Hubble's law as
observational confirmation of the cosmological principle and of the model based
on it.
Contents
1. Cosmological principle and Hubble's law. Reference. How to measure the distances of galaxies? Hubble's discovery. Observational determination of Hubble constant.
2. Hubble's law and the expansion of the Universe.
3. Dimensions of the Universe (determination of
the radius of the
horizon).
4. Continued considerations concerning cosmological speed and
the pace
of expansion (constancy – variability)
5. Age of the Universe and the variability of H factor
Age of the Universe.
H factor varies with time
1. Cosmological principle and the Hubble’s law
Reference
The preceding article recalled
the Copernican principle. This principle has been adopted by the world of
science a priori, as a kind of axiom. Five hundred years ago, the very thought
of the idea constituted a huge breakthrough. Today, it is accepted as something
obvious to the extent that even its breaching by numerous hypotheses and
theories is not thought to be a problem. It is simply not taken any more as a
criterion for evaluation. It has become a common-sense margin and does not
preoccupy the minds involved in "more serious" matters. A little
further along this way and only historians of science will be talking about it.
Is this as it should be that only them? Because nowadays nobody talks about the
movement of galaxies as such. As we all know, today the only subject amongst cosmologists
is the expansion of space. In this context, the cosmological principle is kind
of irrelevant. Is that as it should be? I repeat the question even though it
has become rhetorical in the light of the previous article. Indeed, the very
fact of the existence of the invariant speed c derives directly from this principle. Compliance with the
cosmological principle should be therefore used as a criterion for assessing
cognitive initiatives, for undertaking research in cosmology and maybe not only
in this field of research.
The main consequence of the adoption of
cosmological principle in relation to the dynamics of objects having cosmological
significance, was (see previous article) the hypothesis about the
proportionality of their relative speeds to their mutual distance.
Symbolically, it can be expressed by the equation: v/r = const. It's kind of
anticipation. Is it confirmed by scientific research? Is it possible to
determine the value of this constant? Before we tackle this issue, to get
things in order, let us answer the question: How do they measure the distances of
galaxies?
This
is done primarily through observation of the stars. One of them is the Sun, but
there are various types of stars. There are giants whose radiuses are greater
than the radius of the orbit of Mars, and others that would encompass the whole
Solar System; there are dwarf the size of a large planet, as well as neutron
stars with a radius of the order of ten kilometers and of enormously great
density. Our daily star is stable, but there are stars whose sizes and, of
course, brightness, keeps changing. Among them, some pulsate regularly, some
are more erratic, while others explode: Nova or Supernova.
1.
Special mention should be given to stars called white giants. Thanks to their
particular brightness, they are visible from afar. In vast majority they belong
to the so-called first population. They are the young stars, forming even
today, for example in open clusters. They are all formed from the matter
containing a relatively large amount of heavy elements, the matter creating
spiral arms*. White
giants have been explored in some detail. We know their chemical composition
and characteristics of their spectra, the estimated distance separating them
from planet Earth, thus their absolute brightness. It's a relatively small
distance. Incidentally, our Sun also belongs to the stars of the first
population and is located in one of the spiral arms. Knowledge of the
characteristics of the White Giants enables us to estimate the distances of
galaxies by observing these types of stars, detected there. Star of the second
population, older, are generally much further, most of them outside the disk of
the Galaxy, but also in its center, which makes it much more difficult to calibrate
their distance.
2. Astronomers have paid particular
attention to cepheid variables. Their name comes from the name of the
constellation in which the first one of them was discovered (δ Cephei). It was
discovered in 1784 by an English amateur astronomer John Goodricke. These are
giants pulsating regularly with a constant time intervals (from 1 to 50 days).
At their maximum their brightness is 3-4 times greater than at their minimum.
These characteristics (size and regularity of pulsation) provide the
possibility of using them to measure the distances of galaxies in which they
are perceived. It’s the existence of pulsations that helps in detection of
these objects, even in galaxies which are not necessarily in our immediate
neighborhood. It turns out that there is a close relationship
between the pulsation period and the (absolute) brightness of a star. Brightness
is proportional to the logarithm of the pulse period, which means that the
function M(LogT) is linear. The distance from the star to the observer does not
affect the nature of that relationship. Therefore, knowing the visual brightness
of a star (from measurement), and the period of its pulsation (that is, its
absolute size), it is possible to calculate its distance. Here is the formula:
where: m
– visual brightness, M – absolute brightness, r –distance. See Introductory Information.
Thanks to the existence of this relationship Hubble could determine the
distance of a number of objects, in which he detected cepheid variables. And then it turned out that a
significant number of the "nebulae" which were earlier thought to be
part of the Milky Way, were in fact
separate galaxies, often not smaller than ours. In addition, thanks to the
study of spectra, Hubble derived the radial velocity of distant objects.
Although having a rather small amount of data, he decided, in relation to them,
(as it is simply required by research procedures) to prepare a chart interlinking
the designated parameters (distance and radial velocity). There will be
more about it.
In this context it’s worth mentioning the
class of stars very much resembling cepheid variables by the relationship
between their absolute brightness and the frequency of their pulsation. These
are the stars of RR Lyrae variety (the first one of them was detected in 1901
in the Lyra constellation). Unlike the cepheid variables they are the stars of the second population. In
general, they move pretty fast relative to the Sun, moreover many of them,
perhaps even most of them, are located outside the disk of the Galaxy. They are
characterized by a relatively low concentration of metals. They are white or
yellow-white giants, not as bright as cepheid variables. What’s important is
that they are found in globular clusters (which is rare in case of cepheid
variables). This made it possible to calculate the distances to these objects,
and by the same token to estimate their (averaged) size. As extensive and
relatively bright objects, globular clusters are visible also in other
galaxies. So we have yet another way to estimate distance. By the way, we may
mention the stars of W Virginis type (also of the second population) of similar
(to cepheid variables) characteristics of pulsation, but weaker. Confusing these
stars with cepheid variables had to result in an amplification of estimated
distances. That’s what happened to Hubble. Not having found in the Andromeda
galaxy any stars of RR Lyrae type, he concluded that the galaxy was more
distant than in fact it was. This error was corrected by Baade.
