Redshift FAQ article development

[Mordred]

For an FAQ you need to be really straightforward and uncontroversial. It's difficult. I hope others have time to help you sort it out.

Have you tried calculating the redshift (to first order) within a rapidly spinning system such as a space station (a) as a Special Relativity velocity effect and (b) as due to the effective "gravitational potential" experienced within the spinning system because of the centripetal acceleration? I've always felt that's particularly educational.

That would be educational. No I haven't tried that as of yet. Sounds like a fun challenge lol.

I agree with a FAQ being uncontroversial and straightforward, that is currently my goal in tis article and yes its challenging. I fully expected this project to take a while for those reasons.

I'm still hunting some of the problems Dennis pointed out, I know I'm missing some of the suggested corrections

 

[Mordred]

I changed the Observable unit section. Still working on that section. However so far it reads far better. Can everyone agree on the definition I used for Cosmology based definition as everything measurable in our space-time either directly or indirectly.

 

[Mordred]

EXPANSION AND REDSHIFT
1) Why are all the galaxies accelerating from us?
2) Is Redshift the same as Doppler shift?
3) What is causing the expansion of the universe?
4) Is expansion, faster than light in parts of the Universe, and How does this not violate the faster than light speed limit?
5) What is a Cepheid or standard candle?
6) What do we mean when we say homogeneous and isotropic?
7) How do we measure the distance to galaxies?
8) What is outside the universe?
9) What do we mean when an object leaves our universe?
10) Why is the CMB so vital in cosmology?

These are some of the common questions I will attempt to address in the following article

First we must define some terms and symbols used.

Planck constant: $h\ =\ 6.62606876(52)\ \times\ 10^{-34}\ J\ s$
Gravitational constant: $G\ =\ 6.673(10)\ \times\ 10^{-11}\ m^{3} kg^{-1} s^{-2}$
Speed of light in a vacuum:$c\ =\ 2.99792458\ \times\ 10^{8}\ m\ s^{-1}$

The parsec (symbol: pc) is a unit of length used in astronomy, equal to about 30.9 trillion kilometers (19.2 trillion miles). In astronomical terms, it is equal to 3.26 light-years, and in scientific terms it is equal to 3.09×10¹³ kilometers
Mpc=1 million Parsecs

Universe: A generalized definition of the universe can be described as everything that is. In Cosmology the universe can be described as everything measurable in our space-time either directly or indirectly. This definition forms the basis of the observable universe.

The Observable universe is 46 Billion light years, or 4.3×10²⁶ meters with an age as of 2013, is 13.772 ± 0.059 billion years.
In the hot big bang model we do not think of the universe as starting from a singularity (infinitely, hot, dense point) instead measurements agree space-time as simply expanding. That expansion is homogeneous and isotropic. If you were to take a telescope and look at the night sky, no matter where you look the universe looks the same or homogeneous meaning no preferred location. As you change directions with the telescope you will find that no matter which direction you look the universe looks the same or isotropic meaning no preferred direction. These terms in cosmology are only accurate at certain scales. Below 100Mpc it is obvious that the universe is inhomogeneous and anisotropic. As such objects as stars and galaxies reside in this scale. This also tells us that there is no center of the universe, as a center is a preferred location. These terms also describe expansion. Expansion will be covered in more detail in the Cosmological Redshift section.
Whether or not the universe is finite or infinite is not known. However if it is infinite now so it must be in the beginning.
Common misconceptions arise when one tries to visualize a finite universe such questions include.

"So how do we see farther than 13.772 billion light years?" The answer lies in expansion; as light is traveling towards us, space-time has expanded.
“If the universe is finite what exists outside the Universe?" If you think about this question with the above definition of the universe you will realize that the question is meaningless. One accurate answer in regards to cosmology is nonexistent.
"What makes up the barrier between our universe and outside our universe?" The short answer is there is no barrier.


The CMB, (Cosmic Microwave Background) The CMB is thermal radiation filling the Observable universe almost uniformly, This provides strong evidence of the homogeneous and isotropic measurements and distances. As the universe expanded, both the plasma and the radiation filling it grew cooler. When the universe cooled enough, protons and electrons combined to form neutral atoms. These atoms could no longer absorb the thermal radiation, and so the universe became transparent instead of being an opaque fog. Precise measurements of cosmic background radiation are critical to cosmology, since any proposed model of the universe must explain this radiation. CMB photons were emitted at about 3000 Kelvin and are now 2.73 Kelvin blackbody radiation. Their currently observed energy is 1/1000th of their energy as emitted.

In order to measure an objects motion and distance in cosmology it is important to properly understand redshift, Doppler shift and gravitational redshift. Incorrect usage of any of these can lead to errors in our measurements.

Doppler shift and redshift are the same phenomenon in general relativity. However you will often see Doppler factored into components with different names used, as will be explained below. In all cases of Doppler, the light emitted by one body and received by the other will be red or blueshifted i.e. its wavelength will be stretched. So the color of the light is more towards the red or blue end of the spectrum. As shown by the formula below.

$$\frac{\Delta_f}{f} = \frac{\lambda}{\lambda_o} = \frac{v}{c}=\frac{E_o}{E}=\frac{hc}{\lambda_o} \frac{\lambda}{hc}$$


The Doppler Redshift results from the relative motion of the light emitting object and the observer. If the source of light is moving away from you then the wavelength of the light is stretched out, i.e., the light is shifted towards the red. When the wavelength is compressed from an object moving towards you then it moves towards the blue end of the spectrum. These effects, individually called the blueshift and the redshift are together known as Doppler shifts. The shift in the wavelength is given by a simple formula

(Observed wavelength - Rest wavelength)/(Rest wavelength) = (v/c)

$$f=\frac{c+v_r}{c+v_s}f_o$$



c=velocity of waves in a medium
$$v_r$$ is the velocity measured by the source using the source’s own proper-time clock(positive if moving toward the source
$$v_s$$ is the velocity measured by the receiver using the source’s own proper-time clock(positive if moving away from the receiver)

The above are for velocities where the source is directly away or towards the observer and for low velocities less than relativistic velocities. A relativistic Doppler formula is required when velocity is comparable to the speed of light. There are different variations of the above formula for transverse Doppler shift or other angles.
Doppler shift is used to describe redshift due to inertial velocity one example is a car moving away from you the light will be redshifted, as it approaches you the light and sound will be blueshifted. In general relativity and cosmology, there is a fundamental complication in this simple picture - relative velocity cannot be defined uniquely over large distances. However, it does become unique when compared along the path of light. With relative velocity compared along the path of the light, the special relativity Doppler formula describes redshift for all situations in general relativity and cosmology. It is important to realize that gravity and expansion of the universe affect light paths, and how emitter velocity information is carried along a light path; thus gravity and expansion contribute to Doppler redshift.


