# Redshift FAQ article development

1. Mar 14, 2013

### Mordred

I have been working on an article covering Redshift. The article is in this thread

I would like opinions, recommendations and any other contributions that forum members feel would be good to add.

2. Mar 14, 2013

### Mordred

This is a starting point in explaining Redshift for FAQ purposes.

First we need to distinquish the between RedShift and Doppler shift. In both cases the light emitted by one body and received by the other will be Red/Blue shifted ie its wavelength will be stretched. So the color of the light is more towards the red end of the spectrum. As shown by the relation formula below. But there's a subtle difference,

Δf/f = Δλ/λ = v/c

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. 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)

so long as the velocity v is much less than the speed of light. A relativistic doppler formula is required when velocity is comparable to the speed of light. Doppler shift is used to describe Redshift due to inertial velocity one example is a car moving away from you the light will be Red Shifted, as it approaches you the light and sound will be blueshifted.

The Cosmological Redshift is however a redshift caused by the expansion of space. The wavelength of light increases as it traverses the expanding universe between its point of emission and its point of detection by the same amount that space has expanded during the crossing time. This is a measure of its recessive velocity over DISTANCE. The further away an object is the greater the amount of RedShift. This is given in accordance with Hubbles 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:

Hubbles Law: The greater the distance of measurement the greater the recessive velocity

velocity = H0 × distance.

Velocity represents the galaxy's recessional 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, but the current accepted value is 70 kilometers/second per megaparsec, the latter being a unit of distance in intergalactic space
Any measurement of redshift above the Hubble distance defined as c=H0 = 4300 + or - 400 Mpc will have a recessive velocity of greater than the speed of light. This does not violate GR for a couple of reasons.

1) Recessive velocity is distance dependant
2) Recessive velocity is not inertial velocity, the galaxies are not accelerating from us, the space between us and the galaxy is increasing.

This is given by the form.

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

1+z= λobserved/λemitted or z=λobserved-λemitted/λobserved

the symbol λ is 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

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 first measurement. Indeed expansion occurs the same throughout the cosmos. However Gravity and the strong Nuclear force in Galaxy clusters are 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 between two coordinates are expanding. That expansion is homogeneous and isotropic. In other words, there is no preferred location (Homogenous) and no preferred direction (isotropic). Keep in mind these terms describe the universe on large scales. Indeed below 100 Mpc we know that galaxy clusters, large scale clusters are not homogeneous or isotropic. 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 is the more generic name given to the unknown cause of the acceleration of the universe. 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 $\Lambda$. the greek letter Lambda

The Gravitational Redshift is a shift in the frequency of a photon to lower energy as it climbs out of a gravitational field. The frequency of light will Blueshift as it approaches the gravity well.

Standard candles

Using standard candles to determine the Hubble constant has a long andhonorable history; it's the method used by Hubble himself. The recipe for finnding the Hubble constant is a simple one:

Identify a population of standard candles with luminosity L.
-Measure the redshift z and flux f for each standard candle.
-Compute dL = (L=4¼f)1=2 for each standard candle.
-Plot cz versus dL.
-Measure the slope of the cz versus dL relation when z << 1; this gives
H0.

For cosmological purposes, a standard candle should be bright enough to be detected at large redshifts. It should also have a luminosity which is well determined. One time-honored variety of standard candle is the class of Cepheid variable stars. Cepheids, as they are known, are highly luminous supergiant stars, As they pulsate radially, their luminosity varies
in response, partially due to the change in their surface area, and partially due to the changes in the surface temperature as the star pulsates. Type 1A supenova is a type of Cepheid.

Without the use of Cepheids measuring distances would be extremely problematic especially at extremely large distances, These distances ar too large for the parralex method which is a form of triangulation.

Any comments or idea to add will be greatly appreciated, the more suggestions that are added the more complete this article will become. As such I give permission to everyone to copy/paste or correct any portion of the above starting article to add their contributions.

3. Mar 14, 2013

### marcus

Hi Mordy, you've clearly put a lot of work into this! I'll try to comment very briefly and hope that others will comment as well.
I think you have (not an FAQ exactly but) a kind of GLOSSARY of terms. That kind of thing could be quite useful.

An FAQ tries to be a list of questions that commonly occur to people so that if someone is puzzled they can look down the list and find an actual question corresponding to what they have in mind that's bothering them.

