In one second the light covers about 300 000 kilometers. Since our Universe has a finite age (of about 12 billions of years), the light emitted at the beginning of the expansion has covered the distance corresponding to the number of seconds elapsed since that time but not more. The galaxies located farther than that maximum distance are thus unvisible. In other words our visible Universe is bounded by a horizon.
Contrary to what is often said the existence of that horizon is not directly related to the velocities of the galaxies that emit light. In particular it is wrong to say that too distant galaxies are unvisible because their apparent velocities of recession are larger than the speed of light.
In fact at earlier phases of the Universe the galaxies of the horizon were receding at velocities equal to twice the speed of light and were nevertheless visible. Presently the apparent velocities at the horizon have decreased but are still expected to fall over that speed of light.
Here is a description of the race between the light and the expansion...
Christian Magnan Collège de France, Paris Université de Montpellier II |
PLEASE, EXCUSE ME FOR THE FORMULAE BUT IT IS JUST IMPOSSIBLE TO TREAT THAT SUBJECT BY JUST WAVING THE HANDS....
Imagine an ant crawling on a rubber band and trying to reach a given point. The rubber keeps on stretching. Will the ant succeed in arriving at its destination?
Think of the ant as a light pulse propagating in our expanding Universe. Will the light from one galaxy reach the others?
It is quite impossible to solve the problem without the aid of the mathematical formalism. To this end the first thing to do is to locate the position of the ant on the rubber. Since the ground is expanding under the ant, it is clearly not wise to measure this position by the real distance to some origin, as it will vary with time continuously. Rather we will draw some equidistant ticks along the the rubber while it is at rest and define in this way an intrinsic coordinate system that is independent from the degree of stretch. The corresponding coordinate of a point will then represent its relative distance to the origin with respect to the total length of the rubber.
This relative distance is dimensionless and can be expressed in arbitrary ways, for instance by a fraction between 0 and 1 or by a percentage (a number between 0 and 100). Here, by analogy with an expanding balloon or a globe, where the position of a point is fixed by its latitude and its longitude, it proves convenient (even if that appears strange at first sight) to measure the relative distance by an angle , which will vary between 0 and (or between 0 and 180°). Accordingly we will agree to refer to this distance by the name of "angular relative distance". Now at a given time, the actual total length (let us say in cm) of the rubber, from the relative position 0 up to the relative position will be written as
R = a , | (1) |
Suppose that the ant starts from one end of the rubber (at position 0). When it is located at relative "milepost" , i.e. at a fraction
r = a . | (2) |
Equivalently, this gives also the actual distance of a galaxy found at angular relative distance from us (a being again the so-called radius of the Universe).
The ant advances at speed c and thus covers the real distance
dr = c dt |
d = dr/a c dt/a . | (3) |
The difficulty of the problem is that the length a, the actual scale length of the rubber (conversely, the radius of the Universe), increases with time. However Equation (3) is very easy to be solved for a mathematician once one knows the law of variation of a with respect to time t.
As a first theoretical illustration, let us consider the case where the scale of distance a increases linearly with time (uniform expansion at constant rate). We write
a = a_{0} t / t_{0} , | (4) |
d = (c t_{0} / a_{0} ) (dt/t) . | (5) |
Let us assume that we now see, at time t, the light of a galaxy situated at some particular angular relative distance (which is a constant quantity caracterizing the galaxy that we are observing). Let us denote t_{em} the time at which the light that we now receive has been emitted. Then, by integrating Equation (5), the quantity is given in terms of t and t_{em} by the relation
= Log ( t / t_{em}) , | (6) |
= c t_{0} / a_{0} . | (7) |
This relation shows that in such a (hypothetical) universe all galaxies are theoretically visible. In fact, whatever the galaxy (which is identified by its angular relative distance ) there exists a time t_{em} such that the light emitted at t_{em} will reach us now at time t. According to Relation (6), t_{em} is given by
t_{em} = t exp (- / ) . | (8) |
In this model, the most distant galaxies are visible, provided the light that we see has been emitted early enough. In other words this model has no horizon. And nevertheless the apparent recession velocities of the arbitrarily distant galaxies can exceed the velocity of light. In fact, from Relations (2) and (4) the "true" distance of Galaxy is
r = a_{0} t / t_{0} | (9) |
V dr/dt = a_{0} / t_{0} . | (10) |
Although this exemple is theoretical, it is very instructive because it tells us two things:
The reason why our visible Universe is limited by a horizon is that at earlier epochs it was expanding faster than linearly with time.