The Cosmological Redshift is a redshift attributed to the expansion of space. The expansion causes a Recession Velocity for galaxies (on average) that is proportional to DISTANCE. This recession velocity then produces a Doppler (red) shift proportional to distance (please note that this recession velocity must be converted to a relative velocity along the light path before it can be used in the Doppler formula). The further away an object is the greater the amount of redshift. This is given in accordance with Hubble’s Law. In order to quantify the velocity of this galactic movement, Hubble proposed Hubble's Law of Cosmic Expansion, aka Hubble's law, an equation that states:

Hubble’s Law: The greater the distance of measurement the greater the recessive velocity

Velocity = H₀ × distance.


Velocity represents the galaxy's recessive velocity; H₀ is the Hubble constant, or parameter that indicates the rate at which the universe is expanding; and distance is the galaxy's distance from the one with which it's being compared.

The Hubble Constant The Hubble “constant” is a constant only in space, not in time,the subscript ‘0’ indicates the value of the Hubble constant today and the Hubble parameter is thought to be decreasing with time. The current accepted value is 70 kilometers/second per mega parsec, or Mpc. The latter being a unit of distance in intergalactic space described above.
Any measurement of redshift above the Hubble distance defined as H₀ = 4300±400 Mpc will have a recessive velocity of greater than the speed of light. This does not violate GR because a recession velocity is not a relative velocity or an inertial velocity. It is precisely analogous to a separation speed. If, in one frame of reference, one object is moving east at .9c, and another west at .9c, they are separating by 1.8c. This is their recession velocity. Their relative velocity remains less than c. In cosmology, two things change from this simple picture: expansion can cause separation speeds much greater even than 2c; and relative velocity is not unique, but no matter what path it is compared along, it is always less than c, as expected.

z = (Observed wavelength - Rest wavelength)/(Rest wavelength) or more accurately

1+z= λ_observed/λ_emitted or z=(λ_observed-λ_emitted)/λ_emitted



$$1+Z=\frac{\lambda}{\lambda_o}$$ or $$1+Z=\frac{\lambda-\lambda_o}{\lambda_o}$$

λ₀= rest wavelength
Note that positive values of z correspond to increased wavelengths (redshifts).
Strictly speaking, when z < 0, this quantity is called a blueshift, rather than
a redshift. However, the vast majority of galaxies have z > 0. One notable blueshift example is the Andromeda Galaxy, which is gravitationally bound and approaching the Milky Way.

WMAP nine-year results give the redshift of photon decoupling as z=1091.64 ± 0.47 So if the matter that originally emitted the oldest CMBR photons has a present distance of 46 billion light years, then at the time of decoupling when the photons were originally emitted, the distance would have been only about 42 million light-years away.


Cosmological Redshift is distance dependant as mentioned above, if you were to teleport to the other side of the galaxy where you measured that greater than light recessive velocity, you would find the same expansion rate as your original location relative to an equal distance. Indeed expansion is the same throughout the cosmos. However gravity in galaxy clusters is strong enough to prevent expansion. In other words galaxy clusters are gravitationally bound. In regards to expansion it is important to realize that galaxies are not moving from us due to inertia, rather the space between two coordinates are expanding. This is important in that no FORCE is acting upon the galaxies to cause expansion. As expansion is homogeneous and isotropic then there is no difference in expansion at one location or another. In the LambdaCDM model expansion is attributed to the cosmological constant.

Cosmological Constant is a homogeneous energy density that causes the expansion of the universe to accelerate. Originally proposed early in the development of general relativity in order to allow a static universe solution it was subsequently abandoned when the universe was found to be expanding. Now the cosmological constant is invoked to explain the observed acceleration of the expansion of the universe. The cosmological constant is the simplest realization of dark energy, which the more generic name is given to the unknown cause of the acceleration of the universe. Indeed what we term as "Dark" energy is an unknown energy that comprises most of the energy density of our cosmos around 73%. However the amount of dark energy per m³ is quite small. Some estimates are around about 6 × 10^-10 joules per cubic meter. However their is a lot of space between large scale clusters, so that small amount per m³ adds up to a significant amount of energy in total

Another term often used for the cosmological constant is vacuum energy described originally by the false vacuum inflationary Model by A.Guth. The cosmological constant uses the symbol Λ, the Greek letter Lambda.
This False vacuum inflationary model is one that describes a total energy balance of zero, where gravity is the negative energy. In this model what we term as "Nothing” is really a quantum vacuum with quantum fluctuations described by the Heisenberg uncertainty principle. Virtual particles pop in and out of existence all the time, as a result of expansion those virtual particles quantum tunnel between the false vacuum and the true vacuum, becoming real particles. The full explanation is a little more involved than this quick explanation however this model is often referred to as a "Universe from Nothing" or the "Ultimate free lunch" . Many of our current inflationary models have their roots in this model. However one fundamental problem with all inflationary models is "Runaway expansion" Once the process starts no one has found a mechanism to stop expansion.
One means of relating to expansion is with the use of a grid of squares. Each horizontal and vertical crossing on that grid is a coordinate. In expansion the space between all coordinates not gravitationally bound expand equally. In other words the coordinates do not change, the space between coordinates change. I should also note there is no clear consensus on whether the universe is finite or infinite. If it’s infinite now then it was infinite in the past. Same thing applies with finite. The Big Bang model only describes the Universe from 10 ^-43 seconds and is not considered as starting from a black hole singularity, rather its properly described as a rapid expansion of space-time.

WMAP data confirms that the universe is flat or close to flat.


Gravitational Redshift describes Doppler between static emitter and receiver in a gravitational field. Static observers in a gravitational field are accelerating, not inertial, in general relativity. As a result (even though they are static) they have a relative velocity in the sense described under Doppler. Because they are static, so is this relative velocity along a light path. In fact, the relative velocity for Doppler turns out to depend only on the difference in gravitational potential between their positions. Typically, we dispense with discussion of the relative velocity along a light path for static observers, and directly describe the resulting redshift as a function of potential difference. When the potential increases from emitter to receiver, you have redshift; when it decreases you have blue shift. The formula below is the gravitational redshift formula or Einstein shift in an uncharged, non rotating, spherical mass.
$$\frac{\lambda}{\lambda_o}=\frac{1}{\sqrt{(1 - \frac{2GM}{r c^2})}}$$


G=gravitational constant
c=speed of light
M=mass of gravitational body
r= distance from gravitational body of Mass M

Cosmic Distance ladder, also known as Extragalactic distance scale. Is easily thought of as a series of different measurement methods for specific distance scales. Rather than cover a large range of those distance scales or rungs on the ladder I will cover a few of the essential steps to cosmological distance scales. The first rung on the ladder is naturally.

