aa r X i v : . [ phy s i c s . g e n - ph ] S e p A Globally Unevolving Universe
Meir Shimon School of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel ∗ A scalar-tensor theory of gravity is formulated in which G and particle masses are allowed tovary. The theory yields a globally static cosmological model with no evolutionary timescales, nocosmological coincidences, and no flatness and horizon ‘problems’. It can be shown that the energydensities of dark energy ( ρ DE ) and non-relativistic baryons and dark matter ( ρ M ) are related by ρ DE = 2 ρ M , in agreement with current observations, if DE is associated with the canonical kineticand potential energy densities of the scalar fields. Under general assumptions, the model favors light fermionic dark matter candidates (e.g., sterile neutrinos). The main observed features of theCMB are naturally explained in this model, including the spectral flatness of its perturbations onthe largest angular scales, and the observed adiabatic and gaussian nature of density perturbations.More generally, we show that many of the cosmological observables, normally attributed to thedynamics of expanding space, could be of kinematic origin. In gravitationally bound systems, thevalues of G and particle masses spontaneously freeze out by a symmetry breaking of the underlyingconformal symmetry, and the theory reduces to standard general relativity (with, e.g., all solarsystem tests satisfied). PACS numbers: 95.36.+x, 98.80.-k, 98.80.Es
Introduction .- In standard cosmology physical pro-cesses are regulated by space expansion, i.e. the time-dependent Hubble scale provides the ‘clock’ for the evolv-ing temperature and density of matter and radiation, re-sulting in a sequence of cosmological epochs. This clockis only meaningful if other time scales, e.g. the Plancktime, or the Compton scale, evolve differently.In the current standard (cosmological) model (SM) theconjecture is made that a brief inflationary phase in thevery early universe provides the theoretical frameworkfor resolving the ‘puzzles’ of observed spatial flatness, de-duced super-horizon correlations in the cosmic microwavebackground (CMB) radiation, and absence of primordialcosmic defects [1-3]. Generic inflationary models seem tobe able to explain the nearly flat power spectrum of den-sity perturbations, their gaussian nature, and adiabaticinitial conditions. In addition, they generically predicta unique imprint on the polarization state of the CMBgenerated by inflationary-induced gravitational waves [4-6].Yet, a few puzzling features remain unexplained;baryons account for only ≈
5% of the energy budget ofthe universe, with the rest in some form of dark energy(DE) and dark matter (DM). The latter has so far evadeddetection by terrestrial experiments, and the former doesnot seem to cluster on sub-horizon scales. The nature ofDM is yet unknown, and according to standard lore DEis very likely described by a scalar field with an equationof state (EOS) which very closely mimics vacuum energy,but with an amplitude O (10 ) smaller than naively ex-pected for a cosmological constant [7-9].Although DE and nonrelativistic (NR) matter evolvevery differently in the SM, the current energy density ofthe former is about 2/3 of the total energy budget ofthe universe [10, 11]. Common interpretation of cosmo- logical observations is that the transition from matter-dominated to DE-dominated expansion took place onlyrecently. This provoked anthropic considerations [12-15]to explain the uniqueness of the current epoch. Othersuggestions to explain the small value of DE include mod-els of dynamic DE [16-19], holographic cosmology [20-24],K-essence [25, 26], modified gravity [27, 28], etc. Yet, itis fair to say that none has gained consensus. These andother puzzles, e.g. [29, 30], have been contemplated forquite some time now; generally, they are viewed as eithercoincidences of nature, or as possible indications of theneed for new physics.It should be stressed from the outset