Is the cosmological constant an eigenvalue?
aa r X i v : . [ g r- q c ] F e b General Relativity and Gravitation manuscript No. (will be inserted by the editor)
Is the cosmological constant an eigenvalue?
Giovanni Manfredi the date of receipt and acceptance should be inserted later
Received: date / Accepted: date
Abstract
We propose to reinterpret Einstein’s field equations as a nonlineareigenvalue problem, where the cosmological constant Λ plays the role of the(smallest) eigenvalue. This interpretation is fully worked out for a simple modelof scalar gravity. The essential ingredient for the feasibility of this approach isthat the classical field equations be nonlinear, i.e., that the gravitational fieldis itself a source of gravity. The cosmological consequences and implicationsof this approach are developed and discussed. The Standard Cosmological Model ( Λ CDM) is capable of accurately reproduc-ing most cosmological observations, including primordial nucleosynthesis, thecosmic microwave background radiation, and baryonic acoustic oscillations [1].However, despite its success, the Λ CDM model displays some odd properties.Observable baryonic matter constitutes a tiny 5% of the total mass-energycontent, while the dominant components – cold dark matter (CDM, ≈ Λ , ≈ Giovanni ManfrediUniversit´e de Strasbourg, CNRS, Institut de Physique et Chimie des Mat´eriaux de Stras-bourg, F-67000 Strasbourg, FranceE-mail: [email protected] Giovanni Manfredi which is equivalent to Einstein’s original cosmological constant Λ [4]. How-ever, various studies explore more exotic possibilities where, for instance, darkenergy can vary both in space and in time (quintessence) [5].A peculiarity of the Standard Cosmological Model is that the evolution ofthe universe goes through different phases of acceleration and deceleration, de-pending on which component is dominant at a certain epoch. The very earlyinstants of the universe were characterized by a primordial exponential ex-pansion (inflation), followed by radiation-dominated and a matter-dominatedepochs where the rate of expansion was decreasing (deceleration), and finallyan epoch dominated by the cosmological constant during which the expan-sion again accelerates exponentially. The latter acceleration appears to beginaround the present epoch, which is perceived as an odd coincidence by someauthors.If one changes the composition of the universe, this sequence of acceler-ations and decelerations obviously changes too. Several authors have noticedthat a universe that neither accelerates nor decelerates (‘coasting’) fares ratherwell in explaining many observational data, in particular supernovae luminos-ity distance. A recent review [6] lists at least half a dozen such coasting cos-mologies, the prototype of which is the Milne universe, a universe empty ofmatter and expanding at a constant rate [7]. More recent examples of coast-ing cosmologies include Melia’s R h = ct universe [8], Villata’s lattice universe[9], and Chardin’s Dirac-Milne universe [10,11,12,13]. The latter is a specialscenario where antimatter has a negative gravitational mass and is present inequal amounts as ordinary matter, leading to a gravitationally empty universeat large scales ( >
200 Mpc). For coasting cosmologies, the nondimensionalscale factor grows linearly in time a ( t ) = tt , (1)where the current age of the universe is simply written as: t = 1 /H . Inciden-tally, taking a linear scale factor, as in Eq. (1), resolves most of the coincidencesor oddities of the Λ CDM model, such as why the cosmological constant startsbecoming dominant precisely at the present epoch. It also solves the horizonproblem, because the particle horizon Z t c dt ′ a ( t ′ )diverges as t →
0, implying that any two given places in space were causallyconnected in the past, which removes the need for primordial inflation.It is noteworthy that, for a coasting universe, most cosmological quantitiescan be simply written as a function of one single parameter, namely the Hubbleconstant H : t = 1 /H ≈
14 Gy ,Λ = H /c ≈ × − m − , s the cosmological constant an eigenvalue? 3 ρ = H / (8 πG ) ≈ . / m ,a = cH ≈ . × − ms − , where ρ is the average mass density of the universe and a is Milgrom’sacceleration parameter, used in MOND to obtain a good fit to galaxy rotationcurves without resorting to dark matter. G and c are Newton’s constant andthe speed of light in vacuum, respectively. The numerical values are thoseobtained for H = 70 km s − / Mpc and fit relatively well the accepted values,without needing anything else other than the current Hubble constant.Most existing coasting cosmologies assume, in one manner or another, somefundamental yet still unobserved ‘new physics’. For instance, Melia’s modelpostulates the existence of a dark energy fluid with a peculiar equation ofstate (different from that of the Λ CDM model), while Villata’s lattice universeand Chardin’s Dirac-Milne universe – although fundamentally different – bothimply gravitational repulsion between matter and antimatter.Here, we will present an alternative coasting cosmology that, unlike pre-viously proposed ones, has the advantage of not requiring any additional un-observed components nor modifications of the underlying theory of gravity(eg, antigravity). It merely stems from a new mathematical interpretation ofthe standard equations of gravity, namely Einstein’s field equations of Gen-eral Relativity (GR). It will become clear in the next sections that the crucialproperty needed for this new interpretation is that the equations be nonlinear,ie, that the gravitational field is itself a source of gravity, as is the case forGR. Because of the formidable complexity of Einstein’s field equations, weillustrate this model using a simpler – but still nonlinear – scalar theory ofgravity, one that was proposed by Einstein himself in 1912 [14], en route todiscovering the full GR.
