Mordred

That's not quite correct the FLRW metric isn't used for the lumpiness. It is used to model the evolution history of the entire observable universe in accordance to GR and the thermodynamic ideal gas laws. The metric itself doesn't work well for non uniform distribution it is however well suited for a homogeneous and isotropic energy/mass distribution (uniform). The primary purpose of the FLRW metric is to describe how the universe expands or contracts in accordance with the above. Though it also can be used for a few other details such as the blackbody temperature history . This is the inverse of the scale factor "a" of that metric. The math I posted earlier is mostly the FLRW metric with a bit of GR and the Euler Langrangian. That demonstrates that the three methodologies are compatible with each other.

The two main energy categories used in the Langrangian including Noether is potential energy and kinetic energy. This covers mechanical and quantum energy types. Keep in mind naming energy types is simply convenient labels. The most convenient and near universal labels one can apply being the two I just named as they are used in the Lanqrangians of every gauge group of the Standard model as well as the Langrangian forms describing spacetime. This may help if we were to model the universe using the FLRW metric we tend to set the universe as a perfect fluid with adiabatic expansion. With those settings we further assume a closed system where energy is conserved. FLRW Metric equations \[d{s^2}=-{c^2}d{t^2}+a({t^2})[d{r^2}+{S,k}{(r)^2}d\Omega^2]\] \[S\kappa(r)= \begin{cases} R sin(r/R &(k=+1)\\ r &(k=0)\\ R sin(r/R) &(k=-1) \end {cases}\] \[\rho_{crit} = \frac{3c^2H^2}{8\pi G}\] \[H^2=(\frac{\dot{a}}{a})^2=\frac{8 \pi G}{3}\rho+\frac{\Lambda}{3}-\frac{k}{a^2}\] The following setting describes the energy conservation statement \[T^{\mu\nu}_\nu=0\] gives the energy stress mometum tensor as \[T^{\mu\nu}=pg^{\mu\nu}+(p=\rho)U^\mu U^\nu)\] \[T^{\mu\nu}_\nu\sim\frac{d}{dt}(\rho a^3)+p(\frac{d}{dt}(a^3)=0\] which describes the conservation of energy of a perfect fluid in commoving coordinates describes by the scale factor a with curvature term K=0. the related GR solution the the above will be the Newton approximation. \[G_{\mu\nu}=\eta_{\mu\nu}+H_{\mu\nu}=\eta_{\mu\nu}dx^{\mu}dx^{\nu}\] Thermodynamics Tds=DU+pDV Adiabatic and isentropic fluid (closed system) equation of state \[w=\frac{\rho}{p}\sim p=\omega\rho\] \[\frac{d}{d}(\rho a^3)=-p\frac{d}{dt}(a^3)=-3H\omega(\rho a^3)\] as radiation equation of state is \[p_R=\rho_R/3\equiv \omega=1/3 \] radiation density in thermal equilibrium is therefore \[\rho_R=\frac{\pi^2}{30}{g_{*S}=\sum_{i=bosons}gi(\frac{T_i}{T})^3+\frac{7}{8}\sum_{i=fermions}gi(\frac{T_i}{T})}^3 \] \[S=\frac{2\pi^2}{45}g_{*s}(at)^3=constant\] temperature scales inversely to the scale factor giving \[T=T_O(1+z)\] with the density evolution of radiation, matter and Lambda given as a function of z \[H_z=H_o\sqrt{\Omega_m(1+z)^3+\Omega_{rad}(1+z)^4+\Omega_{\Lambda}}\] Now prior to electroweak symmetry breaking everything is in thermal equilibrium so we can describe this period as a scalar field. As I already have the workup for Higgs inflation handy from another thread I will add it here as an example. The subsequent equation does in fact work the same for chaotic inflation so its essentially identical though the derivatives to arrive to the equation of state does vary slightly. Higgs Inflation Single scalar field Modelling. \[S=\int d^4x\sqrt{-g}\mathcal{L}(\Phi^i\nabla_\mu \Phi^i)\] g is determinant Einstein Hilbert action in the absence of matter. \[S_H=\frac{M_{pl}^2}{2}\int d^4 x\sqrt{-g\mathbb{R}}\] set spin zero inflaton as \[\varphi\] minimally coupled Langrangian as per General Covariance in canonical form. (kinetic term) \[\mathcal{L_\varphi}=-\frac{1}{2}g^{\mu\nu}\nabla_\mu \varphi \nabla_\nu \varphi-V(\varphi)\] where \[V(\varphi)\] is the potential term integrate the two actions of the previous two equations for minimal scalar field gravitational couplings \[S=\int d^4 x\sqrt{-g}[\frac{M_{pl}^2}{2}\mathbb{R}-\frac{1}{2}g^{\mu\nu}\nabla_\mu\varphi \nabla_\nu \varphi-V(\varphi)]\] variations yield the Euler_Langrene \[\frac{\partial \mathcal{L}}{\partial \Phi^i}-\nabla_\mu(\frac{\partial \mathcal{L}}{\partial[\nabla_\mu \Phi^i]})=0\] using Euclidean commoving metric \[ds^2-dt^2+a^2(t)(dx^2+dy^2=dz^2)\] this becomes \[\ddot{\varphi}+3\dot{\varphi}+V_\varphi=0\] \[S=\int d^4 x\sqrt{-g}[\frac{M_{pl}^2}{2}\mathbb{R}-\frac{1}{2}g^{\mu\nu}\nabla_\mu\varphi \nabla_\nu \varphi-V(\varphi)]\] and \[G_{\mu\nu}-\frac{1}{M_{pl}}T_{\mu\nu}\] with flat commoving geometry of a perfect fluid gives the energy momentum for inflation as \[T^\mu_\nu=g^{\mu\lambda}\varphi_\lambda \varphi_\nu -\delta^\mu_\nu \frac{1}{2}g^{\rho \sigma} \varphi_\rho \varphi_\sigma V(\varphi)]\] \[\rho=T^0_0=\frac{1}{2}\dot{\varphi}^2+V\] \[p=T^i_i (diag)=\frac{1}{2}\dot{\varphi}^2-V\] \[w=\frac{p}{\rho}\] \[w=\frac{1-2 V/\dot{\varphi^2}}{1+2V/\dot{\varphi^2}}\] This last equation is the equation of state for a scalar field for both Higgs inflation as well as chaotic inflation. The result gives w=-1 most of us are familiar with. With w=-1 this tells us Lambda (DE) is constant. In thermodynamics it also represents an incompressable fluid. If we're dealing with quintessence then we would have a value greater or lesser than w=-1. In this case DE would vary over time. Anyways what the above shows us is that in cosmology we model our universe under the following assumptions. A perfect fluid with adiabatic and isentropic process where the system is closed (causality via the speed limit of information exchange c further ensures this.) With the further assumption that due to being a closed system we can apply energy conversation. Any conserved quantity must be in a closed system that's one of the golden rules when it comes to any conservation law