3. Distance can be also determined
on the basis of observations of explosions of supernovae. During the explosion
of such a star, its brightness, within a very short time, increases by a factor
of hundreds of millions. Such a star appears all of a sudden in one of the
galaxies. Its gradual waning over time, that is its brightness over time characteristic,
makes it possible to assign it to a particular type in the classification of supernovae,
created on the basis of observation of their appearances in our galaxy, and
thus to determine its absolute brightness. Most suitable for determining
distance are the supernovae of Ia type, which characteristics of brightness changes
are very similar. These supernovae are helpful in determining (significant) distances
of galaxies for which cepheid variable method proves futile. There is a problem,
however, related to their absolute brightness, the value of which has, understandably,
a certain spread. Another problem is the rarity of a supernova. For this
reason, astronomers have not (did not have until recently) sufficient
observational data. For this reason the level of uncertainty in estimating the
absolute magnitudes of these stars was until recently quite significant. More
recently, thanks to the large number of registered supernovae of this type in a
number of distant galaxies (in recent years there has been tremendous progress
in the field of observational techniques), the Ia supernovae have turned out to
be, as the saying goes, the standard candles, enabling fairly accurate determination
of distance. But are they reliable?
There are other methods apart from those mentioned above. However, these
are the most often used, especially when it comes to the cosmological research.
There are also used (where possible) simultaneous measurements by both
aforementioned methods. In relation to particularly remote objects, such as
quasars, there is also used another method, based on gravitational lensing. The
results, though not completely accurate, allow however to draw far-reaching and
fairly reliable conclusions, non-contradictory with the general cosmological
concepts and physical theories describing the microcosm.
Hubble’s discovery
One of
the most important methods of research in astrophysics is the spectral analysis.
It enables us to determine the chemical composition of the examined objects,
their thermodynamic parameters, as well as their movement. So let me explain
how it works. It is known that the lines of a specific spectrum of a moving object
are shifted with respect to the corresponding spectral lines viewed in a laboratory.
The reason behind this is known as the Doppler Effect. Having (from observation)
the magnitude of displacement of line (z) we can determine the speed of
the object. The methods of measuring distance have been just discussed. And
here we have an interesting question: is
there any relationship between the
speed of objects and their distance? [It is about the objects so far away that their
local motions no longer play a role]. This question comes naturally, straight
from the accumulation of observational data, one doesn’t even need to realize that
it is directly associated with the cosmological principle. If there is no
relationship and if we consider a sufficiently large number of objects, then
the area enclosed within the axes speed and distance (OXY) should be covered be
evenly spread points specified by these two parameters, since all pairs of
numbers are possible. Such a result would mean that out of the two models complying
with the cosmological principle and mentioned in the preceding article, the more
acceptable would be the second one. For the record, this model assumes a static,
infinite Universe.
This study was conducted by Edwin Hubble, and
it wasn’t his intention to confirm (or disprove) the cosmological principle. It
was just an interesting, concrete, detailed research topic, which he had
undertaken. First of all, he discovered that all non-local (those not belonging to the local group, that is not
located in the vicinity of our galaxy) objects
move away. In 1924
he discovered that the "spiral nebulae" (that’s how these objects
were called at the time) are galaxies of the same order of magnitude as our
Milky Way galaxy. In these galaxies they were visible (already in his day) cepheid
variables - regularly pulsating giant stars. At that time the relationship between
their period of pulsation and their absolute brightness was already known
(since 1912). Visual brightness depends, of course, on the distance. So Hubble
could determine the distance of
galaxies. He could also compare the shifts of their spectra. So he could plot a
graph. In 1929, he announced the
results of his research. Although he had a rather small set of data (not much
more than a dozen reliable pairs of numbers), it turned out that already in
this very limited set of data, there was quite clear linear relationship
between the speed of a galaxy and its distance. Below is a graph resembling the
one derived by Hubble. Thus his study showed
directly proportional relationship between the relative speed of objects of
cosmological significance, and their mutual distance. Although he didn’t
he decided to publish the work,
confident, and rightly so, that he made the discovery. In the
graph the abscissa represents the ratio r/R (instead of the distance r - this
is not the original diagram). Here R is the radius of the Universe, that is the
distance corresponding to the speed of light - the greatest distance. Its value, shown on the chart, is consistent with the
calculation below. The thing can be justified as follows:
The v/c ratio
is important, if only due to the fact that it determines the shift of the
spectral lines due to Doppler Effect.
Further
observations confirmed the discovery. It is interesting that the discovery
surprised the world of science, and no one (of importance) thought that this is
"just" a confirmation of the Principle known for several centuries.
This observation itself was called Hubble’s law. It is written in the following
form:
v = Hr (1)
Here: H – it is, obviously, the
proportionality coefficient appearing, in anticipation directly resulting from
the cosmological principle, as "const." (see previous article). It is
called the Hubble constant.
Observational determination of Hubble
constant
To deduce the value of this coefficient, one
must know the distance – we have just discussed the method of its determination
- and the speed at which given object moves away. This value can be determined
on the basis of the Doppler shift of spectrum (red-shift). In the preliminary
information I provided the formula for the value of the shift:
Linking together the equations (1) and (2) we get:
So we found the way to determine the Hubble constant.
