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Physics: energy and backreaction
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153 symbol spellings in 147 mathematical passages across 2 selected source files. Subscripts and superscripts have their own entries alongside their base symbols.
The same symbol can have several meanings here. A passage classified as a definition may define another part of its expression. Read the linked context before assigning a meaning.
All 153 entries are available below.
+Introduction passage · 16 passages
Search aliases: +
Introduction passage · Energy_Conserving_Production.tex, line 195
A finite quantum system with an exact energy budget. Regulator and state. Set q=fracsqrtγϕv, τ=At, κ=frackA, ω_hκ(q)=sqrtκ^2+y_h^2(1+q^2). After finite momentum quadrature, label a mode by j and give it a positive weight w_j, including its tower multiplicity. Let C=fracv^2γ A^2 =frac1γλ(A/μ)^2 =frac2S_Eε>0. The prime in the rest of this section denotes d/ dτ.
Exact mathematical passage
q=\frac{\sqrt\gamma\phi}{v},\quad \tau=At,\quad \kappa=\frac{k}{A},\quad \omega_{h\kappa}(q)=\sqrt{\kappa^2+y_h^2(1+q^2)}.
=Introduction passage · 51 passages
Search aliases: =
Introduction passage · Energy_Conserving_Production.tex, line 195
A finite quantum system with an exact energy budget. Regulator and state. Set q=fracsqrtγϕv, τ=At, κ=frackA, ω_hκ(q)=sqrtκ^2+y_h^2(1+q^2). After finite momentum quadrature, label a mode by j and give it a positive weight w_j, including its tower multiplicity. Let C=fracv^2γ A^2 =frac1γλ(A/μ)^2 =frac2S_Eε>0. The prime in the rest of this section denotes d/ dτ.
Exact mathematical passage
q=\frac{\sqrt\gamma\phi}{v},\quad \tau=At,\quad \kappa=\frac{k}{A},\quad \omega_{h\kappa}(q)=\sqrt{\kappa^2+y_h^2(1+q^2)}.
AIntroduction passage · 18 passages
Search aliases: A
Introduction passage · Energy_Conserving_Production.tex, line 195
A finite quantum system with an exact energy budget. Regulator and state. Set q=fracsqrtγϕv, τ=At, κ=frackA, ω_hκ(q)=sqrtκ^2+y_h^2(1+q^2). After finite momentum quadrature, label a mode by j and give it a positive weight w_j, including its tower multiplicity. Let C=fracv^2γ A^2 =frac1γλ(A/μ)^2 =frac2S_Eε>0. The prime in the rest of this section denotes d/ dτ.
Exact mathematical passage
q=\frac{\sqrt\gamma\phi}{v},\quad \tau=At,\quad \kappa=\frac{k}{A},\quad \omega_{h\kappa}(q)=\sqrt{\kappa^2+y_h^2(1+q^2)}.
A^2Introduction passage · 2 passages
Search aliases: A^2
Introduction passage · Energy_Conserving_Production.tex, line 201
ectionA finite quantum system with an exact energy budget Regulator and state. Set q=fracsqrtγϕv, τ=At, κ=frackA, ω_hκ(q)=sqrtκ^2+y_h^2(1+q^2). After finite momentum quadrature, label a mode by j and give it a positive weight w_j, including its tower multiplicity. Let C=fracv^2γ A^2 =frac1γλ(A/μ)^2 =frac2S_Eε>0. The prime in the rest of this section denotes d/ dτ.
Exact mathematical passage
C=\frac{v^2}{\gamma A^2} =\frac{1}{\gamma\lambda(A/\mu)^2} =\frac{2S_E}{\eps}>0.
CIntroduction passage · 7 passages
Search aliases: C
Introduction passage · Energy_Conserving_Production.tex, line 201
ectionA finite quantum system with an exact energy budget Regulator and state. Set q=fracsqrtγϕv, τ=At, κ=frackA, ω_hκ(q)=sqrtκ^2+y_h^2(1+q^2). After finite momentum quadrature, label a mode by j and give it a positive weight w_j, including its tower multiplicity. Let C=fracv^2γ A^2 =frac1γλ(A/μ)^2 =frac2S_Eε>0. The prime in the rest of this section denotes d/ dτ.
Exact mathematical passage
C=\frac{v^2}{\gamma A^2} =\frac{1}{\gamma\lambda(A/\mu)^2} =\frac{2S_E}{\eps}>0.
\ddIntroduction passage · 2 passages
Search aliases: dd d mathrm
Introduction passage · Energy_Conserving_Production.tex, line 206
d τ=At, κ=frackA, ω_hκ(q)=sqrtκ^2+y_h^2(1+q^2). After finite momentum quadrature, label a mode by j and give it a positive weight w_j, including its tower multiplicity. Let C=fracv^2γ A^2 =frac1γλ(A/μ)^2 =frac2S_Eε>0. The prime in the rest of this section denotes d/ dτ.
Exact mathematical passage
\dd/\dd\tau
EIntroduction passage · 5 passages
Search aliases: E
Introduction passage · Energy_Conserving_Production.tex, line 201
ectionA finite quantum system with an exact energy budget Regulator and state. Set q=fracsqrtγϕv, τ=At, κ=frackA, ω_hκ(q)=sqrtκ^2+y_h^2(1+q^2). After finite momentum quadrature, label a mode by j and give it a positive weight w_j, including its tower multiplicity. Let C=fracv^2γ A^2 =frac1γλ(A/μ)^2 =frac2S_Eε>0. The prime in the rest of this section denotes d/ dτ.
Exact mathematical passage
C=\frac{v^2}{\gamma A^2} =\frac{1}{\gamma\lambda(A/\mu)^2} =\frac{2S_E}{\eps}>0.
\epsIntroduction passage · 20 passages
Search aliases: eps ε varepsilon
Introduction passage · Energy_Conserving_Production.tex, line 201
ectionA finite quantum system with an exact energy budget Regulator and state. Set q=fracsqrtγϕv, τ=At, κ=frackA, ω_hκ(q)=sqrtκ^2+y_h^2(1+q^2). After finite momentum quadrature, label a mode by j and give it a positive weight w_j, including its tower multiplicity. Let C=fracv^2γ A^2 =frac1γλ(A/μ)^2 =frac2S_Eε>0. The prime in the rest of this section denotes d/ dτ.
Exact mathematical passage
C=\frac{v^2}{\gamma A^2} =\frac{1}{\gamma\lambda(A/\mu)^2} =\frac{2S_E}{\eps}>0.
hIntroduction passage · 12 passages
Search aliases: h
Introduction passage · Energy_Conserving_Production.tex, line 195
A finite quantum system with an exact energy budget. Regulator and state. Set q=fracsqrtγϕv, τ=At, κ=frackA, ω_hκ(q)=sqrtκ^2+y_h^2(1+q^2). After finite momentum quadrature, label a mode by j and give it a positive weight w_j, including its tower multiplicity. Let C=fracv^2γ A^2 =frac1γλ(A/μ)^2 =frac2S_Eε>0. The prime in the rest of this section denotes d/ dτ.
Exact mathematical passage
q=\frac{\sqrt\gamma\phi}{v},\quad \tau=At,\quad \kappa=\frac{k}{A},\quad \omega_{h\kappa}(q)=\sqrt{\kappa^2+y_h^2(1+q^2)}.
jIntroduction passage · 25 passages
Search aliases: j
Introduction passage · Energy_Conserving_Production.tex, line 200
A finite quantum system with an exact energy budget. Regulator and state. Set q=fracsqrtγϕv, τ=At, κ=frackA, ω_hκ(q)=sqrtκ^2+y_h^2(1+q^2). After finite momentum quadrature, label a mode by j and give it a positive weight w_j, including its tower multiplicity. Let C=fracv^2γ A^2 =frac1γλ(A/μ)^2 =frac2S_Eε>0. The prime in the rest of this section denotes d/ dτ.
Exact mathematical passage
j
kIntroduction passage · 5 passages
Search aliases: k
Introduction passage · Energy_Conserving_Production.tex, line 195
A finite quantum system with an exact energy budget. Regulator and state. Set q=fracsqrtγϕv, τ=At, κ=frackA, ω_hκ(q)=sqrtκ^2+y_h^2(1+q^2). After finite momentum quadrature, label a mode by j and give it a positive weight w_j, including its tower multiplicity. Let C=fracv^2γ A^2 =frac1γλ(A/μ)^2 =frac2S_Eε>0. The prime in the rest of this section denotes d/ dτ.
Exact mathematical passage
q=\frac{\sqrt\gamma\phi}{v},\quad \tau=At,\quad \kappa=\frac{k}{A},\quad \omega_{h\kappa}(q)=\sqrt{\kappa^2+y_h^2(1+q^2)}.
nIntroduction passage · 15 passages
Search aliases: n
Introduction passage · Energy_Conserving_Production.tex, line 209
For each mode use the instantaneous occupation n_j and two real coherence variables u_j,v_j. They obey n_j' · =frac12 W_j u_j, u_j' · =W_j(1+2n_j)-2ω_j v_j, v_j' · =2ω_j u_j, W_j=fracω_j,qq'ω_j. These equations are the oscillator Bogoliubov system in real variables. Quantum-kinetic and scattering descriptions of time-dependent pair production are closely related [source reference]. Their use here is defined directly by the finite oscillator model, not by importing an electromagnetic pair-product…
Exact mathematical passage
n_j'&=\frac12 W_j u_j,\\ u_j'&=W_j(1+2n_j)-2\omega_j v_j,\\ v_j'&=2\omega_j u_j,\qquad W_j=\frac{\omega_{j,q}q'}{\omega_j}. \label{eq:qke}
n_jIntroduction passage · 14 passages
Search aliases: n_j
Introduction passage · Energy_Conserving_Production.tex, line 209
For each mode use the instantaneous occupation n_j and two real coherence variables u_j,v_j. They obey n_j' · =frac12 W_j u_j, u_j' · =W_j(1+2n_j)-2ω_j v_j, v_j' · =2ω_j u_j, W_j=fracω_j,qq'ω_j. These equations are the oscillator Bogoliubov system in real variables. Quantum-kinetic and scattering descriptions of time-dependent pair production are closely related [source reference]. Their use here is defined directly by the finite oscillator model, not by importing an electromagnetic pair-product…
Exact mathematical passage
n_j'&=\frac12 W_j u_j,\\ u_j'&=W_j(1+2n_j)-2\omega_j v_j,\\ v_j'&=2\omega_j u_j,\qquad W_j=\frac{\omega_{j,q}q'}{\omega_j}. \label{eq:qke}
n_j'Introduction passage · 1 passage
Search aliases: n_j'
Introduction passage · Energy_Conserving_Production.tex, line 209
For each mode use the instantaneous occupation n_j and two real coherence variables u_j,v_j. They obey n_j' · =frac12 W_j u_j, u_j' · =W_j(1+2n_j)-2ω_j v_j, v_j' · =2ω_j u_j, W_j=fracω_j,qq'ω_j. These equations are the oscillator Bogoliubov system in real variables. Quantum-kinetic and scattering descriptions of time-dependent pair production are closely related [source reference]. Their use here is defined directly by the finite oscillator model, not by importing an electromagnetic pair-product…
Exact mathematical passage
n_j'&=\frac12 W_j u_j,\\ u_j'&=W_j(1+2n_j)-2\omega_j v_j,\\ v_j'&=2\omega_j u_j,\qquad W_j=\frac{\omega_{j,q}q'}{\omega_j}. \label{eq:qke}
qIntroduction passage · 21 passages
Search aliases: q
Introduction passage · Energy_Conserving_Production.tex, line 195
A finite quantum system with an exact energy budget. Regulator and state. Set q=fracsqrtγϕv, τ=At, κ=frackA, ω_hκ(q)=sqrtκ^2+y_h^2(1+q^2). After finite momentum quadrature, label a mode by j and give it a positive weight w_j, including its tower multiplicity. Let C=fracv^2γ A^2 =frac1γλ(A/μ)^2 =frac2S_Eε>0. The prime in the rest of this section denotes d/ dτ.
Exact mathematical passage
q=\frac{\sqrt\gamma\phi}{v},\quad \tau=At,\quad \kappa=\frac{k}{A},\quad \omega_{h\kappa}(q)=\sqrt{\kappa^2+y_h^2(1+q^2)}.
q'Introduction passage · 9 passages
Search aliases: q'
Introduction passage · Energy_Conserving_Production.tex, line 209
For each mode use the instantaneous occupation n_j and two real coherence variables u_j,v_j. They obey n_j' · =frac12 W_j u_j, u_j' · =W_j(1+2n_j)-2ω_j v_j, v_j' · =2ω_j u_j, W_j=fracω_j,qq'ω_j. These equations are the oscillator Bogoliubov system in real variables. Quantum-kinetic and scattering descriptions of time-dependent pair production are closely related [source reference]. Their use here is defined directly by the finite oscillator model, not by importing an electromagnetic pair-product…
Exact mathematical passage
n_j'&=\frac12 W_j u_j,\\ u_j'&=W_j(1+2n_j)-2\omega_j v_j,\\ v_j'&=2\omega_j u_j,\qquad W_j=\frac{\omega_{j,q}q'}{\omega_j}. \label{eq:qke}
q^2Introduction passage · 4 passages
Search aliases: q^2
Introduction passage · Energy_Conserving_Production.tex, line 195
A finite quantum system with an exact energy budget. Regulator and state. Set q=fracsqrtγϕv, τ=At, κ=frackA, ω_hκ(q)=sqrtκ^2+y_h^2(1+q^2). After finite momentum quadrature, label a mode by j and give it a positive weight w_j, including its tower multiplicity. Let C=fracv^2γ A^2 =frac1γλ(A/μ)^2 =frac2S_Eε>0. The prime in the rest of this section denotes d/ dτ.
Exact mathematical passage
q=\frac{\sqrt\gamma\phi}{v},\quad \tau=At,\quad \kappa=\frac{k}{A},\quad \omega_{h\kappa}(q)=\sqrt{\kappa^2+y_h^2(1+q^2)}.
SIntroduction passage · 5 passages
Search aliases: S
Introduction passage · Energy_Conserving_Production.tex, line 201
ectionA finite quantum system with an exact energy budget Regulator and state. Set q=fracsqrtγϕv, τ=At, κ=frackA, ω_hκ(q)=sqrtκ^2+y_h^2(1+q^2). After finite momentum quadrature, label a mode by j and give it a positive weight w_j, including its tower multiplicity. Let C=fracv^2γ A^2 =frac1γλ(A/μ)^2 =frac2S_Eε>0. The prime in the rest of this section denotes d/ dτ.
Exact mathematical passage
C=\frac{v^2}{\gamma A^2} =\frac{1}{\gamma\lambda(A/\mu)^2} =\frac{2S_E}{\eps}>0.
S_EIntroduction passage · 5 passages
Search aliases: S_E
Introduction passage · Energy_Conserving_Production.tex, line 201
ectionA finite quantum system with an exact energy budget Regulator and state. Set q=fracsqrtγϕv, τ=At, κ=frackA, ω_hκ(q)=sqrtκ^2+y_h^2(1+q^2). After finite momentum quadrature, label a mode by j and give it a positive weight w_j, including its tower multiplicity. Let C=fracv^2γ A^2 =frac1γλ(A/μ)^2 =frac2S_Eε>0. The prime in the rest of this section denotes d/ dτ.
Exact mathematical passage
C=\frac{v^2}{\gamma A^2} =\frac{1}{\gamma\lambda(A/\mu)^2} =\frac{2S_E}{\eps}>0.
tIntroduction passage · 1 passage
Search aliases: t
Introduction passage · Energy_Conserving_Production.tex, line 195
A finite quantum system with an exact energy budget. Regulator and state. Set q=fracsqrtγϕv, τ=At, κ=frackA, ω_hκ(q)=sqrtκ^2+y_h^2(1+q^2). After finite momentum quadrature, label a mode by j and give it a positive weight w_j, including its tower multiplicity. Let C=fracv^2γ A^2 =frac1γλ(A/μ)^2 =frac2S_Eε>0. The prime in the rest of this section denotes d/ dτ.
