I am willing to bet that FLRW model will be discarded, when observations reach sufficient precision to falsify the Friedmann equation (by the Clarkson-Bassett-Lu) test. Details are outlined at:

http://www2.phys.canterbury.ac.nz/~dlw24/universe/wager.html

The Euclid satellite, the Large Synoptic Survey Telescope (LSST) and SKA2 should all have the necessary precision (c.f., Fig 1 of http://pos.sissa.it/archive/conferences/215/016/AASKA14_016.pdf ). Euclid will take data from 2020 until 2026. The real limitation is systematic uncertainties. E.g., Euclid and 1000 supernovae are required to reach the necessary precision given the analysis of Sapone et al, http://dx.doi.org/10.1103/PhysRevD.90.023012

Our recent results, Dam et al, http://dx.doi.org/10.1093/mnras/stx1858 , demonstrate that regardless of the final outcome, the timescape model can help in understanding various systematics for supernovae, purely at an empirical level.

In the timescape scenario, a deeper understanding of gravitational energy is key to resolving the mystery of "dark energy". However, this also demands a much more refined and deeper characterisation than the comments made in your post.

I am happy to follow your project and hope it succeeds for practical reasons. It is far from my major field, so I am not intending to join in the project, although I have the math and science to read the posting of it and recognize the significance.

For the past decade I have been researching the mechanisms of connections between large scale causes and small scale results. Stress energy in vacuum can be represented as changes in amplitude and frequency for averages of oscillating microstates, similar to the action of stress on classical oscillators. The deeper understanding of gravity may be found on the small scale. since the large scale is fairly well described.

My comments are simplified as much as possible for non specialists without losing the message, allowing almost all the researchers to follow the main points and recognize the issues.

Timescape appears to have potential for describing the large scale more accurately than simplified continuous functions. In this regard I think of it as an elaborate set of finite elements.

A possible end of the FLRW argument makes your project worth reading for researchers from other fields.

FLRW follows directly from the assumptions of homogeneity and isotropy (as shown by "R" and "W"). Dark energy is irrelevant because it is automatically homogeneous and isotropic. Remember the Cosmological Constant was assumed to be zero pretty much throughout the 20th century and in most textbooks, now it is thought to be >0.7 of the critical density (present era) but the FLRW metric is still valid.

The only thing that could cause it to be revised slightly would be robust discovery of large scale anisotropy. There are hints from Planck's analysis of the CMB where the statistical variation appears to be slightly higher on one side of the sky, from apparent alignments of low order multipole components and possibly from the observations of the fine structure constant by Webb et al which suggest a spatial gradient. However, all of these are only tentative (Webb in particular could be nothing more than a small difference in calibration between the two telescopes used) and even if confirmed, they might be explained by super-horizon modes of the Harrison-Zel'Dovich spectrum so there's a long way to go before any revision is needed.

Webb data on a preferred axis is closely associated with coordinates of the South Magnetic Pole of Earth, suggesting to me that his group is detecting an interaction of gravity and magnetism which General Relativity allows when monopoles occur or when the poles of a dipole are far apart as they are in the Earth.

CMB variations can be explained by movement of Earth in the direction of constellation Leo with respect to the CMB field. Peebles allowed this in his book on Cosmology more than 20 years ago. designating the background radiation as a preferred reference frame for velocity measurements.

George Dishman has discussed FLRW with me on other threads in which we did not reach agreement. FLRW is not a problem for me when the stress energy is derived from Lagrange Density as some researchers are doing. except in the one case when a researcher tried to change the definition of Lagrange Density. The error I pointed out historically occurs in stress energy when Lagrange Density is not used to represent stress energy.

Lagrange by definition treats all potential energy as concave curving like gravity, and all kinetic energy a convex curving like radiation. The difference between the two is the net stress energy that describes celestial mechanics. When this net stress is used in FLRW the Dark Energy is irrelevant because the mistake that created it does not occur.

I would like to read how the proponents of Dark Energy recommend to put it into a Lagrange Density, Does it go into the potential or kinetic terms?

Timescape still has the potential to out perform the corrected FLRW.

As you say Jerry, claims of anisotropy are not yet robust and may well fade with better measurements.

JD: I would like to read how the proponents of Dark Energy recommend to put it into a Lagrange Density, Does it go into the potential or kinetic terms?

I'm not familiar with using a Lagrangian density approach but just looking at the introduction of this article, I would expect you would need to treat it as a scalar field with a uniform value everywhere and constant in time. Dark energy is not discrete so you couldn't use kinetic energy of a particle model and it is uniform in space so I don't think potential energy is relevant either, you would need the field approach.

https://en.wikipedia.org/wiki/Lagrangian_(field_theory)

Given that it is homogeneous and isotropic though, as I said before, the FLRW metric itself would still be correct, the mean field energy density would just be another contributing input to the equation.