A Glossary, or Handbook of terminology, is obviously different but that's not a bad thing. There could be room for both, I suppose. A glossary could actually be more helpful to some people---a kind of "mini-encyclopedia" with very short explanations of just a few key terms and concepts.

People who work on projects like that rarely have the satisfaction of sole authorship, I believe. It is only to be expected that everything gets chewed over by several writer/editors and revised beyond recognition----or thrown out. But whatever happens your work is a kind of force vector pushing towards a final goal, the emergence of some useful glossary of terms. That's how I see it, though I could easily be wrong.

Along the way you might want to pick up the habit of using the subscript and superscript buttons that PF offers. for example:
1+z= λobserved/λemitted

could be changed using the "X2" button right above the box, into:

1+z= λobservedemitted

And the other thing you say " z=λobserved-λemitted/λobserved "
is not quite right, it needs "emitted" instead of "observed" in the denominator, like this:
z=(λobserved-λemitted)/λemitted

Then you mouse over something you want to make into a subscript and press the X2 button.

My apologies if this is too elementary, in that case just pay it no mind.

4. Mar 14, 2013

### Mordred

Hey Marcus Im glad for any input on this project. My original had a list of questions however I found trying to individually answer common questions ked to a lot of repeating the same things over and over again. So I tried an article that offered a source of answers. Taking what you mentioned about a quick reference of terms offered a viable quick route to the reader. That being said this
is my first FAQ/glossary I've
written. So I am definetely interested in any suggestions.

Yeah I am still working on how to post the various math forms so your advice on how to do so is also helpful. I just discovered the tab at the top on the advance post lol.

I would also like to add an explanation of the expansion rate you have often mentioned on %
The 1/140 % expansion rate. So could use your contribution in that as your best qualified to explain that aspect. Like I mentioned any contributors are more than welcome to copy paste and modify the above developing article

Last edited: Mar 14, 2013
5. Mar 15, 2013

### Chronos

Your need to use linear algebra to properly express this stuff analytically. I parted company with most of my knowledge of linear algebra years ago. It's not terribly useful to most engineers.

6. Mar 15, 2013

### Mordred

EXPANSION AND REDSHIFT

1) Why is all the galaxies accelerating from us ?
2) Is Redshift the same as Dopplershift ?
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 ?

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.

Universe: The Universe in cosmology is defined as the Observable Universe The observable universe from Earth is 46 Billion light years, or 4.3×1026meters with an age as of 2013, is 13.772 ± 0.059 billion years. So how do we see farther than 13.772 billion years, the answer lies in expansion. As light is travelling towards us, spacetime has expanded.
One common Question posters often ask is " What is outside our Universe, this question has no meaning as without space or time you have nonexistence. Also their is no clear concensus on if the Universe is Finite or Infinite. " When an object is said to leave our universe" we mean that the object has crossed the observable universe or rather that it is redshifted to the point of non detectable.

The parsec (symbol: pc) is a unit of length used in astronomy, equal to about 30.9 trillion kilometres(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 kilometres

Mpc=1 million Parsecs

In order to understand Redshift we must first need to distinquish the between RedShift and Doppler shift. In both cases the light emitted by one body and received by the other will be Red/Blue shifted ie its wavelength will be stretched. So the color of the light is more towards the red/blue end of the spectrum. As shown by the relation formula below. But there's a subtle difference, between Dobblershift and redshift.

Δf/f = Δλ/λ = v/c

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 moved towards the blue end of th 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)

so long as the velocity v is much less than the speed of light. A relativistic doppler formula is required when velocity is comparable to the speed of light. Doppler shift is used to describe Redshift due to inertial velocity one example is a car moving away from you the light will be Red Shifted, as it approaches you the light and sound will be blueshifted.

The Cosmological Redshift is however a redshift caused by the expansion of space. The wavelength of light increases as it traverses the expanding universe between its point of emission and its point of detection by the same amount that space has expanded during the crossing time indeed the amount of redshift is the total amount of expansion. This is a measure of its recessive velocity over DISTANCE. The further away an object is the greater the amount of RedShift. This is given in accordance with Hubbles 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:

Hubbles 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 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. The latter being a unit of distance in intergalactic space described above.
Any measurement of redshift above the Hubble distance defined as c=H0 = 4300±400 Mpc will have a recessive velocity of greater than the speed of light. This does not violate GR for a couple of reasons.