We see that for this purpose we need to know the relative displacement of the
spectrum (towards the red), and the distance of the object. The first of these
parameters, we can determine fairly accurately. [This does not guarantee however
that the measured speed is the cosmological speed. We may deal, for example, with
a galaxy receding from us for other, local, reasons (as it is the case with the
galaxy M 31 which is actually moving in our direction). It is true and meaningful
in relation to galaxies in our local cluster.]
The problem
is the distance. The accuracy of its measurement is conditioned by factors that
are not always under our control. Not everything is a function of technological
progress. The objects themselves (cepheid variables, supernovae) may differ
slightly from one another. What must be also considered is the possible
presence between us and the object, of matter which to some degree absorbs
light. This undoubtedly bears an influence on the observable brightness of an
object under our scrutiny. It is therefore necessary to make multiple
measurements, directing telescopes in different directions. Statistical
evaluation cannot be avoided.
The value of the coefficient H
has been estimated at: H = (15-20) km/s/per million light-years, or: H =
(48.9-65.2) km/s/Mps. [Mps - Megaparsec, that is one million parsecs; 1 parsec is the distance of an object which has
a parallax of 1” of arc. This means that it subtends an angle of one second
using the radius of the Earth's orbit as the baseline and amounts to 3.26 light years]
H is therefore just that searched observational parameter, which had to be
determined, as I signalled (anticipation) in the preceding article, in consequence
of the deliberations on the conclusions resulting from the cosmological
principle (model third and fourth). This is not easy (hence the fairly wide
range). For this purpose there have to be measured speed and distance of as
many objects as possible. The outcome is the tilt of the resulting graph v(r) (straight
line). The easiest to measure are the closest objects, but the problem is that
their cosmological speeds are comparable to the speeds of random local movements,
or even lower, and they may be even moving in opposite direction. A good example
is the galaxy M31 in Andromeda, approaching us at a speed of about 300 km/s. And
the further away objects? Determination of their distance is not certain. Besides,
they represent younger Universe, from the times, as we will see further on, when
the value of H factor was different. If it is constant (as a coefficient of proportionality),
it is because Today it is the same in the whole Universe. If so, then there
is indeed the global cosmological time – and here a reminder (from the previous
article), that this is our time, the time that on Our clocks elapsed from the
Big Bang.
So it turns out that very
distant galaxies, all without exception, are moving away from us, and their
radial speeds are proportional to distance (which is clearly manifested in a sufficiently
large data set of measurement). This finding is consistent, of course, with conclusions
drawn from the cosmological principle, and it’s not spoilt by the fact that
some of the closest galaxies, as I mentioned, are even approaching.
In
summary we can say that Hubble’s discovery: a) was an observational
confirmation of the cosmological principle, and even the confirmation of our tentative
model; b) allowed to determine the numerical value of the constant predicted by
that simple model of ours; c) implies that the relative velocities are
constant, or more precisely, that the ratio v/c is constant. The third point
(for now it's just a suggestion), does not fully coincide with the current views,
although it can be taken as a rather insignificant idealization. Further on it
will turn out, however, that it actually contains a sizable load of heuristics.
And therein lays the fundamental progress
that has been made thanks to his discovery. It should be noted that the results
of Hubble’s research, though they can be directly drawn from the
cosmological principle, were not fully compliant with the “existing at the
time” research dynamics. No wonder that this discovery surprised the world of
science. Though it was not associated with the cosmological principle, it
nevertheless constituted a strong heuristic incentive. But it so happened that
the development of cosmology went, in my humble opinion, in the wrong
direction. The new concept of space and the Universe based on the GTR have dominated
cosmological research for the whole century (or longer, if we don’t count my
modest contribution).
Also, in
our time, even though we already know so much, we are occasionally surprised
(despite the introduction of best models, based on the general theory of
relativity). As the most classical example we can mention the darkening of supernovae
– what prompted the "discovery" of dark energy, which was adopted
with enthusiasm (and uncritically), since they were previously scheming with
the cosmological constant, introduced and rejected by Einstein as his
"greatest mistake" (if only in connection with the non-static and
evolutionary Universe). Hubble's discovery was a surprise undermining
"the mandatory" until then models of the static and infinite Universe.
Yet the cosmological constant is still alive and flourishing today... Twists and turns of science. It's really an
interesting contribution to the history of science. And at present? Also the
present time is already history...
2. Hubble’s law and expansion of the Universe
Thoughts related to the rate of expansion. „Interconnective"
approach under censure
So we found that the Universe is expanding, because that is the
direction of movement of the cosmologically relevant objects. Simply put, all
the objects keep on moving away. Already in the preceding article, in
consequence of the adoption of the cosmological principle, we assumed that
there is an upper limit of the speed of objects, which is, of course, equal to
the speed of light (with the resulting conclusion that the linear dimensions of
the Universe are limited.) But still, let us ask, kind of playing fool, not for
the last time anyway: What is the speed of expansion of the Universe? This
“kind of foolish” question is one of the most fundamental. Is it the speed of “the
attack front”? That’s how one can imagine the locus - a set of points moving at
the maximum invariant speed c. As it
will turn out, this “front” is quite significant and has deeper meaning. No
less important is the answer to the question about relative speeds of distant
celestial bodies. Of course, in the context of our discussion it is about the
generalized relative speed of cosmologically relevant objects, not so much
about some specific bodies. So we touch on the problem of the topology of the Universe
which is certainly quite special, and which is undoubtedly quite a unique research
topic. With the topology of the Universe, or rather with a set of premises for
its discovery, I will deal later, in another article.
