Alternative forms of the Friedmann equations

In summary, the conversation discusses different ways of representing the Friedmann equations and including dark energy in them. The preferred method is using the energy density symbol ρ_X for dark energy, while some others use rho sub Λ. The equations are simplified by using a force constant F = c^4/G and a dimensionless scale factor a(t). The equations can also be written in terms of the Hubble parameter H and the Hubble time and area. The critical density, which plays a crucial role in the evolution of spacetime, is calculated to be around 0.85 joule per cubic kilometer, with dark energy being about 70% of that. The alternative forms of the Friedmann equations are presented as being
  • #1


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What is your preferred way of remembering and writing down the Friedmann equations and in what form do you include dark energy in them?

Heres the way that seems best to me. It is a balance between making them uncluttered but not so simplified as to lose intelligibility. For one bit of notation, I refer to the article "Making Sense of the New Cosmology"
by Michael Turner (U Chicago and Fermilab). It is a well-written
recent survey article.

He uses the energy density
ρ_X to stand for the dark energy.
Some others say rho sub Λ
because its associated with the "Cosmological
Constant" but I like rho sub X because
nobody really knows what it is---it's just some
energy that behaves a certain
way, in particular the pressure is at least
roughly equal to minus the energy density
(exactly, if it turns out to be in fact the
cosmological constant.)

Anyway the density and pressure arguments in
the Friedmann equations are

ρ = ρ_m + ρ_X
p = p_m + p_X

Intuitively Fr. equations are about a_t and a_t,t ____often written a-dot and a-doubledot: the first and second time-derivatives of the scale factor a(t). This is a dimensionless factor usually adjusted to equal one at the present moment. To keep time and space commensurable I am going to work with a_ct and a_ct,ct.

It simplifies the equations a bit to use a force constant F = c^4/G.

a_ct,ct/a = -(4π/3)(ρ + 3p)/F

(a_ct/a)^2 = (8π/3)ρ/F - k/a^2

k is the sign of curvature factor and in the most interesting (flat) case is equal to zero. This makes the equations still simpler. And we can take ordinary time derivative of a if we divide the lefthand side by c^2. Doing that, in the k = 0 (flat) case results in these two Friedmann equations:

(1/c^2) a_t,t/a = -(4π/3)(ρ + 3p)/F

(1/c^2)(a_t/a)^2 = (8π/3)ρ/F

This form of the equations is intuitive to me. Pressure and energy density are dimensionally the same type quantity and dividing either one by a force gives one over an area---curvature.
Lots of times people write the equations and set basic constant in them (like c and G) equal to one. So the constants disappear and clutter is reduced. But then, with so much set equal to one, it is harder to parse---at least I find the physics harder to picture. So I wanted to reduce clutter some but not set any basic constants equal to one. This notation F (for the Planck force actually---that's what c^4/G is) provides for an uncluttered intelligible version, I think.

BTW F = 12x10^43 Newtons----very roughly 10^40 metric tonsforce. The same constant c^4/G, or its reciprocal G/c^4,
plays a central role in the Einstein equation, the main equation in GR. So it makes conceptual sense to have it stand out visibly.
Any comment?
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  • #2
solving for the critical density

Guess I'm talking to myself---no replies.

a_ct,ct/a = -(4π/3)(ρ + 3p)/F

(a_ct/a)^2 = (8π/3)ρ/F - k/a^2

we can take ordinary time derivative of a if we divide the lefthand side by c^2. In the k = 0 (flat) case, this results in these two Friedmann equations:

(1/c^2) a_t,t/a = -(4π/3)(ρ + 3p)/F

(1/c^2)(a_t/a)^2 = (8π/3)ρ/F

This second equation can be solved for the critical energy density

By its definition the Hubble parameter H = a_t/a
so the second equation which is the condition for flatness says

(1/c^2) H^2 = (8π/3)ρ/F

Solving gives

ρ_crit = (3/8π) F H^2/c^2

F, the natural unit of force, is 12E43 Newtons and
H is about one over 14 billion years (the Hubble time)
so that H^2/c^2 can be immediately read as one over
(14 billion lightyears)^2, the Hubble area.