Exact mathematical passage
q=\frac{\sqrt\gamma\phi}{v},\quad \tau=At,\quad \kappa=\frac{k}{A},\quad \omega_{h\kappa}(q)=\sqrt{\kappa^2+y_h^2(1+q^2)}.
uIntroduction passage · 7 passages
Search aliases: u
Introduction passage · Energy_Conserving_Production.tex, line 209
For each mode use the instantaneous occupation n_j and two real coherence variables u_j,v_j. They obey n_j' · =frac12 W_j u_j, u_j' · =W_j(1+2n_j)-2ω_j v_j, v_j' · =2ω_j u_j, W_j=fracω_j,qq'ω_j. These equations are the oscillator Bogoliubov system in real variables. Quantum-kinetic and scattering descriptions of time-dependent pair production are closely related [source reference]. Their use here is defined directly by the finite oscillator model, not by importing an electromagnetic pair-product…
Exact mathematical passage
n_j'&=\frac12 W_j u_j,\\ u_j'&=W_j(1+2n_j)-2\omega_j v_j,\\ v_j'&=2\omega_j u_j,\qquad W_j=\frac{\omega_{j,q}q'}{\omega_j}. \label{eq:qke}
u_jIntroduction passage · 6 passages
Search aliases: u_j
Introduction passage · Energy_Conserving_Production.tex, line 209
For each mode use the instantaneous occupation n_j and two real coherence variables u_j,v_j. They obey n_j' · =frac12 W_j u_j, u_j' · =W_j(1+2n_j)-2ω_j v_j, v_j' · =2ω_j u_j, W_j=fracω_j,qq'ω_j. These equations are the oscillator Bogoliubov system in real variables. Quantum-kinetic and scattering descriptions of time-dependent pair production are closely related [source reference]. Their use here is defined directly by the finite oscillator model, not by importing an electromagnetic pair-product…
Exact mathematical passage
n_j'&=\frac12 W_j u_j,\\ u_j'&=W_j(1+2n_j)-2\omega_j v_j,\\ v_j'&=2\omega_j u_j,\qquad W_j=\frac{\omega_{j,q}q'}{\omega_j}. \label{eq:qke}
u_j'Introduction passage · 1 passage
Search aliases: u_j'
Introduction passage · Energy_Conserving_Production.tex, line 209
For each mode use the instantaneous occupation n_j and two real coherence variables u_j,v_j. They obey n_j' · =frac12 W_j u_j, u_j' · =W_j(1+2n_j)-2ω_j v_j, v_j' · =2ω_j u_j, W_j=fracω_j,qq'ω_j. These equations are the oscillator Bogoliubov system in real variables. Quantum-kinetic and scattering descriptions of time-dependent pair production are closely related [source reference]. Their use here is defined directly by the finite oscillator model, not by importing an electromagnetic pair-product…
Exact mathematical passage
n_j'&=\frac12 W_j u_j,\\ u_j'&=W_j(1+2n_j)-2\omega_j v_j,\\ v_j'&=2\omega_j u_j,\qquad W_j=\frac{\omega_{j,q}q'}{\omega_j}. \label{eq:qke}
vIntroduction passage · 10 passages
Search aliases: v
Introduction passage · Energy_Conserving_Production.tex, line 195
A finite quantum system with an exact energy budget. Regulator and state. Set q=fracsqrtγϕv, τ=At, κ=frackA, ω_hκ(q)=sqrtκ^2+y_h^2(1+q^2). After finite momentum quadrature, label a mode by j and give it a positive weight w_j, including its tower multiplicity. Let C=fracv^2γ A^2 =frac1γλ(A/μ)^2 =frac2S_Eε>0. The prime in the rest of this section denotes d/ dτ.
Exact mathematical passage
q=\frac{\sqrt\gamma\phi}{v},\quad \tau=At,\quad \kappa=\frac{k}{A},\quad \omega_{h\kappa}(q)=\sqrt{\kappa^2+y_h^2(1+q^2)}.
v^2Introduction passage · 3 passages
Search aliases: v^2
Introduction passage · Energy_Conserving_Production.tex, line 201
ectionA finite quantum system with an exact energy budget Regulator and state. Set q=fracsqrtγϕv, τ=At, κ=frackA, ω_hκ(q)=sqrtκ^2+y_h^2(1+q^2). After finite momentum quadrature, label a mode by j and give it a positive weight w_j, including its tower multiplicity. Let C=fracv^2γ A^2 =frac1γλ(A/μ)^2 =frac2S_Eε>0. The prime in the rest of this section denotes d/ dτ.
Exact mathematical passage
C=\frac{v^2}{\gamma A^2} =\frac{1}{\gamma\lambda(A/\mu)^2} =\frac{2S_E}{\eps}>0.
v_jIntroduction passage · 4 passages
Search aliases: v_j
Introduction passage · Energy_Conserving_Production.tex, line 209
For each mode use the instantaneous occupation n_j and two real coherence variables u_j,v_j. They obey n_j' · =frac12 W_j u_j, u_j' · =W_j(1+2n_j)-2ω_j v_j, v_j' · =2ω_j u_j, W_j=fracω_j,qq'ω_j. These equations are the oscillator Bogoliubov system in real variables. Quantum-kinetic and scattering descriptions of time-dependent pair production are closely related [source reference]. Their use here is defined directly by the finite oscillator model, not by importing an electromagnetic pair-product…
Exact mathematical passage
n_j'&=\frac12 W_j u_j,\\ u_j'&=W_j(1+2n_j)-2\omega_j v_j,\\ v_j'&=2\omega_j u_j,\qquad W_j=\frac{\omega_{j,q}q'}{\omega_j}. \label{eq:qke}
v_j'Introduction passage · 1 passage
Search aliases: v_j'
Introduction passage · Energy_Conserving_Production.tex, line 209
For each mode use the instantaneous occupation n_j and two real coherence variables u_j,v_j. They obey n_j' · =frac12 W_j u_j, u_j' · =W_j(1+2n_j)-2ω_j v_j, v_j' · =2ω_j u_j, W_j=fracω_j,qq'ω_j. These equations are the oscillator Bogoliubov system in real variables. Quantum-kinetic and scattering descriptions of time-dependent pair production are closely related [source reference]. Their use here is defined directly by the finite oscillator model, not by importing an electromagnetic pair-product…
Exact mathematical passage
n_j'&=\frac12 W_j u_j,\\ u_j'&=W_j(1+2n_j)-2\omega_j v_j,\\ v_j'&=2\omega_j u_j,\qquad W_j=\frac{\omega_{j,q}q'}{\omega_j}. \label{eq:qke}
WIntroduction passage · 1 passage
Search aliases: W
Introduction passage · Energy_Conserving_Production.tex, line 209
For each mode use the instantaneous occupation n_j and two real coherence variables u_j,v_j. They obey n_j' · =frac12 W_j u_j, u_j' · =W_j(1+2n_j)-2ω_j v_j, v_j' · =2ω_j u_j, W_j=fracω_j,qq'ω_j. These equations are the oscillator Bogoliubov system in real variables. Quantum-kinetic and scattering descriptions of time-dependent pair production are closely related [source reference]. Their use here is defined directly by the finite oscillator model, not by importing an electromagnetic pair-product…
Exact mathematical passage
n_j'&=\frac12 W_j u_j,\\ u_j'&=W_j(1+2n_j)-2\omega_j v_j,\\ v_j'&=2\omega_j u_j,\qquad W_j=\frac{\omega_{j,q}q'}{\omega_j}. \label{eq:qke}
wIntroduction passage · 7 passages
Search aliases: w
Introduction passage · Energy_Conserving_Production.tex, line 200
A finite quantum system with an exact energy budget. Regulator and state. Set q=fracsqrtγϕv, τ=At, κ=frackA, ω_hκ(q)=sqrtκ^2+y_h^2(1+q^2). After finite momentum quadrature, label a mode by j and give it a positive weight w_j, including its tower multiplicity. Let C=fracv^2γ A^2 =frac1γλ(A/μ)^2 =frac2S_Eε>0. The prime in the rest of this section denotes d/ dτ.
Exact mathematical passage
w_j
W_jIntroduction passage · 1 passage
Search aliases: W_j
Introduction passage · Energy_Conserving_Production.tex, line 209
For each mode use the instantaneous occupation n_j and two real coherence variables u_j,v_j. They obey n_j' · =frac12 W_j u_j, u_j' · =W_j(1+2n_j)-2ω_j v_j, v_j' · =2ω_j u_j, W_j=fracω_j,qq'ω_j. These equations are the oscillator Bogoliubov system in real variables. Quantum-kinetic and scattering descriptions of time-dependent pair production are closely related [source reference]. Their use here is defined directly by the finite oscillator model, not by importing an electromagnetic pair-product…
Exact mathematical passage
n_j'&=\frac12 W_j u_j,\\ u_j'&=W_j(1+2n_j)-2\omega_j v_j,\\ v_j'&=2\omega_j u_j,\qquad W_j=\frac{\omega_{j,q}q'}{\omega_j}. \label{eq:qke}
w_jIntroduction passage · 7 passages
Search aliases: w_j
Introduction passage · Energy_Conserving_Production.tex, line 200
A finite quantum system with an exact energy budget. Regulator and state. Set q=fracsqrtγϕv, τ=At, κ=frackA, ω_hκ(q)=sqrtκ^2+y_h^2(1+q^2). After finite momentum quadrature, label a mode by j and give it a positive weight w_j, including its tower multiplicity. Let C=fracv^2γ A^2 =frac1γλ(A/μ)^2 =frac2S_Eε>0. The prime in the rest of this section denotes d/ dτ.
Exact mathematical passage
w_j
yIntroduction passage · 10 passages
Search aliases: y
Introduction passage · Energy_Conserving_Production.tex, line 195
A finite quantum system with an exact energy budget. Regulator and state. Set q=fracsqrtγϕv, τ=At, κ=frackA, ω_hκ(q)=sqrtκ^2+y_h^2(1+q^2). After finite momentum quadrature, label a mode by j and give it a positive weight w_j, including its tower multiplicity. Let C=fracv^2γ A^2 =frac1γλ(A/μ)^2 =frac2S_Eε>0. The prime in the rest of this section denotes d/ dτ.
Exact mathematical passage
q=\frac{\sqrt\gamma\phi}{v},\quad \tau=At,\quad \kappa=\frac{k}{A},\quad \omega_{h\kappa}(q)=\sqrt{\kappa^2+y_h^2(1+q^2)}.
y_h^2Introduction passage · 3 passages
Search aliases: y_h^2
Introduction passage · Energy_Conserving_Production.tex, line 195
A finite quantum system with an exact energy budget. Regulator and state. Set q=fracsqrtγϕv, τ=At, κ=frackA, ω_hκ(q)=sqrtκ^2+y_h^2(1+q^2). After finite momentum quadrature, label a mode by j and give it a positive weight w_j, including its tower multiplicity. Let C=fracv^2γ A^2 =frac1γλ(A/μ)^2 =frac2S_Eε>0. The prime in the rest of this section denotes d/ dτ.
Exact mathematical passage
q=\frac{\sqrt\gamma\phi}{v},\quad \tau=At,\quad \kappa=\frac{k}{A},\quad \omega_{h\kappa}(q)=\sqrt{\kappa^2+y_h^2(1+q^2)}.
\gammaIntroduction passage · 10 passages
Search aliases: γ gamma
Introduction passage · Energy_Conserving_Production.tex, line 195
A finite quantum system with an exact energy budget. Regulator and state. Set q=fracsqrtγϕv, τ=At, κ=frackA, ω_hκ(q)=sqrtκ^2+y_h^2(1+q^2). After finite momentum quadrature, label a mode by j and give it a positive weight w_j, including its tower multiplicity. Let C=fracv^2γ A^2 =frac1γλ(A/μ)^2 =frac2S_Eε>0. The prime in the rest of this section denotes d/ dτ.
Exact mathematical passage
q=\frac{\sqrt\gamma\phi}{v},\quad \tau=At,\quad \kappa=\frac{k}{A},\quad \omega_{h\kappa}(q)=\sqrt{\kappa^2+y_h^2(1+q^2)}.
\kappaIntroduction passage · 3 passages
Search aliases: κ kappa
Introduction passage · Energy_Conserving_Production.tex, line 195
A finite quantum system with an exact energy budget. Regulator and state. Set q=fracsqrtγϕv, τ=At, κ=frackA, ω_hκ(q)=sqrtκ^2+y_h^2(1+q^2). After finite momentum quadrature, label a mode by j and give it a positive weight w_j, including its tower multiplicity. Let C=fracv^2γ A^2 =frac1γλ(A/μ)^2 =frac2S_Eε>0. The prime in the rest of this section denotes d/ dτ.
Exact mathematical passage
q=\frac{\sqrt\gamma\phi}{v},\quad \tau=At,\quad \kappa=\frac{k}{A},\quad \omega_{h\kappa}(q)=\sqrt{\kappa^2+y_h^2(1+q^2)}.
\kappa^2Introduction passage · 2 passages
Search aliases: κ^2 kappa
Introduction passage · Energy_Conserving_Production.tex, line 195
A finite quantum system with an exact energy budget. Regulator and state. Set q=fracsqrtγϕv, τ=At, κ=frackA, ω_hκ(q)=sqrtκ^2+y_h^2(1+q^2). After finite momentum quadrature, label a mode by j and give it a positive weight w_j, including its tower multiplicity. Let C=fracv^2γ A^2 =frac1γλ(A/μ)^2 =frac2S_Eε>0. The prime in the rest of this section denotes d/ dτ.
Exact mathematical passage
q=\frac{\sqrt\gamma\phi}{v},\quad \tau=At,\quad \kappa=\frac{k}{A},\quad \omega_{h\kappa}(q)=\sqrt{\kappa^2+y_h^2(1+q^2)}.
\lambdaIntroduction passage · 16 passages
Search aliases: λ lambda
Introduction passage · Energy_Conserving_Production.tex, line 201
ectionA finite quantum system with an exact energy budget Regulator and state. Set q=fracsqrtγϕv, τ=At, κ=frackA, ω_hκ(q)=sqrtκ^2+y_h^2(1+q^2). After finite momentum quadrature, label a mode by j and give it a positive weight w_j, including its tower multiplicity. Let C=fracv^2γ A^2 =frac1γλ(A/μ)^2 =frac2S_Eε>0. The prime in the rest of this section denotes d/ dτ.
Exact mathematical passage
C=\frac{v^2}{\gamma A^2} =\frac{1}{\gamma\lambda(A/\mu)^2} =\frac{2S_E}{\eps}>0.
\muIntroduction passage · 14 passages
Search aliases: μ mu
Introduction passage · Energy_Conserving_Production.tex, line 201
ectionA finite quantum system with an exact energy budget Regulator and state. Set q=fracsqrtγϕv, τ=At, κ=frackA, ω_hκ(q)=sqrtκ^2+y_h^2(1+q^2). After finite momentum quadrature, label a mode by j and give it a positive weight w_j, including its tower multiplicity. Let C=fracv^2γ A^2 =frac1γλ(A/μ)^2 =frac2S_Eε>0. The prime in the rest of this section denotes d/ dτ.
Exact mathematical passage
C=\frac{v^2}{\gamma A^2} =\frac{1}{\gamma\lambda(A/\mu)^2} =\frac{2S_E}{\eps}>0.
\tauIntroduction passage · 4 passages
Search aliases: τ tau
Introduction passage · Energy_Conserving_Production.tex, line 195
A finite quantum system with an exact energy budget. Regulator and state. Set q=fracsqrtγϕv, τ=At, κ=frackA, ω_hκ(q)=sqrtκ^2+y_h^2(1+q^2). After finite momentum quadrature, label a mode by j and give it a positive weight w_j, including its tower multiplicity. Let C=fracv^2γ A^2 =frac1γλ(A/μ)^2 =frac2S_Eε>0. The prime in the rest of this section denotes d/ dτ.
