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Series C: Mathematical and Physical Sciences - Vol. 348
Observational Tests of Cosmological Inflation edited by
T. Shanks A. J. 8anday R.S.Ellis C. S. Frenk and
A. W. Wolfendale Physics Department, University of Durham, UK
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Published in cooperation with NATO Scientific Affairs Division
Proceedings of the NATO Advanced Research Workshop on Observational Tests of Inflation, Durham, UK December 10-14, 1990
ISBN 0-7923-1431-X
Published by Kluwer Academic Publishers, P.O. Box 17, 3300 AA Dordrecht, The Netherlands.
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PREFACE
This book represents the proceedings from the NATO sponsored Advanced Research Workshop entitled "Observational Tests of Inflation" held at the University of Durham, England on the 10th-14th December, 1990. In recent years, the cosmological inflation model has drawn together the worlds of particle physics, theoretical cosmology and observational astronomy. The aim of the workshop was to bring together experts in all of these fields to discuss the current status of the inflation theory and its observational predictions.
The simplest inflation model makes clear predictions which are testable by astronomical observation. Foremost is the prediction that the cosmological density parameter, no, should have a value negligibly different from the critical, Einstein-de Sitter value of 00=1. The other main prediction is that the spectrum of primordial density fluctuations should be Gaussian and take the Harrison-Zeldovich form.
The prediction that n o=l, in patticular, leads to several important consequences for cosmology. Firstly, there is the apparent contradiction with the limits on baryon density from Big Bang nucleosynthesis which has led to the common conjecture that weakly interacting particles rather than baryons may form the dominant mass constituent of the Universe. Secondly, with n o=l, the age of the Universe is uncomfortably short if the Hubble constant and the ages of the oldest star clusters lie within their currently believed limits. The workshop therefore took place at an exciting time for cosmology, with a feeling abroad that either inflation or one of the standard foundations of modern cosmology might have to be surrendered. Interest was heightened by many, new, ground and space-based astronomical advances.
The first day of the workshop reviewed the current theoretical status of the inflation predictions. In succeeding days, the workshop investigated the constraints on no from the cosmological timescale test, from observations of galaxies at high redshift and from the dynamics of galaxy clusters. The implications for the primordial spectrum of density fluctuations from observations of the large-scale galaxy distribution and from measurements of the isotropy of the microwave background were also discussed. The far-reaching impact of inflation on cosmology can be judged by the breadth of topics covered, all of which are reviewed by first rank research workers in this book. Overall, the workshop's conclusion was that inflation remains viable, although relatively small improvements in astronomical data will soon test the basic tenets of the theory. For the time being, inflation is likely to remain the framework for theoretical work in cosmology.
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I should like to take this opportunity to thank fellow members of the International Organising Committee, P.J.E. Peebles (Princeton), D.W. Sciama (Trieste), and A.W. Wolfendale (Durham) and the Local Organising Committee, A.J. Banday, R.S. Ellis and C.S. Frenk (Durham) for all their help in making the workshop possible. Thanks are also due to the conference secretary, Margaret Norman, for her efficient administrative and secretarial skills. We are also grateful to Carol Webster for her help in producing this manuscript. We should also like to thank R.G. Bower, R.L. Guzman, S.J. Lorrimer, I.R. Smail and N.R. Tanvir, for the assistance they gave while the workshop was in progress. Finally we thank the NATO Science Committee for funding assistance.
T. Shanks (Durham) 24th May, 1991
TABLE OF CONTENTS
Predictions of Inflation A.R. Liddle*........................................................... ........................................ 23
Classicality of Density Perturbations in the Early Universe R. Brandenberger, R. Laflamme & M. Mijic................................................. 39
The Influence of Non-Linear Density Fluctuations on the Microwave Sky I. L Sanz & E. Martinez-Gonzalez ................................................................ 47
Quantum Cosmology and the Cosmological Constant I. Moss ............................................................................................................ 53
Lessons from Inflation and Cold Dark Matter P.I.E. Peebles*...................................................................................... ........ 63
The Topology of Galaxy Clustering P. Coles & M. Plionis.................................................................................... 75
Can Non-Gaussian Fluctuations for Structure Formation Arise from Inflation? D.S. Salopek.............................................................................................. ..... 81
Non-Baryonic Dark Matter S. Sarkar*.................................. .................................................................... 91
Are Galactic Halos Made of Brown Dwarfs or Black Holes? B.I. Carr ........................................................................................................ 103
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II. COSMOLOGICAL TIMESCALE TEST
Ages of Globular Clusters P. Demarque*, c.P. Deliyannis & A. Sarajedini .......................................... 111
Globular Cluster Ages and Cosmology A. Renzini* .......... ........................................................................................... 131
The Local Distance Scale: How Reliable Is It? M. W. Feast* ................................................................................................... 147
Distances to Virgo and Beyond M. Rowan-Robinson* .................................................................................... 161
The Luminosity-Line-Width Relations and the Value of Ho M.l. Pierce ..................................................................................................... 173
Observational Status of Ho G.A. Tammann* ............................................................................................. 179
Calibrating Cepheid Sequences in Nearby Galaxies N. Metcalfe & T. Shanks ................................................................................ 187
New D-cr Results for Coma Ellipticals l.R. Lucey, R. Guzman, D. Carter & R.I. Terlevich ..... ................................. 193
Novae and the Distance Scale C.l. Pritchet ... ................................................................................................ 199
A High Resolution, Ground Based Observation of a Virgo Galaxy T. Shanks, N. Tanvir, P. Doe~ C. Dunlop, R, Myers, l. Major, M. Redfern, N. Devaney & P. O'Kane ............................................................................... 205
Globular Clusters as Extragalactic Distance Indicators D.A. Hanes ..................................................................................................... 211
III. HIGH REDSHIFT TESTS OF Do
High-Redshift Tests of no B. Guiderdoni* .............................................................................................. 217
Cosmology with Galaxies at High Redshifts S.l. Lilly ......................................................................................................... 233
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Aligned Radio Galaxies K. C. Chambers ... ............................................................................................ 251
K Band Galaxy Counts and the Cosmological Geometry LL Cowie ...................................................................................................... 257
Selection Effects in Redshift Surveys Y. Yoshii & M. Fukugita .............................................. .................................. 267
An Inflationary Alternative to the Big-Bang F. Hoyle .... ..................................................................................................... 273
IV. GALAXY CLUSTERING, 00 AND THE PRIMORDIAL SPECTRUM
Dynamical Estimates of no from Galaxy Clustering S.D.M. White* ................................................................................................ 279
ROSAT Observations of Clusters of Galaxies H. Biihringer*, W. Voges, H. Ebeling, R.A. Schwarz, A.C. Edge, V.G. Briel and J.P. Henry ...................................................................................... 293
A Deep ROSAT Observation at High Galactic Latitude I. Georgantopoulos, T. Shanks, G. Stewart, K. Pounds, RJ. Boyle & R. Griffiths .......................................................................................................... 309
Large Scale Structure and Inflation J.P. Huchra* ......................................................................................... ......... 315
The Structure of the Universe on Large Scales A.G. Doroshkevich .................................................. ....................................... 327
Testing the Zeldovich Spectrum w.J. Sutherland .............................................................................................. 331
Q on the Scale of 3Mmls D. Lynden-Bell* ............................................................................................. 337
Tests of Inflation Using the QDOT Redshift Survey C.S. Frenk. ..................................................................................................... 355
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Testing Inflation with Peculiar Velocities A. Dekel.. .................................................. ...................................................... 365
The Invisible Cosmological Constant O. Lahav, P.B. Lilje, J.R. Primack & M.J. Rees ............................................ 375
Support for Inflation from the Great Attractor A. Heavens ................................................ ..................................................... 379
The Angular Large Scale Structure y. Hoffinan ..................................................................................................... 385
Is There Any Observational Evidence for Non- Gaussian Primordial Density Fluctuations?
A.L Melott ..................................................................................................... 389
Observations of Microwave Background Anisotropy at Tenerife and Cambridge A.N. Lasenby, R.D. Davies, R.A. Watson, R. Rebolo, C. Gutierrez & J.E. Beckman .... ..................................................................................................... 413
Foreground Effects and the Search for Fluctuations in the CMB Radiation A.J. Banday, M. Giler, B. Szabelska, J. Szabelski & A. W Wolfendale ..................................................................................................... 419
Microwave Background Anisotropies and Large Scale Structure in the Universe G. Efstathiou* ...................................................................... .......................... 425
Discovery of the Small Scale Sky Anisotropy at 2.7cm: Radio Sources or Relic Emission?