In the usual notation, Einstein’s field equations can be written as: G µν + Λg µν = 8 πGc T µν , (2)where G µν ≡ R µν − R g µν is the Einstein tensor, R µν is the Ricci tensor, R isthe curvature scalar, g µν is the metric tensor, and T µν is the energy-momentumtensor, ie, the source of the gravitational field.The cosmological constant Λ has a long history in GR [4]. A positive Λ wasfirst introduced by Einstein to counterbalance the attractive effect of gravity,with the aim of constructing a stationary model of the universe. However, whenit became clear that such a model is by nature unstable, and especially afteraccumulating observational evidence pointed to a non-stationary expandinguniverse, Einstein abandoned the idea of a cosmological constant, which fellinto virtual oblivion for more than half a century. Since 1998, the data ofSN1a supernovae luminosity distance [15,16] suggest that the expansion of the Giovanni Manfredi universe is actually accelerating (or, at least, not decelerating), which may beascribed to a finite and positive cosmological constant, although some recentworks have questioned the accuracy of the data and their interpretation [17].From a theoretical point of view, the introduction of a cosmological con-stant in Einstein’s equations is perfectly legitimate. Indeed, the left-hand sideof Eq. (2) is the most general local, divergence-free, symmetric, rank-two ten-sor that can be constructed solely from the metric and its first and secondderivatives [18]. Without Λ , GR would be in a way ‘incomplete’. So what is Λ ? Here, opinions diverge. Some authors [19] have argued that Λ is just an-other constant of nature, on a par with Newton’s constant G . Within thisview, gravity simply depends on these two fundamental constants, and thereis nothing to be explained here. All we can do is measure to the best accuracythese two constants. Just as we do not worry (at least from a non-quantumpoint of view) about why G takes on a particular value, we should not beconcerned why Λ has its own, very small, numerical value.We do not quite agree with this viewpoint, for a simple reason. Λ is not es-sential to the theory; it can be taken equal to zero and one still gets a perfectlyviable theory of gravity. In contrast, by positing G = 0, we would not evenhave a proper theory of matter and gravity: just a given spacetime geometryuncoupled to the distribution of masses in the universe. In other words, set-ting Λ = 0 changes the solutions to Einstein’s field equations, whereas setting G = 0 changes the nature of the theory itself.Another debate concerns where the cosmological constant term should bewritten in Einstein’s equations – on the left-hand or the right-hand side. If itis written on the left-hand side (lhs), as in Eq. (2), it should be interpreted asa geometrical term, a term that makes spacetime curve even in the absence ofmatter. In contrast, if Λ is placed on the right-hand side (rhs), it should beinterpreted as a source term, part of the energy-momentum tensor, and thuscorrespond to some substance with peculiar properties (dark energy). This isanalogue, in a Newtonian context, to interpreting the inertial forces observedin a rotating reference frame as either real dynamical forces or apparent forcesdue to the transformation to a non-inertial frame.All in all, this appears to be a question of interpretation, void of anyphysical content, except if dark energy varies in space and/or time, whichcannot be reformulated as a cosmological constant. Nevertheless, given thatwe do not know much about dark energy, this debate can have some heuristicinterest.What is proposed here is simply another interpretation of Einstein’s fieldequations as a nonlinear eigenvalue problem , by rewriting Eq. (2) as: G µν − πGc T µν = − Λg µν , (3)or, after defining the nonlinear operator G ( g µν ) ≡ − G µν + πGc T µν , as G ( g µν ) = Λg µν , s the cosmological constant an eigenvalue? 