At this
stage of the discussion, we can imagine that the invariant “horizon” creates -
from the point of view of a single observer - a spherical (in the sense of
locus of absolutely the furthest point from any viewing directions), unsurpassable
limit for countless objects moving away from us, and its speed is at the upper limit
of their speeds in relation to ourselves, that is it is equal to c. This speed I named in the first (preceding)
article, and I will continue to call it the speed of expansion (aware that
generally the matter is approached differently).
In my work I approach many issues differently, in my
own way. It can’t be helped. The model’s characteristics impose an appropriate
set of concepts and definitions. This model differs moreover significantly from
what is now accepted. This doesn’t automatically mean that it represents a
serious alternative for today's convictions, but who knows... On the other
hand, the need to prove that this second way is wrong, may give rise to some
thoughts, some reflection. Or to intensified research? I kindly ask for some
forbearance and a little patience. After all, I haven’t yet explained that
there is a sense of running off the highway onto a bumpy lane running along the
cavernous ridge and leading some... (where?)
This could lead to the thought that all
objects existing in the Universe are "theoretically" visible (we do
not consider here their brightness). Thus, in other words, what is
observable constitutes the whole Universe. Beyond the c borderline nothing exists. You could even say that beyond the
horizon determined by the invariant speed, there is even no space for space. So you can be tempted to
claim that the Universe is the oneness and everythingness, that there are no other
universes. Multiplying them is like multiplying entities beyond necessity. Is this multiplication of some use when we
know so little about our Universe? I would say: That’s just a willy-nilly inertia
of fantasy. But is it just an innocent inertia? Or rather that static and
infinite Universe which fossilised in the minds. Like a stone in the gallbladder, which should be
dissolved (it takes a long time) or the whole gallbladder surgically removed
(although better not to rush with this removal)...
Yet today it is believed (maybe rightly)
that “speeds” (in quotes because it is about the time derivative of the scale
factor) may vary. So we can talk about slowing down (or accelerating)
expansion. Moreover, the visibility of objects means the existence of contact,
which is conditioned by the speed of light. This opens the possibility of the
existence of beings that are beyond the horizon (“their light has not yet reached
us”), the possibility that some parts of the Universe are not visible. As we
can see, it's a different horizon. It is a horizon based on “paradigm of
interconnections”** (that’s how I called it). It is frequently called the horizon
of particles.
So this interconnective approach (“we can
see an object due to the fact that the photons it sends have already reached us,
and we cannot see those entities from which light has not yet arrived”) seems
logical and well-founded. But here we do not take into account the fact (as for
now it has been the fact for a long time) that there was the Big Bang, that at
some point we were all together and we are continuously until today in visual
contact, we’ve seen each other throughout this time. Interesting that this (“interconnective”)
approach was in force even before Hubble
made his discovery, even at a time when it was thought that the Universe is static
and infinite (which was justified at the time). And so it remained. Contrary to
appearances, it is quite important (especially for historians of science), and
perhaps in some ways symptomatic. In the previously published books, as well as
in our articles, this interconnective approach is, expressing it
euphemistically, subject to verification.
Relative speeds and the rate of expansion
So how is it with those relative
speeds? It is of course about the general trend and not merely a selected pair
of objects. Actually one should ask (at this point) otherwise: Is the relative
speed of objects constant over time? Well, until recently, at least intuitively,
it was thought that it decreases due to gravity, just as the body tossed up
slows down, eventually stops, and then falls back. Today, it is widely believed
that it increases due to dark energy. However, it is still
something new, not yet finally confirmed (which did not interfere with granting
the Nobel Prize). [Here it should be noted that it is generally imagined as a
normal movement, and not as a change of scale factor in the expanding space,
contrary to the actually held opinions.] In one of the following articles I
will address this issue, indicating quite another possible reason of the observed
effect (concerning supernovae - the supposed dark energy). For now, it should
be noted that the direct measurement of the possible changes in (cosmological) speeds
is not possible. Even if they occur, they are too slow. The thing is determined
using an indirect approach. The problem is, however, that these findings do not
represent absolute truth, that they may be just a reflection of the current views.
Let us try to present the matter in a more
univocal and general way. For this purpose, we introduce (at least tentatively)
the concept of the coefficient of relative expansion. So as to define it, let’s
imagine two galaxies, at different distance from us. They are moving away from
us at different speeds. Let's ask: What is the difference in their speed per
unit difference of their distances from us? If the speeds difference per unit difference
of distances is large (speed during "distancing" is growing faster),
it means greater rate of expansion. This question can be symbolized as
follows:
Here is the answer based on the Hubble`s law:
As we can see, it is
exactly the Hubble`s factor which is the coefficient of relative expansion and
it defines the rate of expansion. Let’s note that this value does not depend on
which pair of objects we choose. In this sense, it is a constant value.
Constant in space. Is it constant over time? Soon we will see that the H factor
changes with time. Does that automatically mean the change of the relative speeds?
According to our model, these speeds should be constant, while the difference
in distance is widening. Thus, the H factor should decrease with time. We'll
see about it further on. But we can already note that judging by the existence of the upper
limit of the relative speeds, we can conclude that the expansion rate should be
decreasing.
Also,
according to today's view of things, the H factor is the parameter determining
the pace of expansion. Currently the basis for cosmological speculation is the Friedmann
equation derived from the general theory of relativity. So it is about the
expansion of space, and not about the movement as such of specific objects. As
for the H “parameter”, nowadays it is defined as follows:
Here, a is the already mentioned above, the so-called scale
factor, which corresponds to a distance associated with the actual movement of
matter, while a with
a dot above it (in the numerator) is the time derivative corresponding to speed
(in the traditional understanding of motion). This approach is not, however,
consistent with the concept presented in this work. I emphasize: the approach
rather than the final (for ever and ever) determination.
