# What are the important numbers in Cosmology?

#### ChrisVer

Gold Member
a good review on cosmological parameters:
http://arxiv.org/pdf/1401.1389v1.pdf

the 2nd question is vague... it totally depends on which parameter you want to change and there is no answer like "it will change that"...

#### ecastro

Thank you for the article. What I meant in the second question is what will happen to the universe, for example, if the speed of light is slower or faster.

#### Chronos

Science Advisor
Gold Member
For starters, a change in the speed of light would directly affect the particle horizon of the universe [i.e., size of the observable universe]. This, however, is just the tip of the iceberg. The speed of light factor into numerous different physical properties - from quantum physics to cosmology. It affects most aspects of just about everything in the universe - even the basic energy output, life expectancy, and fate of stars.

#### mfb

Mentor
The speed of light is not a fundamental constant. You could not distinguish our universe from one where the speed of light has a different numerical value (in some unit system) but all processes are scaled in time accordingly.
Fundamental constants are always dimensionless (or directly related to dimensionless quantities). The Lambda CDM model has at least 6 but it depends a bit on the way you count them.

#### Chronos

Science Advisor
Gold Member
NIST uses the term fundamental constant for any universal physical quantity believed constant, like the speed of light or gravitational constant. I'm not aware of anything that suggests self tuning between dimensional constants of nature. For example, the fine structure constant, which is dimensionless, is defined by $$\alpha = \frac{\kappa_e e^2}{\hbar c}$$. Given the elementary charge, e, like c, is a dimensional constant, does a change in the value of c force a change in e, or merely alter the value of the fine structure constant? Offhand, I can't think of any dimensionless constant that is derived entirely independent of any dimensional constant.

#### PeterDonis

Mentor
I can't think of any dimensionless constant that is derived entirely independent of any dimensional constant.
You're looking at it backwards. Any dimensional constant depends on two things: (1) a dimensionless constant, and (2) a choice of units.

For example, if we say $c$ has the value $299,792,458 m/s$, we have not just implied a value for the fine structure constant (the dimensionless constant in question). We have also chosen units for distance and time, namely meters and seconds. We can change the value of $c$ without changing the fine structure constant at all, just by picking different units (say feet and seconds, or furlongs and fortnights, or whatever).

But if the fine structure constant is not changed, then no experimental results are changed: the spectral lines of atoms, for example, are exactly the same as they were before. Conversely, we could change the fine structure constant, but keep the numerical value of $c$ the same, at $299,792,458$, by adjusting our units; and yet all the experimental results (spectral lines, etc.) would be changed, even though $c$ is numerically the same.

So it's the dimensionless constants that have physical meaning; they are what determine the experimental results. The dimensional constants are just conveniences; they are artifacts of particular systems of units that humans have developed.

• binbots, mfb and phinds

#### Chronos

Science Advisor
Gold Member
I will agree when I see a derivation of the fine structure constant from first principles.

#### PeterDonis

Mentor
I will agree when I see a derivation of the fine structure constant from first principles.
I agree it would be nice to have one, but I don't think you need one to know that the dimensionless constants are the ones that determine experimental results.

Look at it this way: we can always choose a system of units such that we can eliminate all dimensional constants from the equations. But we can't make any manipulation that will eliminate the dimensionless constants from the equations.

Or look at it this way: consider the set of all possible changes to physical constants. Some of these changes will change both one or more dimensionless constants and one or more dimensional constants. (For example, a change in $c$, with all other dimensional constants staying the same, will also change the fine structure constant.) Other changes will change one or more dimensional constants, but leave all the dimensionless constants the same. (We call these kinds of changes "changes in units".) But there are no changes that change one or more dimensionless constants, but leave all the dimensional constants the same. (There can't be, because every dimensionless constant has an expression in terms of dimensional constants.)

Now, any change in experimental results must correspond to a change in one or more physical constants. And we know that changes in units (the second kind of change above, that changes only dimensional constants but leaves all dimensionless ones the same) do not change any experimental results. So any change in physical constants will change experimental results if and only if it changes one or more dimensionless constants.