Yu. N. Parijskij, B.L Erukhimov, M.G. Mingaliev, A.B. Berlin, N.N. Bursov, N.A.Nizhelskij, M.N. Naugolnaja, v.N. Chernenkov, O. V. Verkhodanov, A. V. Chepurnov & A.A. Starobinsky .............................................. ...................................................... 437
Balloon-Borne Observations of CMB Anisotropies at Intennediate Angular Scales, at Sub-MM and MM Wavelengths
P. de Bernardis, S. Masi, B. Melchiorri & F. Melchiorri ............................ 443
VI. POSTER PAPERS
The Durham/UKST Galaxy Redshift Survey A Broadbent, D. Hale-Sutton, T. Shanks, F.G. Watson, AP. Oates, R. Fong, C.A Collins, H.T. MacGillivray, R. Niclwl & Q.A..
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Parker ............................................................................................................ 447
Time Evolution of Lensed Image Separations T.l. Broadhurst & S. Oliver ........................................................................... 449
Deep Galactic Surveys as Probes of the Large Scale Structure of the Universe O.E. Buryak. M. Demia'nski & A.G. Doroshkevich. ..................................... 453
Intergalactic Absorption in the Spectra of High-Redshift QSOs S. Cristiani & E. Giallongo ........................................................................... 457
A Complete Quasar Sample at Intermediate Redshift F. La Franca, S. Cristiani, C. Barbieri, R.G. Clowes & A Iovino ................ 461
Radio-Luminosity Dependence of the IR-Radio Alignment Effect in High-z Radio Galaxies
l.S. Dunlop & l.A. Peacock ............................................... ............................ 463
Density and Peculiar Velocity Fields in the Region of Dressler's Supergalactic Plane Survey
M.l. Hudson ................................................................................................... 467
Scale Invariance Induced by Non-linear Growth of Density Fluctuations F. Moutarde, l.-M Alimi, F.R. Bouchet & R. Pellat... ................................... 469
The Power Spectrum of Galaxy Clustering l.A. Peacock ................................................................................................... 471
Higher Moments of the IRAS Galaxy Distribution C.A. Scharf .................................................................................................... 475
Collapse of a Protogalactic Cloud S. Yoshioka. .............................................. ...................................................... 477
INDEX OF AUTHORS ............................................................................................. 479
List of Participants
Aragon, A.S., Physics Department, University of Durham, South Road, Durham DHI 3LE, U.K
Banday, A.J., Physics Department, University of Durham, South Road, Durham DHI 3LE, U.K.
Blanchard, A., DAEC, Observatoire de Meudon, 92195 Meudon Cedex, France.
Bohringer, H., Max Planck-Institut fur Extraterrestrische Physik, Giessenbachstrasse, D- 8046, Garching bei Munchen,West Germany.
Borner, G., Max Planck-Institut fur Physik & Astrophysik, Instiut fur Astrophysik, Karl­ Schwarzschild-Strasse 1, 8046 Garching bei Munchen, West Germany.
Bower, R.G., Physics Department, University of Durham, South Road, Durham DHI 3LE, U.K.
Broadhurst, T.J., Theoretical Astronomy Unit, School of Mathematical Sciences, Queen Mary & Westfield, Mile End Road, LONDON EI 4NS, U.K
Buryak, 0., Inst. of Applied Maths, Miusskaya Sq. 4, MOSCOW 125047, USSR.
Cannon, R.D., Anglo-Australian Observatory, Epping Laboratory, PO Box296, Epping, NSW 2121, AUSTRALIA.
Carr, B, J., Theoretical Astronomy Unit, School of Mathematical Sciences, Queen Mary & Westfield, Mile End Road, LONDON EI 4NS, U.K.
Chambers, KC., Sterrewacht, Postbus 9513, 2300 RA LEIDEN, The NETHERLANDS.
Coles, P., Astronomy Centre, School of Mathematical and Physical Sciences, University of Sussex, Falmer, Brighton, BNI9QH, UK
Colless, M.M., Institute of Astronomy, Madingley Road, Cambridge CB3 OHA, U.K.
Cowie, L.L., Institute for Astronomy, University of Hawaii, 2680 Woodlawn Drive, Honolulu, HI 96822, USA.
Cristiani, S., Dipartimento di Astronomia, Universita di Padova, vicolo dell'Osservatorio 5,35122 Padova, ITALY.
Davies, R. D., Nuffield Radio Astronomy Labs., Jodrell bank, Macclesfield, Cheshire SKI I 9DL, U.K
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Dekel, A., Racah Inst. of Physics, The Hebrew Univ. of Jerusalem, JERUSALEM 91904, ISRAEL.
Demarque, P., Yale Univ. Observatory, 260 Whitney Avenue, PO Box 6666, NEW HAVEN, CT06511, USA.
Doroshkevich, A.G., Inst. of Applied Maths, Miusskaya Sq. 4, MOSCOW 125047, USSR.
Efstathiou, G., Dept of Astrophysics, Univ. of Oxford, South Parks Road, OXFORD OXI 3RQ.
Ellis, R.S., Physics Department, University of Durham, South Road, Durham DHI 3LE, U.K.
Feast, M.W., South African Astronomical Observatory, PO Box 9, Observatory 7935, Cape Town, South Africa.
Frenk, C.S., Physics Department, University of Durham, South Road, Durham DHI 3LE, U.K.
Georgantopoulos, I., Physics Department, University of Durham, South Road, Durham DHI 3LE, U.K.
Guiderdoni, B., Institut d'Astrophysique, 98 bis, Boulevard Arago, F-70154, PARIS, France.
Guth, A.H., Centre for Theoretical Physics, Dept of Physics, Massachusetts Inst. of Technology, Cambridge MA02139, USA.
Guzman, R., Physics Department, University of Durham, South Road, Durham DHI 3LE.
Hanes, D., Queen's University, Astronomy Group, Physics Department, Stirling Hall, KINGSTON K7L 3N6, CANADA.
Heavens, A.F., Royal Observatory, Blackford Hill, EDINBURGH EH9 3HJ, U.K.
Hindmarsh, M., Physics Department, George's Place, University of Newcastle, NEWCASTLE UPON TYNE, NEI 7RU, U.K.
Hoffman, Y., TECHNION - Israel Institute of Technology, Dept. of Physics, 32000 Haifa, ISRAEL.
Hoyle, F., 102 Admiral's Way, West Cliff Road, Boumemouth, DORSET, BH2 5HF, U.K.
Huchra, J.P., Smithsonian Observatory, Centre for Astrophysics, 60 Garden Street, CAMBRIDGE, MA02138, USA.
Hudson, M., Institute of Astronomy, Madingley Road, Cambridge CB3 OHA, U.K.
Kaiser, N., CIT A, University of Toronto, McLennan Labs., 60 St. George Street, Toronto, Ontario, M5S IAI, CANADA.
Kraan-Korteweg, R., Astron. Inst. of the Univ. of Basel, Venusstrasse 7, ch-4102, Binningen, Switzerland.
Labav, 0., Institute of Astronomy, Madingley Road, CAMBRIDGE CB3 OHA, U.K.
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Liddle, A.R., Astronomy Centre, School of Mathematical and Physical Sciences, University of Sussex, Falmer, Brighton, BNI9QH, U.K.
Lidsey, J., Room 201, Theoretical Astronomy Unit, School of Mathematical Sciences, Queen Mary & Westfield, Mile End Road, LONDON El 4NS, U.K.
Lilly, SJ., Institute for Astronomy,University of Hawaii, 2680 Woodlawn Drive, Honolulu, HI 96822, USA.
Lorrimer, S., Physics Department, University of Durham, South Road, Durham DH! 3LE, U.K.
Lucey, J.R., Physics Department, University of Durham, South Road, Durham DH! 3LE, U.K.
Lynden-Bell, D., Institute of Astronomy, The Observatories, Madingley Road, Cambridge CB3 OHA, U.K.
Mann, R., Royal Observatory, Blackford Hill, Edinburgh EH9 3HJ, U.K.
Melchiorri, F., Dip Fisica, Univ. La Sapienza, P.zza Aldo Moro, ROMA, ITALY.
Mellier, Y., Observatoire de Toulouse, 14 ave. Edouard Belin, F31400, TOULOUSE, FRANCE.
Melon, A.L., Physics & Astronomy, University of Kansas, Lawrence, KS 66045, USA.
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Metcalfe, N., Physics Department, University of Durham, South Road, Durham DHI 3LE, U.K.
Mingaliev, M., Special Astrophysical Observatory, Nizhny Aekhys, Zelenchukskaya, STAVROPOLOSKYU KRAJ, USSR.
