Cosmic Hysteresis in Cyclic Cosmology

• Cosmic hysteresis is a cyclic cosmology phenomenon where non-retracing pressure-volume loops arise from asymmetric scalar-field dynamics.
• The mechanism exploits the sign change of the Hubble parameter, converting friction into anti-friction and creating varying cycle amplitudes.
• Its study spans multiple frameworks—including braneworld models, f(R) gravity, and loop quantum cosmology—providing key insights on secular cycle evolution.

Cosmic hysteresis, in the cyclic-universe literature also called cosmological hysteresis, denotes the non-vanishing loop integral $\oint p\,dV$ generated over one complete contraction $\rightarrow$ bounce $\rightarrow$ expansion $\rightarrow$ turnaround cycle when the pressure of the dominant field does not retrace the same trajectory as the spatial volume (Sahni et al., 2012). In its standard formulation, the mechanism arises in a non-singular FLRW universe sourced by a single canonical scalar field, for which the sign change of the Hubble parameter converts Hubble damping during expansion into anti-friction during contraction, thereby producing asymmetric equations of state on the two branches (Choudhury et al., 2015). The principal consequence is a secular change in the extrema of the scale factor; depending on the sign of $\oint p\,dV$, the scalar potential, and the gravitational framework, successive cycles can grow, shrink, or exhibit quasi-periodic modulation (Choudhury et al., 2016).

1. Definition and formal structure

In a homogeneous scalar-field cosmology, the field energy density and pressure are

$$\rho=\tfrac12\dot\phi^2+V(\phi), \qquad p=\tfrac12\dot\phi^2-V(\phi).$$

With the comoving volume identified as $V=a^3$, the work performed over one complete cosmic cycle is

$$\oint p\,dV=\oint p\,d(a^3)=3\oint p\,a^2\,da =3\oint\Bigl(\tfrac12\dot\phi^2-V(\phi)\Bigr)a^2\dot a\,dt.$$

This loop integral is the cosmological analogue of the area enclosed by a magnetic hysteresis loop, and its non-vanishing signals that the contraction and expansion histories are not dynamically identical (Choudhury et al., 2015).

The same construction follows from local energy conservation. For the scalar-field fluid,

$$\dot\rho+3H(\rho+p)=0 \quad\Longrightarrow\quad d(\rho a^3)=-p\,d(a^3).$$

Integrated over a full cycle, this relation links the net work to the change in effective mass-energy stored at the turning points of the cycle (Sahni et al., 2012). In this sense, cosmic hysteresis is not merely a graphical feature of a loop in the $(p,V)$ or $\rightarrow$0 plane; it is the thermodynamic bookkeeping device that determines whether consecutive cycles are amplified or damped.

A central point in the early literature is that the mechanism does not require viscous dissipation. The original scalar-field analyses describe it as a purely thermodynamic effect arising from asymmetric pressure histories in an otherwise reversible set of field equations, and they explicitly contrast this with entropy-producing cyclic models (Choudhury et al., 2015).

2. Scalar-field origin of pressure asymmetry

The microscopic source of the asymmetry is the scalar-field equation of motion

$\rightarrow$1

During expansion, $\rightarrow$2, so the term $\rightarrow$3 acts as friction. During contraction, $\rightarrow$4, so the same term acts as anti-friction. This sign flip changes the relative importance of kinetic and potential energy on the two branches of the cycle (Choudhury et al., 2016).

In the expanding phase, the field motion is damped, the slow-roll regime can be approached, and one obtains a soft equation of state with $\rightarrow$5. In the contracting phase, the field is accelerated, the kinetic term can dominate, and one obtains a stiff equation of state with $\rightarrow$6 (Choudhury et al., 2016). Because the pressure is not the same function of $\rightarrow$7 on the two branches, the pressure-volume path fails to retrace itself, and the area $\rightarrow$8 is generically nonzero.