Intuitively F H^2/c^2 is the natural unit of force
over the Hubble area---a gentle pressure. But pressure is the same dimensional type as energy density, so we have the critical density

I think in metric terms it works out to around 0.8 joule per cubic kilometer---the messy part is expressing the Hubble area in square kilometers. It is horrible---something like 1.8E46 square kilometers, and dividing 12E43 Newtons by that and multiplying by 3/8π does in fact give the right thing. But I find it actually easier to remember in terms of the natural force unit, Planck force, and the Hubble area.

Dark energy is observed to be about 0.7 of critical density and so the negative pressure it exerts, so critical to the evolution of spacetime, is easy to express as the corresponding fraction of the (Planck over Hubble) pressure here.

Hoping that any readers will find the formalism simpler and more conceptual than that usual with the Friedmann equations and rho crit.
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  • #3

...Solving gives

ρ_crit = (3/8π) F H^2/c^2

F, the natural unit of force, is 12E43 Newtons and
H is about one over 14 billion years (the Hubble time)
so that H^2/c^2 can be immediately read as one over
(14 billion lightyears)^2, the Hubble area.

Intuitively F H^2/c^2 is the natural unit of force
over the Hubble area---a gentle pressure. But pressure is the same dimensional type as energy density, so we have the critical density

I think in metric terms it works out to around 0.8 joule per cubic kilometer...

The figure of 71 for the Hubble parameter that people seem so confident about nowadays converts to Hubble time 13.77 billion year (overstating the accuracy temporarily) with Hubble distance
1.303E26 meters and Hubble area 1.7E52 square meters.

Recalculating rho_crit for a little better precision: divide Planck force unit c^4/G = 12.1E43 Newton by Hubble area 1.7E52 sq.meters and get

7.1E-9 joule/meter^3

that has to be multiplied by (3/8π) which gives
8.5E-10 joule/meter^3

The dark energy people seem confident is about 70% of that comes out 6E-10 joule/meter^3

And this is confirmed for example [Broken]

I find it easier to remember the prevailing estimate of
rho crit as "0.85 joule per cubic kilometer"

and dark energy as "0.6 joule per cubic kilometer"

when I visualize it that way I don't have powers of ten to
deal with and powers of ten are easy to forget and
6E-10 is so small it is hard to imaging but 0.6 joule is
something I can do with my hand, raise a weight a little ways,
so it is concrete.

Anyway hope somebody enjoys these alternative forms of
Friedmann equations---only very trivially different from
the various standard presentations.
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  • #4
Wow, 0.85 joules per cubic kilometre! Space is pretty empty. I actually prefer units where c=1. It reinforces the idea of time as a dimension (measured in metres, like distances), and the equivalence of mass, energy, momentum, and pressure (measured in kg). But I'm not so comfortable with G=1 or hbar=1.
  • #5
Originally posted by cragwolf
Wow, 0.85 joules per cubic kilometre! Space is pretty empty. I actually prefer units where c=1. It reinforces the idea of time as a dimension (measured in metres, like distances), and the equivalence of mass, energy, momentum, and pressure (measured in kg). But I'm not so comfortable with G=1 or hbar=1.

damgo said something about liking the immediacy of natural comparisons but extended it to hbar = 1
it was in the "dimensionless units thread"

Originally posted by damgo
My QFT text (Peskin&Schroeder) does do that! The very first sentence is "We will work in God-given units, where c=hbar=1."

The problem with not setting them to unity is that you can't eliminate the constants from all your equations then, which is an enormous convenience; less important, you can't do natural comparisons (like momentum to mass to energy) quite as easily.

whatever quantities people like to equate or compare a lot they want to be numerically related with a nice number. so for some people it would be frequency energy and temperature----and they would want boltzmann's k = hbar = 1
(I'm not telling you anything, just completing the idea.)