5 where Λ is the eigenvalue. Formally, nothing is changed in the underlyingequations. However, the proposed interpretation entails that, like in all eigen-value problems, the value of Λ is not arbitrary, but is rather determined bythe boundary conditions of the system under consideration.In the next sections, we will illustrate the consequences of this approachusing a simple toy model of scalar gravity. The equations of GR are very complex to solve except in some idealized andhighly symmetric cases. Very few solutions are known analytically and nu-merical simulations have become feasible only in recent times [20]. In orderto illustrate the idea of an eigenvalue interpretation of gravity, we will use ascalar model that was originally proposed by Einstein in 1912 [14,21,22,23].We start from Poisson’s equation for Newtonian gravity: ∆Φ = 4 πGρ, (4)where Φ ( r , t ) is the gravitational potential and ρ ( r , t ) is the matter density.We want to incorporate in the above equation the idea that the gravitationalenergy itself gravitates, and should therefore appear on the rhs of Eq. (4).The Newtonian gravitational energy density reads as: −|∇ Φ | / πG , but justadding this term (divided by c ) to the rhs would not suffice, because the newequation would imply a different energy density. When the procedure is doneself-consistently [21,22,23], it yields the following equation ∆Φ = 4 πGc ρΦ + |∇ Φ | Φ , (5)which is indeed nonlinear, as expected . Although the velocity of light c ap-pears in Eq. (5), the speed of propagation is infinite in this model, as it isdescribed by an elliptic partial differential equation (PDE). In this sense, theabove scalar model is still Newtonian. We also note that Eq. (5) can be derivedfrom the following Lagrangian: L = c πG ∇ Φ · ∇ ΦΦ + ρΦ. (6)A cosmological constant can be added by considering that the density iscomposed of a matter part ρ m and a vacuum part − ρ Λ (with a minus sign, sothat a positive Λ entails repulsion), with: ρ Λ ≡ c πG Λ. (7) An equation almost identical to Eq. (5) (apart from a factor of 2) can be derived directlyfrom Einstein’s field equations by considering a metric where only the time-time componentdiffers from its Minkowski value and requiring it to approach the Minkowski metric for largespatial distances [21,22]. Giovanni Manfredi
This yields − ∆Φ + 4 πGc ρ m Φ + |∇ Φ | Φ = Λ Φ, (8)already written in eigenvalue format. Here, we would like to stress that theeigenvalue formulation of the equation is possible only when the nonlinearityis taken into account. In the Newtonian limit, Eq. (8) becomes [24,25]: ∆Φ = 4 πGρ m − c Λ, (9)which cannot be cast in an eigenvalue format . Nonlinear eigenvalue problemsof the type of Eq. (8) are frequently found in the mathematical literature, see[27] for a recent review. More details are provided in the Appendix A.A noteworthy feature of this nonlinear scalar model is that it can be lin-earized exactly by setting Ψ = √ Φ , which yields: − ∆Ψ + 2 πGc ρ m Ψ = Λ Ψ. (10) Ψ has the dimensions of a velocity. The similarity of Eq. (10) with the standardSchr¨odinger equation, with Λ/ Ψ , but rather | Ψ | = Φ (see the Appendix Afor some mathematical clarifications). However, we stress that this property(exact linearization) is by no means necessary for the present theory. All thatis needed is that the field equations can be cast in a nonlinear eigenvalueformat, as in Eq. (8).From Eq. (10), one can derive the following first integral:2 πGc Z V ρ m Ψ d r + Z V |∇ Ψ | d r − I S Ψ ∇ Ψ · n dS = Λ Z V Ψ d r , (11)where V is the integration volume and S its boundary, with normal vectordenoted n . When n · ∇ Ψ vanishes on the boundary (see next paragraph), thenEq. (11) can be written as E m + E field = E Λ , where E m , E field , and E Λ arerespectively the total energies due to matter, the gravitational field, and thevacuum. In this case, it is clear that Λ must be non-negative.An eigenvalue is determined by the relevant boundary conditions. In aninfinite medium, the natural boundary condition for Eq. (10) is: Ψ ( | r | → ∞ ) = c [21,22,23]. In the Newtonian limit, this corresponds to the usual inversesquare law for the gravitational force. In a finite volume (e.g., a sphere ofradius R ), the above expression should be replaced by Ψ ( | r | = R ) = c, (12) ∇ Ψ ( | r | = R ) = 0 . (13) Equation (9) implies a modification of Newton’s acceleration in the gravitational fieldgenerated by a pointlike mass m , which becomes: g ( r ) = − Gm/r + c Λ r . Such modificationbecomes sizeable only on cosmological scales. See [26] for further details.s the cosmological constant an eigenvalue? 7