The braneworld analysis of Choudhury and Banerjee adds a phase-space explanation to this thermodynamic picture. As the scalar oscillates around the minimum of $\rightarrow$9 during expansion, its phase becomes randomized by the time the universe reaches turnaround; on recontraction, the field therefore climbs the potential along a different phase-space trajectory. The inequality $\rightarrow$0 is then not a kinematic artifact of plotting variables, but a direct consequence of the field dynamics in a background that changes from damping to anti-damping (Choudhury et al., 2016).

This mechanism also clarifies why cosmic hysteresis is most naturally associated with scalar-field cyclic cosmologies rather than with an arbitrary bouncing background. A bounce and a recollapse are necessary to define a closed thermodynamic cycle, but the nontrivial loop area is generated by the scalar field’s branch-dependent dynamics.

3. Bounce, turnaround, and secular evolution of cycles

A hysteresis loop becomes cosmologically consequential only when the dynamics admits both a nonsingular bounce and a turnaround. In the general cyclic setting, a bounce is characterized by $\rightarrow$1 with $\rightarrow$2 or, equivalently, $\rightarrow$3 with $\rightarrow$4; a turnaround is characterized by $\rightarrow$5 with $\rightarrow$6 or $\rightarrow$7 with $\rightarrow$8 (Sanyal, 9 Jun 2025).

A model-independent relation between the loop area and cycle growth was derived for universes whose turnaround is produced by a term $\rightarrow$9 in the Friedmann equation,

$\rightarrow$0

At turnaround one obtains

$\rightarrow$1

so the sign of $\rightarrow$2 directly fixes whether successive maxima grow or shrink (Sahni et al., 2012). Two special cases emphasized in the literature are $\rightarrow$3 for a negative cosmological constant and $\rightarrow$4 for a closed universe curvature term.

For flat potentials, especially the quadratic potential $\rightarrow$5, the loop is typically negative, $\rightarrow$6, and the amplitude of successive cycles increases. The result is a universe with older and larger successive cycles and an effective arrow of time despite the absence of entropy production in the original scalar-field interpretation (Sahni et al., 2012). For steep potentials such as

$\rightarrow$7

the sign of the loop can change from cycle to cycle, so consecutive cycles can alternately grow and shrink. The resulting modulation was explicitly compared with beats in acoustic systems (Sahni et al., 2012).

The same framework extends to anisotropic cosmology. In Bianchi I,

$\rightarrow$8

so if negative hysteresis drives $\rightarrow$9 upward from cycle to cycle, the anisotropy density $\oint p\,dV$0 decreases correspondingly (Sahni et al., 2012). Cosmic hysteresis therefore acts not only on the overall amplitude of the scale factor but also on the relative importance of anisotropic stress.

4. Braneworld, higher-curvature, and torsion-based realizations

The original phenomenology was quickly embedded in modified-gravity settings that furnish explicit bounce and turnaround mechanisms. In the membrane-paradigm treatment, two representative examples are the Einstein–Gauss–Bonnet braneworld and the Dvali–Gabadadze–Porrati brane (Choudhury et al., 2016). Later work extended the construction to $\oint p\,dV$1 and $\oint p\,dV$2 gravity, where the loop integral is retained but the background dynamics is modified by curvature or torsion corrections (Sanyal, 9 Jun 2025).

Framework	Representative structure	Reported hysteresis behavior
EHGB braneworld	$\oint p\,dV$3	$\oint p\,dV$4; sign and model parameters control growth or shrinkage (Choudhury et al., 2016)
DGP braneworld	Modified Friedmann equation with crossover scale $\oint p\,dV$5	Same hysteresis logic with branch parameter $\oint p\,dV$6 (Choudhury et al., 2016)
Quadratic $\oint p\,dV$7	$\oint p\,dV$8	$\oint p\,dV$9 cycles; $$\rho=\tfrac12\dot\phi^2+V(\phi), \qquad p=\tfrac12\dot\phi^2-V(\phi).$$0; gradual decrease of $$\rho=\tfrac12\dot\phi^2+V(\phi), \qquad p=\tfrac12\dot\phi^2-V(\phi).$$1 and $$\rho=\tfrac12\dot\phi^2+V(\phi), \qquad p=\tfrac12\dot\phi^2-V(\phi).$$2 (Sanyal, 9 Jun 2025)
Reconstructed $$\rho=\tfrac12\dot\phi^2+V(\phi), \qquad p=\tfrac12\dot\phi^2-V(\phi).$$3	Torsion scalar $$\rho=\tfrac12\dot\phi^2+V(\phi), \qquad p=\tfrac12\dot\phi^2-V(\phi).$$4 and reconstructed $$\rho=\tfrac12\dot\phi^2+V(\phi), \qquad p=\tfrac12\dot\phi^2-V(\phi).$$5	Non-zero, negative work per cycle and an arrow of time (Sanyal et al., 17 Feb 2026)