I noticed above that the H factor expresses
the rate of expansion of the Universe. In this work, apart from the rate of expansion
we have also the more intuitive concept of the speed of expansion. They are obviously
not the same. So what about the relative speeds? “Speeds? After all, it's about
space, which puffs up slower and slower or faster and faster.” But I wonder
about speeds. Naive cosmology? ... We'll see further on. Today it is believed
without a shred of doubt that the change applies only to the metrics of space,
its curvature. Until recently it was thought that the "speeds" decrease,
today that they increase. Nobody suspects the Universe of the constancy of
relative speeds which would describe, moreover, the real, the actual relative movement
of objects (and not the growth rate of the scale factor). Ideals are for
philosophers.
– And what about the cosmological principle?
– Of
course, the tendency, whichever way, is to apply to all objects of cosmological
importance (appropriately remote). “If
the rate decreases (increases) to the same extent for each observer, then the
rule is fulfilled. Along with that the Universe is expanding. Its curvature is
decreasing, which means that the “power” of gravity is progressively weakening.
Thus, the gravitational deceleration of expansion is getting weaker, which in
subjective perception could mean its acceleration (even without taking into
account the dark energy).” In that simplified way the matter could be
resolved by a lover of astronomy who would be asking at the same time: “How this curvature (whatever it is) has to
do with observationally ascertained flatness of the geometry of the Universe?”
As one can see, the case is far from closed (not only for amateurs).
For now, we can venture an opinion that the
standard Universe (rather in another standard), in accordance with the concept
preferred here, is as a whole expanding at the speed of light. This expansion
would be actually the Hubble’s expansion. If a point is moving away from us
continuously from the very beginning at the invariant speed c, then today it is away from us to the
maximum extent. The distance at which it is located we shall call the Hubble’s Radius of the Universe. Soon in another article
we will link it the Gravitational Radius
of the Universe. Specific objects are, of course, closer. Is everything that is
closer visible? Is Hubble’s horizon of the Universe coinciding with the interconnective
horizon? Is there (in spite of everything) something beyond the horizon? Here
are some exemplary questions which sooner or later we will have to face. But not
everything at once. However let's already try to answer the question: What
represents the horizon itself? Is it just some “front”, which is the locus of
points, the speed of which is equal to c?
Well, this front is also the place of the Explosion. There-Then in happened. The idea is that straight
after this event this horizon was very close to us, just at our reach. From that
moment it moves away at the speed of light and now it's already a “sphere”
constituting the end of the Universe and it is there where the secret of genesis
is guarded. The farther we can see, the more ancient times unravel in front of
our eyes, all the way to the beginnings at the c border. [And further away?... We could search for the disintegration
of the collapsing universe, before the BB (if it oscillates).] How far away is
the horizon today? How much time has passed since the Explosion? We will attempt
to answer these questions in the continuation of our discussion.
3. Dimensions of the Universe (determination
of the radius of the horizon)
As we have already stated, the
Hubble’s horizon is at the distance corresponding to the maximum speed c. Let us write the Hubble’s law in
relation to this speed limit c:
V = Hr ó r =
v/H →
R = c/H
When v = c, we get: R = c/H. It
is easy to calculate the distance (R). It amounts to 15 billion light years, if
the H factor equals 20 (round number, comfortable for estimations, and also
quite close to the one currently accepted as the most probable). It is believed
that it is slightly larger, which is confirmed by recent CMB (cosmic microwave
background) studies, about which more in another article. They "confirm"
but, which is significant, within the model currently accepted... I would add
that in the context of reflections related to the above-mentioned books, and,
of course, to further articles of this collection, the actual value of the H
factor is not of a decisive importance, moreover, in our considerations “exact”
value of H factor does not matter. Discerning
objects at distances greater than our assumed: 15*** billion light years
(even if one assumes the possibility of their existence), is not possible - not
because the light from there has not yet reached us, but because, further on
there aren’t any material forms. [Unless it would be possible to perceive the Universe
before the Big Bang - today it’s just a fantasy for filmmakers.] In this
context, the natural thing is to accept the thesis that the Horizon of the Universe,
that which reaches furthest and makes the boundary between being and non-being,
is a quasi-sphere of Hubble radius R. “Quasi” because of a specific topology
that Universe certainly represents. There will be more about it.
The objects known to us as the
most distant are called quasars (quasi-stellar objects). The relative shift of
the spectrum towards red (z) with respect to these objects exceeds even the
number 4 (until recently, the record holder was a quasar, for which z = 5.96
and today there is talk about objects for which z is greater than 10). It is
easy to calculate the speed at which the quasar (z = 4) moves away from us.
This amounts to approximately 276923km/s. Is it a constant speed? Was it the
same, let say, 2 billion years ago? Patience. The distance from our quasar is
calculated, obviously, from the Hubble’s law: r = 13.85 billion light-years
(for the constant H=20). It's pretty far, less than half a billion light-years
from the horizon. It is easy to show that even in relation to the object whose
z = 10 or more, we won’t get the value of speed equal to or greater than c, neither we’ll get larger distance
than (or equal to) 15 billion light-years (for H=20). The basis for these
calculations is the relativistic equation****:
4. Continued
considerations concerning the cosmological speed
and thepace of expansion (constancy – variability)
The term currently used
is the "rate of expansion." I defined it above. On my part, I
introduced the concept of "speed expansion" whose semantic sense is
different. The rate of expansion is
expressed by H factor, which informs to what extent the relative (cosmological)
speed increases with distance – in a
collection of various objects (and not depending on time). [It’s irrelevant
whether it’s about the movement as such, or about the variability of scale
factor.] The rate of expansion is simply the upper limit of the relative speeds
and it amounts to c. When studying
the expansion we should first and foremost track any rate changes, that is the
value of the H factor. However, in this chapter we’ll be dealing with the
actual speed of the relative motion of objects, and not with changes of scale
factor a. In addition, we may already propose the constancy in time, though
not of the relative speed itself, but of its ratio to the speed of light b = v/c. We may tentatively call this value “the specific relative speed.” The
point is that the magnitude of the Doppler shift of spectral lines is expressed
by means of this particular b value.
