially flat (akin to an ordinary two- that fills the sky. The photons today have a È10%. The latest measurements of these dimensional flat surface); in this case, all of frequency that corresponds to a temperature perturbations by WMAP and SDSS reveal 0 þ0.039 the usual rules of Euclidean geometry apply. of 2.728 K (11). These photons were released ns 0.977j0.025 (10). Cosmologists use the letter W to designate È400,000 years after the inflationary epoch, Until recently, astronomers were aware of the ratio of the actual mass-energy density in when the universe had cooled to a low several cosmological models that were con- K the universe to this critical value: W r/rc. enough temperature that would allow stable sistent with the known data: an open uni- Although general relativity allows any value (and electrically neutral) atoms to form. verse, with W ; 0.3; an inflationary universe

886 11 FEBRUARY 2005 VOL 307 SCIENCE www.sciencemag.org E INSTEIN’S L EGACY S PECIAL with considerable dark energy (L); an in- energy rather than down it. Over a time period consequences for the subsequent history of our flationary universe without L; and a universe Hj1, this region will grow e3 ; 20 times its universe. For one thing, the colossal expansion in which the primordial perturbations arose original size. If the probability that the field during inflation causes the temperature of the from topological defects such as cosmic will roll up the potential hill during this period universe to plummet nearly to zero, and dilutes S strings. Dark energy (14) is a form of matter is greater than 1/20, then on average the the density of ordinary matter to negligible ECTION 9 with negative pressure that is currently volume of space in which r r0 increases quantities. Some mechanism must therefore believed to contribute about 70% of the total with time (4, 21, 22). The probability of up- convert the energy of the scalar field, f,intoa energy of the observed universe. Cosmic ward fluctuations tends to become large when hot soup of garden-variety matter. strings are long, narrow filaments hypothesized the initial value of f is near the peak at (a) or In most models, inflation ends when f to be scattered throughout space, remnants of a high on the hill near (d), so for most potential oscillates around the minimum of its potential, symmetry-breaking phase transition in the energy functions the condition for eternal as in region (c) of Fig. 1. Quantum-mechanically, early universe (15, 16). [Cosmic strings are inflation is attainable. In that case the volume these field oscillations correspond to a col- topologically nontrivial configurations of fields, of the inflating region grows exponentially, lection of f particles approximately at rest. which should not be confused with the fun- and forever: Inflation would produce an in- Early studies of postinflation ‘‘reheating’’ damental strings of superstring theory. The finity of pocket universes. assumed that individual f particles would latter are usually believed to have lengths An interesting question is whether or not decay during these oscillations like radio- on the order of 10j35 m, although for some eternal inflation makes the big bang unneces- active nuclei. More recently, it has been compactifications these strings discovered (24–28) that these os- might also have astronomical cillations would drive resonances lengths (17).] Each of these models in f’s interactions with other quan- leads to a distinctive pattern of tum fields. Instead of individual f resonant oscillations in the early particles decaying independently, universe, which can be probed these resonances would set up col- today through its imprint on the lective behavior—f would release CMB. As can be seen in Fig. 4 its energy more like a laser than an (18), three of the models are now ordinary light bulb, pouring it ex- definitively ruled out. The full class tremely rapidly into a sea of newly of inflationary models can make a created particles. Large numbers of variety of predictions, but the pre- particles would be created very diction of the simplest inflationary quickly within specific energy-bands, models with large L, shown on the corresponding to the frequency of f’s graph, fits the data beautifully. oscillations and its higher harmonics. This dramatic burst of particle Before and After Inflation creation would affect spacetime Research in recent years has in- itself, as it responded to changes cluded investigations of what in the arrangement of matter and might have preceded inflation and energy. The rapid transfer of ener- how an inflationary epoch might gy would excite gravitational per- have ended. turbations, of which the most Soon after the first inflationary strongly amplified would be those models were introduced, several with frequencies within the reso- physicists (19–21) realized that nance bands of the decaying f once inflation began, it would in Fig. 3. (A to D) The expanding sphere illustrates the solution to the field. In some extreme cases, very all likelihood never stop. Regions flatness problem in inflationary cosmology. As the sphere becomes long-wavelength perturbations can of space would stop inflating, larger, its surface becomes more and more flat. In the same way, the be amplified during reheating, forming what can be called ‘‘pocket exponential expansion of spacetime during inflation causes it to which could in principle even universes,’’ one of which would become spatially flat. leave an imprint on the CMB (29). contain the observed universe. Nonetheless, at any given moment some sary: Might eternal inflation have been truly Brane Cosmology: Sticking Close to portion of the universe would still be under- ‘‘eternal,’’ existing more or less the same way Home going exponential expansion, in a process for all time, or is it only ‘‘eternal’’ to the fu- Although superstring theory promises to called ‘‘eternal inflation.’’ In the model ture once it gets started? Borde and Vilenkin synthesize general relativity with the other depicted in Fig. 1, for example, quantum- have analyzed this question (most recently, fundamental forces of nature, it introduces a mechanical effects compete with the classical with Guth), and have concluded that eternal number of surprising features—such as the motion to produce eternal inflation. Consider a inflation could not have been past-eternal: existence of microscopic strings, rather than region of size Hj1, in which the average value Using kinematic arguments, they showed particles, as the fundamental units of matter, of f is near (b) or (d) in the diagram. Call the (23) that the inflating region must have had along with the existence of several extra average energy density r0. Whereas the clas- a past boundary, before which some alterna- spatial dimensions in the universe. Could our sical tendency of f is to roll slowly downward tive description must have applied. One observable universe really be built from such (red arrow) toward the minimum of its po- possibility would be the creation of the a bizarre collection of ingredients? tential, the field will also be subject to universe by some kind of quantum process. Naı¨vely, one might expect the extra di- quantum fluctuations (green arrows) similar Another major area of research centers on mensions to conflict with the observed behav- to those described above. The quantum the mechanisms by which inflation might have ior of gravity. To be successful, string theory, fluctuations will give the field a certain like- ended within our observable universe. The like general relativity, must reduce to New- lihood of hopping up the wall of potential means by which inflation ends have major ton’s law of gravity in the appropriate limit. In