In the Einstein–Gauss–Bonnet case with space-like extra dimension, the modified Friedmann equation can be written as

$$\rho=\tfrac12\dot\phi^2+V(\phi), \qquad p=\tfrac12\dot\phi^2-V(\phi).$$6

or, in compact form,

$$\rho=\tfrac12\dot\phi^2+V(\phi), \qquad p=\tfrac12\dot\phi^2-V(\phi).$$7

In the DGP model,

$$\rho=\tfrac12\dot\phi^2+V(\phi), \qquad p=\tfrac12\dot\phi^2-V(\phi).$$8

For both classes, the change in the minimum scale factor satisfies $$\rho=\tfrac12\dot\phi^2+V(\phi), \qquad p=\tfrac12\dot\phi^2-V(\phi).$$9 up to a positive function of model parameters, and the conditions for ever-increasing expansion depend on the signature of $V=a^3$0 and the membrane parameters such as $V=a^3$1, $V=a^3$2, and $V=a^3$3 (Choudhury et al., 2016).

The quadratic $V=a^3$4 study sharpens this point by showing that hysteresis does not universally imply growing cycles. Starting from

$V=a^3$5

the modified Friedmann equations become

$V=a^3$6

$V=a^3$7

The work over one cycle is

$V=a^3$8

For the numerical setup

$V=a^3$9

the system undergoes $$\oint p\,dV=\oint p\,d(a^3)=3\oint p\,a^2\,da =3\oint\Bigl(\tfrac12\dot\phi^2-V(\phi)\Bigr)a^2\dot a\,dt.$$0 complete cycles, with average work $$\oint p\,dV=\oint p\,d(a^3)=3\oint p\,a^2\,da =3\oint\Bigl(\tfrac12\dot\phi^2-V(\phi)\Bigr)a^2\dot a\,dt.$$1 and total cumulative work $$\oint p\,dV=\oint p\,d(a^3)=3\oint p\,a^2\,da =3\oint\Bigl(\tfrac12\dot\phi^2-V(\phi)\Bigr)a^2\dot a\,dt.$$2. In that realization, successive bounces and turnarounds are accompanied by a slight secular decrease in both $$\oint p\,dV=\oint p\,d(a^3)=3\oint p\,a^2\,da =3\oint\Bigl(\tfrac12\dot\phi^2-V(\phi)\Bigr)a^2\dot a\,dt.$$3 and $$\oint p\,dV=\oint p\,d(a^3)=3\oint p\,a^2\,da =3\oint\Bigl(\tfrac12\dot\phi^2-V(\phi)\Bigr)a^2\dot a\,dt.$$4, and the loop areas in the $$\oint p\,dV=\oint p\,d(a^3)=3\oint p\,a^2\,da =3\oint\Bigl(\tfrac12\dot\phi^2-V(\phi)\Bigr)a^2\dot a\,dt.$$5 plane range from $$\oint p\,dV=\oint p\,d(a^3)=3\oint p\,a^2\,da =3\oint\Bigl(\tfrac12\dot\phi^2-V(\phi)\Bigr)a^2\dot a\,dt.$$6 to $$\oint p\,dV=\oint p\,d(a^3)=3\oint p\,a^2\,da =3\oint\Bigl(\tfrac12\dot\phi^2-V(\phi)\Bigr)a^2\dot a\,dt.$$7 (Sanyal, 9 Jun 2025).

