Stellar formation / Expansion / Education questions

[TigerDaveJr]

Regarding the creation of the universe and the current model:

Is it assumed that the universe, at the time of creation was finite in size (or at least more finite than it is now) prior to the rapid expansion, or was the protoexistance finite in size in an infinite universe? So, did the universe AND its contents expand, or did a collection of mass within the universe expand, creating the physicality we know today?

I have seen the expansion explained like a balloon. However, if this were true, would not most mass be on the 'outside' of the balloon? Is there content in the middle of the universe, or is there a hollow center that is getting bigger as we get further from the center? I've read that asking about the center is impossible, and that the universe has infinite shape, but if that's true can we say we're expanding? Would there not be an origin point, or is that one of the problems that a physics-uneducated person like myself would be unable to grasp (re: Plato's allegory of the cave).

Can we not use red shift in order to determine the relative center of this expansion? I understand that we observe red shift based upon where we're standing, but should we not be able to calculate from all that where the overall center is? Where are we in regards to this?

Was expansion more like bread dough? Did the pre-expansion material tear? Was that tearing uneven, that left behind general emptiness in some spots and densely clumped matter in others that led to our original star nurseries?

Are galaxies considered expanding or collapsing? I've heard that there's supposed to be black holes in the center, so is this local mass "going down the drain" or is this mass being spun off from the center? Is it both? Do we consider galaxies to be generally "on par" with each other in the creation of more complex atomic structures, or do we expect each birth/nova/collapse/rebirth cycle of stellar material to continually generate more complex material, and that individually from galaxy to galaxy?

Second to last, is it possible, in the same way that we view time against the overall amazingness of deep time, that this initial universal expansion was just one bubble in an even larger sea of expanding pockets that we have yet to get close enough to see the evidence of? Not getting into dimensions, but is our universe just one in an entire "hyper-universe" of immense activity, that we can't directly "observe" in the same way that our tiny blip of existence fits in the concepts of deep time?

Finally and most importantly, where should I be aiming myself educationally in order to learn the answers to these questions, and to ask even more?

 

[Bandersnatch]

Hello, TigerDaveJr. Welcome to PF!

Regarding the creation of the universe and the current model:

Is it assumed that the universe, at the time of creation was finite in size (or at least more finite than it is now) prior to the rapid expansion, or was the protoexistance finite in size in an infinite universe? So, did the universe AND its contents expand, or did a collection of mass within the universe expand, creating the physicality we know today?

The universe was either finite or infinite, and it still is one of those. We cannot say which one it is, but if it's finite, then it has got a very large curvature radius(~88 billion ly was the minimum estimate, iirc).
The key part to understand is that whenever you hear of the universe's expansion, it does mean the entirety of it. It's not about some matter expanding into a preexisting space, but space WITH matter and energy, expanding.

I have seen the expansion explained like a balloon. However, if this were true, would not most mass be on the 'outside' of the balloon? Is there content in the middle of the universe, or is there a hollow center that is getting bigger as we get further from the center? I've read that asking about the center is impossible, and that the universe has infinite shape, but if that's true can we say we're expanding? Would there not be an origin point, or is that one of the problems that a physics-uneducated person like myself would be unable to grasp (re: Plato's allegory of the cave).

The balloon analogy is not perfect, as it creates this erroneous intuition that there is something outside(or inside) the balloon, due to the way we imagine it being a three dimensional object.
The analogy requires you to think of only the surface of the balloon as the universe.
There is no centre to a 2d surface(but there is curvature), and the expansion is still easily observable by comparing the distances between any two points on that surface at two different times.

These two pages go into more detail about the balloon analogy, its aims and limitations, all in layman's terms:
http://www.mso.anu.edu.au/~charley/papers/LineweaverDavisSciAm.pdf (first page is blank)
http://www.phinds.com/balloonanalogy/

Can we not use red shift in order to determine the relative center of this expansion? I understand that we observe red shift based upon where we're standing, but should we not be able to calculate from all that where the overall center is? Where are we in regards to this?

You should see from the above links that it is impossible to define a centre of uniformly expanding space.
You can easily define the centre of the observable universe, which is wherever you are standing.

