Bounce Radius: Definitions and Interpretations

• Bounce radius is a model-dependent finite scale that marks a reversal event, defined variously as the minimal scale factor, Hubble radius, or areal radius across different frameworks.
• In cosmology, it often represents the minimum value of the scale factor in FLRW models, influencing horizon dynamics as traditional Hubble radii diverge at the bounce.
• In black-bounce geometries and kinetic plasma theory, the bounce radius identifies the minimum physical or transverse size essential for regular internal structures and sustained trapping.

“Bounce radius” is not a standardized invariant across the literature. In nonsingular cosmology it can denote the minimum scale factor at the bounce, a physical or comoving Hubble radius, or a curvature scale; in black-bounce geometries it is the minimum areal radius of the spacetime; in kinetic plasma theory it denotes the minimum transverse size required to sustain trapped orbits. The unifying idea is a finite radius-like scale associated with reversal of collapse or contraction, but the precise quantity is model-dependent and must be read from the dynamical variables used in each framework (Rani et al., 31 Jan 2026).

1. Terminological scope and principal definitions

A recurrent feature of the literature is that many papers do not introduce a unique symbol explicitly called “bounce radius.” Instead, they use radius-like quantities adapted to the system under study. In FLRW bounce cosmology, the most common candidates are the minimum scale factor $a_{\min}=a(t_b)$, the physical Hubble radius $R_H=1/|H|$, and the comoving Hubble radius $r_H=(a|H|)^{-1}$ (Nojiri et al., 2016). In Simpson–Visser-type regular spacetimes, the bounce radius is the minimum areal radius, e.g. $\Sigma_{\min}=a$ or $\rho_{\text{bounce}}=q_H$, attained at the throat or interior bounce surface (Bronnikov, 2024). In finite-gyro-radius electron-hole equilibria, the same phrase refers to the minimum transverse size compatible with sustained trapping, bounded by the thermal gyro-radius $r_g$ (Hutchinson, 2021).

An important source of confusion is that some works call the comoving Hubble radius a “Hubble radius,” while others reserve that name for $1/|H|$. The unimodular $F(R)$ study explicitly writes $R_H=1/(aH)$, i.e. its $R_H$ is comoving rather than physical (Nojiri et al., 2016). The Weyl-type $R_H=1/|H|$0 study likewise distinguishes the minimum size $R_H=1/|H|$1 from the Hubble-type radii, and notes a caption-level ambiguity in the plotted “cosmic Hubble radius” $R_H=1/|H|$2 (Rani et al., 31 Jan 2026).

Context	Radius-like quantity	Representative expression
Weyl-type $R_H=1/|H|$3 bounce cosmology	minimum size and Hubble-type radii	$R_H=1/|H|$4, $R_H=1/|H|$5, $R_H=1/|H|$6 (Rani et al., 31 Jan 2026)
Unimodular and related bounce cosmologies	physical or comoving Hubble radius	$R_H=1/|H|$7, $R_H=1/|H|$8 (Nojiri et al., 2016)
LQC matter/deformed matter bounce	minimal scale factor at the bounce	$R_H=1/|H|$9 (Odintsov et al., 2016)
Black-bounce spacetimes	minimum areal radius	$r_H=(a|H|)^{-1}$0, $r_H=(a|H|)^{-1}$1 (Bronnikov, 2024)
Electron-hole equilibria	minimal transverse size	$r_H=(a|H|)^{-1}$2 (Hutchinson, 2021)

2. Minimum-size definitions in bounce cosmology

In FLRW bounce models, the cleanest finite notion of bounce radius is often the minimum value of the scale factor. The Weyl-type $r_H=(a|H|)^{-1}$3 model with quintom signature adopts

$r_H=(a|H|)^{-1}$4

with $r_H=(a|H|)^{-1}$5, $r_H=(a|H|)^{-1}$6, $r_H=(a|H|)^{-1}$7, so the minimum size is

$r_H=(a|H|)^{-1}$8

This is the direct “radius at the bounce” in a spatially flat background, and in that reconstruction it depends only on $r_H=(a|H|)^{-1}$9 and $\Sigma_{\min}=a$0, not directly on $\Sigma_{\min}=a$1, $\Sigma_{\min}=a$2, or $\Sigma_{\min}=a$3 (Rani et al., 31 Jan 2026). The same paper quotes the representative values $\Sigma_{\min}=a$4 for $\Sigma_{\min}=a$5, $\Sigma_{\min}=a$6 for $\Sigma_{\min}=a$7, and $\Sigma_{\min}=a$8 for $\Sigma_{\min}=a$9 (Rani et al., 31 Jan 2026).