Yes 0.85 joule per cubic kilometer does seem empty and yet
all that is is the exact density U needs in order to be flat. Any more
dense and it would crush itself. So the vacuity of the world,
its awesome emptiness, turns out to be a fortunate circumstance (again no news, just completing the thought.)

the rest of this is my own personal musings

How you handle constants and what you set equal to one is largely a matter of taste and my personal taste leads me to want a one-letter symbol for the
c^4/G which appears as the central constant in the Einstein equation.

the main equation of GR says "curvature = (8piG/c^4) energy density"

G_mn = (8piG/c^4) T_mn

But c^4/G is recognizable as a force.

I think this force is more basic and more convenient to use than G.

G appears to say something about two masses, but gravity arises not solely from mass but from all kinds of energy concentrations.

Anytime you have an energy density and divide it by a force you get a curvature (one over the square of distance). So a force is just what is needed to relate energy density and curvature.

So I can scarcely resist rewriting the main GR equation as

G_mn = (8pi/F) T_mn

and then going on to get to know this F (a fundamental proportion in nature) and trying it out in various contexts
as a replacement for the older constant G.

It is the unit of force that arises in the Planck units system--that which gives unit acceleration to the unit (Planck) mass.
And it is 12E43 Newtons or very roughly E40 tons. If you have seen any online places where such a force is mentioned (eg as
a tension in string theory) please pass along the link!

  • #6
F_planck and A_planck

I'm looking for cases where the other Planck quantities besides c, G, and hbar play a central role as fundamental constants.

Of course any Planck quantity can be written out in terms of those three, as the area can be written Ghbar/c^3 and the force can be written c^4/G. But in some cases you get a simpler more intuitive equation if you recognize the Planck quantity involved.

the previous posts showed some cases where I thought using the Planck force F_planck helped make something more intuitive or easier to remember.

the universe's critical energy density ρ_crit for example is easy to express as a pressure in terms of the natural unit of force and the Hubble area A_hubble.
Just have to remember to put a (3/8π) term in front of the F/A.

ρ_crit = (3/8π)F_planck/A_hubble

The Einstein equation uses F_planck as its central constant----relating energy density to curvature.

G_mn = (8pi/F) T_mn

The basic constant in that equation is the force c^4/G, or its reciprocal G/c^4, and that force is the definition of F_planck.

The Friedmann equations which derive from the main GR equation also look a bit simpler in terms of the natural force constant.

I'm also interested in finding cases where A_planck = Ghbar/c^3
serves as a key constant.

Over in Theoretical Physics forum damgo and arivero came up with some pointers for me as to how to find situations where area plays a key role (as in LQG) and some names (such as Ashtekar, Smolin, Penrose, Rovelli,...) to use in keyword search.
I will bring some stuff from that thread ("...what if area is more basic than length").

The area is a nice quantity to have in connection with Planck force because if you multiply the two together you get hbar c.
Neglecting subscripts,

AF = hbar c

and F/A is the Planck energy density, or pressure. Intuitively it is all one picture, so I will sort of merge the threads and get area on hand too.

We came up with some incredible things. The area of an ordinary black hole is quantized in steps of 4 ln(3) A_planck.
There was something about this in Nature in February, I will get the URL.
Also the area of an ordinary black hole is related by the constant 4 A_planck to the hole's entropy.

Here's the Feb 2003 article in Nature by Baez about "quantizing area"

Here is some of what damgo said about area in the other thread:

[[The only thing I can think of right now along these lines is that the idea of using loops turns out to be very very powerful in topology and geometry -- homotopies, holonomies, etc -- even in high-dimensional manifolds. In many of the proofs I've seen, the exact length or dimensions of the loop is irrelevant; what's shows up is the area. Explicit example:

You can get the Riemann tensor -- contains all the curvature information of the manifold and manipulated gives you the left side of the GR field equations -- by considering the effect of parallel transporting a vector around an infinitesimal loop (its holonomy). You find something like

dV_u = area * X_r * Y_s * R_rsuv * V_v

where X and Y are unit vectors that roughly define the 'plane' the loop is in.]]