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Newton’s formulation, gravity can be de- presence of extra dimensions. Gravitational these early moments the departures from the scribed by force lines that always begin and force lines would tend to ‘‘hug’’ the brane, ordinary Einsteinian case can be dramatic. In 2

ECTION end on masses. If the force lines could spread rather than spill out into the ‘‘bulk’’—the particular, the r term would allow inflation

S in n spatial dimensions, then at a radius r spatial volume in which our brane is em- to occur at lower energies than are usually from the center, they would intersect a bedded. Along the brane, therefore, the assumed in ordinary (nonembedded) models, hypersphere with surface area proportional dominant behavior of the gravitational force with potential energy functions that are less to rnj1. An equal number of force lines would still be 1/r2. flat than are ordinarily needed to sustain

PECIAL would cross the hypersphere at each radius, In simple models, in which the spacetime inflation. Moreover, the spectrum of primor- S which means that the density of force lines geometry along our brane is highly symmet- dial perturbations would get driven even would be proportional to 1/rnj1. For n 0 3, ric, such as the Minkowski spacetime of closer to the scale-invariant shape, with 0 this reproduces the familiar Newtonian force special relativity, the effective gravitational ns 1.00 (39, 40). Brane cosmology thus law, F º 1/r2, which has been tested (along field along our brane is found to mimic the leads to some interesting effects during the with its Einsteinian generalization) to re- usual Einsteinian results to high accuracy early universe, making inflation even more markable accuracy over a huge range of (36, 37). At very short distances there are robust than in ordinary scenarios. distances, from astronomical scales down to calculable (and testable) deviations from less than a millimeter (30, 31). standard gravity, and there may also be de- String Cosmology An early response to this difficulty was to viations for very strong gravitational fields, Although theories of extra dimensions estab- assume that the extra spatial dimensions are such as those near black holes. There are also lish a connection between string theory and curled up into tiny closed circles cosmology, the developments of rather than extending to macro- the past few years have pushed scopic distances. Because gravity the connection much further. [For has a natural scale, known as the reviews, see (41–43).] The union K I 3 1/2 ; Planck length, lP ( G/c ) of string theory and cosmology is 10j35 m(whereI is Planck’s barely past its honeymoon, but so constant divided by 2p), physi- far the marriage appears to be a

