An explanation for dark matter and dark energy consistent with the Standard Model of particle physics and General Relativity

Internal galaxy and cluster dynamics employ Newton's gravity, which neglects the field self-interaction effects of General Relativity. This may be why dark matter seems needed to understand these dynamics. While the Universe evolution is treated with General Relativity, the approximations of isotropy and homogeneity effectively suppress field self-interaction effects and this may introduce the need for dark energy to describe the Universe's evolution. Calculations have shown that an increased binding due to self-interaction can indeed account for galaxy and cluster dynamics without dark matter. This increased binding implies an effective weakening of gravity outside the bound system. In this article, such suppression is estimated and its consequence for the Universe's evolution is discussed. Observations are reproduced without need for dark energy.


Introduction
For the last 20 years, observations have shown that the Universe's expansion is now accelerating. The first solid indication came from measurements of the apparent magnitude of supernovae [16,18]. The leading explanations for the phenomenon are either a non-zero cosmological constant Λ, or exotic fields [15]. This article investigates another possibility which does not require Λ, modification of General Relativity (GR) or exotic fields. This alternative is a direct consequence of a mechanism that can explain the missing mass problem without modifying GR or requiring dark matter [5,8]. The phenomenology stems from GR's field self-interaction, which causes GR's non-linear behavior 1 . A well-studied analogous phenomenon exists in Quantum Chromodynamics (QCD) because its Lagrangian has a similar structure to that of GR.
GR's Lagrangian density is: where G is the Newton constant, g µν the metric and R µν the Ricci tensor. The deviation of g µν from a constant reference metric defines the gravity field ϕ µν . Expanding in ϕ µν yields the field Lagrangian [23] 2 : where [ϕ n ∂ϕ∂ϕ] denotes a sum over Lorentz-invariant terms of the form ϕ n ∂ϕ∂ϕ, and M is the system mass. The n > 0 terms cause field self-interactions, i.e. the non-linearities that distinguish GR from Newton's theory. This latter is given by the n = 0 term. QCD's field Lagrangian is: with φ a µ the gluonic field and α s the QCD coupling. In the bracket terms, contractions of the color charge indices a are understood in addition to the sums over Lorentz-invariant terms. As in Eq. (1.2), field self-interactions arise from the terms beside [∂φ∂φ]. Those stem from the color charges carried by the gluonic field. Likewise, GR's self-interaction originates from its field's energy-momentum, the tensor-charge to which gravity couples.
In QCD, self-interaction effects are important because α s is large, typically 0.1 at the transition between QCD's weak and strong regimes [7]. A crucial consequence is an increased binding of quarks, which leads to their confinement. In GR, self-interaction becomes important for GM/L large enough (L is the system characteristic scale), typically for GM/L 10 −3 [8]. As for QCD, this increases the binding compared to Newton's theory. The latter being used to treat the internal dynamics of galaxies or galaxy clusters, self-interaction may account for the missing mass problem [5,6,8]. In Ref. [5] a numerical calculation based on Eq. (1.2) is applied in the static limit to spiral galaxies and clusters. The calculation is non-perturbative, in contrast to e.g. post-newtonian formalism, because in QCD, confinement is an entirely non-perturbative phenomenon unreproducible by a perturbative formalism. The results of Refs. [5,8] indicate that self-interactions increase sufficiently the gravitational binding of large massive systems such that no dark matter nor ad-hoc gravity law modifications are necessary to account for the galaxy missing mass problem. The framework also explains galaxy cluster dynamics and the Bullet cluster observation [3]. Self-interactions are shown to automatically yield flat rotation curves for disk galaxies when they are modeled as homogeneous disks of baryonic matter with exponentially decreasing density profiles, which is a good approximation of the direct observations. In contrast, dark matter halo profiles must be specifically tuned for each galaxy to make its rotation curve flat. An important point for the present article is that the less isotropic and homogeneous a system is, the larger the effect of self-interactions is. For example, this implies a correlation between the missing mass of elliptical galaxies and their ellipticities [5], which was subsequently verified [6].
Besides quark confinement, the other principal feature of QCD is a dearth of strong interaction outside of hadrons (i.e. quark bound states) due to color field confinement. 2 In this article, LGR has been rescaled by 1/M . (For classical systems, L can be rescaled without physical effects). The magnitude of the gravity field φ being proportional to the quantity of matter, φ 2 ∝ M , the rescaled field √ M ϕ = φ is used to emphasize that effectively, self-interaction terms couple as √ GM .
While the confined field produces an interquark force stronger than expected from a theory without self-interaction, such field concentration inside the hadron causes a field depletion outside. Similarly for gravity the increased binding in massive structures mentioned in the previous paragraph must, by energy conservation, weaken gravity at larger scale. This can then be mistaken for a repulsion, i.e. dark energy. Specifically, the Friedmann equation for an isotropic and homogeneous Universe is (for a matter-dominated flat Universe) H 2 = 8πGρ/3, with H the Hubble parameter and ρ the density. Gravity's effective suppression at large scale as massive structures coalesce, means that Gρ effectively decreases with time, which implies a larger than expected value of H at early times, as seen by the observations suggesting the existence of dark energy. GR's self-interaction or the Universe's inhomogeneity have been discussed in the past to explain cosmological observations without requiring dark energy [1]. These approaches are typically perturbative and thus neglect the possibility of non-perturbative phenomena, which are central to the analogous QCD phenomenology. Previous non-perturbative attempts have been inconclusive [2]. The present approach, while remaining within GR's formalism, see section 3, folds the inhomogeneity effects into a generic function D, see section 2, that is then determined from general considerations, like e.g. QCD's structure functions. This allows to identify an explicit mechanism (field trapping) underlying the inhomogeneity effects, and for a non-perturbative treatment. That this approach differs from others using self-interaction or inhomogeneity is illustrated by the direct connection between dark energy and dark matter that this work exposes.
In summary, internal galaxy or cluster dynamics employ Newton's gravity that neglects the self-interaction terms in Eq. (1.2), and this may explain the need for dark matter [5,8]. The Universe evolution equations do use GR, but under the approximations of isotropy and homogeneity, which suppress the effects of the self-interaction terms [5,6]. Gravity's weakening at large distance due to these terms is then neglected, which may be why dark energy seems necessary. This was conjectured in Ref. [5] and the present article investigates this possibility.

