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**Springer** Tracts in Modern Physics Volume 210 Managing Editor: G Höhler, Karlsruhe Editors: J Kühn, Karlsruhe Th Müller, Karlsruhe A Ruckenstein, New Jersey F Steiner, Ulm J Trümper, Garching P Wölfle, Karlsruhe Starting with Volume 165, **Springer** Tracts in Modern Physics is part of the [SpringerLink] service For all customers with standing orders for **Springer** Tracts in Modern Physics we offer the full text in electronic form via [SpringerLink] free of charge Please contact your librarian who can receive a password for free access to the full articles by registration at: springerlink.com If you not have a standing order you can nevertheless browse online through the table of contents of the volumes and the abstracts of each article and perform a full text search There you will also find more information about the series **Springer** Tracts in Modern Physics **Springer** Tracts in Modern Physics provides comprehensive and critical reviews of topics of current interest in 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www-ttp.physik.uni-karlsruhe.de/∼jk Thomas Müller Institut für Experimentelle Kernphysik Fakultät für Physik Universität Karlsruhe Postfach 69 80 76128 Karlsruhe, Germany Phone: +49 (7 21) 08 35 24 Fax: +49 (7 21) 07 26 21 Email: thomas.muller@physik.uni-karlsruhe.de www-ekp.physik.uni-karlsruhe.de Fundamental Astrophysics, Editor Joachim Trümper Max-Planck-Institut für Extraterrestrische Physik Postfach 13 12 85741 Garching, Germany Phone: +49 (89) 30 00 35 59 Fax: +49 (89) 30 00 33 15 Email: jtrumper@mpe.mpg.de www.mpe-garching.mpg.de/index.html Solid-State Physics, Editors Andrei Ruckenstein Editor for The Americas Department of Physics and Astronomy Rutgers, The State University of New Jersey 136 Frelinghuysen Road Piscataway, NJ 08854-8019, USA Phone: +1 (732) 445 43 29 Fax: +1 (732) 445-43 43 Email: andreir@physics.rutgers.edu www.physics.rutgers.edu/people/pips/ Ruckenstein.html Peter Wölfle Institut für Theorie der Kondensierten Materie Universität Karlsruhe Postfach 69 80 76128 Karlsruhe, Germany Phone: +49 (7 21) 08 35 90 Fax: +49 (7 21) 69 81 50 Email: woelfle@tkm.physik.uni-karlsruhe.de www-tkm.physik.uni-karlsruhe.de Complex Systems, Editor Frank Steiner Abteilung Theoretische Physik Universität Ulm Albert-Einstein-Allee 11 89069 Ulm, Germany Phone: +49 (7 31) 02 29 10 Fax: +49 (7 31) 02 29 24 Email: frank.steiner@physik.uni-ulm.de www.physik.uni-ulm.de/theo/qc/group.html Dierck-Ekkehard Liebscher **Cosmology** With 100 Figures ABC Dierck-Ekkehard Liebscher Astrophysikalisches Institut Potsdam An der Sternwarte 16 14482 Potsdam, Germany E-mail: deliebscher@aip.de Library of Congress Control Number: 2005921920 Physics and Astronomy Classification Scheme (PACS): 98.80.-k, 04.20.Cv, 01.55.+b ISSN print edition: 0081-3869 ISSN electronic edition: 1615-0430 ISBN -10 3-540-23261-3 **Springer** Berlin Heidelberg New York ISBN -13 978-3-540-23261-2 **Springer** Berlin Heidelberg New York This work is subject to copyright All rights are reserved, whether the 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package Cover concept: eStudio Calamar Steinen Cover production: design &production GmbH, Heidelberg Printed on acid-free paper SPIN: 11316787 56/3141/jl 543210 Preface This book is intended to explain the theoretical issues of **cosmology** To some extent, it is a complement to presentations that emphasise the progress and success of our observational power and the importance of determining phenomenological parameters Since Einstein solved Newton’s paradox through use of the theory of general relativity, the universe has been described in that framework The present standard model is the result of many observations, which culminated in the celebrated determination of the structure in the microwave background This book will acknowledge this development, of course, but it will emphasise the theoretical aspects of **cosmology** that can be easily forgotten in view of such a success Just as stars and black holes have been found through mere solutions to theory (i.e without observation), **cosmology** too is a scientiﬁc topic, and was so even before anyone knew what to observe The central issue of **cosmology** is the compatibility of diﬀerent areas of physics This compatibility ought not be bounded or restricted by horizons of any kind In contrast, the realm of experience is always limited by the observational or experimental means currently available or possible in principle The compatibility, or universality, of physics is nonetheless indispensable for our trust in physics itself **Cosmology** is an exciting science, even though and indeed because neither its subject nor its epistemological status can be described in a way that is generally accepted On the one hand, **cosmology** is the science of global extrapolation; on the other, it is understood as the science of the gigantic history of the evolving universe This evolution is due to the overall expansion, which pushes many systems away from equilibrium even though the expansion itself is adiabatic Modern **cosmology** is closely connected with the physics of elementary particles and their interactions The energy scale of elementary particles, atoms and molecules is reﬂected in a timescale in the history of the universe The expansion of the universe must have begun from a time when it was so hot, that any value of energy that we might consider at present (from the nuclear binding energy to the grand uniﬁcation scale) has existed as a thermal energy in the distant past In particular, the