cists assumed that lP sets the scale happy one. Inflation, from its in- for these extra dimensions. Just as ception, was a phenomenologically the surface of a soda straw would very successful idea that has been appear one-dimensional when in need of a fundamental theory viewed from a large distance— to constrain its many variations. even though it is really two- String theory, from its inception, dimensional—our space would has been a very well-constrained appear three-dimensional if the mathematical theory in need of a extra dimensions were ‘‘compac- phenomenology to provide con- tified’’ in this way. On scales tact with observation. The match much larger than the radii of the seems perfect, but time will be

extra dimensions, rc,wewould needed before we know for sure fail to notice them: The strength whether either marriage partner of gravity would fall off in its can fulfill the needs of the other. usual 1/r2 manner for distances In the meantime, ideas are stirring d r rc, but would fall off as that have the potential to radically nj1 ¡ 1/r for scales r rc (32). alter our ideas about fundamental The question remained, how- laws of physics. ever, what caused this compac- Fig. 4. Comparison of the latest observational measurements of the For many years the possibility tification, and why this special temperature fluctuations in the CMB with several theoretical models, as of describing inflation in terms of behavior affected only some described in the text. The temperature pattern on the sky is expanded in string theory seemed completely but not all dimensions. multipoles (i.e., spherical harmonics), and the intensity is plotted as a intractable, because the only string Recently, Arkani-Hamed, function of the multipole number l. Roughly speaking, each multipole l vacua that were understood were - l Dimopoulos, and Dvali (33) real- corresponds to ripples with an angular wavelength of 360 / . highly supersymmetric ones, with ized that there is no necessary many massless scalar fields, called l relation between P and rc, and that ex- modifications to the cosmological predictions moduli, which have potential energy func- e periments only require rc 1 mm. Shortly of gravity. In the usual case, when Einstein’s tions that vanish identically to all orders of afterward, Randall and Sundrum (34, 35) equations are applied to a homogeneous and perturbation theory. When the effects of discovered that the extra dimensions could isotropic spacetime, one finds H2 º r j k/a2, gravity are included, the energy density of even be infinite in extent! In the Randall- where k is a constant connected to the such supersymmetric states is never posi- Sundrum model, our observable universe lies curvature of the universe. If instead we lived tive. Inflation, on the other hand, requires on a membrane, or ‘‘brane’’ for short, of on a brane embedded within one large extra a positive energy density, and it requires a three spatial dimensions, embedded within dimension, then H2 º r þ ar2 j k/a2,where hill in the potential energy function. Infla- some larger multidimensional space. The a is a constant (38). tion, therefore, could only be contemplated in key insight is that the energy carried by the Under ordinary conditions, r decreases as the context of nonperturbative supersymmetry- brane will sharply affect the way the grav- the universe expands, and so the new term in breaking effects, of which there was very itational field behaves. For certain space- the effective Einstein equations should have little understanding. time configurations, the behavior of gravity minimal effects at late times in our observ- The situation changed dramatically with along the brane can appear four-dimensional able universe. But we saw above that during the realization that string theory contains not (three space and one time), even in the an inflationary epoch, r ; constant, and in only strings, but also branes, and fluxes,

