Is it possible that the assumption of the FLRW metric is creating a hurdle to the analysis of dark energy?
As long as dark energy distribution is both homogeneous and isotropic on scales larger than about 500 Mpc, it is compatible with FLRW metric since homogeneity and isotropy are the only conditions necessary to derive this metric. If cosmological constant is the dark energy then it trivially satisfies the requirement at all scales.
Of course it's compatible; it suffices to introduce the metric in Einstein's equations, in the presence of a positive cosmological constant, and check that they can be satisfied.
It's the most general metric compatible with expanding, isotropic, homogeneous universe. The metric gij is defined as ds2= gij dxi dxj and line element square ds2 contains a(t)2 explicitly. The shape of this function a(t), called the scale factor, changes in an universe dominated by dark energy having a positive cosmological constant, when compared to the case of where there is no cosmological constant. See: https://ned.ipac.caltech.edu/level5/March08/Frieman/Frieman9.html
ps. In vector analysis scale factor is defined as sqrt(gii), here a(t) is called scale factor according to that definition.
Is the FLRW metric compatible with dark energy?
I suspect the answer is no, though I also suspect there is no such force as dark energy.
My suspicion is based on factors and inferences which include the following.
1) Generally weight, heat and energy (and possibly some other variables) occur per dimension. The first to point that out may have been Galileo about 1638 in connection with Two New Sciences, in his observations about weight relative to the cross-sectional area of supporting bone.
2) A second example of the role of dimension is in the 1838 article of Sarrus and Rameaux, pointing out that heat generation in an animal varies with L^3 while surface area emission of heat varies with L^2.
3) Dimensional capacity is a general principle.
4) Dimensional capacity implicitly appears in Stefan's Law in an intermediate step of its derivation (as in Planck's account of Boltzmann's proof); in Clausius's 3/4 mean path length (1860); Richardson's 4/3 scaling of wind eddies (1926); Kolmogorov's theoretical derivation of Richardson's measured 4/3 scaling; Klieber's measured 3/4 metabolic scaling; the 4/3 fractal envelope of Brownian motion (Lawler 2001). It is also implicit, I think, in J. J. Waterston's 1845 article that first appeared in 1892 at the instance of Lord Rayleigh in the Royal Society's publication.
5) The 4/3 law implies the contemporaneous existence of two reference frames, one of 4 dimensions (3 dim space + one dimension light motion) and one of 3 dimensions (3 dim space by itself).
6) If energy is proportional to distance, then distance in the 3 dim space is 4/3 of the corresponding distance in the 4 dim space. In that case, there is no mysterious force, just a run of the mill everyday consequence of different dimensioned systems sharing weight, heat or energy, considered in a way analogous to the weight - cross-sectional area of bone example of Galileo.
7) If the 4: 3 scaling attributable to dimension accounts for dark energy, then the cosmic scale factor a in the FLRW metric, which brings in Hubble's constant in the treatment of Lambda, must be adjusted or perhaps discarded.
See
1. A. Chubykalo, V. Kuligin, A. Espinoza Spatial curvature as a distorted mapping of Euclidean space. Boson Journal of Modern Physics (BJMP). October 13, 2018, Volume 4, Issue 2, www.scitecresearch.com.
As long as dark energy distribution is both homogeneous and isotropic on scales larger than about 500 Mpc, it is compatible with FLRW metric since homogeneity and isotropy are the only conditions necessary to derive this metric. If cosmological constant is the dark energy then it trivially satisfies the requirement at all scales.
Dr Das Gupta
Your point is important and succinct: "since homogeneity and isotropy are the only conditions necessary to derive" the FLRW metric. Is it possible there is an implicit condition that is overlooked in the derivation? Having regard for the nature of 4/3 scaling which arises from a ratio of dimensions, and which appears to relate to DE, perhaps there is an additional assumed condition used in deriving the FLRW metric, namely, that there is a single spatial reference frame of 3 dimensions? If there are 2 cosmological reference frames, one of 3 dim and one of 4 dim, and if that conflicts with the implicit assumption in the FLRW metric that there is a single 3 dim reference frame, that would raise the question here posed. There are a variety of phenomena which suggest that the ratio of 4 dim to 3 dim is common, universal and scale invariant.
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