processes that are believed to occur at the energies of grand uniﬁcation will only have taken place in a past VI Preface so distant that their consequences will, at best, be observed as relics of that stage Modern **cosmology** is closely connected with the theory of general relativity, and the fundamental question of its relation to quantum theory is reﬂected in the question of the origin of our universe Cosmological models reveal the abstract features of the singularities of general relativity and all their consequences General relativity has to understand other exciting objects, too, but the universe is the simplest object that can be consistently described only by curved space-time, which is the essential invention of general relativity Nevertheless, the notions of **cosmology** can be explained without going too deeply into general relativity or quantum theory, because the simplicity of the overall motion allows us to reduce most of the arguments to phenomenological thermodynamics, and because the general-relativistic equations for the expansion of the universe can be reduced in the case of our cosmological models to equations that are formally equivalent to Newtonian ones This book will introduce the reader to **cosmology** not only with general arguments, but also with the key formulae These formulae will be given with their derivations The reader not only should get an overall impression, but also should be enabled to read more advanced literature without too much trouble to put everything into place The network of basic arguments and basic facts will be presented in a formal context, too In this way, the book will be not only an introduction, but also a kind of revision course Modern **cosmology** rests upon the continuing expansion of the universe, which was identiﬁed ﬁrst by Hubble through an appropriate interpretation of the redshift that increased with the apparent distance Friedmann had already found the correct geometrical form of the cosmic expansion as a solution of the ﬁeld equations of Einstein’s theory of general relativity The expansion triggers the decay of various kinds of equilibria The slow formation of structure by gravitational condensation and the fast nuclear binding reactions can be understood as the result of such decays Two basic observations support the picture of the Friedmann universe First, we observe a comparatively large concentration of deuterium, which can only be explained by some suddenly halted nucleosynthesis in a universal neutron-rich environment The result of primordial nucleosynthesis is explained by reaction kinetics that describe processes that start after cooling and are halted after rarefaction, both of which result from the expansion Second, the heat bath of the universe, now decoupled and cooled by the expansion, forms the microwave background This background is astonishingly isotropic The density perturbations that are reﬂected in its ﬂuctuations are very weak and have a remarkably featureless power spectrum After 1980, a loop of argument was found that makes the arguments about the evolution of the universe very strong (Fig 1) When applied to baryon– antibaryon annihilation in a charge-symmetric universe, the nucleosynthesis paradigm leads to a present baryon concentration that is 10 orders of Preface Inflation Monopole problem Symmetry breakdown Seed suppression Baryon asymmetry Structure evolution Gamow’s conjecture Microwave background Friedmann expansion VII Helium and deuterium Hubble expansion Fig The cosmic loop magnitude lower than the observed value An initial baryon asymmetry must be assumed, and grand uniﬁed theories provide an explanation of how such an asymmetry can result from asymmetric decays of GUT particles In grand uniﬁed theories, symmetries may be spontaneously broken This is ﬁne for the baryon asymmetry, but is bad for the present density of the universe, which would exceed the observed value by far since the breakdown of symmetry leaves too many topological quasi-particles (monopoles) behind How we get rid of these monopoles? We need a dilution in a phase of exponential increase of the size of the universe By coincidence or not, the grand uniﬁed theories contain the concept of a high-temperature vacuum that drives such an inﬂation when the temperature falls below the temperature of symmetry breakdown The universe cools down to nearly absolute zero before the hightemperature vacuum decays into a low-temperature vacuum, releasing its energy density into the degrees of freedom of ordinary GUT particles Now, the zero-point ﬂuctuations that remain lose their coherence and turn into ordinary perturbations, which show correctly the power spectrum inferred from observations Grand uniﬁed theories also keep ready exotic particles VIII Preface that may be candidates for the dark matter that is suspected to exist in galaxies, clusters and large-scale structures The loop is closed by the phenomena in the microwave background and the evolution of structure in the transparent universe This book will follow this cosmic loop After a short introduction to the geometry of space-time in general and the geometry of an expanding universe in particular, we explain the nuclear-synthesis paradigm, follow the argument about inﬂation, describe the inﬂationary scenario, consider the origin of perturbations and follow their evolution to the recombination time and after This closes the loop Finally, we return to the basic questions of consistency and the cosmological singularity, i.e we consider the basic notions of quantum gravity and other extensions of the standard