Direct measurements: Direct measurements form the fundamental distance scale. Units such as the distance from Earth to the sun that are used to develop a fundamental unit called astronomical unit or AU. During the orbit around the sun we can take a variety of measurements such as Doppler shifts to use as a calibration for the AU unit. This Unit is also derived by a method called Parallax.

Parallax. Parallax is essentially trigonometric measurements of a nearby object in space. When our orbit forms a right angle triangle to us and the object to be measured
With the standardized AU unit we can take two AU to form the short leg. With the Sun at a right angle to us the distance to the object to be measured is the long leg of the triangle.

Moving Cluster Parallax is a technique where the motions of individual stars in a nearby star cluster can be used to find the distance to the cluster.

Stellar parallax is the effect of parallax on distant stars . It is parallax on an interstellar scale, and allows us to set a standard for the parsec.

Standard candles A common misconception of standard candles is that only type 1A supernova are used. Indeed any known fundamental distance measurement or stellar object whose luminosity or brightness is known can be used as a standard candle. By comparing an objects luminosity to the observed brightness we can calculate the distance to an object using the inverse square law. Standard candles include any object of known luminosity, such as Cepheid’s, novae, Type 1A supernova and galaxy clusters.

My thanks to the following Contributors, for their feedback and support.

PAllen
Naty1
Jonathon Scott
marcus

Article by Mordred, PAllen

 

[Mordred]

I made a few changes in the beginning section to better cover common questions also to define honmogeneous an isotropic earlier on in the article. Hopefully this gives the article a better flow .

 

[Naty1]

A few minor comments...You may want to note:

"So how do we see farther than 13.772 billion years?"

should be 13.7 billion LIGHT years.

H₀ is the Hubble constant currently.

Something is wrong here:

One accurate answer in regards to cosmology is nonexistent.

That expansion is homogeneous and isotropic. In other words, there is no preferred location (Homogeneous) and no preferred direction (Isotropic). Keep in mind these terms describe the universe on large scales.

This seems a bit convoluted: 'homogeneous and isotropic' are assumptions of space...as you correctly described much earlier. Once such assumptions are made so that we can simplify cosmological model calculations, the uniform expansion of space follows. I am not sure if anyone knows that the actual expansion of space is actually uniform...maybe somebody will comment...

I'd suggest simply saying the uniform expansion of space follows from the assumptions that space is uniform...homogeneous and isotropic.

 

[Mordred]

A few minor comments...You may want to note:



should be 13.7 billion LIGHT years.

H₀ is the Hubble constant currently. .

.


the constant H₀ isn't going to change, The value I gave I added the date of the value.

Velocity represents the galaxy's recessive velocity; H0 is the Hubble constant, or parameter that indicates the rate at which the universe is expanding; and distance is the galaxy's distance from the one with which it's being compared.

The Hubble Constant has been calculated at different values over time, this is essential as the rate of expansion varies over time but the current accepted value is 70 kilometers/second per megaparsec, or Mpc.

Something is wrong here:

One accurate answer in regards to cosmology is nonexistent.

.

.


this statement I agree is questionable. I didn't want to use the word nothing, for obvious reasons. So the only way I could think of was nonexistent.

I'm open to suggestions on better ways to express this.

This seems a bit convoluted: 'homogeneous and isotropic' are assumptions of space...as you correctly described much earlier. Once such assumptions are made so that we can simplify cosmological model calculations, the uniform expansion of space follows. I am not sure if anyone knows that the actual expansion of space is actually uniform...maybe somebody will comment...

I'd suggest simply saying the uniform expansion of space follows from the assumptions that space is uniform...homogeneous and isotropic.

.

Its considered homogeneous and isotropic on the right scales, Out of the 5 introductory to cosmology textbooks I have the value of 100 Mpc or above is often stated. However there has been some dispute on that aspect, Does the value need to be increased ?
Both CMB and Planck confirm the homogeneous and isotropic nature of the universe. So I would think its more than just an assumption to make the maths easier. I'm open to suggestions on better descriptives for the actual homogeneous and isotropic nature of expansion in non gravitationally bound regions.

 

[Mordred]

2.2 On large scales, the universe is isotropic
and homogeneous
What does it mean to state that the universe is isotropic and homogeneous?
Saying that the universe is isotropic means that there are no preferred directions
in the universe; it looks the same no matter which way you point your
telescope. Saying that the universe is homogeneous means that there are no
preferred locations in the universe; it looks the same no matter where you set
up your telescope. Note the very important quali¯er: the universe is isotropic
and homogeneous on large scales. In this context, \large scales" means that
the universe is only isotropic and homogeneous on scales of roughly 100Mpc
or more.
The isotropy of the universe is not immediately obvious. In fact, on small
scales, the universe is blatantly anisotropic. Consider, for example, a sphere
3 meters in diameter, centered on your navel (Figure 2.2a). Within this
sphere, there is a preferred direction; it is the direction commonly referred
to as \down". It is easy to determine the vector pointing down. Just let go
of a small dense object. The object doesn't hover in midair, and it doesn't
move in a random direction; it falls down, toward the center of the Earth.
On signi¯cantly larger scales, the universe is still anisotropic. Consider,
for example, a sphere 3 AU in diameter, centered on your navel (Figure 2.2b).
Within this sphere, there is a preferred direction; it is the direction pointing
toward the Sun, which is by far the most massive and most luminous object
within the sphere. It is easy to determine the vector pointing toward the
Sun. Just step outside on a sunny day, and point to that really bright disk
of light up in the sky.
On still large scales, the universe is still anisotropic. Consider, for example,
a sphere 3 Mpc in diameter, centered on your navel (Figure 2.2c).
This sphere contains the Local Group of galaxies, a small cluster of some 40
galaxies. By far the most massive and most luminous galaxies in the Local
Group are our own Galaxy and M31, which together contribute about 86% of
the total luminosity within the 3 Mpc sphere. Thus, within this sphere, our
Galaxy and M31 de¯ne a preferred direction. It is fairly easy to determine
the vector pointing from our Galaxy to M31; just step outside on a clear

here is a quote from "Introductory to Cosmology" by Barbera Ryden. The drawings references are merely circles.
All of my textbooks describe this in a similar manner

 

[Naty1]

regarding the Hubble parameter:

The Hubble “constant” is a constant only in space, not in time,

the subscript ‘0’ indicates the value of the Hubble constant today and

the Hubble parameter is thought to be decreasing with time.