Was expansion more like bread dough? Did the pre-expansion material tear? Was that tearing uneven, that left behind general emptiness in some spots and densely clumped matter in others that led to our original star nurseries?

You are taking the analogy too far. Of course the universe is not made of dough, so it does not tear like dough does. It is important to limit yourself to only what the analogy is trying to convey(i.e., the expansion of space) and not to go overboard with drawing conclusions from it.

Are galaxies considered expanding or collapsing? I've heard that there's supposed to be black holes in the center, so is this local mass "going down the drain" or is this mass being spun off from the center? Is it both?

Galaxies are stable structures, with no significant amount of mass going down the black hole or escaping.
Sure there might be some rogue star gaining enough speed from random gravitational interactions to fling itself into the intergalactic space, and there tends to be some gas falling down the black hole - mostly because it takes so long to actually get there.
But overall, there is no expansion or collapse. The expansion of space does not affect small scale structures(like galaxies), and the black holes are not the voracious vacuum cleaners of doom that you might sometimes see in the popular media. Most stars stay in pretty much stable orbits around the galactic centre, and it's not going to change much, barring collisions with other galaxies.

Do we consider galaxies to be generally "on par" with each other in the creation of more complex atomic structures, or do we expect each birth/nova/collapse/rebirth cycle of stellar material to continually generate more complex material, and that individually from galaxy to galaxy?

All the galaxies coalesced from the same primordial gas, and the laws of physics governing them are the same, so it stands to reason that they are similar.
The difference is in the time scale. As you look farther away, you see younger galaxies, and the younger the galaxy, the less time its stars have had to go through their life cycles and produce heavier elements.
Generally the longer the universe exists, the more heavy elements it contains(in the early universe there was only hydrogen, helium and some lithium).

Second to last, is it possible, in the same way that we view time against the overall amazingness of deep time, that this initial universal expansion was just one bubble in an even larger sea of expanding pockets that we have yet to get close enough to see the evidence of? Not getting into dimensions, but is our universe just one in an entire "hyper-universe" of immense activity, that we can't directly "observe" in the same way that our tiny blip of existence fits in the concepts of deep time?

It's a kind of a vague and dangerously philosophical-sounding question, but I suppose it asks about the multiverse hypothesis?
As you say, it's not observable, therefore not falsifiable, which makes it an empty question really.
The first half an hour or so of this talk by Lee Smolin:
http://pirsa.org/13020146/
touches on the subject.

Finally and most importantly, where should I be aiming myself educationally in order to learn the answers to these questions, and to ask even more?

I'd recommend starting here:
http://www.astro.ucla.edu/~wright/cosmolog.htm
and going through either/both tutorial or/and FAQ.

Stephen Weinberg's "First three minutes" is a classic book concerning the early expansion of the universe. It's a bit dense at times, and getting somewhat old, but still worth reading.

Alan Guth's "The Inflationary Universe" talks about the birth of the idea of inflation, that is a major(if still somewhat dodgy) part of current cosmology.

Finally, understanding Relativity might be necessary. This popular treatment by Einstein himself is a good start:
http://www.gutenberg.org/files/30155/30155-pdf.pdf

You should be able to understand the ideas without any maths knowledge, but once you dig deeper into cosmology, you'll notice that it's at its heart a mathematical science, requiring you to learn higher mathematics to truly understand what's going on.
Unless you do that, you'll have to do with imperfect analogies, so if you have such an option, take calculus and algebra courses.

Finally, you might find the courses/videos on these sites relevant to your interests:
http://www.perimeterinstitute.ca/video-library (you probably want the "public lectures" section)
http://www.academicearth.org/ (try astronomy section)
https://www.coursera.org/ (actual online courses; physics section covers cosmology as well)
https://www.khanacademy.org/ (not a lot on cosmology, but good for learning maths and basic physics concepts)

 

[Mordred]

Along with the excellent material already mentioned I would add the following article

http://arxiv.org/abs/1304.4446

this article reviews the LCDM model which is the current concordance model. (concordance meaning the most agreed upon by the scientific community)
Although LCDM is a good fit to observational data there are other good fit models to observational evidence.
The paper I posted covers current cosmology without any of the maths so its handy for those that do not know the maths.
As pointed out however to really ubderstand what us going on you will need to understand the math involved.