A torsion-based extension reconstructs exact $$\oint p\,dV=\oint p\,d(a^3)=3\oint p\,a^2\,da =3\oint\Bigl(\tfrac12\dot\phi^2-V(\phi)\Bigr)a^2\dot a\,dt.$$8 functions from prescribed nonsingular bounces, again with a minimally coupled scalar field and the same hysteretic work integral,

$$\oint p\,dV=\oint p\,d(a^3)=3\oint p\,a^2\,da =3\oint\Bigl(\tfrac12\dot\phi^2-V(\phi)\Bigr)a^2\dot a\,dt.$$9

In that setting the loop appears in the $$\dot\rho+3H(\rho+p)=0 \quad\Longrightarrow\quad d(\rho a^3)=-p\,d(a^3).$$0 or $$\dot\rho+3H(\rho+p)=0 \quad\Longrightarrow\quad d(\rho a^3)=-p\,d(a^3).$$1 plane, numerical integration yields a distinctly non-zero, negative net work per cycle, and the analysis interprets the result as thermodynamic memory and a cosmological arrow of time beyond curvature-based theories (Sanyal et al., 17 Feb 2026).

5. Loop quantum cosmology and quasi-periodic beats

Loop quantum cosmology provides a distinct realization in which the singular bounce is replaced by quantum geometry. The effective theory discussed for a spatially closed isotropic spacetime uses the canonical pair $$\dot\rho+3H(\rho+p)=0 \quad\Longrightarrow\quad d(\rho a^3)=-p\,d(a^3).$$2 with $$\dot\rho+3H(\rho+p)=0 \quad\Longrightarrow\quad d(\rho a^3)=-p\,d(a^3).$$3, $$\dot\rho+3H(\rho+p)=0 \quad\Longrightarrow\quad d(\rho a^3)=-p\,d(a^3).$$4, and

$$\dot\rho+3H(\rho+p)=0 \quad\Longrightarrow\quad d(\rho a^3)=-p\,d(a^3).$$5

Two inequivalent loop quantizations are analyzed: the holonomy-based quantization associated with Ashtekar–Pawlowski–Singh–Vandersloot, and the connection-operator quantization associated with Corichi–Karami (Dupuy et al., 2019).

In the holonomy-based case, the effective Friedmann equation is

$$\dot\rho+3H(\rho+p)=0 \quad\Longrightarrow\quad d(\rho a^3)=-p\,d(a^3).$$6

and each cycle contains a single non-singular bounce at $$\dot\rho+3H(\rho+p)=0 \quad\Longrightarrow\quad d(\rho a^3)=-p\,d(a^3).$$7. In the connection-operator quantization,

$$\dot\rho+3H(\rho+p)=0 \quad\Longrightarrow\quad d(\rho a^3)=-p\,d(a^3).$$8

with two distinct bounce densities

$$\dot\rho+3H(\rho+p)=0 \quad\Longrightarrow\quad d(\rho a^3)=-p\,d(a^3).$$9

The connection formulation therefore yields two alternating quantum bounces per cycle rather than one (Dupuy et al., 2019).

Despite these differences, the hysteresis phenomenon is reported to be robust for the quadratic potential. With $(p,V)$0, both quantizations show $(p,V)$1, $(p,V)$2, and a secular increase of the maximum scale factor. The explicit relation

$(p,V)$3

connects the loop area to the change in successive recollapse points (Dupuy et al., 2019). For the $(p,V)$4 potential with $(p,V)$5, both holonomy and connection models exhibit $(p,V)$6 hysteresis cycles before standard inflation sets in, and each cycle increases $(p,V)$7 by $(p,V)$8–$(p,V)$9.

The same paper also identifies a genuinely non-monotonic regime. For the cosh-like potential

$\rightarrow$00

the steepness parameter $\rightarrow$01 can be tuned so that some cycles have $\rightarrow$02 and others $\rightarrow$03. The result is quasi-periodic beats, with small-scale oscillations of period $\rightarrow$04 and an envelope varying on $\rightarrow$05 under curvature-driven recollapse. When a negative cosmological constant is added, the dynamics also displays islands of cluster of bounces separated by accelerated expansion, as well as step-like expansion with multiple turnarounds (Dupuy et al., 2019).