Moore, B., Physics Department, University of Durham, South Road, Durham DHI 3LE, U.K.
Moss, I., Physics Department, George's Place, University of Newcastle, NEWCASTLE UPON TYNE, NEI 7RU, U.K.
Moutarde, F., DAEC, Observatoire de Meudon, 5 PI. J. Janssen, 92195 Meudon Cedex, France.
Oliver, S., Theoretical Astronomy Unit, School of Mathematical Sciences, Queen Mary & Westfield, Mile End Road, LONDON El 4NS, U.K.
Parijskij, Y., Special Astrophysical Observatory, Nizhny Aekhys, Zelenchukskaya, STAVROPOLOSKYU KRAJ, USSR.
Peacock, J.A., Royal Observatory, Blackford Hill, EDINBURGH EH9 3HJ, U.K.
Peebles, P.J.E., Physics Dept., Jadwin Hall, Princeton University, P.O. Box 708, PRINCETON NJ 08544, USA.
Penny, A.J., Rutherford Appleton Laboratory, Chilton, Didcot, OXII OQX, Berks.
Pierce, M.J., Dominion Astrophysical Observatory, 5071 W. Saanich Road, VICTORIA, B.C. V8X 4M6, CANADA.
Piotrkowska, K., Dept of Astrophysics, Nuclear Physics Building, Univ. of Oxford, South Parks Road, OXFORD OXI 3RQ, UK
Pritchet, C.J., Dominion Astrophysical Observatory, 5071 W. Saanich Road, VICTORIA, B.C. V8X 4M6, CANADA.
Rees, M.J., Institute of Astronomy, University of Cambridge, Madingley Road,Cambridge CB3 OHA, U.K.
Renzini, A., Osservatorio Astronomico di Bologna,Via Zamboni 33, 40126, BOLOGNA, Italy.
Rowan-Robinson, M., Dept. of Maths & Astronomy, Queen Mary College, Mile End Road, London El 4NS, U.K.
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Sarkar, S., Dept. of Theoretical Physics, University of Oxford, Keble Road, Oxford OXI 3NP, U.K.
Saich, P., Astronomy Centre, School of Mathematical and Physical Sciences, University of Sussex, Falmer, Brighton, BNI9QH, U.K.
Salopek, D., NASAlFermilab Astrophysics Group, PO Box 500, MS-209, Batavia, ILLINOIS 6051, USA.
Sanz, J.L., Dpto Fisica Moderna, Univ. de Cantabria, Av. de Los Castros s.n., 39005 SANTANDER, SPAIN.
Schade, D., Institute of Astronomy, Madingley Road, Cambridge CB3 OHA, U.K.
Scharf, C., Institute of Astronomy, Madingley Road, Cambridge CB3 OHA, U.K.
Sciama, D.W., International School of Advanced Studies, Strada Costiera II, 34014 TRIESTE, Italy.
Secco, L., Dipartimento di Astronomia, Universita di Padova, vicolo dell'Osservatorio 5, 35122 Padova, ITALY.
Shanks, T., Physics Department, University of Durham, South Road, Durham DHI 3LE, U.K.
Sirousse-Zia, H., Institut Henri Poincare, Laboratoire de Physique Theorique, II, Rue Pierre & Marie Curie, 75231 PARIS Cedex OS, FRANCE.
Smoot, G.F., Space Sciences Lab, Univ. of California, I Cyclotron Road,Berkeley CA94720,USA.
Starobinskii, A., Landau Insitute for Theoretical Physics, Academy of Sciences of USSR, GSP-I 117940, Kosygina Str., Moscow V-334, USSR.
Stewart, E., Dept. of Physics, Univ. of Lancaster, Lancaster, LAI 4YB, U.K.
Stewart, G.C., Dept. of Physics & Astronomy, Univ. of Leicester, University Road, Leicester LEI 7RH, U.K.
Sutherland, W.J., Dept of Astrophysics, Nuclear Physics Building, Univ. of Oxford, South Parks Road, OXFORD OXI 3RQ, U.K.
Tammann, G.A., Astron. Inst. of the Univ. of Basel, Venusstrasse 7, ch4102 Binningen, Switzerland.
Tanvir, N.R., Physics Department, University of Durham, South Road, Durham DHI 3LE, U.K.
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Taylor, A.N., Theoretical Astronomy Unit, School of Mathematical Sciences, Queen Mary & Westfield, Mile End Road, LONDON E1 4NS, U.K.
Treumann, R.A., Institut fur Extraterrestriche Physik, Max-Planck-Institut fur Physik und Astrophysik, 8046 Garching b. Munchen, West Germany.
White, S.D.M., Institute of Astronomy, Madingley Road,Cambridge, CB3 OHA, U.K.
Wolfendale, A.W., Physics Department, University of Durham, South Road, Durham DH1 3LE, U.K.
Yoshii, Y., National Astronomical Observatory, Mitaka, Tokyo 181, JAPAN.
Yoshioka, S., Department of Physics, Tokyo University of Mercantile Marine, Koto-ku, Tokyo 135, JAPAN.
FUNDAMENTAL ARGUMENTS FOR INFLATION
ALAN H. GUTH Center for Theoretical Physics, Laboratory for Nuclear Science and Department of Physics, Massachusetts Institute of Technology,
Cambridge, Massachusetts, 0£139 and
Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, Massachusetts 02138
ABSTRACT. The mechanism of inflation is described, and the fundamental arguments in favor of inflation are summarized. It is claimed that the inflationary model provides a very plausible explanation for (1) the large number of particles in the universe, (2) the Hubble expansion, (3) the large-scale uniformity of the universe, (4) the nearness of the universe to a critical density, and (5) the absence of magnetic monopoles.
1 IntrC'duction
I try to be modest about my own role in the development of inflation, and I think I am aided in these efforts by the fact that I have a lot to be modest about. When I began working on cosmology, essentially all the key ideas of inflation had already been discovered. On the particle physics side, there had been much investigation of grand unified theories and other spontaneously broken gauge theories; important properties such as phase transitions, false vacua, and the decay of false vacua were already understood. On the cosmology side, the shortcomings of the standard big bang theory had been studied, and the horizon and flatness problems were both known. My role, then, was mainly just to pull these ideas together into a coherent model. I see no cause, on the other hand, for any of us to be modest about the inflationary universe theory itself. It is a dramatic development of modern physics, providing for the first time a theory that accounts for the origin of essentially all of the matter and energy in the universe. The model is certainly a major achievement in cosmology if it is correct, and if it turns out to be wrong, it will be a disappointment to a large number of people.
In this article I will try to explain the fundamental arguments for inflation. To put it another way, I will try to explain why many people presently believe in the inflationary model, even though the astronomers have not yet found enough matter to make no = 1.
Fig. 1 shows the first (and most naive) fundamental argument for inflation. The graph shows the number of articles per year related to inflation, as tabulated from the SPIRES database at the Stanford Linear Accelerator Center. It is mainly a particle physics database, so some of the more astrophysical papers on the subject are probably not represented. In any case, one sees that the inflationary model has stirred up a lot of interest.
T. Shanks et al. (eds.). Observational Tests of Cosmological Inflation. 1-21. @ 1991 Kluwer Academic Publishers.
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a 80 81 82 83 84 85 86 87 88 89 90
YEAR
Figure 1: A histogram of articles concerning the inflationary universe model. The graph shows all articles in the SPIRES database that either refer to any of the three basic in­ flationary universe papers ([Guth, 1981; Linde, 1982; or Albrecht & Steinhardt, 1982]), or contain the string "inflation" in the title.
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2 The Mechanism of Inflation
Before explaining the arguments for inflation, however, I will first explain how the infla­ tionary model works.
The mechanism of inflation depends on scalar fields, so I will begin by briefly summa­ rizing the role of scalar fields in particle physics. To begin with, the reader should recognize that in the context of modern particle physics, all fundamental particles are described by fields. The best known example is the photon. The classical equations describing the electromagnetic field were written down in the 1860's, but then in the early 20th century physicists learned that the underlying laws of nature are quantum and not classical. The quantization of the electromagnetic field can be carried out in a very straightforward way. For simplicity one can consider the fields inside a box, letting the size of the box approach infinity at the end of the calculation. One can then think of the electromagnetic fields inside the box as a mechanical system. The electromagnetic field is written as a sum of normal modes, and the coefficients of the normal mode functions can be taken as the dynamical degrees of freedom of the system. It turns out that each of these coefficients obeys the equations of a harmonic oscillator, and there is no interaction between the coefficients. The system is quantized by the same rules that one uses to quantize the hydrogen atom or the harmonic oscillator, and the result is that each normal mode has evenly spaced energy levels. In this case, we interpret each energy level above the ground state as the occupation of the mode by a photon. Thus, the photon is interpreted as the quantized excitation of a field.