Thus, there is an option, at least theoretically, that c may vary, which would also entail the (proportional) changeability
of v value. If c does not change, then
the relative speed v is constant. Also to this issue we will come back at a
right moment.
The
discovered by Hubble, and anticipated by ourselves, proportionality of relative
speed and distance implies that if we “re-tracked the film”, we would find that
the Universe shrinks and all the heavenly bodies, all galaxies are approaching
one another, to eventually, at one time, becoming a point, or rather a drop, so
as to consistently exclude singularity. By the way, even a droplet of one kilometre
would be something very peculiar in comparison with the enormity of what we
experience looking at the sky. Thus the Universe has its beginning: the Great
Explosion. George Gamow called it the “Big Bang”. In short, it is often written
as BB or GE. Was it an absolute beginning? I think rather a distinguished moment
of a continuous oscillation.
Do we really need to get all
together sometime in the distant future? Or will there come the time when the
“film” will be played backwards? Is the development of the Universe cyclical?
Such a suggestion has already appeared in the preceding article dealing with
the cosmological principle, for instance in the passage: “The mere existence of a
universal movement, including the relationship between the speed of objects and
their mutual distances, would suggest, either: 1. The existence of absolute
beginning (once in the past) or: 2. Continuous drive from an infinitely distant
past to the final end, when everything shrinks to a point, or else: 3. The
cyclicity of changes What's better?.” Regardless, let’s
note that if the Universe is expanding into infinity, the relative speed of the
galaxies can increase (acceleration), remain constant, or even gradually decreases,
though asymptotically, to zero (coming to zero at infinity). Is therefore no possibility
of a reversal, of a chance for the Universe to return to the “starting point”,
so that its development has a cyclical nature? This is the option preferred by
some inner need, probably not only mine. Is it right? How does this relate to
the postulate of the constancy of relative speeds? Or maybe for this reversal
to take place, the speed should decrease appropriately fast? Decrease??? And in case it decreases, what would happen with
the horizon? Where the horizon, and where the galaxies? In this situation horizon
should be much further than the most distant quasars. “Zone of Silence”? I
think it's incoherent. And what comes from observation? That those furthest are
quite close to the horizon, that is, the radial speed is quite close to the
speed of light. Does this answer the above questions? It rather makes them
stronger. These are just some of the questions which we will try to answer (at
least some of them) so as to… provoke the next avalanche of questions, and
thanks to their content once again lift the veil of eternal mystery. Science
develops when questions prevail over answers. And today? I think the opposite
is true.
According to the concept presented here,
Hubble's law is about the proportionality of the relative speeds of objects to
their mutual distance. If this is true today, it is always true, which means that
time is not here a parameter in relation to the general spatial trend (Hubble's
law). This does not, however, apply to any possible changes (in time) of relative
speeds, regardless of the fairly (already) categorical judgment about the
constancy of the specific speed. [I do not mean slowdown as a result of gravity
(Friedmann), neither do I mean acceleration due to dark energy.] The point is
that in another moment the relative speeds may be different, but then
everywhere, because of possible changes in the speed of light. For no other
reason. If they are changing, they should do so as not to violate the Hubble’s
law, which if valid today, is valid forever. After all, it expresses the cosmological
principle, which constitutes (?) the intellectual basis for the generally
accepted vision of the World, a vision that has a good chance to coincide with the
objective reality. However, this principle applies to the particular state of
the Universe, and therefore not its evolution. Time plays no role here. The
Hubble constant itself is constant in space. Or is it also constant in time? In
a moment we will find out that the variation of Hubble factor lies in
the essence of the law itself.
Indeed, relative velocities
may change, after all we proposed only the constancy of specific speeds (v/c).
And if the relative speed actually changes? In this situation, the changeability
of the relative speed would mean variability of the invariant speed c. If it actually declines, it is
rather probable that it tends to zero. And in this (let’s say) zero moment
there will be an inversion and the Universe would begin to shrink. Here lies
the meaning of the oscillation of the Universe - not in a slowdown as a result
of gravity (based on the Friedmann equations) in an image of the stone thrown
upwards. Let’s add that some observations seem to indicate that the invariant c changes (changeability of the fine
structure constant). For now, however, the case is not clear. But in spite of
all, some basis for such modelling exist. Add to this that c is an electrodynamic constant and, as such, in certain specific
circumstances may be different (I omit here the influence of the medium on
speed). The mechanism of changes of the actual invariant, changes of
cosmological character, may not relate to it in any way. Then we would have the
problem concerning measurement of these changes (“what
would serve as a basis "). We will come back to this topic.