Field depletion outside massive structures
Energy conservation 3 implies that the increased binding energy in massive non-isotropic systems, e.g. galaxies or clusters, should decrease gravity's influence outside these systems. The consequence on the Universe's dynamics can be folded in a depletion factor D. It naturally appears in the Universe's evolution equations once the approximations that the Universe is isotropic and homogeneous are relaxed, see section 3. When self-interaction effects are small D 1, and when gravity's field is trapped in massive systems D → 0. Different D factors would affect matter (D M ), radiation (D R ) and dark energy (D Λ ). Electromagnetic radiation does not clump and couples weakly to gravity, so D R 1. Similarly, D Λ = 1. for the early Universe, while D M would decrease as the Universe evolved to a (locally) inhomogeneous state. Thus, D M depends on time, i.e. on the redshift z, and this dependence is driven by large structure formation. In particular significant field trapping, i.e. the transition from D M (z) 1 to D M (z) < 1, occurs when galaxies formed and became massive enough, in the range 2 z 10. D M (z) then continues to change as groups and clusters form. D M may not always decrease with time even if the structures' masses increase, since it also depends on the homogeneity and symmetry of the structures. For example, some galaxies had filament shapes for z 2. They then grew to more symmetric shapes: disks or ellipsoids, and the ellipsoid/disk galaxy ratio is continuously increasing [9,14,17]. This implies that D M may rise at small z. In all, D(z) can be constrained by its expected values at z = 0 and large z, and by the timeline of structure formation. This results in the band shown in Fig. 1, see appendix A. The functional form for D(z) is physically motivated and the width of the band reflects the uncertainties of our knowledge of the formation and evolution of large structures, and on the relative amount of matter associated with each structure type. The factor D(z) shown in Fig. 1 explains the supernova observations without modifying GR or requiring dark energy, see Fig. 2. The next two sections detail the steps leading to this result.

