A Small Patch Hypothesis in Cosmology

TL;DR

The paper proposes the Small Patch Hypothesis, arguing that a positive cosmological constant $\Lambda$ imposes a fixed observational horizon $l_{\Lambda} \sim \Lambda^{-1/2}$, rendering horizon/flatness puzzles as selection effects within a much larger, conformally older Universe. It reframes the origin of near-scale invariance, Gaussianity, and adiabaticity as outcomes of observing a small causal patch in equilibrium rather than signatures of an inflationary mechanism, and explores a concrete illustrative logarithmic infrared spectrum $P_{\zeta}(k)=\frac{A_s}{1-(n_s-1)\ln(k/k_0)}$ to demonstrate consistency with data while allowing an ultra-large IR divergence scale $l_c$. The work analyzes the implications for perturbation coherence via horizon-entry geometry and uses a maximum-entropy argument to motivate Gaussianity and adiabaticity, with the cosmological constant setting a geometric bound that makes the observable sector appear smooth even if the global spacetime is highly inhomogeneous. Current data show no strong preference between the standard red-tilted spectrum and the logarithmic alternative, but future 21-cm observations could distinguish them, offering a low-energy, geometry-based complement or alternative to inflation. Overall, the Small Patch Hypothesis highlights how causal boundedness and entropy considerations in a large hosting Universe can reproduce key cosmological features without invoking high-energy inflationary physics.

Abstract

If our observable Universe is only a tiny region of a vastly larger and conformally older spacetime, then the usual formulations of the classical flatness and horizon problems of the Hot Big Bang can be reinterpreted as artifacts manifesting an observational selection effect; we occupy a small causal domain of a much larger causally-connected and possibly non-flat spacetime. A sufficiently large positive cosmological constant, $Λ$, sets the future asymptotic horizon scale of the observable Universe, $\sim$$Λ^{-1/2}$, thereby implying that the observable Universe may simply be a minute patch of a far larger pre-existing one, hereafter a Small Patch Hypothesis. Importantly, this observational bound is purely geometric; regardless of when the Universe is observed, the maximum accessible scale is finite and fixed by $Λ$, independent of inflationary dynamics, anthropic arguments, or assumptions about the global hosting spacetime. In this sense, inflation becomes one viable realization of the proposed Small Patch Hypothesis. Here, one particular non-inflationary alternative is considered for illustrative purposes in which a primordial spectrum grows logarithmically toward large scales, and in fact diverges at some finite $k_{c}$. If $k_{c}\ll Λ^{-1/2}$, then our local cosmic patch probes only the linear regime and appears exceptionally smooth. Over the comparatively narrow observable window, this power spectrum mimics a slightly red-tilted, inflation-like spectrum. Rather than introducing high-energy new fields, this perspective frames large-scale homogeneity, isotropy, Gaussianity, adiabaticity, and the observed thermodynamic Arrow of Time as possible consequences of restricted observational access to a much larger Universe in equilibrium, rather than signatures of a unique early-Universe mechanism. [abridged]