Loop Quantum Cosmology formulations often package the same idea as $\rho_{\text{bounce}}=q_H$0. In the deformed matter-bounce scenario,

$\rho_{\text{bounce}}=q_H$1

and $\rho_{\text{bounce}}=q_H$2 is the minimal radius. Because the deformation is negligible near the bounce for the parameter regime studied, the paper concludes that $\rho_{\text{bounce}}=q_H$3 to an excellent approximation in its normalization (Odintsov et al., 2016). The $\rho_{\text{bounce}}=q_H$4CDM bounce scenario uses the same normalization idea: the LQC bounce occurs at $\rho_{\text{bounce}}=q_H$5, while the curvature scale is encoded by the critical density $\rho_{\text{bounce}}=q_H$6 (Cai et al., 2014).

The ghost-condensate matter-bounce literature makes the same conceptual distinction explicit: since $\rho_{\text{bounce}}=q_H$7, neither $\rho_{\text{bounce}}=q_H$8 nor $\rho_{\text{bounce}}=q_H$9 is finite at the bounce, so the operationally meaningful “radius” is $r_g$0, together with the curvature scale set by $r_g$1 (Lin et al., 2010). This suggests that, in cosmological usage, “bounce radius” most often denotes the minimum physical size whenever the Hubble radii diverge at the bounce.

3. Hubble-radius formulations and horizon dynamics

A second, widely used meaning of bounce radius is the Hubble radius. Here the distinction between physical and comoving quantities is essential: $r_g$2 These scales control horizon crossing and the perturbative chronology of contracting, bouncing, and expanding phases (Nojiri et al., 2016).

The behavior of these radii is strongly model-dependent. In the Weyl-type $r_g$3 reconstruction,

$r_g$4

so

$r_g$5

Both diverge at $r_g$6, and both are even in time: $r_g$7 The paper identifies this as the symmetric behavior of the radius scales around the bounce (Rani et al., 31 Jan 2026).

By contrast, the superbounce in unimodular $r_g$8 gravity has

$r_g$9

so the physical Hubble radius vanishes at the bounce because $1/|H|$0, while the comoving radius also goes to zero for $1/|H|$1 (Nojiri et al., 2016). Matter-bounce models show yet another pattern: the comoving Hubble radius decreases during a long contracting phase and increases during expansion, which is the standard exit-and-re-entry structure required for nearly scale-invariant perturbations (Brandenberger, 2012).

Type-IV singular and symmetric bounces do not share that perturbative advantage. In the singular Type-IV case,

$1/|H|$2

so both radii diverge at the bounce and vanish far away from it, making perturbation generation near the bounce non-scale-invariant in the model as studied (Odintsov et al., 2015). The symmetric bounce has $1/|H|$3 at $1/|H|$4 and both fall to zero for large $1/|H|$5, which prevents the usual exit-and-re-entry story (Nojiri et al., 2016).

Asymmetric bounce-to-dark-energy constructions sharpen this point further. In the ghost-free $1/|H|$6 model, $1/|H|$7 because the exponential late-time factor is negligible in contraction but important in expansion; the comoving Hubble radius diverges both in the far contracting past and at the bounce, while late-time acceleration makes it decrease again (Nojiri et al., 2022). The Chern–Simons-corrected $1/|H|$8 scenario uses the same logic: $1/|H|$9 diverges in deep contraction and at the bounce, then decreases during the late accelerating era (Odintsov et al., 2021).

4. Weyl $F(R)$0 gravity and the quintom bounce

The most explicit recent treatment of bounce radius in this sense appears in the Weyl-type $F(R)$1 model with a massive Weyl vector and power-law non-metricity sector

$F(R)$2

after imposing the FLRW ansatz and the simplifying condition $F(R)$3 (Rani et al., 31 Jan 2026). The effective density and pressure are

$F(R)$4

$F(R)$5

with

$F(R)$6

For the bounce ansatz

$F(R)$7

the Hubble rate and its derivative are

$F(R)$8

The bounce conditions follow immediately: $F(R)$9 The minimum-size identification is therefore

$R_H=1/(aH)$0

while the causal scales are encoded in the divergences

$R_H=1/(aH)$1

The amplitude of the divergence scales as $R_H=1/(aH)$2 for $R_H=1/(aH)$3 and $R_H=1/(aH)$4 for $R_H=1/(aH)$5 (Rani et al., 31 Jan 2026).

The dynamical interpretation is quintom-like. Near the bounce, the null energy condition is violated, $R_H=1/(aH)$6 crosses the phantom divide $R_H=1/(aH)$7, and $R_H=1/(aH)$8 while

$R_H=1/(aH)$9

Hence $R_H$0 is ill-defined exactly at $R_H$1 but crosses $R_H$2 on either side (Rani et al., 31 Jan 2026). The same work reconstructs an effective two-scalar description in which the quintessence-like kinetic term becomes negative near the bounce and the phantom-like kinetic energy becomes maximally positive, matching the NEC-violating regime. Stability analysis via the adiabatic index indicates instability near the bouncing point, whereas the energy conditions indicate dark-energy dominance (Rani et al., 31 Jan 2026).

