version française
cliquez ici


WHAT DETERMINES THE POSITION OF THE COSMOLOGICAL HORIZON?



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....

 


SECTIONS

1. The parable of the ant on the rubber band
2. Defining the angular relative distance
3. The ant conscientiously pursues his way
4. The case of a linear expansion rate
5. Galaxies running faster than light
6. Where is located the horizon in our Universe?
7. The cosmological redshift
8. The distance-redshift relation

THE PARABLE OF THE ANT ON THE RUBBER BAND

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?

DEFINING THE ANGULAR RELATIVE DISTANCE

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)
where a (which is defined in this way) is a length fixing the actual scale of distance on the rubber. You may rightly think that a represents what is called the radius of the Universe and that R is the distance of the farthest galaxy in our Universe (located at the anticenter).

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 ( / ) of the total length of the rubber, its "true" distance to the origin is
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 CONSCIENTIOUSLY PURSUES HIS WAY

The ant advances at speed c and thus covers the real distance
dr = c dt  
during time dt. (Analogously a crest of light travels at speed c from point r to point r+dr.)  From Definition (2), the relative (angular) distance corresponding to this true distance dr is
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.

THE CASE OF A LINEAR EXPANSION RATE

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 = a0 t / t0 , (4)
in noting a0 the radius of the universe at time t0 . Equation (3) now reads:
d = (c t0 / a0 ) (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 tem 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 tem by the relation
= Log ( t / tem) , (6)
where we have denoted
= c t0 / a0 . (7)
Incidentally, just to give the order of magnitude of that quantity, the present instantaneous rate of expansion of our Universe is expressed by a constant    that is roughly of order unity.

GALAXIES RUNNING FASTER THAN LIGHT

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 tem such that the light emitted at tem will reach us now at time t. According to Relation (6),  tem is given by
tem = 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 = a0 t / t0 (9)
and its "velocity" is thus
 dr/dt = a0 / t0 . (10)
Since we have noticed before that a0 / t0 is of order of c, we conclude that V may becomes larger than the speed of light. This is not in contradiction with the theory of the relativity, which states that no velocity can exceed the speed of light, because the "velocities" in question are only "apparent" velocities that do not correspond to actual displacements of the galaxies with respect to the "cosmological rubber" but that are due to the expansion of space, or rubber, itself.

Although this exemple is theoretical, it is very instructive because it tells us two things:

  1. galaxies may recede from us at "velocities" larger than the speed of light and be nevertheless visible;
  2. the mere fact that our universe is expanding does not explain the existence of a horizon.

The reason why our visible Universe is limited by a horizon is that at earlier epochs it was expanding faster than linearly with time.

WHERE IS FOUND THE HORIZON IN OUR UNIVERSE?

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)
where the quantity A (which has the dimension of a time) characterises the model entirely. For our Universe A could be of the order of some tens of billions years. The length cA is equal to the radius of the univers at maximum expansion. We notice that this maximum of expansion (just before the expansion turns into contraction) occurs when the horizon is found at position = . This means that at that time the Universe is uncovered in its totality with all the galaxies being visible. In particular the light from the anticenter, the most distant galaxy located at angular position = , can at last reach us.

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)
This formula indicates that the expansion proceeded at infinite rate at the big-bang, when was close to 0. It also shows that at such an epoch the horizon at a given point was infinitely close to that point since its distance is given by  , which tends to zero. In other words, all points are disconnected at the origin of time in this model. This is the so-called "horizon problem", which expresses that one cannot understand how disconnected points can build a universe that seems otherwise homogeneous. One can conclude that the connexion between the points of our Universe was established "before" it became a model described by Equations (11) and (12).

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 = , (14)
according to Definition (2). Therefore its apparent recession velocity is
W = (dr / dt) = (da /dt) a (1/a) (da / dt) . (15)
We then get, not unexpectedly, a "Hubble law"
W = H r  ,    where    H (1/a) (da / dt)  , (16)
expressing that the recession velocity W of a galaxy is proportional to its distance r . According to Formulae (11) and (12) one finds easily that the Hubble constant is
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   =  , we immediately get from (15)
W(horizon) = (da / dt) = c sin / ( 1 - cos ) . (18)
This expression is very interesting. It shows that the "horizon velocity" (if you allow me this expression) is roughly of the order of the speed of light. It shows also that near the origin of the universe, this velocity is W = 2 c. When the universe gets older, the horizon velocity decreases and remains larger than c untill = 2.33 (or 134°).

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 2 (see C.W. Misner, K.S. Thorne and J.A. Wheeler, Gravitation, Freeman & Company, 1973, page 738). With such a value we would get W = 1.3 c, so that the apparent recession velocity at horizon would be in fact 30% larger than the speed of light. This value is perhaps not the correct one, but anyway it illustrates the possibility for the horizon galaxies to exceed that speed of light.

The reverse situation, when the horizon galaxies run slower than the speed of light, is met when the Universe gets older ( > 2.33). This confirms that the horizon is not located at the place where the velocity of "escape" becomes equal to that of the light. Eventually the universe will be entirely explored when = , at the time of maximum expansion. Formula (18) shows that the apparent recession velocity of the last galaxies made visible at the horizon will then be equal to zero.

THE COSMOLOGICAL REDSHIFT

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  em the wavelength of the light at emission, when the radius of the universe is aem , and  rec the wavelength at reception, when the radius of the universe is arec , we have the simple redshift relation
 rec /  em = arec / aem . (19)
In astrophysics one very often introduces the "redshift"
z / = arec / aem - 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).

THE DISTANCE-REDSHIFT RELATION

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 tem 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 t + t is just equal to the distance c t covered during that lapse of time by a ray of light anywhere in the Universe, provided that this distance is measured in terms of the angular relative distance, i.e. relatively to the scale of distance a (or the radius of the Universe). In other words,
= 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)
The real radial distance of that galaxy in length units is
r = a = ( rec - em ) a . (23)
By knowing the values of the parameters rec and em , we can determine the age of the universe at emission and at reception and also the values of the radius of the Universe at those times by simply applying the formulae (11) and (12). Therefore, from Equation (19), the ratio the wavelength of the light at reception to that at emission is simply
 rec /  em = (1 - cos rec ) / (1 - cos em ) . (24)

The quantity  rec is the present value of  , and will be simply denoted  . Thus  em = -  . We can then write Equations (23) and (24) as
r = a

 rec /  em = (1 - cos ) / [(1 - cos ( - )] .
(25)
Those equations express the general relation between the "wavelength expansion" ( rec /  em) and the true radial distance r of the galaxy found at relative distance .

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)
This gives immediately Hubble's relation between the redshift z and the distance l = a of the observed galaxy as:
z = H l , (27)
where a is always given by Expression (11) and the Hubble constant H by Expressions (16) and (17).


   Cosmology (in french, sorry )
   Science and Life (in french, sorry )
   Christian Magnan's homepage (in french, sorry )


Last update: 4th March, 1999

URL :  http://www.lacosmo.com/horizon.html