Hubble Flat-Space Limit in Cosmology

• Hubble flat-space limit is defined as the simultaneous limit H → 0 and K → 0, where cosmological metrics converge to Minkowski space and restore standard flat-space observables.
• It employs a scaling prescription that holds the ratio E/H fixed, enabling the extraction of flat-space scattering amplitudes from de Sitter quantum field correlators.
• This limit unifies perspectives from quantum field theory, holography, and observational cosmology, offering practical insights into causal boundaries and empirical curvature constraints.

The Hubble flat-space limit delineates the regime where cosmological processes become indistinguishable from those in Minkowski space. This concept has multiple technical manifestations: as an operational limit in cosmological models with vanishing curvature and Hubble parameter; as a precise scaling prescription for extracting flat-space S-matrix elements from correlation functions in de Sitter backgrounds; and as an effective limit on the observability of the universe tied to the Hubble radius in spatially flat Robertson–Walker spacetimes. The Hubble flat-space limit provides a unifying interface between cosmology, quantum field theory in curved spacetime, and holographic correspondences.

1. Definition and Mathematical Formulation

The Hubble flat-space limit can be defined as the simultaneous limit $H \to 0$ (vanishing Hubble parameter) and vanishing spacetime curvature $K \to 0$. In the context of de Sitter or FRW cosmologies, the flat-space metric emerges in this regime. A canonical formulation appears in quantum field theory on de Sitter space, where, in conformal time $\tau$ with the metric

$$ds^2 = \frac{1}{(-H\tau)^2}\left(-d\tau^2 + d\mathbf{x}^2\right),$$

the Hubble flat-space limit is executed as $H \to 0$ and external energies $E \to 0$, with the scale-invariant ratio $\alpha = E/H$ held fixed. In this prescription, all nontrivial dependence on the expansion is washed out, yielding the flat Minkowski metric and restoring the standard energy-conservation properties of the flat-space S-matrix (Kristiano et al., 27 Nov 2025).

In terms of causal structure, the Hubble radius $R_H = c/H$ sets the maximum proper distance within which signals can reach the observer at present time. In spatially flat Friedmann–Robertson–Walker (FRW) models, the limit $K\to 0$ and $H \to 0$ yields Minkowski space with no cosmological horizon, whereby the propagation of null geodesics and causal relationships reduce to those of special relativity (Bikwa et al., 2011, Acquaviva et al., 2013).

2. Causal and Observational Limits in Flat-Space Cosmologies

For spatially flat FRW spacetimes, photon geodesics are bounded by the Hubble radius $R_H(t_0)$ at the observation epoch $t_0$. The geodesic analysis demonstrates that no photon observed at $t_0$ could have originated from beyond $R_H(t_0)$; the maximal proper distance attained by any such photon is strictly $R_{\gamma,\,\mathrm{max}} < R_H(t_0) \equiv c/H_0$ (Bikwa et al., 2011). In the $\Lambda$CDM cosmology, this bound is numerically quantified: the proper radius of the observable universe is $\approx 0.45\, c t_0$ for $t_0$ the present cosmic time, whereas $R_H(t_0) \sim c/H_0 \sim 13.5$ Glyr.

The limit $H \rightarrow 0$ removes this bound, restoring the global causal connectivity characteristic of flat Minkowski space. Flat-space models constructed in the unimodular gravity framework (with evolving mass parameters and no explicit expansion of the metric) recover the empirical Hubble law while remaining within the strictures of special relativity and adiabatic quantum field evolution (Jain, 2012).

3. Hubble Flat-Space Limit in QFT and the S-matrix

In quantum field theory and cosmological correlators, the Hubble flat-space (HFS) limit prescribes a scaling that connects the de Sitter S-matrix and the analytic structure of the flat-space S-matrix. In de Sitter, tree-level correlators for, e.g., four-point scalar exchange, depend both on Mandelstam invariants $(s, t, ...)$ and the total external energy $E$ due to broken time-translation invariance. The HFS limit, defined by

$$H \to 0, \quad E \to 0, \quad \frac{E}{H} = \alpha \text{ finite},$$

transmutes the de Sitter amplitude into an integral transform of the flat-space amplitude. Explicitly, at tree-level for exchange of a heavy mass $m$,

$$S_{2\to2}^{\mathrm{dS}}(s;m) = \alpha^{2-d} E^2 \int_0^{\infty} d\xi\, \xi^{d-3} e^{-i \xi} \mathcal{M}_{\text{flat}}(s \xi^2; \alpha m ),$$

where $\mathcal{M}_{\text{flat}}$ is the flat-space scattering amplitude. The flat-space amplitude itself is recovered by a normalization and the singular piece of the integral, reflecting that in the HFS limit, amplitudes depend only on Mandelstam invariants and not on the total energy $E$ (Kristiano et al., 27 Nov 2025). Corrections in $H$ are suppressed and can be systematically accounted for. This procedure is crucial for extracting physically meaningful flat-space observables from inflationary correlators and de Sitter quantum gravity (Kristiano et al., 27 Nov 2025).

