Mordred

The first is just your spherical coordinates see http://mathinsight.org/spherical_coordinates The a(t) is that other dimensionless constant called the scale factor. https://en.m.wikipedia.org/wiki/Scale_factor_(cosmology)

Let me get back to you on this, I'd rather see how well Stephaneww picks up on the above first hehe as my reply will step into a very useful equation.

Here this will save some time on critical density with regards to Equations of state. https://www.google.ca/url?sa=t&source=web&rct=j&url=http://www.astronomy.ohio-state.edu/~dhw/A5682/notes4.pdf&ved=0ahUKEwjzzbmGobPXAhVO22MKHeYwA4IQFgggMAE&usg=AOvVaw0xjZcR1b8OcNFbrriXGEr2 I also have my own article on universe geometry that details more into the FRW metric. http://cosmology101.wikidot.com/universe-geometry Page two covers how the line metric of the FLRW metric works but doesn't cover the fluid equations themself as per the first link. http://cosmology101.wikidot.com/geometry-flrw-metric/ The FLRW is probably the easiest approach to modelling in Cosmology and to understanding the more complex aporoaches under other field theories. this is a good overview descriptive of how the FLRW metric works. https://arxiv.org/abs/astro-ph/0409426

Fair kk no prob but will take a bit to keep the explanation as Newtonian as possible. Ok lets start with a homogeneous and isotropic state of a given radius. ( no preferred location nor direction) In essence a uniform distribution, this is how Lambda would be modelled as a scalar field with no preferred direction.. Now ti understand the [math]w=\frac{p}{\rho}[/math] relation one must understand that pressure is force per unit volume. This obviously requires momentum, however in the case of Lambda one of no preferred direction ie random Brownian motion. Matter fields with low momentum exert no pressure, w=0. Radiation exerts higher pressure due to high kinetic energy. Now in the ideal gas laws under statistical mechanics, you often see pressure as described as particles bouncing of walls, in this case the treatment is number of particles that will cross a given coordinate axis in a given unit of time. w is just a dimensionless constant for the ratio of proprtionality of the last descriptive. The FLRW metric itself primarily concerns itself with how the volume of the universe evolves. (as per) the bold above but we can simply use the change in radius and apply another dimensionless constant for how the volume itself evolves via Hubbles parameter (careful it is only constant everywhere at a given moment in time) That being the scale factor. a(t) This change in volume will naturally be modelled in a vector like fashion as were concerned if it is expanding or contracting. As matter causes collapse its natural to keep this positive with w being negative being expanding. (negative pressure, contracting volume of gas naturally increases pressure of a vacuum). if were good with the above, we can show this under math to explain equation 9 better. (ps this treatment above is essentially similar to particles in a box under QM)

Well keep in mind the FRW metric is a simplification of GR. You essentially have the above correct but the reason I suspect you don't follow why is that math your not following. Don't feel bad on that as some of the key relations above is mentioned but not fully described. So I need some more feedback, Is it the stress tensor equations that your not understanding ie the Minkowskii Stress equation given as equation 3 which this article defines as the perfect fluid energy tensor.? Secondly have you worked much with statistical mechanics ? ie in this case an adiabatic and istropic fluid? (ie are you familiar with how to mathematically model such using statistical mechanics. thirdly equation, 9 has a mathematical proof that can be heuristically described via Newtonian physics involving the shell theorem. If I recall Liddle has it described in one of his books. I can dig it up as it better explains how the critical density formula works in which equation 9 works with. (how familiar are you with the FLRW metric? GR? statistical mechanics and shell theorem?) I need a guideline to what level of treatment to provide. Edit I think I may have posted the Liddle proof of the above previously will check

Exactly so there are numerous QFT treatments to handle Levi-civita that will be vital to your modelling. Ie Newton approximation, Schwartzchild metric etc... As regards to the article I found it highly infomative.

