UNIVERSE AND STABILITY

G. UNIVERSE AND STABILITY

The size of the universe is determined to be 10 billion light years. There may be 100 billion galaxies in the universe. The dimension of our galaxy, the Milky Way, is of the order of 100,000 light years (1 light year is equivalent to roughly 6 trillion miles and 1 megaparsec equals the distance light travels in 3.26 million years). Galaxies tend to group together in clus- ters and the clusters in super clusters.

Since the gravitational forces within the universe exceed those determined just with the observable matter, the imbalance is attributed to dark matter (black holes, white stars, etc.,) unseen with existing technology. The amount of dark matter controls the gravitational force and, hence, balances the forces due to the motion of the galaxies. When balanced, it is a flat universe, but when galaxies expand, it is an open universe. When it collapses upon itself, it is called a closed universe. It is thought that there is 90% dark matter in the universe. The current evidence seems to suggest that the universe is expanding with increasing velocity, which ap- pears to contradict the Big Bang theory that suggests a slowing velocity. The expanding uni- verse is believed to be open so that a light beam will never return to its point of origination. The density of the open universe is believed to roughly equal 1.3 times the mass of a single H atom. If that density were instead equal to 130 times the mass of an H atom, it is believed that the universe would be closed.

New theories suggest a pouring of energy from “virgin” vacuum into the bubble of the universe containing dark matter. Since this violates the First law, the existence of an anti- energy in virgin vacuum has been proposed. Thus, energy input to a bubble produces an out- ward pressure that causes the universe to expand with increasing velocity. While the Big Bang problem reduces to a fixed mass with dark matter and fixed energy, the new approach suggests adiabatic throttling into “virgin” vacuum (no dark matter). The new theory also implies a non- adiabatic closed system (i.e., fixed dark matter and other observable mass) universe.

A recent image obtained by Chandra X-ray Observatory shows a cosmic "cloud sys- tem” with pressure fronts in the system and they show a colder 50 million degree central re- gion embedded in an elongated cloud of 70 million degree gas all of which is muddied in an "atmosphere" of 100 million degree gas. The size of the cosmic cloud system containing hun- dreds of galaxies is six million light years across. There is enough gaseous matter to make more galaxies. The galaxies may collide and merge over billions of years and release a large amount of energy, which can heat the cluster gas.

The universe is also presumed to have a “nonzero” substance called “aether” with non-zero levels of energy, pressure, and density; it is also the medium of transmission of light and electromagnetic radiation. From the thermodynamics perspective, a question arises if we can we apply the stability criteria developed so far to the “cosmic clouds” or to the universe with a modified real gas state equation with a generalized power law for attractive forces. The following is a hypothetical analysis of the authors. First, we will assume that the universe obeys the state equation

P = R´T/(v–b) – a/T n v m , (54) where P denotes the repulsive pressure. It is proportional to energy content per unit volume of

free space minus the reduction in pressure due to attractive forces between matter. The variable

b refers to the volume of the matter per unit mass, a is the attractive force constant, R´ is a pro- portionality constant that is not necessarily equal to universal gas constant, and the “pseudo- temperature” T is proportional to the sum of the average translational and rotational energies. The attractive force model between clouds is assumed be similar to the model derived for real gases (Appendix of Chapter 6) except that the power law is modified. Since the boundary of the universe is at zero pressure, any small positive pressure will expand the universe and vice versa. Thus if P= 0, forces are balanced or the universe is flat. From Eq. (54),

T (n+1) = (a (v-b)/ (R’v m )) ≈ = (a / (R’v m-1 )) if v»b.