6. Thermodynamic interpretation, conceptual issues, and broader astrophysical usages

One persistent conceptual issue concerns whether cosmic hysteresis should be regarded as dissipationless or as a source of irreversibility. The scalar-field braneworld and model-independent analyses emphasize that no entropy is produced and that the arrow of time emerges in a dissipationless cosmology because the asymmetry is created by the sign change of the Hubble-friction term rather than by viscous transport (Choudhury et al., 2015). The quadratic $\rightarrow$06 analysis, by contrast, interprets the non-vanishing work loop as a thermodynamic signature of irreversible cyclic evolution, describes the cumulative work as a steadily decreasing curve that demonstrates irreversible energy dissipation into the higher-curvature degrees of freedom, and states that the effect provides a built-in mechanism for entropy production (Sanyal, 9 Jun 2025). This suggests that the thermodynamic reading of the loop is model-dependent.

A second common misunderstanding is that cosmic hysteresis necessarily means monotonic amplification of the cycle amplitude. The literature does not support that simplification. Negative $\rightarrow$07 in flat-potential cyclic models produces larger successive cycles, but steep potentials can yield either sign and generate beats, while higher-curvature realizations can instead damp the oscillation amplitude (Sahni et al., 2012).

Outside cyclic cosmology, the phrase “cosmic hysteresis” and closely related hysteresis language also appear in several astrophysical settings that involve loop-like, history-dependent evolution of other observables.

Domain	Hysteretic quantity	Reported feature
Cosmic-ray solar modulation	Flux–flux loops or effective modulation parameters	Proton/electron loop amplitude $\rightarrow$08 at $\rightarrow$09 GV; helium/proton $\rightarrow$10 at $\rightarrow$11 GV; 22-year and Forbush-event loops (Lipari et al., 2023)
Solar wind cycle	Permutation entropy $\rightarrow$12 versus sunspot number	Hysteresis over Solar Cycle 23, indicating multistability (Suyal et al., 2011)
Astrophysical dynamos	Magnetic-energy or field-amplitude bifurcation loops	Bistability, subcritical branches, and chaotic transients in $\rightarrow$13, Babcock–Leighton, and $\rightarrow$14 dynamos (Oliveira et al., 2020); (Vashishth et al., 2021); (Vashishth, 2024)
Curved-spacetime kinetic theory	Entropy-rate loop $\rightarrow$15	Curvature-induced memory described as gravitational hysteresis (Moti et al., 2024)

These usages are technically distinct from the scalar-field pressure-volume mechanism of cyclic cosmology. In the AMS-02 analysis, hysteresis refers to loop-like relations between fluxes of different cosmic-ray species or rigidities over the solar cycle, requiring a two-parameter generalization of the force-field approximation and reflecting opposite-charge drift effects or local-interstellar-spectrum shape differences (Lipari et al., 2023). In the solar-wind study, hysteresis is seen in permutation entropy as the system traverses the ascending and descending phases of Solar Cycle 23 (Suyal et al., 2011). In dynamo theory, hysteresis denotes bistability between decaying and strong-field attractors, as in the $\rightarrow$16 dynamo with thresholds $\rightarrow$17 and $\rightarrow$18, or in Babcock–Leighton and large-scale $\rightarrow$19 models with subcritical branches (Oliveira et al., 2020); (Vashishth et al., 2021); (Vashishth, 2024). A separate kinetic-theory construction defines “gravitational hysteresis” through the non-return of the entropy-production rate after transport around a closed spacetime loop, with

$\rightarrow$20

set by integrated spacetime curvature (Moti et al., 2024).

Within cosmology proper, however, the core concept remains precise: a non-singular cyclic universe with a scalar field can carry memory from one branch of a cycle to the other through the sign reversal of Hubble friction, and that memory is quantitatively encoded in the thermodynamic loop integral $\rightarrow$21.