# See light that is red-shifting z > 5.4

JesseM
Re: Redshifting

It is really simple. If you look at the graph, ordinate represents average distance between galaxies, or scalefactor - a. Wherever slope of the curve is more then 45 deg relative to abcissa, universe is accelerating.
Ah, so "a" stands for the "scale factor" and not the acceleration--does that mean the vertical axis of the graph is "a", and that the scale factor at a given time is proportional to the distance of some randomly-selected test particle at that time? But if so, why would they use the term "accelerating" to mean that the slope (proportional to the velocity of a test particle if the above is correct) is greater than 45 degrees? This section of a wikipedia article seems to say that the terminology of the expansion "accelerating/decelerating" is based on the sign of the deceleration parameter which can be defined either in terms of the Hubble constant and its first derivative or the scale factor "a" and its first and second derivatives...are you saying there's some mathematical argument that shows that a positive deceleration parameter (decelerating expansion) implies a slope closer to horizontal than 45 degrees, while a negative deceleration parameter (accelerating expansion) implies a slope closer to vertical than 45 degrees? i.e. that for any FLRW solution, if $$\dot{a}$$ is less than 1 then $$-\frac{\ddot{a} a }{\dot{a}^2}$$ will always be positive, while if $$\dot{a}$$ is greater than 1 then $$-\frac{\ddot{a} a }{\dot{a}^2}$$ will always be negative?

edit: never mind, I see you corrected yourself. I guess the "deceleration parameter" has no simple intuitive physical meaning in terms of the motion of a test particle though, it's just defined that way because it simplifies various cosmological equations?

JesseM
Re: Redshifting

Re: the non-intuitiveness of defining accelerating/decelerating expansion in terms of the "deceleration parameter"--looking on google books, I came across this textbook which denotes the scale factor as R(t), and indicates that the terminology of accelerating/decelerating expansion just refers to whether the second derivative $$\ddot{R}(t)$$ is positive or negative, but that if it's positive that always means the deceleration parameter is negative:
$$q(t) = \frac{-R(t)}{[\dot{R}(t)]^2} \ddot{R}(t)$$

Note the negative sign in this equation; this implies that if $$\ddot{R}(t)$$ is positive (i.e. if the expansion is accelerating) then the deceleration parameter will be negative.
I guess this does make sense in any universe that isn't collapsing, since in an expanding universe both $$R(t)$$ and $$\dot{R}(t)$$ should be positive. I do wonder whether cosmologists would say the expansion was "accelerating" or "decelerating" in a collapsing universe where both $$\ddot{R}(t)$$ and $$\dot{R}(t)$$ were negative so the deceleration parameter was negative too.

edit: never mind, I forgot that $$\dot{R}(t)$$ is squared in the equation for deceleration parameter, so it doesn't matter if it's positive or negative. And $$R(t)$$ is always positive by definition, so positive $$\ddot{R}(t)$$ always implies negative deceleration parameter and vice versa.

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Re: Redshifting

I forgot that $$\dot{R}(t)$$ is squared in the equation for deceleration parameter, so it doesn't matter if it's positive or negative. And $$R(t)$$ is always positive by definition, so positive $$\ddot{R}(t)$$ always implies negative deceleration parameter and vice versa.
Yes, exactly. Basically, if you are looking for physical meaning of deceleration parametar, it is that when second derivative changes sign function changes from being concave to being convex, or so called inflection point.

That is what Ich was trying to say, he just swaped $\dot a$ and $\ddot a$ :

Essentially, we're talking about $\dot a$ being positive or not.

Ich
Re: Redshifting

That is what Ich was trying to say, he just swaped $\dot a$ and $\ddot a$
Exactly, thanks for correcting that. I also didn't anticipate that "a" could be misinterpreted as acceleration, which of course it could.

Concerning the maths:
For a comoving particle, $\dot D = \dot a r$ and $\dot D = \ddot a r$. So the coordinate acceleration of a paricle is proportional to $\ddot a$. Accelerating/decelerating expansion = accelerating/decelerating particle.
For small distances (some 0.1 Hubble radii), this relation also holds for particles with peculiar motion. The reason is that the accelerating/decelerating expansion of the universe is exactly the same as accelerating/decelerating test particles, with exacly the same cause: the gravitational repulsion/attraction of the homogeneous fluid those particles are embedded in.

JesseM
Re: Redshifting

Another question--does wikipedia have an incorrect definition of the scale factor in this article? They write the equation $$l_p = l_t a(t)$$ where lp is the distance (say, to some galaxy) at present and lt is the distance at some other arbitrary time t. This can be rearranged as $$a(t) = \frac{l_p}{l_t}$$. However, p. 363 of this textbook indicates that the scale factor is defined with the distance at time t in the numerator and the distance at present in the denominator:
Consider two galaxies at some particular reference time, say at the present moment $$t_0$$. Let their separation be $$d_0$$. At another time t they are separated by a distance d(t). We define the scale factor R to be the ratio of these distances

$$R = \frac{d(t)}{d_0}$$

Ich
Re: Redshifting

Hmmm...

I really have problems understanding what they mean. The sentence
Wikipedia said:
It relates the comoving distances for an expanding universe with the distances at a reference time arbitrarily taken to be the present.

$$l_p = l_t \; a(t)$$

where $\! l_t$ is the comoving distance at epoch $\! t$, $\! l_p$ is the distance at the present epoch $\! t_p$ and $\! a(t)$ is the scale factor.
could almost be correct if they chose nonstandard definitions.
If Lt is in fact a comoving distance taken at some reference time, then a(tp)*Lt is Lp, the distance now (tp).
However, they write a(t), which is wrong in this context. Further, you'd normally define Lp as the comoving distance, and relate it with the distance at a different time t -as you said.

