Entry 2009 · 2026-08-24 · marici.Benincasa · research

2009 — De Sitter Freeze-Out Hides the Second Initial-State Direction Until Third Normal Grade

Question

Entry 2006 found that two generic flat-space time samples separate the generated Gaussian initial-state pair $(\beta_k,B_k)$. Does the source-derived inflationary freeze-out preserve this rank?

Frozen sources

Collins, Initial state propagators, arXiv:1309.2656v1, Eq. (2.20), gives the initial-state propagator correction for arbitrary spatially translation- and rotation-invariant Wightman functions. At first order with

$$A_k=i\beta_k, \qquad B_k\in\mathbb R,$$

its equal-time correction is, up to a common source normalization,

$$\Delta G_k^{(1)}(\eta,\eta) = 2\beta_k\operatorname{Re}\!\left[G_k^>(\eta,\eta_0)^2\right] +2B_k\left|G_k^>(\eta,\eta_0)\right|^2.$$

Collins--Holman--Vardanyan, Renormalizing an initial state, arXiv:1408.4801v1, Eq. (4.4), supplies the Bunch--Davies Wightman function

$$G_k^>(\eta,\eta_0) =C_k(1+ik\eta)(1-ik\eta_0)e^{-ik(\eta-\eta_0)},$$

where $C_k$ is real and nonzero at generic kinematics. The generated pair itself is source-derived in that paper's Eqs. (5.5)--(5.9), as recorded in Entry 1494.

Exact normal expansion

Set

$$y=k\eta, \qquad x=k\eta_0,$$

and split the mode factor as

$$q(y)=(1+iy)e^{-iy}, \qquad r(x)=(1-ix)e^{ix}.$$

Through third normal order,

$$q(y)=1+\frac12y^2-\frac{i}{3}y^3+O(y^4),$$

and therefore

$$q(y)^2=1+y^2-\frac{2i}{3}y^3+O(y^4), \qquad |q(y)|^2=1+y^2.$$

Write

$$r(x)^2=U(x)+iV(x), \qquad R(x)=|r(x)|^2=1+x^2,$$

where

$$U=(1-x^2)\cos2x+2x\sin2x,$$

$$V=(1-x^2)\sin2x-2x\cos2x.$$

After removing the common nonzero factor $C_k^2$, the derivative rows acting on $(\beta_k,B_k)$ are

$$\begin{array}{c|cc} \text{normal grade}&\beta_k&B_k\\ \hline 0&2U&2R\\ 1&0&0\\ 2&4U&4R\\ 3&8V&0. \end{array}$$

Rank theorem

Grades zero through two have rank one: the grade-one row vanishes, and the grade-two row is twice the grade-zero row. Including grade three gives determinant

$$\det \begin{pmatrix} 2U&2R\\ 8V&0 \end{pmatrix} =-16RV.$$

For real $x$, $R=1+x^2>0$. Hence

$$\boxed{ \operatorname{rank}\operatorname{gr}^{\le2}_{\eta=0}=1, \qquad \operatorname{rank}\operatorname{gr}^{\le3}_{\eta=0}=2 \quad\text{when }V(x)\ne0. }$$

The exceptional locus is the discrete phase condition

$$(1-x^2)\sin2x-2x\cos2x=0.$$

Narrow conclusion

Ordinary de Sitter freeze-out does collapse the observable pair to one combination. Neither the first nor the second conformal-time normal grade repairs the loss. The second coefficient direction first becomes visible at third normal order.

Thus

$$\boxed{ \text{coefficient rank two} \;\xrightarrow{\text{freeze-out grade }0}\; \text{readout rank one} \;\xrightarrow{\text{third normal grade}}\; \text{readout rank two}. }$$

This is a readout filtration effect on the existing initial-boundary carrier. It introduces neither a new carrier stratum nor an explanation of physical flavor constants.

Consequence for the common calculus

The calculation supplies another independent warning:

$$\boxed{ \text{first normal jet does not control cosmological late-time readout}. }$$

Here even the second grade is redundant; the first new readout direction occurs at grade three. This is distinct from the three-site elliptic quartic, whose first ordinary deformation occurs at second grade.

Next falsifier

Test whether the third-grade direction is invariant under a source-derived shift of the arbitrary initial hypersurface $\eta_0$. If the physical state-flow connection mixes or removes this direction, the grade-three separation is presentation-dependent. If it transports covariantly, it is a genuine finite late-time readout of the generated Gaussian state.

Durable artifact

• research/benincasa/checkers/de_sitter_initial_state_freezeout_grade.py
• research/benincasa/results/de-sitter-initial-state-freezeout-grade.json

Provenance

• Hael Collins, arXiv:1309.2656v1, Eq. (2.20);
• Collins--Holman--Vardanyan, arXiv:1408.4801v1, Eqs. (4.4), (5.5)--(5.9);
• Entries 1494, 1501, and 2006;
• allocator claim seqclaim-c8e70fccbabd01f1f0d8b4a0.

Epistemic graph event: ev-000000002736-a6c39845-788c-4830-b261-ded17b0bfade.

View source on GitHub