# Dynamically evolving universe

 AstroNuclPhysics ® Nuclear Physics - Astrophysics - Cosmology - Philosophy Gravity, black holes and physics

Chapter 5
GRAVITATION AND THE GLOBAL STRUCTURE OF THE UNIVERSE :
RELATIVISTIC  COSMOLOGY
5.1. Basic starting points and principles of cosmology
5.2. Einstein's and deSitter's universe. Cosmological constant.
5.3. Fridman's dynamic models of the universe
5.4. Standard cosmological model. Big Bang.
5.5. Microphysics and cosmology. Inflationary universe.
5.6. The future of the universe. time arrow. Dark matter. Dark energy
5.7. Anthropic principle and existence of multiple universes 5.8. Cosmology and physics

5.3. Fridman's dynamic models of the universe

Obviously, the real universe, at least at its current stage of development, cannot be described by any of the models based on the assumption of static, because in Einstein's model there is no redshift of light from distant galaxies, and in de Sitter's model space cannot contain any matter or radiation. To model the real universe, it is therefore necessary to abandon the assumption of static (which is incompatible with current astronomical knowledge) and create a more general time - dynamic cosmological model.

Metrics of a non-stationary homogeneous isotropic universe
We will therefore consider a homogeneous isotropic universe, which
will generally not be stationary. The metric of three-dimensional space (ie the spatial part of the interval) in such a case will again have a homogeneous isotropic general shape (5.4), but the radius of curvature a here will generally be a function of time a(t) :

 (5.21)

Recall that originally in relation (5.4) we marked the value of the curvature of the space "R" (R : -> a) as "a". However, in an application to a dynamically expanding universe, a(t) will become important as a scale expansion factor. The length element (5.21) is sometimes modified to a shape in Cartesian coordinates, in which it is proportional to the corresponding Euclidean expression (however, this is for illustration only, it is not relevant for cosmological analysis). This can be done by introducing a new coordinate r using the transformation r ® r/(1 + r2/a2). The spatial metric (5.21) then has the form

 .

It is natural to choose a spatial reference system to reflect the isotropy of space as well as the distribution and motion of matter. Thus, the most suitable is a locally "parallel - concurrent" reference system moving in each place of space together with the matter contained therein. The coordinates are "carried - drifting" along with the expansion. The local velocity of matter in such a system is thus zero everywhere, the reference system is formed by the matter filling the universe itself. All motion of mass is expressed by deformation of reference system (cosmological "mollusc"). It is therefore appropriate to transform the spherically symmetric coordinate system by replacing the original radial coordinate r with a parallel (concurenz) coordinate: r(t) ® r(t).a(t). Or, because the radius of curvature a can be used as a natural unit for distance measurement, it is advantageous to introduce new dimensionless coordinates r ® r/a, in which the length element has the shape

 (5.21 ')

The distance dl between any nearby points is thus proportional to the a(t) - scale factor, so that increase or decrease a(t) over time means increasing or decreasing all distances in the system - expanding or shrinking all matter carried by space (cf. discussion in §5.4, passage "What expands during the expansion of the universe? (And what does not expand)").
It is advisable to choose the time coordinate so that the space metric is the same at all points and in all directions at all times. In order for all directions to be equivalent, the components goa of the metric tensor must be zero in this reference frame. The space-time metric will therefore have the form ds2 = goo dx°2 + dl 2. The coefficient goo is a function of only x°, so goo = -1 (= -c2) can be achieved by a suitable choice of the time coordinate. The time coordinate x°, which we can denote by t , then gives its own time at each point in space. The space-time interval here will have the simple form ds2 = -c2 dt2 + dl 2.
The spacetime geometry of a homogeneous isotropic universe can therefore be generally written in the form of the so-called Robertson-Walker-Fridman metric :