#### Garth

Science Advisor
Gold Member
I will agree when I see a derivation of the fine structure constant from first principles.
The way I see it, the fine structure constant is one of the "first principles".

Another one, 'in my book', is c. The concept of space-time requires there to be a relationship between the measurement of time dimensions and space. We call this c: the 'rate of exchange' that converts time units into length units and vice versa.

The value you give to c simply tells you the ratio between the definition of the units of time and length that you are choosing to use.

Garth

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#### mfb

Mentor
The important ascept of the dimensionless constants is their independence of a unit system.
Take the proton to electron mass ratio, for example - not really fundamental, but it is still a dimensionless constant. If you change this, physics does change because you change the size of the hyperfine structure splitting relative to the other transitions, for example. And no matter how you measure masses, within our universe you will always get the number 1836.15267 (and every sufficiently advanced species in the universe will recognize that number if transmitted in a recognizable way).
If you change the length of a meter, the value of many dimensional constants (like the speed of light) changes, but all dimensionless constants stay the same.

The fine-structure constant can be expressed in terms of ratios of transition frequencies in the hydrogen atom, for example, or as function of the (also dimensionless) g-factor of electrons and muons - and its value is independent of the length of a "meter" and other arbitrary definitions.

#### PeterDonis

Mentor
The concept of space-time requires there to be a relationship between the measurement of time dimensions and space.
I would say that the concept of spacetime means that, fundamentally, the time and space dimensions have the same units. But we humans have historically used different units for them, and in those weird systems of units, once we discover spacetime and its fundamental nature, we end up having to add a conversion factor, $c$. That conversion factor is an artifact of our weird units, not a fundamental property of spacetime.

Similarly, in quantum field theory, fundamentally, spacetime and energy-momentum have units that are the inverse of each other (because they're canonically conjugate). But we humans have historically used units that don't have this property, so once we discover QFT, we have to add this weird conversion factor, $\hbar$, to our units. That's an artifact of our weird units, not a fundamental property of quantum fields.

#### Garth

Science Advisor
Gold Member
I absolutely agree Peter; within space-time c is dimensionless, as with ℏ.

Garth

#### ecastro

Thank you for the discussion. If I got it right, are these constants quantities that have the same effect everywhere (no matter how it is represented numerically)?
Next, is there such an equation (at least in our terms) that would tell the consequences if these "constants" are not the same? Although I used the speed of light as an example, I need to focus on the Hubble's constant, Lambda, and Omega.

#### Chronos

Science Advisor
Gold Member
So, I perceive the consensus view is c as a 'fundamental' constant?

#### Garth

Science Advisor
Gold Member
So, I perceive the consensus view is c as a 'fundamental' constant?
You have to start somewhere and define constants by which the others are to be measured.

If c is a property of spacetime - as I would understand it to be - then it is defined to be a 'fundamental' constant.

With this convention, if the speed of light in vacuo was measured to vary then that would be a measurement of the variation in the ratio between the standard of length and the standard of time.

In GR, c is constant, and the conservation of energy-momentum predicts that with free-falling steel rulers and atomic clocks, L (atomic radii) and T (atomic frequencies) and M (atomic masses) are all constant. A measurement of a variation in c would refute the theory.

Garth

#### yogi

I will agree when I see a derivation of the fine structure constant from first principles.
Since you asked, here is a curious perspective:

Alpha is dimensionless because it Is the ratio of two angular momentums - mrc/(h/4pi) where m is the electron mass, and r is the radius obtained by building the electron from its mass energy (9.1 x 10^-31 kgm) to create a shell of radius (1.41 Fermi, i.e., half the classical electron radius), and c is the velocity of light.