Here is some of what arivero said there on the same (area) topic:

[[Let me point out that an area appears also in any situation of symmetry breaking. While/if the unifyed coupling is dimension less, the efective broken theory gets a coupling corrected by inverse mass square. So, for instance, Fermi constant for weak interactions.]]

Another quote from damgo:
Originally posted by damgo
It's a common idea among the 'classical' quantum gravity (Penrose etc) and all the quantum geometry (aka loop quantum gravity aka nonperturbative quanutm gravity) people. cf Penrose, spin foams, Lee Smolin, Abhay Ashtekar...


I will try to get more relevant stuff gathered here later.
  • #7

Originally posted by marcus
I'm looking for cases where the other Planck quantities besides c, G, and hbar play a central role as fundamental constants.
I will try to get more relevant stuff gathered here later.

The Quantization of Area Baez Nature 03 Dreyer 02 Motl 02 Hod 98 Ashtekar Baez 98 Rovelli Smolin 95 Rovelli 98 LQG review

The last is a review article about LQG, otherwise all these are
specifically about quantization of area where A_planck = Ghbar/c^3 appears as a fundamental area constant.
In the formulas it is usually written as the square of the Planck length----something like "L_planck^2" or somesuch thing.
Anyway this area is making its way in and proving useful.
I got the urls from the article in Nature and am shoing the links
because I am impressed how much is available online about this.
You can access all these thru the Nature article.

ALPHA the fine stucture constant (approx 1/137.036...)

While we are on the subject of Planck force F and Planck area A, I should mention that alpha is simply the value of the coulomb constant in Planck units

k_coulomb = alpha FA/e^2

So if you have two charges of a 10^14 electrons each and they are 10^34 Planck length apart then to find the force you just
multiply alpha FA/e^2 by 10^28 e^2 and divide by 10^68 A
and get alpha 10^-40 F ----which is about 1/137 of a ton. That is just to see how it works, so you know that the coulomb constant really is
alpha FA/e^2
with no funny conventions. You can work thru it in standard SI units if you wish. And that is the same as
alpha hbarc/e^2

This is just to illustrate that the fundamental constants Planck force and Planck area are comfortable with a lot of different things---with the main GR equation and the Friedmann equations of cosmology, and in calculating the critical density, and with the fine structure constant alpha, and with quantum gravity and the quantization of area of black holes... I am on the lookout for places where they fit in comfortably as fundamental constants. Please point out any similar situations you happen to think of.
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  • #8
I'm looking for cases where F_planck and A_planck make sense and are convenient in stating physical laws. Here is another case---it may be OK or may be too much of a stretch.

First an example. E_sun = the mass-energy of the sun.

(remember that in the Einstein equation describing gravity it is the energy density that matters, concentrations of energy are what curve space so we go directly to the joules the sun represents rather than its kilograms of inertia)

Divide this amount of energy by the Planck force and you get
a length which is 1.5 kilometers. Call it L_sun.

Light passing within distance R of the sun is bent by an angle
which is 4L_sun/R radians.

(this is the prediction of 1916 GR that Eddington checked in 1919
by observing Pleiades during eclipse---they were displaced by the predicted angle)

A normal black hole with mass equal to the sun would have radius 2L_sun. That is, 3 kilometers.

this length you get by dividing a things rest energy by the natural force unit is sometimes called the "gravitational length". I found that phrase used in Halliday Resnick some edition. The usual formula for the length is GM/c^2.

Orbit stuff is often easy to calculate using the gravitational length. It is a useful handle on a thing's gravity.
L_jupiter = 1.4 meters.
Often it is easier to remember the L, visually or in meters, than to remember the conventional mass in (a large number of) kilograms.