 (5.22)

where a(t) is the scale factor, k is a parameter of the type of curvature of space and r is the radial coordinate in the above-mentioned "parallel-concurent" reference system, in spherical coordinates (r,J,j) with the origin r=0 at any point. All time dependence - the evolution of the universe - is contained here only in the scale factor a(t).
As early as the mid-1930s, H.P.Robertson and A.G.Walker showed, that the expression of type (5.22) is the most general metric that can describe an expanding homogeneous and isotropic universe.
To determine the time evolution of a (t), we use Einstein's equations, where we substitute the necessary components of the Ricci curvature tensor Rik and scalar curvature R, which we calculate from metric (5.22) (using the methods from §2.4). The non-zero components here are :
Rtt = -3.ä/a ; Rrr = [(a.ä + 2..a
2)/c2 + 2k]/(1-kr2) ; RJJ = r2.[(a.ä + 2..a2)/c2+ 2k] ; Rjj = r2.[(a.ä + 2..a2)/c2+ 2k].sin2J ;
R = 6.[ä/(c
2.a) + .a2/(c2.a2) + k/a2] ,
where each dot above a represents a derivative according to time t .

Fridman´s equations of universe evolution
Equation
of evolution of the universe resulting from general Einstein equation (5.7) Rik - 1/2 gik R - L.gik = 8pTik , where we substitute the components of the tensor of energy-momentum Tik, the components of metric tensor gik and the Ricci curvature tensor Rik for metric (5.22) (calculated above). The energy-momentum tensor Tik of the cosmological "gas" (3.3) in everywhere locally quiescent reference system has non-zero components only Too = r. c2, T11 = T22 = T33 = -p , where in a homogeneous and isotropic universe the density r and pressure p can be functions only of time t. Einstein's equations (5.7) for the metric (5.22) then lead to two differential equations after adjustment - they are important

 Fridman's equations :
 (5.23a) (5.23b)

which are the basis of relativistic cosmology. Together with the equation of state p = p(r) of a cosmological substance (gas, liquid, plasma, radiation), these equations make it possible to determine a, p, r as a function of time t , ie to determine the evolution of the universe. Each dot above a denotes the derivative by time. Both of these equations are related by identity

 d ( r .c 2 a 3 ) / dt   =   - p. d (a 3 ) / dt   , (5.23c)

which is an expression of the local law of conservation of energy .
Orthographic note :
In the literature, the author's name is usually written "Friedmann", but the Petersburg's author of cosmological equations was actually written "Fridman" ( ).
Fridman's equations (5.23) are an exact solution of Einstein's equations of general relativity for a homogeneous and isotropic universe with Robertson-Walker metric (5.22), filled with a substance with density r, showing pressure p (according to the corresponding equation of state), with energy-momentum tensor (5.3).

In relativistic cosmology, instead of absolute values of time changes of the scale factor . a = da/dt, ä = d2a/dt2, relative quantities derived from them are often introduced, which can be (at least in principle) directly measured from astronomical observations. The Hubble constant H *) is used as a measure of the relative rate of change of the radius of curvature, i.e., the rate of expansion (or compression ); a more apt name should be Hubble speed :

 H  =def  .a(t) / a(t)   . (5.24)

*) The quantity H is called a "constant" only in the sense that it is the same for all places (it does not depend on the coordinates); in general, however, it can be a function of time H (t). A more apt name is therefore the Hubble parameter. The current value of the Hubble constant Ho was estimated mostly in the range Ho » (50 ¸ 100) km s-1/megaparsec. Newer astronomical measurements give a more precise value :

 H 0  »   67 km s -1 / megaparsec  . (5.24´)

In addition to the "classical" Hubble constant H, the so-called reduced dimensionless Hubble constant h, normalized to a speed of 100 km s-1/Mpc, is sometimes used :

 h = def H 0 / 100 km s-1 / Mpc  . (5.24´´)