I have trouble extracting any of those concepts from the current description you posted.

I'm open to suggestions on better descriptives for the actual homogeneous and isotropic nature of expansion in non gravitationally bound regions.

It's space that is described this way, not expansion. Such a description of space leads to the uniform expansion...or uniform distance increases, if you prefer...

see what others may comment...if anything...

 

[PAllen]

It's space that is described this way, not expansion. Such a description of space leads to the uniform expansion...or uniform distance increases, if you prefer...

see what others may comment...if anything...

In principle, you could have space that is (at any cosmological time slice) neither homogenous nor isotropic, yet experiences homogenous and isotropic expansion (preserving the spatial asymmetries). Alternatively, you could homogenous, isotropic space (at some cosmological time), the experiences anisotropic and / or inhomogeneous expansion; space would then loose its symmetries due to the asymmetric expansion.

The only real connection I see is that if space retains these symmetries, then the expansion must have them. Every other implication (if ... then) statement I can think of relating spatial and expansion symmetries is false (except of course, the contrapositive of the above).

 

[Mordred]

regarding the Hubble parameter:

The Hubble “constant” is a constant only in space, not in time,

the subscript ‘0’ indicates the value of the Hubble constant today and

the Hubble parameter is thought to be decreasing with time.

I have trouble extracting any of those concepts from the current description you posted.
/QUOTE]

that descriptive is better than the one I used.

It's space that is described this way, not expansion. Such a description of space leads to the uniform expansion...or uniform distance increases, if you prefer...

see what others may comment...if anything...

This descriptive sounds good, however is it accurate ? consider the cause of expansion, as vacuum energy as described in the inflationary model. There is no evidence of variations in the vacuum energy. The vacuum energy is also homogenous and isotropic. As this energy is constant, homogenous and isotropic and the rate of expansion requires the cosmological constant. That would mean expansion is also homogenous and isotropic, the difference in expansion rates is a sum of energy densities. However the vacuum energy density in all areas in space remain the same. So in these terms, outside of gravitationally bound regions describing the 100Mpc takes that into consideration.
I'm willing to go either way on the two decriptives but it seems to me that expansion can be accurately described as homogenous and isotropic on the right scales.


Take for example the De-Sitter universe. this is a universe where matter is removed.

the rate of expansion is defined as h$\propto\sqrtλ$

this shows that its homogenous and isotropic.

I will use your Hubble constant descriptive its more accurate than what I have.

 

[Mordred]

I added your suggested change to he Hubble constant thanks for pointing that out Naty1

 

[Naty1]

Pallen:

In principle, you could have space that is (at any cosmological time slice) neither homogenous nor isotropic, yet experiences homogenous and isotropic expansion (preserving the spatial asymmetries).

now THAT would be an interesting cosmological model. [LOL]

If there is a mainstream model that does not assume symmetrical space and mass/energy, I have not yet come across it.

The only real connection I see is that if space retains these symmetries, then the expansion must have them.

That's one nice way to express what I was trying to describe. I do not see how you can derive the FLRW cosmological model nor any other that is practical without such an assumption up front. If you do not make such an assumption it seems you are stuck with numerical approximations as solutions...like trying to apply the FLRW model to a lumpy galaxy and finding out expansion may not even be forecast..

If you watch Leonard Susskind derive even a Newtonian model of space [Youtube Cosmology Lecture #2, and it is quite cool and simple by the way] you'll note the FIRST thing he does after drawing a variable diameter sphere of matter/energy is to ASSUME homogeneous and isotropic mass[energy] density for that sphere...and all others of any size.

 

[Mordred]

I can't think of any models where one stays homogenous and isotropic while the other doesn't either..The part about the universe retaining assymetries I have read about once but can't recall where.

 

[PAllen]

Pallen:


now THAT would be an interesting cosmological model. [LOL]

If there is a mainstream model that does not assume symmetrical space and mass/energy, I have not yet come across it.

I said "in principle" not "in a mainstream model". The point was to clarify the logical coupling of concepts. Note that both of the following are both correct ways of looking at it:

- If any spatial slice is homogenous and isotropic, and expansion is homogenous and isotropic, then all spatial slices are homogenous and isotropic.

- If all spatial slices are homogenous and isotropic, then expansion must be homogenous and isotropic.

 

[Naty1]

The part about the universe retaining assymetries I have read about once but can't recall where.

Funny you should mention that:

I just posted a link to a NY Times [newspaper] article earlier today on
new Planck Satellite data...

Universe as an Infant: Fatter Than Expected and Kind of Lumpy
https://www.physicsforums.com/showthread.php?t=680161

That title could be a description of my wife! [No,no, I did NOT say that!]

 

[Mordred]

lol even more funny is I was reading it while you posted this lol

 

[Mordred]

I've been considering if I should add critical density and space-time geometry to the article. In one camp the FAQ is already large. In the other it would encourage more ppl to use it as a reference.

 

[PAllen]

I've been considering if I should add critical density and space-time geometry to the article. In one camp the FAQ is already large. In the other it would encourage more ppl to use it as a reference.

I lean towards leaving it out. Another FAQ could be created on critical density, open/closed geometry, flat or not, etc.

Supporting this further, is that before finalizing this FAQ, another step is collecting references. Almost all FAQ in PF provide references to more detailed treatment of the issues. As this one is big, I would see a minimum of six, possibly 10 references being required.

 

[Mordred]

I lean towards leaving it out. Another FAQ could be created on critical density, open/closed geometry, flat or not, etc.

Supporting this further, is that before finalizing this FAQ, another step is collecting references. Almost all FAQ in PF provide references to more detailed treatment of the issues. As this one is big, I would see a minimum of six, possibly 10 references being required.

Yeah I agree on leaving that out, the references in regards to the FAQ on PF is something I wasn't aware of. Should be easy enough to find some decent articles for reference.
I'll look around for some decent articles, and try to find sites that will stay so that the links don't become broken or unusable.