The latter condition on the gradient implies that the gravitational force van-ishes on the boundary, which is automatically satisfied in an infinite medium,but has to be imposed explicitly in a finite volume.Equations (10), (12), and (13) constitute a Cauchy problem for an ellipticPDE. This problem is notoriously ill-posed [28], meaning that it does not gen-erally possess a solution for arbitrary values of the parameters and boundaryconditions. Then, the eigenvalues Λ are determined precisely as those valuesfor which the Cauchy problem does have a solution that satisfies the requiredboundary conditions. Further clarifications are found in the Appendix A.We further claim that, on a cosmological scale, the above boundary condi-tions should be applied on the Hubble sphere, R = cH , where the recessionvelocity is equal to the speed of light. Indeed, in a series of papers [29,8,30],Melia has shown that no photon emitted since the Big Bang singularity andobserved now can have traveled a distance larger than R . Thus, the Hubbleradius constitutes an apparent gravitational horizon, where the escape veloc-ity is equal to c (in contrast to its Schwarzschild counterpart, however, thishorizon is not static but expanding). The boundary conditions (12)-(13) arecompatible with a particle with velocity equal to c emitted in the past travel-ing outwards up to R , where it reverses its path and starts traveling towardsus. To illustrate the above ideas, we performed numerical simulations of Eq.(10) in spherical symmetry (ie, all quantities depend on r = | r | only), forsome matter distributions ρ m ( r ). For given boundary conditions, the eigen-value problem usually yields a whole spectrum of solutions, just like for theordinary Schr¨odinger equation. Here, we shall only consider the smallest eigen-value, ie, the equivalent of the ‘ground state’ of the system.Two typical examples are shown in Fig. 1. The results are expressed inunits in which 2 πG/c = 1 and space is normalized to an arbitrary length R .The left and right panels differ only in the total volume considered. For thecase on the left panel, the various energy terms are: E m = 2 . E field = 0 . E Λ = 2 .
65. For the case on the right panel: E m = 1 . E field = 0 . E Λ = 2 .
13. Both sets of values satisfy Eq. (11) with good accuracy. Thevacuum density (in units of c / πGR ) is respectively ρ Λ = 0 .
367 and ρ Λ =0 . Λ = 1 .
47 and Λ = 0 .
139 (in these units, ρ Λ = Λ/ Λ and ρ Λ go like 1 /V . Giovanni Manfredi
Fig. 1
Potential function Ψ ( r ) normalized to c (solid lines), matter density ρ m ( r ) (dashedline) and vacuum density ρ Λ (dotted line), as a function of the radius r normalized to areference value R . Both ρ m ( r ) and ρ Λ have been divided by the peak value ρ m (0). The leftand right panels differ only in the total volume considered. ρ m = const . , an immediate solution of Eq. (10) is: Ψ = c, Λ = 4 πGc ρ m , ρ Λ = ρ m . (14)In this case, the vacuum density perfectly cancels the mass density, yieldinga gravitationally empty universe. Taking ρ m ≈ yields the correctorder of magnitude for the cosmological constant, as we saw in Sec. 1. Foran almost homogeneous distribution with fluctuations (Fig. 2), the result issimilar: the vacuum term cancels on average the matter distribution (as long asgradients of the gravitational potential can be neglected). The above reasoningis very simple, almost trivial, but nevertheless powerful. We have not modifiedthe field equations of gravity nor have we introduced any extra components.We have only reinterpreted the equations as an eigenvalue problem, positedthe correct boundary conditions, and hence deduced the eigenvalue.Therefore, if we adopt Eq. (8) or (10) as the basis for a cosmological model,then it follows that the total density (matter + vacuum) of the universe is zeroat each epoch. A gravitationally empty universe was first proposed by Milne[7], but presented the obvious drawback of ignoring the effect of observed mat-ter. More recently, Benoit-L´ey and Chardin [10] proposed the so-called ‘Dirac-Milne’ cosmology, where antimatter has negative active gravitational mass andis present in an equal amount as matter, so that the universe is gravitationallyempty. This is an appealing proposal, but rests on a yet unverified fundamentalhypothesis about antimatter, although this may soon change, with forthcom-ing laboratory measurements of the gravitational acceleration of antihydrogenatoms being expected in the next few years [31,32,33]. s the cosmological constant an eigenvalue? 9 Fig. 2