picture Potsdam February **2005** Dierck-Ekkehard Liebscher Contents Basics 1.1 The Cosmos, the Universe and the Metagalaxy 1.2 Explanation and Evolution 1.3 The Cosmological Principle 1.4 Observation and Measurement 1.5 Newton’s Paradox 1.6 Olbers’ Paradox 1.7 Lambert’s Solution 1.8 Einstein’s Solution 1.9 Friedmann’s Solution and the Hubble Expansion 1.10 Gamow’s Conjecture 1.11 Jeans’ Problem 1.12 Lemaˆıtre’s Problem References 1 8 10 11 12 16 18 19 22 Relativity 2.1 Relativity of Simultaneity: Solving Fresnel’s Paradox 2.2 The Equivalence Principle, Deﬂection of Light and Geometry 2.3 General Relativity: Solving Galileo’s Paradox 2.4 Positive Curvature: Solving Newton’s Paradox References 25 25 Expansion 3.1 The Friedmann Equations 3.2 The Worlds of Constant Curvature 3.3 Worlds with Matter Only 3.4 Barotropic Components of Matter 3.5 Friedmann’s Universe: Solving Olbers’ Paradox 3.6 The Cosmological Singularity References 53 53 58 65 67 69 74 77 Cosmometry 79 4.1 The Past Light Cone 79 4.2 Horizons 81 34 42 46 50 X Contents 4.3 The Cosmological Redshift 4.4 Distance Deﬁnitions in the Expanding Universe 4.5 Distance Determinations 4.6 Determinations of the Expansion Rate 4.7 Curvature Determinations 4.8 Gravitational Lenses 4.9 Quasar Absorption Forests References 83 88 92 98 101 104 111 115 Matter and Radiation 5.1 The Average Mass Density 5.2 Counting Procedures 5.3 The Microwave Background Radiation 5.4 The Photon Bath and the Notion of the Ether 5.5 The Homogeneous Large-Scale Structure 5.6 Correlations and Power Spectra 5.7 Fractal Structure References 119 119 123 125 127 129 131 137 139 Standard Synthesis 6.1 The Gamow Universe 6.2 The Standard Process and the Decay of Equilibrium 6.3 The Primordial Nucleosynthesis 6.4 Weakly Interacting Particles 6.5 The Problem of the Baryons References 143 143 146 154 160 161 166 Inﬂation 7.1 The Implications of the Monopole Problem 7.2 The Vacuum 7.3 The Inﬂaton Field 7.4 Homogeneous Inﬂation 7.5 Concomitant Solutions to Fundamental Problems 7.6 Inhomogeneities and Inﬂation 7.7 Variations References 167 167 168 170 173 176 182 184 188 Structure Formation in the Opaque Universe 8.1 Perturbations in General 8.2 The Relativistic Approach and the Evolution on Large Scales 8.3 Inhomogeneities and Inﬂation 8.4 The Evolution of Small Scales and the Microwave Background References 189 189 194 199 205 212 278 13 Machian Aspects 13.3 Time without Time One way out of this situation is the argument of Barbour [5] that we have gone only half the way if we give up absolute position and orientation in space but not in time Barbour returns to a concept of motion that takes the time as a pure variation in the conﬁguration of a system and measures the time by this variation only This concept is familiar through Maupertuis’s, Fermat’s and Jacobi’s principles: the orbit of a motion is a kind of shortest path, and the measure of the path reﬂects the potential of gravitation and the inertia, but does not contain the time explicitly In classical mechanics, the motion through conﬁguration space is determined by the minimum of the integral S= dt E − Epotential 2Ekinetic This integral does not depend on time; t is a freely exchangeable parameter The kinetic energy yields a metric for the conﬁguration space, ds2Maupertuis = 2(E − Epotential )Ekinetic dt2 When a solution has been found, the condition Ekinetic + Epotential = E yields the ﬂow of time Ephemeris time is determined explicitly in this way The discovery is that this procedure can be applied to the theory of general relativity Here, we deﬁne a superspace The elements of this superspace are the states of a space, that are represented by a metric ds2 = hik dxi dxk of the space The gravitational ﬁeld in fourdimensional space-time has the form ds2 = −(N − Ni N i )dt2 + 2Ni dxi dt + hij dxi dxj It is a curve in superspace (Chap 12), which represents the history of the universe In analogy to the familiar type of mechanics, we deﬁne the Maupertuis metric in the superspace [2, 6] The corresponding action integral is obtained as S= dλ d3 x 3R Gabcd ∂gab − N(a|b) ∂λ ∂gcd − N(c|d) ∂λ This yields the complete solution of the problem of how the metrics of space are constructed exclusively relative to each other, and of how they vary along the path in superspace and determine the time ﬂow like an ephemeris time as well A particular point is that one can possibly use this integral to interpret a 13.5 The Assumed and the Explained 279 canonical quantum ﬁeld theory in which the Wheeler–de Witt equation does not contain the time but yields only probability amplitudes in superspace From this point of view, Mach’s principle supports the three-dimensional concept of the canonical quantisation of the gravitational ﬁeld, and does not reach beyond the theory of general relativity 13.4 Measure by Mass In view of the initial problem, one may ask for more: after all, the metric ds2 = gik dxi dxk of GRT gives an expression for the orientation of an inertial system, which should be determined not with respect to itself but with respect to matter, i.e the distribution of the energy-momentum tensor The Einstein equations state this immediately, Rik − gik R = κTik From these equations, the metric, i.e the conﬁguration of an inertial reference system, is not determined solely by the matter distribution We have to choose boundary conditions to ﬁx the geometry for a given matter distribution This is the back door through which an equivalent to absolute space comes back in Hence, it is important to look for conditions that allow us to proceed without choosing boundary conditions, i.e