Gravitational entropy and the flatness, homogeneity and isotropy puzzles

We suggest a new explanation for the observed large scale flatness, homogeneity and isotropy of the universe. The basic ingredients are elementary and well-known, namely Einstein's theory of gravity and Hawking's method of computing gravitational entropy. The new twist is provided by the boundary conditions we recently proposed for"big bang"type singularities dominated by conformal matter, enforcing $CPT$ symmetry and analyticity. Here, we show that, besides allowing us to describe the big bang, these boundary conditions allow new gravitational instantons, enabling us to calculate the gravitational entropy of cosmologies which include radiation, dark energy and space curvature of either sign. We find the gravitational entropy of these universes, $S_g \sim S_\Lambda^{1/ 4} S_r$, where $S_\Lambda$ is the famous de Sitter entropy and $S_r$ is the total entropy in radiation. To the extent that $S_g$ exceeds $S_\Lambda$, the most probable universe is flat. By analysing the perturbations about our new instantons, we argue it is also homogeneous and isotropic on large scales.

Pictures of the earth from space reveal it to be remarkably round and smooth, with a curvature radius much larger than the distances we explore in our everyday lives.Hence, on those scales, it is often a good approximation to treat the earth's surface as flat.The explanation of this flatness involves thermodynamics.First, gravity pulls matter inwards, so its potential energy is minimised in a spherical configuration.Second, relative motions of the earth's constituent atoms generate friction: as a mountain is pulled downwards or a rock falls, gravitational potential energy is converted into heat.Even if the earth is regarded as a closed, conservative system, there are vastly more ways of distributing its internal energy among its ∼ 10 50 atoms as heat than there are of creating more complicated, less spherical geometries.Taken together, gravity, friction and the earth's many atoms explain why it is locally flat [1].
In this Letter, we provide a similar, entropic explanation for the observed flatness, homogeneity and isotropy of the cosmos.Our argument rests on a new calculation of gravitational entropy, along the lines advocated by Hawking and others in the context of black hole thermodynamics [2][3][4].The calculation is made possible by our new approach to the boundary conditions for cosmology, implementing CP T symmetry and analyticity at the bang, quantum mechanically, to solve many puzzles [5][6][7][8].We outline the connection to Penrose's classical Weyl curvature hypothesis [9] in the conclusion.Using these new boundary conditions, we showed that cosmologies with radiation, dark energy and curvature are periodic in imaginary proper time, with a Hawking temperature given by that of the corresponding de Sitter spacetime [7].This is a strong hint that the solutions should be interpreted thermodynamically.Here, in an appropriate time slicing we find the solutions describe new gravitational instantons, one for each value of the macroscopic parameters, allowing us to calculate the gravitational entropy.For a given, positive dark energy density, with radiation included the gravitational entropy can be arbitrarily large.As the total entropy is raised, the most likely universe becomes progressively flatter.Furthermore, inhomogeneous or anisotropic perturbations are suppressed.Hence, to the extent that the total entropy exceeds the de Sitter value, the most probable universe is not only flat, but also homogeneous and isotropic on large scales.
The partition function of a statistical ensemble can be represented by a path integral over configurations that are periodic in imaginary time.Hawking and collaborators first used these methods to investigate black hole thermodynamics, a topic which has since burgeoned into holographic studies [10,11] and experimental tests using analog systems [12,13].We shall follow Hawking et al.'s original approach here [14][15][16][17].In the cosmological setting, it is an excellent approximation to treat the radiation as a relativistic fluid, in local thermal equilibrium at a temperature declining inversely with the scale factor.The instantons we present are saddle points of the path integral for gravity: real, Euclidean-signature solutions to the Einstein equations for cosmologies with dark energy, radiation and space curvature.The associated semiclassical exponent iS/ is real, large and positive, and may be interpreted as the gravitational entropy.
