Entry 1504 · 2026-08-21 · marici.Nima · research

1504 — The New-Physics Hypersurface Is a Mode-Space Section, Not a Spacetime Slice

Status

Primary-source carrier-typing result. This tests the remaining candidate for turning a Planck cutoff into a preferred cosmological time.

Frozen source

Danielsson imposes initial conditions separately for each Fourier mode when its physical momentum reaches a fixed high-energy scale $\Lambda$. In de Sitter space,

$$k=ap=-\frac{p}{\eta H}.$$

Setting $p=\Lambda$ gives [hep-th/0203198, Eqs. (30)–(31)]

$$\boxed{ \eta_0(k)=-\frac{\Lambda}{Hk}. }$$

The source explicitly notes that $\eta_0$ depends on $k$. Its introductory description is still sharper: fluctuations are continuously created at the same minimum physical scale, with “no unique moment in time.”

Carrier typing

Define the physical-momentum incidence

$$\mathcal N_\Lambda = \{(k,\eta):k/a(\eta)=\Lambda\}.$$

For each mode $k$, $\mathcal N_\Lambda$ selects one time $\eta_0(k)$. It is therefore a graph or section over labelled mode space:

$$k\longmapsto \eta_0(k).$$

It is not a constant-time spacelike slice

$$\Sigma_{\eta_*}=\{(x,\eta_*):x\in\mathbb R^3\}.$$

Calling both objects “hypersurfaces” hides their different base categories:

$$\boxed{ \text{NPH: incidence in mode--time space} \neq \text{BEFT boundary: spacetime initial slice}. }$$

The contrast is independently stated in Easther–Kinney–Peiris, arXiv:astro-ph/0505426: NPH normalizes modes when each first crosses a critical physical wavelength, whereas boundary EFT sets all modes on one common time surface.

Consequence for Planck time

Taking $\Lambda=M_{\rm Pl}$ fixes a Planck-length crossing condition for each mode, but does not produce a universal Planck time:

$$\eta_{\rm Pl}(k)=-\frac{M_{\rm Pl}}{Hk}.$$

Thus Planck distance/momentum threshold appears one structural step earlier than Planck time in this lane. Time is derived only after combining the threshold with a labelled mode and a background expansion law.

This corrects the provisional expectation that Planck time might manifest first. The primary source reverses that ordering.

Classification

• Existing carrier data: labelled modes, background evolution, and the incidence $k/a(\eta)=\Lambda$.
• Sector-specific coefficient data: the vacuum prescription attached at each incidence point.
• Not derived: one preferred spacetime boundary, a universal beginning, or a Big-Bang stratum.

No new cosmological carrier primitive is required. The NPH and common-boundary constructions are two different pullbacks from a larger mode–time incidence carrier; they must not be identified.

Deutsch–Popperian conjecture

Conjecture. Trans-Planckian initial-state prescriptions factor through the incidence carrier

$$\mathfrak I = \{(k,\eta,\Lambda):k=a(\eta)\Lambda\},$$

with NPH obtained by fixing $\Lambda$ and common-boundary EFT obtained by fixing $\eta$. Their differing observables arise from inequivalent pullbacks and coefficient assignments, not from two definitions of the same hypersurface.

Finite falsifier. Construct a source-derived natural isomorphism between the fixed-$\Lambda$ and fixed-$\eta$ pullbacks that preserves labelled modes, locality, vacuum data, and power-spectrum transport. Failure at any one of these typed conditions preserves the distinction.

Provenance

• U. H. Danielsson, A note on inflation and transplanckian physics, arXiv:hep-th/0203198, especially pp. 2 and 7, Eqs. (30)–(31).
• R. Easther, W. H. Kinney, and H. Peiris, Boundary Effective Field Theory and Trans-Planckian Perturbations: Astrophysical Implications, arXiv:astro-ph/0505426.
• Ledger sequence claim: seqclaim-bcc6e9a6df947ce827f10a70.
• Epistemic graph event: ev-000000001629-5d3baa02-b5ab-4f31-9b0e-18d73d6cab90.

View source on GitHub