An alternate analogy to the balloon analogy is to use a 3d grid coordinate. Each vertical, horizontal and z crossing forms a coordinate.
As expansion occurs none of the coordinates change nor do any of the angles between x,y or z change.
The space between x,y or z simply increases.
I always prefer the grid analogy over the balloon or raison bread analogies mainly because it can also show how curvature affects the sum of angles which relates to the universes geometry. Which also helps in understanding commoving and proper coordinates as well as light
cones.
my signature includes a link to
a handy calculator called lightcone 6.0. Thus calculator will help you in various maths involved in the expansion history of the universe past, present or future.

 

[TigerDaveJr]

So, the most important part of the balloon analogy is to get away from the concept that the balloon itself is a 3d construct in 3d space and only focus on the plane expansion into an immense curvature. This makes much more sense for the sites where I'm reading the universe is flat.

So, when we're looking at the outlier clusters of material that are getting faster away from us, that is only local perspective, yes? And that, from their location, we're getting faster away from them. Aside from local cluster material (as I've read regarding blue-shifted objects) every group should be able to look at every other object and see the same thing: we're all expanding from each other. And because of this, there can't be a definable center. So as you said earlier, from personal perspective the center is ... where you happen to be at, but this isn't something like a ball or any other 3d object in that we can identify the central core, etc. If there are no definable edges, there can be no definable equidistant from all edges.

I hope that's on the right track because that really cleared up a lot of questions I had. I definitely need to get into the higher maths to see this at work. Your assistance has been invaluable - thank you.

 

[Mordred]

There is no center or preferred location. Expansion occurs every where expect gravitationally bound regions equally.

Im having trouble getting the latex working from my phone on a print out history from the calculator however this thread has some print outs

https://www.physicsforums.com/showthread.php?t=692240

 

[marcus]

Hi Mordy, here is the 10 step history that Jorrie's calculator opens with:


$${\scriptsize\begin{array}{|c|c|c|c|c|c|}\hline R_{0} (Gly) & R_{\infty} (Gly) & S_{eq} & H_{0} & \Omega_\Lambda & \Omega_m\\ \hline 14.4&17.3&3400&67.9&0.693&0.307\\ \hline \end{array}}$$ $${\scriptsize\begin{array}{|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|} \hline a=1/S&S&T (Gy)&R (Gly)&D_{now} (Gly)&D_{then}(Gly)&D_{hor}(Gly)&V_{now} (c)&V_{then} (c) \\ \hline 0.001&1090.000&0.0004&0.0006&45.332&0.042&0.057&3.15&66.18\\ \hline 0.003&339.773&0.0025&0.0040&44.184&0.130&0.179&3.07&32.87\\ \hline 0.009&105.913&0.0153&0.0235&42.012&0.397&0.552&2.92&16.90\\ \hline 0.030&33.015&0.0902&0.1363&38.052&1.153&1.652&2.64&8.45\\ \hline 0.097&10.291&0.5223&0.7851&30.918&3.004&4.606&2.15&3.83\\ \hline 0.312&3.208&2.9777&4.3736&18.248&5.688&10.827&1.27&1.30\\ \hline 1.000&1.000&13.7872&14.3999&0.000&0.000&16.472&0.00&0.00\\ \hline 3.208&0.312&32.8849&17.1849&11.118&35.666&17.225&0.77&2.08\\ \hline 7.580&0.132&47.7251&17.2911&14.219&107.786&17.291&0.99&6.23\\ \hline 17.911&0.056&62.5981&17.2993&15.536&278.256&17.299&1.08&16.08\\ \hline 42.321&0.024&77.4737&17.2998&16.093&681.061&17.300&1.12&39.37\\ \hline 100.000&0.010&92.3494&17.2999&16.328&1632.838&17.300&1.13&94.38\\ \hline \end{array}}$$

 

[Mordred]

Thanks Marcus working from a phone gets fustrating lol.