 

[PAllen]

Yeah I agree on leaving that out, the references in regards to the FAQ on PF is something I wasn't aware of. Should be easy enough to find some decent articles for reference.
I'll look around for some decent articles, and try to find sites that will stay so that the links don't become broken or unusable.

Here is a technical reference sent to me by PM, from back when we were debating whether Doppler was distinct from other redshift's in GR.

http://arxiv.org/abs/1111.6704

It is modern, evenhanded, and I think represents modern consensus. In particular, on page 4, it endorses the view that all spectral shifts in GR are correctly regarded as Doppler effect for curved spacetime, but also clarifies (as I agree) that this does not imply they can be regarded as purely kinematic in origin.

[Interesting to me was the demonstration that over large distances in open, non-flat RW models, you cannot split the Doppler into kinematic vs non-kinematic components even for distant co-moving observers (this paper only considers co-moving observers, which they call fundamental observers).]

 

[Mordred]

Yeah I have a copy of this article in my database. I also found it interesting. For the expansion portion Ned Wrights tutorial is probably one of the better references. He also has a basic redshift article in his FAQ section.
I'm searching my database for another arxiv article on redshift. There was one done at a more entry level.

 

[Mordred]

Here is a good visualization from NASA done as a youtube vid
http://m.youtube.com/#/watch?v=sc0_f3e_qwE&desktop_uri=/watch?v=sc0_f3e_qwE

This site has a decent power point slide for visualizing cosmic distance ladder.
http://terrytao.wordpress.com/2010/10/10/the-cosmic-distance-ladder-ver-4-1/
actually this one is more entry level.
http://calgary.rasc.ca/downloads/distance_ladder.pdf

 

[Mordred]

In self critique.

I need to fix the sequence this article reads. I'm thinking of moving Hubble prior to all shift descriptives.

The Cosmological section also needs fixing. The descriptive of false vacuum is out of place there.


Of key note a short descriptive of spectrography for the Cosmic distance ladder section.

Anyways going to take me a bit to work on that.

 

[Mordred]

As PAllen mentioned we need good FAQ articles to support this article. So suggest away I will pore through them lol.

 

[Naty1]

Here is a good visualization from NASA done as a youtube vid
http://m.youtube.com/#/watch?v=sc0_f...=sc0_f3e_qwE

Funny, NASA shows here a flying saucer picking up a cow in a spotlight ray.

Do they know something we do not?

 

[Naty1]

For references,
you have Ned Wright listed already,
how about Lineweaver and Davis?
I'll get the links...

 

[Naty1]

PAllen's link in his post #90 got me looking back in my notes from earlier discussions:
Here are comments for your consideration which I think are consistent with the abstract:

[I'm unsure if all three are from Chalnoth.]

Chalnoth: it is perfectly-valid to talk about the redshift either as coming from the motions of galaxies, or as coming from the stretching of space. ..

The observed redshift will be equal to the total amount of ‘expansion’ between the emission and absorption of the photon, regardless of what the rate of that expansion was at different times.

There is no simple way to differentiate one kind of redshift from another. Gravitational and Doppler redshifts are spectroscopically identical to cosmological redshift. They cannot be told apart without knowing something about the emitting source. If you happen to know the source is either a black hole or neutron star, you can approximate the gravitational redshift contribution. If it is a galaxy you can approximate its kinematical redshift by examining redshift of opposing arms [assuming your view is not parallel to its rotational axis].


PS: 'Hubbles constant' should always be 'Hubble's constant'

 

[Naty1]

For Lineweaver and Davis:

the professional paper is here:
http://arxiv.org/abs/astro-ph/0310808

There is also an abbreviated Scientific American article by them...I keep losing a link...

///////////////

Expanding Space: the Root of all Evil?
http://arxiv.org/PS_cache/arxiv/pdf/0707/0707.0380v1.pdf
From "Conclusions":

Despite (and perhaps in part because of) its ubiquity,
the concept of expanding space has often been articulated
poorly and formulated in contradictory ways.
That addressing this issue is important must be placed
beyond doubt, as the phrase ‘expansion of space’ is
in such a wide use—from technical papers, through to
textbooks and material intended for school students or
the general public—that it is no exaggeration to label
it the most prominent feature of Big Bang cosmologies.
In this paper, we have shown how a consistent description
of cosmological dynamics emerges from the idea
that the expansion of space is neither more nor less
than the increase over time of the distance between
observers at rest with respect to the cosmic fluid.

Also: FYI: for your text??
The Hubble radius where v = c is about 16bly away…at distances beyond, space is now expanding at greater than c relative to our frame of reference. [Z here is??]

Expansion has slowed enormously since year 380,000 when the CMB light got loose and started on its way.

 

[Naty1]

I've changed my thinking about this document a bit...

I think the only way to shorten this proposed document would be to change the organization...I'll leave it to Mordred to decide if the extra effort is worth it.

My own difficulty with my own suggestion here is that it may not end up being any shorter, but maybe will be more focused...worse, I even add one question, below!

First, place the questions in a different order to get definitions, that is, basics, upfront:
say, 5,6,10 up front...then maybe my FLRW model idea...

order the rest of the questions so answers will build in a logical order.

Maybe even add one question: What is the FLRW [Lambda CDM] model and why is it important? [see below]

EXPANSION AND REDSHIFT
1) Why are all the galaxies accelerating from us?
2) Is Redshift the same as Doppler shift?
3) What is causing the expansion of the universe?
4) Is expansion, faster than light in parts of the Universe, and How does this not violate the faster than light speed limit?
5) What is a Cepheid or standard candle?
6) What do we mean when we say homogeneous and isotropic?
7) How do we measure the distance to galaxies?
8) What is outside the universe?
9) What do we mean when anm object leaves our universe?
10) Why is the CMB so vital in cosmology?

Next, answer the questions one by one in order...this can be used to eliminate a lot of superfluous explanations like virtual particles...may help focus replies??

finally, think about what is missing: for example, I would suggest in #10 including that all measurements are based on widely separated observers being at rest with respect to their local CMB. That is a universal standard. It's a cornerstone basis for measuring distance to galaxies,for example.

Also, 'distance measures' are based on the scale factor as a solution to EFE within the FLRW cosmology model. Use a different model, use a different scale factor, use a different distance metric, get different answers.