That doesn't work sorry to say if a volume changes but the total energy remains constant then accordingly the energy density decreases. The only way energy density would remain constant is if energy is added to the system as the volume increases

I believe you mean total energy remains the same. As the volume increases the energy density would as well. However consider this detail. Does total energy remain the same if the cosmological constant is constant? Indeed the energy gets incredibly high along with the temperature which will correspond to the inverse of the scale factor. As to whether the conservation of mass energy applies to the universe as a whole. Well there are arguments in both corners. If your curious here is a useful formula to calculate the Hubble parameter at a given cosmological redshift \[H_z=H_o\sqrt{\Omega_m(1+z)^3+\Omega_{rad}(1+z)^4+\Omega_{\Lambda}}\] Here is an example argument stating why energy conservation wouldn't apply https://bigthink.com/starts-with-a-bang/expanding-universe-conserve-energy/ However one can easily find counter arguments that energy conservation does apply. For example I've read a recent paper from Allen Guth that it does apply. Needless to say its still debatable

Fair enough, one often sees different claims of our universe being in a BH or WH. The aforementioned difficulty in having a homogeneous and isotropic universe is one piece of evidence against the possibility.

More food for thought the majority of BHs rotate. So how does one arrive at a homogeneous and isotropic universe that resides in a rotating BH ? Even if the BH isn't rotating having a homogeneous and isotropic universe would be difficult.

Your welcome, a couple of Your welcome numerous articles will often state that one can use the critical density formula to calculate the energy density of Lambda. \[\rho_{crit} = \frac{3c^2H^2}{8\pi G}\] Which if you use the Hubble parameter value today will give an energy density of roughly \[6.0*10^{-10} joules/m^3\] However there is an interesting side note. The Hubble parameter is higher in the past than today hence I rarely call it Hubble constant. Now if this formula is used to calculate the energy density of Lambda this would then imply a far higher energy density at the pre-inflation period just after the initial moment of the BB. If this is true then it is the equation of state for Lambda that is constant and not the energy density itself. This is something I have been thinking about for some time. As I question whether the critical density formula is a valid method to calculate the critical density of Lambda. It may simply be accurate only during the Lambda dominant epoch we are currently in as a rough calculation for Lambda energy density. If one examines how the critical density formula is derived its derivative arises using matter with the corresponding equation of state. Originally its use was to define the point where the universe would switch from expansion to contraction. Which is another reason I question its validity with regards to Lambda. The main point however is that we cannot directly measure the energy density of the vacuum we can only infer its energy density from its influence.