In contemporary particle theory, all elementary particles are described in this way. There is an electron field to correspond to an electron, a quark field to correspond to a quark, a neutrino field to correspond to a neutrino, etc. Among the different types of fields, the simplest is the scalar field- a field that has the same value to any Lorentz observer. The quantized excitation of a scalar field is a spinless particle. Although spinless particles that are regarded as elementary have yet to be observed, they are nonetheless a key ingredient to a number of important theories. In particular, the Glashow-Weinberg-Salam model of the electroweak interactions makes use of a scalar field, called the Higgs field, to cause a symmetry in the theory to be spontaneously broken. (If this symmetry were not broken, then electrons and neutrinos would both be massless, and would be indistinguishable.) This Higgs field corresponds to a neutral spinless Higgs particle, which will hopefully be observed at the SSC (Superconducting Super-Collider), if not before. Grand unified theories make use of similar Higgs fields, but at a much higher mass scale, to spontaneously break the grand unified symmetry which relates electrons, neutrinos, and quarks.
There is much current interest in superstring theories, which actually go beyond the pattern of the field theories that were described above. In these theories the fundamental object is not a field, but is really a string-like object, which has length but no width. These theories are believed to behave as field theories, however, at energy scales well below the Planck scale (Mp == 1/.../G = 1.02 X 1019 GeV, where G is Newton's constant, and I use units for which n == c == 1). In fact, these field theories contain a large number of scalar fields. Thus, the particle physics motivation for believing that scalar fields exist is quite strong.
4
<Pt Scalar Field
Figure 2: The potential energy function for the scalar field 4>, appropriate for the new inflationary universe model.
The electromagnetic field has a potential energy density given by
v = J.-(E2 + ffl) , 811'
(1)
but the potential energy density of the scalar field has a wider range of possibilities. It is restricted by the criterion that the field theory be renormalizable (Le., the requirement that the field theory leads ultimately to finite answers), but there are still several free parameters involved in specifying the potential. The form of the scalar field potential appropriate for the new inflationary universe model is shown in Fig. 2. The potential has a minimum at a value of the scalar field 4> that is nonzero, a property that is generic for Higgs fields. The potential also has a plateau centered at 4> = cPr, a feature that Higgs fields do not necessarily have. This property is necessary, however, in order for the new inflationary scenario to be possible.
A particle physicist defines the word "vacuum" to mean the state of lowest possible energy density, so it is the state in which 4> lies at the minimum of the potential, labeled 4>t on the diagram. Notice that the energy density of this state is shown as zero, which is equivalent to the statement that the cosmological constant is either zero or immeasurably small. The explanation for this fact has been a long-standing mystery to particle physicists, although now we have at least a possible solution [Coleman, 1988] in the context of a crude understanding of quantum gravity. Note that even if the cosmological constant has a value that is cosmologically significant, the effect on this diagram would be completely imperceptible. For clarity, I will refer to the vacuum in this situation as the "true vacuum."
5
_____ ~_4~} { True Vacuum
dV
........ -----.. F
dW= pdV
Figure 3: A thought experiment to calculate the pressure of the false vacuum. As the piston chamber filled with false vacuum is enlarged, the energy density remains constant and the energy increases. The extra energy is supplied by the agent pulling on the piston, which must pull against the negative pressure of the false vacuum.
The state in which the scalar field, in some region of space, is perched at the top of the plateau of the potential energy diagram is called the "false vacuum." Note that the energy density of this state is pc, and is therefore fixed by the laws of physics. The false vacuum is of course not stable, since eventually the scalar field will no doubt evolve toward the minimum of the potential. Nonetheless, if the plateau is broad and flat enough, it will take a long time for the scalar field to roll off the hill. Thus the false vacuum is metastable, and can be very long-lived, by the time scales of the early universe.
The crucial property of the false vacuum is that the energy density is positive, but on short time scales it cannot be lowered. This explains the etymology of the name "false vacuum." Here "false" is being used to mean temporary, and "vacuum" means the state of lowest possible energy density.
It is now possible to construct a simple energy-conservation argument to determine the
6
pressure of the false vacuum. Imagine a chamber filled with false vacuum, as shown in Fig. 3. Since the energy density of the vacuum is fixed at Pc, the energy inside the chamber is given by U = pcV, where V is the volume of the chamber. Now suppose the piston is pulled outward, increasing the volume by dV. Unlike any normal substance, the false vacuum will maintain a constant energy density despite the increase in volume. The change in energy is then dU = pcdV, which must be equal to the work done, dW = pdV, where p is the pressure. Thus, the pressure of the false vacuum is given by
p = -pc· (2)
If one puts this relation between pressure and energy density into the general form of the energy-momentum tensor, one finds
(3)
where gl''' = diag[-l, 1, 1, 1] is the Lorentz metric. Thus, the energy-momentum tensor of the false vacuum is Lorentz-invariant, as one would expect, since the state is Lorentz­ invariant. That is, if a region of space has a scalar field with a constant value <Pc, to another Lorentz observer it will also look like a region of space with a scalar field of value <Pc. One could in fact have used this Lorentz-invariance as an alternative derivation of Eq. (2).
Finally, we are ready to discuss the gravitational effects of this very peculiar energy­ momentum tensor. Starting with the Einstein field equations
'Ill''' - !gl''''Il = 81rGTI''' - Agl''' , 2
(4)
one sees that the effect of the energy-momentum tensor of Eq. (3) is precisely the same as having a temporarily nonzero value of the cosmological constant A, related by
A = 81rGpc . (5)
To see the effect on a Robertson-Walker universe, note that the scale factor R( t) obeys the equation of motion
.. 41r R = -3"G(p + 3p)R. (6)
In the present universe, the pressure term is a small relativistic correction. If the universe was ever dominated by false vacuum, however, then the pressure term has the opposite sign, and overcomes the gravitational attraction caused by the energy density term:
.. 81r R= -GpcR.
3 (7)
The force of gravity actually becomes repulsive, and the expansion rate of the universe is accelerated. The general solution to Eq. (7) is
(8)
where CI and C2 are arbitrary constants, and the exponential rate is given by
(9)
7
After some time the growing term will dominate, and the expansion becomes a pure expo­ nential.
While the new inflationary scenario proposed by Linde [1982] and Albrecht & Steinhardt [1982] assumes a potential energy function of the form shown in Fig. 2, Linde [1983a, 1983b] has shown that such a severe restriction is not necessary. In a model known as chaotic inflation, Linde showed that inflation can work for a scalar field potential as simple as V(4)) = >'4>4, provided that one makes some assumptions about the initial conditions. He proposed that the scalar field begins in a chaotic state, so that there are some regions in which the value of 4> is a few times larger than the Planck mass Mp. These regions must exceed some minimal size, which is estimated to be several times H-l, where H denotes the Hubble constant. Then 4> rolls down the hill of the potential energy diagram, and a straightforward calculation indicates that there is an adequate amount of inflation. The Hubble "constant" is not a constant in this case, but it is slowly varying, so the expansion can be called "quasi-exponential".
3 The Inflationary Universe Scenario
The inflationary universe scenario begins with a patch of the universe somehow settling into a false vacuum state. The mechanism by which this happens has no influence on the later evolution. The following three possibilities have been discussed in the literature:
1. Supercooling from high temperatures. This was the earliest suggestion [Guth, 1981; Linde, 1982; Albrecht & Steinhardt, 1982]. If we assume that the universe began very hot, as is traditionally assumed in the standard big bang model, then as the universe cooled it presumably went through a number of phase transitions. For many types of scalar field potentials, supercooling into a false vacuum occurs naturally. This sce­ nario has the difficulty, however, that there is no known mechanism to achieve the desired pre-inflationary thermal equilibrium state. Calculations [Starobinsky, 1982; Guth & Pi, 1982; Hawking, 1982; Bardeen, Steinhardt, & Turner, 1983] show that the scalar field must be very weakly coupled in order for quantum-induced density perturbations to be sufficiently small, and consequently the scalar field would require much more than the available time to relax to thermal equilibrium. It has been shown, however, that true thermal equilibrium is not really necessary: a variety of random configurations give results that are very similar to those of thermal equi­ librium [Albrecht, Brandenberger, & Matzner, 1985; Kung & Brandenberger, 1990; Brandenberger, 1991; Goldwirth, 1991].