to look for an integral formulation of Einstein’s equations This can be done for space-times with closed space sections, i.e for space-times that are related to the Einstein model Independent of other ﬁndings along this route, however, the result can only be a general selection rule for solutions of the Einstein equations This concept is strange in the context of other ﬁeld theories [2, 15, 16, 24] 13.5 The Assumed and the Explained The historical development from Newton’s mechanics to the GRT can be understood as a repeated withdrawal of absolute elements from physics Special relativity removed absolute simultaneity and replaced the formal product of space and time with an inseparable union The determination of absolute velocities that appeared to be feasible through a combination of mechanics and electrodynamics was found to be impossible, and the relativity of velocity was restored The GRT explained the local relativity of accelerations as a consequence of the local equivalence of inertia and gravitation The fact that all objects fall with the same acceleration allows the determination of relative accelerations only, that is, of tidal forces [12] If Mach’s principle is expected to lead beyond the GRT, it must be understood as a programme to explain absolute features of the GRT as the result of some generalised 280 13 Machian Aspects dynamics The absolute element in the GRT is the (local) Lorentz invariance of inertial reference frames, the existence (not the actual form) of a unique metric of space-time that ﬁxes the validity of the special relativity theory on an inﬁnitesimal scale From this point of view, Mach’s principle is a programme to replace the local Lorentz invariance with a more general invariance, and to understand the local Lorentz invariance as a result of a classical breakdown of symmetry that is observed in small regions of space-time (on a cosmic scale) only Such a programme corresponds to what we have learned from relational mechanics and has a ﬁeld-theoretical background now The simplest generalisation of Lorentz invariance is conformal invariance This expects that all ﬁeld equations and equations of motion not only are also valid irrespective of Lorentz transformations, but not change with rescaling, in particular with a rescaling of mass The mass is immediately dependent on the conﬁguration of the universe We start with elementary arguments We know that the natural unit of mass, the Planck mass MPlanck = ¯c h ≈ 10−5 g , G is a combination that contains the velocity of light, even though the notion of mass was born in Newtonian mechanics and does not depend on the special-relativistic geometry of space-time If, instead, we construct the natural unit of mass as a combination of the gravitational constant, the Planck constant and the cosmological constant2 Λ ≈ Λcrit ≈ 10−52 m−2 , we obtain the Eddington mass [31, 36] MEddington = h4 Λ ≈ × 10−25 g G2 This value is an astonishingly good approximation to the mass scale of nucleons Are we allowed to interpret this result as an indication that the size of the universe represented by Λ is reﬂected in the masses of the stable particles, which are nearly zero in comparison with the characteristic masses of uniﬁed quantum ﬁeld theories (mp ≈ 10−15 MGUT ≈ 10−19 MPlanck )? With a vanishing cosmological constant, the Eddington mass vanishes too A constructive theory for such a relation, however, does not exist, and it remains an anecdote only A theory that contains the reduction of an a priori conformal invariance to Lorentz invariance usually requires a scalar ﬁeld that describes the scaling that is obtained through the breakdown of a symmetry Such a theory is a scalar–tensor theory for the gravitational ﬁeld, in contrast to the GRT, which is a pure tensor theory The Brans–Dicke theory [10] and the Hoyle–Narlikar theory [13, 14] are the main representatives The cosmological constant can be determined today by various methods Our −2 general argument may use Λcrit = 3RH 13.5 The Assumed and the Explained 281 Irrespective of the observational limits, however, a conformal theory is too short an extrapolation of the GRT; it does not change anything in relation to the existence of a causal structure, and needs purely conventional calculations only The next step is to determine not only the local scale but also the very existence of the light cone when a symmetry breakdown is induced by the universe This may be achieved by a theory that concerns a priori only the aﬃne connection, and assumes the relation between two space-time volumes to be the simplest dynamical element [9, 18, 21, 22, 30] In such a theory, there is no causality beyond the limits of a region that is small on a cosmic scale, and the existence of time and of causal order is a consequence of the existence and symmetry of the universe The metric ﬁeld does not exist without the matter in the universe that surrounds us.3 Even without a particular theory of this type, characteristic eﬀects can be identiﬁed If the universe diﬀers from the ideal state that leads to local Lorentz invariance, if it is not ideally isotropic (and our galaxy disturbs the ideal universe with its potential of 10−6 ), ﬁeld components must exist that propagate with velocities slightly diﬀerent from the abstract absolute velocity that characterises Lorentz invariance We can test for component-dependent signal propagation [1] The general Euler–Lagrange equations of second order for a set of ﬁelds ΦA , CAB ik ΦB , ik = − ∂2L ∂L ∂L ΦB , i − , g, i + ∂ΦA , i ∂ΦB ∂ΦA , i ∂g ∂ΦA (13.5) determine the propagation properties of a signal through the coeﬃcients of the second-order derivatives CAB ik = ∂2L ∂ΦA ,i ∂ΦB ,k Let us suppose a situation where the ﬁeld is discontinuous on a wavefront z(x0 , , x3 ) = const , for instance ΦA (z > 0) = ΦA (z < 0) + ϕA z We obtain, through the ﬁeld equations, the condition CAB ik z,i z,k φB = Es wă are meiner Meinung nach unbefriedigend, wenn es eine denkbare Welt ohne Materie gă abe Das g -Feld soll vielmehr durch die Materie bedingt sein, ohne dieselbe nicht bestehen kă onnen Das ist der Kern dessen, was ich unter der Forderung von der Relativită at der Tră agheit verstehe.’ (In my opinion it would be dissatisfying, if there were a conceivable world without matter The g µν -ﬁeld should rather be determined by the matter, and not be able to exist without it This is the heart of what I understand by the demand of the relativity of inertia.) Einstein writes this in a letter to de Sitter (24.3.1917) However, he then believes that this in an argument for the inclusion of the cosmological constant 282 13 Machian Aspects This is a linear homogeneous equation for the discontinuity amplitudes φB Non-trivial solutions exist on hypersurfaces with normals z,k that obey det(CAB ik z,i z,k ) = In a Lorentz-invariant theory, this condition is simpliﬁed to yield g ik z,i z,k = The system (13.5) of equations is Lorentz-invariant in the shock-wave approximation if the set of coeﬃcients C factorises into a product of the form CAB ik = aAB g ik If the Lorentz invariance is weakly disturbed, we expect CAB ik = aAB g ik + AB ik The determinant condition for the existence of a non-trivial shock amplitude yields, to ﬁrst order in , the equation (g ik z,i z,k )(N −1) (g lm + aAB AB lm )z,l z,m = To this order, all ﬁeld components except one propagate on the common light cone, but the one exception deﬁnes its own cone In every subsequent order, a further component leaves the common light cone The propagation becomes dependent on a kind of generalised isospin (Fig 13.2) Only a real theory can decide which ﬁeld components yield the maximal deviation and whether one should observe an anisotropy of the mass (which has been shown for some components to be smaller than 10−24 ) Hence, the task would be to identify the ﬁeld components that show anomalous propagation in lower orders and are observable The concept of a spontaneous breakdown of symmetry was ﬁrst introduced in the form of the Higgs mechanism (Chap 7) Can this mechanism be made responsible for the symmetry breakdown considered in this chapter? A theory that uses a Higgs mechanism [11, 25, 26] refrains from using the universe to deﬁne orientation It remains in this sense local, and it has to explain the longrange coherence of the symmetry-breaking ﬁeld This ﬁeld then represents the former absolute space in a new form The concept of ﬁelds that determine the current physical laws with universe-wide coherence leads to the inﬂationary universe, in all its forms From this point of view, the inﬂationary universe is the anti-Mach universe per se References 283 Fig 13.2 Metaphor of the splitting of the light cone with increasing perturbation References Audretsch, J., Bleyer, U., Lă ammerzahl, C.: Testing Lorentz invariance with atomic beam interferometry, Phys Rev A 47 (1993), 4632–4640 281 Baierlein, R F., Sharp, D H., Wheeler, J A.: Three-dimensional geometry as carrier of information about time, Phys Rev 126 (1962), 1864–1865 278, 279 Barbour, J B.: Forceless Machian dynamics, Nuovo Cimento B 26 (1975), 16–22 275 Barbour, J B.: Absolute or Relative Motion? Vol 1, The Discovery of Dynamics, Cambridge University Press (1990) 274 Barbour, J B.: The End of Time, Weidenfeld & Nicholson, London (1999) 278 Barbour, J B.: The emergence of time and its arrow from timelessness, in J J Halliwell, J P`erez-Mercader, W H Zurek (eds.): Physical origins of time asymmetry, Proceedings, Cambridge University Press (1992) 278 Barbour, J B., Bertotti, B.: Gravity and inertia in a Machian framework, Nuovo Cimento B 38 (1977), 1–27 275 Barbour, J B., Pﬁster, H (eds.): Mach’s Principle From Newtons Bucket to Quantum Gravity, Birkhă auser, Boston (1994) 274, 283, 284 Bleyer, U., Liebscher, D.-E.: Mach’s principle and local causal structure, in [8], 293-307 (1994) 281 284 13 Machian Aspects 10 Brans, C H., Dicke, R H.: Mach’s principle and a relativistic theory of gravitation, Phys Rev 124 (1961), 925–935 280 11 Carlini, A., Greensite, J.: Why is spacetime Lorentzian? Phys Rev D 49 (1994), 866–878 282 12 Ehlers, J.: Machian ideas and general relativity, in [8], 458-473 (1994) 279 13 Hoyle, F., Narlikar, J V.: Action at a Distance in Physics and Cosmology, Freeman, New York (1974) 280 14 Hoyle, F., Narlikar, J V.: Mach’s principle and the creation of matter, Proc R Soc London A 273 (1963), 1–11 280 15 Isenberg, J., Wheeler, J A.: Inertia here is ﬁxed by mass-energy there in every W model universe, in Pantaleo, M., DeFinis, F (eds.): Relativity, Quanta and **Cosmology** in the Development of the Scientiﬁc Thought of Albert Einstein, Johnson Reprint Corp., New York, 267–293 (1979) 279 16 Isenberg, J A.: Wheeler–Einstein–Mach space-times, Phys Rev D 24 (1981), 251–256 279 17 Kasper, U., Liebscher, D.-E.: On the post-newtonian approximation of theories of gravity, Astron Nachr 295 (1974), 11–17 277 18 Lanczos, C.: Signal propagation in a positive deﬁnite Riemannian space, Phys Rev 134 (1964), 475480 281 ă 19 Lense, J., Thirring, H.: Uber den Einﬂuß der Eigenrotation der Zentralkă orper auf die Bewegung der Planeten und Monde nach der Einsteinschen Gravitationstheorie, Phys Z 19 (1918), 156–163 274 20 Liebscher, D.