Matter density ρ m ( r ) (solid line) and vacuum density ρ Λ (dotted line), as a functionof radius. The potential Ψ (not shown) is basically flat in this case. The present model also implies a gravitationally empty universe, with thenegative part coming from a negative vacuum energy that automatically can-cels out the positive matter density. However, this is achieved without intro-ducing any new physical hypotheses, only by reformulating the field equationsas an eigenvalue problem.For an empty universe, the scale factor a ( t ) is linear in time, as in Eq.(1), so that the expansion is neither accelerating nor decelerating (coasting).As the matter density ρ m gets diluted during the expansion and decreases as a − ( t ), the vacuum density ρ Λ also decreases following the same law, so thatthey cancel each other at each instant. Importantly, for a coasting universe,the age of the universe is always t = 1 /H ( t ), where H ( t ) = ˙ a/a is the Hubbleparameter. In contrast, in Λ CDM, this relationship is only valid at the presentepoch, which is often seen as a peculiar coincidence demanding an explication.4.2 Structure formationUsually, gravitational structure formation in a homogenous universe is studiedwith Newtonian gravity, although some results that use the full Einstein’sequations were obtained recently [20]. The Newtonian limit of the presentscalar model is given by Eq. (9), which can be rewritten as: ∆Φ = 4 πG ( ρ m − ρ Λ ) . (15)As we have seen, the vacuum densities decrease as the cube of the nondimen-sional scale factor a ( t ), so that: ρ Λ ( t ) = ρ Λ /a ( t ), where ρ Λ is a constantrepresenting the present vacuum density.For a spherically symmetric universe, where all quantities depend only onthe radius r , the Newtonian equation of motion isd rdt = − ∂Φ∂r . (16) Using comoving co-ordinates r = a ( t )ˆ r, (17) dt = a ( t ) d ˆ t, (18)the scaled equation of motion is thend ˆ r dˆ t + ˙ a dˆ r dˆ t = − a ∂ ˆ Φ∂ ˆ r , (19)where ˆ Φ (ˆ r, ˆ t ) is the scaled gravitational potential. As the density must scaleas ˆ ρ m (ˆ r, ˆ t ) = a ( t ) ρ m ( r, t ) in order to preserve the total mass, we scale thegravitational field as ˆ Φ (ˆ r, ˆ t ) = a ( t ) Φ ( r, t ), so that Poisson’s equation remainsinvariant in the comoving variables: ∆ ˆ r ˆ Φ = 4 πG (ˆ ρ m − ρ Λ ) . (20)The system of Eqs. (19)-(20) was solved numerically using an N-body codein a previous work [12], as an approximation to the Dirac-Milne cosmology. Werecall that the Dirac-Milne universe is constituted of matter and antimatterin equal amounts. Antimatter is supposed to possess a negative active gravi-tational mass and to be repelled by both matter and antimatter itself, so thatit spreads almost uniformly across the universe (albeit being expelled from re-gions of large matter overdensities, ie, galaxies). As a first approximation, onecan assume that the negative-mass antimatter is spread uniformly everywherewith constant density, which leads exactly to the scaled Poisson’s equation(20), where, in that case, ρ Λ would represent the density of antimatter incomoving co-ordinates.In the context of the present work, Eq. (20) is exact (apart from beinga Newtonian limit). Therefore, the simulations reported in Refs. [12,13] alsoapply to the present case. The main result of those simulations was that gravi-tational structures form rather quickly and closely resemble those observed forthe Λ CDM universe. In addition, structure formation slows down and stopsaround the present epoch, also in agreement with the standard cosmologicalmodel.Thus, the present ‘eigengravity’ model appears to be consistent with theformation and evolution of gravitational structures in our universe.4.3 Dark matterDark energy, or the cosmological constant, was devised to understand the be-havior of the universe on a very large scale, but, due to its extremely smallvalues, it has basically no effect locally. Galactic dynamics should be under-stood entirely in terms of standard attractive gravity. In addition, given the lowvelocities and weak gravitational fields that are involved, Newtonian gravityshould constitute a perfectly acceptable approximation. s the cosmological constant an eigenvalue? 11
Fig. 3