I have made up a list of conventions regarding the FLRW model we use in addressing the above questions...took me quite a while, I think, when learning about cosmology to realize how arbitrary yet extremely useful] such conventions are:

Since FLRW is the ‘standard [cosmological] model’, listing a few conventions within the model to illustrate its UNIQUE characteristics could be helpful:
being at rest [“Comoving”] with respect to the CMBR is what defines the universal cosmological time parameter utilized;
Superluminal distances are are result of the FLRW model metric , those FLRW distances ARE great circles and space geodesics on the balloon model especially when you think of the balloon surface as a time of radius ‘r’ [approximating a constant, fixed cosmological time;
the FLRW metric starts after the initial inflationary epoch;

The Universe is assumed homogeneous (space has the same metric properties (measures) at all points) and isotropic (space has the same measures in all directions). The present consensus is that the isotropic model, in general, gives an adequate [approximate] description of the present state of the [large scale] Universe.

the LCDM is a ‘fine-tuned version’ of the general FLRW model where the observationally based model parameters are chosen for the best fit to our universe;

the most common distance measure, comoving distance defines the chosen connecting curve to be a curve of constant cosmological time;

operationally, neither comoving distances nor proper [instantaneous] distances can be directly measured by a single earth-bound observer, etc,etc.

Also, if a grid is employed as in the current paper, let's say the scale factor is the distance between adjacent coordinate points...and also let's consider saying Hubble parameter H = a'[t]/a the rate of change of the scale factor with respect to time/ the scale factor...

Lots of complicated/detailed explanations and lotsa additional work...

Anyway, maybe there are some helpful ideas here...

 

[Mordred]

Yeah I am currently working on the sequence of the overall article. You have some good suggestions in the above. I will probably pull out the false vacuum explanation. That will help shorten it down.

 

[Mordred]

If you feel up to it Naty1. It would be easier to see many of your proposals above if you could
quote the article. Copy it to microsoft word. That way we don't lose the math latex forms.
Then write in your changes. The cosmological constant section should only be about the constant. So that section needs changing. We can probably drop false vacuum replace in the right section your FLRW related quotes. On infinity I have the same described twice so that can be deleted in the cosmological constant section.
Hubble section needs to be prior to redshift may involve moving cosmological section up.

Those are some of the changes I see that definitely need addressing along with the reorganization of questions to a good sequence.

If your willing to try your editted version we can compare between your editted version and mine and pick the best one.

 

[Mordred]

EXPANSION AND REDSHIFT
1) What is outside the universe?
2) What is causing the expansion of the universe?
3) Is expansion, faster than light in parts of the Universe, and How does this not violate the faster than light speed limit?
4) What do we mean when an object leaves our universe?
5) What do we mean when we say homogeneous and isotropic?
6) Why is the CMB so vital in cosmology?
7) Why is the LambdaCDM so vital to cosmologists?
8) Why are all the galaxies accelerating from us?
9) Is Redshift the same as Doppler shift?
9) How do we measure the distance to galaxies?
10) What is a Cepheid or standard candle

These are some of the common questions I will attempt to address in the following article
First we must define some terms and symbols used.

Planck constant: $h\ =\ 6.62606876(52)\ \times\ 10^{-34}\ J\ s$
Gravitational constant: $G\ =\ 6.673(10)\ \times\ 10^{-11}\ m^{3} kg^{-1} s^{-2}$
Speed of light in a vacuum:$c\ =\ 2.99792458\ \times\ 10^{8}\ m\ s^{-1}$

The parsec (symbol: pc) is a unit of length used in astronomy, equal to about 30.9 trillion kilometers (19.2 trillion miles). In astronomical terms, it is equal to 3.26 light-years, and in scientific terms it is equal to 3.09×10¹³ kilometers
Mpc=1 million Parsecs

Universe: A generalized definition of the universe can be described as everything that is. In Cosmology the universe can be described as everything measurable in our space-time either directly or indirectly. This definition forms the basis of the observable universe. The Hot Big Bang model does not describe prior to 10^-43 seconds. The LambdaCDM or $\Lambda$CDM model is a fine tuned version of the general FLRW (Freidmann Lemaitre Robertson Walker) metrics, where the six observationally based model parameters are chosen for the best fit to our universe.

The Observable universe is 46 Billion light years, or 4.3×10²⁶ meters with an age as of 2013, is 13.772 ± 0.059 billion years.
In the hot big bang model we do not think of the universe as starting from a singularity (infinitely, hot, dense point) instead measurements agree space-time as simply expanding. That expansion is homogeneous and isotropic. If you were to take a telescope and look at the night sky, no matter where you look the universe looks the same or homogeneous meaning no preferred location. As you change directions with the telescope you will find that no matter which direction you look the universe looks the same or isotropic meaning no preferred direction. These terms in cosmology are only accurate at certain scales. Below 100Mpc it is obvious that the universe is inhomogeneous and anisotropic. As such objects as stars and galaxies reside in this scale. This also tells us that there is no center of the universe, as a center is a preferred location. These terms also describe expansion. Expansion will be covered in more detail in the Cosmological Redshift section. Whether or not the universe is finite or infinite is not known. However if it is infinite now so it must be in the beginning.
Common misconceptions arise when one tries to visualize a finite universe such questions include.

"So how do we see farther than 13.772 billion light years?" The answer lies in expansion; as light is traveling towards us, space-time has expanded.
“If the universe is finite what exists outside the Universe?" If you think about this question with the above definition of the universe you will realize that the question is meaningless. One accurate answer in regards to cosmology is nonexistent.
"What makes up the barrier between our universe and outside our universe?" The short answer is there is no barrier.


The CMB, (Cosmic Microwave Background) The CMB is thermal radiation filling the Observable universe almost uniformly, This provides strong evidence of the homogeneous and isotropic measurements and distances. As the universe expanded, both the plasma and the radiation filling it grew cooler. When the universe cooled enough, protons and electrons combined to form neutral atoms. These atoms could no longer absorb the thermal radiation, and so the universe became transparent instead of being an opaque fog. Precise measurements of cosmic background radiation are critical to cosmology, since any proposed model of the universe must explain this radiation. CMB photons were emitted at about 3000 Kelvin and are now 2.73 Kelvin blackbody radiation. Their currently observed energy is 1/1000th of their energy as emitted.

In order to measure an objects motion and distance in cosmology it is important to properly understand redshift, Doppler shift and gravitational redshift. Incorrect usage of any of these can lead to errors in our measurements.