1) Recessive velocity is distance dependant
2) Recessive velocity is not inertial velocity, the galaxies are not accelerating from us, the SPACE between us and the galaxy is increasing.

This is given by the form.

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

1+z= λobservedemitted or z=(λobservedemitted)/λemitted

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 occurs the same throughout the cosmos. However Gravity and the strong Nuclear force in Galaxy clusters are 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 between two coordinates are expanding. This is important in that no FORCE is acting upon the galaxies to cause expansion. 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. Indeed below 100 Mpc we know that galaxy clusters, large scale clusters are not homogeneous or isotropic.
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 is the more generic name 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 $\Lambda$. 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 Heisenburg uncertainty principle. Virtual particles pop in and out of existence all the time, As expansion occurs 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 has its 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 the 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 te coordinates do not change, the space between coordinates change. I should also note their is no clear concensus on whether the universe is finite or infinite. If its infinite now then it was infinite in the past. Same with Finite. The Big Bang model only describes the Universe from 10-43 seconds and is not considered as starting from a blackhole singularity, rather its properly described as a rapid expansion of spacetime.

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

The Gravitational Redshift is a shift in the frequency of a photon to lower energy as it climbs out of a gravitational field. The frequency of light will Blueshift as it approaches the gravity well.

Standard candles

Using standard candles to determine the Hubble constant has a long and honorable history; it's the method used by Hubble himself. The recipe for finnding the Hubble constant is a simple one:

Identify a population of standard candles with luminosity L.
-Measure the redshift z and flux f for each standard candle.
-Compute dL = (L=4¼f)1=2 for each standard candle.
-Plot cz versus dL.
-Measure the slope of the cz versus dL relation when z << 1; this gives
H0.

The CMB, (Cosmic Microwave Background) also provides a good reference point in regards to Homogeneous and Isotropic confirmation, and of distances. However that is best described in greater detail than this article.

For cosmological purposes, a standard candle should be bright enough to be detected at large redshifts. It should also have a luminosity which is well determined. One time-honored variety of standard candle is the class of Cepheid variable stars. Cepheids, as they are known, are highly luminous supergiant stars, As they pulsate radially, their luminosity varies
in response, partially due to the change in their surface area, and partially due to the changes in the surface temperature as the star pulsates. Type 1A supenova is a type of Cepheid.

Without the use of Cepheids measuring distances would be extremely problematic, especially at extremely large distances, These distances are far too large for the parallex method which is a form of triangulation.

Last edited: Mar 15, 2013
7. Mar 15, 2013

### Mordred

I've made some changes to the above article including a bit more detail on expansion, What types of questions the article attempts to answer. I'll be adding more as I go I am considering adding spacetime Geometry to the article in regards to flat or curved spacetime but am not sure if it would make the article too lengthy

8. Mar 15, 2013

### marcus

Mordy,
I don't want to seem to encourage you because I am not the gatekeeper about sticky quality and FAQ quality, the mentors are. So I'd feel bad if I encouraged you and the stuff wasn't used and you were disappointed. But. There's a but.

I actually think that is a pretty good list of very basic FAQ questions. they do reflect what many newcomers are actually wondering about. or things they ought to be asking and will realize they should be asking when they see the question.

So the "But" is that I personally think you are doing something worthwhile and are learning by it. There's some chance, I suppose, that someone will see your list and utilize all or some of it, and supply some different answers etc. use some of your wording perhaps and some from somebody else. That would be nice, it would be nice to see some use made of it.
But I CAN'T SPECULATE about that possibility. I just personally think it looks like a worthwhile activity.

I won't help, except very indirectly.

You asked me earlier why it is that there is an exact equivalence like this:

H0 = 70 km/s per Mpc = 1/140 percent per million years.

Well, you should learn how to use the world's best online calculator, Google box.
Try calculating the RECIPROCAL of H0, namely 1/H0 = 1/(70 km/s per Mpc)

Paste "1/(70 km/s per Mpc)" into the Google box. Try it. (without the quotes)

Then type an equal sign, to activate the calculator. Try it.

So what you have typed in is 1/(70 km/s per Mpc)=

You should get 14 billion years. Let me know if it does not work for you and you do not get the answer 14 billion years.