I'll try to find a better wording for the article, or maybe you'd like to correct it?

JesseM
Re: Redshifting

I'll try to find a better wording for the article, or maybe you'd like to correct it?
I had a go at correcting it, but feel free to modify further...

Ich
Re: Redshifting

Better, but comoving distance is something different. The scale factor relates proper distance (Lp) with comoving distance (Lc): Lp= a*Lc.
Comoving distance is proper distance now.

JesseM
Re: Redshifting

Better, but comoving distance is something different. The scale factor relates proper distance (Lp) with comoving distance (Lc): Lp= a*Lc.
Comoving distance is proper distance now.
Thanks, I edited it to distinguish between proper distance and comoving distance, and also edited the comoving distance article which discussed the proper distance but was missing that simple equation.

JesseM
Re: Redshifting

Better, but comoving distance is something different. The scale factor relates proper distance (Lp) with comoving distance (Lc): Lp= a*Lc.
Comoving distance is proper distance now.
Hmm, but p. 263 of this book seems to say proper distance is something different from the scale factor times the comoving distance:
First it is important to remember that r in (10.23) is a comoving coordinate. If an observer here on Earth is at r=0 and a distant galaxy is at r=re, then the observer remains at r=0 and the distant galaxy remains at r=re. The term r is thus better thought of as a label than as a distance. The coordinate distance, dC, is given by (10.20). If the light emitted by a galaxy with comoving radial coordinate re is observed by us at the present time t0, then the present coordinate distance to the galaxy is given by

dC(t0) = R(t0) re

where R(t0) is the present value of the scale factor. The coordinate distance to the galaxy changes because R(t) changes, not because the galaxy has a large velocity through space away from us.

What is the actual distance to the galaxy? ... In the present context it is easiest to use the proper distance. To measure the proper distance to a galaxy, imagine that there is a chain of observers between us and the galaxy. Each observer measures the distance between himself and his immediate neighbor in the direction of the galaxy at the same cosmic time t. If we then add up all these small distance elements the result is the proper distance dP to the galaxy at cosmic time t ... There are thus three expressions for the proper distance to an object, depending on the curvature of the universe:

dP = R(t) sin-1 r --- spherical
dP = R(t) r --- flat
dP = R(t) sinh-1 r --- hyperbolic

Note that the proper distance is equal to the coordinate distance only in the case of a flat (i.e. k=0) space.
edit: On the other hand, p. 11-12 of this book distinguish between the "co-moving coordinate" r of a given galaxy and the function $$\chi(r)$$ which is multiplied by the scale factor to get the proper distance (defined at the top of p. 11 as the actual ruler distance), i.e. $$d_{proper}(r, t) = R(t) \chi(r)$$, with $$\chi(r)$$ working out to equal r when k=0 (flat universe), sin-1(r) when k=1 (spherical) and sinh-1(r) when k=-1 (hyperbolic). This mirrors the previous book but I am not sure whether r or $$\chi(r)$$ would normally be defined as the "comoving distance".

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Ich
Re: Redshifting

It depends on the coordinates you choose. What you have to do is to account for space curvature, where circumference/radius != pi. You may either scale the circumference or the radius in your coordinates, but you can't have both be "proper" coordinates if space is curved.
In http://en.wikipedia.org/wiki/Friedmann–Lemaître–Robertson–Walker_metric#General_metric", you scale the radius, and that is what the book does. You have to unscale r to get proper radial distance, but you can use r*dphi directly to get tangential proper distance.
In Hyperspherical coordinates, you scale the circumference, and r measures proper radial distance.
I almost exclusively use hypersherical coordinates, so there's no ambiguity between r now and proper distance now.

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Re: Redshifting

So we can see light that is red-shifting z > 5.4 which means it would be moving faster than light speed away in relation to us. How can we see this? We couldn't see the object's light while the distance between us is increasing at more than light speed right? I realize the light we are seeing is billions of years old but having a z of more than 5.4 doesn't make sense to me. The only thing I can figure is that the space fabric itself is expanding while the light is traveling through it elongating the light more than when it started. Could anyone shed light on this. I feel really bad for making that pun.
cosmic microwave background is at z = 1100

Re: Redshifting

cosmic microwave background is at z = 1100
Yeah. But apparently light observed currently around 5.4 will reach us redshifted to infinity, so it won't. The cosmic radiation background would account for objects physically impossible to get light from anymore.

Re: Redshifting

One simple way to understand why Hubble constant is decreasing: consider galaxy 1 Mpc away. It is receding from us at 71 km/s. Now, if value of Hubble constant remains the same, once it is 2 Mpc away it should be receding at 142 km/s, and so on. Our universe is accelerating in expansion, but not all that much.
Let me see if I understand this. 1Mpc away an object would be moving away at 71 km/s due to the expansion of space. 1 billion years from now a different object 1 Mpc away would be moving away at a lower rate, <71 km/s. Does this value ever go to 0 or <0?

Re: Redshifting

Let me see if I understand this. 1Mpc away an object would be moving away at 71 km/s due to the expansion of space. 1 billion years from now a different object 1 Mpc away would be moving away at a lower rate, <71 km/s. Does this value ever go to 0 or <0?
Yes, that is correct. In empty (or near-empty) universe it would approach 0 as $$t\rightarrow\infty$$. However in acclelerated expansion model, it should approach asymptotic value around 60.