The $\Lambda$ and the CDM as integration constants

Notoriously, the two main problems of the standard $\Lambda$CDM model of cosmology are the cosmological constant $\Lambda$ and the cold dark matter, CDM. This essay shows that both the $\Lambda$ and the CDM arise as integration constants in a careful derivation of Einstein's equations from first principles in a Lorentz gauge theory. The dark sector of the universe might only reflect the geometry of a spontaneous symmetry breaking that is necessary for the existence of a spacetime and an observer therein.


I. INTRODUCTION
General relativity is the local version of special relativity.Gravity is thus understood to be a gauge theory of the Lorentz group.The basic variable is then a Lorentz connection 1-form ω a b , which defines the covariant derivative D, and thereby the curvature 2-form R a b = dω a b + ω a c ∧ ω c b subject to the 3-form Bianchi identity DR a b = 0 inherited from the Jacobi identity of the Lorentz algebra.
Since the beginning [1], the role of translations in the inhomogeneous Lorentz group has been elusive.What has been clear is that, in order to recover the dynamics of general relativity, some extra structure is required besides the connection ω a b .The standard approach since Kibble's work [2] has been to introduce the coframe field e a , another 1-form valued in Lorentz algebra.Only recently, the more economical possibility of introducing solely a scalar field τ a , was put forward by Z lośnik et al [3].Only then is gravity described by variables that are fully analogous to the fields of the standard Yang-Mills theory.
The symmetry-breaking scalar τ a has been called the (Cartan) Khronon because it encodes the foliation of spacetime.The theory of Z lośnik et al is pre-geometric in the sense that there exist symmetric solutions (say τ a = 0) where there is no spacetime.Only in a spontaneously broken phase τ 2 < 0, there emerges a coframe field e a = Dτ a and further, if the coframe field is non-degenerate, a metric tensor g µν dx µ ⊗ dx ν = η ab e a ⊗ e b .In terms of the 2 fundamental fields, the Lorentz connection and the Khronon scalar, the theory realises the idea of observer space [4].
A serendipitous discovery was that in the broken phase the theory does not quite reduce to general relativity, but to general relativity with dust [3].The presence of this "dust of time" could explain the cosmological observations without dark matter.In this essay, we shall elucidate how this geometrical dark matter appears as an integration constant at the level of field equations.In addition, we consider the next-to-simplest model by introducing the cosmological Λ-term.This will require another symmetry-breaking field, the (Weyl) Kairon σ a , which turns out to impose unimodularity.
The conclusion we wish to present is that a minimalistic gauge theory of gravity includes both the Λ and CDM, and they both enter into the field equations as integration constants in the broken phase.

II. DARK MATTER
Let us first make the case for dark matter.In the original, quite dense article [3] the result was derived by a Hamiltonian analysis that may not be easy to follow in details.Therefore, we believe the simple derivation below could be useful.
The SO(4,C) action can be written in the quadratic form [3], and the variations with respect to the two fields yield (what have been called "the infernal equations"), The anti/self-dual projections of a field X a b in the adjoint representation are denoted as ± X a b and defined by the property ǫ ad bc There emerges a formal solution to (2a), To make further progress we will assume τ 2 < 0, so that we can call Dτ a = e a and have the coframe field at hand.Then, since ǫ ad bc We have recovered the Einstein field equations for the self-dual curvature, sourced by a yet unknown 3-form M a .
It remains to show that this source term behaves as idealised dust.By combining (3) with the (2b) we see that − (τ [a M b] ) = 0.At this point, we can pick the simplifying gauge τ a = τ δ a 0 , wherein it becomes apparent that the spatial 3-forms M I = 0 vanish.By construction (3) we have DM a = 0, which yields two further constraints, ω I 0 ∧ M 0 = 0 and dM 0 = 0.The former implies that M 0 is a spatial 3-form, M 0 = (iρ/2) * e 0 for some function ρ, and the latter implies that this function ρ dilutes with the spatial volume.Thus ρ indeed effectively describes the energy density of dust.
Though the derivation was particularly transparent with the gauge choice τ a = τ δ a 0 , the conclusion naturally holds in any other gauge.We also have checked that coupling matter with (1) would not change the form of M a .

IV. CONCLUSION
Rather than unknown particles or modified gravity, the dark sector of the universe could be the manifestation of a spontaneous symmetry breaking that underpins the emergence of a metric spacetime.In a rigorous derivation of the Einstein's field equations from a more fundamental, pre-geometric theory, both the Λ and the CDM appear as integration constants.Though spontaneous symmetry breaking has been considered as the origin for the difference between time and space [9,10], similar results to ours have not, to our knowledge, been arrived at in less minimalistic settings.
Of the 12 real components of the complexified Lorentz connection ω a b , the 6 self-dual pieces + ω a b account for the spin connection as usual, whereas 3 (the boosts − ω I 0 in the τ a = τ δ a 0 gauge) give rise to the spatial triad through e a = Dτ a in the broken phase.It remains to be seen whether the 3 remaining anti-self-dual rotations − ω I J could be related to the SU(2) L connection in the particle sector, and whether the scalars τ a and σ a could be related to the Higgs field.Another speculation is that our formulation might provide an improved starting point for loop quantum gravity that is currently suffering from a "covariance crisis" [11].
To conclude, we propose a theory behind the two main parameters of the standard ΛCDM model of cosmology [12].
• 1 st Λ problem, the sensitivity of gravity to vacuum energy, is resolved [13].
• 2 nd Λ problem, the observed value of Λ, is related to the age of the Universe1 .
• 3 rd Λ problem, the coincidence that m 2 P Λ ∼ ρ, has a rationale in their dual origin.
In particular, from the construction of the 3-form M a we have that ∂τ ∼ m P /ρ, and from (7) we read ∂ µ σ µ ∼ 1/ √ Λ.The duality τ ∼ σ could explain the cosmic coincidence.