If you know the reciprocal of a quantity, you know the quantity itself===it is just one over its reciprocal!

That means that H0 = 1/(14 billion years).

Now everybody knows H0 is the type of quantity that if you multiply a distance by it you get a speed. That type of quantity has to be a reciprocal time, because if you divide a distance by a time you get a speed. It's the only thing arithmetic you can do to a distance to turn it into a speed. Something you multiply distance by, to get speed, can only be the reciprocal of a time.

So regardless of whatever quaint units they traditionally give it to you with, H0 really is equal to one over a time. And Google calculator tells you the time is 14 billion years. That is called the "Hubble time".

A possible way to go from there is to take that fraction H0 = 1/(14 billion years) and divide the numerator and the denominator by 14,000.

For the numerator you get 1/14000 = 1/140 percent.
For the denominator you get 14 billion years divided by 14000, which is a million years.

That is a fractional rate of increase, an INSTANTEOUS rate it is now, which if it would continue steady for a million years would make a distance increase by 1/14000 of its length. That is the same as 1/140 of one percent.

So that is the fraction of its length that a distance grows by in a million years.

Last edited: Mar 15, 2013
9. Mar 15, 2013

### Mordred

Yeah the sticky thread aspect is something I will ask the MODS on. No matter the results writing this article has been enjoyable and provided a good learning aid for myself snd hopefully others. I'll try the methodology you just mentioned.
I do hope others like the article and use it as a simple forum based reference. I am close to completion of the article as I do not want it to get too lengthy. Neither did I want to bog it down with too much mathematics.
Thanks for your explaination and assistance thus far

10. Mar 15, 2013

### marcus

Maybe that is not the best explanation. Let's try a different route.
You asked me earlier why it is that there is an exact equivalence like this:

H0 = 70 km/s per Mpc = 1/140 percent per million years.

Well, you should learn how to use the world's best online calculator, Google box.
Try calculating the RECIPROCAL of 1/H0 = 1/(70 km/s per Mpc)

Paste "1/(70 km/s per Mpc)" into the Google box. Try it. (without the quotes)

Then type an equal sign, to activate the calculator. Try it.

So what you have typed in is 1/(70 km/s per Mpc)=

You should get 14 billion years. Let me know if it does not work for you and you do not get the answer 14 billion years.

That means that H0 = 1/(14 billion years)

So then
multiply that by 14 billion light years. That will give the speed that a 14 billion lightyear distance is expanding. BUT THAT IS THE SPEED OF LIGHT!

Something that goes one light year in one year is going at speed of light and
1 lightyear/1 year = c.

14 billion lightyears/14 billion years = c.

So 14 billion lightyears is the distance that is expanding at the speed of light.

What is 1/140 of one percent of 14 billion lightyears?

It is a million lightyears. that is therefore the distance that is growing at 1/140 of one percent of the speed of light.

So think about that distance. What percentage of its length does it increase by in a million years?

11. Mar 16, 2013

### Jonathan Scott

This sort of explanation leads to a lot of confusion, and I find it unhelpful.

Gravitational redshift isn't something that happens to photons. A photon has constant frequency in any observer's static frame of reference. Gravitational redshift is to do with comparing results seen by different observers. An observer at a different potential will see a photon to have a different frequency compared with one generated locally by a similar process (for example a spectral line).

You can see the constant frequency aspect by considering any repetitive signal being sent along a fixed path, regardless of potential. Clearly, the number of iterations of the signal passing any point in a given amount of time must be the same. Photons work in the same way.

12. Mar 16, 2013

### Mordred

Thanks Marcus for your explanation.
Yeah I agree Jonathon that statement is misleading.
I am using my phone this weekend I'll change that line on Monday. I had wanted to expand on gravitational redshift
a bit anyways.
Thanks for your input.
If anyone else can spot incorrect or misleading lines in the article I would definetely appreciate any constructive feedback. Also if anyone would like questions and answers added to the article. That would also be helpful

Last edited: Mar 16, 2013
13. Mar 18, 2013

### Mordred

EXPANSION AND REDSHIFT

1) Why is all the galaxies accelerating from us ?
2) Is Redshift the same as Dopplershift ?
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 ?

