Why is a fine-tuned universe a problem?

[sethoflagos]

On 2/28/2022 at 3:59 AM, Airbrush said:

If anyone understands these 6 constants, please summarize them for us.  I think Michio Kaku said the apparent fine-tuning suggests a multiverse of universes with all the wrong parameters, but OUR universe is the lucky one!

Fine-tuned universe - Wikipedia

"Martin Rees formulates the fine-tuning of the universe in terms of the following six dimensionless physical constants.^[2][17]

• N, the ratio of the electromagnetic force to the gravitational force between a pair of protons, is approximately 10³⁶. According to Rees, if it were significantly smaller, only a small and short-lived universe could exist.^[17]
• Epsilon (ε), a measure of the nuclear efficiency of fusion from hydrogen to helium, is 0.007: when four nucleons fuse into helium, 0.007 (0.7%) of their mass is converted to energy. The value of ε is in part determined by the strength of the strong nuclear force.^[18] If ε were 0.006, a proton could not bond to a neutron, and only hydrogen could exist, and complex chemistry would be impossible. According to Rees, if it were above 0.008, no hydrogen would exist, as all the hydrogen would have been fused shortly after the Big Bang. Other physicists disagree, calculating that substantial hydrogen remains as long as the strong force coupling constant increases by less than about 50%.^[15][17]
• Omega (Ω), commonly known as the density parameter, is the relative importance of gravity and expansion energy in the universe. It is the ratio of the mass density of the universe to the "critical density" and is approximately 1. If gravity were too strong compared with dark energy and the initial metric expansion, the universe would have collapsed before life could have evolved. If gravity were too weak, no stars would have formed.^[17][19]
• Lambda (Λ), commonly known as the cosmological constant, describes the ratio of the density of dark energy to the critical energy density of the universe, given certain reasonable assumptions such as that dark energy density is a constant. In terms of Planck units, and as a natural dimensionless value, Λ is on the order of 10⁻¹²².^[20] This is so small that it has no significant effect on cosmic structures that are smaller than a billion light-years across. A slightly larger value of the cosmological constant would have caused space to expand rapidly enough that stars and other astronomical structures would not be able to form.^[17][21]
• Q, the ratio of the gravitational energy required to pull a large galaxy apart to the energy equivalent of its mass, is around 10⁻⁵. If it is too small, no stars can form. If it is too large, no stars can survive because the universe is too violent, according to Rees.^[17]
• D, the number of spatial dimensions in spacetime, is 3. Rees claims that life could not exist if there were 2 or 4 dimensions of spacetime nor if the number of time dimensions in spacetime were anything other than 1.^[17] Rees argues this does not preclude the existence of ten-dimensional strings.^[2]

Fine-tuned universe - Wikipedia

How many of these statements can be paraphrased as:

'Significant variation in property X would reduce the ability of the universe to meet the spatial volume and particle diversity requirements of the 2nd Law of Thermodynamics.'

All of them I think.

I have a nagging suspicion that the cart may be being put before the horse. Perhaps the universe simply does what it is obliged to do to meet the 2nd Law. The settings of the various coupling constants etc then become less of a 'lucky lottery ticket' and more of a forced asymptotic approach to optimum values.

   

[Mordred]

14 hours ago, sethoflagos said:

. The settings of the various coupling constants etc then become less of a 'lucky lottery ticket' and more of a forced asymptotic approach to optimum values.

   

Hrrm interesting thought. Technically the Langrangian paths of the particle interactions will follow the path of least action. So there may very well be some truth in that statement.

[sethoflagos]

10 hours ago, Mordred said:

Hrrm interesting thought. Technically the Langrangian paths of the particle interactions will follow the path of least action. So there may very well be some truth in that statement.

I think what started me thinking along these lines is the density parameter of the universe being so very close to 1.

Not only is it a curious value for a random number, but I see no clear reason why omega shouldn't vary with time. Unless of course the balance of its various components is being continuously and dynamically adjusted in order to drive it toward unity. I've not a clue regarding a possible mechanism, but it just has a sort of 2nd Law feel to it.

[Mordred]

There is little to no reason for the density parameter to change as one can accurately treat expansion as a closed adiabatic perfect fluid.

lets put some math to that using The FLRW metric.

the GR form of the FLRW equation is \[(\frac{\dot{a}}{a})^2=\frac{8\pi G}{3}\frac{\epsilon(t)}{c^2}-\frac{k c^2}{R^2_0}\frac{1}{a^2(t)}\]

k=0 , curvature

\[\frac{\epsilon(t)}{c^2}\] is the energy density in the Babera Ryden notation  as opposed to mass density \[\rho\] the reason will become clear later on

\[\rho_c(t)=\frac{\epsilon(t)}{c^2}=\frac{3H^2(t)}{8\pi G}\]

\[H=\frac{\dot{a}}{a}\]

critical density value present day value approx 70 km/sec/Mpc

\[\rho_c]+\frac{\epsilon_c}{c^2}=\frac{3H_0^2}{8\pi G}=9.2*10^3 g cm^3\]using the 70 km/sec/Mpc

\[H^2=\Omega H^2-\frac{kc^2}{R^2_0a^2(t)}\Rightarrow1-\Omega(t)=\frac{kc^2}{H^2(t)a^2(t)R^2_0}\]

if \[\Omega=1\] then it equals one at all times since the RHS of the last equation always vanishes

for the flat case for the \[\Omega>1,\Omega<1\] the value may change however never change sign ie positive curvature will change but never become negative curvature