So I
will stand by the affirmation (even not a supposition in the sense of
postulate, as I put it above) that the relative velocity is constant over
time, at least when it comes to the value of v/c. That’s because, as we
shall see, it leads to the model which is quite coherent (and to my taste). Moreover,
the assumption of variability of relative speeds (acceleration or delay of
expansion), while at the same time assuming a priori the constancy of invariant
c, would require finding (or invoking)
physical reasons behind it, and also sorting out the resulting problems - the
necessity of such a presentation of things, that they would not infringe on
Hubble's law, would not undermine the cosmological principle and, most
importantly, that they would be consistent with observations. Of course, here
we think about movement in the Newtonian sense. Despite appearances, the
"gravitational pull which slows down expansion or repulsion by dark
energy", are really very troublesome things, and numerous attempts which I
undertook to describe accelerated or delayed expansion came to naught because
they lead either to a clear contradiction, or to an incoherent image of reality
due to excessive scheming and speculation, along with the danger of multiplying
entities beyond need. Such presentation of the matter, that is resignation from
acceleration and deceleration, is actually supported by the fact that the
visible Universe, in a global scale, is homogeneous. We can think about attraction
or repulsion only in relation to local objects. Thus, at least for the moment,
it makes sense to insist on the constancy of speed (in any case, relative to c). If this will lead to contradictory
results, we can always pull back and take a different path or return to the
more frequented route.*****
“The above stated motivations in favour of the thesis about the constancy
of relative speed of expansion are not convincing. Now, the measurement of the
speed of distant objects is based on spectral analysis. Such was the speed of
the given quasar at the time when it was sending those photons, which have just
reached us. And today? Its speed can be different - higher or lower.”
Is that a
fair doubt? Yes, but seemingly. We'll see about that later. For now, look at
the last sentence (before the quote in italics). Besides, it is worth noting
for the record that at the moment of explosion all objects of the Universe had
to be together, formed one integrated whole, and with all of them, the whole
time from BB (Big Bang), we are in visual contact. As for cosmological objects,
to see them, we do not have to wait for the proverbial photons. [Something
else, the events having nothing to do with cosmology, for example, an explosion
of supernova.] This is an extremely important occurrence. It is, moreover,
consistent with the Hubble’s law. However, many seem to be oblivious to this circumstance. But the conclusions which may be
drawn from its apprehension in an appropriate context of thoughts, may be of
considerable heuristic importance. With this statement I anticipated the facts,
because the acceptance of the fact that the Big Bang occurred came only after
the discovery of CMB radiation.
5. Age of the Universe and
the variability of the H factor.
Age of the Universe
Let's choose a random
galaxy at r distance from us, which is moving away at speed v. The Universe
expands, so let us ask: When the distance between us was equal to zero? It
doesn’t matter how we looked at the time. When (how many years ago), we were all,
the Universe, in one point (or to avoid singularities, within the “sphere” of a
very small size)? Well, the time needed to get us back there is equal to: t = r/v.
As you can see, we assume that the relative velocity is constant. This
assumption, even if uttered “in whisper” is not contrary to the general, even current,
views - in terms of the raw material, base. [If the speed varies with any
changes of c (and not, for example,
under the influence of dark energy), then to determine the time exactly, we
should know the nature of changes of the invariant c. And for that it’s a bit too early.] Let’s assume that another
galaxy is located two times farther than the first one (2r). Its speed is,
therefore, equal to 2v. It’s not difficult to notice that it took the same time.
No wonder, since back then we were all together. When was it? The best way to find
out is to use the radius of the horizon and the speed of light: t = r/v = R/c. We can immediately see that:
t = 1/H (*)
And so we get the physical sense of the H factor as the
inverse of the age of the Universe. Thus we immediately get the age.
Actually, we already know it. Since the horizon is at a distance of about 15
billion light years away (assuming the tentatively agreed value of the H factor
to be 20), and the radius R is the distance corresponding to the speed equal to
c. How much time the light would
need to move away from us fifteen billion light years? Of course, the time
equal to the number of years. This number is obviously an example, just as our
assumed value of the H factor. I remind you that we received it based on the
assumption of the constancy of relative speeds. Here it is worth noting that
according to the general opinion, the true age of the Universe is different
than the "ideal", estimated on the basis of the Hubble’s law. This
"ideal" is sometimes called the Hubble age. This supposedly true
results from the Friedmann equation and takes account of dark energy (and
therefore of the cosmological constant), and the characteristics of the CMB radiation.
According to the latest data it amounts to 13.8 billion years. Incidentally, it
is interesting that Einstein rejected the cosmological constant, when after
Hubble’s discovery the Friedmann equation became the basic equation of cosmology.
He came to conclusion that the introduction of L constant was his biggest mistake. Well, the twists and
turns of the history of science.
The Universe,
according to the currently binding view is therefore younger. The reason for
this lies in the rate of expansion which, according to the currently accepted
“standard” model, was formerly greater than it is today. According to this
model, the expansion rate gradually decreased due to universal gravitation and
after 7 billion years since the Big Bang it increases more and more because of
dark energy. Incidentally, this is a serious inconsistency, which I already
addressed in the first article. I will come back to it later and will propose
more consistent solution of this issue.
I think
that there are fundamental reasons for which the speed of expansion of the Universe
is equal to c. Our calculations
are approximate, even in relation to our models that understandably present an
approximate image of reality which is unambiguous, an idealization of all
models combined. So let’s not be afraid to use idealizations in the search for the
objective truth. But this is not the only argument. It is not just about
aesthetics.
We have started above with the assumption
that the relative speed of specified two galaxies (of the cosmological
significance) is constant over time (in any case in relation to c). In the past, even distant past,
their relative speed was therefore the same. It follows that the distance,
determined by the Hubble’s law, of these two objects depends only on the values
H factor. Also the current size of the Universe is determined by the current
value of H factor. Change of the distance (the Universe is expanding)
suggests change over time of this factor, incidentally, determined on the basis
of observation presenting the current state of affairs******. So we can assume
(at least hypothetically) that the distance determined on the basis of
observation is the real, current distance (when not taking into account the
uncertainty as to the value of H). It is the real, not “historical”, distance,
based on contact (via photons) between us and the object. But we have to
remember that the determination of H is possible on the basis of measurements concerning
objects of cosmological importance, that is distant objects. And this slightly overstates
the result, since we are looking into the past, and formerly - according to our
findings above (concerning the physical sense of the H factor) - the value of H
was higher. We will come back to this statement in a moment.