Top panel: Gravitational potential as a function of the galaxy radius r normalizedto R g = 200 kpc. Bottom panel: Baryonic mass density (dashed line,) and rotation velocity(solid line) as a function of the galaxy radius. The thin doted line represents the equivalent ρ Λ . Here, Φ is normalized to v g and the velocity to v g = 500 km / s. Densities are representedin arbitrary units in order to show their profiles. Nevertheless, it has been known for a long time that the rotation velocitiesof stars in the outer region of most galaxies are far too large compared to thevisible mass of the galaxy [34]. If Newtonian gravity holds, those stars couldnot be trapped in the gravitational well of the galaxy and should instead fly offtangentially under the action of the centrifugal force. Dark matter (usually inthe form of large spherical halos surrounding the galaxy) is thus postulated inorder to compensate for the missing mass. However, although many possiblecandidates have been evoked in the past, it is still unclear what particles couldmake up such invisible dark matter.A second important reason for suspecting that the total mass in the uni-verse is far larger than the visible one is related to structures formation. Withonly the baryonic mass present, there would not be enough time for the uni-verse to develop the intricate cosmological structures (galaxies, clusters ofgalaxies, superclusters) that we observe today. In the context of the modelproposed here, we have seen in the preceding Sec. 4.2 that structure formationdoes occur on the expected time scale, even in the absence of any extra ‘dark’mass.The elusiveness of dark matter has led some researchers to speculate thatthe reason for the inaccuracy of the predictions for the rotation velocities results not from the presence of some unknown substance, but rather froma modification of Newton’s inverse-square law at low accelerations. The mostestablished of these theories is Milgrom’s MOND, which performs quite well atpredicting such rotation curves [3]. The only adjustable parameter in MONDis the acceleration a below which Newton’s theory fails. By fitting MOND’slaw to rotation curve data, one finds a ≈ . × − m / s . As was alreadynoticed by Milgrom [2], the order of magnitude of this acceleration is very closeto cH = c/t = c /R , where t = 1 /H is the age of a coasting universe and R = c/H is the radius of Hubble’s sphere or cosmological horizon. In otherwords, a is approximately the acceleration necessary to bring a body fromzero velocity at the Big Bang up to the speed of light at the present time.The above value of a hints at the interesting possibility that dark mattermay in fact be the local manifestation of a global (cosmological) effect. Here,we will use this idea to show that the effect usually attributed to dark mattercan be seen as a consequence of imposing certain boundary conditions on theuniverse.We solve Eq. (8) in the vicinity of a galaxy and set the boundary conditionsat the border of a spherical region of radius R g containing such galaxy. Asusual, the field ψ , which has the dimension of a velocity, must approach c on theHubble radius R , where it has zero gradient. Therefore, on the radius R g ≪ R that we consider, we can estimate the gradient of ψ as: ∇ ψ ( R g ) ≈ c/R ,which is of the order of Milgrom’s acceleration a divided by c . This simpleconsideration supports the idea that the existence of a critical acceleration a is the local result of a distant boundary condition.We rewrite Eq. (10) in nondimensional units, by normalizing velocities (i.e., Ψ ) to v g , distances to R g , and densities to ρ g (the matter density within thegalaxy). This yields: − ∆Ψ + K ˆ ρ m Ψ = ΛR g Ψ, (21)where K = 2 πGR g ρ g /c is a dimensionless number and ˆ ρ m = ρ m /ρ g . In theseunits, the boundary conditions are: Ψ ( R g ) = 1 and ∇ Ψ ( R g ) = ( c/v g )( R g /R ).A solution of Eq. (21) is presented in Fig. 3. We have taken as parameters: R g = 200 kpc, v g = 500 km s − , and R = 4 . K = 0 .
56 and ∇ Ψ ( R g ) = 0 .
027 in normalized units. The chosen massdensity profile is displayed in Fig. 3, bottom panel: it starts at ρ g ( r = 0) =100 M ⊙ / pc at the center of the galaxy and then falls off rapidly.The rotation velocity, defined as v ( r ) = √ r ∇ Φ , grows inside the galaxycore, where the density is approximately constant, then reaches a plateau upto a distance R g = 200 kpc from the galaxy center. Despite the crudenessof the model and the many approximations that were made, the numericalvalues are rather reasonable: the potential well (top panel of Fig. 3) has adepth ∆Φ ≈ . v g = 1 . × km s − and the rotation velocity plateaus at v ≈ . v g = 100 km s − . s the cosmological constant an eigenvalue? 13 Interestingly, if we compute the matter and vacuum energies [see Eq. (11)]: E m = 2 πGc Z V ρ m Ψ d r ,E Λ = Λ Z V Ψ d r , we find E m = 0 .