Now for adiabatic fluid first law of thermodynamics

\[dE-PDV+DQ\]

the change in internal energy equates to the sum of PDV work and added heat/energy however there is no place for heat/energy to come from or leave the system

therefore\[DE+pdV=0\Rightarrow \dot{E}+p\dot{V}=0\]

for a commoving sphere \[V=\frac{4\pi}{3}r^3_sa^3{t}\]

\[\dot{V}=\frac{4\pi}{3} r^3_s(3a^2\dot{a})=V \frac{\dot{a}}{a}\]

\[E=V_\epsilon\]

\[\dot{E}=V\dot{\epsilon}+\dot{V_{\epsilon}}\]\[=V\dot{\epsilon}+3\frac{\dot{a}}{a}\epsilon\]

with \[\dot{E}+P\dot{V}=0 \]we get \[V\dot{\epsilon}+3\frac{\dot{a}}{a}\epsilon+3\frac{\dot{a}}{a}P=0\]

thus

\[\dot{\epsilon}+3\frac{\dot{a}}{a}(\epsilon+P)=0\]

which is the same as the fluid equation standard notation

\[\dot{\rho}+3\frac{\dot{a}}{a}(\rho+P)=0\]

 there's the first law of thermodynamics as its a closed system according to this examination conservation of energy would apply however this doesn't examine quantum fluctuations or the cosmological constant. 

 

 

 

Edited December 9, 2022 by Mordred

[studiot]

6 hours ago, Mordred said:

There is little to no reason for the density parameter to change as one can accurately treat expansion as a closed adiabatic perfect fluid.

lets put some math to that using The FLRW metric.

the GR form of the FLRW equation is

(a˙a)2=8πG3ϵ(t)c2−kc2R201a2(t)

 

k=0 , curvature

 

ϵ(t)c2

is the energy density in the Babera Ryden notation  as opposed to mass density

ρ

the reason will become clear later on

 

 

ρc(t)=ϵ(t)c2=3H2(t)8πG

 

 

H=a˙a

 

critical density value present day value approx 70 km/sec/Mpc

 

ρc]+ϵcc2=3H208πG=9.2∗103gcm3

using the 70 km/sec/Mpc

 

 

H2=ΩH2−kc2R20a2(t)⇒1−Ω(t)=kc2H2(t)a2(t)R20

 

if

Ω=1

then it equals one at all times since the RHS of the last equation always vanishes

 

for the flat case for the

Ω>1,Ω<1

the value may change however never change sign ie positive curvature will change but never become negative curvature

 

Now for adiabatic fluid first law of thermodynamics

 

dE−PDV+DQ

 

the change in internal energy equates to the sum of PDV work and added heat/energy however there is no place for heat/energy to come from or leave the system

therefore

DE+pdV=0⇒E˙+pV˙=0

 

for a commoving sphere

V=4π3r3sa3t

 

 

V˙=4π3r3s(3a2a˙)=Va˙a

 

 

E=Vϵ

 

 

E˙=Vϵ˙+Vϵ˙

=Vϵ˙+3a˙aϵ

 

with

E˙+PV˙=0

we get

Vϵ˙+3a˙aϵ+3a˙aP=0

 

thus

 

ϵ˙+3a˙a(ϵ+P)=0

 

which is the same as the fluid equation standard notation

 

ρ˙+3a˙a(ρ+P)=0

 

 there's the first law of thermodynamics as its a closed system according to this examination conservation of energy would apply however this doesn't examine quantum fluctuations or the cosmological constant. 

 

 

 

Very interesting comparison. +1

[sethoflagos]

10 hours ago, Mordred said:

There is little to no reason for the density parameter to change as one can accurately treat expansion as a closed adiabatic perfect fluid.

IFF omega = 1 exactly. 

11 hours ago, Mordred said:

 

 

 

H2=ΩH2−kc2R20a2(t)⇒1−Ω(t)=kc2H2(t)a2(t)R20

 

if

Ω=1

then it equals one at all times since the RHS of the last equation always vanishes

 

Not sure if this will display correctly, but did we lose a minus sign somewhere? If so it, appears that any divergence from 1 would increase with time making it a metastable solution at best. 

This was how I saw the problem presented somewhere, but it's now clear from your explanation that one can view it from the other direction - zero curvature on the largest scales imposes omega =1. It still looks as if something is being driven towards an asymptote.

Your post is going to take me a while to absorb. You've mentioned this analogy between FLRW and gas expansion before. Many thanks for taking the time to present it so clearly. 

 

[Mordred]

Yeah looks like a typing error will have to double check that but yes any divergence would lead to a curvature divergents. Hence it's still viable our universe has a slight curvature. That's still viable for both Plus or minus curvature.

[sethoflagos]

On 12/9/2022 at 6:14 PM, Mordred said:

Yeah looks like a typing error will have to double check that but yes any divergence would lead to a curvature divergents. Hence it's still viable our universe has a slight curvature. That's still viable for both Plus or minus curvature.

If the analogy holds into the 2nd Law I get the result:

                                                     (a/a_ref)^[3(k-1)] >= Tref/T

where k is the isentropic constant, T is the bulk temperature of the universe and the _ref subscript indicates values at some arbitrary reference time.

This is the sort of thing I was thinking about as the 2nd Law setting constraints on the rate of expansion.

I've uploaded my working as an attachment if anyone's interested.

Univ. Expansion.docx

[Mordred]

Your work sheet looks good I don't see any problems or mistakes in that analysis well done.