2. Tunneling from "nothing". These ideas are of course very speculative, since they involve a theory of quantum gravity that does not actually exist. The basic idea, however, seems very plausible. If geometry is to be described by quantum theory, then the geometry of space can presumably undergo quantum transitions. One can then imagine an initial state of absolute nothingness- the absence of matter, energy, space, or time. The state of absolute nothingness can presumably undergo a quantum transition to a small universe, which then forms the initial state for an inflationary
8
scenario. Variations of these ideas have been studied by Tryon [1973], Vilenkin [1982, 1985], Linde [1983c, 1984a, 1984b], and Hartle & Hawking [1983].
3. Random fluctuations in chaotic cosmology. In Linde's [1983a, 1983b] chaotic cosmol­ ogy, it is assumed that the scalar field ~ begins in a random state in which all possible values of ~ occur. Inflation then takes place in those regions that have appropriate values of ~, and these inflated regions dominate the universe at later times. In these models it is not necessary for the scalar field potential energy function V( ~) to have a plateau, but as in other models it must be very flat (i.e., weakly self-coupled) in order to minimize the density perturbations that result from quantum fluctuations.
Regardless of which of the above mechanisms is assumed, one expects that the correlation length of the scalar field just before inflation is of the order of the age of the universe at the time. For typical grand unified theory parameters, this gives a correlation length of about 10-24 cm.
The patch then expands exponentially due to the gravitational repulsion of the false vacuum. In order to achieve the goals of inflation, we must assume that this exponential expansion results in an expansion factor ~ 1025 • For typical grand unified theory numbers, this enormous expansion requires only about 10-32 sec of inflation. During this inflationary period, the density of any particles that may have been present before inflation is diluted so much that it becomes completely negligible. At the same time, any nonuniformities in the metric of space are smoothed by the enormous expansion. The explanation for this smoothness is identical to the reason why the surface of the earth appears to be flat, even though the earth is actually round- any differentiable curve looks like a straight line if one magnifies it enough and looks at only a small segment. The correlation length for the scalar field is stretched by the expansion factor to become at least about 10 cm. If the duration of inflation is more than the minimal value, which seems quite likely, then the final correlation length could be many orders of magnitude larger. There appears to be no upper limit to the amount of inflation that may have taken place.
The false vacuum is not stable, so it eventually decays. If the decay occurs by the usual Coleman-Callan process [Coleman, 1977; Callan & Coleman, 1977] of bubble nucleation, then the randomness of the bubble nucleation process would produce gross inhomogeneities in the mass density [Hawking, Moss, & Stewart, 1982; Guth & Weinberg, 1983]. This problem is avoided in the new inflationary scenario [Linde, 1982; Albrecht & Steinhardt, 1982] by introducing a scalar field potential with a flat plateau, as was shown in Fig. 1. This leads to a "slow-rollover" phase transition, in which quantum fluctuations destabilize the false vacuum, starting the scalar field to roll down the hill of the potential energy diagram. These fluctuations are initially correlated only over a microscopic region, but the additional inflation that takes place during the rolling can stretch such a region to be large enough to easily encompass the observed universe.
When the phase transition takes place, the energy that has been stored in the false vacuum is released in the form of new particles. These new particles rapidly come to thermal equilibrium, resulting in a temperature with kT ~ 1014 GeV. At this point the scenario rejoins the standard cosmological model.
9
The baryons are produced [see, for example, Kolb & Turner, 1983] by baryon noncon­ serving processes after inflation. Any baryons that may have been present before inflation are simply diluted away by the enormous expansion factor. Thus, inflationary cosmology requires an underlying particle theory, such as a grand unified theory, in which baryon number is not conserved.
It is a dramatic feature of inflationary models that essentially all of the matter, energy, and entropy of the observed universe is produced by the expansion and subsequent decay of the false vacuum. (I used the qualifier "essentially" to acknowledge the fact that a small patch of false vacuum is necessary to start inflation. For typical grand unified theory parameters, with a mass scale of order 1014 GeV, the required volume of radius rv H-l has a mass of order 10 kg.) For this reason I sometimes refer to the process of inflation as the ultimate free lunch.
4 The Eternal Nature of Inflation
A fascinating feature of inflation, which in my opinion is also important in evaluating the plausibility of inflation, is the fact that inflation is eternal- if inflation ever begins, then it will never stop [Vilenkin, 1983; Starobinsky, 1982, 1986; Linde, 1986, 1987; Aryal & Vilenkin, 1987].
To understand the endlessness of inflation, one first notices that the decay of the false vacuum, like the decay of many other unstable systems, is an exponential process. S.-Y. Pi and I [Guth & Pi, 1985] have verified the exponential decay law in a simplified but exactly soluble model of a slow-rollover phase transition, in which the potential is taken as V(</» = _!p.2</>2. For the case of chaotic inflation, on the other hand, one might think that the scalar field would roll inexorably down the hill in the potential energy diagram, completing the decay in a finite time. Linde [1986] has shown, however, that if the scalar field starts at a sufficiently high value, then it can be sustained by quantum fluctuations, with again an exponential decay law. As long as the false vacuum endures it drives an exponential expansion, and for reasonable parameters the rate of expansion is much faster than the rate of decay. Thus, even though the false vacuum is decaying, the total volume of the false vacuum region actually increases with time.
As time goes on, pieces of the false vacuum region are constantly undergoing decay. As each piece decays, it releases energy and thereby sets into motion a hot big bang universe. Other regions of false vacuum, however, continue to exponentially expand, so the false vacuum never disappears.
The infinity of universes produced in this way rapidly become causally disconnected, so there is no way for us to verify, even in principle, that the other universes exist. Nonetheless, I feel that the eternal character of inflation makes it a more plausible theory. In the absence of this feature, there is some difficulty in deciding whether the initial conditions required for inflation are sufficiently plausible. Since there is no established theory of initial conditions, questions of this sort can easily lead to inconclusive answers. Given the endlessness of inflation, however, the question becomes much less significant. Just as most of us accept the claim that complicated DNA molecules originated through random processes sometime during the history of the earth, we can also accept the claim that a
10
region of false vacuum originated through random processes sometime during the history of spacetime. Just as primitive DNA molecules increased their abundance by replication, one patch of false vacuum would inflate to produce an infinity of universes, one of which could be the universe in which we live.
5 Evidence for Inflation
5.1 BIGNESS OF THE UNIVERSE
To most students of cosmology, one of the most startling features of the universe is its incredible size-- the observable universe contains approximately 1090 particles. Since the standard big bang model (without inflation) contains no mechanism to produce such a huge amount of entropy, the model requires us to assume that essentially all of these particles were here at the start.
The inflationary model, on the other hand, can actually explain where such a vast number of particles can come from. Particles are produced by the expansion and decay of the false vacuum. Since the expansion is exponential, it makes sense to write 1090 = e201 = (e69)3. Thus, an exponential expansion of 69 e-foldings is sufficient to turn a single particle into 1090 particles. Inflation therefore reduces the problem of explaining the 1090 particles to the problem of explaining why there were more than 69 e-foldings of inflation. In fact it is easy to construct underlying particle theories that will give far more than 69 e-foldings of inflation. The suggestion is that even though the observed universe is incredibly large, it is only an infinitesimal fraction of the entire universe.
5.2 HUBBLE EXPANSION
Although the standard cosmological model is called the big bang, the theory in fact contains no description whatever of the "bang". It is really a theory of the aftermath of a bang, describing how the matter expands and cools, coagulating to form galaxies and other visible structures in the universe.
With inflation, however, we have for the first time a theory of the bang itself- the outward thrust of the big bang can be attributed to the repulsive gravity of the false vacuum. As we noticed in discussing Eq. (7), the false vacuum leads to a gravitational repulsion that in turn leads to exponential expansion. This uniform expansion agrees precisely with the pattern of expansion discovered by Hubble.
5.3 HOMOGENEITY/ISOTROPY
My third argument for inflation is known as the homogeneity or isotropy problem, and it is also called the horizon problem. The problem is that the extreme uniformity that is observed on very large scales, particularly in the cosmic background radiation, cannot be explained without inflation.
Observationally, the effective temperature of the cosmic background radiation is known to be isotropic to about one part in 103 , and even this anisotropy can be accounted for
11
by the assumption that the earth is moving through the background radiation. If one removes the dipole component that can be attributed to the earth's motion, then the residual anisotropy is known to be less than one part in 104 [Wilkinson, 1986]. The limits on anisotropies continue to improve, and the latest results will be summarized later in this conference by G. Smoot and F. Melchiorri. The extreme uniformity in the observations is very difficult to understand in the context of the standard cosmological model, in which the horizon distance (i.e., the distance that a light pulse could have traveled since the initial singularity) is rather short. The existence of horizons in cosmology was first discussed by Rindler [1956], and the horizon problem was discussed by Weinberg [1972] and by Misner, Thorne, & Wheeler [1973].