032 and E Λ = 0 . Ψ on the boundary of theintegration volume.A purely Newtonian rotation curve would instead fall off as r − / . The dif-ferent behavior observed in our case originates from the boundary conditionon ∇ ψ , reflecting the fact that the gravitational potential Φ = Ψ must ap-proach c on the Hubble sphere. This boundary condition imposes a negativeeigenvalue Λ , which translates into an attractive vacuum density ρ Λ , depictedin Fig. 3 (bottom panel) as a thin dotted line. This additional density, extend-ing to and beyond 100 kpc, plays a similar role as the dark matter halo in thestandard theory. This is in line with the above interpretation that ‘dark mat-ter’ is in fact the local manifestation of a global (cosmological) effect, whichappears through the imposition of a suitable boundary condition. In this work, we proposed a new interpretation of the gravitational field equa-tions as a nonlinear eigenvalue problem. This interpretation relies on a three-fold conjecture:1. Any gravitational field equation that incorporates a self-field term (ie,where the gravitational field is itself a source of gravity) can be cast math-ematically in the form of a nonlinear eigenvalue problem;2. The cosmological constant Λ can be interpreted as the smallest (‘groundstate’) eigenvalue for this problem;3. The value of Λ is determined by the boundary conditions imposed on thefield equation.In order to illustrate the features of this interpretation, we applied it toa scalar toy model of gravity, which is still Newtonian but where the gravi-tational field sources itself. Interestingly, this model can be linearized exactlyand, when this is done, takes the form of a standard Schr¨odinger equation,with the cosmological constant as the eigenvalue. Just like the Schr¨odingerequation, the eigenvalue is determined by the choice of the boundary condi-tions. Nevertheless, we emphasize that this property of the scalar toy model(ie, exact linearization), although appealing and possibly suggestive, is not re-quired for the present theory. In other words, even if the field equations were intrinsically nonlinear and not reducible to a set of linear equations (exceptsas an approximation), our approach would still retain its validity.This approach was then tested against some of the most topical issuesin current cosmology. We could conclude that our approach: (i) provides thecorrect order of magnitude for Λ , (ii) is compatible with structure formationon a cosmological scale, and (iii) is compatible with the effects of Dark Matteron a local scale, particularly the shape of the galaxy rotation curves.Of course, the above results were obtained with a simple semi-Newtonianscalar model (although this is true also for standard approaches) and no at-tempt was made to quantitatively compare these results with observationaldata. Hence, they have to be taken as a first heuristic step to check the valid-ity of our model.We also emphasize that the model put forward here is purely classical andthe reference to eigenvalues and Schr¨odinger equations is only a mathematicalanalogy, albeit a precise one. Nevertheless, the present approach may turn outto be useful in making contact with issues in quantum gravity. In particular,an eigenvalue interpretation of the cosmological constant has been proposedby some authors in the framework of the Wheeler-DeWitt equation [35,36,37,38,39]. The link to the present work remains to be established.The present ‘eigengravity’ approach could be extended in several direc-tions. First, the relevant field equation (10) should be made Lorentz invariant,possibly by simply replacing the Laplacian operator with the Dalambertian:1 c ∂ Ψ∂t − ∆Ψ + 2 πGc ρ m Ψ = Λ Ψ, (22)which has the structure of a Klein-Gordon equation. A connection betweenGR and Klein-Gordon and Schr¨odinger-like equations was made in a recentwork [40].Secondly, the present ideas should be tested against the full GR, or atleast a better approximation than the toy model used here. More generally,one could explore the highly nontrivial question of whether Einstein’s fieldequations could be linearized exactly in a similar fashion (of course, they can belinearized as an approximation, which is often used in many contexts, not leastgravitational wave propagation). However, we stress again that the propertyof exact linearization is not crucial for the present theory.Finally, a lot more work is needed to firmly establish whether the presentapproach is quantitatively compatible with observations, particularly on cos-mological structure formation and galactic rotation curves. Acknowledgments
I wish to thank Gabriel Chardin for his thorough reading of the manuscriptand many insightful comments. I also thank Omar Morandi and Raffaele Chi-appinelli for helping with some mathematical issues. Needless to say, I amsolely responsible for the errors or inaccuracies that may still remain in thispaper. s the cosmological constant an eigenvalue? 15
A Mathematical digressions
A.1 Nonlinear eigenvalue problems