As Phi for All mentioned is that we honestly cannot accurately define any region of spacetime as absolute nothing. For example under QM the minimum energy due to quantum fluctuations is \[E=\frac{1}{2}\hbar w\] This is referred to as zero point energy ZPE for short https://en.m.wikipedia.org/wiki/Zero-point_energy

How do you have a field excitation without a field ? One can describe any geometry as a field. Let's take an example let's toy model a hypothetical universe one that is critically flat. As spacetime is a field theory by all definitions. In this case one would measure zero curvature at all locations. The only source of uncertainty in this example would be systematic measurement errors. We quantize the amount of performed all the time. Energy is defined as the ability to perform work. Any time a force is exerted work has been done.

Your welcome you will find that this will help understand a large range of physics related topics. Examples being particle creation/annihilation virtual, real and quasi. Gravity aka spacetime curvature. (Apply Newtons Shell theorem) in essence it's an identical phenomenal. Aharom Bohm effect just to name a few related examples Quite accurate

Here is food for thought under the aforementioned QFT. What we of as particles are essentially localized field excitations. The fields themselves pervade all of spacetime. This makes sense as a field is a set of values at every geometric location even if the value is zero. So at any point in our universe is there anything we can honestly term as nothing ? Now energy is simply the ability to perform work. In essence it is simply a property much like mass being the property of resistance to inertia change or acceleration. Now ask yourself this question. If you have a perfectly uniform field where every point has precisely the same value. Are you able to measure the amount of energy at any point ? It would be much like trying to measure voltage on two points on the same conducting wire. You would read zero volts as you have no potential differences between the two points. When you think about it energy of a field results from non uniformity. Aka quantum fluctuations. Those fluctuations affect each other in constructive and destructive interference. The zero energy universe model aka universe from nothing in essence details this. You take this one step further. If every object was moving at the same direction and at the same velocity. Then you would believe every object is stationary (aka one of the statements involving relativity).

The Cosmological principle is still fully valid. Nothing I have heard or read with regards to the James Webb data counters that AFIAK. Everything I stated with regards to this thread still stands

just an fyi one under development though its been around awhile for gravity is quantum geometrodynamics. It like any quantum gravity theory is still not renormalizable

its not as simple a matter as mere calculations in order to renormalize gravity you must eliminate any divergence. a common method used is via a regulator operator. However with gravity we do not know any upper bound ( ultraviolet boundary). Good example being the singularity of a BH.

Deuterium BBN reference https://www.astro.uvic.ca/~jwillis/teaching/astr405/astr405_lecture4.pdf equation 18 \[\frac{\mathcal{n}_D}{\mathcal{n}_p\mathcal{n}_n}=6(\frac{m_nk\tau}{\pi\hbar^2})^{-3/2}exp(\frac{Q_D}{k\tau}\]

Bump, still examining this still trying to figure out thermal equilibrium dropout of several other particles and relevant atoms via Saha equations

not necessarily mass is simply resistance to inertia change. The video is likely referring to thermal equilibrium and the subsequent thermal dropout due to inflation of the electroweak symmetry breaking. Likely Higgs inflation

It all depends on the mass of the BH. It isn't reliant on the gradient. I'm well aware your not strong on the mathematics so I won't try to post the ZAMO mathematics (zero angular momentum observer) with regards to frame dragging. The details of such can be found here https://www.roma1.infn.it/teongrav/onde19_20/kerr.pdf

attending a school is the best way short of that one an work through numerous textbooks choose ones with practice problems.

Careful the overdot on the scale factor is a time derivative for velocity. Will look over the rest later on. Here https://en.m.wikipedia.org/wiki/Time_derivative I don't believe your intention is to modify the fluid equations above The fluid equations directly derive from the thermodynamic laws via the effective equations of state (cosmology). I'm not clear on this statement of yours \[a_{obs0} = a_{real0} = a_{apparent0} = 1\] the scale factor is simply a constant of proportionality as I'm sure your aware the scale factor today is set at 1. the scale factor at some point in the past is simply the ratio of radius of the observable universe then as opposed to today. For example a scale factor of 0.5 the radius of the observable universe would be half what it is today. So I do not know where your getting the extra scale factor terms such as apparent. Nor the need to normalize the above terms as that would remove the usefulness. However the equations above wouldn't work. As I mentioned the overdot is the velocity term of the scale factor in the equation above. If your goal is to match observational data you will also need the full Friedmann equation with the cosmological term. You will still have nonlinearity in point of detail until the Lambda dominant era the universe expansion was slowing down until the Lambda term became dominant a close examination of this will show this occurs roughly at the universe age of 7 Gyrs. The value will vary according the the cosmological parameter dataset used. Via the lookback time equation you have above which is similar enough to the one I posted earlier. None of this still addresses numerous other problems such as the nucleosynthesis, electroweak symmetry breaking, which both rely on thermal equilibrium relations and expansion not to mention the fine structure constant and other related coupling constants which have relations involving radius for their effective strength. Nor have you mentioned any particular cause for shrinking matter and how the rates of the shrinking will correspond to a varying rate of change in expansion rates. The rates of shrinkage would have to correspondently shrink. I fail to think of any viable mechanism as to how that could possibly work. That includes via gravity or any of the fundamental forces. However my feelings as to your model viability is secondary. The point of detail is that you are making the effort to properly model and in that regard I will still assist as one learns from correctly modelling even when the model is wrong. LOL truth be told a good theorist physicist will do everything in their power to prove their own models wrong. That's how they become robust to begin with. One other thing to keep in mind. Even if you fully get the mathematics to work and match the curves I mentioned (they correspond to actual datasets otherwise the theory wouldn't be viable to begin with). You will still need to figure out the cause of shrinkage and what controls that states behavior over time. As well as the cause of its history of variations. That will also need to be mathematically modelled.

