# Thread: Universe's size and Dark energy value!

1. ## Universe's size and Dark energy value!

Relations between radius of universe and dark energy density

http://2.bp.blogspot.com/_udXTGLmmdK...s400/fig11.jpg
fig11. Relations between radius of universe and dark energy density

* mass density of ordinary matter = 1 proton/5m3
* Proton mass= 1.67264 X 10-27kg
* G =6.6726 X 10-11 m3/s2kg
* 1J = 6.242 x 1018 eV
* $n_-=n_+=n, m_-=(23.3/4.6)m_+=(5.06522)m_p$
* $\bar r_{ - + } = \frac{R}{{3.27273}}$
* If $\bar r_{ - + } = \frac{R}{{2.17879}}$ $(U_T \approx 0)$, dark energy density has a 1/3 smaller than $\bar r_{ - + } = \frac{R}{{3.27273}}$.

In a WMAP, observed value $\Lambda = 2.14 \pm 0.13 \times 10^{ -3} eV$

Dark energy density :
$
\rho _{de} = 2.09 \times 10^{ - 47} [_{ - 0.465}^{ + 0.557} ]GeV^4
$

Ridius of the Universe :
$
R_{UNI} = 96.76[_{ - 11.44}^{ + 12.13} ]Gly = 85.32 \sim 108.89Gly
$

( If $\bar r_{ - + } = \frac{R}{{2.17879}}$,
$
R_{UNI} = 118.8[_{ - 14.0}^{ + 14.9} ]Gly = 104.8 \sim 133.7 Gly
$
)

From Neil J. Cornish,
the universe' radius is at least 24Gpc(78Gly).

(2003, Neil J. Cornish, "Constraining the Topology of the Universe", http://arxiv.org/abs/astro-ph/0310233v1 )

[Proof]
In negative mass hypothesis, dark energy is corresponding to that positive potential term in total potential energy.

*Potential energy between positive mass and positive mass has - value:$U = \frac{{ - G(m_ +) (m_ +) }}{r} = 1U_ -$
*Potential energy between negative mass and positive mass has + value:$U = \frac{{ - G( - m_ - )(m_ +) }}{r} = 1U_ +$
*Potential energy between negative mass and negative mass has - value:$U = \frac{{ - G( - m_ - )( - m_ - )}}{r} = 1U_ -$

When the number of negative mass is n_- , and the number of positive mass is n_+ , total potential energy is given as follows.

$
U_T = \sum\limits_{i,j}^{i = n_ - ,j = n_ + } {(\frac{{Gm_{ - i} m_{ + j} }}{{r_{ - + ij} }})}
$

$
+\sum\limits_{i,j,i > j}^{i,j = n_ - } {(\frac{{ - Gm_{ - i} m_{ - j} }}{{r_{ - - ij} }})}
+ \sum\limits_{i,j,i > j}^{i,j = n_ + } {(\frac{{ - Gm_{ + i} m_{ + j} }}{{r_{ + + ij} }})}---(78)
$

$
U_T = (n_ - \times n_ + )(\frac{{Gm_ - m_ + }}{{\bar r_{ - + } }})$

$
+ (\frac{{n_ - (n_ - - 1)}}{2}(\frac{{ - Gm_ - m_ - }}{{\bar r_{ - - } }}) + \frac{{n_ + (n_ + - 1)}}{2}(\frac{{ - Gm_ + m_ + }}{{\bar r_{ + + } }})) ---(79)
$

In equation (79)
$
E_{de}=U_{de} = (n_ - \times n_ + )(\frac{{Gm_ - m_ + }}{{\bar r_{ - + } }})
$

If radius of the universe is 60Gyr, ordinary matter density is about proton 1ea/5m3. So, m+ = mp,

$
m_ - = km_ + \simeq (\frac{{23.3}}{{4.6}})m_ + = (5.06522)m_p$

(because that dark matter has about (23.3/4.6) times ordinary matter in WMAP)

From equation (95)

$
\bar r_{ - +} =(60Gyr/3.27273)= 1.73447 X 10^{26}m
$

From analysis of V-5,

If $U_T \ge 0$, $n_ - \approx n_ +$, Therefore, Define, $n_ - = n_ + = n$

$
V = \frac{{4\pi R^3 }}{3} = \frac{{4\pi \times (5.67648 \times 10^{26} )^3 }}{3} = 7.66171 \times 10^{80} m^3
$

$
n = \frac{{\rho V}}{{m_p }} = \frac{{(1m_p /5m^3 )V}}{{m_p }} = 1.53234 \times 10^{80}
$

( 1080 is about total proton number of our universe).

$
U_{de} = (kn^2 )(\frac{{Gm_p^2 }}{{\bar r_{ - + } }})
$

$
U_{de} = (5.06522)n^2 \frac{{(6.6726 \times 10^{ - 11} )(2.79772 \times 10^{ - 54} )}}{{1.73447 \times 10^{26} }}J
$

$
U_{de} = (n^2 ) \times 5.45168 \times 10^{ - 90} J = 1.28009 \times 10^{71} J
$

1J = 1kg(m/s)2 = 6.242 X 1018 eV

$
U_{de} = 7.99031 \times 10^{89} eV
$

$
\rho _{de} = \frac{{U_{de} }}{V} = \frac{{7.99031 \times 10^{89} eV}}{{7.66171 \times 10^{80} m^3 }} = \frac{{1.04289 \times 10^{ - 6} GeV}}{{cm^3 }}
$

Planck Unit transformation(1cm =0.5063 x 1014GeV-1 )

$
\rho _{de} = \frac{{1.04289 \times 10^{ - 6} GeV}}{{1.29784 \times 10^{41} GeV^{ - 3} }} = 0.80355 \times 10^{ - 47} GeV^4
$

$
\rho _{de} = 0.80355 \times 10^{ - 47} GeV^4
$

Observation value is $\rho _{obs} \approx 10^{ - 47} GeV^4$

If R=90Gyr, $\rho _{de} = 1.808 \times 10^{ - 47} GeV^4$(refer to fig11).

$
\rho _{de} \approx \rho _{obs}
$

[Proof end]

In Quantum Field Theory, the energy density of the vacuum is estimated as 1070GeV4, which is about 10117 orders of magnitude large than the observation value 10-47GeV4.

Dark energy density :
$
\rho _{de} = 2.09 \times 10^{ - 47} [_{ - 0.465}^{ + 0.557} ]GeV^4
$

Ridius of the Universe :
$
R_{UNI} = 96.76[_{ - 11.44}^{ + 12.13} ]Gly = 85.32 \sim 108.89Gly
$

=========
Hypothesis of Dark Matter and Dark Energy with Negative Mass :
http://vixra.org/abs/0907.0015