that the onlycosmological observables are dimensionless quantities onthe past light cone . From this perspective, cosmologicalmodels are practically redundant in the sense that theytypically describe the evolution of dimensional quantitiesin the entire spacetime, and consequently most of therich dynamics characterizing these models is not readilyamenable to comparison with observations.We show that the above cosmological puzzles may beattributed to our conventional system of units which isbased on constant dimensional quantities, namely, nat-ural units, such as Planck length, and Compton wave-length. In contrast to other scalar-tensor theories, e.g.[31-35], we propose a ‘single-clock’ cosmological modelcharacterized by Minkowski background metric with noglobal background evolution. By projecting certain di-mensionless quantities derived from the model on thepast light cone a few well known SM results are repro-duced, and new insight is gained. All this is achieved in aframework that does not recourse to GUT scale physics,which is many orders of magnitudes beyond our currentexperimental reach. Rather, the proposed framework re-places the dynamical SM with a much simpler kinematic model, the latter is only required to satisfy self-consistentboundary conditions on the past light cone – essentiallythe observable universe – thereby explaining, e.g., the ob-served ratio of DE and NR matter, and estimating themass of a fermionic cold DM (CDM) candidate to belikely below 1
GeV . Theoretical Framework .- A scalar-tensor theory ofgravity, linear in the curvature scalar, R , can be summa-rized by the following action given in ~ units [36, 37] S/ ~ = Z (cid:20) F ( ϕ K ) R − g IJ ( ϕ K ) ϕ Iµ ϕ J,µ − V ( ϕ K )+ L M ( ϕ K ) / ( ~ c ) (cid:3) × √− gd x (1)where the integration measure is d x = cdt · d x , sum-mation convention is implied on both greek and capi-tal Latin letters, the N scalar fields ϕ K are labeled by I, J, K = 1 , , ...., N , and ~ and c have their usual mean-ing. The potential V is an explicit function of the scalarfields. In general, the matter Lagrangian L M accountsfor the entire mass-energy contributions to the energydensity of the universe, i.e. DE, DM, baryons, electrons,neutrinos, and radiation. In addition to the spacetimemetric g µν we introduce the dimensionless g IJ which isa ‘metric in field space’. R is calculated from g µν , and f µ ≡ f ,µ is the derivative of a function f . We allow L M to explicitly depend on ϕ K .The Einstein-Hilbert (EH) action of conventional gen-eral relativity (GR) is recovered by setting all scalarfields to constant values. Consistency with GR requires F ( ϕ K ) ≡ κ − and V = Λ /κ , where κ ≡ πG ~ /c = O ( l P ), and G , l P , and Λ are the gravitational constant,Planck length, and cosmological constant, respectively.Eq. (1) describes GR in dynamic units, with ϕ k non-minimally coupled to gravitation. In general, scalar fieldscan make both negative and positive contributions to theenergy density; while the former can replace the gravi-tational energy density, thereby ‘offset’ the standard ex-pansion, the latter can be identified with DE, in whichcase L M accounts only for the remaining contributionsto the total energy density.Applied to a homogeneous and isotropic background,this theory admits classical solutions characterized by auniversal evolution of all quantities with the same phys-ical units, namely, global dimensionless ratios are fixed,indeed a cornerstone of the approach adopted here. Inthis special case the theory reduces to a single-scalarmodel where ϕ I ≡ λ I ϕ , with dimensionless coupling con-stants λ I . For V = λϕ and F ≡ βϕ , where β < λ >
