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# Dark energy 3

Posted Jul24-08 at 12:41 AM by stevebd1
Updated Aug16-08 at 07:26 AM by stevebd1

(continue from Part 2)

-Speed of light (c)- 299,792,458 m/s ≈ 3x10^8 m/s

-The Hubble constant (H)- 70 km/s/Mpc
The rate at which the velocity of recession of galaxies increases with distance from us (for every megaparsec (3,261,630 Lys) an object will increase by 70 km s^-1 in velocity) according to the Hubble law. The law proposed by Edwin Hubble in 1929 claiming a linear relation between the distance (r) of galaxies from us and their velocity of recession (v), deduced from the redshift in their spectra. The figure linking velocity with distance is the Hubble constant. Hence v = Hr and H = v/r.

For the density parameter equations, this is converted to units of s^-1 by dividing 70 km s^-1 by the number of km's in a megaparsec, which is 3.09x10^19 km. The result is 2.26x10^-18 s^-1 (which is also the inverse of Hubble time- 1 divided by 4.425x10^17 seconds).

$$H=\frac{70\ \text{km/s}}{\text{Mpc\ (in\ km)}}= 2.26\times10^{-18}\ \text{s}^{-1}$$

(Note how the Hubble constant also provides the approx. age of the universe, if you reverse the acceleration rate, you can figure out when all matter was in the same place i.e. when the big bang took place).

$$H_t=\frac{1}{2.26\times10^{-18}}=4.425\times10^{17}\ \text{s}= 14\ \text{billion\ years}$$

where Ht is Hubble time.

Density parameters
The results of the calculations below provide omega (Ω), a measurement for the density of universe.

$$\Omega=\frac{\text{density of universe}}{\text{critical density}}$$

Incorporating the critical density, the equation becomes-

$$\Omega=\frac{\rho}{\left(3H^2/8\pi G\right)}}=\frac{8 \pi G}{3H^2}\times\rho$$

where ρ is density in kg/m^3, G is the gravitational constant and H is the inverse of Hubble time in units of second^-1.

In the calculations below, Ω is the sum of two parts; normalized matter (baryonic matter (Ωb) and dark matter (Ωdm)) and normalized vacuum energy (dark energy (ΩΛ)), these together give the total omega figure, (Ωb + Ωdm) + ΩΛ = Ωtotal.

Ω < 1 = hyperbolic (saddle shaped), open universe, will keep expanding until the temperature reaches absolute zero and the universe ends in a big freeze.

Ω > 1 = spherical, closed universe, will contract until the universe collapses in to the big crunch (this could induce another big bang, this is called an oscillating universe).

Ω = 1 = flat universe, will keep expanding but temperature will not reach absolute zero. Hovers between the two extremes.

(The best estimate for the density of the universe puts it slightly below the critical density, if Omega doesn't equal 1 exactly, then the next realistic option would be an open universe.)

Normalized matter-

Omega baryonic (Ωb)

$$\Omega_b=\frac{8 \pi G}{3H^2}\ \times\ \rho_b=\frac{8\ \times\ 3.14159\ \times\ 6.6742\times10^{-11}}{3\ \times\ \left(2.26\times10^{-18}\right)^2}\ \times\ 0.040\times10^{-26}$$

= 0.044 for baryonic matter (matter composed of protons, neutrons and electrons)

Omega dark matter (Ωdm)

$$\Omega_{dm}=\frac{8 \pi G}{3H^2}\ \times\ \rho_{dm}=\frac{8\ \times\ 3.14159\ \times\ 6.6742\times10^{-11}}{3\ \times\ \left(2.26\times10^{-18}\right)^2}\ \times\ 0.202\times10^{-26}$$

= 0.222 for dark matter

Ωb + Ωdm = 0.044 + 0.222 = 0.266

Normalized vacuum energy-
(incorporating cosmological constant equation. By rearranging the cosmological constant equation shown in Part 1, we get Λc^2 = 8πG x density of Λ, therefore Λc^2 replaces 8[itex]\pi[/tex]G x density of Λ in the omega equation).

Omega dark energy (ΩΛ)

$$\Omega_\Lambda =\frac{\Lambda c^2}{3H^2}=\frac{1.252\times10^{-52}\ \times\ (3\times10^8)^2}{3\ \times\ (2.26\times10^{-18})^2}$$

= 0.732 for dark energy

Summary
Ωb + Ωdm + ΩΛ = 0.998 implies a flat universe (space is Euclidean or an 'Einstein De Sitter Universe'. The figures are not final and are likely to change but not by much (between ± 5-10% for luminous and dark matter, ± 5% for dark energy).

Baryonic matter
Heavy elements- 0.2%
Stars- 0.4%
Free hydrogen and helium- 3.8%

Dark matter- 22.2%
Cold dark matter (WIMPs, MACHOs, axions, cosmic strings?), hot dark matter (fast moving neutrinos), light and warm dark matter

Dark energy- 73.2%
Cosmological constant
Quintessence
?

One theory for dark energy is that quantum fluctuations lead to the appearance and disappearance of virtual pairs of particles which continuously pop into and then out of existence. They cannot be measured directly but they produce energy that has an affect on the overall density of universe and the curvature of space.

(Continue- The Cosmological Constant Problem)

Λ as Planck Units

Planck length (lP)- 1.616x10^-35 m, Planck mass (mP)- 2.176x10^-8 kg, Planck density (lP3/mP)- 5.156x10^96 kg/m^3, Λ as density- 1.685x10^-25 kg/m^3

Λ as Planck Units- 1.685x10^-25/5.156x10^96 = 3.268x10^-122 in Planck units
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