888 11 FEBRUARY 2005 VOL 307 SCIENCE www.sciencemag.org E INSTEIN’S L EGACY S PECIAL which can be thought of as higher-dimensional strings would be produced (17). These could of vacuum for our universe; instead, the generalizations of magnetic fields. The com- be fundamental strings, or branes with one observable universe is viewed as a tiny speck bination of these two ingredients makes it spatial dimension. The CMB data of Fig. 4 within a multiverse that contains every possible to construct string theory states that rule out the possibility that these strings are possible type of vacuum. If this point of S break supersymmetry and that give non- major sources of density fluctuations, but view is right, then a quantity such as the ECTION trivial potential energy functions to all the they are still allowed if they are light enough electron-to-proton mass ratio would be on scalar fields. so that they do not disturb the density the same footing as the distance between our One very attractive idea for incorporating fluctuations from inflation. String theorists planet and the sun. Neither is fixed by the inflation into string theory is to use the are hoping that such strings may be able to fundamental laws, but instead both are positions of branes to play the role of the provide an observational window on string determined by historical accidents, restricted scalar field that drives inflation. The earliest physics. only by the fact that if these quantities did version of this theory was proposed in 1998 A key feature of the constructions of not lie within a suitable range, we would by Dvali and Tye (44), shortly after the inflating states or vacuumlike states in string not be here to make the observations. This possibility of large extra dimensions was theory is that they are far from unique. The idea—that the laws of physics that we proposed in (33). In the Dvali-Tye model, number might be something like 10500 observe are determined not by fundamental the observed universe is described not by a (48–50), forming what Susskind has dubbed principles, but instead by the requirement single three-dimensional brane, but instead the ‘‘landscape of string theory.’’ Although that intelligent life can exist to observe by a number of three-dimensional branes, the rules of string theory are unique, the low- them—is often called the anthropic princi- which in the vacuum state would sit on top energy laws that describe the physics that we ple. Although in some contexts this princi- of each other. If some of the branes were can in practice observe would depend ple might sound patently religious, the displaced, however, in a fourth spatial strongly on which vacuum state our universe combination of inflationary cosmology direction, then the energy would be in- was built upon. Other vacuum states could and the landscape of string theory gives creased. The brane separation would be a give rise to different values of ‘‘fundamen- the anthropic principle a scientifically viable function of time and the three spatial tal’’ constants, or even to altogether different framework. coordinates along the branes, and so from types of ‘‘elementary’’ particles, and even A key reason why the anthropic approach the point of view of an observer on the different numbers of large spatial dimen- is gaining attention is the observed fact that brane, it would act like a scalar field that sions! Furthermore, because inflation is the expansion of the universe today is could drive inflation. At this stage, however, generically eternal, one would expect that accelerating, rather than slowing down under the authors needed to invoke unknown the resulting eternally inflating spacetime the influence of normal gravity. In the con- mechanisms to break supersymmetry and to would sample every one of these states, each text of general relativity, this requires that give the moduli fields nonzero potential an infinite number of times. Because all of the energy of the observable universe is energy functions. these states are possible, the important dominated by dark energy. The identity of In 2003, Kachru, Kallosh, Linde, and problem is to learn which states are probable. the dark energy is unknown, but the simplest Trivedi (45) showed how to construct com- This problem involves comparison of one possibility is that it is the energy density of plicated string theory states for which all infinity with another, which is in general not the vacuum, which is equivalent to what the moduli have nontrivial potentials, for a well-defined question (51). Proposals have Einstein called the cosmological constant. which the energy density is positive, and for been made and arguments have been given To particle physicists it is not surprising that which the approximations that were used in to justify them (52), but no conclusive solu- the vacuum has nonzero energy density, the calculations appeared justifiable. These tion to this problem has been found. because the vacuum is known to be a states are only metastable, but their lifetimes What, then, determined the vacuum state very complicated state, in which particle- can be vastly longer than the 14 billion years for our observable universe? Although many antiparticle pairs are constantly materializing that have elapsed since the big bang. There physicists (including the authors) hope that and disappearing, and fields such as the was nothing elegant about this construction— some principle can be found to under- electromagnetic field are constantly under- the six extra dimensions implied by string stand how this choice was determined, there going wild fluctuations. From the standpoint theory are curled not into circles, but into are no persuasive ideas about what form of the particle physicist, the shocking thing is complicated manifolds with a number of such a principle might take. It is possible that the energy density of the vacuum is so internal loops that can be threaded by several that inflation helps to control the choice of low. No one really knows how to calculate different types of flux, and populated by a state, because perhaps one state or a subset the energy density of the vacuum, but naı¨ve hodgepodge of branes. Joined by Maldacena of states expands faster than any others. estimates lead to numbers that are about and McAllister, this group (46) went on to Because inflation is generically eternal, the 10120 times larger than the observational construct states that can describe inflation, in state that inflates the fastest, along with the upper limit. There are both positive and which a parameter corresponding to a brane states that it decays into, might dominate negative contributions, but physicists have position can roll down a hill in its potential over any others by an infinite amount. Pro- been trying for decades to find some reason energy diagram. Generically the potential gress in implementing this idea, however, why the positive and negative contributions energy function is not flat enough for success- has so far been nil, in part because we cannot should cancel, so far to no avail. It seems ful inflation, but the authors argued that the identify the state that inflates the fastest, and even more hopeless to find a reason why number of possible constructions was so large in part because we cannot calculate proba- the net energy density should be nonzero, that there may well be a large class of states bilities in any case. If we could calculate the but 120 orders of magnitude smaller than for which sufficient inflation is achieved. decay chain of the most rapidly inflating its expected value. However, if one adopts Iizuka and Trivedi (47) showed that successful state, we would have no guarantee that the the anthropic point of view, it was argued inflation can be attained by curling the extra number of states with significant probability as early as 1987 by Weinberg (53) that an dimensions into a manifold that has a special would be much smaller than the total number explanation is at hand: If the multiverse kind of symmetry. of possible states. contained regions with all conceivable val- A tantalizing feature of these models is Another possibility, now widely dis- ues of the cosmological constant, galaxies that at the end of inflation, a network of cussed, is that nothing determines the choice and hence life could appear only in those

