This preprint is on the right track.

Notoriously, the two main problems of the standard CDM model of cosmology are the cosmological constant and the cold dark matter, CDM. This essay shows thatboth the 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.

Priidik Gallagher, Tomi Koivisto, "The Λ and the CDM as integration constants" arXiv (March 9, 2021).

From the body text:

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, 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 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 Zlo´snik et al. 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)Khrononbecause it encodes the foliation of spacetime. The theory of Zlo´snik 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 theKhrononscalar, the theory realises the idea of observer space.

A serendipitous discovery was that in the broken phase the theory does not quite reduce to general relativity, but to general relativity with dust. 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)Kaironσ^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. . . .

the similar magnitude of the observed energy densities due to the Λ and due to the CDM could be the result of their common origin in the conformal geometry of the observer space.

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, 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 = τδ^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”.

To conclude, we propose a theory behind the two main parameters of the standard ΛCDM model of cosmology.

• 1st Λ problem, the sensitivity of gravity to vacuum energy, is resolved.

• 2nd Λ problem, the observed value of Λ, is related to the age of the Universe.

• 3rd 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 ∼ √p m(P)/ρ, and ∂(µ)σ^µ ∼ 1/√Λ. The duality τ ∼ σ could explain the cosmic coincidence.

## 2 comments:

It is a mystery to me, what it means *physically* for these entities to "be" constants of integration. An integration constant is a parameter in the solution to an indefinite integral. It means there is a family of solutions to the integral, indexed by the possible values of the constant.

So here that means that the solutions to their pre-general-relativistic theory have numbers associated with them, that correspond to dark matter density and dark energy density in lambda CDM. But what are these numbers physically?

One way that a solution to a physics equation can have a number associated with it, is if it represents a conserved quantity like energy or momentum (see Noether's theorem). That would mean that different values of "DM" and "DE" correspond to different initial conditions of something in the theory.

Another possibility is that these numbers would be physical constants, e.g. coefficients of new interaction terms.

Really, if you think about the role of numbers in physics, they generally appear either in the physical state, or in the equation which describes how the state evolves over time. other. Maybe their mathematics doesn't say which, but I think this will have to be one or the other.

Good points. I agree that it really deserves better articulation and thought.

The money quote, in my mind is: "A serendipitous discovery was that in the broken phase the theory does not quite reduce to general relativity, but to general relativity with dust. The presence of this “dust of time” could explain the cosmological observations without dark matter."

There was a physics forum thread in GR talking about the technical meaning of "dust" in GR and I should review it.

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