4. Flat-Space Limits, Holography, and Near-Horizon Geometry

The Hubble flat-space limit has significant implications in holography, particularly in the context of the static patch of de Sitter space. As the de Sitter radius $R_{dS} = 1/H$ diverges $(H \to 0)$, the static de Sitter metric reduces, in the near-horizon region, to the Rindler metric,

$ds^2 = -\rho^2 dt^2 + d\rho^2,$

which is the canonical flat-space geometry for uniformly accelerated observers. In holographic dualities such as those involving double-scaled SYK (DSSYK), the Hubble flat-space limit enables identification of the dual boundary quantum system with a bulk $1+1$-dimensional QCD (the 't Hooft model), providing precise correspondence between singlet spectra, Regge trajectories, and Rindler-mesonic excitations (Miyashita et al., 22 Jun 2025). The boundary-to-bulk mapping in this limit displays how dynamical confinement and spectral information in the dual quantum system map onto flat-space fields and interactions in the emergent Rindler geometry.

5. Observational and Model-Independent Bounds: Cosmological Chronometers

Empirical constraints on spatial curvature and the applicability of the flat-space limit have been derived using Friedmann-independent methods. Chronometer analyses exploit differential age measurements of passively evolving galaxies, yielding $H(z)$ without assuming the Friedmann equation. Through comparison to luminosity distances from SNe Ia and angular distances, these methods set model-independent lower bounds on the curvature radius $R_c$ that are at least as large as the Hubble radius $D_H = c/H_0$, directly supporting near-flatness on Hubble scales (Rich, 2011). Quantitatively, $R_c > 0.3\, D_H$ from low-$z$ data, and $R_c > D_H$ from measurements at $z \simeq 1.5$, although these bounds are much weaker than standard cosmic microwave background constraints. This approach operationalizes the Hubble flat-space limit as a physical assertion about the large-scale spatial geometry of the universe, independent of the underlying gravitational field equations.

6. Model Selection and Precision Cosmology in the Flat-Space Limit

Flat-$\Lambda$CDM cosmologies provide a framework in which the Hubble flat-space limit is implicitly applied. Model selection analyses using cosmic chronometer data and type Ia supernovae distances, assessed via information-theoretic criteria such as the Akaike Information Criterion (AIC), consistently favor spatially flat models over non-flat alternatives, with the best-fit $H_0$ agreeing with indirect flat-space projections ($H_0 \simeq 68.7\pm3.1$ km\,s$^{-1}$\,Mpc$^{-1}$) (Thakur et al., 2023). These findings underscore that, even with current precision, the flat-space limit remains an excellent approximation for phenomenological cosmology, despite persistent anomalies such as the Hubble tension between local and global $H_0$ measurements.

7. Controversies and Caveats

The status of the Hubble sphere $R_H = c/H$ as a strict horizon or observational limit has been debated. While geodesic analyses confirm that, in spatially flat FRW models with $w > -1$, no observed photon path has ever exceeded the present Hubble radius (Bikwa et al., 2011, Acquaviva et al., 2013), this is not a true causal horizon in the rigorous sense: in "phantom" energy models ($w < -1$), the Hubble sphere can decrease and null rays may cross it multiple times, demonstrating that the Hubble radius is not a universal causal boundary (Lewis et al., 2012). Rather, it serves as a "trapping horizon" or practical limit relevant to the current cosmological epoch, but the ultimate causal domain is set by the particle and event horizons defined by the global evolution of $a(t)$. Furthermore, the correspondence between flat- and curved-space S-matrix elements in the HFS limit can be altered by subleading corrections, UV completions, or breakdown of effective field theory at high energy (Kristiano et al., 27 Nov 2025).

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References:

• "Friedmann-free limits on spatial curvature" (Rich, 2011)
• "How does the Hubble Sphere limit our view of the Universe?" (Lewis et al., 2012)
• "Photon Geodesics in FRW Cosmologies" (Bikwa et al., 2011)
• "Ray tracing in FLRW flat space-times" (Acquaviva et al., 2013)
• "A flat space-time model of the Universe" (Jain, 2012)
• "Investigating The Hubble Tension Through Hubble Parameter Data" (Thakur et al., 2023)
• "DSSYK at Infinite Temperature: The Flat-Space Limit and the 't Hooft Model" (Miyashita et al., 22 Jun 2025)
• "Imprints of flat space analyticity in de Sitter S-matrix" (Kristiano et al., 27 Nov 2025)