Exact mathematical passage
q=\frac{\sqrt\gamma\phi}{v},\quad \tau=At,\quad \kappa=\frac{k}{A},\quad \omega_{h\kappa}(q)=\sqrt{\kappa^2+y_h^2(1+q^2)}.
\phiIntroduction passage · 3 passages
Search aliases: ϕ phi
Introduction passage · Energy_Conserving_Production.tex, line 195
A finite quantum system with an exact energy budget. Regulator and state. Set q=fracsqrtγϕv, τ=At, κ=frackA, ω_hκ(q)=sqrtκ^2+y_h^2(1+q^2). After finite momentum quadrature, label a mode by j and give it a positive weight w_j, including its tower multiplicity. Let C=fracv^2γ A^2 =frac1γλ(A/μ)^2 =frac2S_Eε>0. The prime in the rest of this section denotes d/ dτ.
Exact mathematical passage
q=\frac{\sqrt\gamma\phi}{v},\quad \tau=At,\quad \kappa=\frac{k}{A},\quad \omega_{h\kappa}(q)=\sqrt{\kappa^2+y_h^2(1+q^2)}.
\omegaIntroduction passage · 12 passages
Search aliases: ω omega
Introduction passage · Energy_Conserving_Production.tex, line 195
A finite quantum system with an exact energy budget. Regulator and state. Set q=fracsqrtγϕv, τ=At, κ=frackA, ω_hκ(q)=sqrtκ^2+y_h^2(1+q^2). After finite momentum quadrature, label a mode by j and give it a positive weight w_j, including its tower multiplicity. Let C=fracv^2γ A^2 =frac1γλ(A/μ)^2 =frac2S_Eε>0. The prime in the rest of this section denotes d/ dτ.
Exact mathematical passage
q=\frac{\sqrt\gamma\phi}{v},\quad \tau=At,\quad \kappa=\frac{k}{A},\quad \omega_{h\kappa}(q)=\sqrt{\kappa^2+y_h^2(1+q^2)}.
\omega_{h\kappa}Introduction passage · 1 passage
Search aliases: ω_hκ omega kappa
Introduction passage · Energy_Conserving_Production.tex, line 195
A finite quantum system with an exact energy budget. Regulator and state. Set q=fracsqrtγϕv, τ=At, κ=frackA, ω_hκ(q)=sqrtκ^2+y_h^2(1+q^2). After finite momentum quadrature, label a mode by j and give it a positive weight w_j, including its tower multiplicity. Let C=fracv^2γ A^2 =frac1γλ(A/μ)^2 =frac2S_Eε>0. The prime in the rest of this section denotes d/ dτ.
Exact mathematical passage
q=\frac{\sqrt\gamma\phi}{v},\quad \tau=At,\quad \kappa=\frac{k}{A},\quad \omega_{h\kappa}(q)=\sqrt{\kappa^2+y_h^2(1+q^2)}.
\omega_jIntroduction passage · 8 passages
Search aliases: ω_j omega
Introduction passage · Energy_Conserving_Production.tex, line 209
For each mode use the instantaneous occupation n_j and two real coherence variables u_j,v_j. They obey n_j' · =frac12 W_j u_j, u_j' · =W_j(1+2n_j)-2ω_j v_j, v_j' · =2ω_j u_j, W_j=fracω_j,qq'ω_j. These equations are the oscillator Bogoliubov system in real variables. Quantum-kinetic and scattering descriptions of time-dependent pair production are closely related [source reference]. Their use here is defined directly by the finite oscillator model, not by importing an electromagnetic pair-product…
Exact mathematical passage
n_j'&=\frac12 W_j u_j,\\ u_j'&=W_j(1+2n_j)-2\omega_j v_j,\\ v_j'&=2\omega_j u_j,\qquad W_j=\frac{\omega_{j,q}q'}{\omega_j}. \label{eq:qke}
\omega_{j,q}Introduction passage · 3 passages
Search aliases: ω_j,q omega
Introduction passage · Energy_Conserving_Production.tex, line 209
For each mode use the instantaneous occupation n_j and two real coherence variables u_j,v_j. They obey n_j' · =frac12 W_j u_j, u_j' · =W_j(1+2n_j)-2ω_j v_j, v_j' · =2ω_j u_j, W_j=fracω_j,qq'ω_j. These equations are the oscillator Bogoliubov system in real variables. Quantum-kinetic and scattering descriptions of time-dependent pair production are closely related [source reference]. Their use here is defined directly by the finite oscillator model, not by importing an electromagnetic pair-product…
Exact mathematical passage
n_j'&=\frac12 W_j u_j,\\ u_j'&=W_j(1+2n_j)-2\omega_j v_j,\\ v_j'&=2\omega_j u_j,\qquad W_j=\frac{\omega_{j,q}q'}{\omega_j}. \label{eq:qke}
aUse in source · 13 passages
Search aliases: a
Use in source · Energy_Conserving_Production.tex, line 90
ildeρ=A^4 S_E(x), S_E(10^-4)=2.18498795564, where the exact discrete degeneracies are retained. The kinetic reservoir is K=fracv^2 A^22γ, ρ_ V=fracμ^44λ, frac Kρ_ V=frac2γ(frac Aμ)^2. The stored reference has λ_0=1 and a_0:=fracA_0μ_0 · =1.93844860142×10^9, fracK_0ρ_ V · =4.69697872544×10^17, R_0:=fracwidetildeρ_0K_0 · =2.6272875374×10^20. The original occupation projection used η_0=R_0^-1=3.80620691783×10^-21. It made the reported density equal to K_0, but did not supply equations that remove t…
Exact mathematical passage
a_0:=\frac{A_0}{\mu_0}&=1.93844860142\times10^9,\\ \frac{K_0}{\rhoV}&=4.69697872544\times10^{17},\\ R_0:=\frac{\widetilde\rho_0}{K_0} &=2.6272875374\times10^{20}.
A^4Use in source · 1 passage
Search aliases: A^4
Use in source · Energy_Conserving_Production.tex, line 78
Write the energy evaluated with the minimum masses as widetildeρ=A^4 S_E(x), S_E(10^-4)=2.18498795564, where the exact discrete degeneracies are retained. The kinetic reservoir is K=fracv^2 A^22γ, ρ_ V=fracμ^44λ, frac Kρ_ V=frac2γ(frac Aμ)^2. The stored reference has λ_0=1 and a_0:=fracA_0μ_0 · =1.93844860142×10^9,
Exact mathematical passage
\widetilde\rho=A^4 S_E(x),\qquad S_E(10^{-4})=2.18498795564, \label{eq:SE}
A_0Use in source · 2 passages
Search aliases: A_0
Use in source · Energy_Conserving_Production.tex, line 90
ildeρ=A^4 S_E(x), S_E(10^-4)=2.18498795564, where the exact discrete degeneracies are retained. The kinetic reservoir is K=fracv^2 A^22γ, ρ_ V=fracμ^44λ, frac Kρ_ V=frac2γ(frac Aμ)^2. The stored reference has λ_0=1 and a_0:=fracA_0μ_0 · =1.93844860142×10^9, fracK_0ρ_ V · =4.69697872544×10^17, R_0:=fracwidetildeρ_0K_0 · =2.6272875374×10^20. The original occupation projection used η_0=R_0^-1=3.80620691783×10^-21. It made the reported density equal to K_0, but did not supply equations that remove t…
Exact mathematical passage
a_0:=\frac{A_0}{\mu_0}&=1.93844860142\times10^9,\\ \frac{K_0}{\rhoV}&=4.69697872544\times10^{17},\\ R_0:=\frac{\widetilde\rho_0}{K_0} &=2.6272875374\times10^{20}.
a_0Use in source · 2 passages
Search aliases: a_0
Use in source · Energy_Conserving_Production.tex, line 90
ildeρ=A^4 S_E(x), S_E(10^-4)=2.18498795564, where the exact discrete degeneracies are retained. The kinetic reservoir is K=fracv^2 A^22γ, ρ_ V=fracμ^44λ, frac Kρ_ V=frac2γ(frac Aμ)^2. The stored reference has λ_0=1 and a_0:=fracA_0μ_0 · =1.93844860142×10^9, fracK_0ρ_ V · =4.69697872544×10^17, R_0:=fracwidetildeρ_0K_0 · =2.6272875374×10^20. The original occupation projection used η_0=R_0^-1=3.80620691783×10^-21. It made the reported density equal to K_0, but did not supply equations that remove t…
Exact mathematical passage
a_0:=\frac{A_0}{\mu_0}&=1.93844860142\times10^9,\\ \frac{K_0}{\rhoV}&=4.69697872544\times10^{17},\\ R_0:=\frac{\widetilde\rho_0}{K_0} &=2.6272875374\times10^{20}.
A_0^4Use in source · 2 passages
Search aliases: A_0^4
Use in source · Energy_Conserving_Production.tex, line 109
[Conditional abundance-preserving family] Fix ρ_ V, γ, the tower order, and x=Λ/A. Let the target density be the original projected value ρ_T=η_0 A_0^4 S_E=K_0. For any 0<ε≤1, choose λ=η_0ε^2, frac Aμ=fraca_0sqrtε, fracΛμ=xfraca_0sqrtε. Then the unprojected Gaussian family satisfies widetildeρ=ρ_T, K=fracK_0ε, fracwidetildeρK=ε. No common occupation multiplier is used in this construction. At fixed ρ_ V, μ^4=4λρ_ V, so widetildeρ=4ρ_ Vλ(A/μ)^4S_E. Substitute [source reference]. The factors ε^2 a…
Exact mathematical passage
\rho_T=\eta_0 A_0^4 S_E=K_0
a_0^4Use in source · 2 passages
Search aliases: a_0^4
Use in source · Energy_Conserving_Production.tex, line 352
[Invariant curvature power counting] At fixed ρ_ V and γ, the family [source reference] leaves N_H unchanged for each fixed depth: N_H =fracγη_0 a_0^416π^2∑_h=1^H d_hy_h^4 =fracγ64π^2ρ_ V∑_h=1^H d_hm_h^4. The factor λ(A/μ)^4 equals η_0a_0^4 by [source reference]. Equivalently, substitute μ^4=4λρ_ V into [source reference]. The physical masses are fixed along the family by [source reference].
Exact mathematical passage
\mathcal N_H =\frac{\gamma\eta_0 a_0^4}{16\pi^2}\sum_{h=1}^H d_hy_h^4 =\frac{\gamma}{64\pi^2\rhoV}\sum_{h=1}^H d_hm_h^4.
a_jUse in source · 5 passages
Search aliases: a_j
Use in source · Energy_Conserving_Production.tex, line 227
Precise subtraction convention. For complex mode functions a_j satisfying a_j''+ω_j^2a_j=0 and the normalization a_joverlinea_j'-overlinea_ja_j'=i, fix the variables by n_j=frac|a_j'|^2+ω_j^2|a_j|^22ω_j-frac12, u_j=ω_j|a_j|^2-frac|a_j'|^2ω_j, v_j=-2Re(overlinea_ja_j'). Use the finite energy E=frac C2(q')^2+ V(q) +∑_jw_j[frac|a_j'|^2+ω_j^2|a_j|^22 -fracω_j2]. The subtracted quantity depends on q, so its derivative must also be included in the force. This gives [source reference]. The bracket in […
Exact mathematical passage
a_j
a_j'Use in source · 3 passages
Search aliases: a_j'
Use in source · Energy_Conserving_Production.tex, line 227
Precise subtraction convention. For complex mode functions a_j satisfying a_j''+ω_j^2a_j=0 and the normalization a_joverlinea_j'-overlinea_ja_j'=i, fix the variables by n_j=frac|a_j'|^2+ω_j^2|a_j|^22ω_j-frac12, u_j=ω_j|a_j|^2-frac|a_j'|^2ω_j, v_j=-2Re(overlinea_ja_j'). Use the finite energy E=frac C2(q')^2+ V(q) +∑_jw_j[frac|a_j'|^2+ω_j^2|a_j|^22 -fracω_j2]. The subtracted quantity depends on q, so its derivative must also be included in the force. This gives [source reference]. The bracket in […
Exact mathematical passage
a_j\overline{a_j'}-\overline{a_j}a_j'=i
a_j''Use in source · 1 passage
Search aliases: a_j''
Use in source · Energy_Conserving_Production.tex, line 227
Precise subtraction convention. For complex mode functions a_j satisfying a_j''+ω_j^2a_j=0 and the normalization a_joverlinea_j'-overlinea_ja_j'=i, fix the variables by n_j=frac|a_j'|^2+ω_j^2|a_j|^22ω_j-frac12, u_j=ω_j|a_j|^2-frac|a_j'|^2ω_j, v_j=-2Re(overlinea_ja_j'). Use the finite energy E=frac C2(q')^2+ V(q) +∑_jw_j[frac|a_j'|^2+ω_j^2|a_j|^22 -fracω_j2]. The subtracted quantity depends on q, so its derivative must also be included in the force. This gives [source reference]. The bracket in […
Exact mathematical passage
a_j''+\omega_j^2a_j=0
a_\starUse in source · 2 passages
Search aliases: a_⋆ star
Use in source · Energy_Conserving_Production.tex, line 181
There is also a quantitative background cost. Using the stored benchmark inputs, not a new observational fit, a_⋆=10^-6, Ω_r0=9×10^-5, Ω_ V=0.68991, we obtain fracKρ_r(a_⋆)=frac0.003600547325ε. The ratio is 0.0360 at ε=0.1, 0.360 at 0.01, and 3.60 at 0.001. Taking ε arbitrarily small is therefore not a cost-free way to preserve the old cosmological history. These ratios are not observational exclusion bounds. They identify a background component that the next cosmological calculation must includ…
Exact mathematical passage
a_\star=10^{-6},\quad \Omega_{r0}=9\times10^{-5},\quad \Omega_{\mathrm V}=0.68991,
C_HUse in source · 2 passages
Search aliases: C_H
Use in source · Energy_Conserving_Production.tex, line 390
n Integrating this local second-adiabatic-order energy over the full momentum range gives E^(2)_H =frac(q')^296π^2fracq^21+q^2 ∑_h=1^H d_hy_h^2, because ∫_0^∞fracκ^2 dκ(κ^2+M^2)^5/2 =frac13M^2. It therefore corresponds to the local kinetic correction δ C_H(q)=frac148π^2fracq^21+q^2 ∑_h=1^H d_hy_h^2. At large |q|, δ C_H/C for ε=0.01 is approximately 0.121 at depth 14, 0.971 at 15, 7.77 at 16, and 62.2 at 17.
Exact mathematical passage
\delta C_H(q)=\frac{1}{48\pi^2}\frac{q^2}{1+q^2} \sum_{h=1}^H d_hy_h^2.
dUse in source · 5 passages
Search aliases: d
Use in source · Energy_Conserving_Production.tex, line 55
The obstruction being addressed. The source model is a real background scalar coupled to a finite tower of real scalar fields. In the massless-base benchmark, U(ϕ) · =fracλ4(ϕ^2-v^2)^2, · μ^2 · =λ v^2, m_h · =Λ 2^h, · d_h · =2^h-1, M_h^2(ϕ) · =m_h^2(1+γfracϕ^2v^2), · γ · =16. The general covariant construction and its limitations are given in the source monograph [source reference]. This study fixes the dyadic order-one tower used in the stored benchmark. The quartic λ is not a spectral eigenval…
Exact mathematical passage
U(\phi)&=\frac{\lambda}{4}(\phi^2-v^2)^2, &\mu^2&=\lambda v^2,\\ m_h&=\Lambda 2^h, &d_h&=2^{h-1},\\ M_h^2(\phi)&=m_h^2\left(1+\gamma\frac{\phi^2}{v^2}\right), &\gamma&=16.
d_hUse in source · 5 passages
Search aliases: d_h
Use in source · Energy_Conserving_Production.tex, line 55
The obstruction being addressed. The source model is a real background scalar coupled to a finite tower of real scalar fields. In the massless-base benchmark, U(ϕ) · =fracλ4(ϕ^2-v^2)^2, · μ^2 · =λ v^2, m_h · =Λ 2^h, · d_h · =2^h-1, M_h^2(ϕ) · =m_h^2(1+γfracϕ^2v^2), · γ · =16. The general covariant construction and its limitations are given in the source monograph [source reference]. This study fixes the dyadic order-one tower used in the stored benchmark. The quartic λ is not a spectral eigenval…
Exact mathematical passage
U(\phi)&=\frac{\lambda}{4}(\phi^2-v^2)^2, &\mu^2&=\lambda v^2,\\ m_h&=\Lambda 2^h, &d_h&=2^{h-1},\\ M_h^2(\phi)&=m_h^2\left(1+\gamma\frac{\phi^2}{v^2}\right), &\gamma&=16.