For the Berthelot type of equation, n=1, m=2, T= ±(a / (R’v)) 1/2 . The flat universe solutions seem to suggest both positive and negative temperatures. While we

rule out negative temperatures for planetary matters, we are not sure about the dark matter or “aether”. Remember that if T<0, the first term in Eq. (54) causes attraction while the second

term causes repulsion. Since T = T(v) for a flat universe, u= u(T) and u 0 (a=0,T)–u(T) = (2 (aR´/v)) 1/2 or 2 R´T. Further, h = u +Pv = u, since P =0. We now discuss what happens during expansion or contraction if the universe follows Eq. (54) at fixed values of u. One consequence is that an increase in volume will form liquid like or condensed subsections (i.e., regions in which the matter/dark matter/clouds is closely packed) within the universe. Alternately, is may be possible to form vapor like (or expanded) subsections in which the matter is loosely packed. An example of the stability of the universe follows. Note that a true stability model of the universe must include all planets in all the gal- axies, comets, asteroids, space dust, cosmic clouds, etc.

q. Example 17 We will address the stability of mater in the universe through a crude “elementary” thermodynamic analysis. First, we will assume that the universe obeys the state equation

P = R’T/(v–b) – a/T n v m . (A) Where the attractive force model is assumed to be similar to the model derived for

real gases (Appendix of Chapter 6) except that the power law is modified.. Due to in- creased pressure in the beginning, a cloud of mass starts expanding. We will derive the criterion for the stability of the system at a specified internal energy and mass in terms of the temperature and volume. What should be the range of values of n for which thermal stability maybe affected? What will be the change in average tem- perature with change in volume as the universe expands or contracts? Next, for the purpose of illustration, what will be the relation for T if the universe/cosmic clouds behaves as a Berthelot gas (m=2, n=1)?

Solution Using Eq. (A)

(B) Recall that

( m ∂P/∂T) v = R’/(v–b) + n a/T

n+1

du = c v dT + (T ( ∂P/∂T) v – P) dv. (C) Note that this expression yields only change in the value of u for the universe due to

a change in the random energy (the dT term) and ipe (the dv term) At a specified value of u,

( ∂T/∂v) u =– ((T ( ∂P/∂T) v – P)/c v (u,v)). (D) Substituting Eq. (B) in Eq. (D) we obtain the relation

(E) Similarly, (

= – a (n+1)/(T ∂T/∂v) n v m u c v (u,v)).

(F) Differentiating Eq. (F) with respect to v

∂s/∂v) m u = P/T =R’/(v–b) – a/(T

n+1

v ).

(G) Substituting Eq. (D) in Eq. (G) we have

n+2 ∂ m s/ ∂v = –R’/(v – b) + ma/(T v ) + (a(n + 1)/(T v )) ( ∂T/∂v)

2 2 2 n+1 m+1

u.

∂ 2 s/ 2 ∂v =–R’/(v – b) 2 + m a/(T n v m+1 ) – (a (n + 1)/(T n+1 v m )) 2 /c v (u,v)). (H)

Mechanical Stability:

2 2 (2n+2) For a system to be stable, (2n+1) ∂ s/ ∂v < 0. Multiplying Eq. (H) by T v and ap- plying this criterion,

maT (n+1) v m < (a 2 (n + 1) 2 v/(c v (u,v))) + R’ v (2m+1) T (2n+2) /(v – b) 2 . Assuming that m =2, n =0 (i.e., assuming that the universe behaves as a VW gas,

(I) which is a quadratic equation in terms of T. For a VW gas c v =c v0 (T), and

2aT v 2 < (a 2 v/(c v (u,v))) + R’ v 5 T 2 /(v – b) 2 ,

2aT v 2 < (a 2 v/(c (T))) + R’ v 5 T v0 2 /(v – b) 2 .

(J)

A quantitative criterion for stability can be provided by Eq. (J). (The current tem- perature in the universe is about 3 deg. K.) For a VW gas

u–u ref =c v0 (T – T ref )– a((1/v) – (1/v ref )). (K) Assume that u ref = –c v0 T ref + a/v ref , i.e., u=c v0 T– a /v, so that

(L) T =(u + a /v)/c v0 .

(M) Note that the temperature as defined in real gas state equations (e.g., Eq. (M)) is pro-

portional average energy per molecule. Amo Penzios and Robert Wilson of Bell Labs found that the universe emits radiation with a pattern similar to a blackbody radiation of a container at temperature 3 K (more precisely 2.726 K). The universe has cooled

to the current temperature from 10 32 K due to expansion from the Big Bang which occurred about 12 billion years back. Thus σ

where σ SB is the Stefan- Boltzmann constant (5.67 ×10 -11 kWm -2 K -4 ). Once the temperature is known, u can be determined using Eq. (M). The temperature decreases with increasing volume and

4 SB T univ =u rad,

vice versa.