Accounting for field depletion in evolution of the Universe
The Einstein field equation, with S µν the energy-momentum, together with assuming that the Universe is homogeneous and isotropic yields the Friedmann equation: . Suzuki  The band is the present work (Universe containing only baryonic matter, with gravity field partially trapped in massive systems due to field self-interactions) and has no free parameters adjusted to the γ-ray or supernovae data.
where K is the space curvature sign, a the Robertson-Walker scale factor and ρ the density. Structure formation causes spatial inhomogeneities and then field trapping. Terms in R µν and S µν that vanish in the isotropic and homogeneous case now appear with the formation of structures. As shown in appendix B, these terms can be regrouped in a general factor D(z) modifying Eq.
with H 0 ≡ . a 0 /a 0 . The densities of matter, radiation and a possible Λ evolve as usual: Defining The "screened" density fractions Ω * α have the form Ω * α = Ω α D α where Ω α corresponds to the traditional definition: Ω α ≡ 8πG The Ω * α are not directly comparable with the mass-energy census of the Universe. They are relevant to densities assessed from the Universe dynamical evolution. With the definition of Ω * α , ρ is explicitly independent of D(z). Eq. (3.3) taken at present time yields: , as usual. Note that due to the D M term in Eq. (3.7), that Ω M = 1, Ω R 0 and Ω Λ = 0 does not imply Ω K = 0. This does not necessarily disagree with the WMAP assessment that Ω K ≈ 0 [10] since it depends on the Universe dynamical evolution, which is modeled differently in Eq. (3.7).
Eq. (3.3) yields the usual expression for D L , the luminosity distance of a source with redshift z, except that z−dependent density fractions Ω * α now enters D L : with x ≡ 1/(1 + z). Likewise, the Universe age is given by: . (3.9)

Supernova observations
Explaining supernova observations with GR's self-interaction is the focus of this article. The residual apparent magnitude with the last term corresponding to the empty Universe case, is computed using Eqs. (A.1) and (3.8), with Ω R ≈ 0 and Ω Λ = 0. It agrees well with the data from large-z γ-ray bursts [19] and supernovae [4,11,13,20], see Fig. 2. There is no free parameter in the band shown in Fig. 2: all the parameters in D M are constrained by observations of large structure formation, see Appendix.

Age of the Universe
For H 0 = 68±1 km/s/Mpc [15], Eq. (3.9) yields a Universe age of 13.2±1.7 Gyr, compatible with the age of 13.7 ± 0.2 Gyr from ΛCDM, for the same H 0 value and the ΛCDM values Ω M = 0.32, Ω Λ = 0.68. That the parameters entering Eq. (3.9) are constrained only by the timeline of structure formation rather than being fit e.g. to the data in Fig. 2, causes the relatively large uncertainty of ±1.7 Gyr.

Large structure formation
In a Universe without gravity self-interaction or dark matter, large structures do not have time to coalesce. What happens in the present framework can be sketched as follows: As D M (z) departs from 1, i.e. as gravity weakens globally, it strengthens locally and structure formation is accelerated compared to a Universe without self-interaction or dark matter. Since D M (z) evolves following the large structure formation, gravity strengthens locally with the same timeline. Since strengthening reproduces the dynamics of galaxies and clusters [5], the local effect of self-interaction is equivalent to the effect of dark matter. Furthermore, the position of the peak of the matter power spectrum is given by k eq = H 0 2Ω * M (0)/a eq , with a eq the scale parameter at z eq . Assuming Ω Baryon = Ω M (no dark matter) and using Ω * M = Ω M D M yield Ω * M (0) 0.3, i.e. k eq 0.014, in agreement with observations [21]. This indicates that the present approach is compatible with large structure formation.