EDIT removed

This idea has been proposed before but will take time to dig up the papers. Particularly since it never gained much credence to begin with. Give me time to see if I can up the papers

I'd prefer to review the current studies before replying on this topic. I have some resources available via colleague's etc. However I will state we never place too much faith in any cosmic distance measure methodology. Every method of the cosmic distance ladder has its range of applicable use and accuracy. This includes redshift, parallax, Stellar parallax, mass/luminosity, Tully Fisher. This also includes standard candles, there have been well published boundaries as to some of the inherent properties as long as I can remember. So a review of current findings I will need to familarize with.

in circles couldn't resist

Well depending on how you follow charge flow lightning can be viewed as starting from the ground towards the sky lol. (I lost track of how many high school exams this came up) this comment reminds me of a particular book written in the Star trek series where Enterprise encounters a race that claimed the Universe moves and not the ship. When Kirk argues against it, Spock states "Your forgetting one thing Captain, it obviously works for them" They repeat this theme in one of their movies, with Scotty stating "it never occured to me to think of space moving" when discussing how to use a transporter to deliver Kirk back to Enterprise under warp.

yes but we know that the perspective of a photon frame of reference is an invalid reference frame and no such perspective is possible. I know this is what your referring to but wanted to make this clearer to other readers. Massless particles follow null geodesics with ds^2=0 line elements (seperation distance) this is the detail that literally leads to the "null" portion under definition. As stated I know your already familiar with this but the above could mislead discussion into "what is the viewpoint of a photon" discussion. So I wanted to prevent that sidelight from occuring

gotta love auto edits lmao

Why didn't you include the article your quoting from? Giving a page number without the reference is useless to us

Accurate, as mergers disrupt galaxy formation so any galaxies that avoid mergers do indeed have more time to develop dominant Spiral arms and can indeed form with stability earlier than galaxies that have undergone mergers. The Milky way itself is an older galaxy. Also keep in mind that the higher density past causes a larger number of supernovas, quasars and BH events as Stars etc burn out far faster than the younger stars today. Such events contributes to nebulae collapse and aids in the development of galaxy rotation. This could very well be the case in regards to the galaxy of the OP discussion.

To add to the above the biggest challenge supporting DM is in fact the early large scale structure formation. No other model including MOND could account for it. This is one of the primary reasons why two of the main competing models fail. MOND and Spin and torsion via Poplowskii. Currently we have some maps of DM distribution as well as the Millenium simulation. The latter though it is a simulation strongly supports DM in so far as to its influence on nucleosynthesis and the metalicity we see in direct observation. We are even developing specialized equipment specifically designed to measure DM distributions. Granted mostly through the indirect evidence ie x rays and mass luminosity relations. Originally Zwicky used the mass/luminosity relations to discover the need for DM to account for the galaxy rotation curves and why there is no Kepler decline previously predicted. Only one mass distribution can possibly prevent Kepler decline. That being a uniform mass of a isothermal halo. Baryonic matter distributes on a disk/bulge distribution. Which will automatically cause a Kepler decline l, the calculations are intense I've personally spent several years studying Spiral galaxy formation using virial and Density wave theorem coupled with Jeans instability. Even went so far as to program N-body codes (though only 250 particle count) any more and a typical home computer will bog down and take forever to complete a single calculation loop. edit last two posts are cross post. the above is to add to Strange reply Essentially correct, but keep in mind we never ignore any body of evidence regardless of how few those observations are. Spiral arms form specifically due to Density waves of matter under rotation. It is a direct application of Newtons f=ma though many miss that. A central potential force under rotation will naturally cause different rates of velocity of particles with the heavier particles moving slower than the lighter particles. This gradually creates traffic jams as the lighter particles overlap the heavier particles left behind. An easy at home method to clearly demonstrate this is fill a sink full of water, then toss in some semi bouyant particulates of varying mass and unplug your sink. You see spiral arms forming as the water drains in the whirlpool.... Same phenomena, Density wave theory is also the primary theory for Saturns rings... It is actually surprising how fast this process occurs as it only takes a miniscule number of complete rotations to start forming Spiral arms.

How often have I stressed using proper physics definitions when discussing physics. Excellent answer Strange

I read your white paper, the goal is applaudible I wish you well in it.