\dot\phiUse in source · 1 passage
Search aliases: dotϕ dot phi
Use in source · Energy_Conserving_Production.tex, line 66
For a prescribed locally linear crossing, put A=fracsqrtγ |dotϕ_⋆|v, x=fracΛ A=10^-4, y_h=fracm_hA=x2^h. The assumed single-crossing occupation is f_h(k)=exp [-π(fracm_hA+frack^2Am_h)]. Such a prescribed crossing does not account for depletion of its source. The distinction between prescribed production, particle-mass growth, and self-consistent reaction is also central in established preheating calculations [source reference].
Exact mathematical passage
A=\frac{\sqrt\gamma\,|\dot\phi_\star|}{v},\qquad x=\frac\Lambda A=10^{-4},\qquad y_h=\frac{m_h}{A}=x2^h.
\dot\phi_\starUse in source · 1 passage
Search aliases: dotϕ_⋆ dot phi star
Use in source · Energy_Conserving_Production.tex, line 66
For a prescribed locally linear crossing, put A=fracsqrtγ |dotϕ_⋆|v, x=fracΛ A=10^-4, y_h=fracm_hA=x2^h. The assumed single-crossing occupation is f_h(k)=exp [-π(fracm_hA+frack^2Am_h)]. Such a prescribed crossing does not account for depletion of its source. The distinction between prescribed production, particle-mass growth, and self-consistent reaction is also central in established preheating calculations [source reference].
Exact mathematical passage
A=\frac{\sqrt\gamma\,|\dot\phi_\star|}{v},\qquad x=\frac\Lambda A=10^{-4},\qquad y_h=\frac{m_h}{A}=x2^h.
\mathcal EUse in source · 6 passages
Search aliases: E mathcal
Use in source · Energy_Conserving_Production.tex, line 234
onvention For complex mode functions a_j satisfying a_j''+ω_j^2a_j=0 and the normalization a_joverlinea_j'-overlinea_ja_j'=i, fix the variables by n_j=frac|a_j'|^2+ω_j^2|a_j|^22ω_j-frac12, u_j=ω_j|a_j|^2-frac|a_j'|^2ω_j, v_j=-2Re(overlinea_ja_j'). Use the finite energy E=frac C2(q')^2+ V(q) +∑_jw_j[frac|a_j'|^2+ω_j^2|a_j|^22 -fracω_j2]. The subtracted quantity depends on q, so its derivative must also be included in the force. This gives [source reference]. The bracket in [source reference] equa…
Exact mathematical passage
\mathcal E=\frac C2(q')^2+\mathcal V(q) +\sum_jw_j\left[\frac{|a_j'|^2+\omega_j^2|a_j|^2}{2} -\frac{\omega_j}{2}\right]. \label{eq:hamiltonian}
eUse in source · 2 passages
Search aliases: e
Use in source · Energy_Conserving_Production.tex, line 150
The canonical coupling of level h is g_h=γλ(fracm_hμ)^2. At the stored peak level h=13, this becomes g_rm peak=0.1535680410 ε. The following numbers are algebraic consequences of the parameter choice, not fitted dynamical outputs.
Exact mathematical passage
g_{\rm peak}=0.1535680410\,\eps.
\mathcal E^{(2)}_HUse in source · 1 passage
Search aliases: E^(2)_H mathcal
Use in source · Energy_Conserving_Production.tex, line 378
Vacuum response from unproduced levels. The observed depth sensitivity has a useful analytic warning sign. In a sufficiently adiabatic mode, n^(2)≃frac(ω')^216ω^4, ω n^(2)≃fracy^4q^2(q')^216ω^5. Integrating this local second-adiabatic-order energy over the full momentum range gives E^(2)_H =frac(q')^296π^2fracq^21+q^2 ∑_h=1^H d_hy_h^2, because ∫_0^∞fracκ^2 dκ(κ^2+M^2)^5/2 =frac13M^2. It therefore corresponds to the local kinetic correction δ C_H(q)=frac148π^2fracq^21+q^2 ∑_h=1^H d_hy_h^2. At lar…
Exact mathematical passage
\mathcal E^{(2)}_H =\frac{(q')^2}{96\pi^2}\frac{q^2}{1+q^2} \sum_{h=1}^H d_hy_h^2, \label{eq:adiabatice}
\eps^{-2}Use in source · 1 passage
Search aliases: eps^-2 ε^-2 eps varepsilon
Use in source · Energy_Conserving_Production.tex, line 130
mily satisfies widetildeρ=ρ_T, K=fracK_0ε, fracwidetildeρK=ε. No common occupation multiplier is used in this construction. At fixed ρ_ V, μ^4=4λρ_ V, so widetildeρ=4ρ_ Vλ(A/μ)^4S_E. Substitute [source reference]. The factors ε^2 and ε^-2 cancel, leaving η_0 A_0^4S_E. Equation [source reference] gives K=K_0/ε. Division proves the final identity.
Exact mathematical passage
\eps^{-2}
\eps^2Use in source · 2 passages
Search aliases: eps^2 ε^2 eps varepsilon
Use in source · Energy_Conserving_Production.tex, line 110
[Conditional abundance-preserving family] Fix ρ_ V, γ, the tower order, and x=Λ/A. Let the target density be the original projected value ρ_T=η_0 A_0^4 S_E=K_0. For any 0<ε≤1, choose λ=η_0ε^2, frac Aμ=fraca_0sqrtε, fracΛμ=xfraca_0sqrtε. Then the unprojected Gaussian family satisfies widetildeρ=ρ_T, K=fracK_0ε, fracwidetildeρK=ε. No common occupation multiplier is used in this construction. At fixed ρ_ V, μ^4=4λρ_ V, so widetildeρ=4ρ_ Vλ(A/μ)^4S_E. Substitute [source reference]. The factors ε^2 a…
Exact mathematical passage
\boxed{\lambda=\eta_0\eps^2,\qquad \frac A\mu=\frac{a_0}{\sqrt\eps},\qquad \frac\Lambda\mu=x\frac{a_0}{\sqrt\eps}.} \label{eq:family}
\expUse in source · 1 passage
Search aliases: exp
Use in source · Energy_Conserving_Production.tex, line 71
For a prescribed locally linear crossing, put A=fracsqrtγ |dotϕ_⋆|v, x=fracΛ A=10^-4, y_h=fracm_hA=x2^h. The assumed single-crossing occupation is f_h(k)=exp [-π(fracm_hA+frack^2Am_h)]. Such a prescribed crossing does not account for depletion of its source. The distinction between prescribed production, particle-mass growth, and self-consistent reaction is also central in established preheating calculations [source reference].
Exact mathematical passage
f_h(k)=\exp\!\left[-\pi\left(\frac{m_h}{A}+\frac{k^2}{Am_h}\right)\right]. \label{eq:gaussian}
fUse in source · 2 passages
Search aliases: f
Use in source · Energy_Conserving_Production.tex, line 71
For a prescribed locally linear crossing, put A=fracsqrtγ |dotϕ_⋆|v, x=fracΛ A=10^-4, y_h=fracm_hA=x2^h. The assumed single-crossing occupation is f_h(k)=exp [-π(fracm_hA+frack^2Am_h)]. Such a prescribed crossing does not account for depletion of its source. The distinction between prescribed production, particle-mass growth, and self-consistent reaction is also central in established preheating calculations [source reference].
Exact mathematical passage
f_h(k)=\exp\!\left[-\pi\left(\frac{m_h}{A}+\frac{k^2}{Am_h}\right)\right]. \label{eq:gaussian}
f_hUse in source · 1 passage
Search aliases: f_h
Use in source · Energy_Conserving_Production.tex, line 71
For a prescribed locally linear crossing, put A=fracsqrtγ |dotϕ_⋆|v, x=fracΛ A=10^-4, y_h=fracm_hA=x2^h. The assumed single-crossing occupation is f_h(k)=exp [-π(fracm_hA+frack^2Am_h)]. Such a prescribed crossing does not account for depletion of its source. The distinction between prescribed production, particle-mass growth, and self-consistent reaction is also central in established preheating calculations [source reference].
Exact mathematical passage
f_h(k)=\exp\!\left[-\pi\left(\frac{m_h}{A}+\frac{k^2}{Am_h}\right)\right]. \label{eq:gaussian}
gUse in source · 3 passages
Search aliases: g
Use in source · Energy_Conserving_Production.tex, line 146
The canonical coupling of level h is g_h=γλ(fracm_hμ)^2. At the stored peak level h=13, this becomes g_rm peak=0.1535680410 ε. The following numbers are algebraic consequences of the parameter choice, not fitted dynamical outputs.
Exact mathematical passage
g_h=\gamma\lambda\left(\frac{m_h}{\mu}\right)^2.
g_hUse in source · 1 passage
Search aliases: g_h
Use in source · Energy_Conserving_Production.tex, line 146
The canonical coupling of level h is g_h=γλ(fracm_hμ)^2. At the stored peak level h=13, this becomes g_rm peak=0.1535680410 ε. The following numbers are algebraic consequences of the parameter choice, not fitted dynamical outputs.
Exact mathematical passage
g_h=\gamma\lambda\left(\frac{m_h}{\mu}\right)^2.
g_{\rm peak}Use in source · 2 passages
Search aliases: g_rm peak rm
Use in source · Energy_Conserving_Production.tex, line 150
The canonical coupling of level h is g_h=γλ(fracm_hμ)^2. At the stored peak level h=13, this becomes g_rm peak=0.1535680410 ε. The following numbers are algebraic consequences of the parameter choice, not fitted dynamical outputs.
Exact mathematical passage
g_{\rm peak}=0.1535680410\,\eps.
HUse in source · 6 passages
Search aliases: H
Use in source · Energy_Conserving_Production.tex, line 343
Why a smaller coupling is not a complete repair. A radiative quantity invariant along the abundance family. The source monograph gives the finite-depth one-loop curvature contribution with its subtraction-scale logarithms. To isolate power counting, define the nonnegative diagnostic N_H:=fracγλ16π^2 ∑_h=1^H d_h(fracm_hμ)^4. This omits logarithms and finite counterterms. It is neither a scheme-independent lower bound on the renormalized mass nor a proof that locking is impossible.
Exact mathematical passage
\mathcal N_H:=\frac{\gamma\lambda}{16\pi^2} \sum_{h=1}^H d_h\left(\frac{m_h}{\mu}\right)^4. \label{eq:curvatureproxy}
iUse in source · 3 passages
Search aliases: i
Use in source · Energy_Conserving_Production.tex, line 168
The reservoir cannot be ignored. [Residual-energy obligation] Suppose an event begins and ends at the same scalar potential value, expansion and external losses are negligible during that event, and the receiving sector gains ε K_i. Energy conservation requires K_f=(1-ε)K_i. In particular, weak transfer in that event cannot also bring the background to rest at the same potential value without another energy-transfer mechanism. Subtract the initial and final conserved total energies. The potentia…
Exact mathematical passage
\eps K_i
KUse in source · 10 passages
Search aliases: K
Use in source · Energy_Conserving_Production.tex, line 83
Write the energy evaluated with the minimum masses as widetildeρ=A^4 S_E(x), S_E(10^-4)=2.18498795564, where the exact discrete degeneracies are retained. The kinetic reservoir is K=fracv^2 A^22γ, ρ_ V=fracμ^44λ, frac Kρ_ V=frac2γ(frac Aμ)^2. The stored reference has λ_0=1 and a_0:=fracA_0μ_0 · =1.93844860142×10^9,
Exact mathematical passage
K=\frac{v^2 A^2}{2\gamma},\qquad \rhoV=\frac{\mu^4}{4\lambda},\qquad \frac K{\rhoV}=\frac{2}{\gamma}\left(\frac A\mu\right)^2. \label{eq:kinetic}
k^2Use in source · 1 passage
Search aliases: k^2
Use in source · Energy_Conserving_Production.tex, line 71
For a prescribed locally linear crossing, put A=fracsqrtγ |dotϕ_⋆|v, x=fracΛ A=10^-4, y_h=fracm_hA=x2^h. The assumed single-crossing occupation is f_h(k)=exp [-π(fracm_hA+frack^2Am_h)]. Such a prescribed crossing does not account for depletion of its source. The distinction between prescribed production, particle-mass growth, and self-consistent reaction is also central in established preheating calculations [source reference].
Exact mathematical passage
f_h(k)=\exp\!\left[-\pi\left(\frac{m_h}{A}+\frac{k^2}{Am_h}\right)\right]. \label{eq:gaussian}
K_0Use in source · 6 passages
Search aliases: K_0
Use in source · Energy_Conserving_Production.tex, line 90
ildeρ=A^4 S_E(x), S_E(10^-4)=2.18498795564, where the exact discrete degeneracies are retained. The kinetic reservoir is K=fracv^2 A^22γ, ρ_ V=fracμ^44λ, frac Kρ_ V=frac2γ(frac Aμ)^2. The stored reference has λ_0=1 and a_0:=fracA_0μ_0 · =1.93844860142×10^9, fracK_0ρ_ V · =4.69697872544×10^17, R_0:=fracwidetildeρ_0K_0 · =2.6272875374×10^20. The original occupation projection used η_0=R_0^-1=3.80620691783×10^-21. It made the reported density equal to K_0, but did not supply equations that remove t…
Exact mathematical passage
a_0:=\frac{A_0}{\mu_0}&=1.93844860142\times10^9,\\ \frac{K_0}{\rhoV}&=4.69697872544\times10^{17},\\ R_0:=\frac{\widetilde\rho_0}{K_0} &=2.6272875374\times10^{20}.
K_fUse in source · 1 passage
Search aliases: K_f
Use in source · Energy_Conserving_Production.tex, line 169
The reservoir cannot be ignored. [Residual-energy obligation] Suppose an event begins and ends at the same scalar potential value, expansion and external losses are negligible during that event, and the receiving sector gains ε K_i. Energy conservation requires K_f=(1-ε)K_i. In particular, weak transfer in that event cannot also bring the background to rest at the same potential value without another energy-transfer mechanism. Subtract the initial and final conserved total energies. The potentia…
Exact mathematical passage
K_f=(1-\eps)K_i.