0, the EH action is recovered from Eq. (1) by setting ϕ = (8 βπG ) − and λ = 8 πβ G Λ.The field equations derived from variation of Eq. (1) with respect to g µν , and ϕ are [36, 37] F · G νµ = T νM,µ + (cid:18) ϕ µ ϕ ν − δ νµ ϕ α ϕ α (cid:19) − δ νµ V + F νµ − δ νµ F αα (cid:3) ϕ + 12 F ,ϕ R − V ϕ = −L ; ϕM T νM,µ ; ν = L M,ϕ ϕ µ (2)where G νµ is Einstein’s tensor, g IJ λ I λ J ≡ f νµ ≡ [( f ) ,µ ] ; ν , with f ; µ denoting covariant derivatives of f , (cid:3) f is the covariant Laplacian, ( T M ) µν ≡ √− g δ ( √− g L M ) δg µν , andthe last of Eqs. (2) implies that energy-momentum isnot generally conserved. Indeed, energy-momentum isclearly not conserved when G , Λ, and particle masses,are time-dependent. Homogeneous and Isotropic Spacetimes .- Theline element describing a homogeneous and isotropicspacetime in cosmic coordinates is ds = − ( cdt ) + a (cid:18) dr − Kr + r d Ω (cid:19) , (3)where d Ω ≡ dθ + sin θdφ is a differential solid angle, a ( t ) is a scale factor, and K is a spatial curvature con-stant. Eqs. (2) are then3 F ( H + Ka ) = ρ M + 12 ˙ ϕ + V − H ˙ F − F ( ˙ H − Ka ) = ρ M (1 + w M ) + ˙ ϕ + ¨ F − H ˙ F ¨ ϕ + 3 H ˙ ϕ − F ,ϕ R + V ϕ = L ,ϕM , (4)where R = 6(2 H + ˙ H + Kc a ), H ≡ ˙ a/a , and ( T M ) νµ = ρ M · diag ( − , w M , w M , w M ), i.e. there are no shearand momentum-flow in a homogeneous background, and w M ≡ P M /ρ M is the EOS of matter.A homogeneous and isotropic model (with ρ M = −L M >
0) admits nonvanishing constant ϕ , only when βRϕ >
0, by virtue of a symmetry breaking of the un-derlying conformal symmetry where βRϕ / βRϕ ≤
0, i.e. R ≤
0, onlydynamical ϕ is a nontrivial solution. Combining thetraces of the first two of Eqs. (2) results in L M = T .A solution, of the form ˙ a + Kc = 0 (that amounts to G = G ii = R = 0, i.e. flat spacetime), with ϕ scalingas ϕ = ϕ /a (with subscript ‘0’ denoting present valueshere and throughout) results in the following constraints ρ M + H ( 12 + 6 β ) ϕ + V = 0 ρ M (1 + w M ) + H (1 + 8 β ) ϕ = 0 − H ϕ + V ϕ = L ϕM , (5)where L ϕM ≡ L M,ϕ and V ϕ ≡ V ,ϕ . Under the conditionthat the energy density associated with the i’th species is ρ i ∝ a − , (irrespective of EOS), and therefore ρ M ∝ a − and L M ∝ a − , the last of Eqs. (5) becomes ρ M ( − w M ) = L M,ϕ ϕ. (6)This condition implies that there is essentially no globalevolution, and in case of a NR matter L M = − ρ M if ρ M ∝ ϕ . In fact, combining the derivative of the first ofEqs. (5) with the last is only consistent if β = − / w M ≈
0. From the first two of Eqs.(5), we see that ρ DE = 2 ρ M (7)where ρ M stands for the NR (DM and baryonic) mat-ter. Remarkably, this relation conforms well with bestfit estimates from SM, e.g. [10, 11]. In the model ex-plored here the ratio ρ DE /ρ M is fixed and there is no‘coincidence problem’ in the current epoch. The appar-ent coincidence in the SM stems from arbitrarily fixingdimensional units; doing so at any past or future timewould have resulted in the same ρ DE /ρ M ≈
2. From theabove discussion it is clear that − K = O (Λ) = O ( H ).In a stationary universe, unlike in the SM, the fact thatthe curvature radius is not much larger than the Hub-ble radius has no direct observational consequences sincethere is no a-priori expectation for the coherence scale ofCMB perturbations, i.e. O (1 ◦ ) does not imply K = 0.In our model, the EOS of DE and NR matter are − / − w DE = − w DE is not an observable. It is therefore not surprisingthat in other conventions, such as the one adopted here,the EOS might be different.We note that a = √− Kt is Milne’s solution to emptyspacetime in cosmic coordinates [41-44]. Unlike Milne’soriginal work that was based on special relativity andtherefore implicitly assumed that l P , l Λ (Λ ≡ l − ), and l C = ~ / ( mc ) are constant, the framework proposed hereaccounts for the entire energy density in the universe. Kinematics of the Observed BackgroundUniverse .