The LCDM Cosmological Model in Simplified Math

[Jorrie]

If we restrict ourselves to a homogeneous, spatially flat LCDM universe model, the Friedman equations can be written a very simple form. Marcus posted several threads in a collaborative effort to develop the simplified approach. Here is a summary of the parts that seem the most informative, to me at least.

Part 1. The Hubble constant

Arguably the most important equation of the model is the evolution of the expansion rate over cosmological time. In other words, how the Hubble constant H has changed over time. If one knows this function, most of the other LCDM equations can be derived from it, because it fixes the expansion dynamics.

(1.1) $\large H^2 - \frac{\Lambda c^2}{3} = \frac{8\pi G}{3 c^2} \rho$

Here H is the fractional expansion rate at time t, Λ is Einstein's cosmological constant, G is Newton's gravitational constant and ρ is the changing concentration of matter and radiation (at time t) expressed as a mass density. This density includes dark matter, but no 'dark energy', because Λ appears as a curvature on the left side of the equation.

As you can check, the right-hand side gives SI units of 1/s², also the units of H², as it is the square of an instantaneous fractional growth rate. Since Λ represents a constant curvature, its SI units would be reciprocal area 1/m² and multiplying by c² again gives a 1/s² quantity. Hence both sides' units agree.

It is convenient to replace Λc²/3 with the square of a constant growth rate H_∞², representing the square of the Hubble constant of the 'infinite future', when cosmic expansion will effectively have reduced matter density to zero.

(1.2) $\large H^2 - H_\infty^2 = \frac{8\pi G}{3 c^2} \rho$

Since we can measure the present value of H, labeled H₀ (H-naught) and also how it has changed over time, it allows us to use Einstein's GR and his cosmological constant to determine the value of H_∞. If we assume that present radiation energy is negligible compared to other forms (as is supported by observational evidence), then we can express eq. (1.2) as:

(1.3) $\large H^2 - H_\infty^2 = (H_0^2 - H_\infty^2) S^3$

H₀ is the present observed rate of expansion per unit distance, which tells us that all large scale distances are presently increasing by 1/144 % per million years. This gives us a Hubble radius of 14.4 billion light years (Gly). S is the 'stretch factor' by which wavelengths of all radiation from galaxies have increased since they were emitted.^[1]

The fractional distance growth rate H is declining and leveling out at the constant value H_∞. The point of this equation is to understand how it is changing over time and how this effects the expansion history. But let's imagine that H remains constant (as it will in the distant future). Then, as you can check, the size a(t) of a generic distance would increase as e^Ht. The time interval 1/H would then be a natural time-scale of the expansion process. In that length of time distances would increase by a factor of e = 2.718. For constant H, the time 1/H can be called the "e-fold time", by analogy with "doubling time". An e-fold is like a doubling except by a factor of 2.718 instead of 2.

In the long run the universe's expansion process will be exponential at nearly the constant rate H_∞, so eventually all large scale distances will undergo an e-fold expansion every 17.3 Gy. Or stated differently, all distances will eventually grow at about H_∞ = 1/173 % per million years.

The 17.3 Gy 'e-fold time' is a natural time scale set by Einstein's cosmological constant. An informal study by a group of Physics-Forums contributors suggested that the 17.3 Gy time-span could be a natural timescale for the universe. For lack of an 'official name' for it, the group called it a 'zeit'. The longterm expansion rate, the reciprocal of the e-fold time, is therefore H_∞ = 1 per zeit.

Here is a graph of the normalized H changing over time, expressed in zeits.

The blue dot represents our present time, 0.8 zeit and a Hubble constant of 1.2 zeit^-1. The long term value of H approaches 1.
One light-zeit is 17.3 Gly in conventional terms. If the current rate H₀ were to continue unchanged, distances would expand by a factor 'e' every 14.4/17.3 = 0.832 zeit. H₀ = 17.3/14.4 =1.201 per zeit.^[2] Our present time is 13.8/17.3 ~ 0.8 zeit.

We can easily normalize equation (1.3) to the new (zeit) scale by dividing through by H_∞ (which then obviously equals 1).