QM is a good topic on how common sense can often lead one to the wrong conclusions. Phi for All gave an excellent answer

For the w=-1 equation of state this scholarpedia article has sufficient coverage to get you started http://www.scholarpedia.org/article/Cosmological_constant It is the same you will find in most intro textbooks of cosmology sorry it took so long busy RL schedule

As a stable mean lifetime massless particle the photon has an effective infinite range. See http://hyperphysics.phy-astr.gsu.edu/hbase/Forces/exchg.html

See equation 1.7 for the above https://www.google.ca/url?sa=t&source=web&rct=j&url=http://www.cpt.univ-mrs.fr/~rovelli/IntroductionLQG.pdf&ved=0ahUKEwjYz5nkm6_XAhVO-GMKHcgKCnMQFggdMAA&usg=AOvVaw1gqghLG0TDxskcgiMdoID9

Excellent your getting the correct tensor products involved now. I don't see any errors in the above well done. I've always enjoyed Caroll's articles regardless of topic. His papers are always incredibly well written. I'm also glad to see your applying the boundary cutoffs. I would still like to see you involve the Levi-Cevita with the i,j,k summation. This will give you the complete orthogonal group SO(1.3) and giving the full group with the curvature. (Under relativity the Levi-Cevita affine connection) is the one that corresponds to [latex] G_{\mu\nu}[/latex] where as the Kronecker affine connection i,j is the Minkowskii Euclid. [latex]\eta_{\mu\nu}[/latex] Mainly as this will complete your understanding to arrive at [latex]SO(1.3)=SU(2)\otimes SU(2)/\mathbb{Z}[/latex] where the Z denotes parity, helicity and chirality following the right hand rule under lie groups. Note this is also applicable to the LQG treatments In essence the antisymmetric [latex][L^i,L^j]=i\hbar\epsilon^{i,j}_kL^k[/latex] this is where you should arrive if your doing everything correctly lol how the latex displays the superscript and subscript of the last looks a bit off, the k subscript should be after the i,j superscript but that is trivial. Wonder why it changes the superscript subscript sequence? Anyways the total antisymnetric Levi -Civita is [latex]\epsilon^{i,j}_k[/latex] for some reason I can't keep subsript k at the end of that statement following after the superscript i,j The Kronecker connection being [latex]\epsilon^{i,j}[/latex] though more commonly its written as [latex]\delta_{i,j}[/latex] you will see variations on which symbols are used but thats trivial I used the presentation as per Rovelli under covariant LQC

In a sense Strange has a valid point. Early structure formation arises from regions where particles effectively drop out of thermal equilibrium. Dark matter is strongly supported as being one of the earliest matter particles to do so. However as we know so little about DM we can only estimate when this will occur. Unfortunately we cannot lab test the properties of DM to even estimate its total mass to apply a calculation as to when it will drop out of thermal equilibrium. IF DM drops out of equilibrium early enough then structure formation can occur far earlier, the rates can then be estimated via Jeans Instability and Density wave theory as to the time for Spiral galaxy formation, however there is two major problems. Lack of mass value for DM and the Dark ages beyond the surface of last scattering, using light we can only see so deep into the CMB opacity regions where baryonic matter starts to decouple and form atoms. Evidence supports DM as the original anistropy seeds and not baryonic matter. (baryonic matter decouples far too late)

No WMAP, COBE and Planck all agree with the cosmological principle despite the pop media coverage of those studies. This includes the great attractor etc. The consensus is that 100 Mpc may not be a sufficient volume where the principle becomes applicable below this scale obviously you have LSS anistropy but just like looking at an ocean with waves the larger the volume, the less the waves on the surface matters in terms of applying a uniformity. Some papers suggest upgrading the scale to 120 or as high as 150 Mpc. This is the trick many fail to see, usually from efforts to self teach. Barbera Ryden "Introductory to Cosmology" has an excellent descriptive of the size to uniformity relations. Cosmologists obviously understand the gist of the principle as being one of scale. Think of the ocean analogy the further you are away from the surface and the larger the view, the more uniform the ocean appears. The closer you are the smaller the view the more chaotic the surface appears. The Principle is the same. Secondly 1/1000th of a degree difference in the temperature variation studies of the CMB certainly supports the uniformity. The planck anistropy of evil was a dipole anistropy calibration error. (Matt Roose Introductory to cosmology) has an excellent coverage of dipole anistropy including the basic calibration formula to account for Earths motion effects on redshift. Which is written long before Planck even published its data. Never trust pop media or heuristic explanations, they will mislead you every time...