- Incoming radial null geodesics in Milnespacetime (Eq. 3 with a = √− Kt ) are sinh η = −√− Kr ,where η ≡ ln( t ) is conformal time. Setting ξ ≡ − η where ξ is the local ‘rapidity parameter’, the dimen-sionless ‘velocity’ and ’Lorentz factor’ are, respectively, β ≡ v/c ≡ tanh ξ = √− Kr/ √ − Kr , and γ ≡ cosh ξ =(1 − Kr ) / . Recasting Milne spacetime in ‘light-coneconformal coordinates’ t ′ = t cosh ξ, r ′ = ct sinh ξ, (8) energy densities now scale as t − = ( t ′ − r ′ ) − , i.e. thelatter are inhomogeneous in the new frame, which is infact the Minkowski frame. Note that this transforma-tion is singular at t = 0, essentially the natural choicefor the ‘origin’ of (cosmic) time. Since the coordinatetransformation of Eq. (8) mixes time and radial coor-dinates the energy-momentum tensor in Minkowski co-ordinates includes new off-diagonal terms that representradial momentum flow in this frame. This is the ori-gin of the cosmological redshift of radiation emitted at a(Minkowski) spacetime point as measured by an observerat the origin. Specifically, in Minkowski coordinates T t ′ r ′ = (1 − Kr ) / √− Kr ( ρ + P ). In the limit − Kr ≪ β ≈ √− Kr , which is perceived by the observer as re-cession of the distant emitter with apparent velocity v = H r , i.e., the Hubble law with H = c √− K . Trans-forming to the appropriate Minkowski coordinates it isthen found that v ≈ H ′ r ′ where H ′ = t ′− , i.e. t ′ = H − at present. As a result t/t = e η = p (1 + β ) / (1 − β ),i.e. the frequency redshifts ∝ p (1 + β ) / (1 − β ) wherethe velocity monotonically increases with r . Similarly,energy densities scale as ρ ∝ [(1 + β ) / (1 − β )] . Since bydefinition ρ ≡ ρ (1 + z ) , this implies that the distance-redshift relation reads √− Kr = z (1 + z/ / (1 + z ). In-deed, this differs from the corresponding SM relation;however, cosmological (e.g. luminosity, angular diame-ter) distances are not observables, rather they are model-dependent inferred quantities. For local observations thisrelation reduces to √− Kr ≈ z , which coincides with theSM result.In the light-cone conformal coordinates the universe isinfinitely old. This does not conflict with observations,e.g. that stars have not exhausted their nuclear fuel,because the model universe must be of finite age only incosmic, not conformal, coordinates, and indeed it is (byconstruction). Unlike the SM, our model conforms withthe Perfect Cosmological Principle, according to whichthere are no preferred cosmic (spacetime) events. Thismakes the light-cone conformal coordinates aestheticallyappealing as well as readily amenable to interpretationas shown above. Clearly, the transformation η = ln( t/t )maps a finite to semi-infinite time interval. This is notpossible in the SM in which η has an origin, η = 0, sincethe big bang marks a genuine- rather than a coordinate-singularity. Linear Perturbations and the CMB .- Any glob-ally stationary (and infinitely old) universe must be foundin its maximum entropy state. In the following, we con-sider entropy densities S , i.e. entropy per volume ofdiameter ( − K ) − / . It is easily shown that with fieldmass √− K DE saturates the holographic entropy (den-sity) ‘bound’, S DE = O (Λ − l − P ) = (10 ). Second toDE is the CMB with S CMB = O (10 ), and indeed theCMB black body (BB) energy distribution is not nec-essarily indicative of a hot and dense past epoch of theuniverse as is usually thought, but rather a reflectionof the requirement that in a stationary model the CMBshould be nearly at its maximum entropy state. DMparticles (if not ultralight) and baryons contribute muchlower entropy and are therefore less constrained by en-tropy considerations; yet, their overall density distribu-tion is rather homogeneous, consistent with a stationary,maximum entropy configuration.In our model, the entire sky sphere has always been,and will always be, the observable causal horizon, and theobserved CMB correlations on scales & ◦ are thereforewell within our causal horizon. Thus, by construction,there is no ‘horizon problem’.The essentially stationary background cosmologicalmodel described here clearly has to break down at ‘sub-horizon’ scales, as has been observed in, e.g. redshiftevolution of galaxy metallicity and number counts andby the very observations of reionization itself. Our so-lution of the full system of perturbation equations [45]yields the gravitational potential φ k ∝ exp( ± iω k η ) × Q k with the dispersion relation ω k ≡ c q k − K , and Q k are the eigenfunctions of the Laplacian in 3D curvedspace. This implies that the gravitational potential os-cillations propagate at the speed c/ √ O ( ~ √− K/c ) = O ( ~ √ H /c ), i.e. ‘horizon’ scaleCompton wavelength. Indeed, linear perturbations ofthe open and stationary background admit oscillatoryrather than growing modes for dimensionless perturbedquantities, e.g. Newtonian gravitational potentials, andfractional density perturbations [45]. This stability prop-erty distinguishes our model from other static models,e.g. [46].In our model there is no preferred period for the be-ginning of structure formation, and the only conceivablescenario is that, embedded in the evolving ϕ phase, thereare broken-symmetry phases of ϕ = constant which lo-cally set the standard typical timescale for gravitationalcollapse t coll = O (( Gρ ) − / ). Assuming w M = 0, thetrace of the first of Eqs. (2) implies that a constant ϕ is only possible if R > t →
0, i.e. the mass termin Eq. (1) is positive and ϕ is constant by virtue of aconformal symmetry breaking. For R > static equi-librium (real) solution of the last of Eqs. (4) with β < ϕ stat = − q | β | R λ sinh h sinh − (cid:16) A | β | R q λ | β | R (cid:17)i where ρ M ≡ Aϕ . In the limit A √ λ ( | β | R ) ≪ ϕ stat ≈ − A | β | R , anapproximation justified by virtue of the facts that λ = O ( l P l − ) and that CDM particles cannot be extremelylight [45]. This solution is consistent with R = O ( t − coll )locally.Although the fundamental physical quantities may set-tle at different constant values in different gravitationallybound systems this leaves no detectable signatures, e.g.the dimensionless ratio of electromagnetic and gravita-tional forces is universal. Since in our model the universe is infinite the shape ofthe matter power spectrum is determined not by quan-tum fluctuations of the vacuum, but by an entirely differ-ent mechanism: The CMB power spectrum on the largestangular scales is flat due to Poisson noise in photon num-ber per solid angle on the sky. The rms fractional num-ber perturbation (of CMB photons) is inversely propor-tional to the angular scale, i.e. ( ∆ nn ) rms ∝ Ω − , andtherefore l C Tl ∼ constant (where C Tl is the CMB an-gular power spectrum of temperature perturbations atmultipole number l ) by virtue of Parseval’s identity andthe fact that for a BB ∆ n/n ∝ ∆ T /T . Since bothbaryons and CDM contributions are linear in ϕ , whichvery weakly self-couples, i.e. λ = O (10 − ), perturba-tions are gaussian. In addition, adiabatic perturbationsrender the (infinitely old) universe globally stationary.From the perturbation equations the observed ‘acous-tic’ oscillations (explained in the SM as due to plasmaoscillations at recombination) could be understood in ourmodel as oscillations of the gravitational potential on cos-mological scales.Since in the present model the ‘Hubble rate’, Γ H = O ( c ( − K ) / ), is at least O (10 ) times larger than theCompton rate, Γ C ; for all practical purposes the uni-verse is optically thin. Therefore, the CMB tempera-ture perturbation (∆ T /T ) γ = δρ γ /ρ γ = O ( φ ) is ob-tained from the collisionless Boltzmann equation. Since − k K φ = δρ/ρ in the limit k ≫ − K [45], where ρ is theNR energy density, η M = η B (cid:16) ρ CDM ρ B m B m CDM (cid:17) is thematter-to-photon ratio, that is the ratio of matter, i.e.CDM and baryons, and photon number densities, where η B ≡ n B /n γ = O (10 − ), δρ/ρ = δn/n , and δn/n ∝ n − / . Poisson equation with the SM-inferred η B then re-sults in M B /M CDM = O (cid:16) η B ρ B ρ CDM ( Kk max ) (cid:17) where the mat-ter power spectrum P ( k ) peaks at k max = O (100 M pc ).Adopting ρ B /ρ CDM ≈ / M CDM equals a few MeV. Due to the strong M CDM ∝ k max dependence this mass may well be a factor of fewsmaller or larger. An example for fermionic candidatein this mass range is a sterile neutrino. This estimate,which is a consistency requirement of our model, favorsfermionic light DM candidates over the classical & GeV