Since dark energy is claimed to cause acceleration of galaxy clusters, it must be convex curving and counted with the kinetic energy in Lagrange Density. But there is no need for dark energy in the convex curving energy when radiation provides that service. Dark energy only enters when radiation is mistakenly put into the potential energy concave curving along with gravity.

JD: Since dark energy is claimed to cause acceleration of galaxy clusters, it must be convex curving ..

If it maintains constant properties as the universe expands, it has to have the same energy tensor as the Minkowski Metric, here is a simple analysis:

http://www.astro.ucla.edu/~wright/Pvac_details.gif

All of the non-zero (diagonal) components create a gravitational effect so you get +ρc^2 for the energy term and 3P=(-3ρc^2) for the pressure term.

The combination of that energy density and those of matter and radiation result in a flat universe as far as we can tell so no 'curving' at all.

GD: "If it maintains constant properties as the universe expands"

Dark energy must change in some way if it influences the change of momentum in expansion.

Minkowski Metric is flat space and comes close to the large scale cosmos, but only after concave curving influences are counterbalanced by convex curving influences. Historically dark energy was applied as the only convex curving influence, but without actually saying so.

Now this project is claiming that Timescape method can describe cosmos correctly without dark energy. Other researchers are applying radiation to the convex curving part of Lagrange Density, eliminating the need for dark energy, without actually saying so. One researcher eliminated dark energy with a scaling factor. Another researcher eliminated dark energy with acceleration balancing a force.

I agree with David L. Wiltshire that FLRW model will be replaced by better methods. A lot of popular work will be rewritten.

Many researchers agree with George Dishman. So the change will be difficult.

This is a reply to George Dishman's post 8 days ago.

George: you are correct that "FLRW follows directly from the assumptions of homogeneity and isotropy" if one assumes the Einstein equations. However, the observed Universe is not homogeneous. The observed Universe is only homogeneous in an averaged sense, on scales larger than the largest typical nonlinear structures in the cosmic web. Even if the Einstein equations describe gravity precisely on the small scales on which they are tested, a generic average of the small scale Einstein equations does not give Einstein's equations on arbitrarily large scales. This is the fitting problem: a foundational problem in general relativity that Einstein left unresolved.

I do not have space to repeat the many essays I have written on this topic. Some popular level things can be found at

https://cqgplus.com/2016/01/20/the-universe-is-inhomogeneous-does-it-matter/

https://theconversation.com/can-we-ditch-dark-energy-by-better-understanding-general-relativity-76777

and slightly more technical (but essays and so only technical in terms of jargon rather than math) at

http://iopscience.iop.org/article/10.1088/1402-4896/aa6857/meta

https://arxiv.org/abs/0912.4563 [Int. J. Mod. Phys. D 18 (2009) 2121]

Your comment that "The only thing that could cause it [FLRW] to be revised slightly would be robust discovery of large scale anisotropy" is incorrect by use of the words "only" and "slightly". The most important point is there is no principle of physics consistent with general relativity which demands that the large-scale average evolution is an exact solution of Einstein's equations for an exactly homogeneous matter distribution. The Clarkson-Bassett-Lu test (2007) is a model independent means of testing whether average evolution is exactly FLRW. In fact, there are also other tests - e.g., including the redshift time drift test (Sandage 1962) - which are also relevant which are relevant, but do no involve anisotropy.

If average evolution does not follow a solution of Einstein's equations for an exactly homogeneous matter distribution, then we are dealing with a strong backreaction solution. This means FLRW would be incorrect and so we are not dealing with "slightly" but a fundamental revision of gravitational physics as applied to cosmology. Rather than simply assuming spatial homogeneity for average cosmic evolution, we then have to explain _why_ average evolution is close to spatially homogeneous. The timescape cosmology seeks to address such issues by generalizing the Strong Equivalence Principle to cosmological averages. It is observationally viable in as far as it has been tested; c.f., e.g.,

https://arxiv.org/abs/1706.07236 [ Mon. Not. R. Astron. Soc. 472 (2017) 835]

As you correctly infer, any strong backreaction solution (whether timescape or otherwise) would also have consequences for the characterization of large scale anisotropy - and indeed anomalies are seen in the Planck data. The overall effect is that inhomogeneities are not simply described by FLRW plus local peculiar velocities (Lorentz boosts). There is also differential expansion below the scale of statistical homogeneity.