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very rare regions where the value is small, Meanwhile, several major puzzles persist. 0.05, and H 0 70 km sj1 Mpcj1, with the remaining parameters fixed as follows. ‘‘Inflation with L’’: WDM because otherwise the huge gravitational Now that physicists and astronomers are 0 0 0 (dark matter) 0.23, WL 0.72, and optical depth ECTION repulsion would blow matter apart without confident that W 1 to high accuracy, the parameter t 0 0.17; ‘‘Inflation without L’’: WDM 0

S 0 0 0 allowing it to collect into galaxies. question remains of just what type of matter 0.95, WL 0, t 0.06; ‘‘Open universe’’: WDM 0.25, W 0 0, t 0 0.06. The 1-SD error bars include both The landscape of string theory and the and energy is filling the universe. Ordinary L observational uncertainty and ‘‘cosmic variance,’’ the evolution of the universe through the land- matter, such as the protons, neutrons, and intrinsic quantum uncertainty in the predictions, as scape are of course still not well understood, electrons that make up atoms, contributes calculated from the ‘‘Inflation with L’’ model. With

PECIAL and some have argued (54) that the land- just 4% to this cosmic balance. Nearly one- our current ignorance of the underlying physics,

S none of these theories predicts the overall amplitude scape might not even exist. It seems too quarter of the universe’s mass-energy is of the fluctuations; the ‘‘Inflation with L’’ curve was early to draw any firm conclusions, but some form of ‘‘dark matter’’—different in normalized for a best fit, and the others were clearly the question of whether the laws of kind from the garden-variety matter we see normalized arbitrarily. 19. P. J. Steinhardt, in The Very Early Universe,G.W. physics are uniquely determined, or whether around us, and yet exerting a measurable Gibbons,S.W.Hawking,S.T.C.Siklos,Eds.(Cambridge they are environmental accidents, is an issue gravitational tug that shapes the way gal- Univ. Press, New York, 1983), p. 251. too fundamental to ignore. axies behave. Particle physicists have offered 20. A. Vilenkin, Phys. Rev. D 27, 2848 (1983). 21. A. D. Linde, Phys. Lett. B 175, 395 (1986). many candidates for this exotic dark matter, 22. For a review, see A. H. Guth, Phys. Rep. 333-334, Conclusions but to date no single contender has proved 555 (2000). During the past decade, cosmology has fully convincing (55). 23. A. Borde, A. H. Guth, A. Vilenkin, Phys. Rev. Lett. 90, unquestionably entered the domain of high- Even more bizarre is the dark energy now 151301 (2003). 24. L. A. Kofman, A. D. Linde, A. A. Starobinsky, Phys. precision science. Just a few years ago known to contribute about 70% to W.This Rev. Lett. 73, 3195 (1994). several basic cosmological quantities, such dark energy is driving a mini-inflationary 25. Y. Shtanov, J. H. Traschen, R. H. Brandenberger, Phys. as the expansion parameter, H,andthe epoch today, billions of years after the initial Rev. D 51, 5438 (1995). 