5. Black-bounce geometries and black-hole interior bounces

In black-bounce spacetimes, bounce radius is not a cosmological scale but the minimum areal radius of the geometry. The Simpson–Visser prescription replaces the spherical radius by

$R_H$3

so the minimum occurs at $R_H$4 and is

$R_H$5

If this minimum lies in a static region it is a wormhole throat; if it lies inside an event horizon it is a regular bounce in the black-hole interior (Bronnikov, 2024). This is the canonical geometric meaning of “bounce radius” in the regular-black-hole literature.

The same structure persists in halo-embedded solutions. In the M60-calibrated Simpson–Visser metric,

$R_H$6

and the dark-matter halo modifies only the lapse $R_H$7, not the areal radius. Consequently,

$R_H$8

remains the bounce radius even in the presence of the halo (Junior et al., 19 Jun 2026). In generalized $R_H$9-$R_H=1/|H|$00 black-bounce metrics,

$R_H=1/|H|$01

so $R_H=1/|H|$02 is again the bounce radius, while the thresholds

$R_H=1/|H|$03

and $R_H=1/|H|$04 separate regular-black-hole, horizonless double-ring, and no-photon-sphere regimes (Nascimento et al., 27 Oct 2025).

A charged version appears in the Reissner–Nordström geometry corrected by a bounce parameter. There

$R_H=1/|H|$05

so the bounce radius is again the minimal areal radius. The coordinate horizon becomes

$R_H=1/|H|$06

and the coordinate photon-sphere radius becomes

$R_H=1/|H|$07

showing that the bounce parameter lowers the coordinate radii while leaving the areal photon-sphere radius unchanged (Javed et al., 2023).

A distinct but related usage appears in the semiclassical Schwarzschild-interior analysis. In Kantowski–Sachs form, the areal radius is $R_H=1/|H|$08, and the bounce ansatz imposes

$R_H=1/|H|$09

This $R_H=1/|H|$10 is the minimum areal radius reached inside the black hole, with an explicit example giving $R_H=1/|H|$11 for suitable curvature-quadratic couplings (Bolokhov et al., 2018). By contrast, regular-center alternatives explicitly reject the bounce/throat interpretation and instead keep $R_H=1/|H|$12 with $R_H=1/|H|$13 as a regular center rather than a minimum-radius surface (Bronnikov, 2024).

6. Other specialized uses and interpretive cautions

Outside gravitation, “bounce radius” can refer to a kinetic size threshold. In multidimensional electron-hole equilibria with finite gyro-radius, the relevant quantity is the minimal transverse size that can sustain trapped parallel motion. For a Gaussian profile, the practical bound is

$R_H=1/|H|$14

equivalently $R_H=1/|H|$15, and widths $R_H=1/|H|$16 do not persist beyond roughly a quarter of a bounce period (Hutchinson, 2021). Here the word “bounce” comes from parallel bounce motion, not cosmological reversal or spacetime regularization.

A related conceptual shift occurs in black-hole evolution through a cosmological bounce. There the operational “bounce radius” is the minimum proper Hubble radius during contraction,

$R_H=1/|H|$17

not the minimum scale factor (Corman et al., 2022). The ratio $R_H=1/|H|$18 controls horizon behavior: if $R_H=1/|H|$19, the black-hole apparent horizon persists; if $R_H=1/|H|$20, it merges with the cosmological horizon and temporarily disappears during contraction (Corman et al., 2022).

A third gravitational usage appears in relativistic spherical collapse with a vacuum-like ground state. There the gravitational bounce radius is

$R_H=1/|H|$21

the turning-point radius induced by the transition to a constant ground-state density $R_H=1/|H|$22 (Gaztanaga et al., 29 May 2025). This is neither a Hubble radius nor a scale factor, but the minimum physical radius of a collapsing closed FLRW patch.

The central interpretive caution is therefore straightforward: the phrase “bounce radius” does not by itself specify a unique observable. In FLRW cosmology it often means $R_H=1/|H|$23 because $R_H=1/|H|$24 makes $R_H=1/|H|$25 and $R_H=1/|H|$26 diverge at the bounce; in horizon-based analyses it may instead denote $R_H=1/|H|$27 or $R_H=1/|H|$28; in black-bounce geometries it is the minimum areal radius; and in kinetic plasma theory it is a minimum trapping width. Any technical use of the term is meaningful only after the underlying dynamical variable has been fixed by the model.