Doppler shift and redshift are the same phenomenon in general relativity. However you will often see Doppler factored into components with different names used, as will be explained below. In all cases of Doppler, the light emitted by one body and received by the other will be red or blueshifted i.e. its wavelength will be stretched. So the color of the light is more towards the red or blue end of the spectrum. As shown by the formula below.

$$\frac{\Delta_f}{f} = \frac{\lambda}{\lambda_o} = \frac{v}{c}=\frac{E_o}{E}=\frac{hc}{\lambda_o} \frac{\lambda}{hc}$$

The Cosmological Redshift is a redshift attributed to the expansion of space. The expansion causes a Recession Velocity for galaxies (on average) that is proportional to DISTANCE.
A key note is expansion is the same throughout the cosmos. However gravity in galaxy clusters is strong enough to prevent expansion. In other words galaxy clusters are gravitationally bound. In regards to expansion it is important to realize that galaxies are not moving from us due to inertia, rather the space between two coordinates are expanding. One way to visualize this is to use a grid where each vertical and horizontal joint is a coordinate. The space between the coordinates increase rather than the coordinates changing. This is important in that no FORCE is acting upon the galaxies to cause expansion. As expansion is homogeneous and isotropic then there is no difference in expansion at one location or another. In the $\Lambda$CDM model expansion is attributed to the cosmological constant described later on. The rate a galaxy is moving from us is referred to as recession velocity. This recession velocity then produces a Doppler (red) shift proportional to distance (please note that this recession velocity must be converted to a relative velocity along the light path before it can be used in the Doppler formula). The further away an object is the greater the amount of redshift. This is given in accordance with Hubble’s Law. In order to quantify the velocity of this galactic movement, Hubble proposed Hubble's Law of Cosmic Expansion, aka Hubble's law, an equation that states:

Hubble’s Law: The greater the distance of measurement the greater the recessive velocity

Velocity = H₀ × distance.

Velocity represents the galaxy's recessive velocity; H₀ is the Hubble constant, or parameter that indicates the rate at which the universe is expanding; and distance is the galaxy's distance from the one with which it's being compared.

The Hubble Constant The Hubble “constant” is a constant only in space, not in time,the subscript ‘0’ indicates the value of the Hubble constant today and the Hubble parameter is thought to be decreasing with time. The current accepted value is 70 kilometers/second per mega parsec, or Mpc. The latter being a unit of distance in intergalactic space described above.
Any measurement of redshift above the Hubble distance defined as H₀ = 4300±400 Mpc will have a recessive velocity of greater than the speed of light. This does not violate GR because a recession velocity is not a relative velocity or an inertial velocity. It is precisely analogous to a separation speed. If, in one frame of reference, one object is moving east at .9c, and another west at .9c, they are separating by 1.8c. This is their recession velocity. Their relative velocity remains less than c. In cosmology, two things change from this simple picture: expansion can cause separation speeds much greater even than 2c; and relative velocity is not unique, but no matter what path it is compared along, it is always less than c, as expected.

z = (Observed wavelength - Rest wavelength)/(Rest wavelength) or more accurately

1+z= λ_observed/λ_emitted or z=(λ_observed-λ_emitted)/λ_emitted

$$1+Z=\frac{\lambda}{\lambda_o}$$ or $$1+Z=\frac{\lambda-\lambda_o}{\lambda_o}$$

λ₀= rest wavelength
Note that positive values of z correspond to increased wavelengths (redshifts).
Strictly speaking, when z < 0, this quantity is called a blueshift, rather than
a redshift. However, the vast majority of galaxies have z > 0. One notable blueshift example is the Andromeda Galaxy, which is gravitationally bound and approaching the Milky Way.

The rate of expansion is expressed in the $\Lambda$CDM model in terms of
The scale factor, cosmic scale factor or sometimes the Robertson-Walker scale factor parameter of the Friedmann equations represents the relative expansion of the universe. It relates the proper distance which can change over time, or the comoving distance which is the distance at a given reference in time.

d(t)=a(t)d_o

where d(t) is the proper distance at epoch (t)
d₀ is the distance at the reference time (t_o)
a(t) is the comoving angular scale factor. Which is the. ..
r(t) is the comoving radial scale factor. Which is...

Proper distance =R(t)r

the notation R(t) indicates that the scale factor is a function of time and its value changes with time. R(t)<1 is the past, R(t)=1 is the present and R(t)>1 is the future.

z+1=1/1+z

H(t)=change in a(t)/R(t)

Expansion velocity
v=change in a(t)r

This shows that Hubble's constant is time dependant.


WMAP nine-year results give the redshift of photon decoupling as z=1091.64 ± 0.47 So if the matter that originally emitted the oldest CMBR photons has a present distance of 46 billion light years, then at the time of decoupling when the photons were originally emitted, the distance would have been only about 42 million light-years away.

Cosmological Constant is a homogeneous energy density that causes the expansion of the universe to accelerate. Originally proposed early in the development of general relativity in order to allow a static universe solution it was subsequently abandoned when the universe was found to be expanding. Now the cosmological constant is invoked to explain the observed acceleration of the expansion of the universe. The cosmological constant is the simplest realization of dark energy, which the more generic name is given to the unknown cause of the acceleration of the universe. Indeed what we term as "Dark" energy is an unknown energy that comprises most of the energy density of our cosmos around 73%. However the amount of dark energy per m³ is quite small. Some estimates are around about 6 × 10^-10 joules per cubic meter. However their is a lot of space between large scale clusters, so that small amount per m³ adds up to a significant amount of energy in total. In the De_Sitter FLRW metric (matter removed model)
this is described in the form.