(1.4) $\large H^2 - 1 = (1.2012 -1) S^3 = 0.443 S^3$

or

(1.5) $\large H^2 = 0.443 S^3 + 1$

This remarkably simple equation forms the basis of a surprisingly large number of modern cosmological calculations (at least for flat space), as will be discussed in Part 2.

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End-notes:

[1] 'Stretch factor' S = 1/a, where a is the scale factor, as used in the LightCone calculator. S is also simply related to cosmological redshift z by S=z+1; S is the direct factor by which wavelengths have been increased, while z is a fractional change in wavelengths.

[2] The traditional unit of the Hubble constant as used by Edwin Hubble is kilometers per second per Megaparsec. From an educational p.o.v. it was an unfortunate choice, because it seems to indicate a recession speed, while it is really a fractional rate of increase of distance. It is a distance divided by a distance, all divided by time. So its natural unit is 1/time, or simply time^-1.

 

[Jorrie]

Part 2: Cosmological Time and Distance

We have seen how one of the simplest ways of expressing the fractional expansion rate of the universe, i.e. the Hubble parameter H, is this equation (1.5 from Part 1):

$H = \sqrt{0.443 S^3+1}$ zeit^-1

where S is the inverse of the expansion (or scale) factor a, i.e. S=1/a, here labeled the "stretch factor". By convention, the present stretch is S = 1 and hence the present H = √(1.443) ~ 1.2 zeit^-1. Recall that 1 zeit is 17.3 Gy, the long term time scale of the universe. Presently, our time is at about 0.8 zeit (13.8 Gy). If light from some distant galaxy that we observe today has twice the emitted wavelength, we have S = 2 and H = √(1 + 0.443 x 2³) = 2.13 zeit^-1.

One zeit can be called the natural e-fold time of the universe, in other words the time that it will take cosmological distances to increase by a factor e = 2.718 (in the long term, when H remains constant). The reason why this factor is not 2 (i.e. a doubling time) is that the cosmic expansion is not linear over time, but is accelerating.

This brings us to the relationship between H and cosmological time, which is given by this rather simple equation:

(2.1) $t = \frac{1}{3}\ln(\frac{H+1}{H-1})$ zeit.

Let us put our present H=1.2 zeit^-1 in here and check it with a calculator:
$t = \frac{1}{3}\ln(\frac{2.2}{0.2})\approx 0.8$ zeit or ~13.8 Gy, which is what we expect.

That galaxy at S=2 with H = 2.13 zeit^-1 emitted the light that we observe today at
$t = \frac{1}{3}\ln(\frac{3.131}{1.131})\approx 0.34$ zeit, or ~ 5.9 Gy after the start of the present expansion._[1] The light from that galaxy obviously took 0.8 - 0.34 = 0.46 zeit (~8 Gy) to reach us - it is the so-called 'lookback time'.

Now that we have a feeling for the relationships, what is better than viewing it on graph paper? Here you see all three parameters mentioned so far against time.
https://www.physicsforums.com/attachments/h-vs-t-in-zeit-png.85158/
You can see our present time (0.8 zeit) clearly where the red and blue curves cross, i.e. where both S and a are one. You can also see how H starts high and asymptotically approaches unity. Time=2 zeit is far into the future - it translates to (34.6-13.8) = 20.8 Gy from now. The graphs were plotted using LightCone7zeit, a variant of LightCone 7, which works in the natural time units (zeit).

I have sneaked in the curve of the scale factor (a), but did not give a direct relationship of it against time yet. You get it by substituting S=1/a into eq. 4.1 and then solve together with eq. 4.2, which is not a simple exercise at all. Fortunately, mathematics comes to the rescue and we have a nifty solution for a(t), i.e. the scale factor as a function of time:

(2.2) $a(t) = \frac{\sinh^{2/3}(\frac{3}{2}t)}{1.3}$

where sinh is the hyperbolic sine (or hypersine for short) function. The hypersine is obviously related to the natural logarithm ln(x) and the natural exponent (e^x). The factor 1.3 is just scaling to make a(t_now) = 1 at present (t = 0.8 zeit), as is the convention. The importance of a(t) is that its curve shows how a decelerating expansion gradually changed over to an accelerating expansion. The inflection point is (visually judged to be) between 0.4 and 0.5 zeit, around a(t) = 0.6.

