arXiv:2105.07827v2 [gr-qc] 25 May 2021 esit n hst bevtos isen after Einstein, to observations. Universe to solu- contracting thus formal or and the redshifts expanding related an he for ap- tions time: little the was at wrote that preciated 1927 paper in groundbreaking Lemaˆıtre, who another Georges also to apparently unkown were papers Sitter’s Friedmann’s de observa- an and solution. is Einstein’s there between whether difference years out tional those find in to question trying main was The 1989]. 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Friedmann version of the Hamiltonian constraint Friedmann’s equations.8 Starting point is Ein- in general relativity [Kiefer 2012]. stein’s field equation From (3) and (4), one can derive a third equa- tion, 1 8πG − ρ˙ +3H(ρ + p)=0, (5) Rµν gµν R +Λgµν = 2 Tµν . (1) 2 c where H :=a/a ˙ is the Hubble parameter (its eval- Observations indicate that the Universe is approx- uation at the present day is called Hubble constant, imately isotropic around our position. These come denoted by H0). The combination ρ+p occuring in mainly from the anisotropy spectrum of the Cosmic (5) is called inertial mass density. In these Fried- Microwave Background (CMB). Adopting the Cos- mann equations, we have followed the modern prac- mological Principle (“all places in the Universe are tice of including the cosmological constant Λ into alike”), one is led to assume (approximate) isotropy the density ρ (although this was already suggested around every position. One can then mathemati- by Schr¨odinger in 1919), because it contributes an cally prove that our Universe must also be (approx- ‘energy density of the vacuum’ ρΛ := Λ/8πG. Its imately) homogeneous. The geometry of a homo- equation of state reads pΛ = −ρΛ, so from (5) we geneous and isotropic spacetime is characterised by see that ρΛ is constant. For barotropic equations the line element of state p = wρ, w =6 −1, we find from (5) the solution dr2 3(1+w) ds2 = −c2dt2 + a2(t) + r2dΩ2 , (2) ρa = constant, (6) 1 − kr2 which includes as particular cases: − where a(t) is the scale factor. For the parameter • dust (p =0) −→ ρ ∝ a 3, k, we have the possible choices k = 0 (flat spa- • radiation (p = ρ/3) −→ ρ ∝ a−4, tial geometry), k =1 (positive curvature), k = −1 − (negative curvature); only the latter two cases were • stiff matter (p = ρ) −→ ρ ∝ a 6. treated by Friedmann. Current observations favour By the kinematic relation a0/a =1+ z, with a0 as a spatially flat Universe, although there is still a the present scale factor, we can relate ρ to the ob- controversy [Di Valentino 2020]. A given value for servable redshift z of objects. The case of radiation k does not fix the topology of our (spatial) Uni- is relevant for the early Universe, while stiff mat- verse, and it is a most intriguing question to deter- ter so far seems unrealistic. Today, the Universe is mine the cosmic topology from observations [Lu- dominated by dust (about one third) and vacuum minet 2015]. energy (about two thirds), leading to the temporal Inserting the ansatz (2) into (1), one is led to evolution Friedmann’s equations.9 The first equation is the 3Ω H 1/3 3 Λ restriction of the general Raychaudhuri equation to a(t)= a 0 0 sinh2/3 t , (7) 0 Λ 2 3 a homogeneous and isotropic Universe, r ! 4πG where Ω0 is today’s matter density in terms of the a¨ = − (ρ +3p)a, (3) critical density, observationally determined to be 3 about 1/3. Observations also indicate that the age where ρ and p denote energy density and pressure of our Universe is about 13.8 billion years. For large of matter, respectively. If matter obeys the strong times, the evolution law (7) asymptotes to de Sit- 10 energy condition ρ +3p ≥ 0, (3) leads to concave ter space. From observations of the CMB, there solutions for a(t), that is, to a world model with are strong indications that our Universe underwent a singular origin. The second Friedmann equation a quasi-exponential expansion (with very large Λ) reads already very early in its history, a phase called in- 8πG a˙ 2 = ρa2 − k. (4) flation. Inflation offers the means to explain the 3 origin of structure in the Universe. In contrast to (3), this equation only contains tem- Instead of barotropic equations of state, one often poral derivatives up to first order, so it has the employs dynamical matter models, typically with interpretation of a constraint. In fact, it is the a scalar field φ. In the Friedmann limit, this field 9From here on, we set c = 1. 10 For late time expansion with constant positive ρΛ one speaks of dark energy, but there is also the possibility that the effective vacuum energy density varies with time, in which case one speaks of quintessence. The latter is thought to originate from matter sources and is often modelled by means of a time-dependent scalar field φ.