The Hubble constant H was already introduced in §5.1, relation (5.2), as a coefficient of proportionality in Hubble's law of expansion, which determines by how much the velocity [km/s] of a distant cosmic object (galaxy) increases when its distance increases by 1 megaparsec.: v = H . r, where r is the distance and v is the retraction speed. The velocity of the distance is astronomically measured on the basis of the Doppler effect using the redshift z = Dl/l - the relative increase of the wavelength l of radiation of this galaxy: for z <1 it is approximately v ~ c . z; the exact relationship between the retraction speed and the redshift is v = c . [(z+1)2 -1] / [(z+1)2 +1].
Furthermore, the so-called deceleration parameter q is introduced

 q  = def a. ä / . a 2 (5.25)

characterizing the slowing or acceleration of expansion or contraction. Using the quantities H and q, Fridman's equations (5.23a, b) can be expressed in the form

 k.c2/a2  =  8pGr/3 - H2 + L.c2/3 ,   k.c2/a2  =  (2 q - 1) H2 + L.c2  . (5.23'a, b)

Dynamics and evolution of cosmological models
Using Fridman's equations, it is possible to analyze the dynamics of the expansion of a(t) of the universe, depending on the mass density of matter r, its equation of state p(r) and the cosmological constant L. Let us first notice the simpler case L = 0 , without the cosmological constant :
From equation (5.23'a) it can be seen, that the sign of 8pG.r/3 - H2, ie the relationship between mass density and expansion rate, decides which of the variants k = 1, 0, -1 can be realized.
The case k = 1 corresponding to a closed universe occurs when 8p G.r/3 > H2, ie when the mean mass density r in space is greater than a certain "critical density" rcrit

 r crit  =  3 H2 / 8p G  . (5.26)

Based on currently observed galaxy distance rates (Hubble constants), this critical density is approximately 8.10-30 g/cm3, which corresponds to only about 5 hydrogen atoms per 1 m3.
If r < rcrit => k = -1 - it is an open universe, in the boundary case r = rcrit we have k = 0 - it corresponds to the Euclidean universe, which is also open. Equation (5.23b) shows that the equivalent criterion for the curvature character of the Fridman universe is the value of the deceleration parameter q: in a closed universe it is q> 1/2, in an open universe q <1/2 and in the Euclidean universe it corresponds to q = 1/2. These variants will be analyzed in more detail below.

In the limit case r = rcrit will be k = 0 , the universe has an infinitely large radius of curvature - it is a model of an open universe with flat space (Euclidean). Spacetime metrics have a simple shape here

 ds 2   =   - c 2 dt 2 + a 2 (t) . (dx 2 + dy 2 + dz 2 )  , (5.27)

whereas the time-varying coefficient a(t) does not express the curvature of space, but is only a scale factor.
It should be noted that although the Euclidean metric of three-dimensional space is obtained for the r = rcrit , the whole four-dimensional space-time is not flat here! Only certain special sections (hypersurfaces) of spacetime are flat, corresponding to the same proper time of all particles filling the universe.
Fridman's equations have an accurate analytical solution in the case of a universe filled with an "ideal fluid" with the equation of state

 p  =  w. c 2 . r   , (5.28)

where p is the pressure, r is the mass-energy density of the fluid (in a locally quiescent reference frame) and w is the state constant. The energy-momentum tensor Tik was given in §5.1, relation (5.3). The time dependence of the scale function a(t) is then obtained by substituting p into Fridman's equation (5.23a). In our case k = 0 has a simple form

 a (t)  =  a 0 . t 2 / [3 (w + 1)]   , (5.29)

where a0 is the corresponding integration constant depending on the initial conditions. We can distinguish two basic cases here :
w = 0 corresponds to incoherent dust that does not create any pressure, p = 0. The universe is controlled by matter, where the pressure is negligible due to the density of matter - it corresponds to the late stages of evolution. Then it follows from the general solution (5.29) that the distance between every two points increases with time according to the law

 a (t)  =  a 1 . t 2/3   ,  a 1 = ...... (5.30)

w = 1/3 - radiation dominant universe. In the early stages of the evolution of the universe, which is to be considered the maximum pressure p =r.c 2 /3 of (5.29) for the expansion we get the time dependence of the shape

 a (t)  =  a 2 . t 1/2   ,  a 2 = ...... (5.31)

In both cases space with dominant agent or radiation at r = rcrit dependence a = a(t) here has the shape of a parabola (Fig.5.3) - radius of curvature a increases monotonically from zero (a singularity!) at t = 0 to infinity when t®¥.