Consider, for example, two microwave antennae pointing in opposite directions. Each is receiving radiation that is believed to have been emitted (or last scattered) at the time of hydrogen recombination, tr , about 105 years after the big bang, when the temperature Tr was about 4000oK. (At earlier times the plasma that filled the universe was opaque to this radiation.) At the time of emission, these two sources were separated from each other by many horizon distances. To estimate how many, let us assume that n ~ 1, and that the universe can be approximated as being matter-dominated during the relevant time period. Then the Robertson-Walker scale factor is given by
R(t) = bt2/3 , (10)
for some constant b. The scale factor specifies the ratio of a physical distance to a coordinate distance. Since the physical speed of light in our units is one, the coordinate speed of light is 1/ R(t). The coordinate distance between the source of the cosmic background radiation and us is then given by
_lto dt' b-1( 1/3 1/3) rcoord - tr bt,2/3 = 3 to - tr , (11)
where to denotes the present time. The coordinate value of the horizon distance at time tr is given by
_ t r ~ _ -11/3 lhor,coord - 10 bt12/ 3 - 3b tr . (12)
Thus, the number of horizon distances separating the two sources in opposite directions is given by
N = 2rcoord = 2 [(to)1/3 -1] = 2 [(Tr)1/2 -1] ~ 75, lhor,coord tr To
(13)
where we used the fact that RT ~ const and To ~ 2.74°K. The problem is to understand how two regions over 75 horizon distances apart came to be at the same temperature at the same time. (In the above discussion we have assumed for simplicity that n = 1 and that the universe was completely matter-dominated. More detailed calculations [Guth, 1983a], however, show that the problem is if anything a little worse- the two regions were at least 90 horizon distances apart.)
The horizon problem is not an inconsistency in the standard model, but represents in­ stead a lack of explanatory power. If the universe is assumed to have begun homogeneously,
12
then it will continue to evolve homogeneously. The problem is that the very striking large­ scale homogeneity of the universe is not explained or predicted by the model, but instead must simply be assumed.
In the inflationary model, on the other hand, the horizon distance is stretched dur­ ing inflation by the overall inflationary expansion factor, which we denote by Z. At all subsequent times the entire observed universe is much smaller than the horizon distance. Furthermore, the inflationary model does more than simply enlarge the horizon distance­ it actually provides a mechanism to create the observed large-scale homogeneity. In the inflationary model the size of the observable universe at times before the GUT phase tran­ sition was smaller than it would have been in the standard scenario by a factor of Z. If Z ~ 1025, then the entire observable universe would have been within its horizon before inflation; it would have become homogeneous at this time by normal thermal processes, and then this very small homogeneous region would have been stretched by inflation to become large enough to encompass the observed universe. The region would then remain homogeneous as it continued to evolve.
5.4 FLATNESS
Inflation can also solve the "flatness" problem, which was first highlighted by Dicke & Peebles [1979]. That is, inflation can explain why the mass density of the early universe was so close to the critical value.
The critical mass density Pc is defined as that mass density which is just barely sufficient to eventually halt the expansion of the universe. Today the crucial ratio n == pI Pc (where p is the mass density of the universe) is known to lie in the range
0.1 ~ n ~ 2. (14)
Despite the breadth of this range, the value of n at early times is highly constrained, since n = 1 is an unstable equilibrium point of the standard model evolution. Thus, if n is every e:r:actiy equal to one, it will remain exactly equal to one forever. If, on the other hand, n is slightly greater than one in the early universe, then it will rapidly rise toward infinity. If it is slightly less than one, it will rapidly fall toward zero. In particular, it can be shown that n - 1 grows as
n _ 1 {t (during radiation-dominated era) ()( t2/ 3 (during matter-dominated era) .
(15)
At t = 1 sec, for example, which was the beginning of the processes of big bang nucleosyn­ thesis, n must have been equal to one to an accuracy of one part in 1015. If we extrapolate further to t = 10-35 sec, the typical time scale for grand unified theory cosmology, we find that n must have been equal to one to an accuracy of one part in 1049 . Standard cosmology provides no explanation for this fact- it is simply assumed as part of the initial conditions.
In the inflationary model, however, n is driven during the period of inflation very rapidly toward one, as
n - 1 ()( e-2Ht , (16)
where H is the Hubble constant during inflation. Thus, in an inflationary model one can begin with a value of n far from unity, and inflation will drive n toward one with spectacular
13
swiftness. This mechanism is so effective, in fact, that it is expected to overshoot by a wide margin. This leads to the cleanest prediction of inflation, which is that even today the value of n should differ from one by no more than about one part in 104 • This discrepancy from one is a quantum effect, comprising the long wavelength tail of the density fluctuations that are possibly responsible for galaxy formation. The magnitude of these fluctuations is not fixed by theory, since we do not presently know enough about the parameters of particle physics models to calculate this number from first principles. Thus the magnitude must be estimated from observation, using the scale-invariant spectrum predicted by inflation to extrapolate from much shorter wavelengths.
So far I have described the flatness problem under the assumption that the cosmological constant A is identically zero. If it is nonzero, then the role of inflation can be described by specifying that it drives the universe to a state of geometric flatness, corresponding to
A n + 3H2 = 1. (17)
It is useful to regard the quantity on the left-hand side of this equation as neff, with the A/(3H2) term regarded as the vacuum energy contribution. Using this definition, inflation always drives the universe to neff = 1.
5.5 ABSENCE OF MAGNETIC MONOPOLES
Finally, the inflationary model can cure the "magnetic monopole problem." In the context of grand unified theories, cosmologies without inflation generally lead to huge excesses of superheavy ('" 1016 GeV) 't Hooft-Polyakov ['t Hooft, 1974; Polyakov, 1974; for a review, see Coleman, 1983] magnetic monopoles. These monopoles are produced at the grand unified theory phase transition, when the GUT Higgs fields acquire their nonzero values. The rapidity of the phase transition implies that the correlation length of the Higgs fields is very short, and the fields therefore [Kibble, 1976] become tangled in a high density of knots- these knots are the magnetic monopoles. For typical grand unified theories the expected mass density of these magnetic monopoles would exceed [Zel'dovich & Khlopov, 1978; Preskill, 1979; Guth & Tye, 1980] the mass density of everything else by a factor of about 1012 • A value this high would imply that the expansion rate of the universe would slow to its present value in only 30,000 years, which is rather clearly excluded by observation.
The monopole problem is easily solved in the context of an inflationary model by ar­ ranging for the Higgs field to acquire its nonzero expectation value either before or during the inflationary era. The monopole density would then become negligible as it is diluted by the inflation. Some monopoles would still have been produced by thermal fluctuations after reheating, but this production is suppressed by a large Boltzmann factor- it would be negligible in the minimal SU(5) model [Lazarides, Shafi, & Trower, 1982; Guth, 1983b], and presumably in most other models as well.
14
6 Conclusion
In evaluating the plausibility of a theory, it is always reasonable to ask if there are other theories that might provide alternative explanations of the same phenomena. In the case of inflation, I think it is fair to say that there are no direct competitor theories. For the most part, we are choosing between inflation and the possibility that these questions are not ready to be answered. Nonetheless, I would like to comment on two alternative points of view that I sometimes hear mentioned.
The first pertains to the flatness problem in particular. Some people have said that perhaps we are simply living in a flat universe, and that the flatness should be accepted as a fundamental law in its own right, not to be explained in terms of other laws. In response to this suggestion there are two points that I like to emphasize. First, one must recognize that the closed, open, and flat cases of the Robertson-Walker metric should not be viewed as being of equal a priori probability. The Robertson-Walker metric is in fact well-defined for any real value of the parameter k, and it is only by rescaling the coordinates and the scale factor that one achieves the three standard cases of k = +1, -1 and O. Thus the three cases really correspond to values of k that are positive, negative, or precisely zero. When phrased in this manner, the special case k = 0 sounds like it should have a probability of zero. Nonetheless, the laws of physics are not usually thought to be random, so this probability estimate is not a convincing answer to someone who claims that k = 0 has the status of a new law of physics. To such a person I would argue that the proposed law of physics is peculiarly vague. The ratio
(18)
is a quantity that must be defined by averaging over some region of space, and in fact we know that because of density fluctuations it will certainly not in general be equal to one. The most that one can hope for is that it will tend to one as the volume used for averaging approaches infinity, but such a statement is much more vague than any of the laws of physics that we normally discuss. In particular, this statement could never be falsified by measurements made in any finite volume, no matter how accurate those measurements were.