K_iUse in source · 2 passages
Search aliases: K_i
Use in source · Energy_Conserving_Production.tex, line 168
The reservoir cannot be ignored. [Residual-energy obligation] Suppose an event begins and ends at the same scalar potential value, expansion and external losses are negligible during that event, and the receiving sector gains ε K_i. Energy conservation requires K_f=(1-ε)K_i. In particular, weak transfer in that event cannot also bring the background to rest at the same potential value without another energy-transfer mechanism. Subtract the initial and final conserved total energies. The potentia…
Exact mathematical passage
\eps K_i
MUse in source · 2 passages
Search aliases: M
Use in source · Energy_Conserving_Production.tex, line 55
The obstruction being addressed. The source model is a real background scalar coupled to a finite tower of real scalar fields. In the massless-base benchmark, U(ϕ) · =fracλ4(ϕ^2-v^2)^2, · μ^2 · =λ v^2, m_h · =Λ 2^h, · d_h · =2^h-1, M_h^2(ϕ) · =m_h^2(1+γfracϕ^2v^2), · γ · =16. The general covariant construction and its limitations are given in the source monograph [source reference]. This study fixes the dyadic order-one tower used in the stored benchmark. The quartic λ is not a spectral eigenval…
Exact mathematical passage
U(\phi)&=\frac{\lambda}{4}(\phi^2-v^2)^2, &\mu^2&=\lambda v^2,\\ m_h&=\Lambda 2^h, &d_h&=2^{h-1},\\ M_h^2(\phi)&=m_h^2\left(1+\gamma\frac{\phi^2}{v^2}\right), &\gamma&=16.
mUse in source · 7 passages
Search aliases: m
Use in source · Energy_Conserving_Production.tex, line 55
The obstruction being addressed. The source model is a real background scalar coupled to a finite tower of real scalar fields. In the massless-base benchmark, U(ϕ) · =fracλ4(ϕ^2-v^2)^2, · μ^2 · =λ v^2, m_h · =Λ 2^h, · d_h · =2^h-1, M_h^2(ϕ) · =m_h^2(1+γfracϕ^2v^2), · γ · =16. The general covariant construction and its limitations are given in the source monograph [source reference]. This study fixes the dyadic order-one tower used in the stored benchmark. The quartic λ is not a spectral eigenval…
Exact mathematical passage
U(\phi)&=\frac{\lambda}{4}(\phi^2-v^2)^2, &\mu^2&=\lambda v^2,\\ m_h&=\Lambda 2^h, &d_h&=2^{h-1},\\ M_h^2(\phi)&=m_h^2\left(1+\gamma\frac{\phi^2}{v^2}\right), &\gamma&=16.
M^2Use in source · 1 passage
Search aliases: M^2
Use in source · Energy_Conserving_Production.tex, line 385
iently adiabatic mode, n^(2)≃frac(ω')^216ω^4, ω n^(2)≃fracy^4q^2(q')^216ω^5. Integrating this local second-adiabatic-order energy over the full momentum range gives E^(2)_H =frac(q')^296π^2fracq^21+q^2 ∑_h=1^H d_hy_h^2, because ∫_0^∞fracκ^2 dκ(κ^2+M^2)^5/2 =frac13M^2. It therefore corresponds to the local kinetic correction δ C_H(q)=frac148π^2fracq^21+q^2 ∑_h=1^H d_hy_h^2. At large |q|, δ C_H/C for ε=0.01 is approximately 0.121 at depth 14, 0.971 at 15, 7.77 at 16, and 62.2 at 17.
Exact mathematical passage
\int_0^\infty\frac{\kappa^2\,\dd\kappa}{(\kappa^2+M^2)^{5/2}} =\frac{1}{3M^2}.
m_hUse in source · 5 passages
Search aliases: m_h
Use in source · Energy_Conserving_Production.tex, line 55
The obstruction being addressed. The source model is a real background scalar coupled to a finite tower of real scalar fields. In the massless-base benchmark, U(ϕ) · =fracλ4(ϕ^2-v^2)^2, · μ^2 · =λ v^2, m_h · =Λ 2^h, · d_h · =2^h-1, M_h^2(ϕ) · =m_h^2(1+γfracϕ^2v^2), · γ · =16. The general covariant construction and its limitations are given in the source monograph [source reference]. This study fixes the dyadic order-one tower used in the stored benchmark. The quartic λ is not a spectral eigenval…
Exact mathematical passage
U(\phi)&=\frac{\lambda}{4}(\phi^2-v^2)^2, &\mu^2&=\lambda v^2,\\ m_h&=\Lambda 2^h, &d_h&=2^{h-1},\\ M_h^2(\phi)&=m_h^2\left(1+\gamma\frac{\phi^2}{v^2}\right), &\gamma&=16.
M_h^2Use in source · 1 passage
Search aliases: M_h^2
Use in source · Energy_Conserving_Production.tex, line 55
The obstruction being addressed. The source model is a real background scalar coupled to a finite tower of real scalar fields. In the massless-base benchmark, U(ϕ) · =fracλ4(ϕ^2-v^2)^2, · μ^2 · =λ v^2, m_h · =Λ 2^h, · d_h · =2^h-1, M_h^2(ϕ) · =m_h^2(1+γfracϕ^2v^2), · γ · =16. The general covariant construction and its limitations are given in the source monograph [source reference]. This study fixes the dyadic order-one tower used in the stored benchmark. The quartic λ is not a spectral eigenval…
Exact mathematical passage
U(\phi)&=\frac{\lambda}{4}(\phi^2-v^2)^2, &\mu^2&=\lambda v^2,\\ m_h&=\Lambda 2^h, &d_h&=2^{h-1},\\ M_h^2(\phi)&=m_h^2\left(1+\gamma\frac{\phi^2}{v^2}\right), &\gamma&=16.
m_h^2Use in source · 1 passage
Search aliases: m_h^2
Use in source · Energy_Conserving_Production.tex, line 55
The obstruction being addressed. The source model is a real background scalar coupled to a finite tower of real scalar fields. In the massless-base benchmark, U(ϕ) · =fracλ4(ϕ^2-v^2)^2, · μ^2 · =λ v^2, m_h · =Λ 2^h, · d_h · =2^h-1, M_h^2(ϕ) · =m_h^2(1+γfracϕ^2v^2), · γ · =16. The general covariant construction and its limitations are given in the source monograph [source reference]. This study fixes the dyadic order-one tower used in the stored benchmark. The quartic λ is not a spectral eigenval…
Exact mathematical passage
U(\phi)&=\frac{\lambda}{4}(\phi^2-v^2)^2, &\mu^2&=\lambda v^2,\\ m_h&=\Lambda 2^h, &d_h&=2^{h-1},\\ M_h^2(\phi)&=m_h^2\left(1+\gamma\frac{\phi^2}{v^2}\right), &\gamma&=16.
m_h^4Use in source · 1 passage
Search aliases: m_h^4
Use in source · Energy_Conserving_Production.tex, line 352
[Invariant curvature power counting] At fixed ρ_ V and γ, the family [source reference] leaves N_H unchanged for each fixed depth: N_H =fracγη_0 a_0^416π^2∑_h=1^H d_hy_h^4 =fracγ64π^2ρ_ V∑_h=1^H d_hm_h^4. The factor λ(A/μ)^4 equals η_0a_0^4 by [source reference]. Equivalently, substitute μ^4=4λρ_ V into [source reference]. The physical masses are fixed along the family by [source reference].
Exact mathematical passage
\mathcal N_H =\frac{\gamma\eta_0 a_0^4}{16\pi^2}\sum_{h=1}^H d_hy_h^4 =\frac{\gamma}{64\pi^2\rhoV}\sum_{h=1}^H d_hm_h^4.
\maxUse in source · 2 passages
Search aliases: max
Use in source · Energy_Conserving_Production.tex, line 290
The DOP853 integration uses relative tolerance 2×10^-10 and absolute tolerance 2×10^-13. The largest total-energy residual in the principal run is max_τ|frac E(τ) E(0)-1| =7.34×10^-11. The maximum hyperboloid residual is 3.73×10^-13. The largest instantaneous mode-energy fraction is 0.648634, and the endpoint fraction is 0.126747. The final position is q=6.65567, rather than the externally forced value q=10.
Exact mathematical passage
\max_\tau\left|\frac{\mathcal E(\tau)}{\mathcal E(0)}-1\right| =7.34\times10^{-11}.
\max_\tauUse in source · 1 passage
Search aliases: max_τ max tau
Use in source · Energy_Conserving_Production.tex, line 290
The DOP853 integration uses relative tolerance 2×10^-10 and absolute tolerance 2×10^-13. The largest total-energy residual in the principal run is max_τ|frac E(τ) E(0)-1| =7.34×10^-11. The maximum hyperboloid residual is 3.73×10^-13. The largest instantaneous mode-energy fraction is 0.648634, and the endpoint fraction is 0.126747. The final position is q=6.65567, rather than the externally forced value q=10.
Exact mathematical passage
\max_\tau\left|\frac{\mathcal E(\tau)}{\mathcal E(0)}-1\right| =7.34\times10^{-11}.
\mathcal NUse in source · 3 passages
Search aliases: N mathcal
Use in source · Energy_Conserving_Production.tex, line 343
Why a smaller coupling is not a complete repair. A radiative quantity invariant along the abundance family. The source monograph gives the finite-depth one-loop curvature contribution with its subtraction-scale logarithms. To isolate power counting, define the nonnegative diagnostic N_H:=fracγλ16π^2 ∑_h=1^H d_h(fracm_hμ)^4. This omits logarithms and finite counterterms. It is neither a scheme-independent lower bound on the renormalized mass nor a proof that locking is impossible.
Exact mathematical passage
\mathcal N_H:=\frac{\gamma\lambda}{16\pi^2} \sum_{h=1}^H d_h\left(\frac{m_h}{\mu}\right)^4. \label{eq:curvatureproxy}
n^{(2)}Use in source · 1 passage
Search aliases: n^(2)
Use in source · Energy_Conserving_Production.tex, line 373
Vacuum response from unproduced levels. The observed depth sensitivity has a useful analytic warning sign. In a sufficiently adiabatic mode, n^(2)≃frac(ω')^216ω^4, ω n^(2)≃fracy^4q^2(q')^216ω^5. Integrating this local second-adiabatic-order energy over the full momentum range gives E^(2)_H =frac(q')^296π^2fracq^21+q^2 ∑_h=1^H d_hy_h^2, because ∫_0^∞fracκ^2 dκ(κ^2+M^2)^5/2 =frac13M^2. It therefore corresponds to the local kinetic correction δ C_H(q)=frac148π^2fracq^21+q^2 ∑_h=1^H d_hy_h^2. At lar…
Exact mathematical passage
n^{(2)}\simeq\frac{(\omega')^2}{16\omega^4},\qquad \omega n^{(2)}\simeq\frac{y^4q^2(q')^2}{16\omega^5}.
\mathcal N_HUse in source · 3 passages
Search aliases: N_H mathcal
Use in source · Energy_Conserving_Production.tex, line 343
Why a smaller coupling is not a complete repair. A radiative quantity invariant along the abundance family. The source monograph gives the finite-depth one-loop curvature contribution with its subtraction-scale logarithms. To isolate power counting, define the nonnegative diagnostic N_H:=fracγλ16π^2 ∑_h=1^H d_h(fracm_hμ)^4. This omits logarithms and finite counterterms. It is neither a scheme-independent lower bound on the renormalized mass nor a proof that locking is impossible.
Exact mathematical passage
\mathcal N_H:=\frac{\gamma\lambda}{16\pi^2} \sum_{h=1}^H d_h\left(\frac{m_h}{\mu}\right)^4. \label{eq:curvatureproxy}
\overline{a_j}Use in source · 2 passages
Search aliases: overlinea_j overline
Use in source · Energy_Conserving_Production.tex, line 227
Precise subtraction convention. For complex mode functions a_j satisfying a_j''+ω_j^2a_j=0 and the normalization a_joverlinea_j'-overlinea_ja_j'=i, fix the variables by n_j=frac|a_j'|^2+ω_j^2|a_j|^22ω_j-frac12, u_j=ω_j|a_j|^2-frac|a_j'|^2ω_j, v_j=-2Re(overlinea_ja_j'). Use the finite energy E=frac C2(q')^2+ V(q) +∑_jw_j[frac|a_j'|^2+ω_j^2|a_j|^22 -fracω_j2]. The subtracted quantity depends on q, so its derivative must also be included in the force. This gives [source reference]. The bracket in […
Exact mathematical passage
a_j\overline{a_j'}-\overline{a_j}a_j'=i
\overline{a_j'}Use in source · 1 passage
Search aliases: overlinea_j' overline
Use in source · Energy_Conserving_Production.tex, line 227
Precise subtraction convention. For complex mode functions a_j satisfying a_j''+ω_j^2a_j=0 and the normalization a_joverlinea_j'-overlinea_ja_j'=i, fix the variables by n_j=frac|a_j'|^2+ω_j^2|a_j|^22ω_j-frac12, u_j=ω_j|a_j|^2-frac|a_j'|^2ω_j, v_j=-2Re(overlinea_ja_j'). Use the finite energy E=frac C2(q')^2+ V(q) +∑_jw_j[frac|a_j'|^2+ω_j^2|a_j|^22 -fracω_j2]. The subtracted quantity depends on q, so its derivative must also be included in the force. This gives [source reference]. The bracket in […
Exact mathematical passage
a_j\overline{a_j'}-\overline{a_j}a_j'=i
pUse in source · 2 passages
Search aliases: p
Use in source · Energy_Conserving_Production.tex, line 150
The canonical coupling of level h is g_h=γλ(fracm_hμ)^2. At the stored peak level h=13, this becomes g_rm peak=0.1535680410 ε. The following numbers are algebraic consequences of the parameter choice, not fitted dynamical outputs.
Exact mathematical passage
g_{\rm peak}=0.1535680410\,\eps.
q''Use in source · 1 passage
Search aliases: q''
Use in source · Energy_Conserving_Production.tex, line 219
The reaction force required by the same finite Hamiltonian is Cq''+ V'(q) +∑_j w_jω_j,q(n_j+fracu_j2)=0. The coherence term is essential. An occupation-only force does not conserve the energy of [source reference].
Exact mathematical passage
\boxed{Cq''+\mathcal V'(q) +\sum_j w_j\omega_{j,q}\left(n_j+\frac{u_j}{2}\right)=0.} \label{eq:reaction}
RUse in source · 2 passages
Search aliases: R
Use in source · Energy_Conserving_Production.tex, line 90
ildeρ=A^4 S_E(x), S_E(10^-4)=2.18498795564, where the exact discrete degeneracies are retained. The kinetic reservoir is K=fracv^2 A^22γ, ρ_ V=fracμ^44λ, frac Kρ_ V=frac2γ(frac Aμ)^2. The stored reference has λ_0=1 and a_0:=fracA_0μ_0 · =1.93844860142×10^9, fracK_0ρ_ V · =4.69697872544×10^17, R_0:=fracwidetildeρ_0K_0 · =2.6272875374×10^20. The original occupation projection used η_0=R_0^-1=3.80620691783×10^-21. It made the reported density equal to K_0, but did not supply equations that remove t…
Exact mathematical passage
a_0:=\frac{A_0}{\mu_0}&=1.93844860142\times10^9,\\ \frac{K_0}{\rhoV}&=4.69697872544\times10^{17},\\ R_0:=\frac{\widetilde\rho_0}{K_0} &=2.6272875374\times10^{20}.
rUse in source · 2 passages
Search aliases: r
Use in source · Energy_Conserving_Production.tex, line 181
There is also a quantitative background cost. Using the stored benchmark inputs, not a new observational fit, a_⋆=10^-6, Ω_r0=9×10^-5, Ω_ V=0.68991, we obtain fracKρ_r(a_⋆)=frac0.003600547325ε. The ratio is 0.0360 at ε=0.1, 0.360 at 0.01, and 3.60 at 0.001. Taking ε arbitrarily small is therefore not a cost-free way to preserve the old cosmological history. These ratios are not observational exclusion bounds. They identify a background component that the next cosmological calculation must includ…
Exact mathematical passage
a_\star=10^{-6},\quad \Omega_{r0}=9\times10^{-5},\quad \Omega_{\mathrm V}=0.68991,
R_0Use in source · 1 passage
Search aliases: R_0
Use in source · Energy_Conserving_Production.tex, line 90
ildeρ=A^4 S_E(x), S_E(10^-4)=2.18498795564, where the exact discrete degeneracies are retained. The kinetic reservoir is K=fracv^2 A^22γ, ρ_ V=fracμ^44λ, frac Kρ_ V=frac2γ(frac Aμ)^2. The stored reference has λ_0=1 and a_0:=fracA_0μ_0 · =1.93844860142×10^9, fracK_0ρ_ V · =4.69697872544×10^17, R_0:=fracwidetildeρ_0K_0 · =2.6272875374×10^20. The original occupation projection used η_0=R_0^-1=3.80620691783×10^-21. It made the reported density equal to K_0, but did not supply equations that remove t…
Exact mathematical passage
a_0:=\frac{A_0}{\mu_0}&=1.93844860142\times10^9,\\ \frac{K_0}{\rhoV}&=4.69697872544\times10^{17},\\ R_0:=\frac{\widetilde\rho_0}{K_0} &=2.6272875374\times10^{20}.