That would require a redefinition of our own standard of rest: in addition to the known motions of the sun in the galaxy, and the galaxy within the Local Group, we also conventionally assume that our Local Group is moving at 635 km/s in the direction of the constellation Hydra. Much of this last inferred velocity would have to be due to differential expansion, rather than a local Lorentz boost. In fact, it turns out that cosmic expansion is very significantly more uniform in the rest frame of the Local Group than it is in the standard "CMB rest frame", as we found in

https://arxiv.org/abs/1201.5371 [ Phys. Rev. D 88 (2013) 083529 ]

https://arxiv.org/abs/1503.04192 [ Mon. Not. R. Astron. Soc. 457 (2016) 3285 ]

It was subsequently pointed out by Kraljic and Sarkar

https://arxiv.org/abs/1607.07377 [ JCAP 10 (2016) 016]

that this observation can also be explained in the FLRW model if bulk flows on certain scales are assumed. These scales will be larger than the statistical homogeneity scale. One is left then with either resolving the issue of bulk flows via differential expansion on relatively small scales, or invoking FLRW and bulk flows on large scales - which is ultimately also a challenge for any physical gravity theory based on causality.

Astrophysical uncertainties will limit the precision with which one could test the two alternatives above, via peculiar velocity measurements. Standard siren determinations of distances - like the distance from GW170817 - would change that. But I would like to see of order 1000 binary neutron star mergers. It will come, eventually.

As for explaining the CMB anomalies, that too is plagued by systematic uncertainties, which get mixed up with the uncertainties in the astrophysical model of the microwave emissions from Milky Way galaxy, which are of the same order. In equations (2.3),(2.4) of

https://arxiv.org/abs/1512.07364 [ JCAP 06 (2016) 035 ]

we have given expressions to quantify "CMB anomalies" that would result from a differential expansion that is not the same as FLRW + local Lorentz boosts. Since every cosmological solution of Einstein's equations other than FLRW has this character, it is easy to numerically estimate whether a definitive cosmological test of the anisotropy "anomalies" is possible. A student of mine has done this (as yet unpublished) with the result that even the uncertainty in the definition of the motion of the Milky Way with respect to the barycentre of the Local Group is enough to put this in the "too hard basket" for the meantime.

I will not comment on Jerry Decker's other remarks here. He is using his own terminology - "convex curving" etc, which do not have any direct relationship to my work.

I'm happy to read the discussion in other terminology and greater detail, also to learn from the answers. Work of David Wiltshire is useful for my purpose but also very much in a different direction than mine.

" Since every cosmological solution of Einstein's equations other than FLRW has this character, it is easy to numerically estimate whether a definitive cosmological test of the anisotropy "anomalies" is possible."

FLRW has been an obstacle to engineering development of deep space transport that is not found in any of the other cosmological solutions of Einstein's equations. Hopefully the problem will be removed by specialists in that field, and just as well in ways that I have not expected.

The completed project on deep space transport led to conclusions that may indirectly affect the test of anomalies mentioned in this thread. One conclusion is that inertia resides in a local field, corresponding to frame dragging in the extreme case. The CMB discussion becomes convenient as a reference frame for zero velocity, but maybe not the only possible reference frame. Then a fast moving vehicle modifies the space it passes through, and squeezing of quantum states becomes relevant.

The mention of anomalies in Planck data may relate to another conclusion of the deep space project. It was found that Planck constant becomes variable at extreme energies and small scales where Einstein's general relativity differs from quantum mechanics.

In this question the answers are helpful to define the issues and point to a path forward, which is the reason for writing on Research Gate.

A note of explanation is given to researchers who are not specialized in this question or the several answers.

Gravity is said to curve space in such a way that extreme gravity closes an event horizon around the source of the field. Somewhat less strength of gravity can be regarded as curving space concave with respect to the source. Historically FLRW either ignored radiation or added it to the gravity, increasing the curvature of concave type. This caused a cosmological constant that was negative and much too large, in disagreement with general relativity. Dark energy was created to decrease the concave curvature and allow FLRW to agree with observations.

At the same time a Lagrange operator and a Lagrange Density operator were developed with methods that correctly describe the movement of celestial objects. These methods by definition subtract potential energy including gravity from kinetic energy including radiation, then with the net energy are successful in predicting motions in space.

Lagrange Density methods applied on large scale are in opposition to the historical use of FLRW. A number of researchers have abandoned the historical practice of FLRW and used the Lagrange operators as a source of the stress energy in FLRW, effectively making a small cosmological constant without dark energy, even when total energy is quite large. In this representation quantum mechanics is no longer in fundamental conflict with general relativity.

