1. Introduction
The Standard Cosmological Model entails a space-time metric with line element [
1–
4]
(1)
where
c is the vacuum speed of light relative to a local Lorentz frame,
t is cosmic time,
a(
t) is the time-dependent scale factor of the Universe, and
r,
θ,
φ are spherical coordinates of a spatially flat (Euclidean) 3-space. A particle of proper mass
m has the Hamiltonian [
5,
6]
(2)
where
pr,
pθ, and
pφ are canonical momenta conjugate to canonical coordinates
r,
θ, and
φ, respectively. The azimuthal angular momentum
pφ is conserved because
H does not depend on
φ, but
H is not conserved unless all three momenta vanish. If
pφ = 0, the polar angular momentum
pθ is conserved because
H no longer depends on
θ. If
pθ =
pφ = 0, the radial momentum
pr is conserved because
H no longer depends on
r. Defining
pr ≡
p,
H then reduces to
(3)
showing how a particle with momentum loses energy to the expanding Universe [
5].
A logical question, then, is
where does that energy go? A logical answer is that it goes to the Hamiltonian of the Universe, which, in a one-dimensional minisuperspace model, can be expressed in geometrized units as the conserved quantity [
7]
(4)
with canonical momentum
P, canonical coordinate
x, effective mass
M, and potential
(5)
where
k is the spatial curvature constant (±1 or 0),
α is a constant such that
a =
α1/3x2/3, Λ is Einstein’s cosmological constant,
ρmo and
ρro are energy densities of matter and radiation, respectively, at some initial time
to, and
xo =
x(
to). The first term of
𝒰 is eliminated by metric (
1), for which
k = 0. The second term gives exponentially accelerated expansion of a vacuum dominated Universe, with Λ attributed to dark energy due to vacuum fluctuations, a quantum effect. The third term gives
x ∝
t, and
a ∝
t2/3, the Einstein-de Sitter scale factor of a matter dominated Universe. The fourth term gives
a ∝
t1/2 in a radiation dominated Universe. The constraint
ℋ = 0 gives the same differential equation for the expansion factor
a(
t) as is obtained from the Standard Cosmological Model [
4,
7].
However,
ℋ is defective, because
the Standard Cosmological Model is defective. ρmo only includes the rest mass
m of an object whose relativistic mass is Hamiltonian (
3) with momentum
p. This deficiency is corrected here by revising
ℋ to the form
(6)
where
a2 =
α2/3x4/3 in terms of the cosmic coordinate
x, thereby conserving
K by making
∂K/
∂t = 0. The sum over
n is taken over all objects
mn,
whose momenta pn create an interaction with the Universe [
7],
so they are no longer treated like test particles.
p denotes all the
pn, which remain constant because
K is independent of the radial coordinates
rn.
2. Canonical Equations
Hamiltonian (
6) is an example of scientific induction [
8], from which rigorous mathematical deduction, in the form of Hamilton’s equations, gives the coordinate velocities
(7)
dx/
dt =
∂K/
∂P =
P/
M. Defining
ℓn ≡
a rn as the proper or physical distance of object
n from a real or hypothetical observer and dropping the subscript
n, give the recession velocity
(8)
the Hubble velocity increased by positive
p and decreased by negative
p.
Using an overdot to denote
d/
dt and substituting
in (
6), give
(9)
indicating that
is a monotonically increasing function of
x, and
is positive definite. But this mathematical acceleration does not necessarily imply physical acceleration. Expressing (
9) in terms of the scale factor
a through the relation
x =
α−1/2a3/2 gives
(10)
where the first term on the right (
K α a−1) is a monotonically decreasing function of
a, while the second and third terms are monotonically increasing, so
can be positive or negative. Mathematics is simpler in terms of the canonical coordinate
x [
7], but physics is clearer in terms of the expansion factor
a. The time derivative of (
10) gives the acceleration
(11)
Defining the conserved quantities
(12)
conserved because the
mn and
pn are constant; Hamiltonian (
6) allows a momentum dominated epoch at very early times, a
matter dominated epoch at intermediate times, and a
vacuum dominated epoch at later times. Acceleration (
11) is positive in the momentum era, negative in the matter era, and positive again in the vacuum era.
3. Momentum Dominated Epoch
Momentum domination occurs when
Λ
and the
mn can be neglected in (
10) and (
11), giving
(13)
(14)
vanishes at scale factor
a1 = 2
𝒫/
K, the first inflection point, with
for
a <
a1 and deceleration for
a >
a1.
vanishes at scale factor
ao =
𝒫/
K and is positive definite for
a >
ao but imaginary for
a <
ao, which is thereby forbidden by (
13) as physically impossible. Thus, the momentum factor
𝒫 rules out a singularity by requiring the Universe to be created spatially flat with initial scale factor
ao > 0—the ultimate example of inflation—after which no further inflation is needed to achieve flatness or size. But (
14) indicates that accelerated expansion continues for
ao <
a <
a1, after which deceleration sets in.
Equation (
13) can be integrated to give
(15)
with the initial time being defined so that
a =
ao when
t = 0. Equation (
15) confirms the impossibility of the scale factor
a(
t) being less than
ao, since that would make
t imaginary.
The momentum factor 𝒫, defined in (12), is essential for these results and is independent of the signs of the pn in (6). This momentum symmetry, together with the time reversal symmetry of Einstein’s equation, allows a contracting Universe to undergo a smooth bounce at the minimum scale factor ao and then rebound from it as if it had been created at t = 0.
Unlike Hamiltonian ℋ of (4), Hamiltonian K of (6) need not vanish. K > 0 is necessary for accelerating expansion () in the momentum dominated era (ao < a < a1).
The recession velocity (
8) reduces to the form
(16)
indicating that
|
p|
is assumed to be so large that the motion of the receding object approximates that of a massless particle moving at luminal speed during this era. In this respect, it is like a radiation era but with repulsive radiation, rather than the attractive radiation of Hamiltonian (
4). But the end result is the same, as the Universe expands and matter domination sets in.
4. Matter and Vacuum Dominated Epochs
When the
pn can be neglected for
t ≥
τ (the starting time of matter domination), (
10) gives
(17)
For
t ≥
τ, (
17) is then readily integrated to give
(18)
Defining
T as the transition time from matter domination to vacuum domination, it follows that, for
τ ≤
t ≤
T and
, (
18) gives the Einstein-de Sitter scale factor
(19)
with
and
in the matter dominated era. For
, (
18) gives
(20)
the de Sitter scale factor, with
and
in the vacuum dominated era, so called because
Λ, which occurs naturally in Einstein’s equation, is a classical property of the vacuum, whose quantum fluctuations are not invoked here because virtual particle-antiparticle pairs created spontaneously from the vacuum can have positive or negative energy [
9], making it uncertain whether such vacuum fluctuations can explain
Λ, since their contributions to positive and negative
Λ
may be canceled.
5. Conclusions
The initial singularity of the Standard Model comes from neglecting the conserved momenta pn in the relativistic mass terms of Hamiltonian (6). When the pn are included, Einstein’s equation forbids a singularity, thereby disproving the singularity theorems [1, 3, 10]. This quantum leap in cosmology is achieved within the framework of general relativity, through the classical mechanism of momentum, without quantization or any non-Einsteinian effects. It does not improve on Einstein’s theory, but proves that Einstein’s theory is much better than it was thought to be. Other models based on a nonsingular bounce followed by expansion are not strictly Einsteinian, because they invoke other mechanisms [11] in lieu of the pn, whereas this galactic momentum is the essential mechanism of nonsingular Einsteinian cosmology.