Anyways as you can see the calculator is a handy tool for seeing the expansion history.
It also has graphing capabilities as shown in the other thread. As well as a couple more row options.
Marcus has posted numerous examples of its usage on PF.
The above pinned thread "See 80 billion years into..." above has a large collection of examples
skip near the end though as there
is numerous older versions lol.

Here is a link to the main page under development. PF members such Jorrie and Marcus are fine tuning.

http://cosmocalc.wikidot.com/start

There is several links to how to use the calculator. Including a user manual, a tutorial manual and advanced info for the math usage.

 

[marcus]

So, the most important part of the balloon analogy is to get away from the concept that the balloon itself is a 3d construct in 3d space and only focus on the plane expansion into an immense curvature. This makes much more sense for the sites where I'm reading the universe is flat.

So, when we're looking at the outlier clusters of material that are getting faster away from us, that is only local perspective, yes? And that, from their location, we're getting faster away from them. Aside from local cluster material (as I've read regarding blue-shifted objects) every group should be able to look at every other object and see the same thing: we're all expanding from each other. And because of this, there can't be a definable center. So as you said earlier, from personal perspective the center is ... where you happen to be at, but this isn't something like a ball or any other 3d object in that we can identify the central core, etc. If there are no definable edges, there can be no definable equidistant from all edges.

I hope that's on the right track because that really cleared up a lot of questions I had. I definitely need to get into the higher maths to see this at work. Your assistance has been invaluable - thank you.

Hi TigerDaveJr, your post#1 asked reasonable and clearly worded questions which to me seem kind of exemplary newcomer questions. That is one reason Bandersnatch was able to make such an excellent answer in post#2. I was impressed by its conciseness accuracy and completeness so I quoted post#2 to have for reference in the "balloon analogy" sticky thread, here:
https://www.physicsforums.com/showthread.php?p=4386996#post4386996
Thanks both for this very useful exchange. IMO it's a real contribution.

 

[Mordred]

Hi TigerDaveJr, your post#1 asked reasonable and clearly worded questions which to me seem kind of exemplary newcomer questions. That is one reason Bandersnatch was able to make such an excellent answer in post#2. I was impressed by its conciseness accuracy and completeness so I quoted post#2 to have for reference in the "balloon analogy" sticky thread, here:
https://www.physicsforums.com/showthread.php?p=4386996#post4386996
Thanks both for this very useful exchange. IMO it's a real contribution.

I couldn't agree more

 

[Mordred]

So,

So, when we're looking at the outlier clusters of material that are getting faster away from us, that is only local perspective, yes? And that, from their location, we're getting faster away from them. Aside from local cluster material (as I've read regarding blue-shifted objects) every group should be able to look at every other object and see the same thing: we're all expanding from each other. And because of this, there can't be a definable center. So as you said earlier, from personal perspective the center is ... where you happen to be at, but this isn't something like a ball or any other 3d object in that we can identify the central core, etc. If there are no definable edges, there can be no definable equidistant from all edges.

I hope that's on the right track because that really cleared up a lot of questions I had. I definitely need to get into the higher maths to see this at work. Your assistance has been invaluable - thank you.

you may find this article will be handy in regards to redshift, cosmic distance measures and expansion.

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.
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 off the vacuum surrounding 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= the radial coordinate (measured as the circumference, divided by 2pi, of a sphere centered around the massive body)

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 distance coordinate for calculating proper distance between objects at the same epoch (time)
r(t) is the comoving radial scale factor. Which is distance coordinates for calculating proper distances between objects at two different epochs (time)

$$Proper distance =\frac{\stackrel{.}{a}(t)}{a}$$

The dot above a indicates change in.

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.

$$H(t)=\frac{\stackrel{.}{a}(t)}{a(t)}$$

Expansion velocity
$$v=\frac{\stackrel{.}{a}(t)}{a}$$

This shows that Hubble's constant is time dependant.



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 where absolute luminosity is the amount of energy emitted per second.

Luminosity is often measured in flux where flux is

$$f=\frac{L}{4\pi r^2}$$

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

 

[Bandersnatch]

I hope that's on the right track

It's spot on.
Once you wrap your head around it, it makes perfect sense. Have fun learning!