-E.: Inertia-free mechanics and the bi-metric procedure, Astron Nachr 302 (1981), 137–142 277 21 Liebscher, D.-E.: Purely aﬃne ﬁeld theories, Ann Phys (Leipzig) 45 (1988), 200–204 281 22 Liebscher, D.-E., Yourgrau, W.: Classical spontaneous breakdown of symmetry and the induction of inertia, Ann Phys (Leipzig) 36 (1979), 20–24 281 23 Mach, E.: Die Mechanik in ihrer Entwicklung, Leipzig (1883) 273 24 Maltsev, V K., Markov, M A.: On the integral formulation of the Mach’s principle in a conformally ﬂat space, Trudy FIANa (1976), 9–23 279 25 Ne’eman, Y., Sijacki, Dj.: Gravity from symmetry breakdown of a gauge aﬃne theory, Phys Lett B 200 (1988), 489–494 282 26 Peldan, P.: Gravity coupled to matter without the metric, Phys Lett B 248 (1990), 62–66 282 27 Pﬁster, H., Braun, K H.: Induction of correct centrifugal force in a rotating mass shell, Class Quant Grav (1985), 909–918 274 28 Reinhardt, M.: Mach’s principle – a critical review, Z Naturforsch 28a (1973), 529537 274 29 Schră odinger, E.: Die Erfă ullbarkeit der Relativită atsforderungen der klassischen Mechanik, Ann Phys (Leipzig) 77 (1925), 325336 275 30 Schră odinger, E.: Space-Time Structure, Cambridge University Press (1950) 281 31 Treder, H.-J.: Elementare Kosmologie, Akademie-Verlag, Berlin (1975) 280 32 Treder, H.-J.: Die Relativită at der Tră agheit, Akademie-Verlag, Berlin (1972) 275 ă 33 Treder, H.-J.: Uber die Prinzipien der Dynamik von Einstein, Hertz, Mach und Poincar´e, Akademie-Verlag, Berlin (1974) 274, 275 34 Treder, H.-J.: Philosophische Probleme des physikalischen Raums, AkademieVerlag, Berlin (1974) 274 References 285 35 Yourgrau W., van der Merwe, A.: Did Ernst Mach ‘miss the target’ ? Synthese 18 (1968), 234–250 274 36 Zeldovich, Ya B.: The cosmological constant and the theory of elementary particles, Sov Phys Uspekhi 11 (1968), 381–393 (Uspekhi Fiz Nauk 95 (1968), 209–230) 280 14 Anthropic Aspects 14.1 Cosmological Principle and Cosmological Result **Cosmology** attempts to understand the observable metagalaxy as a representative part of the universe and as a necessary consequence of the laws that govern physical processes in this universe How far have we got? Just through the claim that the metagalaxy represents the universe, one characteristic degree of freedom can be isolated – unexpectedly at ﬁrst This is the expansion parameter With some success, cosmological models have been constructed that describe appropriately the large-scale behaviour beyond the observed structures and individual celestial objects The stability considerations of the next step show, however, that in simple models the evolution of the expansion parameter is too unstable to yield a large-scale homogeneous picture The characteristic indication is the stability of calculations into the past, which seems always to yield a cosmological singularity It remains to be discussed whether a reference to the necessity for a satisfactory quantum theory of gravity really defuses the problem [5], or whether the cosmological singularity is intertwined with the inﬁnity problems of elementary-particle physics [7, 8] If we consider only phenomena in the post-quantum universe, it is the reasoning about the inﬂationary stage that appears to be the solution to the problem of explaining the history of the universe through dynamics – irrespective of the initial conditions The stage of inﬂationary expansion provides an argument for why we observe large-scale homogeneity today, for why an intermediate state of extremely low temperature can be considered as the initial state of the universe in spite of the extremely large temperatures before, and for why we can forget about the details of the time before that This picture, however, requires us to accept large negative values for the pressure When large negative pressures are accepted, the evolution of the universe is determined and restricted by the subtleties of microscopic physics Just because of these restrictions, the state of the universe and the objects in it is a kind of observation and measurement of microphysical quantities in the early universe that may be beyond our reach in earthbound laboratories Of course, there are also observations that can be compared with more precise microphysical measurements Such a comparison simply states the Dierck-Ekkehard Liebscher: **Cosmology** STMP 210, 287–291 (2004) c Springer-Verlag Berlin Heidelberg 2004 288 14 Anthropic Aspects consistency of the evolution of the universe with our familiar physics This was the goal of constructing a cosmos Our success in answering the question of the necessity of evolution into the presently observed state has been paid for by giving up the concept that the metagalaxy is representative of the real universe From the interplay of quantum gravitation and uniﬁed ﬁeld theories, it is obvious that inﬂation did not start everywhere and simultaneously; that inﬂation did not reach the same order of magnitude everywhere; and that the symmetry breakdown did not everywhere produce the same group (at present SU (3) × SU (2) × U (1)), the same particle masses, or the same coupling constants for the lighter particles (particles with m MGUT ) An attempt to calm ourselves down with the conception that the metagalaxy lies wholly in a quasi-homogeneous region that is suﬃciently inﬂated can only be a cold comfort In the end this conception admits that far beyond the horizon, equally important regions exist where not only the evolution of cosmic objects has proceeded at diﬀerent rates to diﬀerent states but also the objects of the local physics are made up diﬀerently Even if one does not doubt the universal