If r > rcrit , is k = 1 - this is the closed universe. Equation (5.23a) for k = 1 , L = 0 has the form .a2 + 1 = a2.8p G.r/3. If the universe is filled with incoherent dust, ie p = 0, it follows from equation (5.23c) r.a3 = const.; since the volume of a closed universe is V = 2p2 a3, the sum of mass in the whole space is constant:

 r . 2 p 2 a 3  =  M  = const. =   2 p 2 a o 3 r o   , (5.32)

where ao and ro are the radius and density of the mass of the universe at some fixed point in time to.
The function a (t) is often expressed in parametric form. After introducing a new "time" variable h, by substituting dh = a.dt, the solution of equation (5.23a) can be written in parametric form

 a = (4GM / 6 p c 2 ). (1 - cos h ) ,  t = (4GM / 6 p c 2 ). ( h - sin h )  . (5.33)

The graphical representation of the time dependence a = a(t) is thus a cycloid (Fig.5.3a), which describes a fixed point on a circle of radius

 a max   =  4GM / 3 p c 2 (5.34)

when rolling along a straight line (time axis t); the parameter h is the rolling angle. The density of the mass changes according to the law r = 3/a2max(1 - cos h)3 = 6H2/8pG(1+cos h). In Friedman's model of a closed universe filled with dust with density r > rcrit , evolution looks so (Fig.5.2) that at the beginning t = 0 the universe is based on an initial singular state a = 0 with zero volume and infinite mass density, gradually expanding to to dimension a = amax , and then shrinks again to the point a = 0 - terminal singularities.
According to the left part of Fig.5.2, the evolution of the universe is often modeled by an inflating balloon (and then shrinking), on the surface of which galaxies or clusters of galaxies are drawn. In such an inflation of the balloon, all points of its surface move away from each other at a speed proportional to their distance from each other, in accordance with Hubble's law (5.2). A critical assessment and refinement of this model will be discussed in the following §5.4, passage "What is actually expanding?".

Fig.5.2. Temporal evolution of a closed universe.
Left : The closed Fridman universe can be imagined as a three-dimensional sphere that gradually "inflates" from zero radius (initial singularity at time t = 0) to a certain maximum radius, and then shrinks back to a point (final singularity). All distances Dl between any objects (galaxies or clusters of galaxies) increase or decrease in proportion to the radius of curvature as the universe expands or contracts.
Middle : A space-time diagram of a closed universe nested in a fictional five-dimensional space.
Right : An illustration of expanding and contracting matter during the evolution of the universe.

However, in the stages a ® 0, ie at the beginning and at the end of evolution, the assumption of the equation of state of incoherent dust is not realistic. Conversely, a substance are inevitably becomes ultrarelativistic, so closer to reality is the state equation p = r.c2/3. Equation (5.23c) then gives r .a4 = const. and the solution of equation (5.23a) is here

 a  =  a ~ . sin h  ,   t  =  a ~ . (1 - cos h ) / c

(the graph is a semicircle), where a ~ = Ö(8p G. r. a4/3c4) = const. = Ö(8p G.ro ao4/3c4) . The global nature of evolution will be the same as in the previous case - no pressure of matter filling the closed universe is able to prevent singular points a = 0.

