# Lightcone 1.0 basic redshift article development

1. May 6, 2013

### Mordred

Developing a basic explanatory manual for the Light cone 1.0 calculator. This is as a supplement to give a basic understanding on what the terms used in the calculator mean. The user manual is separate as is the advanced manual which shows the math forms used in the calculator.

The CMB, (Cosmic Microwave Background) The CMB is thermal radiation filling the Observable universe almost uniformly. The CMB provides an excellent reference point in distance measurements and corresponds to a stretch of 1090

Doppler shift and redshift are the same phenomenon in general relativity. However you will often see Doppler factored into components with different names used. 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.

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

However the only form of redshift we need to concern ourselves with is called the cosmological redshift.

The Cosmological Redshift is a redshift attributed to the expansion of space. The expansion causes a Recession Velocity for galaxies (on average) that is proportional to DISTANCE.
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 (red) shift proportional to distance. 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. 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
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 blue shift 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.

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)do

where d(t) is the proper distance at epoch (t)
d0 is the distance at the reference time (to)
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.

Luminosity: absolute luminosity is the amount of energy emitted per second.
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.

Last edited: May 6, 2013
2. May 6, 2013

3. May 1, 2015

### virgil1612

Hello,

I just started using the calculator, and I have some problems understanding all the columns. For example now I'm trying to understand the meaning of those 3 velocities. Could you provide some explanation for that?

Thanks, Virgil.

4. May 1, 2015

### marcus

Hi Virgil, Mordy is not around at the moment so I'll reply at least partially. Before answering just want to remind you distance expansion is not like ordinary motion of things thru space. Nobody gets anywhere by it. Nobody travels in any definable direction approaching some destination---everybody just becomes farther apart. So a RECESSION SPEED is not the speed some galaxy is moving.
It is the speed that the distance between it and our galaxy is growing in size.

Of the three speeds in Lightcone, two are just recession speeds now and at the time of emission, vnow and vthen

If we are receiving some light now, as we speak, then the distance to the source galaxy was increasing (at time it emitted the light) at speed vthen and the distance to that galaxy IS increasing NOW at speed vnow

That doesn't tell you all the different recession speeds that distance had in between times while the light was traveling!

The stretch that occurred (of wavelengths and distances) while the light was in transit is the cumulative effect of the whole history of expansion speeds.

So it's interesting to fix attention on one sample distance and watch it change size and plot its whole history over a long span of time.

That is what the other thing does. You pick a particular distance, just for definiteness----one that is 14.4 Gly at the present moment. It grows over time in proportion as the SCALE FACTOR grows, a(t). But a(t) is normalized to equal 1 at present so it is just a number, not any particular distance measured in Gly. So we multiply a(t) by R0 = 14.4 Gly. to get the history of a particular distance which happens to be 14.4 Gly at the present moment.

The prime on a' denotes the slope of the curve a(t). a'(t)R0 is the speed that sample distance is increasing at time t.

So if you tell Lightcone to do that, it will give you the whole history of expansion speeds of that one sample distance over a long span of time

5. May 1, 2015

### marcus

Here's an example. I checked the "linear steps" box to get regular sized steps of S down from S=3 to S=1 (present day) and selected the number of decimal places I wanted it to show in the various columns selected. You can see the sample distance growth speed at first slows down.
To some time between 7.3 and 7.9 billion years.
And then it starts to pick up.
$${\scriptsize\begin{array}{|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|r|} \hline a=1/S&S&T (Gy)&a'R_{0} (c) \\ \hline 0.333&3.000&3.3&0.9995\\ \hline 0.345&2.900&3.5&0.9867\\ \hline 0.357&2.800&3.6&0.9741\\ \hline 0.370&2.700&3.8&0.9617\\ \hline 0.385&2.600&4.0&0.9495\\ \hline 0.400&2.500&4.3&0.9376\\ \hline 0.417&2.400&4.5&0.9262\\ \hline 0.435&2.300&4.8&0.9153\\ \hline 0.455&2.200&5.1&0.9051\\ \hline 0.476&2.100&5.5&0.8957\\ \hline 0.500&2.000&5.9&0.8875\\ \hline 0.526&1.900&6.3&0.8807\\ \hline 0.556&1.800&6.8&0.8757\\ \hline 0.588&1.700&7.3&0.8729\\ \hline 0.625&1.600&7.9&0.8730\\ \hline 0.667&1.500&8.6&0.8768\\ \hline 0.714&1.400&9.4&0.8852\\ \hline 0.769&1.300&10.3&0.8996\\ \hline 0.833&1.200&11.3&0.9218\\ \hline 0.909&1.100&12.4&0.9542\\ \hline 1.000&1.000&13.8&1.0000\\ \hline \end{array}}$$
actually the time of slowest growth speed comes right around year 7.6 billion
so about halfway between the 7.3 and the 7.9
IN CALCULUS IT IS CALLED AN INFLECTION POINT where a curve stops being convex and becomes concave (looking down on it)
the slope stops decreasing and starts getting steeper.
We can get Lightcone to plot the curve of the scale factor curve a(t) (which is proportional to the growing size of the distance so has the same shape). It will have an inflection point about halfway between 5 Gly and 10 Gly---that is about where 7.6 is.