range for weakly interactive massive particles (WIMP’s).Another appealing property of sterile neutrinos for aninfinitely old universe is their stability.In addition, the flat C l implies a flat 3D power spec-trum of density perturbations P ( k ) by virtue of the cos-mological principle. On scales smaller than ∼ k − corr ≈ k − max the density perturbations are not statistically inde-pendent and their correlation typically damps P ( k ) byan extra ∝ k − factor [45] which approximately agreeswith SM predictions, e.g. [47-49].Since Γ C ≪ Γ H cosmic reionization at redshift z ≈ ∝ t − , the op-tical depth to Compton scattering is τ ∝ R dt/t , i.e. for-mally diverges at t = 0, rendering the singularity in theenergy densities practically unobservable. Although theuniverse is infinite, only a finite fraction of it is observ-able. We show in [45] that the observed Silk (diffusion)damping in the CMB can be explained as a projection ef-fect, rather than diffusion transversal to the line of sightas in the SM. It is also shown in [45] that CMB polar-ization is sourced by velocity gradients in the transversaldirection to the line of sight. Summary .- Although the SM is an extremely usefultheoretical framework for the description of the struc-ture and evolution of the universe, it is based on certain conventions that give rise to several puzzling features.Clearly, energy-momentum conservation in the realm ofstandard GR results in a dynamically evolving universe,but with apparent ‘coincidences’ that seem to single outthe current epoch in several ways. As has been shownabove, the coincidence problem associated with the cur-rent near equality of the energy densities of DE and mat-ter can be explained away by invoking a dynamic sys-tem of natural units, i.e. scalar fields. Moreover, theircurrently observed ratio can be obtained if the sum ofthe canonical kinetic and potential energies associatedwith the model scalar fields is identified as DE, and bothbaryons and DM particles are described by NR fermionicfields. The present model applies to scenarios in whichCDM is mainly composed of e.g. sterile neutrinos ratherthan WIMP’s.More generally, the framework proposed here makesuse of dynamical l P , l C , l Λ , etc., to effectively offsetspacetime expansion. From this perspective all standard‘early universe’ processes, e.g. phase transitions and pro-duction of topological defects, inflation, BBN, recombi-nation, etc., never take place (and indeed none of theseis practically observable); the background universe as wecurrently see it has always been the same. The present model provides a kinematical alternativeto the dynamical
SM. Consequently, our model exhibitsno evolution, no inflation to set up the initial condi-tions, etc. Instead, a kinematical model only requiresself-consistent boundary conditions, e.g. M CDM is likelysub-
GeV . The two models need only agree on dimen-sionless past light cone observables , i.e observed angu-lar sizes, angular correlations, redshifts, etc., indeed a‘boundary condition’. The proposed model explains awide spectrum of cosmological observables in terms ofself-consistency requirements, rather than via their dy-namical evolution from ‘initial conditions’ [45].More generally, cosmology may have already provided us with a unique low energy window to the underlyingconformal nature of gravity that we hitherto ignored byforcing our broken-phase solar system experience on cos-mological scales. This seems to have also led to the mis-interpretation of cosmological redshift as due to expand-ing space, rather than to varying masses in the unbrokencosmic phase.Finally, the absence of curvature singularity, and thefact that our model is infinite, may have far reach-ing implications on a broad spectrum of cosmologicalparadigms, such as quantum cosmology, the multiverse,chaotic inflation, and (the absence of) cosmic phase tran-sitions in the very early universe, and ‘early universe’theories in general.
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