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:$c\ =\ 2.99792458\ \times\ 10^{8}\ m\ s^{-1}$

Universe: The Universe in cosmology is defined as the Observable Universe The observable universe from Earth is 46 Billion light years, or 4.3×1026meters with an age as of 2013, is 13.772 ± 0.059 billion years. So how do we see farther than 13.772 billion years, the answer lies in expansion. As light is travelling towards us, spacetime has expanded.
One common Question posters often ask is " What is outside our Universe, this question has no meaning as without space or time you have nonexistence. Also their is no clear concensus on if the Universe is Finite or Infinite. " When an object is said to leave our universe" we mean that the object has crossed the observable universe or rather that it is redshifted to the point of non detectable.

The parsec (symbol: pc) is a unit of length used in astronomy, equal to about 30.9 trillion kilometres(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 kilometres

Mpc=1 million Parsecs

In order to understand Redshift we must first need to distinquish the between RedShift and Doppler shift. In both cases the light emitted by one body and received by the other will be Red/Blue shifted ie its wavelength will be stretched. So the color of the light is more towards the red/blue end of the spectrum. As shown by the relation formula below. But there's a subtle difference, between Dobblershift and redshift.

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

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 moved towards the blue end of th 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 medium
$$v_r$$ is velocity of the reciever to the medium (positive if moving toward the source, negative if moving away)
$$v_s$$ is the velocity of the source to the medium (positive if moving away from the reciever, negative if other direction)

The above are for velocities where the source is directly away or towards the observer and fo low velocities less than relativistic velocities. A relativistic doppler formula is required when velocity is comparable to the speed of light. Their are different variations of the above formula for Tansverse 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 Red Shifted, as it approaches you the light and sound will be blueshifted.

The Cosmological Redshift is however a redshift caused by the expansion of space. The wavelength of light increases as it traverses the expanding universe between its point of emission and its point of detection by the same amount that space has expanded during the crossing time indeed the amount of redshift is the total amount of expansion. This is a measure of its recessive velocity over DISTANCE. The further away an object is the greater the amount of RedShift. This is given in accordance with Hubbles 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:

Hubbles 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 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. The latter being a unit of distance in intergalactic space described above.
Any measurement of redshift above the Hubble distance defined as c=H0 = 4300±400 Mpc will have a recessive velocity of greater than the speed of light. This does not violate GR for a couple of reasons.

1) Recessive velocity is distance dependant
2) Recessive velocity is not inertial velocity, the galaxies are not accelerating from us, the SPACE between us and the galaxy is increasing.

This is given by the form.

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

1+z= λobservedemitted or z=(λobservedemitted)/λemitted

$$1+Z=\frac{\lambda}{\lambda_o}$$ 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 occurs the same throughout the cosmos. However Gravity and the strong Nuclear force in Galaxy clusters are 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 between two coordinates are expanding. This is important in that no FORCE is acting upon the galaxies to cause expansion. 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. Indeed below 100 Mpc we know that galaxy clusters, large scale clusters are not homogeneous or isotropic.
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 is the more generic name 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 $\Lambda$. 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 Heisenburg uncertainty principle. Virtual particles pop in and out of existence all the time, As expansion occurs 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 has its 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 the 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 te coordinates do not change, the space between coordinates change. I should also note their is no clear concensus on whether the universe is finite or infinite. If its infinite now then it was infinite in the past. Same with Finite. The Big Bang model only describes the Universe from 10-43 seconds and is not considered as starting from a blackhole singularity, rather its properly described as a rapid expansion of spacetime.

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

The Gravitational Redshift Gravitational redshift or Einstein shift is the process by which electromagnetic radiation originating from a source that when observed in a region of a weaker gravitational field its frequency is redshifted. When observed in a stronger gravitational field its frequency is blueshifted. The formula below is the gravitational redshift formula or Einstein shift in an unchanged, 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

when the signal is 90 degrees to the observer we need to use the Transverse Redshift formula

$$1+z=\frac{1}{\sqrt{1-v^2/c^2}}$$

Standard candles

Using standard candles to determine the Hubble constant has a long and honorable history; it's the method used by Hubble himself. The recipe for finnding the Hubble constant is a simple one:

Identify a population of standard candles with luminosity L.
-Measure the redshift z and flux f for each standard candle.
-Compute dL = (L=4¼f)1=2 for each standard candle.
-Plot cz versus dL.
-Measure the slope of the cz versus dL relation when z << 1; this gives
H0.