CMB and BAO
Predicting the CMB and BAO is complex and model-dependent, involving many parameters. Like for large structure formation, a detailed investigation is beyond the scope of this first investigation. However, one can observe that since the CMB main acoustic peak position depends on the Universe dynamical evolution, its calculation should involve Ω * M rather than Ω M : θ Ω * M /z rec (with z rec 1100 the redshift at the recombination time). The resulting θ 0.8 • , agrees with observations [10].

Other consequences
Field trapping naturally explains the cosmic coincidence, i.e. that in the ΛCDM model, dark energy's repulsion presently nearly compensates matter's attraction, while repulsion was negligible in the past and attraction is expected be negligible in the future. In the present approach, structure formation depletes attraction and compensating it with a repulsion is unnecessary. The QCD analogy to the cosmic coincidence is that instead of accounting for the color field confinement in hadrons, one would introduce an exotic repulsive force to nearly counteract the strong force outside hadrons.

Summary
The structure of GR's Lagrangian induces field self-interactions. They are unaccounted for in galaxies and cluster dynamical studies, which use Newton's gravity. Accounting for them locally strengthens gravity's binding, thereby making dark matter superfluous. Energy conservation demands that the stronger binding is balanced by a weakening of gravity outside the system. However, the Universe evolution equations, when derived assuming homogeneity and isotropy, neglect such weakening since homogeneity and isotropy suppress the self-interaction effects, Similar short-range increased binding and long-range suppression are well-known in QCD, another self-interacting force whose Lagrangian is similar to that of GR.
In this article, evolution equations accounting for these effects are obtained from Einstein's field equations, and gravity's weakening is assessed. Large−z γ-ray bursts and supernova data are thus explained without dark energy. No free parameter were adjusted to these data: the effect of gravity's weakening is derived from our knowledge of large structure formation, as discussed in the appendix. Other direct consequences of this approach are an explanation for the missing mass in galaxy and cluster dynamics without requiring dark matter , flat rotation curves for disk galaxies and the Tully-Fisher relation. The direct connection between dark matter and dark energy provided by this approach eliminates the cosmic coincidence problem.