Oh, and thanks for the accolades, you guys.

 

[marcus]

It's spot on.
Once you wrap your head around it, it makes perfect sense. Have fun learning!
Oh, and thanks for the accolades, you guys.

Now that TigerDave's question is well and truly answered, let me ask you about your 88 Gly figure for the lower limit of radius of curvature (in your post#2). I have the same rough impression ≈ 100 Gly.

Do you have a recent reference that you like? There seem to be a lot of 95% confidence intervals for Ω_k. Presumably one wants to take the square root of the absolute value of the most negative and divide the Hubble radius R by it:
R/|Ω_k|^.5

I posted some links to Planck estimates of Ω_k here:
https://www.physicsforums.com/showthread.php?p=4385618#post4385618
I don't especially like them, just what I was able to come up with. You may have others to compare. Some of those I quoted in that post were:
==quote==

...using observations of the CMB alone:

100Ω_K= −4.2^+4.3_-4.8 (95%; Planck+WP+highL);
100Ω_K= −1.0^+1.8 _-1.9 (95%; Planck+lensing + WP+highL)

...by the addition of BAO data. We then find

100Ω_K = −0.05^+0.65_-0.66 (95%; Planck+WP+highL+BAO)
100Ω_K = −0.10^+0.62_-0.65 (95%;Planck+lensing+WP+highL+BAO)
==endquote==
This was from page 40 of http://arxiv.org/abs/1303.5076

 

[Jorrie]

...by the addition of BAO data. We then find

100Ω_K = −0.05^+0.65_-0.66 (95%; Planck+WP+highL+BAO)
100Ω_K = −0.10^+0.62_-0.65 (95%;Planck+lensing+WP+highL+BAO)
==endquote==
This was from page 40 of http://arxiv.org/abs/1303.5076

How would one find the maximum likelihood value from these ranges? 100Ω_K = -0.10 - 0.65 + 0.62 = -0.4?

It is obviously so close to Ω=1 that it might not matter. Putting Ω = 1.004 into LightCone yields the following minor changes from flat space:
Age: 13.7543 vs 13.7872 Gy flat
Event Hor: 16.476 vs 16.472 Gly flat
Particle Hor: 46.081 vs 46.279 Gly flat

Sorry, goofed with a zero, but have already posted; it is even closer, 100Ω_K = -0.10 - 0.65 + 0.62 = -0.13
Age: 13.776 vs 13.7872 Gy flat
Event Hor: 16.473 vs 16.472 Gly flat
Particle Hor: 46.214 vs 46.279 Gly flat

 

[marcus]

What happens if we just take the central value 100Ω_k = -0.1?

Then we have to divide the Hubble radius R=14.4 Gly by the square root of 0.001 which is 0.0316
so 14.4/.001^.5 = 455 Gly

Ugh

Let's not include BAO, that represents counting galaxies. Let's just base everything on observations of the Cosmic Microwave Background. Pick the best-tasting cherries. It's 1AM here. I'm off to bed. :-D

 

[Jorrie]

What happens if we just take the central value 100Ω_k = -0.1?

Then we have to divide the Hubble radius R=14.4 Gly by the square root of 0.001 which is 0.0316
so 14.4/.001^.5 = 455 Gly

What I find fascinating is how a non-zero curvature would evolve over time. If I understand correctly, a non-zero 'curvature density parameter' Ω_k evolves with a^-2, meaning the radius of curvature R_k evolves with a^-1. Using the above value for the present, R_k = 455 Gly, then at the time of the CMB, it was around 500 trillion light years and at a = 100, it will be only around 4.5 Gly. This points towards an extreme curvature (small radius) as a -> infinity.

Or do I have the relationship wrong?

 

[Bandersnatch]

Now that TigerDave's question is well and truly answered, let me ask you about your 88 Gly figure for the lower limit of radius of curvature (in your post#2). I have the same rough impression ≈ 100 Gly.

So I've spent a good few hours trying to track down the source I got it from, but with no luck. But even if I did, that would do us little good, as I'm pretty sure that was not the most recent of papers. It would stand to reason that the constraints on the minimum radius improved with more data from WMAP streaming in.