Secondly, I would like to comment on the anthropic principle, which is sometimes offered as an explanation of flatness and perhaps some other properties ofthe universe. This prin­ ciple holds roughly that some of the properties of the universe must be the way they are, or else intelligent life would never have evolved to observe it. To me, however, anthropic argu­ ments are unsatisfactory for two reasons. First, they presume a knowledge, which I think is lacking, of the minimal conditions necessary for intelligent life. Anthropic arguments are therefore convenient to "explain" what is known, but I am not aware of any predictions. Second, even if we had a complete understanding of life and its evolution, I would say that anthropic arguments entail the risk of bypassing the important questions. For example, I would guess that some properties of the water molecule are necessary to support life, so one might say that these properties are "explained" by the anthropic principle. Yet we would all agree that these properties are also dictated by the laws of quantum theory, and most of us would agree that the quantum approach is the more productive.
15
Most of our fellow scientists have avoided the anthropic principle completely. I am quite sure that I have never seen a book on anthropic chemistry, that I have never seen a book on anthropic nuclear physics, and that Jackson's book (even the newer and thicker edition) does not contain a chapter on the implications of the anthropic principle for electromagnetic theory. If used in biology, the anthropic principle would allow scientists to sidestep all questions about the evolution of life by arguing that otherwise we would not be here to ask the question- but I doubt very much that this argument would be acceptable to most biologists. If we are to pursue cosmology with the same standards and methods of argument that are used in the other sciences, then the anthropic principle should be avoided whenever possible.
To summarize, I think it is clear that the inflationary universe model explains many of the most salient features of the observed universe. It explains (1) why the universe contains a huge number of particles, (2) why the universe is undergoing Hubble expansion, (3) why the cosmic background radiation is so isotropic, (4) why the early universe had a mass density so extraordinarily close to the critical value, and (5) why the universe is not filled with magnetic monopoles. Of these arguments, I find items (3) and (4) to be the most persuasive, because of the precision involved. The isotropy of the cosmic background radiation has an accuracy better than one part in 104 , and the value of n at t = 1 sec is measured indirectly to an accuracy of 15 decimal places. Inflation also makes predictions: it predicts the value of n, and it makes a slightly less rigid prediction for the shape of the spectrum of the primordial density perturbations. It is still unclear whether these predictions will be borne out by observation, but we can look forward to hearing the latest results on these questions at this conference.
Acknowledgements. This work was supported in part by funds provided by the U. S. Department of Energy (D.O.E.) under contract #DE-AC02-76ER03069.
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DISCUSSION:
Borner: Are there possible observations that could disprove the inflationary idea?
Guth: In general, I think ideas die when they are replaced by a better idea. It's very rare that an idea is given up in exchange for absolutely nothing. In the case of inflation, I would say that if observations could convincingly show that n + (A/3H2) is not equal to one, then I think the theory would have to be given up. The density fluctuations, on the other hand, are not so clear-cut. If the density fluctuations were shown to have a primordial spectrum that is not Harrison-Zeldovich, then there are ways to modify the theory. I would still say, however, that measurements of the density fluctuations are crucial. If inflation is correct, they will help us to refine the theory. If inflation turns out not to be correct, then these measurements may help to point the way to the new theory that will replace it.
Rowan-Robinson: Surely inflation cannot predict a universe like the one we observe from arbitrary initial conditions? Specifically, if the inflation factor is Z, inflation predicts uniformity and flatness only if the initial inhomogeneity and deviation from flatness are < Z. In a sense, even though Z may be a large number, this still makes our universe a member of a set of measure zero. (George Ellis' argument).
Guth: Yes, inflation does not work for arbitrary initial conditions. For example, one can always start with a closed universe with all quantities of Planck scale, and such a universe will collapse on the order of a Planck time, without ever reaching the GUT scale for inflation to begin. However, I think it is the wrong question to ask whether inflation can work for arbitrary initial conditions. Inflation is a theory of evolution, and I think it should be judged by the same criteria that are used for other theories of evolution. In particular, I have never heard anyone criticize Darwin's theory of the evolution on the grounds that it doesn't work for arbitrary initial conditions. Inflation is not a complete theory of the universe, but simply tries to describe the evolution of the universe from about 10-35 sec
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to maybe 10-30 sec. So inflation depends on having some other theory of creation, which will describe how the initial conditions for inflation could have been achieved. I think inflation should be judged on whether it gives a plausible explanation of how the universe evolved to where it is now, starting from initial conditions that seem reasonable- more precisely, initial conditions that can hopefully be explained later by some theory of creation, or perhaps some theory that merely describes the period immediately before inflation.
Your question seems to refer specifically to the idea that space may have a fractal structure, with inhomogeneities persisting down to arbitrarily small length scales. We have to keep in mind, however, that the universe is quantum mechanical. This means that to have an excitation of wavelength A requires an energy of order lie/A. Thus it would require an infinite energy density to have excitations down to arbitrarily small wavelengths, so I think it's reasonable to assume that this is impossible. As long as the energy density is finite, there must be some wavelength below which there are no excitations at all. There are, however, also quantum zero point fluctuations, and the story about them is a bit more complicated. These zero point fluctuations do persist down to arbitrarily small wavelength, at least in the context of a quantum field theory, but the fluctuations are not arbitrary­ they are completely controlled by the theory. When one quantizes a scalar field in de Sitter space, one finds that the zero point fluctuations of extremely short wavelength at one time evolve to become the long wavelength zero point fluctuations at a later time. Furthermore, this simple evolution is dictated by the fact that the zero point fluctuations are de Sitter invariant. For the real theory, we in principle have to deal with the zero point fluctuations of quantum gravity on very short wavelengths, and here we of course have no rellable theory. Nonetheless, it seems reasonable to hope that these zero point fluctuations are also de Sitter invariant, since the classical theory is, and then the process of inflation will stretch these perturbations to become the usual de Sitter invariant zero point fluctuations on longer wavelengths- which are completely tame.
Rowan-Robinson: You need an anthropic argument to explain why we do not find ourselves in a region of false vacuum today.
Guth: Yes, I guess I do. But I guess that's the only application of the anthropic principle that I am happy with. I would say that it avoids each of the two objections to anthropic arguments that I mentioned in my talk. It does not rely on any knowledge of what it takes to produce intelligent life, except for the rather safe assumption that life cannot form in the inflating false vacuum. Furthermore the properties of the universe (its constituents, its mass density, density fluctuations, everything) are still determined by the mechanism of inflation, so this anthropic argument is not an attempt to bypass such an explanation.
Smoot: What level of isotropy and homogeneity is needed for inflation to begin? Then can you estimate the probability of inflation starting by comparing the needed conditions to the Planck scale?
Guth: It's hard to quantify, but one can solve for the evolution of a sphere of false vacuum surrounded by true vacuum. If the sphere is larger than one horizon volume, inflation will begin. So we assume that the necessary condition for inflation is approximate homogeneity and isotropy on the scale of the horizon. If we assume that the universe
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survives long enough to reach the energy scale of the false vacuum, then the horizon length is the typical length scale, and inflation looks plausible. A detailed calculation depends, however, on assumptions about the initial conditions, and different authors have reached different conclusions.
There is the associated question of whether the universe ever reaches the false vacuum energy scale. IT the patch of universe under discussion resembles an open universe, then it will expand forever and there is no problem. IT it is closed, however, then some mechanism is needed to make it plausible for the patch to survive long enough- one possibility is a Planck-scale episode of inflation, which can be totally independent of the inflation to occur later. (The most recent episode of inflation cannot, however, be at the Planck scale, because too much gravitational radiation would be produced.) Another possibility is the creation of the universe by quantum processes, with a quantum transition directly into the false vacuum state.
Rees: The mass within the horizon of a "true-vacuum" Einstein-de Sitter region increases linearly with time. So will our descendents eventually see the edge of our domain?
Guth: Yes, in principle our ancestors will at some time in the distant future be able to see the edge of our domain. It is possible even that the domain is closed, in which case the wall will hit them in the face. The time for this to happen, however, is proportional to the cube of the overall expansion factor, and so it will not happen until the extremely distant future.
Sarkar: You suggested that the entropy of the universe is large. How would you respond to the remark by Roger Penrose that the observed entropy ('" 1010 per baryon) is actually completely negligible relative to the natural value of the gravitational entropy ('" 1040 per baryon)? Does inflation shed any light on the very special initial conditions which seem to be required for the gravitational field? In other words, how did the universe survive from the Planck scale down to the energy scale where inflation occurred.