Nonlinear eigenvalue problems occur frequently in the mathematical literature. A veryreadable review was published recently [27]. More extensive discussions can be found inthe two monographs [41,42]. A nonlinear eigenvalue problem can be written generally as: F ( λ, u ) = 0, where λ is the eigenvalue, F is a nonlinear function, and u usually belongsto a Banach space. In our case, the problem takes the special form: F ( u ) = λu . A typicalexample is the p-Laplacian eigenvalue problem with Dirichlet boundary conditions:div( |∇ u | p − ∇ u ) + λ | u | p − u = 0 , in Ω, (23) u = 0 , on ∂Ω (24)where Ω ⊂ R , and p > u , just as Eq. (8) is homogeneous in Φ . this means that if u is a solution, then Cu is also asolution, for any real or complex number C . This is a property shared with linear equations. A.2 Relevant functional spaces
Let us consider the following linear eigenvalue problem, which corresponds to Eq. (10) in1D, with 2 πG/c = 1 and λ = Λ/ − u xx + ρu = λu, x ∈ I ≡ [0 , π ] , (25) u (0) = u (2 π ) = 1 , (26) u x (0) = u x (2 π ) = 0 , (27)with ρ ≥ R I ρ ( x ) dx < ∞ , and the subscript stands for differentiation. Elliptic PDEsare usually defined in Sobolev spaces [44], because such spaces guarantee the existence ofthe derivatives, at least in a weak sense. Here, the appropriate space seems to be H , whichis also a Hilbert space equipped with the inner product ( u, v ) = R I uvdx + R I u x v x dx andthe related norm k u k = p ( u, u ).We first consider the simple case where ρ = 0 (no matter density). Then, the eigenfunc-tions of Eq. (25) are cosines: ϕ n = cos( nx ), with eigenvalues λ n = n . Then, if a solution u ( x ) exists for the full equation (25) with non-vanishing ρ , it can be represented as: u ( x ) = P n a n ϕ n ( x ) P n a n , (28)which satisfies the required boundary conditions ( a n are real numbers). Of course, we havenot proven that such a solution actually exists, which is a nontrivial mathematical problem.In an infinite space ( I = R ), the norm generally diverges, because of the boundarycondition on the first derivative (27). However, this point should not be considered crucial:for instance, non-integrable wave functions are routinely used as solutions of the standardSchr¨odinger equation to describe propagating plane waves. In addition, as we have seen inthe main text, these boundary conditions are to be applied on the Hubble sphere, ie in afinite volume, so this problem should not actually arise. A.3 Elliptic equations with Cauchy boundary conditions
An elliptic PDE, such as Eq. (23), with Dirichlet boundary conditions, constitutes a well-posed problem. In contrast, if one takes Cauchy boundary conditions, the related problem is6 Giovanni Manfrediill-posed [28]. Cauchy boundary conditions correspond to specifying both the function andits normal derivative on the boundary, e.g. u = ∂u/∂n = 0 on ∂Ω . In that case, the problemdoes not always have a solution. Our Eqs. (10), (12), and (13) fall in this category.However, as an eigenvalue problem, the problem makes perfect sense. The eigenvalue λ is determined precisely by the requirement that the problem does possess a solution forCauchy boundary conditions. We can illustrate this on a simple problem in 1D. Let usconsider the following first-order nonlinear differential equation: u u x + u = λu, x ∈ [0 , , (29) u (0) = 0 , u x (1) = √ / . (30)The problem is obviously overdetermined, as a 1D differential equation admits only oneboundary condition. Hence, for fixed λ , Eq. (29) does not generally admit a solution re-specting both boundary conditions (30). This is easily verified by checking that a solutionof Eq. (29) is u ( x ) = u √ x , with u = ± p λ − x = 0 but not (for arbitrary λ ) in x = 1. But if one treats Eq. (29) as an eigenvalueproblem, the second boundary condition becomes a constraint that fixes the eigenvalue, inthis case λ = 2. This is precisely what happens for our problem: adding the extra boundarycondition on the gradient of Ψ on the boundary of the domain determines the value of Λ .As a further ‘physical’ example, let us consider the 1D heat equation with Cauchyboundary conditions: T t = T xx − S ( x ) T + λT, x ∈ [0 , L ] , (31) T ( x,
0) = T , (32) T (0 , t ) = T ( L, t ) = T , (33) T x (0 , t ) = T x ( L, t ) = 0 , (34)where T ( x, t ) is the local temperature at an instant t , − S ( x ) T is a heat sink, and λT is a heatsource. The above problem corresponds to a system initially at temperature T everywhere,which evolves under the action of the sinks and sources of heat. The boundary conditionsprescribe that the temperature must remain equal to T at x = 0 and x = L [Eq. (33)] andthat the heat flux must vanish at the boundaries [Eq. (34)]. This is not physically realizable,of course: if no heat can escape the system, then the temperature at the boundaries cannot befixed arbitrarily, but will be determined by the interplay of the sinks and sources. However, if λ is not fixed but rather considered as an eigenvalue, then the Cauchy boundary conditionscan indeed be satisfied. Physically, this means that the source term λT is tuned precisely soas to keep the temperature equal to T at the two boundaries. This determines the value of λ . We also note that a steady-state solution of Eq. (31) corresponds to a solution of ourmodel equation (10), in a 1D planar geometry. Indeed, the numerical results shown in thiswork were obtained by propagating the field Ψ ( r, t ) according to a heat-type equation likeEq. (31), so that the solution relaxes naturally to the lowest-value eigenfunction. References
1. J. Rich,