One of us has given a formal argument, based on Picard-Lefschetz theory, that a path integral for a quantum gravitational transition amplitude can never yield a positive semiclassical exponent [18][19][20].However, for a statistical ensemble, a formal argument indicates precisely the opposite.Consider, for example, the partition function Z(β) = Tr(e −βH ).The time reparameterization invariance of general relativity means that the Hamiltonian H vanishes on physical states [21].Thus, Z = e S simply counts the number of states.If Z is approximated by a saddle, the semiclassical exponent must be positive.In this Letter, we treat background spacetimes with where n is the lapse, which may be set constant by reparameterizing t, and a(t) is the scale factor.Comoving 3-space is assumed to be maximally symmetric, with metric γ ij (x) and Ricci scalar 6κ.For κ > 0, it is S 3 , with volume V = 2π 2 κ −3/2 .For κ < 0, we assume a compact subspace of H 3 , whose volume is 2π 2 |κ| −3/2 times a topology-dependent constant.For ease of presentation, we generally leave the constant implicit.
The partition function may be represented formally as Z =Tr( dne −iHn ), with the integral enforcing the contraint that the total Hamiltonian H = 0 [21].In the first (background) approximation, the Hamiltonian for gravity, H g depends only on the scale factor a whereas, due to conformal invariance, the Hamiltonian for the radiation, H r is independent of a. (Using conformal time, the Hamiltonian is dimensionless.)Hence we can trace over the radiation at some temperature T = β −1 , using Tr r (e −βHr ) = e −βFr = e Sr−βUr where S r is the total entropy in the radiation and U r , expressed as a function of S r , is the associated energy in the conformal frame.Analytically continuing to Lorentzian time, β = i n, with n real, we obtain Z = e Sr Tr g ( dne −i(Hg+Ur)n ).S r is an adiabatic invariant.It serves as the key parameter, analogous to the Earth's number of atoms in our opening paragraph, controlling the gravitational ensemble.
We perform the gravitational trace Tr g in the semiclassical (small ) approximation.For general relativity coupled to radiation and a cosmological constant, the exponent in the path integral is Here, L P l ≡ (8πG ) 1 2 , ȧ ≡ da/dt and V is the comoving volume.The proper radiation density is ρ r = L −2 P l r/a 4 (with r ≡ U r L 2 P l /V ) and the dark energy density is ρ Λ = L −2 P l λ. (r has dimensions of length squared, λ of inverse length squared, a of length and n is dimensionless.)Since the exponent is odd under n → −n, saddles occur in pairs with equal and opposite semiclassical exponents.
The equations of motion following from (2) are: respectively the Friedmann equation and the trace of the Einstein equations, R = 4λ, to which conformal invariant matter, like radiation, does not contribute.The set of Lorentzian cosmologies, described by κ, r and λ are illustrated in Fig. 1 i).The evolution of the scale factor in (3a) is that of a particle moving in a potential, with r signifying its energy.For κ > 0 and r = 9κ 2 /(4λ) ≡ r s , we have Einstein's static universe (ESU) (which [22] shows is thermodynamically stable).For κ < 0, r > 0 and κ > 0, r > r s , solutions L ± a (for κ ≷ 0) run "above" the potential.For κ > 0, r < r s , the "inner" and "outer" solutions are L + i and L + o .The line element (1) may be rendered Euclidean in two ways: i) W , the usual Wick rotation, which sets n = −iN with N real and ii) a CW rotation combining W with a conformal rotation a(t) = ib(t), with b(t) real.The first yields a metric with "all positive" signature, the second "all negative."As we shall now explain, the second rotation is more generally applicable in cosmology.It is easily seen from Fig. 1, ii), that W only yields cosmological instantons for positive κ and r ≤ r s .These cosmologies are intermediate between de Sitter and the ESU, both of which have static descriptions.Hence, they may be considered as "close to equilibrium."On the other hand, the CW rotation yields instantons for all κ, positive or negative, and positive r and λ.Since all these cosmologies, except the ESU, have a horizon and a Hawking temperature [7], they presumably have a gravitational entropy.The CW rotation yields an instanton with positive entropy, for all parameter values: where there is overlap with the W instantons, the two entropies agree.