What is then the situation for our real Universe? One does not know its history with certainty but a very good idea of its structure is probably given by the standard expanding homogeneous model of Friedmann based on Einstein's gravitation theory. It happens that the formulae describing this universe model directly yield the solution of the problem of the visibility of the galaxies. In fact those formulae express the radius a of the universe and its age t precisely in terms of the angular relative distance of the horizon, which we will denote . Let us recall those famous formulae:
a = c (A/ 2) (1 - cos ) | (11) |
t = (A/ 2) ( - sin ) , | (12) |
According to Formulae (11) and (12), the speed of increase of the length a is given by the velocity factor
W da/dt = c sin / (1 - cos ) | (13) |
What is the apparent recession velocity of a visible galaxy? This quantity is not a well-defined concept. It is clear that in some way the "distance" of a receding galaxy increases with time - and this implies a kind of velocity - but the problem is to define the term "distance". At wich time and how will we measure this quantity? As the most convenient definition we will take for the distance the "static" distance of the galaxy at the time of reception of the light if the Universe stopped its expansion suddenly. Algebraically speaking this means that the distance r of a galaxy with angular relative distance is
r = a , | (14) |
W = (dr / dt) = (da /dt) a (1/a) (da / dt) . | (15) |
W = H r , where H (1/a) (da / dt) , | (16) |
H = (2/A) sin ( 1 - cos )^{-2} . | (17) |
We can now write the apparent recession velocity of the galaxies found at the horizon. Recalling that they are located at radial angular position
W(horizon) = (da / dt) = c sin / ( 1 - cos ) . | (18) |
Thus the apparent recession velocities of visible galaxies can actually be larger than the speed of light.
What is the present situation in our Universe? Its age and real structure are too uncertain to fix the precise value of the parameter . Some earlier estimates gave
The reverse situation, when the horizon galaxies run slower than the speed of light, is met when the Universe gets older
We have here at our disposal all the necessary tools for calculating the redshift of light due to the expansion. Let us take advantage of this opportunity.
Let us imagine now two ants advancing on the rubber one behind the other at the same speed. If the length of the rubber band remained constant the distance between the ants would also kept constant. But if the rubber stretches, the distance between the ants will just follow the variation of the length of the rubber in the same proportions. This is the explanation of the redshift of light in an expanding universe. The distance between two crests of light waves propagating through space is the wavelength of the light and this wavelength will increase in the same proportions as the dimensions of the universe. The formula that resumes this law is extremely simple as it only expresses that the wavelength of the radiation is proportional to the radius a of the universe.
If we denote
_{rec} / _{em} = a_{rec} / a_{em} . | (19) |
z / = a_{rec} / a_{em} - 1 . | (20) |
An illustration of the redshift formula is provided by the cosmic background radiation at 2.7 K. This radiation was emitted about 100,000 years after the big-bang under the form of visible light corresponding to a temperature of about 4500 degrees Kelvin. As the Universe expanded over the last 15 billions of years, those waves also expanded and are now received under the form of millimetric wavelengths. The expansion factor between the emission and the reception (now) is of the order 1500. The universe is 1500 times larger, the wavelengths are 1500 times larger (according to the redshift formula) and the corresponding temperature of the radiation is 1500 times smaller (the temperature of the radiation is inversely proportional to the wavelengths).
In the beginning Hubble's law stated that the redshifts of the nearest galaxies are proportional to their distances. But what is the exact general relation between redshift and distance? The above equations enable us to answer that question.
Suppose that we observe today some galaxy. The problem amounts to finding the time t_{em} at which the light that we now receive has been emitted in the past. From the previous equations we can see that the distance covered by the light is readily known thanks to the parameter introduced in the equations (11)-(12) and giving the distance from us to the horizon. Indeed, Equation (3) shows that the increase of from one time t to a following one
= c t / a(t) . | (21) |
It is thus convenient to measure the galaxy that we observe today in terms of its angular relative distance, which we will denote . This relative radial distance is constant for a given galaxy. According to the law of propagation of light which we have just stated and which is expressed by Formula (21), the total angular length covered by light between emission and reception (i.e. the relative radial distance of the galaxy) is equal to the increase in the value of the parameter (the relative radial distance to the horizon) during that time, or
= _{rec} - _{em} . | (22) |
r = a = ( _{rec} - _{em} ) a . | (23) |
_{rec} / _{em} = (1 - cos _{rec} ) / (1 - cos _{em} ) . | (24) |
The quantity
_{rec} is the present value of , and will be simply denoted . Thus
r = a _{rec} / _{em} = (1 - cos ) / [(1 - cos ( - )] . | (25) |
For nearby galaxies, we can develop those expressions up to the first order and obtain in this way the well-known Hubble law. We write:
/ = ( 1 - cos ) / ( 1 - cos ) = sin / ( 1 - cos ) . | (26) |
z = H l , | (27) |
Last update: 4th March, 1999
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