H_o$\propto\sqrt\Lambda$

Another term often used for the cosmological constant is vacuum energy described originally by the false vacuum inflationary Model by A.Guth. The cosmological constant uses the symbol Λ, the Greek letter Lambda.
The dark energy density parameter is given in the form:
$\Omega_\Lambda$ which is approximately 0.685

The Doppler Redshift results from the relative motion of the light emitting object and the observer. If the source of light is moving away from you then the wavelength of the light is stretched out, i.e., the light is shifted towards the red. When the wavelength is compressed from an object moving towards you then it moves towards the blue end of the spectrum. These effects, individually called the blueshift and the redshift are together known as Doppler shifts. The shift in the wavelength is given by a simple formula

(Observed wavelength - Rest wavelength)/(Rest wavelength) = (v/c)

$$f=\frac{c+v_r}{c+v_s}f_o$$

c=velocity of waves in a medium
$$v_r$$ is the velocity measured by the source using the source’s own proper-time clock(positive if moving toward the source
$$v_s$$ is the velocity measured by the receiver using the source’s own proper-time clock(positive if moving away from the receiver)

The above are for velocities where the source is directly away or towards the observer and for low velocities less than relativistic velocities. A relativistic Doppler formula is required when velocity is comparable to the speed of light. There are different variations of the above formula for transverse Doppler shift or other angles. Doppler shift is used to describe redshift due to inertial velocity one example is a car moving away from you the light will be redshifted, as it approaches you the light and sound will be blueshifted. In general relativity and cosmology, there is a fundamental complication in this simple picture - relative velocity cannot be defined uniquely over large distances. However, it does become unique when compared along the path of light. With relative velocity compared along the path of the light, the special relativity Doppler formula describes redshift for all situations in general relativity and cosmology. It is important to realize that gravity and expansion of the universe affect light paths, and how emitter velocity information is carried along a light path; thus gravity and expansion contribute to Doppler redshift

Gravitational Redshift describes Doppler between static emitter and receiver in a gravitational field. Static observers in a gravitational field are accelerating, not inertial, in general relativity. As a result (even though they are static) they have a relative velocity in the sense described under Doppler. Because they are static, so is this relative velocity along a light path. In fact, the relative velocity for Doppler turns out to depend only on the difference in gravitational potential between their positions. Typically, we dispense with discussion of the relative velocity along a light path for static observers, and directly describe the resulting redshift as a function of potential difference. When the potential increases from emitter to receiver, you have redshift; when it decreases you have blue shift. The formula below is the gravitational redshift formula or Einstein shift in an uncharged, non rotating, spherical mass.
$$\frac{\lambda}{\lambda_o}=\frac{1}{\sqrt{(1 - \frac{2GM}{r c^2})}}$$

G=gravitational constant
c=speed of light
M=mass of gravitational body
r= distance from gravitational body of Mass M

Cosmic Distance ladder, also known as Extragalactic distance scale. Is easily thought of as a series of different measurement methods for specific distance scales. Previous in the article we discussed the various forms of Redshift. These principles are used in conjunction with the following methods described below. Modern equipment now allows use spectrometry. Spectrographs of an element give off a definite spectrum of light or wavelengths. By examining changes in this spectrum and other electromagnetic frequencies with the various forms of shifts caused by relative motion, gravitational effects and expansion. We can now judge an objects luminosity

Luminosity is often measured in flux where flux is

f=L/4pi r^2
inverse square law needed here...

However cosmologists typically use a scale called magnitudes. The magnitude scale has been developed so that a 5 magnitude change corresponds to a differents of 100 flux.
Rather than cover a large range of those distance scales or rungs on the ladder I will cover a few of the essential steps to cosmological distance scales. The first rung on the ladder is naturally.

Direct measurements: Direct measurements form the fundamental distance scale. Units such as the distance from Earth to the sun that are used to develop a fundamental unit called astronomical unit or AU. During the orbit around the sun we can take a variety of measurements such as Doppler shifts to use as a calibration for the AU unit. This Unit is also derived by a method called Parallax.

Parallax. Parallax is essentially trigonometric measurements of a nearby object in space. When our orbit forms a right angle triangle to us and the object to be measured
With the standardized AU unit we can take two AU to form the short leg. With the Sun at a right angle to us the distance to the object to be measured is the long leg of the triangle.

Moving Cluster Parallax is a technique where the motions of individual stars in a nearby star cluster can be used to find the distance to the cluster.

Stellar parallax is the effect of parallax on distant stars . It is parallax on an interstellar scale, and allows us to set a standard for the parsec.

Standard candles A common misconception of standard candles is that only type 1A supernova are used. Indeed any known fundamental distance measurement or stellar object whose luminosity or brightness is known can be used as a standard candle. By comparing an objects luminosity to the observed brightness we can calculate the distance to an object using the inverse square law. Standard candles include any object of known luminosity, such as Cepheid’s, novae, Type 1A supernova and galaxy clusters.

My thanks to the following Contributors, for their feedback and support.

PAllen
Naty1
Jonathon Scott
marcus

Article by Mordred, PAllen

 

[Mordred]

Thee order in this copy makes more sense and reads smoother overall. I also added a couple of lines into the Cosmic disntance ladder. I removed the false vacuum descriptive. So overall length is a little better. Still looking at fitting some of your Ideas into the article Nay1

 

[Mordred]

Ok I've got this article modified as far as I can currently think of needed changes. I added the FLRW explanations into the end of the Hubble section.

 

[Naty1]

From your paper above:

The scale factor, cosmic scale factor or sometimes the Robertson-Walker scale factor parameter of the Friedmann equations which is a function of time which represents the relative expansion of the universe. It relates the proper distance (which can change over time, unlike the comoving distance which factors out the expansion of the universe, giving a distance that does not change in time due to the expansion of space) between a pair of objects, e.g. two galaxies, moving with the Hubble flow in an expanding or contracting FLRW universe at any arbitrary time to their distance at some reference time . The formula for this is:

d(t)=a(t)do

That appears to be the Wikipedia explanation which I have always had trouble relating to...I did not check to see if you explained proper and comoving distance...that takes some getting used to...

Instead, Consider:
Previously you described expansion terms of grid coordinates...[Is it still included??] expanding distances with the same fixed coordinate identities... I like that,,,,what I like better is the way Leonard Susskind sets up that grid...he just assigns the scale factor a[t] to each edge of the grid..

now THAT is a way anybody [I think] can understand the expansion of the scale factor over time and directly relate it to the expansion of distance in the universe...If you want to see Leonard Susskind develop that idea in a video it's Susskind Lecture 2, Cosmology, Youtube...if you watch much of the video you'll realize he uses a Newtonian model to develop the FLRW formulas...no matter, it's valid for GR too.

http://www.youtube.com/watch?v=ERjkSbdn6-4&list=PLB64419BFD176F2FD&index=41

Derivation of the FRW cosmological model from energy conservation. A Newtonian Cosmological Model…
/////////////

Also,and you may have chosen to omit this, and I already posted it above, but I did not know the Hubble parameter H =a'[t]/a[t] for a long time...I stumbled across it in Wikipedia somewhere.

It gives some insight into the relationship between the Hubble formula you do include [and also the change in H over time!] to the metric expansion within FLRW model developed from the EFE. [as a[t] is part of the metric...no need to discuss all this muck in your paper...]

this will be my last comment...or you'll be revising forever! LOL
good job.