validity of quantum theory, the properties of atoms and molecules may depend so sensitively on the values of the elementary masses and coupling constants that the qualitative phenomenology of nature is sensitive to these constants, i.e that this phenomenology measures the constants On scales so much larger than the horizon, the universe will be inhomogeneous, not only in terms of small variations of density, peculiar velocity or structure, but in a grossly phenomenological way in the qualitative composition of its bound systems, from atoms to stars On these scales, in an extreme case, the universe may have even diﬀerent dimensions What does this mean? If quantum gravitation, inﬂation and symmetry breakdown are the ﬁnal way to solve the problems related to the expanding universe, we might have to abandon the Cusanus principle Thus the whole chain of conclusions that leads to the homogeneous model, inﬂation and a quantum stage of the universe falls to pieces This may be a reason to look for another way out, for instance by relating the assumption of homogeneity and the initial-value problem to a quasi-static stage at a density below the Planck density.1 14.2 Life as an Observational Datum Why does the observation that there are observers have such far-reaching consequences? The existence of carbon-based life2 in a late enough stage of The Eddington-LeMaˆıtre universe (Sect 3.5), with interplays by inﬂation, is the simplest background of such a programme Stanislaw Lem describes, in his Star Diaries, the visit of the protagonist to creatures living in an environment of boiling sulphur who demonstrate to each 14.2 Life as an Observational Datum 289 the evolution of the universe – often called the ‘existence of intelligent life’3 – puts limits on the existence, age and lifetime of main-sequence stars and supernovae, and on the possibility of the production of heavy elements in spite of the bottleneck described in Chap Even the start of the primordial nuclear synthesis depends sensitively on the masses of the proton and neutron and the diﬀerence between those masses Deuteron synthesis must be able to start, but not too early, in order to have helium, but not too much helium The mass of the electron and the Sommerfeld constant determine the structure of atoms Virtual variations of these constants would make atoms too loose or too rigid for the stability of the many molecules we know The strength of the strong coupling is just as important Only in a narrow region of its value can the wealth of chemical elements that are used by the architecture of life coexist The most important point here is the position of the resonances of carbon and oxygen, as we have seen in Chap Also, the strength of gravitation decides the age and lifetime of the stars, and in particular the question of the coexistence of a long age with recycling of stellar material that releases heavy elements into free space [9] If we consider our own existence in the framework of the evolution of the universe, we live beyond doubt in a peculiar stage The universe had to cool down to allow large-scale structure and to allow stars to form and to explode, before stars such as our sun and planetary systems such as the solar system could form from the debris, with planets that could host life [3] The universe must not be too old In the far future, when stellar fuel is exhausted, life will not be able to develop in the familiar form We can live only in a particular stage of the evolution of the universe and observe it This is the content of an interpretation that is called the weak anthropic principle [1, 4, 6]: the values of all observed cosmological and physical parameters are bounded by the observation that we exist as observers In this context, the well-known large Dirac numbers (for instance the 1080 protons within the horizon) can also be included The point is that the popular misunderstanding of probability makes the observed state of the universe feel so improbable that there should be a strong anthropic principle stating that the universe has to have just those properties that admit the development of life during a certain stage of its history [6] This is, of course, a concept that cannot possibly be falsiﬁed, so it cannot contribute to scientiﬁc explanation other that the universe has to admit their life because life in another form would be obviously unthinkable One could also refer also to The Black Cloud by Fred Hoyle or Out of the sun by Arthur C Clarke, which discuss this topic, as many other books A professor is asked whether he believes that there is intelligent life in the universe ‘There is an optimistic and a pessimistic answer,’ he states ‘The optimistic answer is ‘Yes, have a look at me’, the pessimistic is ‘Why should we expect intelligent life in the universe? We would be happy to ﬁnd some on earth.’’ 