R_0^{-1}Use in source · 1 passage
Search aliases: R_0^-1
Use in source · Energy_Conserving_Production.tex, line 97
R_0:=fracwidetildeρ_0K_0 · =2.6272875374×10^20. The original occupation projection used η_0=R_0^-1=3.80620691783×10^-21. It made the reported density equal to K_0, but did not supply equations that remove this energy from the evolving scalar.
Exact mathematical passage
\eta_0=R_0^{-1}=3.80620691783\times10^{-21}. \label{eq:eta}
\operatorname{Re}Use in source · 1 passage
Search aliases: Re operatorname
Use in source · Energy_Conserving_Production.tex, line 228
Precise subtraction convention. For complex mode functions a_j satisfying a_j''+ω_j^2a_j=0 and the normalization a_joverlinea_j'-overlinea_ja_j'=i, fix the variables by n_j=frac|a_j'|^2+ω_j^2|a_j|^22ω_j-frac12, u_j=ω_j|a_j|^2-frac|a_j'|^2ω_j, v_j=-2Re(overlinea_ja_j'). Use the finite energy E=frac C2(q')^2+ V(q) +∑_jw_j[frac|a_j'|^2+ω_j^2|a_j|^22 -fracω_j2]. The subtracted quantity depends on q, so its derivative must also be included in the force. This gives [source reference]. The bracket in […
Exact mathematical passage
n_j=\frac{|a_j'|^2+\omega_j^2|a_j|^2}{2\omega_j}-\frac12, \quad u_j=\omega_j|a_j|^2-\frac{|a_j'|^2}{\omega_j}, \quad v_j=-2\operatorname{Re}(\overline{a_j}a_j').
\rhoVUse in source · 9 passages
Search aliases: rhoV ρ_ V rho mathrm
Use in source · Energy_Conserving_Production.tex, line 83
Write the energy evaluated with the minimum masses as widetildeρ=A^4 S_E(x), S_E(10^-4)=2.18498795564, where the exact discrete degeneracies are retained. The kinetic reservoir is K=fracv^2 A^22γ, ρ_ V=fracμ^44λ, frac Kρ_ V=frac2γ(frac Aμ)^2. The stored reference has λ_0=1 and a_0:=fracA_0μ_0 · =1.93844860142×10^9,
Exact mathematical passage
K=\frac{v^2 A^2}{2\gamma},\qquad \rhoV=\frac{\mu^4}{4\lambda},\qquad \frac K{\rhoV}=\frac{2}{\gamma}\left(\frac A\mu\right)^2. \label{eq:kinetic}
\rmUse in source · 2 passages
Search aliases: rm
Use in source · Energy_Conserving_Production.tex, line 150
The canonical coupling of level h is g_h=γλ(fracm_hμ)^2. At the stored peak level h=13, this becomes g_rm peak=0.1535680410 ε. The following numbers are algebraic consequences of the parameter choice, not fitted dynamical outputs.
Exact mathematical passage
g_{\rm peak}=0.1535680410\,\eps.
TUse in source · 2 passages
Search aliases: T
Use in source · Energy_Conserving_Production.tex, line 109
[Conditional abundance-preserving family] Fix ρ_ V, γ, the tower order, and x=Λ/A. Let the target density be the original projected value ρ_T=η_0 A_0^4 S_E=K_0. For any 0<ε≤1, choose λ=η_0ε^2, frac Aμ=fraca_0sqrtε, fracΛμ=xfraca_0sqrtε. Then the unprojected Gaussian family satisfies widetildeρ=ρ_T, K=fracK_0ε, fracwidetildeρK=ε. No common occupation multiplier is used in this construction. At fixed ρ_ V, μ^4=4λρ_ V, so widetildeρ=4ρ_ Vλ(A/μ)^4S_E. Substitute [source reference]. The factors ε^2 a…
Exact mathematical passage
\rho_T=\eta_0 A_0^4 S_E=K_0
UUse in source · 1 passage
Search aliases: U
Use in source · Energy_Conserving_Production.tex, line 55
The obstruction being addressed. The source model is a real background scalar coupled to a finite tower of real scalar fields. In the massless-base benchmark, U(ϕ) · =fracλ4(ϕ^2-v^2)^2, · μ^2 · =λ v^2, m_h · =Λ 2^h, · d_h · =2^h-1, M_h^2(ϕ) · =m_h^2(1+γfracϕ^2v^2), · γ · =16. The general covariant construction and its limitations are given in the source monograph [source reference]. This study fixes the dyadic order-one tower used in the stored benchmark. The quartic λ is not a spectral eigenval…
Exact mathematical passage
U(\phi)&=\frac{\lambda}{4}(\phi^2-v^2)^2, &\mu^2&=\lambda v^2,\\ m_h&=\Lambda 2^h, &d_h&=2^{h-1},\\ M_h^2(\phi)&=m_h^2\left(1+\gamma\frac{\phi^2}{v^2}\right), &\gamma&=16.
u_j^2Use in source · 1 passage
Search aliases: u_j^2
Use in source · Energy_Conserving_Production.tex, line 246
[Finite energy conservation and positivity] Assume C>0, w_j>0, y_j>0, and V≥0 is smooth. Start the modes in the instantaneous vacuum n_j=u_j=v_j=0. Equations [source reference]--[source reference] preserve (1+2n_j)^2-u_j^2-v_j^2=1, and conserve E=frac C2(q')^2+ V(q) +∑_jw_jω_j n_j. Every occupation remains nonnegative. Consequently 0≤∑_jw_jω_j n_j≤ E(0). Differentiate the left side of [source reference] and substitute [source reference]; all terms cancel. The initial value is one. The continuous…
Exact mathematical passage
(1+2n_j)^2-u_j^2-v_j^2=1, \label{eq:hyperboloid}
\mathcal VUse in source · 5 passages
Search aliases: V mathcal
Use in source · Energy_Conserving_Production.tex, line 219
The reaction force required by the same finite Hamiltonian is Cq''+ V'(q) +∑_j w_jω_j,q(n_j+fracu_j2)=0. The coherence term is essential. An occupation-only force does not conserve the energy of [source reference].
Exact mathematical passage
\boxed{Cq''+\mathcal V'(q) +\sum_j w_j\omega_{j,q}\left(n_j+\frac{u_j}{2}\right)=0.} \label{eq:reaction}
\mathrm VUse in source · 1 passage
Search aliases: V mathrm
Use in source · Energy_Conserving_Production.tex, line 181
There is also a quantitative background cost. Using the stored benchmark inputs, not a new observational fit, a_⋆=10^-6, Ω_r0=9×10^-5, Ω_ V=0.68991, we obtain fracKρ_r(a_⋆)=frac0.003600547325ε. The ratio is 0.0360 at ε=0.1, 0.360 at 0.01, and 3.60 at 0.001. Taking ε arbitrarily small is therefore not a cost-free way to preserve the old cosmological history. These ratios are not observational exclusion bounds. They identify a background component that the next cosmological calculation must includ…
Exact mathematical passage
a_\star=10^{-6},\quad \Omega_{r0}=9\times10^{-5},\quad \Omega_{\mathrm V}=0.68991,
\mathcal V'Use in source · 1 passage
Search aliases: V' mathcal
Use in source · Energy_Conserving_Production.tex, line 219
The reaction force required by the same finite Hamiltonian is Cq''+ V'(q) +∑_j w_jω_j,q(n_j+fracu_j2)=0. The coherence term is essential. An occupation-only force does not conserve the energy of [source reference].
Exact mathematical passage
\boxed{Cq''+\mathcal V'(q) +\sum_j w_j\omega_{j,q}\left(n_j+\frac{u_j}{2}\right)=0.} \label{eq:reaction}
v_j^2Use in source · 1 passage
Search aliases: v_j^2
Use in source · Energy_Conserving_Production.tex, line 246
[Finite energy conservation and positivity] Assume C>0, w_j>0, y_j>0, and V≥0 is smooth. Start the modes in the instantaneous vacuum n_j=u_j=v_j=0. Equations [source reference]--[source reference] preserve (1+2n_j)^2-u_j^2-v_j^2=1, and conserve E=frac C2(q')^2+ V(q) +∑_jw_jω_j n_j. Every occupation remains nonnegative. Consequently 0≤∑_jw_jω_j n_j≤ E(0). Differentiate the left side of [source reference] and substitute [source reference]; all terms cancel. The initial value is one. The continuous…
Exact mathematical passage
(1+2n_j)^2-u_j^2-v_j^2=1, \label{eq:hyperboloid}
\widetilde\rhoUse in source · 4 passages
Search aliases: widetildeρ widetilde rho
Use in source · Energy_Conserving_Production.tex, line 78
Write the energy evaluated with the minimum masses as widetildeρ=A^4 S_E(x), S_E(10^-4)=2.18498795564, where the exact discrete degeneracies are retained. The kinetic reservoir is K=fracv^2 A^22γ, ρ_ V=fracμ^44λ, frac Kρ_ V=frac2γ(frac Aμ)^2. The stored reference has λ_0=1 and a_0:=fracA_0μ_0 · =1.93844860142×10^9,
Exact mathematical passage
\widetilde\rho=A^4 S_E(x),\qquad S_E(10^{-4})=2.18498795564, \label{eq:SE}
\widetilde\rho_0Use in source · 1 passage
Search aliases: widetildeρ_0 widetilde rho
Use in source · Energy_Conserving_Production.tex, line 90
ildeρ=A^4 S_E(x), S_E(10^-4)=2.18498795564, where the exact discrete degeneracies are retained. The kinetic reservoir is K=fracv^2 A^22γ, ρ_ V=fracμ^44λ, frac Kρ_ V=frac2γ(frac Aμ)^2. The stored reference has λ_0=1 and a_0:=fracA_0μ_0 · =1.93844860142×10^9, fracK_0ρ_ V · =4.69697872544×10^17, R_0:=fracwidetildeρ_0K_0 · =2.6272875374×10^20. The original occupation projection used η_0=R_0^-1=3.80620691783×10^-21. It made the reported density equal to K_0, but did not supply equations that remove t…
Exact mathematical passage
a_0:=\frac{A_0}{\mu_0}&=1.93844860142\times10^9,\\ \frac{K_0}{\rhoV}&=4.69697872544\times10^{17},\\ R_0:=\frac{\widetilde\rho_0}{K_0} &=2.6272875374\times10^{20}.
xUse in source · 4 passages
Search aliases: x
Use in source · Energy_Conserving_Production.tex, line 66
For a prescribed locally linear crossing, put A=fracsqrtγ |dotϕ_⋆|v, x=fracΛ A=10^-4, y_h=fracm_hA=x2^h. The assumed single-crossing occupation is f_h(k)=exp [-π(fracm_hA+frack^2Am_h)]. Such a prescribed crossing does not account for depletion of its source. The distinction between prescribed production, particle-mass growth, and self-consistent reaction is also central in established preheating calculations [source reference].
Exact mathematical passage
A=\frac{\sqrt\gamma\,|\dot\phi_\star|}{v},\qquad x=\frac\Lambda A=10^{-4},\qquad y_h=\frac{m_h}{A}=x2^h.
y^4Use in source · 1 passage
Search aliases: y^4
Use in source · Energy_Conserving_Production.tex, line 373
Vacuum response from unproduced levels. The observed depth sensitivity has a useful analytic warning sign. In a sufficiently adiabatic mode, n^(2)≃frac(ω')^216ω^4, ω n^(2)≃fracy^4q^2(q')^216ω^5. Integrating this local second-adiabatic-order energy over the full momentum range gives E^(2)_H =frac(q')^296π^2fracq^21+q^2 ∑_h=1^H d_hy_h^2, because ∫_0^∞fracκ^2 dκ(κ^2+M^2)^5/2 =frac13M^2. It therefore corresponds to the local kinetic correction δ C_H(q)=frac148π^2fracq^21+q^2 ∑_h=1^H d_hy_h^2. At lar…
Exact mathematical passage
n^{(2)}\simeq\frac{(\omega')^2}{16\omega^4},\qquad \omega n^{(2)}\simeq\frac{y^4q^2(q')^2}{16\omega^5}.
y_hUse in source · 2 passages
Search aliases: y_h
Use in source · Energy_Conserving_Production.tex, line 66
For a prescribed locally linear crossing, put A=fracsqrtγ |dotϕ_⋆|v, x=fracΛ A=10^-4, y_h=fracm_hA=x2^h. The assumed single-crossing occupation is f_h(k)=exp [-π(fracm_hA+frack^2Am_h)]. Such a prescribed crossing does not account for depletion of its source. The distinction between prescribed production, particle-mass growth, and self-consistent reaction is also central in established preheating calculations [source reference].
Exact mathematical passage
A=\frac{\sqrt\gamma\,|\dot\phi_\star|}{v},\qquad x=\frac\Lambda A=10^{-4},\qquad y_h=\frac{m_h}{A}=x2^h.
y_h^4Use in source · 1 passage
Search aliases: y_h^4
Use in source · Energy_Conserving_Production.tex, line 352
[Invariant curvature power counting] At fixed ρ_ V and γ, the family [source reference] leaves N_H unchanged for each fixed depth: N_H =fracγη_0 a_0^416π^2∑_h=1^H d_hy_h^4 =fracγ64π^2ρ_ V∑_h=1^H d_hm_h^4. The factor λ(A/μ)^4 equals η_0a_0^4 by [source reference]. Equivalently, substitute μ^4=4λρ_ V into [source reference]. The physical masses are fixed along the family by [source reference].
Exact mathematical passage
\mathcal N_H =\frac{\gamma\eta_0 a_0^4}{16\pi^2}\sum_{h=1}^H d_hy_h^4 =\frac{\gamma}{64\pi^2\rhoV}\sum_{h=1}^H d_hm_h^4.
y_jUse in source · 3 passages
Search aliases: y_j
Use in source · Energy_Conserving_Production.tex, line 245
[Finite energy conservation and positivity] Assume C>0, w_j>0, y_j>0, and V≥0 is smooth. Start the modes in the instantaneous vacuum n_j=u_j=v_j=0. Equations [source reference]--[source reference] preserve (1+2n_j)^2-u_j^2-v_j^2=1, and conserve E=frac C2(q')^2+ V(q) +∑_jw_jω_j n_j. Every occupation remains nonnegative. Consequently 0≤∑_jw_jω_j n_j≤ E(0). Differentiate the left side of [source reference] and substitute [source reference]; all terms cancel. The initial value is one. The continuous…
Exact mathematical passage
y_j>0
zUse in source · 2 passages
Search aliases: z
Use in source · Energy_Conserving_Production.tex, line 283
Numerical results and tests that failed. The principal run takes ε=0.01, depth 16, 48 Gauss--Legendre momentum nodes per level, and κ=sqrty_h z, 0≤ z≤3.5, q(0)=-10, q'(0)=1, 0≤τ≤20. Here A is the nominal reference scale used to set the incoming speed. The actual speed at q=0 is an output of the coupled solution, not an imposed crossing condition. There are 768 quadrature modes, representing 65,535 internal scalar degrees of freedom. The modes begin in the specified instantaneous vacuum. We use V…
Exact mathematical passage
\kappa=\sqrt{y_h}\,z,\quad 0\le z\le3.5,\quad q(0)=-10,\quad q'(0)=1,\quad 0\le\tau\le20.
z_{\max}Use in source · 1 passage
Search aliases: z_max max
Use in source · Energy_Conserving_Production.tex, line 331
At depth 16, changing z_max from 2.5 to 3.5 to 4.5 to 6 changes the maximum mode-energy fraction from approximately 0.476 to 0.649 to 0.762 to 0.871. Moving the initial state from q=-10 to -15 or -20 also changes the endpoint result. Thus the failed scientific gate is genuine. Conservation has passed; regulator and state robustness have not.