H factor varies with time
Above we noted that H is constant in space, that is
the same everywhere, in accordance, as a matter of fact, with the cosmological principle.
But it is different with respect to time. Suffice to note that H changes with
time (decreases) because (even) at a constant relative speed, it the distance increases
(the denominator in the fraction expressing H in Hubble’s law) – galaxies recede.
The fact that the H factor is changing in time, results also, and immediately,
form the formula (*). After all, the time from the beginning of everything passes
away and the number which expresses it is getting bigger. Time is the only
quantity that cannot be a constant parameter, it does not stop and goes only
forward. We can assume that it was always like that, because our time cannot be
considered as exceptional, even if elsewhere the clock shows a different time. It
is about the universal, cosmic time. Formula (*) indicates that the graph changes of H factor is a
hyperbole (it’s about an inversely proportional relationship), if we don’t take
into account any possible changes of the invariant c. It is about the magnitude
of this value at our place, the value which changes, though of course at a pace
too slow to attempt any measurement of the change in a reasonable time span
(like million years). It is possible, however, to determine the magnitude of
this value in a distant past, thanks to the observation of very distant
objects, where time, according to our view, flows more slowly because of their
relativistic speed (judging by the discussion conducted in this work) – there
will be more about it later, in one of the following articles.
By the way, let’s note that the H factor decreases in proportion
to time, while the distance increases proportionally to time. Thus the
relative speed, according to the Hubble’s law does not change. Any “misgivings”
about the actual variation of speed would be therefore unwarranted.
*) The cosmogony of galaxies, and thus also the
creation of spiral arms will be discussed in particular in the essay entitled “How
the galaxies came into being”. It will be also explained why the stars from
spiral arms contain, for the most part, a relatively large amount of metals (that’s
how the astrophysicists call all elements heavier than helium). Here,
anticipating the case, I will mention that according to the model proposed there,
the spiral arms formed as secondary objects, when the proto-galactic object was
already full of stars (today belonging to the so-called second population, or
in fact the third, according to today's trend).
**) “The interconnective paradigm”, coinciding with
today's understanding of the issue, accepts the existence of "the interconnective
horizon" (That’s how I have called the horizon of particles). It is the
distance covered by photons coming from the farthest object that we can still
see, because to see it, we have to wait for these photons. The “interconnective”
approach, characterizing the current state of cosmological opinions is based on
the paradigm of observability; put it simply: “we see thanks to photons which
came from those places.” This implies the possibility of the existence of
objects outside the visible Universe. This doctrine (commonly accepted outright
as an axiom), in relation to the cosmological issues “forgets” that sometime in
the past, “we were all together”, that there was the Big Bang, which has been
confirmed observationally. From that moment on, “we are all without exception
in visual contact with each other” and there is no need for some “photon
messengers” to see object of a cosmological significance. Thus, the observed Universe
is Everything. The horizon itself is a kind of topological manifold. And in
this manner, I think, the notion of Horizon should be generally treated. The
generally accepted nomenclature uses the term "cosmological horizon",
which coincides with the horizon of particles. It is determined on the basis of
GTR. Currently (since quite recently) the cosmological constant has entered the
scene.
In another article I will present this
“classic” description of things in a systematic way, so as to confront it with the
approach used in this work. By the way, it is interesting that Grigory Perelman
(b. 1966 - Russia) proved the famous Poincaré’s hypothesis in a surprising (for
mathematicians-topologists) way – he is a mathematical physicist, and the basis
for proof were considerations of a cosmological character. There will be more
about the topology of the Universe.
***) In all the calculations
based on Hubble’s law I don’t take into consideration any corrections and
clarifications resulting from the general theory of relativity. Of course, I am
also disregarding (as non-existent) the effects associated with the hypothetical
dark energy. For this reason the adopted today as certain 13.7-8 billion years
as the age of the Universe - I utterly reject. For two reasons. 1. For me of
primary importance is the qualitative aspect, as well as clarity and
transparency of arguments, even at the cost of precision demanded by mathematical
requirements and even if they don’t comply with the current opinions; 2. To
this day, the problem of the age of stars in globular clusters have not been
fully resolved, until recently it was estimated at 15 billion years; 3. These
articles are the result of individual studies, that is conducted solely by one
person, and constituting an arrogant attempt to create a more or less
all-encompassing model of the Universe not based on GTR and not taking into
account the wrong, in my humble opinion, interpretation of the dimming of
supernovae (dark energy). In an essay under the telling title: Horizontal Disaster
I justify my reproachful attitude.
****) The full
derivation of this formula you will find, among others, in my book: Elementary
introduction to the special theory of relativity a bit (...) differently (in
Polish language)
*****) As we
will find out later, the results won’t be contradictory. In the essay on
neutrinos I presented the physical cause of the observed (in disbelief),
absolute flatness of the geometry of the Universe, why there is no global
attraction, neither repulsion, that the flatness problem is an apparent problem,
and the original cause is ignorance of the existence of the duality of gravity.
******) It does not matter that this concerns objects very far away, and
therefore, that it takes very long assumed time before the light which they
emit reaches us. As we will soon find out, the interconnective problem
associated with photon journey will be resolved in a rather surprising way.
