[Paper Review] Reciprocity invariance of the Friedmann equation, Missing Matter and double Dark Energy

J. Alberto Vázquez, M. P. Hobson+3 more|arXiv (Cornell University)|Aug 13, 2012

Cosmology and Gravitation Theories1 references3 citations

TL;DR

This paper proposes a 'missing matter' component with w_X = -2/3 to restore reciprocity invariance in the Friedmann equation under conformal time duality, introducing a double dark energy model. Though constraints allow Ω_{X,0} ≈ -0.11 ± 0.14 and w_X ≈ -1.02 ± 0.20, Bayesian analysis disfavors the model by ~1.5 log-units, and posterior distributions show strong bimodality, suggesting tension with standard LCDM despite consistency.

ABSTRACT

The current concordance model of cosmology is dominated by two mysterious ingredients: dark matter and dark energy. In this paper, we explore the possibility that, in fact, there exist two dark-energy components: the cosmological constant \Lambda, with equation-of-state parameter w_\Lambda=-1, and a `missing matter' component X with w_X=-2/3, which we introduce here to allow the Friedmann equation written in terms of conformal time \eta to be form-invariant under the reciprocity transformation a(\eta) o 1/a(\eta) of the scale factor. Using recent cosmological observations, we constrain the present-day energy density of missing matter to be \Omega_{X,0}=-0.11\pm 0.14. This is consistent with the standard LCDM model, but constraints on the energy densities of all the components are considerably broadened by the introduction of missing matter; significant relative probability exists even for \Omega_{X,0}\sim 0.2, and so the presence of a missing matter component cannot be ruled out. Nonetheless, a Bayesian model selection analysis disfavours its introduction by about 1.5 log-units of evidence. Foregoing our requirement of form invariance of the Friedmann equation under the reciprocity transformation, we extend our analysis by allowing w_X to be a free parameter. For this more generic `double dark energy' model, we find w_X= -1.02\pm 0.20 and \Omega_{X,0}= 0.08\pm 0.57, which is again consistent with LCDM, although once more the posterior distributions are sufficiently broad that the existence of a second dark-energy component cannot be ruled out. Moreover, the two-dimensional posterior in the (w_X,\Omega_{X,0})-plane is strongly bimodal with both peaks offset from the standard LCDM corresponding to (-1,0), although the latter is still admissible; this bimodality is in contrast to the correctly-centred unimodal posterior obtained when analysing simulated observations from a LCDM model.

Motivation & Objective

• To address the lack of reciprocity invariance in the Friedmann equation under conformal time duality by introducing a new dark-energy-like component.
• To investigate whether a missing matter component with w_X = -2/3 can restore form invariance of the Friedmann equation under the transformation a(η) → 1/a(η).
• To test the viability of a double dark energy model with both Λ and X components using cosmological observations.
• To assess whether the existence of a second dark-energy component can be ruled out or supported by current data, using Bayesian model selection.

Proposed method

• Introduce a new component X with equation-of-state parameter w_X = -2/3 to ensure the Friedmann equation remains form-invariant under the reciprocity transformation a(η) → 1/a(η).
• Derive the Friedmann equation in terms of conformal time η and impose invariance under the duality transformation to constrain w_X.
• Use recent cosmological data to constrain the present-day energy density Ω_{X,0} under the fixed w_X = -2/3 assumption.
• Relax the w_X = -2/3 constraint and allow w_X to be a free parameter in a generalized 'double dark energy' model.
• Perform Bayesian model selection to compare the evidence for the standard LCDM model versus the extended model with X.
• Analyze posterior distributions in the (w_X, Ω_{X,0}) plane to assess model compatibility and detect potential bimodal structures.

Experimental results

Research questions

• RQ1 Can the Friedmann equation be made invariant under the reciprocity transformation a(η) → 1/a(η) by introducing a new component with w_X = -2/3?
• RQ2 What is the observational constraint on the present-day energy density Ω_{X,0} of the missing matter component under the reciprocity-invariance condition?
• RQ3 How does the inclusion of a second dark-energy component affect the posterior distributions of cosmological parameters?
• RQ4 Is the existence of a second dark-energy component disfavored by Bayesian evidence compared to the standard LCDM model?
• RQ5 Does the posterior distribution in the (w_X, Ω_{X,0}) plane exhibit bimodal structure inconsistent with the standard LCDM model?

Key findings

• The constraint on the present-day energy density of missing matter is Ω_{X,0} = -0.11 ± 0.14 when w_X is fixed at -2/3, consistent with LCDM.
• The posterior distribution for Ω_{X,0} allows significant probability even for Ω_{X,0} ≈ 0.2, indicating the component cannot be ruled out observationally.
• When w_X is allowed to vary freely, the best-fit value is w_X = -1.02 ± 0.20 and Ω_{X,0} = 0.08 ± 0.57, again consistent with LCDM.
• The two-dimensional posterior in the (w_X, Ω_{X,0}) plane is strongly bimodal, with both peaks offset from the standard LCDM point (-1, 0).
• The bimodal structure contrasts sharply with the unimodal, correctly centered posterior obtained when analyzing simulated LCDM data, indicating potential tension.
• Bayesian model selection disfavors the introduction of the missing matter component by approximately 1.5 log-units of evidence.