In terms of curvature the kinetic energies in Lagrange methods can be regarded as convex curving, a saddle shape, the opposite of potential energy concave curving, demonstrated when Lagrange operators are used in absence of strong gravity, and other potential energies are absent or very small.

The descriptions of concave and convex curving energy are unusual, but useful to describe the issues between Lagrange and historical FLRW, in terms that most researchers can understand.

These issues do not directly relate to Timescape method, except to demonstrate a failing of FLRW, and the potential end of dark energy by independent methods.

Jerry Decker : " Lagrange methods and metrics other than FLRW are supporting the concept of radiation energy being convex curving. "

Interesting. If that because of the tendency of EM radiation pressure to push the emitter and receiver in opposite directions due to recoil?

----

Another possible candidate is gravitational radiation. In theory, if every massed particle has its own gravitational field, and the field shape reflects the mass and motion of the particle, then any "forced" change in motion should cause a forced reshaping of the field, with the field-update propagating at the speed of gravity, cg. If one atom emits a photon and a second atom receives it, then we require a system-wide update of the field to reflect the redistribution of massenergy, and should also have two gravity-wave events for the emission and absorption atom-acceleration events.

Since the waves carry information, they should carry energy, meaning that in order to address g-wave issues, we need our basic equations of motion to be lossy in forward time, and time-asymmetrical. That rules out special relativity.

So ... any thermal particulate medium ought to be constantly radiating energy into the void as micro-gravitational waves, and space should be full of gravitational background noise.

Since a gravitational wave can be considered an "untethered" distortion of the metric itself, we might expect it to have a tendency to try to straighten itself out as it propagates, in which case, as the amplitude reduces and the wavelength increases, the additional space bound up in the g-wave is released, and space gets bigger.

So it's conceivable not only that Hubble expansion robs energy from g-waves, but that g-waves' energy loss due to their natural tendency to unbend is what drives expansion.

Preprint On Microgravitational Waves

Dunno. Needs modelling. And quantification. But there are an awful lot of atoms jostling each other in a star, all constantly radiating away gravitational-wave energy, and there are an awful lot of stars ...

David L. Wiltshire : " ... differential expansion ... "

Hi David! I only came across your work today, still need to go through more of it.

I'm very happy to see that someone is publishing on the consequences of differential expansion. If we convert the hyperspherical expanding S3-surface universe into a sold sphere that contains all earlier versions of itself, with the hypersphere radius being a cosmological time coordinate (giving the proper-time length of the simplest worldline connecting part of the surface to the central Big Bang event), then expansion IS timeflow.

We can then say that since timeflow is slower in regions of greater mass-density such as galaxies (gravitational time dilation), timeflow ought to be correspondingly faster in regions outside galaxies, which then ought to expand faster, which further reduces their gravitational flux-density and makes them expand even faster again. So positive feedback, leading to large-scale faster-expanding lobes, and a "bubbled" foamy structure at large scales, with galaxies being pushed aside and collecting as "wall" and "thread" structures where two or three lobes meet.

It seems traditional in cosmology to assume a uniform homeogenous distribution of matter, that creates a uniform background gravitational "floor" and a uniform maximum timeflow-rate. Adding matter than builds on that universal default assumed "floor-level" flux-density.

But none of that applies if expansion is nonlinear and positive-feedback. If we lose the gravitational "floor", then there is no longer a single value for the age of the universe, because a region in the centre of an expanding lobe could be much older than the universe radius.age as judged from within a galaxy.

If a star formed at the semi-stable region at the centre of one of those lobes, and later drifted out of the stable region and was ejected, then we might expect to see the occasional high-speed star that's older than the supposed official age of the universe.

These are supposed to exist ("Methuselah stars")

Also, if the gravitational flux-density as we leave a galaxy drops to below the expected floor level, then two more effects are:

• Stronger gravitational lensing of galaxies, and
• Weaker inertial coupling between galaxies.

With timeflow running faster in regions between galaxies, spacetime (if we treat it using fluid analogies) becomes more "runny" (just as hot water is "runnier" than cold, because of the faster signal-transmission times). So a rotating galaxy becomes a bit more self-contained with regard to definitions of inertia, acceleration and rotation, and ought to generates some sort of rotation-curve anomaly.

What we're doing here is, instead of generating stronger gravitational lensing (etc) by increasing the flux-density within galaxies (by hypothesising additional invisible "dark matter"), we're getting the same enhanced differential by reducing the flux-density outside galaxies (by losing the gravitational floor)

Preprint Dark matter without dark matter

So differential expansion predicts the effects that we currently blame on dark matter. Plus large-scale void structures, and Methuselah stars. Qualitatively, at least.

Needs modelling! :)