Last edited: May 1, 2015
6. May 1, 2015

### marcus

Another version of essentially the same scale factor a(t) curve---just look at the right half which is the expanding part, not the contracting mirror image.

this is where you use a different time scale on the x-axis. But the same y-axis. the scale is you use a time unit of 17.3 billion years. So the present (13.8 billion) comes out to be 0.8
and the scale factor is normalized to equal 1 at present, so you can see a(now) = a(0.8) = 1
The inflection point comes about halfway between 0.4 and 0.5 on the x axis. that is where year 7.6 billion is, on this time scale.
the curve is basically a hyperbolic trig function that happens to match the way the universe grows, the history of how distances have increased.

Last edited: May 1, 2015
7. May 1, 2015

### virgil1612

Thank you Marcus. The thing I don't get is that even at their minimum value, those velocities are very close to c. How would I reproduce the case of a Galaxy in the Coma Cluster for example. For a distance of roughly 300 Mpc they would produce a recession velocity of under 30000 km/s and I haven't been able to simulate that in the calculator.

8. May 1, 2015

### marcus

All the growth history curves, for other distances, look the same as the one in the figure. they are just scaled proportionally.
The curve which Lightcone can make is for the distance which equals 14.4 Gly at the present (S=1)

If you want the distance history curve for a distance that is 1/10 that size, like 1.44 Gly at the present, you just have to divide all the distances in the sample by 10. and divide all the SPEEDS by 10.

when you scale a curve down, it scales the slopes down in the same proportion.

9. May 1, 2015

### virgil1612

Decreasing Supper isn't equivalent to having my photons coming from a closer source?

10. May 1, 2015

### marcus

the sample "history" column is different from all the other columns. so it is potentially confusing.
I hesitate to say anything because you may be getting confused. I don't understand your question. Maybe the right thing to do is to ignore the
a'R0 history column and get a really solid understanding of how the rest works
It makes tables
Supper and Slower establish the limits of the table. You control the number of rows.
Each row represents a different batch of light, or photons that arrive today. Each row they are coming from a different time T, when the scale factor a is different and experiencing a different stretch S. Each row they come from a different distance (shown as it was then and as it is now)

Along the Dthen curve they can be thought of as all following that curve, because passing thru at that distance is subject to the same conditions as being emitted right then, as the older one is passing thru.

But logically all the rows are separate .. Separate flashes of light from different times, distances, all of which arrive at the present day.

the a'R0 column (and curve if you plot it) is different.
it is a way of showing THE SLOPE OF THE a(t) curve which is the size of distances. It is about the history of one sample distance. So it is about the geometry, not about photons.

The multiplying by the fixed distance R0 is basically to scale a and its slope a' up enough so they register. a(t) is less than ONE for all times before the present. I think its slope would not be visible on typical plots that Lightcone makes. Not sure if there was anything else Jorrie could have done to show the history of the size of the universe (or rather the size of a representative sample distance)

Last edited: May 1, 2015
11. May 1, 2015

### virgil1612

Yeah, I guess I should think more about this. My thinking about Supper was that when I decrease it I'm decreasing the streching so I'm looking back at a more recent point in time so I'm looking to a point situated at a smaller distance.
I'll keep thinking about it, hopefully I will eventually understand.

Virgil.

12. May 2, 2015

### Jorrie

The Coma cluster is at about 100 Mpc (~320 Mly), redshift around 0.023, which is no problem to handle in Lightcone 7, but you obviously have to use it in "one-shot" mode, because you have only one 'stretch' S to work with.

Enter S_upper = 1.023 and S_step = 0 for a one-shot calculation. Leave S_lower at default, because Lightcone does not like S_lower to equal S_upper. You should see only one row output with, amongst other things, the cosmic time T when the light from Coma we observe now was emitted, how far Coma is 'now', how far it was 'then' and also the recession speeds of Coma 'now' and 'then'.

13. May 2, 2015

### virgil1612

Sorry, I've written Mpc instead of Mly. Thank you, I will give it a try.

Virgil.

14. May 5, 2015

### Mordred

I'm back was extremely busy, I see this post has been answered, you've always explained the calculate best Marcus.

15. May 5, 2015

### Mordred

Play around with setting the range of search and stretch range along with number of steps. You can really hone into specific moments that way