If r < rcrit , then k = -1 - it is an open universe. If it is filled with dust, the solution of equation (5.23a) is

 a  =  â. (cosh h - 1) ,   t  =  â. (sinh h - h ) / c  ,

where â = 8p G. r a3/3c2 = const. = 8p G. ro ao3/3c2). The dependence a = a(t) here has the shape of a hyperbola (Fig.5.3a) - the radius of curvature a grows monotonically from zero (singularity!) at t = 0 to infinity at t®¥. A similar picture is obtained when the effect of pressure is included; the extreme case p = r .c2/3 is a solution

 a  =  a ~ . sinh h  ,   t  =  a ~ . (cosh h - 1) / c   .

Thus, Friedman's open universe also has a singularity, but only one thing - the initial one.

Fig.5.3. Evolution of cosmological models (time course of radius
a of space) depending on the value of the cosmological constant L and the density of mass distribution r .
(a
E and L E in the figure on the right denote the values of the radius of the universe and the cosmological constant corresponding to Einstein's cosmological model)

In addition, when a non-zero cosmological constant L is included, a certain additional force appears in the universe (repulsive for L > 0 and attractive at L <0), which accelerates or slows down the expansion or contraction of the universe. This force does not depend on the weight and increases with distance. From the point of view of the global evolution of the universe, the effective energy of the vacuum, generated by the cosmological member, has an important property (different from the material form of matter) - it does not dilute or thicken during the expansion or contraction of the universe, it maintains a constant value. The solution of equations (5.23) then at L ¹ 0 leads to the following possibilities :
If L <0 , gravity always ultimately prevails and the evolution of the universe has a course according to Fig.5.3b at any mass density r. This variant probably does not apply in space. More varied possibilities of space evolution arise at L > 0 - are shown in Fig.5.3c :
- If the cosmological constant L < LE is less than the Einstein value (5.15) L E = 4p G.r/c2, for supercritical density r > rcrit evolution of the universe to proceed roughly (qualitatively) as for L = 0.
- At L > LE the a(t) increases from zero to infinity, but at a certain stage the expansion slows down significantly for a time - there is a kind of "quasi-static the phase" during which the attractive forces are balanced by the repulsive ("indecisive" universe); later dominated by the repulsive force *). The duration Tst of this quasi-static phase, during which the radius of curvature of the universe is maintained approximately at the value of the radius of the Einstein static model (5.16) a = aE, is longer the smaller the difference L - LE : Tst ~ ln [L/(L - LE)].
- With L®LE , the universe enters the state of Einstein's static universe mentioned in the previous paragraph. However, this Einstein model is unstable because the slightest perturbation of density will lead to expansion.
For r > rcrit and L = LE there are two other solutions :
1. In the infinitely distant past t®-¥ was a = aE, in the future unlimited expansion (unlikely variant) ;
2. The universe come out of the state a(0) = 0 at time t = 0, after which it expands and asymptotically (at infinitely distant future t®¥) reaches a radius a®aE.
For L > 0 exists, in addition to the special possibilities mentioned, there is also solution, according to which at t = -¥ universe was infinite radius, then the contraction took place up to a certain minimum value of amin, after which unbounded expansion occurs (an unlikely scenario).
The mentioned peculiarities of cosmological models with non - zero cosmological constant are used from time to time in attempts to overcome the supposed or real difficulties of relativistic cosmology (internal difficulties and inconsistencies with the results of observations) - cf. §5.5 "Microphysics and Cosmology. Inflation Universe.", or §5.6, section "Dark energy and accelerated expansion of the universe".