Thermal Stability:

Consider the relations

( ∂ c v / ∂ v ) T = TP ( ∂ / ∂ T 2 ) v ,

( ∂P/∂T) v = R’/(v-b) + na/T n+1 v m , m>0, and ( ∂ 2 P/ ∂T 2 ) v = - n(n+1)a/T n+2 v m , m>0. Note that a can be evaluated in terms of force field constants as described in Chapter

6 in context of the VW equation of state. Hence, ∂c v / ∂v = - n(n+1)a /T n+2 v m . In order that c v >c v0 , this slope must be negative. Otherwise, c v <c v0 as the volume is decreased

and the slope may become negative causing thermal instability, thereby creating a sun and a freezing system with a positive entropy generation. Therefore, n(n+1) > 0. If n>

0, n(n+1) is always positive. If (n+1) < 0 i.e, n <–1, then n(n+1) > 0. Thermal stabil- ity is affected in the universe if –1< n<0. It is possible to create a warming planet and

a freezing planet simultaneously as the volume is decreased if -1<n<0 below certain volumes.

Adiabatic Throttling of the Universe:

As the volume of the adiabatic universe changes (in the context of the Big Bang the- ory) in the empty space, energy must remain constant. Recall from Chapter 7, Eqs. (165) and (166), that for adiabatic throttling at constant internal energy ( ∂T/∂v) u = –(T

∂P/∂T – P)/c v = –T 2 ( ∂/∂T(P/T)) u /c v = - (n+1) a/(c v T n v m ). If c v > 0. Therefore, the av- erage temperature decreases with an increase in volume and vice versa for any value

of n (the inverse happens if c < 0). If c is constant, one can integrate to obtain T (n+1) v v = (n+1) 2 a/(c v (m-1) v (m-1) )+ C (n+1) where m ≠ 1 and C is a constant. Then (T (n+1) -

C (n+1) ) = ((n+1) 2 a/(c v (m-1) v (m-1) )). Per this model, at constant values of u, as v → 0, T → ∞ and T → constant as v → ∞, since attractive forces become negligible. If m=2, n=1 (Berthelot), then at constant values of u, T 2 =(4 a/(c v))+ C 2 . Then (T v 2 -C 2 ) =(4

a/(c v v)) for adiabatic throttling of the universe. For continuum C = T ig , (T 2 - (T ig ) 2 ) =(4 a/(c v v)).

Remarks Recall Eq. (C). If a fixed mass system undergoes a reversible adiabatic process, du = - δw = -P dv = c v dT + (T ( ∂P/∂T) v – P) dv and, hence, dT = -T ( ∂P/∂T) v dv/c v . Using

Eq. (B), dT= -(RT/(v–b) + n a/T n v m )dv /c. While the numerator is always positive, a negative specific heat implies that dT>0 if dv >0 (expansion process). If the process is isometric, one can show that dT = δq/c v so that heat rejection will cause dT>0 imply-

ing negative specific heats. For a continuum, the RMS of the microscopic thermal part of the energy (te+re+ve) is proportional to the temperature. Similarly, if one makes an approximation that the RMS otational energy for a dense cluster of objects is also proportional to the tem- perature, then it is possible to use the Maxwell–Boltzmann type of distribution for the various speeds of the objects within a cluster. We quote “Because universal gravitation is, well, universal, every body in the uni- verse attracts every other body. A true model of the system would not only include the solar system but also the thousands of asteroids, billions of comets, and every spec of dust in the solar system, every space probe that has ever been launched, and every passing atom of hydrogen, but it would also have to take into account things like the gravitational attraction of Alpha Centauri, M31, and the most distant quasar,” http://math.bu.edu/INDIVIDUAL/jeffs/stability.html .