290 14 Anthropic Aspects The basic question all the time is, given that the state of the universe is what is observed, with life and observers and all that, has this happened by chance (more precisely, is this due to particular initial conditions for the evolution of the universe), or is it a necessary consequence of the evolution of any cosmological model (see Chap 1) The inﬂationary models of the universe propose that the observable metagalaxy developed from a far smaller part of the universe than one would expect if one were to follow a simple Einstein–Gamow model into the past.4 For a long time, proponents of inﬂation insisted that inﬂation implies that the density of matter in the universe has exactly the critical value that can be calculated from the observed recession velocity of galaxies They know better now, and try to cover up their mistake by christening the cosmological constant dark energy The inﬂationary models provide arguments for how the observed large-scale structure has evolved from zero-point quantum ﬂuctuations They also claim that the assumption of homogeneity and isotropy is a viable approximation only because the metagalaxy evolved from an extremely small part of the pre-inﬂationary universe, while the universe could be rather inhomogeneous again on scales far beyond the horizon We not get information from these regions, of course The present values of the elementary masses and coupling constants therefore obtain a truly accidental character One is tempted to construct models where these values are state variables and evolve with the universe [10, 11] One may compare the weak anthropic principle with an aspect of quantum theory When a measurement is interpreted in quantum theory, one has to take the conditions that the measuring apparatus imposes into account In our case, the imposed condition is our own existence However, humanity is not a measuring apparatus in the classical sense but – in this context – is again subject to observation The consistency of processes in nature implies that our existence cannot contradict the cosmological parameters We are an observational statement; the amazing fact is that this qualitative observation leads to so many quantitative conclusions The weak anthropic principle is of methodical use; we should not forget, however, that there are many observational questions that are irrelevant to life If we accept the concept that the universe might be structured in a rather complicated way and that it is not necessarily homogeneous and isotropic on a large scale in order to be so in the metagalaxy, the conditions for the evolution of life need to be met only in the latter small part If the overall state is homogeneously chaotic, there are again so many diﬀerent paths of evolution on the small (i.e metagalaxy-sized) scale that our metagalaxy is not a preferred one All other evolutionary scenarios exist with the same At T = TPlanck , the Einstein–Gamow model expects R[TPlanck ] ≈ 10−5 m In an inﬂationary model, this temperature is attained only before inﬂation Therefore lPlanck For we have to take inﬂation into account, and this yields R[TPlanck ] a model with positive spatial curvature, the temperature TPlanck may never be reached [2] References 291 right, but they exist far beyond our horizon From this point of view, the strong anthropic principle loses its content totally References Barrow, J D., Tipler, F J.: The Anthropic Cosmological Principle, Oxford University Press (1986) 289 Blome, H J., Priester, W.: Big bounce in the very early universe, Astron Astrophys 250 (1991), 43–49 290 Baez, C.: Is life improbable? Found Phys 19 (1989), 91–95 289 Balashov, Yu V.: Multifaced anthropic principle, Comments on Astrophys 15 (1990), 19–28 289 Bondi, H.: Cosmology, 2nd edn., Cambridge University Press (1960) 287 Carter, B.: Large number coincidences and the anthropic principle in cosmology, in M S Longair (ed.): Confrontation of Cosmological Theories with Observational Data, Proceedings, IAU Symposium No 63, Reidel, Dordrecht (1974) 289 Einstein, A.: Spielen Gravitationsfelder im Aufbau der materiellen Elementarteilchen eine wesentliche Rolle? SBer Preuss Akad Wiss (1919), 349356 287 Einstein, A.: Bietet die Feldtheorie Mă oglichkeiten fă ur die Lă osung des Quantenproblems? SBer Preuss Akad Wiss (1923), 359–364 287 Rosental, I L.: Big Bang – Big Bounce: How Particles and Fields Drive Cosmic Evolution, Springer, Heidelberg (1988) 289 10 Rothman, T., Ellis, G F R.: Smolin’s natural selection hypothesis, Q J R Astron Soc 34 (1993), 201–212 290 11 Smolin, L.: Did the universe evolve? Class Quant Grav (1992), 173–191 290 Index inﬂation 167 inhomogeneities aberration 26 absorber statistics 113 anthropic principle 289 Jeans barotropic matter 67 baryon asymmetry 161 breakdown of symmetry 170 canonical GRT 266 cosmological principle 5, 287 cosmological singularity 74 cosmos curvature 46 decoupling 223 distance 88 domain walls 255 Einstein 11 Einstein’s equations 43 equivalence principle 34 ether 127 explanation ﬂuctuations 182 fractal structure 137 Fresnel 27 Friedmann 12 Friedmann’s equations Galileo 26 Gamow 16, 143 gravitational lenses higher dimensions horizon 81 Hubble 12 53 104 243 182, 199 18, 224 kinetic equation 146 Lambert 10 Lemaˆıtre 19 light cone 80, 108, 177 Mach 273 metagalaxy microwave background minisuperspace 271 monopoles 260 neutrinos 162 Newton 8, 215 non-linear evolution Olbers 125 233 peculiar velocities 231 perturbations 132, 189 power spectra 131 primordial synthesis 154 proper time 32 quantum **cosmology** 265 quasi-particles 253 recombination redshift 83 125, 205 standard model 70, 176 strings 250 structure formation 189 294 Index tensor calculus textures 260 topology 253 universe 38 vacuum 168 virial theorem 227 Wheeler-deWitt equation 269 ... from Springer Violations are liable for prosecution under the German Copyright Law Springer is a part of Springer Science+Business Media springeronline.com c Springer- Verlag Berlin Heidelberg 2005. .. Number: 20059 21920 Physics and Astronomy Classification Scheme (PACS): 98.80.-k, 04.20.Cv, 01.55.+b ISSN print edition: 0081-3869 ISSN electronic edition: 1615-0430 ISBN -10 3-540-23261-3 Springer. . .Springer Tracts in Modern Physics Springer Tracts in Modern Physics provides comprehensive and critical reviews of

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