[(9.12 x 10^-31)(3 x 10^8)(1.41 x 10^-15)]/[5.27 x 10^-35] = 0.0073 = 1/137

#### mfb

Mentor
Thank you for the discussion. If I got it right, are these constants quantities that have the same effect everywhere (no matter how it is represented numerically)?
Which constants? The dimensionless ones or those with dimensions?
Next, is there such an equation (at least in our terms) that would tell the consequences if these "constants" are not the same? Although I used the speed of light as an example, I need to focus on the Hubble's constant, Lambda, and Omega.
Our world would look differently, but it is hard enough to model the world with our constants, so models with other constants probably need more work.
So, I perceive the consensus view is c as a 'fundamental' constant?
I see consensus for the opposite - unless you count "1" as fundamental constant.

#### PeterDonis

Mentor
unless you count "1" as fundamental constant.
Well, it is a dimensionless number that's the same everywhere. #### mfb

Mentor
I found a new constant:
m = e m for objects with mass m and the new constant e=1. As we can see, this constant has the same value everywhere in the universe, and therefore should be considered a fundamental constant of nature.

#### mfb

Mentor
Is the speed of light derived, or is the vacuum permittivity derived?
$\epsilon_0 = \frac{1}{c^2 \mu_0}$
Or the vacuum permeability?
$\mu_0 = \frac{1}{c^2 \epsilon_0}$

And what about unit systems where they are all 1?

• Tanelorn

#### Vanadium 50

Staff Emeritus
Science Advisor
Education Advisor
There is no answer to this question. One doesn't say permittivity has a fundamentalness of 4 and permeability has a fundamentalness of 3.

• Tanelorn and mfb

#### Tanelorn

Thanks. Are these characteristics of space or are they characteristics of light i.e. particles or photons?

#### marcus

Science Advisor
Gold Member
Dearly Missed
Is the speed of light the most fundamental characteristic, or are the vacuum permittivity ε0, and vacuum permeability μ0, more fundamental characteristics of space from which c is derived?

c2 = 1/(ε0μ0).

http://en.wikipedia.org/wiki/Speed_of_light
Tanelorn, wouldn't you allow for an element of TASTE in determining which of several quantities is more fundamental. Maybe any of five can be derived algebraically from the others but there is one that can make the derivations more simple or elegant. there is some aesthetic element, in theory building. And it is possible to arrange the equations several different ways.

Personally I think permit'y and per'bility are constants belonging to ELECTROMAGNETISM. And c the basic speed quantity happens to be the speed of electromagnetic waves but it also plays a role in gravity and in E=mc2 and in Special Rel geometric effects, like the speed limit where the objects can be more general than waves of light. So I would choose to regard c as more fundamental. Because it gets into and is basic to a greater variety of kinds of physics.

BTW I hope you take a look at the Wikipedium about the PROPOSED CHANGE IN THE METRIC SYSTEM, that could be voted in in 2018 at the next big meeting.
http://en.wikipedia.org/wiki/Proposed_redefinition_of_SI_base_units

It proposes to define the metric units by assigning exact values to a handful of physical quantities.

So it will no longer be possible to measure planck's constant because it will be assigned an exact value as part of a strategy to define the kilogram.
Nor will it be possible to measure the charge on the electron because it will be assigned an exact value as part of a scheme to define metric units like the Ampere and the Coulomb. the Amp will be defined as so and so many ELECTRON CHARGES PER SECOND. Solid state devices can count electrons flowing thru a junction so why not?

I think this gets at what you were asking about. What physical quantities are fundamental?
One sign that a quantity might be fundamental is if the Metric governors decide to assign it an exact value as a technique for defining units. The criterion doesn't always work, convenience is a consideration, but it's something to take note of.

It is already impossible to measure the natural unit speed (the speed of light in vacuum) because it has been assigned the exact value of 299792458 meters per second, as a way of defining the meter.
Here's the more general proposed scheme for redefining the metric units. You can see which natural quantities besides c get assigned exact numerical values. Avogadro number, electron charge etc... Wikipedia graphic contributed by someone named "Wikipetzi". Thanks.

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• Tanelorn

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