Exact mathematical passage
z_{\max}
|Use in source · 6 passages
Search aliases: |
Use in source · Energy_Conserving_Production.tex, line 66
For a prescribed locally linear crossing, put A=fracsqrtγ |dotϕ_⋆|v, x=fracΛ A=10^-4, y_h=fracm_hA=x2^h. The assumed single-crossing occupation is f_h(k)=exp [-π(fracm_hA+frack^2Am_h)]. Such a prescribed crossing does not account for depletion of its source. The distinction between prescribed production, particle-mass growth, and self-consistent reaction is also central in established preheating calculations [source reference].
Exact mathematical passage
A=\frac{\sqrt\gamma\,|\dot\phi_\star|}{v},\qquad x=\frac\Lambda A=10^{-4},\qquad y_h=\frac{m_h}{A}=x2^h.
|^2Use in source · 2 passages
Search aliases: |^2
Use in source · Energy_Conserving_Production.tex, line 228
Precise subtraction convention. For complex mode functions a_j satisfying a_j''+ω_j^2a_j=0 and the normalization a_joverlinea_j'-overlinea_ja_j'=i, fix the variables by n_j=frac|a_j'|^2+ω_j^2|a_j|^22ω_j-frac12, u_j=ω_j|a_j|^2-frac|a_j'|^2ω_j, v_j=-2Re(overlinea_ja_j'). Use the finite energy E=frac C2(q')^2+ V(q) +∑_jw_j[frac|a_j'|^2+ω_j^2|a_j|^22 -fracω_j2]. The subtracted quantity depends on q, so its derivative must also be included in the force. This gives [source reference]. The bracket in […
Exact mathematical passage
n_j=\frac{|a_j'|^2+\omega_j^2|a_j|^2}{2\omega_j}-\frac12, \quad u_j=\omega_j|a_j|^2-\frac{|a_j'|^2}{\omega_j}, \quad v_j=-2\operatorname{Re}(\overline{a_j}a_j').
\timesUse in source · 20 passages
Search aliases: × times
Use in source · Energy_Conserving_Production.tex, line 44
sCopperrule1.25in1pt The crossing-energy obstruction, an abundance-preserving parameter family, and a quantum-mode calculation with explicit energy exchange. Komorebi P.hfill October 2026 Result and boundary. The original prescribed-background benchmark exceeds its crossing reservoir by 2.6272875374×10^20. This study constructs a finite dynamical system in which that kind of overdraw is excluded by an exact conservation law. A numerical implementation agrees with an independent oscillator repres…
Exact mathematical passage
2.6272875374\times10^{20}
\deltaUse in source · 2 passages
Search aliases: δ delta
Use in source · Energy_Conserving_Production.tex, line 390
n Integrating this local second-adiabatic-order energy over the full momentum range gives E^(2)_H =frac(q')^296π^2fracq^21+q^2 ∑_h=1^H d_hy_h^2, because ∫_0^∞fracκ^2 dκ(κ^2+M^2)^5/2 =frac13M^2. It therefore corresponds to the local kinetic correction δ C_H(q)=frac148π^2fracq^21+q^2 ∑_h=1^H d_hy_h^2. At large |q|, δ C_H/C for ε=0.01 is approximately 0.121 at depth 14, 0.971 at 15, 7.77 at 16, and 62.2 at 17.
Exact mathematical passage
\delta C_H(q)=\frac{1}{48\pi^2}\frac{q^2}{1+q^2} \sum_{h=1}^H d_hy_h^2.
\etaUse in source · 8 passages
Search aliases: η eta
Use in source · Energy_Conserving_Production.tex, line 97
R_0:=fracwidetildeρ_0K_0 · =2.6272875374×10^20. The original occupation projection used η_0=R_0^-1=3.80620691783×10^-21. It made the reported density equal to K_0, but did not supply equations that remove this energy from the evolving scalar.
Exact mathematical passage
\eta_0=R_0^{-1}=3.80620691783\times10^{-21}. \label{eq:eta}
\eta_0Use in source · 7 passages
Search aliases: η_0 eta
Use in source · Energy_Conserving_Production.tex, line 97
R_0:=fracwidetildeρ_0K_0 · =2.6272875374×10^20. The original occupation projection used η_0=R_0^-1=3.80620691783×10^-21. It made the reported density equal to K_0, but did not supply equations that remove this energy from the evolving scalar.
Exact mathematical passage
\eta_0=R_0^{-1}=3.80620691783\times10^{-21}. \label{eq:eta}
\eta_0^{1/4}Use in source · 1 passage
Search aliases: η_0^1/4 eta
Use in source · Energy_Conserving_Production.tex, line 137
What is and is not preserved. The density and the normalized distributions in m/A and k/A are preserved. The physical mass scale is not the old one: fracμμ_0=η_0^1/4sqrtε, frac AA_0=fracΛΛ_0=η_0^1/4 =7.85458237208×10^-6. Thus the physical tower masses are independent of ε along this family, but differ from the original projected benchmark. Number density increases by the inverse of the common mass factor, while the mass density is held fixed. Occupations themselves are not preserved: they are no…
Exact mathematical passage
\frac\mu{\mu_0}=\eta_0^{1/4}\sqrt\eps,\qquad \frac A{A_0}=\frac\Lambda{\Lambda_0}=\eta_0^{1/4} =7.85458237208\times10^{-6}. \label{eq:physicalscale}
\LambdaUse in source · 5 passages
Search aliases: Λ Lambda
Use in source · Energy_Conserving_Production.tex, line 55
The obstruction being addressed. The source model is a real background scalar coupled to a finite tower of real scalar fields. In the massless-base benchmark, U(ϕ) · =fracλ4(ϕ^2-v^2)^2, · μ^2 · =λ v^2, m_h · =Λ 2^h, · d_h · =2^h-1, M_h^2(ϕ) · =m_h^2(1+γfracϕ^2v^2), · γ · =16. The general covariant construction and its limitations are given in the source monograph [source reference]. This study fixes the dyadic order-one tower used in the stored benchmark. The quartic λ is not a spectral eigenval…
Exact mathematical passage
U(\phi)&=\frac{\lambda}{4}(\phi^2-v^2)^2, &\mu^2&=\lambda v^2,\\ m_h&=\Lambda 2^h, &d_h&=2^{h-1},\\ M_h^2(\phi)&=m_h^2\left(1+\gamma\frac{\phi^2}{v^2}\right), &\gamma&=16.
\Lambda_0Use in source · 1 passage
Search aliases: Λ_0 Lambda
Use in source · Energy_Conserving_Production.tex, line 137
What is and is not preserved. The density and the normalized distributions in m/A and k/A are preserved. The physical mass scale is not the old one: fracμμ_0=η_0^1/4sqrtε, frac AA_0=fracΛΛ_0=η_0^1/4 =7.85458237208×10^-6. Thus the physical tower masses are independent of ε along this family, but differ from the original projected benchmark. Number density increases by the inverse of the common mass factor, while the mass density is held fixed. Occupations themselves are not preserved: they are no…
Exact mathematical passage
\frac\mu{\mu_0}=\eta_0^{1/4}\sqrt\eps,\qquad \frac A{A_0}=\frac\Lambda{\Lambda_0}=\eta_0^{1/4} =7.85458237208\times10^{-6}. \label{eq:physicalscale}
\lambda_0Use in source · 1 passage
Search aliases: λ_0 lambda
Use in source · Energy_Conserving_Production.tex, line 89
es as widetildeρ=A^4 S_E(x), S_E(10^-4)=2.18498795564, where the exact discrete degeneracies are retained. The kinetic reservoir is K=fracv^2 A^22γ, ρ_ V=fracμ^44λ, frac Kρ_ V=frac2γ(frac Aμ)^2. The stored reference has λ_0=1 and a_0:=fracA_0μ_0 · =1.93844860142×10^9,
Exact mathematical passage
\lambda_0=1
\mu^2Use in source · 1 passage
Search aliases: μ^2 mu
Use in source · Energy_Conserving_Production.tex, line 55
The obstruction being addressed. The source model is a real background scalar coupled to a finite tower of real scalar fields. In the massless-base benchmark, U(ϕ) · =fracλ4(ϕ^2-v^2)^2, · μ^2 · =λ v^2, m_h · =Λ 2^h, · d_h · =2^h-1, M_h^2(ϕ) · =m_h^2(1+γfracϕ^2v^2), · γ · =16. The general covariant construction and its limitations are given in the source monograph [source reference]. This study fixes the dyadic order-one tower used in the stored benchmark. The quartic λ is not a spectral eigenval…
Exact mathematical passage
U(\phi)&=\frac{\lambda}{4}(\phi^2-v^2)^2, &\mu^2&=\lambda v^2,\\ m_h&=\Lambda 2^h, &d_h&=2^{h-1},\\ M_h^2(\phi)&=m_h^2\left(1+\gamma\frac{\phi^2}{v^2}\right), &\gamma&=16.
\mu^4Use in source · 3 passages
Search aliases: μ^4 mu
Use in source · Energy_Conserving_Production.tex, line 83
Write the energy evaluated with the minimum masses as widetildeρ=A^4 S_E(x), S_E(10^-4)=2.18498795564, where the exact discrete degeneracies are retained. The kinetic reservoir is K=fracv^2 A^22γ, ρ_ V=fracμ^44λ, frac Kρ_ V=frac2γ(frac Aμ)^2. The stored reference has λ_0=1 and a_0:=fracA_0μ_0 · =1.93844860142×10^9,
Exact mathematical passage
K=\frac{v^2 A^2}{2\gamma},\qquad \rhoV=\frac{\mu^4}{4\lambda},\qquad \frac K{\rhoV}=\frac{2}{\gamma}\left(\frac A\mu\right)^2. \label{eq:kinetic}
\mu_0Use in source · 2 passages
Search aliases: μ_0 mu
Use in source · Energy_Conserving_Production.tex, line 90
ildeρ=A^4 S_E(x), S_E(10^-4)=2.18498795564, where the exact discrete degeneracies are retained. The kinetic reservoir is K=fracv^2 A^22γ, ρ_ V=fracμ^44λ, frac Kρ_ V=frac2γ(frac Aμ)^2. The stored reference has λ_0=1 and a_0:=fracA_0μ_0 · =1.93844860142×10^9, fracK_0ρ_ V · =4.69697872544×10^17, R_0:=fracwidetildeρ_0K_0 · =2.6272875374×10^20. The original occupation projection used η_0=R_0^-1=3.80620691783×10^-21. It made the reported density equal to K_0, but did not supply equations that remove t…
Exact mathematical passage
a_0:=\frac{A_0}{\mu_0}&=1.93844860142\times10^9,\\ \frac{K_0}{\rhoV}&=4.69697872544\times10^{17},\\ R_0:=\frac{\widetilde\rho_0}{K_0} &=2.6272875374\times10^{20}.
\piUse in source · 5 passages
Search aliases: π pi
Use in source · Energy_Conserving_Production.tex, line 71
For a prescribed locally linear crossing, put A=fracsqrtγ |dotϕ_⋆|v, x=fracΛ A=10^-4, y_h=fracm_hA=x2^h. The assumed single-crossing occupation is f_h(k)=exp [-π(fracm_hA+frack^2Am_h)]. Such a prescribed crossing does not account for depletion of its source. The distinction between prescribed production, particle-mass growth, and self-consistent reaction is also central in established preheating calculations [source reference].
Exact mathematical passage
f_h(k)=\exp\!\left[-\pi\left(\frac{m_h}{A}+\frac{k^2}{Am_h}\right)\right]. \label{eq:gaussian}
\pi^2Use in source · 4 passages
Search aliases: π^2 pi
Use in source · Energy_Conserving_Production.tex, line 343
Why a smaller coupling is not a complete repair. A radiative quantity invariant along the abundance family. The source monograph gives the finite-depth one-loop curvature contribution with its subtraction-scale logarithms. To isolate power counting, define the nonnegative diagnostic N_H:=fracγλ16π^2 ∑_h=1^H d_h(fracm_hμ)^4. This omits logarithms and finite counterterms. It is neither a scheme-independent lower bound on the renormalized mass nor a proof that locking is impossible.
Exact mathematical passage
\mathcal N_H:=\frac{\gamma\lambda}{16\pi^2} \sum_{h=1}^H d_h\left(\frac{m_h}{\mu}\right)^4. \label{eq:curvatureproxy}
\rhoUse in source · 3 passages
Search aliases: ρ rho
Use in source · Energy_Conserving_Production.tex, line 109
[Conditional abundance-preserving family] Fix ρ_ V, γ, the tower order, and x=Λ/A. Let the target density be the original projected value ρ_T=η_0 A_0^4 S_E=K_0. For any 0<ε≤1, choose λ=η_0ε^2, frac Aμ=fraca_0sqrtε, fracΛμ=xfraca_0sqrtε. Then the unprojected Gaussian family satisfies widetildeρ=ρ_T, K=fracK_0ε, fracwidetildeρK=ε. No common occupation multiplier is used in this construction. At fixed ρ_ V, μ^4=4λρ_ V, so widetildeρ=4ρ_ Vλ(A/μ)^4S_E. Substitute [source reference]. The factors ε^2 a…
Exact mathematical passage
\rho_T=\eta_0 A_0^4 S_E=K_0
\rho_rUse in source · 1 passage
Search aliases: ρ_r rho
Use in source · Energy_Conserving_Production.tex, line 186
There is also a quantitative background cost. Using the stored benchmark inputs, not a new observational fit, a_⋆=10^-6, Ω_r0=9×10^-5, Ω_ V=0.68991, we obtain fracKρ_r(a_⋆)=frac0.003600547325ε. The ratio is 0.0360 at ε=0.1, 0.360 at 0.01, and 3.60 at 0.001. Taking ε arbitrarily small is therefore not a cost-free way to preserve the old cosmological history. These ratios are not observational exclusion bounds. They identify a background component that the next cosmological calculation must includ…
Exact mathematical passage
\boxed{\frac{K}{\rho_r(a_\star)}=\frac{0.003600547325}{\eps}.} \label{eq:radiation}
\rho_TUse in source · 2 passages
Search aliases: ρ_T rho
Use in source · Energy_Conserving_Production.tex, line 109
[Conditional abundance-preserving family] Fix ρ_ V, γ, the tower order, and x=Λ/A. Let the target density be the original projected value ρ_T=η_0 A_0^4 S_E=K_0. For any 0<ε≤1, choose λ=η_0ε^2, frac Aμ=fraca_0sqrtε, fracΛμ=xfraca_0sqrtε. Then the unprojected Gaussian family satisfies widetildeρ=ρ_T, K=fracK_0ε, fracwidetildeρK=ε. No common occupation multiplier is used in this construction. At fixed ρ_ V, μ^4=4λρ_ V, so widetildeρ=4ρ_ Vλ(A/μ)^4S_E. Substitute [source reference]. The factors ε^2 a…
Exact mathematical passage
\rho_T=\eta_0 A_0^4 S_E=K_0
\phi^2Use in source · 1 passage
Search aliases: ϕ^2 phi
Use in source · Energy_Conserving_Production.tex, line 55
The obstruction being addressed. The source model is a real background scalar coupled to a finite tower of real scalar fields. In the massless-base benchmark, U(ϕ) · =fracλ4(ϕ^2-v^2)^2, · μ^2 · =λ v^2, m_h · =Λ 2^h, · d_h · =2^h-1, M_h^2(ϕ) · =m_h^2(1+γfracϕ^2v^2), · γ · =16. The general covariant construction and its limitations are given in the source monograph [source reference]. This study fixes the dyadic order-one tower used in the stored benchmark. The quartic λ is not a spectral eigenval…
Exact mathematical passage
U(\phi)&=\frac{\lambda}{4}(\phi^2-v^2)^2, &\mu^2&=\lambda v^2,\\ m_h&=\Lambda 2^h, &d_h&=2^{h-1},\\ M_h^2(\phi)&=m_h^2\left(1+\gamma\frac{\phi^2}{v^2}\right), &\gamma&=16.