Guth: The comment of Penrose that you cite is quite old, and was made in the days when people assumed that baryon number was exactly conserved. From that point of view the baryon number of the visible universe, about lOBO, was assumed to be present from the start. It was in this context that Penrose pointed out that the entropy of these baryons would have been much higher if they were all put into one black hole. In the present context in which baryons are believed to be produced by baryon nonconserving processes, I don't see that Penrose's argument has any relevance. The first step is to explain the entropy of the visible universe, '" 1090, and inflation can do this. The next step is to explain how each 1010 units of entropy can produce one unit of baryon number, and grand unified theories can presumably do this. And if inflation has spread the entropy uniformly through space, then the baryons will also form uniformly, which is consistent with what we observe.
Rees: Although I'm perhaps less antipathetic to anthropic arguments than you are, I think the strongest reason for regarding them as inadequate comes from the large-scale anisotropy (tl.T IT < 10-4 ). There seems no purely anthropic reason why our existence requires the universe to be so smooth on scales larger than superclusters.
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Guth: Yes, I agree.
Hoyle: Are the partial differential equations connecting inflationary regions with those of false vacuum actually solved? Or are the regions so big that the equations in different regions are effectively chopped off from each other?
Guth: The equations that describe a spherically symmetric inflationary universe that forms within a larger region of false vacuum are not difficult to write down and solve. And since the de Sitter region expands so quickly that if forms causal horizons, these newly forming universes will affect only a finite region of comoving coordinate space. It is then easy to piece together solutions of this type to form the schematic fractal picture of the decaying false vacuum that I showed in my talk. However, in a more realistic picture the decay of the false vacuum would be random, the regions might not have spherical symmetry, and the decaying regions of false vacuum would sometimes overlap. This situation could get very complicated, and I would not be at all capable of solving these equations.
Starobinsky: Comment: First, I would like to remind the audience about two of my papers on the Universe originating from the maximally symmetric (de Sitter) initial state (this stage of exponential expansion was later called "inflation" by Prof. A. Guth). The first paper (1979) contains a direct observational test of the existence of such a state in the past - the search for a stochastic background of gravitational waves with the char­ acteristic flat spectrum generated during the de Sitter stage. The second paper (1980) contains an internally self-consistent model of the Universe beginning from the de Sitter stage and ending in the radiation - dominated stage. This model is purely geometrical, it is based on a variant of the so-called higher-derivative theory of gravity. However, it is remarkable that it was proved later that, from the gravitational point of view, this model is completely mathematically equivalent to inflationary models based on the Einstein gravity interacting with a minimally coupled scalar field with some specific potential (the latter is actually intermediate between those used in "new" and "chaotic" models). Thus, from the gravitational point of view, there exists no principal differences between these models.
Second, let me add a response to the question of Prof. Rees, too. It seems to me that in case of many low-density universes separated by still inflating domains borders of the latter ones are inaccessible for us (just as we can't hit a particle moving with a constant acceleration even by means of photons, if it is sufficiently far from us initially). Moreover, if constant energy density hypersurfaces are defined to be the hypersurfaces of constant time or the synchronous system of reference is used (as is usually done in cosmology), these inflating domains and other universes beyond them always lie in the past for us.
PREDICTIONS OF INFLATION
ANDREW R. LIDDLE Astronomy Centre, Division of Physics and Astronomy, University of Sv.ssez, Falmer, Brighton BNl 9QH, U. K.
ABSTRACT. Many models exist based upon the inflationary universe paradigm. The predictions of these models are outlined, and it is emphasised that the differing models can have substantially different implications. Consequently, inflation seems more likely to fall from favour through us being restricted to unappealing models rather than being explicitly ruled out. Whether or not the universe
is at the critical density remains the most solid testable prediction of inflation.
1 Inflation and its Aims
After ten years, inflation (for a selection of reviews of differing emphasis, see [1,2,3,4]) remains a very attractive solution to a number of cosmological conundrums, and its influence pervades, sometimes explicitly and sometimes implicitly, a considerable segment of modern cosmology (for instance, it is often used as a justification for a scale invariant spectrum in structure formation simulations [5]). One is therefore entitled to ask how much observational evidence actually points in favour of the inflationary cosmology, and further one should of course be very interested in knowing the types of evidence capable of ruling out the scenario. This, in a nutshell, is the purpose of this conference.
As this article is very much aimed at providing an introduction and setting for topics which will be examined in more detail in the remainder of this volume, I intend to keep my presentation almost entirely non-mathematical, with pointers to the relevant parts of the literature should the reader be interested in following up the details. After a brief reminder of the aims of inflation, I will classify the different types of inflationary model before proceeding with a discussion of the predictions (some fairly solid, others much less so) that these models make. One point which will become clear during the article is that a huge number of models have sprung from the original inflationary paradigm. These models are capable of a whole range of predictions, and this means that inflation as a whole shows considerable robustness in the face of observations. Hence the immediate moral is that it is very hard to slay inflation with a single observation.
I'll begin by defining what I mean by inflation. Inflation proposes to solve a number of cosmological problems by invoking a very rapid expansion of the universe in its earliest stages [6,7]. The definition of inflation I will use is a period in the early universe where the scale factor a obeys the condition a (t) > O. (For example, conventional exponential inflation a '" eHt has a '" H 2eHt .) Equivalently, inflation is an epoch in which the separation between points that have been in causal contact may increase faster than the speed of light.
The hot big bang model is unable to answer several problems whose origin is outwith
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T. Shanks et al. (eds.), Observational Tests o/Cosmological Inflation, 23-38. © 1991 Kluwer Academic Publishers.
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the influence of the model. A selection of these are
• The Flatness Problem: Why is n (the ratio of the universe's density to the critical density that makes the universe flat) so close to one, when in the big bang model n = 1 is an unstable critical point?
• The Horizon Problem: Why are points at the opposite ends of the universe, which have never experienced causal contact, in thermal equilibrium at the same tempera­ ture?
• Homogeneity and Isotropy: Why does the universe have these properties (on sufficiently large scales that dynamics has been unable to move matter around signif­ icantly)?
• Relic Abundances: What stopped massive stable relics, such as monopoles, coming to dominate the universe?
As discussed in the previous paper by Guth [8], a sufficiently long period of inflation, mul­ tiplying the scale factor by at least 1030, can provide an explanation for all these problems.
2 Inflationary Models
Before discussing the predictions that the inflationary model makes, it is necessary to detail the mechanism by which inflation is envisaged to arise. For this, only one observation is required; that if the universe is dominated by an energy density the solutions to Einstein's equations are an exponential expansion of the scale factor, satisfying the condition for inflation. Such solutions are well known from studies of space-times with a cosmological constant [9]. However, much more significantly for our purposes, such an expansion arises if the universe is dominated by the potential energy of some quantum field. This is normally taken to be a scalar field (though some work has been done where inflation is driven by a vector field [10]); ideally this would be a field associated with a symmetry breaking in the early universe as such fields are ubiquitous in current particle theories. Recent trends have however favoured the introduction of scalar fields for the sole purpose of providing a satisfactory inflationary model. In any case, the potential energy of the scalar mimics a cosmological constant and leads to inflation.
For the purposes of this article, the plethora of inflationary models can be slotted into two categories. This is a rather lopsided classification, with nearly all models belonging in the first category, but useful because the two categories lead to rather different predictions.
2.1 SLOW-ROLLING INFLATION
Members: New inflation [7], chaotic inflation [11], power law inflation [12,13]' R2 (and other super-Einstein) inflation [14], double inflation [15] ...
In slow-rolling inflation, a scalar field is displaced from the minimum of its potential. Under a wide range of circumstances [16], its kinetic energy becomes negligible and its evolution is dominated by its potential energy, thus mimicking a cosmological constant and
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providing inflation. The scalar field 'slow-rolls' to the minimum of its potential, bringing inflation to an end - see figure 1.
V(u)
o
o u Figure 1: Slow rolling down a potential.
This class includes almost all currently favoured examples; the variants are plentiful due to the wide choice of possible potentials and the different identities of the scalar, which for instance may be a general scalar introduced solely to provide inflation, or connected with extra compact spatial dimensions, or a consequence of extra terms in the gravitational action. The expansion of space may be approximately exponential, or a power law, or something intermediate. A small subset of the options, where V( a) is the potential of the scalar field a, are
• One scalar field, V( a) = tm2a2: Standard chaotic inflation scenario, with a massive scalar field. Apparently experimentally viable for m < 1014 GeV.
• One scalar field, V(a) = VoC/A<T: Exact power law inflation solutions a ex: t2//A2 •