So that is another instance of where the force constant plays a role.

Near the beginning of the thread I mentioned that the critical density of the universe is equal to (3/8pi) times this force over the Hubble area. That is what works out to 0.85 joules per cubic kilometer. It seems to be the average density of the universe---that is, the observations seem to suggest the universe actually is flat or nearly!

rho_crit = (3/8pi) F_planck/A_hubble = 0.85 joule/cubic km.

there seem to be a lot of places it comes up in a natural way,
not all in one specialized department, so I should probably
try to list them. more later.
  • #9
my take on Planck units

sorry if the perspective here is narrowly personal
this is my own intuition about Planck scale

you visit Baez site and you find out what Planck scale means to him as a research physicist----scale where you have to put GR and QM together, where the Compton and Schwarzschild lengths coincide.

or someone else says the Planck units are just meaningless algebraic combinations of G, hbar, and c. They are the units in terms of which those constants have values equal to one.

someone else says laughingly that they are the "god-given" units,
or that they are good for studying the really fundamental structures in nature
this is all right but what I want is something that is intuitive and perceptual, accessible to a non-physicist
one problem is that G and hbar are not perceptual quantities
the primitive physical quantities you perceive are ones such as length, force, speed, acceleration
(the fact that you can catch a ball probably means the eye can judge acceleration as well as speed---it certainly can judge speed)

inertia (mass) is a secondary construct based on force and acceleration----mechanical energy and momentum are even higher level abstract quantities, built up from the directly perceptual ones.
(although admittedly blood sugar level is sensed directly,
blood oxygen level too, there are nerves specialized to
sense energy reserves and deficits in the body, temperature too)

Anyway G as a ratio of gravitation to inertia or however you like to think of it is fairly abstract----more so than length or force. And hbar as action, or angular momentum, or a ratio of energy to frequency, is too.

But the units can be constructed from speed c, force F and length L (for instance) just as well as they can from c, G, hbar.
Algebraically there is no *a priori* reason to prefer one set of generators to another.

So I want to see what happens if one tries to assimilate some really perceptual Planck units like c, F, L
(on which the whole set can be based as easily as it can on c, G, hbar)
and if one says things like:

Well c is all around you---walk out the door and most of the energy and info in your world is whizzing around at about that speed and coming from the sun at that speed etc. So it is pervasive.

And F? That is the main constant in the GR model of gravity
G_mn = (8π/F)T_mn
curvature = (8π/F) energy density
It has to be a force because curvature is one over area-----a force is the only thing you can divide energydensity by that gives curvature. So the fact that you are standing on the ground feeling your weight in the soles of your feet is F. And the fact that the planet youre on is circling the sun you see up there is F.
Well it is 12E43 Newtons and that is just the right force to make it all work so let's not be too astonished by the size...

And L? Well for example FL^2 is a combination of energy and length which is in every bit of light.
The quantum energy of a bit of light multiplied by its vacuum angular wavelength is always the same for every photon in the world. It's just the rule of how light energy is packaged. So this quantity FL^2-----the Planck force combined with Planck area----is all around us repeated over and over in countless identical cases. So its fairly basic----supplies the energy in the food we eat etc.
the calories in the peanut butter and jelly was FL^2, and the bread too (Mr Robin Parsons reported that his bicycle was broken
so that he was having to subsist on peanut butter and jelly, so this is an FL^2 case in point)

unfortunately I just got a call and have to go ASAP so cannot proofread. The discussion of L, or L^2 Planck area doubtless needs improvement, but I'll just post as is. You see what I am driving at. I want the basis of Planck units to be very concrete without depending too much on high physical theory and to be related to the sorts of quantities one perceives directly. Hope this does not seem too idiosyncratic---it might be useful to others besides myself.

of course F, Planck force, is c^4/G
and L^2 or A the area is G hbar/c^3
in terms of the conventional c G hbar that they are usually based on.
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