Ⅱ-6. Particle Physics

RIKEN Accel. Prog. Rep. 53

R = 12H02 and its implications to gravity and cosmology
Y. Akiba ∗1
In the preceding article,1) I report a model (QSTmodel) that can yield the formulas of the standard
model (SM) parameters. The model implies that theproduct of the Hubble constant H0 and the Plancktime tpl is H0 tpl = 2 × (6π)−48 = ϵ0 . This relation isderived as follows.
In general relativity, the action is written as∫∫ √√4
S = L −gdx = L det(gµν )d4 x,

which means that ϕ satisties the Laplace equation.
The general solution of ϕ isϕ(⃗r) = 1 − Σi

We note that ϕ2 = e2u = −g00 . In general relativity, g00 ≃ −1 − 2ϕG , where ϕG is the gravitational
potential. This means that ϕ = ϕG . Thus, the relation
R = 0 implies the Coulombic gravitational potential.
If we take into account the fact that H0 ̸= 0, thepotential equation for gravity becomes non-linear.

where gµν is the metric and L is the Lagrangian densityof the system.
In the QST model, this is expressed as√(1 + L̂) −g = 1 + L̂ + L̂g3D∆ϕ + 6H02 e2H0 t (ϕ − 1/ϕ) = 0.
One may question the validity of the QST model,
since the model implies that the Hubble constant is aconstant, which would mean that spacetime expands
exponentially at a constant rate. This contradicts
standard Big Bang cosmology. In the Big Bang theory,
H0 is not constant, and the scale of the universe is zeroat the Big Bang, which supposed to have occurred 13.8
billion years ago.
We note that there is no direct evidence that the Big
Bang occured 13.8 billion years ago, because the absolute scale of the time coordinate in the past is impossible to determine. The distance to an object far away
can only be measured by the red shift z. The valueof z is then translated to the time based on standard
cosmology by using the Friedmann equations. However, there is no experimental method to verify thatthe translation from z to t is correct.
The cosmic microwave background (CMB) is usuallyconsidered as the direct evidence of the Big Bang. I
argue that it is not. CMB is definitive evidence that
there was an era of z = 1100 when the spacetime scaleis 1/1100 of the present one and the temperature of
the Universe was 3000 K. In the standard cosmology, z = 1100 translates to 13.8 billion years ago and
380,000 years after the Big Bang. However, the only
basis for this time scale is the Friedmann equations. If
we assume that H0 is constant, the only change is thatthe time of CMB formation becomes about 1 trillion
years ago instead of 13.8 billion years ago.
The agreement between the abundance of 4 He andthe Big Bang nucleosynthesis (BBN) model is considered another evidence of the Big Bang. In standard
cosmology, BBN occured 3 minutes after the Big Bang.
However, it is impossible to determine the absolute
time when BBN occured. There is no observation that
contradicts with the assumpition that H0 is constant.

where L̂ is the elementary action (EA), and L̂g3D = 3ϵ0
is the EA corresponding to the gravity.
√−g = 1 + L̂g3D = 1 + 3ϵ0
Now the spacetime metric of the Hubble expansion isds2 = −dt2 + e2H0 t (dx2 + dy 2 + dz 2 ).
√For t = tpl , we have −g = e3H0 t = 1 + 3H0 tpl . Fromthe comparison, we have H0 tpl = ϵ0 . The Ricci scalar
curvature R and the 3D curvature K of this metric are
R = 12H02 ,K = 0.
Since H0 = ϵ0 /tpl is a constant, R is also a constant in
the QST model.
Now, we generalize the metric to allow local changesof the scale with a constraint that Ricci curvature
should remain constant.
ds2 = −e2u(x,y,z) dt2 + e2H0 t (dx2 + dy 2 + dz 2 ),R = 12H02 .

The Ricci curvature of this metric is
R = −2(∆u + (∇u)2 )e−2H0 t + 12H02 e−2u = 12H02 .
We can show that Coulomb gravity can be derived
from this metric. When we take the approximation
H0 ≃ 0, we have
R = −2(∆u + (∇u)2 ) = 0.
Now, we define ϕ = eu ; thus, we have
∆ϕ = ∆(eu ) = (∆u + (∇u)2 )ϕ = 0,∗1

rsi.
|⃗r − r⃗i |References
1) Y. Akiba, in this report.

RIKEN Nishina Center- 87 -