The CW rotation actually follows from the boundary conditions we proposed in Refs.[5][6][7].There, we considered scale factors with zeros, i.e., conformal singularities, of a special type.All variables are classified according to whether they are even or odd about the conformal zero, and we insist upon analyticity there.In particular, a(t) is odd so it has a simple analytic zero at t = 0 and becomes imaginary when we Wick rotate about t = 0. So, as well as continuing n to −iN with N real, we set a(t) = ib(t) with b(t) real.Conformal invariant matter, which is all we consider here, does not couple to a(t) at all and hence it is insensitive to the conformal rotation.However, the gravitational action ( 2) is affected by it.For κ > 0 and r < r s , the cases intermediate between de Sitter and ESU, we find a positive semiclassical exponent requires N > 0, the usual sign.However, for κ < 0 or for κ > 0 and r > r s , a positive exponent requires N < 0. The sign of N indicates the orientation of Euclidean time with respect to the spacelike slices.Hence, our Lorentzian ensemble uses two different Euclidean descriptions: one for the radiation and one for gravity.We believe this to be reasonable since we regard the Lorentzian description as fundamental and the combined radiation/gravity ensemble is far from equilibrium, with the radiation temperature T r totally distinct from the Hawking temperature.We emphasize that our e iθ e ±iθ/4 e iθ ±ie ∓iθ/4 Table I: Analytic forms for Euclidean instantons (Fig. 1, ii) and iii)), periodic in t with period unity, from which the Lorentzian solutions (Fig. 1, i)) are obtained by analytic continuation.Here, sn(z, m), dn(z, m) are Jacobi elliptic functions and K(m), E(m) are complete elliptic integrals [23].W denotes Wick rotation and CW combined conformal and Wick rotations.Superscript ± indicates κ ≶ 0; m satisfies m/(1 + m) 2 = λr/(3κ) 2 .For 0 < r ≤ r s , m = e −α with 0 ≤ α < ∞, and for r ≥ r s , m = e iθ with 0 ≤ θ < π. z ± are the two branch points pictured in Fig. 2. new instantons would not be accessible if one insisted upon using a single Wick rotation to describe these coupled systems, at totally different temperatures.Indeed, our work seems to indicate that taking a Euclidean description to be fundamental (see, for example, the recent approach of Ref. [24], reviewed in Ref. [25]) may preclude one from describing a realistic cosmology.
Our new instantons are displayed in Table I.For r > 0, they are periodic in 0 < t ≤ 1, with topology S 3 × S 1 or (compactified H 3 ) × S 1 .We choose N so the period in t is unity and the semiclassical exponent is positive.In fact, one can solve (3a) for ȧ/N and substitute back into (2).Upon rescaling a via a = cz, one obtains where c is chosen so that z + z − = 1.The integral runs over a closed contour, representing one period of the solution.The lapse n has been eliminated but the orientation of the z integration contour on the integrand's Riemann surface gives the same sign ambiguity.The Laurent series of the integrand about z = ∞ contains only even powers.So, if one analytically continues to avoid that singularity, there is no contribution associated with it.Hence, the contour may be deformed to run between two branch points, shown in Fig. 2 and in the final column of Table I.The behavior of the solutions as a function of the parameters is most easily understood from this picture.I), for θ = π/2.Zeros are dark blue, poles white.Right: branch cuts in the integrand of iS on the Riemann z-sphere (see (4) and ensuing text).
For example, the W + instanton and the CW + r<rs instantons are defined by the same branch points (see leftmost panel of Fig. 2).For the W + instanton, the solution runs around the cut (just above and then just below it) on the real z-axis.For the CW + r<rs instanton, we deform the contour from the imaginary z-axis into the right half z-plane and around the cut, obtaining the same result.As r is increased to r s , the branch points annihilate and the integral vanishes, as expected since the ESU has no horizon.Increasing r further, branch points appear above and below z = 1, moving apart on the unit circle as r is increased, reaching z = e ±iπ/4 at infinite r.For κ negative and r < r s we have the CW − i and the CW + o solutions shown in Fig. 1 iii).They, too, are defined by the same branch points.As the third panel of Fig. 2 shows, the solution for CW − i runs between the two "inner" branch points ±ie −α/4 .Whereas the solution for CW − i runs between the "outer" branch points ±ie α/4 , via the point at infinity.Again, a contour deformation shows that the two semiclassical exponents are equal.For this reason we do not show CW − o in Table I.Fig. 3 shows how the complex t-plane and z-plane pictures are related.The left panel shows the double periodicity of a solution in the t-plane.The fundamental domain (shown in dashed lines) is a rectangle with opposite sides identified, i.e., a torus.The black vector Figure 4: Gravitational entropy S g as a function of the entropy in radiation S r .The dotted blue line shows the maximum size of the curvature term in the Friedmann equation, relative to the total energy density [26].