The CMB, (Cosmic Microwave Background) also provides a good reference point in regards to Homogeneous and Isotropic confirmation, and of distances. However that is best described in greater detail than this article.

For cosmological purposes, a standard candle should be bright enough to be detected at large redshifts. It should also have a luminosity which is well determined. One time-honored variety of standard candle is the class of Cepheid variable stars. Cepheids, as they are known, are highly luminous supergiant stars, As they pulsate radially, their luminosity varies
in response, partially due to the change in their surface area, and partially due to the changes in the surface temperature as the star pulsates. Type 1A supenova is a type of Cepheid.

Without the use of Cepheids measuring distances would be extremely problematic, especially at extremely large distances, These distances are far too large for the parallex method which is a form of triangulation.

Last edited: Mar 18, 2013
14. Mar 18, 2013

### Mordred

This is probably the extent that I'll add for this article, leaswise I cannot think of what else to add. If anyone spots incorrect or misleading statements or has suggestions I would appreciate it.

Even if the article may or may not be used as a FAQ article I enjoyed writing it so wouldn't mind if its checked for accuracy

15. Mar 18, 2013

### PAllen

I would like to argue, a la Synge 1960 and Bunn and Hogg, that Doppler and redshift, for SR, GR (including dynamic gravity like binary stars, static gravity around a dominant source, and cosmology - expansion) are a single phenomenon. The following single description covers all of these cases (first proven by Synge in 1960, and re-popularized by Bunn and Hogg):

- parallel transport emitter 4-velocity along light path to receiver. Then, in the local frame of receiver, compute Doppler using transported 4-velocity and light propagation vector.

This produces the right answer and explains all cases. Gravitational redshift and cosmpological redshift are simply special cases of Doppler that admit computational shortcuts (due to a family of static observers in the former case, and isotropy and homogeneity in the latter case). Note, in particular, that a parallel transported 4-velocy will never exceed c in any local frame. Recession velocity greater than c is really a generalization of the simple phenomenon in SR that in a given frame, two sources moving away from each other can separate at arbitrarily close to 2c (while have relative velocity - to each other - via parallel transport (trivial in SR) < c). In GR, such s separation speed can greatly exceed 2c, but it is wrong IMO to consider it a relative velocity. In GR relative velocity is non-unique due to path dependence of parallel transport, but no matter what the path, the timelike (<c) character of a vector doesn't change via transport. In the case of Doppler, there is a distinguished path - the light path from emitter to receiver which carries information about the emitter's motion to the receiver.

Also, note, that recession velocity is a coordinate dependent quantity, while the Doppler definition I gave is coordinate independent.

Last edited: Mar 18, 2013
16. Mar 18, 2013

### Mordred

I looked up the paper written by
Bunn and Hogg.

http://arxiv.org/abs/0808.1081

I haven't read this before so I am studying what your stating. Do you recommend a better paper I would like to read it as well.

17. Mar 18, 2013

### Chronos

Bunn and Hogg use incremental doppler shift to justify kinematical redshift. It works naively, but, suffers serious difficulty in nearly all cosmological models. That disqualifies it from an occams razor point of view, IMO.

18. Mar 18, 2013

### Mordred

I looked into that article I posted. Then looked at several counter papers. I agree with Chronos their viewpoint has too many problems. The article above is written with the intent of a FAG reference. In that regard I prefer to stick to the mainstay teachings of the hot Big Bang, rather than the various counter models. Such as models that argue against Dark energy, dark matter and other counter models.

19. Mar 18, 2013

### PAllen

What difficulty? In SR, Doppler results from comparison of 3 vectors: emitter 4-velocity, light propagation vector, and target 4-velocity. In GR, where comparison of vectors is path dependent, specify only that the comparison is done via parallel transport along the light path from emitter to receiver (thus light transporting emitter motion to receiver). Occams razor says, if there is one explanation for all redshift, why invent three? "Incremental Doppler" was an interpretational statement for the above mathematical fact coined by Bunn and Hogg. However, the mathematical fact was proven in all generality by Synge in 1960, in his book "Relativity: the General Theory".

20. Mar 18, 2013

### PAllen

These other things are irrelevant. The single definition of Doppler is true as long as you have a pseudo-Riemannian manifold. There is no conflict between Big Bang and single explanation of redshift.