The scale factor a(t) = 1/S is the ratio of distances today compared to some time in the past (or in the future, for that matter). This said, how do we find the proper distance of a galaxy (in light-zeit or lzeit for short) if we know its redshift z (or its stretch S)? Because of the non-linear expansion, there is no direct analytical solution for the distance (that I know of). We need to numerically integrate in small steps, but fortunately the definite integral is rather neat:

(2.3) $D_{now} = \int_1^S{\frac{dS}{H}} = \int_1^S{\frac{ds}{\sqrt{0.443S^3+1}}}$

We have substituted H from eq. 1.5 in here, so that we have a distance in terms of the observable S = 1/a = z+1. You can use your favorite integrator, but a rather cool web based one is available at

http://www.numberempire.com/definiteintegralcalculator.php.

For our previous sample galaxy at S=2 (a(t) = 0.5), we have S = 1/a(t) = 1.667. I entered 1/sqrt(0.443x^3+1) into the Function box, 1 into "From", 2 into "To" and then clicked "Compute" - it returned the answer 0.64 lzeit (about 11 Gly). So we receive the light when the proper distance to the galaxy is D_now = 0.64 lzeit, but when the light was emitted, the proper distance was only half of that, i.e. D_then = 0.32 lzeit (5.5 Gly).

Contrast the last two figures with the 0.46 zeit (8 Gy) lookback time that we calculated above. It is clear that we cannot just multiply lookback time by c to get either of the proper distances. In fact if we do that, we get a quite meaningless distance. Here is a graph of D_now and D_then.

https://www.physicsforums.com/attachments/dnow-dthen-vs-t-png.85296/
The red D_then curve is also known as the cosmological lightcone (which is where the LightCone calculator gets its name from). To the left of 0.8 zeit it represents our past lightcone and to the right of 0.8 zeit our future lightcone. Near Time = 0 the universe was very dense and objects we now observe were actually very close to our space locality. Distances then grew much faster than the progress that photons could make in our direction. At around T=0.23 zeit, the fractional expansion rate dropped enough to allow photons to make progress and eventually reach our telescopes.

To summarize, we have now looked at a further three rather easy equations that allow us to calculate the most common values for the standard cosmological model.

Part 3 will deal with the Hubble radius and some horizons (of the cosmological and particle varieties).

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Endnotes

[1] To verify that the approximations are valid, I used the Lightcone calculator (any one of the two in my sig.) for a 'one-shot' calculation with S_upper=2 and S_step=0

 

[Jorrie]

Part 3: Important Cosmological Distances

A question that often comes up is: "how big is the observable universe?"

The question can have more than one answer, depending on the context, so cosmologists have given it a technical name and a precise definition. It is called the 'particle horizon', here indicated by D_par, essentially with the following definition: it is the proper distance that a massless particle, emitted at cosmological time zero, could have bridged until the time it is observed, as judged by the observer.^[1] We cannot presently observe anything farther than that, but as time goes on, we will be able to "see farther" (but not necessarily "see more").

We have seen the integral for determining the proper distances to a source with stretch S, as seen by us today (eq. 2.3 of section 2): $D_{now} = \int_1^S{\frac{dS}{H}}$.

The only difference between the equation for the particle horizon and proper distance today is in the limits of the integral - we have to integrate from today (S=1) all the way to S approaching infinity (i.e. time approaching zero).

3.1 $D_{par} = \int_1^{\infty}{\frac{dS}{H}}$.

Putting the equation: 1/sqrt(0.443*x^3+1) and values from: '1' to: 'inf' into http://into%20http://www.numberempire.com/definiteintegralcalculator.php gives the value D_par = 2.73 lzeit (47.15 Gly). The correct value from LighCone7z is 2.67 lzeit. The reason for the difference is that we necessarily integrate all the way into the radiation dominated era, for which the simplified equations do not cater. The LightCone7 calculators do cater for radiation energy. However, it is still remarkably close to the correct value for such a simple equation.