 

[marcus]

. What I find fascinating is how a non-zero curvature would evolve over time. If I understand correctly, a non-zero 'curvature density parameter' Ω_k evolves with a^-2,...

This much I do think is right, at least the curvature does evolve with a^-2. Now remember that curvature is big when the radius is small. The curvature is by definition 1/R_k². I believe the radius of curvature evolves with a.

So in the future when a=100, the radius of curvature would be 100 times what it is today.

R_k(a=100) = 100R_k(now)

It would be good to check with George about this or some other knowledgeable person. But that's what I think is the case. There are problems with using Ω_k because it expresses the curvature (a reciprocal area, or reciprocal square time) in a somewhat roundabout way in relation to today's critical density. And that reference quantity changes over time, which could mess up the simple proportionality if one wants to track the curvature over a long enough interval of time that the critical density might change significantly. I'll make a one-time use of it to get the radius of curvature just for NOW and then from thenceforth I'll just use the proportionality with the scalefactor a.

 

[marcus]

. What I find fascinating is how a non-zero curvature would evolve over time. If I understand correctly, a non-zero 'curvature density parameter' Ω_k evolves with a^-2,...

This much I do think is right, at least the curvature does evolve with a^-2. Now remember that curvature is big when the radius is small. The curvature is by definition 1/R_k². I believe the radius of curvature evolves with a.

So in the future when a=100, the radius of curvature would be 100 times what it is today.

R_k(a=100) = 100R_k(now)

It would be good to check with George about this or some other knowledgeable person. But that's what I think is the case. There are problems with using Ω_k because it expresses the curvature (a reciprocal area, or reciprocal square time) in a somewhat roundabout way in relation to today's critical density. And that reference quantity changes over time, which could mess up the simple proportionality if one wants to track the curvature over a long enough interval of time that the critical density might change significantly. I'll make a one-time use of it to get the radius of curvature just for NOW and then from thenceforth I'll just use the proportionality with the scalefactor a.

 

[Jorrie]

This much I do think is right, at least the curvature does evolve with a^-2. Now remember that curvature is big when the radius is small. The curvature is by definition 1/R_k². I believe the radius of curvature evolves with a.

I think you are right - if not zero, curvature was much closer to zero (critical density) in the past and will evolve away from zero in the future. Since critical density was much higher in the past, it means curvature was higher in the past and will be smaller in the future. I think this is where I did the head-flip to 1/a for the radius, instead of just a. One should analyze actual densities in this case and not normalized density ratios, like Ω.

 

[marcus]

Recalling a previous post with a link to Planck estimates of CURRENT Ω_k:
https://www.physicsforums.com/showthread.php?p=4385618#post4385618
Some of those quoted in that post were:
==quote from page 40 of http://arxiv.org/abs/1303.5076 ==
...using observations of the CMB alone:

100Ω_K= −4.2^+4.3_-4.8 (95%; Planck+WP+highL);
100Ω_K= −1.0^+1.8 _-1.9 (95%; Planck+lensing + WP+highL)
==endquote==

We could take the second of these and calculate the radius of curvature in two cases, the
CENTRAL VALUE for the present-day Ω_K namely -0.01
and also the MOST NEGATIVE VALUE namely -0.029

and then we could calculate the radius of curvature corresponding to these two cases.

(central value) R_k= 14.4/(.01)^.5 = 144 billion lightyears
(most negative) R_k= 14.4/(.029)^.5 = 84.5597116 billion lightyears

WOW! There is Bandersnatches 88 Gly radius of curvature!
Basically we get 85 Gly and we are using March 2013 Planck report numbers so of course slightly different from whatever his 88 was based on but very likely the same type of thing. Namely some 95% confidence interval for presentday Ω_k taking the MOST NEGATIVE end of the confidence interval---therefore corresponding to the most positive curvature and thus to the SHORTEST RADIUS of curvature.
In our case the confidence interval was [-0.029, 0.008] they almost all tend to be lopsided towards negative thus favoring positive curvature. this is no exception.