Three basic dynamics of the scale factor - summary
For the astrophysically plausible scenario of evolution of our universe (according to the standard model of physical cosmology - §5.4), three basic dynamics should be applied in individual stages of evolution - three time dependences of the scale factor a = a(t), resulting from solving of Fridman equations (5.23) :
when the material content of the universe is dominated by radiation and relativistic particles, arousing pressure
p=r.c2/3. The time dependence of the scale function a(t) is then
a(t)  ~  t 1/2  .
This situation was in the early hot universe
(after inflation expansion) and lasted (in "pure form") for about 50,000 years.
- The era of matter ,
when the energy density of a substance
(non-relativistic particles, gas, dust) is greater than the energy of radiation. The time dependence of the scale function a(t) is here
a(t)  ~  t 2/3  .
Both of these dependences have a similar parabolic shape according to Fig.5.3a
(k >= 0). Proportionality coefficients depend on the specific substance content. The era of matter in the universe is still going on now ...
- Vacuum energy stage - cosmological constant
when the density of matter and radiation is negligible compared to the vacuum energy generated by the cosmological member
L.gik in the gravitational equations. The time dependence of the scale function a(t) is then exponential
a(t)  ~  e
H . t  ~  e Ö(L/3) . t
and in fact corresponds to the special case of de Sitter's model. In §5.5 "
Microphysics and Cosmology. Inflation Universe." we will see that this situation probably occurred in a very early universe, just after the Big Bang in a time range aprox. 10-36 ÷ 10-32 sec. - inflationary expansion of the early universe. And then it may happen again in the very distant future - accelerated expansion of the universe, when the density of matter and radiation drops to very low concentrations due to the expansion of the universe and "dark energy", generated by the cosmological constant, begins to dominate (§5.6, section "Dark energy and accelerated expansion of the universe ").?..

Relative Omega - parameterization of cosmological models
Instead of absolute values of mass density and cosmological constant, their dimensionless
relative ratios W with respect to their respective significant (critical) values are often used to model the evolution of the universe. The ratio of the actual value of the mass density r with respect to the critical density is introduced :

 W M   = def r / r crit     (=   r . 8 p G / 3 H 2 ) (5.36)

and the ratio of the actual value of the cosmological constant L to the Einstein value :

 W L   = def L / L E     (= L. c 2 / 3H 2 )  . (5.37)

Fridman's equation (5.23a) expressed by the parameters W is as follows :

 k.c 2 / H 2 a 2  =  W M + W L - 1  . (5.38)

The deceleration parameter q, introduced by equation (5.25), is expressed by the parameters W :

 q  =  W M / 2 - W L    . (5.39)

For WL = WM/2 the expansion takes place at a constant speed, at WL < WM/2 the expansion slows down, at WL > WM/2 the expansion accelerates.
Using dimensionless W values, it is easier to test the evolution of a cosmological model. E.g. for a flat model, WM + WL = 1. It is also possible to construct clear graphs of the behavior of cosmological models in coordinates, on the axes of which the values WM and WL are plotted .
Sometimes an even more detailed W- parameterization of the cosmological model is introduced. The general mass density r (rM) is divided into a substance formed by non-relativistic particles rm (especially baryons - also denoted by rB) and relativistic particles and radiation rrad (in the early stages of the universe it is high-energy gamma radiation, also denoted by rg). With W -option further expresses the curvature parameter k . The basic Fridman equation (5.23a) for the rate of expansion of the universe is then written in the form :

 (5.40)

where Wxxx are the contributions of the individual components of matter ~ energy to the dynamics of expansion: Wrad from relativistic particles and radiation, Wm from non-relativistic matter, Wk from the curvature of space and WL from the cosmological constant - "vacuum energy". The parameter H0 » 67 km s1 /Mpcs is the current value of the Hubble constant.

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Current note: According to recent astronomical observations of distant supernovae, there are some indications that the expansion of the universe is currently accelerating, that in addition to ordinary and dark (non-radiant) matter, there is also so-called dark energy in universe, which shows "antigravity". Thus, the evolution of the universe seems to follow the curve in Fig.5.3c, the case of L > LE (see §5.6 "The Future of the Universe. The Arrow of Time.", the passage "Dark Energy and the Accelerated Expansion of the Universe"), corresponds to WL > WM/2 ...

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Vojtech Ullmann