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depends, of course, only on time. In the case of a approximated by a homogeneous and isotropic spa- massless field, it corresponds in (4) to the choice of tial part, is successfully described by Friedmann’s 2 a density ρφ = φ˙ /2. equations, but small inhomogeneities must be taken Beyond the Friedmann approximation. Be- into account in order to understand properties of yond the immediate and obvious utility of the the CMB in the framework of cosmological per- Friedmann equations for cosmological applications turbation theory. Furthermore, a precise under- there are several important and promising direc- standing of the formation of galaxies and clusters tions for future development that build on Fried- of galaxies requires the numerical treatment of the mann’s achievements. For lack of space we here Einstein equations (1) and their Newtonian limit. mention only two of these, namely (i) their use Incorporating inhomogeneities is likewise crucial for taking first steps towards a theory of quantum for a better understanding of the origin of the uni- gravity, and (ii) the generalization of the isotropic verse, because inhomogeneities scale like a−6, like ansatz (2) in order to search for a fundamental sym- stiff matter, and thus dominate very close to the metry of Nature. Big Bang singularity. This is a crucial issue for in- When adapting the ansatz (2) to a quantum flationary cosmology, which hinges on the ansatz mechanical context one speaks of the so-called min- (2). Finally, there remain difficult issues related to isuperspace approximation, in which the full su- defining a generally covariant averaging procedure perspace of geometrodynamics, being the moduli in Einstein’s theory that would provide a rigorous space of all three-metrics modulo spatial diffeomor- basis for the Friedmann approximation [Buchert phisms, is restricted to few homogeneous degrees of 2015]. freedom such as the scale factor a. This limit was first discussed by DeWitt in his pioneering paper on On the more mathematical side, a key in- canonical quantum gravity [DeWitt 1967]. This is sight came from the Belinski-Khalatnikov-Lifshitz a huge simplification because key technical issues (BKL) analysis [Belinski 1970, 1982] of the generic such as the non-renormalizability of perturbative behaviour of solutions of Einstein’s equations near quantum gravity can be ignored in this approxi- a spacelike singularity. There, one generalizes the mation. Furthermore, various conceptual issues of ansatz (2) to quantum gravity and quantum cosmology can be studied. Namely, the direct canonical quantiza- ds2 = −dt2 + a2(t)dx2 + b2(t)dy2 + c2(t)dz2, tion of the second Friedmann equation (4) leads to a special case of the Wheeler-DeWitt equation [Kiefer 2012, Calcagni 2017], here given for the case thus giving up isotropy, but retaining spatial ho- of a massless homogeneous scalar field, mogeneity. A surprising result to come out of 2 2 2 this analysis is the appearance of chaotic oscilla- 4G~ ∂ ∂ ~ ∂ 3π a − − ka Ψ(a,φ) = 0, tions of the metric coefficients a,b,c as one ap- 3πa2 ∂a ∂a a3 ∂φ2 4G proaches the singularity. This result indicates that where a particular factor ordering has been the ‘near singularity limit’ of the metric exhibits a adopted. The wave function Ψ(a, φ) is a simple ex- far more complicated behaviour than inspection of, ample of the ‘wave function of the universe’. It is, say, the Schwarzschild metric would suggest, thus in particular, possible to analyse the behaviour of also showing the limitations of the assumption of Ψ near the singularity where a → 0. The Wheeler- isotropy. DeWitt equation has no external time parameter, The BKL analysis has been generalised in many but one can employ the scale factor a so as to track directions, in particular also to accommodate inho- the evolution of the matter degrees of freedom with mogeneities [Krasinski 1997]. Furthermore, closer respect to this “intrinsic time”. (Note that the min- study of the BKL limit has revealed evidence for isuperspace Wheeler-DeWitt equation is hyperbolic a huge infinite-dimensional symmetry of indefinite with respect to a.) Key open issues concern the Kac-Moody type, vastly generalizing the known du- physical interpretation of Ψ, the construction of ality symmetries of supergravity and string theory. a suitable Hilbert space, and the meaning of ob- This novel symmetry can possibly serve as a guid- servables; for a survey and further discussion, see ing principle towards unifying the fundamental in- [Kiefer 2012]. teractions [Damour 2002]. The other extension concerns the inclusion of non-homogeneous degrees of freedom. On the phe- We thank Alexander Kamenshchik for his com- nomenological side, the evolution of our Universe, if ments on our manuscript.
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