In order to get an idea how the particle horizon have evolved over the history of our universe, we divide by the stretch factor S before the integral and then integrate from S at time 't' to S approaching infinity.

3.2 $D_{par} = \frac{1}{S}\int_S^{\infty}{\frac{dS}{H}}$

as shown in the red curve below:
https://www.physicsforums.com/attachments/d_hor-d_par-png.85644/

The blue curve (D_Hor) is another form of horizon, the cosmological communications horizon, which is about how distant an observer can be and still receive a signal that we transmit today. Contrast this to D_par, which is about information reaching us from the past. As you can see, D_Hor flattens out at one lzeit and today, t=0.8 zeit, we have a communications horizon of about 0.95 lzeit, or about 16.5 Gly.

The equation is essentially the same as for D_par, but we have to work from today, S=1, to the infinite future, where S = 0 (since 'a goes to infinity and S=1/a).

3.3 $D_{Hor} = \frac{1}{S}\int_0^{S}{\frac{dS}{H}}$

So, we can receive signals from sources that presently are less than 46 Gly from us, but we can only communicate with receivers that are less than some 16 Gly from us. The reason is that the expansion decelerated in the past due to the radiation and then matter domination, but is presently accelerating due to the cosmological constant domination.

The Hubble Radius

Finally, we need to look at a very important radius, the Hubble radius. It is the distance at which the recession rate of an object equals the speed of light. In our natural units, it is simply the inverse of the Hubble constant H at the time. Since H is changing over time, so does the Hubble radius R. I prefer to just use H and R for these parameters, but more formally one should use H(t) and R(t). H(t) is also called the 'time variable Hubble value', but note: not the 'Hubble parameter', which has a different meaning in cosmology.^[2]

Since they are the inverse of each other, graphically, they look like below, shown together with the lightcone (D_then) curve of part 2.

https://www.physicsforums.com/attachments/h-r_h-d_then-png.85319/

Since R is a distance in lzeit, the maximum of the red curve happens exactly where it crosses the blue Hubble radius (R) curve, because this is where the proper recession rate equaled the local speed of light. Since R was increasing rather sharply at that time, in-bound photons immediately thereafter found themselves inside our 'Hubble sphere' and started to make some headway towards us.^[3]

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End-notes:

[1] Like many other distances, the value of D_par is model (and model parameter) dependent and cannot be directly measured, but is calculated from other observed parameters, using the model and parameters that fit observational data best.
[2] The Hubble parameter 'h' is a dimensionless ratio used in some cosmological equations, expressing the value H₀ as a fraction of 100 km/s/Mpc. The present 'best' value is h = 0.68, so that (e.g.) the present cosmological time can be expressed as 20.3h Gy. As the value of H₀ is refined with better and better measurements, numerical values in some papers do not have to change - the new value for h compensates for that.
[3] Observers at different places in the universe all have their individual Hubble spheres, which my or may not overlap.

 

[marcus]

Nice work! Also I like the use of Lightcone7zeit "preview" mode to produce a low-profile graph that fits compactly into text---interrupts the text less.

 

[Chronos]

Does this include an integration constant?

 

[Jorrie]

If you are asking about the definite integrals for the various distances, then the answer is no, because the limits are set.

 

[marcus]

It would be nice if the three part essay could be uploaded to PF Insight and then this thread could serve as the comment and discussion thread. I noticed that works well in the case of Lisi's Insight piece. The discussion thread becomes more casual and open because the work itself is safely stashed elsewhere.

I noticed the obvious fact that 0.44 is a universal number. It does not depend on what the present is. And you can change 17.3 billion years all over the place. that just changes the zeit timescale. 0.44 zeit *whatever the zeit happens to be* is the time in history when acceleration kicks in.

Humble enough fact. And it is slightly smudged by the small effects of varying the radiation fraction. But nice that up to small (fraction of percent) blurring the 0.44 is a universal. Or am I missing something?

This is a really neat piece of work, Jorrie. I think people can learn from it.

 

[Chronos]

Jorrie, thnanks for pointing out the obvious brain vapors I suffered there.