26. D. Boyanovsky et al., Phys. Rev. D 51, 4419 (1995). flatness parameter, W, were known only to round of inflation. Today’s accelerated ex- 27. D. I. Kaiser, Phys. Rev. D 53, 1776 (1996). within a factor of 2. Now new observations pansion is far less fast than the earlier 28. For a review, see L. A. Kofman, in COSMO-97: using WMAP, SDSS, and the high-redshift inflationary rate had been, but the question Proceedings,L.Roszkowski,Ed.(WorldScientific, Singapore, 1998), pp. 312–321. type Ia supernovae measure these and other remains why it is happening at all. Could the 29. B. A. Bassett, D. I. Kaiser, R. Maartens, Phys. Lett. B crucial quantities with percent-level accura- dark energy be an example of Einstein’s 455, 84 (1999). cy. Several of inflation’s most basic quan- cosmological constant? Or maybe it is a 30. Clifford Will, Theory and Experiment in Gravitational 0 ; Physics (Cambridge Univ. Press, New York, ed. 2, 1993). titative predictions, including W 1 and ns variation on an inflationary theme: Perhaps 31. C. D. Hoyle et al., Phys. Rev. D 70, 042004 (2004). 1, may now be compared with data that are some scalar field has been sliding down its 32. See, for example, J. Polchinski, String Theory (Cam- discriminating enough to distinguish in- potential energy hill on a time scale of bridge Univ. Press, New York, 1998), vol. 1. 33. N. Arkani-Hamed, S. Dimopoulos, G. Dvali, Phys. Lett. flation from many of its theoretical rivals. billions of years rather than fractions of a B 429, 263 (1998). So far, every measure has been favorable to second (14). Whatever its origin, dark 34. L. Randall, R. Sundrum, Phys. Rev. Lett. 83,4690 inflation. energy, much like dark matter, presents a (1999). Even with the evidence in favor of fascinating puzzle that will keep cosmolo- 35. For a review, see L. Randall, Science 296, 1422 (2002). 36. J. Garriga, T. Tanaka, Phys. Rev. Lett. 84, 2778 (2000). inflation now stronger than ever, much work gists busy for years to come. 37. S. B. Giddings, E. Katz, L. Randall, J. High Energy Phys. remains. Inflationary cosmology has always 3, 23 (2000). been a framework for studying the inter- References and Notes 38. P. Binetruy, C. Deffayet, D. Langlois, Nucl. Phys. B 1. A. H. Guth, Phys. Rev. D 23, 347 (1981). 565, 269 (2000). connections between particle physics and 2. A. D. Linde, Phys. Lett. B 108, 389 (1982). 39. R. Maartens, D. Wands, B. A. Bassett, I. P. C. Heard, gravitation—a collection of models rather 3. A. Albrecht, P. J. Steinhardt, Phys. Rev. Lett. 48, 1220 Phys. Rev. D 62, 041301 (2000). than a unique theory. The next generation (1982). 40. For a review, see D. Langlois, http://arxiv.org/abs/gr- qc/0410129. of astronomical detectors should be able to 4. A. D. Linde, Particle Physics and Inflationary Cosmol- ogy (Harwood, Philadelphia, 1990). 41. F. Quevedo, Class. Quant. Grav. 19, 5721 (2002). distinguish between competing inflationary 5. E. W. Kolb, M. S. Turner, The Early Universe (Addison- 42. A. D. Linde, http://arxiv.org/abs/hep-th/0402051. models, whittling down the large number of Wesley, Reading, MA, 1990). 43. C. P. Burgess, Pramana 63, 1269 (2004). options to a preferred few. One important 6. A. R. Liddle, D. H. Lyth, Cosmological Inflation and 44. G. R. Dvali, S.