\phi^4Use in source · 1 passage
Search aliases: ϕ^4 phi
Use in source · Energy_Conserving_Production.tex, line 424
instant G. Felder, L. Kofman, and A. Linde. Instant Preheating. Physical Review D 59, 123523 (1999). arXiv:hep-ph/9812289. bibitemqke A. M. Fedotov, E. G. Gelfer, K. Yu. Korolev, and S. A. Smolyansky. On the kinetic equation approach to pair production by time-dependent electric field. arXiv:1008.2098 (2010). bibitemmpr C. Molina-Par is, P. R. Anderson, and S. A. Ramsey. One-loop λϕ^4 theory in Robertson--Walker spacetimes: adiabatic regularization and analytic approximation. arXiv:gr-qc/9908037…
Exact mathematical passage
\lambda\phi^4
\OmegaUse in source · 1 passage
Search aliases: Ω Omega
Use in source · Energy_Conserving_Production.tex, line 181
There is also a quantitative background cost. Using the stored benchmark inputs, not a new observational fit, a_⋆=10^-6, Ω_r0=9×10^-5, Ω_ V=0.68991, we obtain fracKρ_r(a_⋆)=frac0.003600547325ε. The ratio is 0.0360 at ε=0.1, 0.360 at 0.01, and 3.60 at 0.001. Taking ε arbitrarily small is therefore not a cost-free way to preserve the old cosmological history. These ratios are not observational exclusion bounds. They identify a background component that the next cosmological calculation must includ…
Exact mathematical passage
a_\star=10^{-6},\quad \Omega_{r0}=9\times10^{-5},\quad \Omega_{\mathrm V}=0.68991,
\omega'Use in source · 1 passage
Search aliases: ω' omega
Use in source · Energy_Conserving_Production.tex, line 373
Vacuum response from unproduced levels. The observed depth sensitivity has a useful analytic warning sign. In a sufficiently adiabatic mode, n^(2)≃frac(ω')^216ω^4, ω n^(2)≃fracy^4q^2(q')^216ω^5. Integrating this local second-adiabatic-order energy over the full momentum range gives E^(2)_H =frac(q')^296π^2fracq^21+q^2 ∑_h=1^H d_hy_h^2, because ∫_0^∞fracκ^2 dκ(κ^2+M^2)^5/2 =frac13M^2. It therefore corresponds to the local kinetic correction δ C_H(q)=frac148π^2fracq^21+q^2 ∑_h=1^H d_hy_h^2. At lar…
Exact mathematical passage
n^{(2)}\simeq\frac{(\omega')^2}{16\omega^4},\qquad \omega n^{(2)}\simeq\frac{y^4q^2(q')^2}{16\omega^5}.
\omega^4Use in source · 1 passage
Search aliases: ω^4 omega
Use in source · Energy_Conserving_Production.tex, line 373
Vacuum response from unproduced levels. The observed depth sensitivity has a useful analytic warning sign. In a sufficiently adiabatic mode, n^(2)≃frac(ω')^216ω^4, ω n^(2)≃fracy^4q^2(q')^216ω^5. Integrating this local second-adiabatic-order energy over the full momentum range gives E^(2)_H =frac(q')^296π^2fracq^21+q^2 ∑_h=1^H d_hy_h^2, because ∫_0^∞fracκ^2 dκ(κ^2+M^2)^5/2 =frac13M^2. It therefore corresponds to the local kinetic correction δ C_H(q)=frac148π^2fracq^21+q^2 ∑_h=1^H d_hy_h^2. At lar…
Exact mathematical passage
n^{(2)}\simeq\frac{(\omega')^2}{16\omega^4},\qquad \omega n^{(2)}\simeq\frac{y^4q^2(q')^2}{16\omega^5}.
\omega^5Use in source · 1 passage
Search aliases: ω^5 omega
Use in source · Energy_Conserving_Production.tex, line 373
Vacuum response from unproduced levels. The observed depth sensitivity has a useful analytic warning sign. In a sufficiently adiabatic mode, n^(2)≃frac(ω')^216ω^4, ω n^(2)≃fracy^4q^2(q')^216ω^5. Integrating this local second-adiabatic-order energy over the full momentum range gives E^(2)_H =frac(q')^296π^2fracq^21+q^2 ∑_h=1^H d_hy_h^2, because ∫_0^∞fracκ^2 dκ(κ^2+M^2)^5/2 =frac13M^2. It therefore corresponds to the local kinetic correction δ C_H(q)=frac148π^2fracq^21+q^2 ∑_h=1^H d_hy_h^2. At lar…
Exact mathematical passage
n^{(2)}\simeq\frac{(\omega')^2}{16\omega^4},\qquad \omega n^{(2)}\simeq\frac{y^4q^2(q')^2}{16\omega^5}.
\Omega_{\mathrm V}Use in source · 1 passage
Search aliases: Ω_ V Omega mathrm
Use in source · Energy_Conserving_Production.tex, line 181
There is also a quantitative background cost. Using the stored benchmark inputs, not a new observational fit, a_⋆=10^-6, Ω_r0=9×10^-5, Ω_ V=0.68991, we obtain fracKρ_r(a_⋆)=frac0.003600547325ε. The ratio is 0.0360 at ε=0.1, 0.360 at 0.01, and 3.60 at 0.001. Taking ε arbitrarily small is therefore not a cost-free way to preserve the old cosmological history. These ratios are not observational exclusion bounds. They identify a background component that the next cosmological calculation must includ…
Exact mathematical passage
a_\star=10^{-6},\quad \Omega_{r0}=9\times10^{-5},\quad \Omega_{\mathrm V}=0.68991,
\omega_j^2Use in source · 3 passages
Search aliases: ω_j^2 omega
Use in source · Energy_Conserving_Production.tex, line 227
Precise subtraction convention. For complex mode functions a_j satisfying a_j''+ω_j^2a_j=0 and the normalization a_joverlinea_j'-overlinea_ja_j'=i, fix the variables by n_j=frac|a_j'|^2+ω_j^2|a_j|^22ω_j-frac12, u_j=ω_j|a_j|^2-frac|a_j'|^2ω_j, v_j=-2Re(overlinea_ja_j'). Use the finite energy E=frac C2(q')^2+ V(q) +∑_jw_j[frac|a_j'|^2+ω_j^2|a_j|^22 -fracω_j2]. The subtracted quantity depends on q, so its derivative must also be included in the force. This gives [source reference]. The bracket in […
Exact mathematical passage
a_j''+\omega_j^2a_j=0
\Omega_{r0}Use in source · 1 passage
Search aliases: Ω_r0 Omega
Use in source · Energy_Conserving_Production.tex, line 181
There is also a quantitative background cost. Using the stored benchmark inputs, not a new observational fit, a_⋆=10^-6, Ω_r0=9×10^-5, Ω_ V=0.68991, we obtain fracKρ_r(a_⋆)=frac0.003600547325ε. The ratio is 0.0360 at ε=0.1, 0.360 at 0.01, and 3.60 at 0.001. Taking ε arbitrarily small is therefore not a cost-free way to preserve the old cosmological history. These ratios are not observational exclusion bounds. They identify a background component that the next cosmological calculation must includ…
Exact mathematical passage
a_\star=10^{-6},\quad \Omega_{r0}=9\times10^{-5},\quad \Omega_{\mathrm V}=0.68991,
\sumUse in source · 8 passages
Search aliases: ∑ sum
Use in source · Energy_Conserving_Production.tex, line 219
The reaction force required by the same finite Hamiltonian is Cq''+ V'(q) +∑_j w_jω_j,q(n_j+fracu_j2)=0. The coherence term is essential. An occupation-only force does not conserve the energy of [source reference].
Exact mathematical passage
\boxed{Cq''+\mathcal V'(q) +\sum_j w_j\omega_{j,q}\left(n_j+\frac{u_j}{2}\right)=0.} \label{eq:reaction}
\sum_{h=1}^HUse in source · 4 passages
Search aliases: ∑_h=1^H sum
Use in source · Energy_Conserving_Production.tex, line 343
Why a smaller coupling is not a complete repair. A radiative quantity invariant along the abundance family. The source monograph gives the finite-depth one-loop curvature contribution with its subtraction-scale logarithms. To isolate power counting, define the nonnegative diagnostic N_H:=fracγλ16π^2 ∑_h=1^H d_h(fracm_hμ)^4. This omits logarithms and finite counterterms. It is neither a scheme-independent lower bound on the renormalized mass nor a proof that locking is impossible.
Exact mathematical passage
\mathcal N_H:=\frac{\gamma\lambda}{16\pi^2} \sum_{h=1}^H d_h\left(\frac{m_h}{\mu}\right)^4. \label{eq:curvatureproxy}
\sum_jUse in source · 4 passages
Search aliases: ∑_j sum
Use in source · Energy_Conserving_Production.tex, line 219
The reaction force required by the same finite Hamiltonian is Cq''+ V'(q) +∑_j w_jω_j,q(n_j+fracu_j2)=0. The coherence term is essential. An occupation-only force does not conserve the energy of [source reference].
Exact mathematical passage
\boxed{Cq''+\mathcal V'(q) +\sum_j w_j\omega_{j,q}\left(n_j+\frac{u_j}{2}\right)=0.} \label{eq:reaction}
\inftyUse in source · 1 passage
Search aliases: ∞ infty
Use in source · Energy_Conserving_Production.tex, line 385
iently adiabatic mode, n^(2)≃frac(ω')^216ω^4, ω n^(2)≃fracy^4q^2(q')^216ω^5. Integrating this local second-adiabatic-order energy over the full momentum range gives E^(2)_H =frac(q')^296π^2fracq^21+q^2 ∑_h=1^H d_hy_h^2, because ∫_0^∞fracκ^2 dκ(κ^2+M^2)^5/2 =frac13M^2. It therefore corresponds to the local kinetic correction δ C_H(q)=frac148π^2fracq^21+q^2 ∑_h=1^H d_hy_h^2. At large |q|, δ C_H/C for ε=0.01 is approximately 0.121 at depth 14, 0.971 at 15, 7.77 at 16, and 62.2 at 17.
Exact mathematical passage
\int_0^\infty\frac{\kappa^2\,\dd\kappa}{(\kappa^2+M^2)^{5/2}} =\frac{1}{3M^2}.
\intUse in source · 1 passage
Search aliases: ∫ int
Use in source · Energy_Conserving_Production.tex, line 385
iently adiabatic mode, n^(2)≃frac(ω')^216ω^4, ω n^(2)≃fracy^4q^2(q')^216ω^5. Integrating this local second-adiabatic-order energy over the full momentum range gives E^(2)_H =frac(q')^296π^2fracq^21+q^2 ∑_h=1^H d_hy_h^2, because ∫_0^∞fracκ^2 dκ(κ^2+M^2)^5/2 =frac13M^2. It therefore corresponds to the local kinetic correction δ C_H(q)=frac148π^2fracq^21+q^2 ∑_h=1^H d_hy_h^2. At large |q|, δ C_H/C for ε=0.01 is approximately 0.121 at depth 14, 0.971 at 15, 7.77 at 16, and 62.2 at 17.
Exact mathematical passage
\int_0^\infty\frac{\kappa^2\,\dd\kappa}{(\kappa^2+M^2)^{5/2}} =\frac{1}{3M^2}.
\int_0^\inftyUse in source · 1 passage
Search aliases: ∫_0^∞ int infty
Use in source · Energy_Conserving_Production.tex, line 385
iently adiabatic mode, n^(2)≃frac(ω')^216ω^4, ω n^(2)≃fracy^4q^2(q')^216ω^5. Integrating this local second-adiabatic-order energy over the full momentum range gives E^(2)_H =frac(q')^296π^2fracq^21+q^2 ∑_h=1^H d_hy_h^2, because ∫_0^∞fracκ^2 dκ(κ^2+M^2)^5/2 =frac13M^2. It therefore corresponds to the local kinetic correction δ C_H(q)=frac148π^2fracq^21+q^2 ∑_h=1^H d_hy_h^2. At large |q|, δ C_H/C for ε=0.01 is approximately 0.121 at depth 14, 0.971 at 15, 7.77 at 16, and 62.2 at 17.
Exact mathematical passage
\int_0^\infty\frac{\kappa^2\,\dd\kappa}{(\kappa^2+M^2)^{5/2}} =\frac{1}{3M^2}.
\simeqUse in source · 1 passage
Search aliases: ≃ simeq
Use in source · Energy_Conserving_Production.tex, line 373
Vacuum response from unproduced levels. The observed depth sensitivity has a useful analytic warning sign. In a sufficiently adiabatic mode, n^(2)≃frac(ω')^216ω^4, ω n^(2)≃fracy^4q^2(q')^216ω^5. Integrating this local second-adiabatic-order energy over the full momentum range gives E^(2)_H =frac(q')^296π^2fracq^21+q^2 ∑_h=1^H d_hy_h^2, because ∫_0^∞fracκ^2 dκ(κ^2+M^2)^5/2 =frac13M^2. It therefore corresponds to the local kinetic correction δ C_H(q)=frac148π^2fracq^21+q^2 ∑_h=1^H d_hy_h^2. At lar…
Exact mathematical passage
n^{(2)}\simeq\frac{(\omega')^2}{16\omega^4},\qquad \omega n^{(2)}\simeq\frac{y^4q^2(q')^2}{16\omega^5}.
\leUse in source · 5 passages
Search aliases: ≤ le
Use in source · Energy_Conserving_Production.tex, line 109
[Conditional abundance-preserving family] Fix ρ_ V, γ, the tower order, and x=Λ/A. Let the target density be the original projected value ρ_T=η_0 A_0^4 S_E=K_0. For any 0<ε≤1, choose λ=η_0ε^2, frac Aμ=fraca_0sqrtε, fracΛμ=xfraca_0sqrtε. Then the unprojected Gaussian family satisfies widetildeρ=ρ_T, K=fracK_0ε, fracwidetildeρK=ε. No common occupation multiplier is used in this construction. At fixed ρ_ V, μ^4=4λρ_ V, so widetildeρ=4ρ_ Vλ(A/μ)^4S_E. Substitute [source reference]. The factors ε^2 a…
Exact mathematical passage
0<\eps\le1
\geUse in source · 3 passages
Search aliases: ≥ ge
Use in source · Energy_Conserving_Production.tex, line 245
[Finite energy conservation and positivity] Assume C>0, w_j>0, y_j>0, and V≥0 is smooth. Start the modes in the instantaneous vacuum n_j=u_j=v_j=0. Equations [source reference]--[source reference] preserve (1+2n_j)^2-u_j^2-v_j^2=1, and conserve E=frac C2(q')^2+ V(q) +∑_jw_jω_j n_j. Every occupation remains nonnegative. Consequently 0≤∑_jw_jω_j n_j≤ E(0). Differentiate the left side of [source reference] and substitute [source reference]; all terms cancel. The initial value is one. The continuous…
Exact mathematical passage
\mathcal V\ge0
\starUse in source · 3 passages
Search aliases: ⋆ star
Use in source · Energy_Conserving_Production.tex, line 66
For a prescribed locally linear crossing, put A=fracsqrtγ |dotϕ_⋆|v, x=fracΛ A=10^-4, y_h=fracm_hA=x2^h. The assumed single-crossing occupation is f_h(k)=exp [-π(fracm_hA+frack^2Am_h)]. Such a prescribed crossing does not account for depletion of its source. The distinction between prescribed production, particle-mass growth, and self-consistent reaction is also central in established preheating calculations [source reference].
Exact mathematical passage
A=\frac{\sqrt\gamma\,|\dot\phi_\star|}{v},\qquad x=\frac\Lambda A=10^{-4},\qquad y_h=\frac{m_h}{A}=x2^h.
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