A Construction of D M (z)
The depletion function D M (z) is constrained by the timeline of structure formation: At z eq 3400, the Universe is isotropic, D M (z eq ) = 1. From 15 z 0, galaxies form and evolve to their present shapes. At z 10, about 10% of the baryonic matter has coalesced into highly asymmetric protogalaxies [12,22], with field trapping in galaxies becoming important. At z 2, galaxies evolve to more symmetric shapes. Mergers increase the elliptical/disk galaxy ratio [9,14,17]. Both effects release some of the trapped field [6,8]. From 10 z 2, galaxies form groups and protoclusters. At z 6, most of the baryonic matter is in these structures, with field trapped between pairs of galaxies. From 2 z 1.2, protoclusters evolve to clusters. For z 1.2, clusters start arranging themselves in more homogeneous superstructures: filaments and sheets, releasing some of the field trapped between clusters.
D M (z) is separated into a galactic part and a groups/clusters/supercluster part. (Such separation is an approximation since the baryonic contents of galaxies and intracluster medium interact.) This gives: where g stands for galaxy, and c for cluster, group or supercluster. R i (with i = g or c) is the fraction of the baryonic mass contained in the structure i at z = 0. (Such fractions vary with z but this is factored in D i (z)). D i (z) is the depletion function for the system i and can be adequately modeled by a Bose-Einstein (BE) function, with an additional exponential term to represent the release of trapped field as galaxies become more symmetric and as superclusters form. A BE function is well-suited because a structure grows linearly with z and when its mass m(z) becomes larger than a critical mass m crit (that depends on the structure geometry) at z crit , field lines quickly collapse and the field is trapped. Hence, for a single growing system, D(z) is essentially a step-function, D(z) = ε for z≥z crit 0 for z<z crit , where ε is the system baryonic mass divided by the Universe baryonic mass. Since the considered systems, e.g. galaxies, have different masses and shapes, the overall D i (z) is the convolution of the step-function with the mass and shape distributions. The result is close to a BE function.
Galaxies presently contain a baryonic mass fraction R g = 0.15 ± 0.10. The galaxy depletion function is: Approximating that galaxies grow most of their mass between z g,b = 15 and z g,e = 3 (b stands for "begin" and e for "end"), and then evolve to their more symmetric shapes, the BE function is centered at z g0 = (z g,b + z g,e )/2 = 9 ± 1. The uncertainty is assumed. τ g characterizes the transition width, with BE(z 0 − τ ) ≈ BE(z 0 )/2. Setting 2τ g = z g0 − z g,e , i.e. BE(z g,e ) ≈ 0.1 and the trapping has essentially ended, yields τ g = 3 ± 0.5. At small z, galaxies become more symmetric: at z 3, the ratio of elliptical over disk galaxy populations is negligible, growing to about 50% at z = 0 [12,22]. This results in a restrengthening of gravity. The z−span over which it happens being small, formalizing it with a linear or an exponential function yields similar results. The advantage of the exponential function is that D g remains analytic, which is why A g e −Bgz was chosen for Eq. (A.2). Considering that most of the field is trapped in disk galaxies while it is mostly released in elliptical ones, one has D g (z = 0) 0.5, which corresponds to A g 0.4. However, elliptical galaxies usually belong to clusters and the released intragalactic field may be retrapped between galaxy pairs. Choosing A g = 0.1 ± 0.1 accounts for this. It yields D g (z = 0) 0.2 ± 0.1. Choosing B g = (4 ± 1)z ge makes restrengthening significant only for z 0.1.
If groups and clusters contained perfectly homogeneous and isotropically distributed gas, extragalactic field would not be trapped inside groups or clusters. Then, D c (z) would represent the field trapped between groups or clusters rather than inside these structures, and one would have R c = 1 − R g . Instead, the relation is used to account for gas anisotropy. Assuming that this effect concerns a mass similar to that which has already coalesced in galaxies yields β = R g . Nevertheless, most of field trapping occurs between the groups and clusters. Groups and clusters are now arranging themselves in superstructures more homogeneous than a scattered distribution. Thus, some of the trapped field is being released and D c (z) has the same form as D g (z): with z c0 (z c,b + z c,e )/2 and τ c (z c0 − z c,e )/2. Setting z c,b = 10 ± 1 (when group/cluster start forming) and z b,e = 1.2 (when superclusters start forming) yields z c0 = 5.6 ± 1 and τ c = 2.2±0.5. As for galaxies, B c = 4z g,e = 4.8±1.6. A c is difficult to assess. A c = 0.3±0.87 is tentatively chosen. Finally, α in Eq. (A.1) accounts for possible field trapping for z 15. This must be a small effect, so α = 0.9 ± 0.1 is assumed.

B Evolution equations for an inhomogeneous, anisotropic Universe
Assuming that the Universe is isotropic and homogeneous, the Einstein Eq. (3.1) yields: S ik = 1 2 (ρ − p)a 2 g ij (δ j k + θ j k ), (B.8) where δ k j is the Kronecker delta, and α, β ij , γ ij and θ ij are functions representing the anisotropic components of R µν and S ij , i.e. the components vanishing when isotropy and the Robertson-Walker metric are assumed. Combining Eqs. where ω ≡ g il (β l j + θ l j + θ l k β k j )(g −1 ) ij i.e. it represents an average of the anisotropy factors. This is the modified Friedmann equation, Eq. (3.3), in which D ≡ ρ(1+ 3ω+α 4 )+ 3 4 p(α−ω) . In this appendix, for simplicity no distinction is made between non-relativistic matter and radiation, while the radiation anisotropy factors ω R and α R are negligible. Thus, when p ρ (early Universe), 3p(ω R −α R ) ≈ 0 and when ρ p (latter Universe), 3p(ω M −α M ) ρ(4 + 3ω M + α M ). (ω M and α M are the anisotropy factors for non-relativistic matter.) Accounting for this simplifies the depletion factor to: In the main text, the fact that anisotropy factors are different for matter, radiation and Λ is formalized by making D, ρ and p vectors.