I would like you to consider the following in terms of Luminosity distance. You can see from the equations above in my previous post that the evolution of matter, radiation and Lambda has significance in distance measurements as well as expansion rates. Furthermore the common formulas you often see for redshift, luminosity, universe age etc do not include those details. example above. here is how luminosity distance relates with the evolution of the above. the energy flux being the measured energy per unit time per unit area of the detector. the luminosity distance is then defined on the radius of the sphere centered on the source in which the absolute luminosity would give the observed flux. \[\mathcal{F}=\frac{\mathcal{L}}{4\pi d^2_L}\] as light travels on null geodesic ds^2=0 \[ds^2=dt^2-at^2[\frac{dr^2}{1-k r^2}+r^2(d\theta^2+sin^2\theta d\phi^2)]\] with k=0 and the various contributions above this gives \[\frac{dr}{1+a_o^2H_o^2 r^2\Omega_k}=\frac{1}{a_o^2H_o^2}\frac{dz}{(1+dz)^2(1+dz\Omega_M)-z(2+z)\Omega_\Lambda}\] which determines the coordinate distance (not proper distance) as a function of redshift for \[r=r(z,H_0,\Omega+M\Omega_\Lambda)\] energy becomes \[E_O=\frac{E}{1+z}\] rate of photon arrival will be time delayed via \[dt_o=(1+z)dt\] \[\mathcal{F}=\frac{\mathcal{L}}{4\pi a^2_Or^2(z)}=\frac{\mathcal{L}}{4\pi d^2_L}\] gives luminosity distance as a function of redshift \[H_O d_L=(1+z)|\Omega_k|^{-1/2}sinn[|\Omega_k|^{-1/2}\int^z_0\frac{d\acute{z}}{\sqrt{(1+\acute{z}^2)(1+\acute{z}\Omega_M)-\acute{z}(2+\acute{z})\Omega_\Lambda}}]\] where sinn(x)=x k=o,sin(x) if k=1, sinh(x) if k=_1 leads to \[H_Od_L=z+\frac{1}{2}(1-\frac{\Omega_M}{2}+\Omega_\Lambda)z^2+\mathcal{O}(z^3)\] where H evolves as \[H_z=H_o\sqrt{\Omega_m(1+z)^3+\Omega_{rad}(1+z)^4+\Omega_{\Lambda}}\] temperature as a function of redshift gives \[T=T_O(1+z)\] the above is a methodology by Juan Garcıa-Bellido in his numerous papers, though you can find similar solutions in Bunn and Hoggs (distance measures) and Lineweaver and Davies. Here is the Hogg paper Distance measures in cosmology https://arxiv.org/pdf/astro-ph/9905116.pdf in this paper you can see similar a similar treatment using E(z) equation 14. The paper further covers angular diameter distance which further relates to luminosity distance. One of the details you should note in the last paper is that many of these factors do not have identical rates of change. see the graphs in the paper above for an example. Those should further highlight the non linearity logarithmic rates of change.

I have a very simple policy. When I see a posting that the author cannot be bothered to ensure its legible and easy to read. Then I cannot be bothered with that posting. I am positive numerous other readers feel the same way.

I have read numerous Cambridge materials as well. Some of the better examples of various math treatments in numerous fields such as the ones you mentioned and others are contained within them. I have always found them handy

Thanks for the info as I tend to collect good literature I may pick up a copy

The cosmological contribution is a kinetic energy term if you take the critical density term. When you solve the critical density formula you will find it will give an answer that will be approximately 10^{-10} joules/cubic metre. much like photons can power a solar sail even though photons have no inavariant mass the photon has momentum.