indicates a period: if the t-integral for iS is taken on any contour connecting two points separated by a period, the same value results.Hence iS is a topological invariant.The right panel shows the Riemann z-sphere, with its two square root branch cuts.Since there are two Riemann sheets, a second copy of the Riemann sphere is joined to the first along the right and left edges of the branch cuts.By "opening" the cuts, one again finds a torus.Since a(t) maps the fundamental t-plane domain to the z-plane, one to one, the two tori are identified.
In the flat limit, κ → 0, the CW + r>rs and CW − a solutions tend to b(t) = (r/λ) 1/4 e −iπ/4 sn(2K( 12 )e iπ/4 t, −1).Unsurprisingly, the curvature is dynamically unimportant in the limit.What is slightly less obvious is that the Euclidean solution for b(t) is actually the same function as the Lorentzian solution for a(t) so the descriptions are self-dual.The semiclassical exponent diverges as V ∼ κ −3/2 and we obtain for the gravitational entropy where S r is the total entropy in radiation, S Λ = 24π 2 λ −1 L −2 P l is the de Sitter entropy, C = 8K(1/2)/ √ 3 ≈ 8.56 and D = 2 1/4 Γ(1/4) 2 /(3 3/4 π) ≈ 2.18 [26].Fig. 4 shows the gravitational entropy S g , in units of the de Sitter entropy S Λ , versus the total entropy in radiation S r in units of S Λ , for κ > 0 (solid line) and κ < 0 (dashed line).As mentioned, the gravitational entropy diverges with the volume in the flat limit.Hence, in the regime where S g dominates the total entropy, the larger the total entropy, the smaller κ must be.For example, from Fig. 4, if the gravitational entropy exceeds the de Sitter entropy by a thousand, the curvature term in the Friedmann equation is at most a hundredth the size of the combined radiation and dark energy terms.
We now ask whether, in backgrounds where Ω κ is always small, our new instantons represent local maxima of the entropy for observable perturbations.On wavelengths larger than the Hubble radius, tensor perturbations h T describe anisotropies.
Our results point to a new and remarkably economical explanation for the large scale geometry of the universe.However, much remains to be done.We have only sketched the Lorentzian gravity-matter ensemble and its path integral for which we expect our instantons provide relevant saddles.Such an ensemble could be defined by tracing over the initial data.Our CP T -symmetric boundary conditions are closely related to those studied in the context of Penrose's "Weyl curvature hypothesis" [9,28,29].Newman has shown that classical Einstein gravity coupled to a perfect radiation fluid, with conformal singularities of just the type we consider, has a well-posed Cauchy problem for which the conformal 3geometry and radiation density on the singular 3-surface provide initial data [30][31][32].As in our analysis, all dynamical quantities are either even or odd in the conformal factor, the natural time variable.Of course, the standard model is not exactly conformal in the U V .However, we have recently shown that the leading quantum corrections to the trace anomaly may be cancelled by dimension zero fields, without introducing any new particles.The same fields cancel the leading corrections to the vacuum energy, and could seed scale-invariant large-scale density perturbations [8].Finally, our calculations must be extended to include non-conformal matter, including massive fields and black holes, to provide a more complete analysis of the thermodynamics of the universe.

Figure 1 :
Figure 1: Cosmological solutions for: i) a(t) in Lorentzian time (see Eqs. (1),(3), with n real); ii) a(t) after a Wick rotation W , setting n = −iN , with N real; iii) b(t) after a conformal and a Wick rotation CW , setting a(t) = ib(t), n = −iN with b(t) and N real.

Figure 2 :
Figure 2: Branch cuts in the complex z-plane (see (4)).In each case, only two of four branch points are shown (relative to the unit circle, shown dotted).

Figure 3 :
Figure 3: Double periodicity and topology.Left: contours of |b(t)| in the complex t-plane for CW + a (TableI), for θ = π/2.Zeros are dark blue, poles white.Right: branch cuts in the integrand of iS on the Riemann z-sphere (see (4) and ensuing text).