-H. H. Tye, Phys. Lett. B 450, 72 (1999). Large-Scale Structure (Cambridge Univ. Press, New 45. S. Kachru, R. Kallosh, A. Linde, S. P. Trivedi, Phys. Rev. goal is the high-precision measurement of York, 2000). D 68, 046005 (2003). polarization effects in the CMB, which 7. A. H. Guth, The Inflationary Universe: The Quest for a 46. S. Kachru et al., J. Cosmol. Astropart. Phys. 0310, allows the possibility of uncovering the New Theory of Cosmic Origins (Addison-Wesley, 013 (2003). Reading, MA, 1997). 47.N.Iizuka,S.P.Trivedi,Phys. Rev. D 70, 043519 traces of gravity waves originating from 8. R. C. Tolman, Phys. Rev. 39, 320 (1932). (2004). inflation. Gravity waves of the right pattern 9. D. H. Lyth, A. Riotto, Phys. Rep. 314, 1 (1999). 48. R. Bousso, J. Polchinski, J. High Energy Phys. 0006, would be a striking test of inflation, and 10. M. Tegmark et al., Phys. Rev. D 69, 103501 (2004). 006 (2000). 11. D. J. Fixsen et al., Astrophys. J. 473, 576 (1996). 49. L. Susskind, http://arxiv.org/abs/hep-th/0302219. would allow us to determine the energy 12. D. N. Spergel et al., Astrophys. J. Suppl. 148, 175 50. R. Bousso, J. Polchinski, Sci. Am. 291, 60 (September density of the ‘‘false vacuum’’ state that (2003). 2004). drove inflation. The new cosmological obser- 13.Forareview,seeV.F.Mukhanov,H.A.Feldman, 51. A. D. Linde, D. Linde, A. Mezhlumian, Phys. Rev. D 49, R. H. Brandenberger, Phys. Rep. 215, 203 (1992) 1783 (1994). vations also offer physicists one of the best and also (6). 52. See, for example, J. Garriga, A. Vilenkin, Phys. Rev. D resources for evaluating the latest develop- 14. For a review, see R. P. Kirshner, Science 300,1914 64, 023507 (2001) and also (56). ments in idea-rich (but data-poor) particle (2003). 53. S. Weinberg, Phys. Rev. Lett. 59, 2607 (1987). 54. T. Banks, M. Dine, E. Gorbatov, J. High Energy Phys. theory, where much of the current research 15. A. Vilenkin, E. P. S. Shellard, Cosmic Strings and other Topological Defects (Cambridge Univ. Press, New 0408, 058 (2004). has been aimed at the high-energy frontier, York, 1994). 55. For a review, see J. P. Ostriker, P. Steinhardt, Science well beyond the range of existing acceler- 16. U.-L. Pen, U. Seljak, N. Turok, Phys. Rev. Lett. 79, 300, 1909 (2003). ators. Perhaps the interface between string 1611 (1997). 56. M. Tegmark, http://arxiv.org/abs/astro-ph/0410281. 17. For a review, see J. Polchinski, http://arxiv.org/abs/ 57. We thank G. Dvali, H. Liu, M. Tegmark, S. Trivedi, and theory and cosmology will lead to new hep-th/0412244. H. Tye for helpful comments on the manuscript. This predictions for the astronomers to test. Wheth- 18. We thank M. Tegmark for providing this graph, which work was supported in part by funds provided by the er such tests are successful or not, physicists shows the most precise data points for each range of U.S. Department of Energy (D.O.E.) under cooperative l from recent observations, as summarized in (10). research agreement no. DF-FC02-94ER40818. are certain to learn important lessons about the The cosmic string prediction is taken from (16). The 0 0 nature of space, time, and matter. other curves were all calculated for ns 1, Wbaryon 10.1126/science.1107483

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