---
title: Cosmic Hysteresis in Cyclic Cosmology
url: https://www.emergentmind.com/topics/cosmic-hysteresis
type: topic
---

# Cosmic Hysteresis in Cyclic Cosmology

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 [1203.0395]. 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 [1512.08360]. 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 [1603.02805].

## 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 [1512.08360].

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 [1203.0395]. In this sense, cosmic hysteresis is not merely a graphical feature of a loop in the $(p,V)$ or $(w,a)$ 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 [1512.08360].

## 2. Scalar-field origin of pressure asymmetry

The microscopic source of the asymmetry is the scalar-field equation of motion
\[
\ddot\phi+3H\dot\phi+V'(\phi)=0.
\]
During expansion, $H>0$, so the term $3H\dot\phi$ acts as friction. During contraction, $H<0$, 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 [1603.02805].

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 $p\approx-\rho$. In the contracting phase, the field is accelerated, the kinetic term can dominate, and one obtains a stiff equation of state with $p\approx+\rho$ [1603.02805]. Because the pressure is not the same function of $a$ on the two branches, the pressure-volume path fails to retrace itself, and the area $\oint p\,dV$ 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 $V(\phi)$ 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 $p_{\rm exp}\neq p_{\rm cont}$ 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 [1603.02805].

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 $H=0$ with $\dot H>0$ or, equivalently, $\dot a=0$ with $\ddot a>0$; a turnaround is characterized by $H=0$ with $\dot H<0$ or $\dot a=0$ with $\ddot a<0$ [2506.08109].

A model-independent relation between the loop area and cycle growth was derived for universes whose turnaround is produced by a term $-A\,a^{-n}$ in the Friedmann equation,
\[
H^2=\kappa\rho-Aa^{-n}, \qquad \kappa\equiv \frac{8\pi G}{3}.
\]
At turnaround one obtains
\[
\boxed{
\bigl(a_{\rm max}^{(i)}\bigr)^{3-n}
-
\bigl(a_{\rm max}^{(i-1)}\bigr)^{3-n}
=
-\,\frac{\kappa}{A}\oint p\,dV
}
\]
so the sign of $\oint p\,dV$ directly fixes whether successive maxima grow or shrink [1203.0395]. Two special cases emphasized in the literature are $n=0$ for a negative cosmological constant and $n=2$ for a closed universe curvature term.

For flat potentials, especially the quadratic potential $V(\phi)=\tfrac12 m^2\phi^2$, the loop is typically negative, $\oint p\,dV<0$, 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 [1203.0395]. For steep potentials such as
\[
V(\phi)=V_0\bigl(\cosh(\lambda\phi)-1\bigr),
\]
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 [1203.0395].

The same framework extends to anisotropic cosmology. In Bianchi I,
\[
3H^2=8\pi G\,\rho+\Lambda+\Sigma/a^6,
\]
so if negative hysteresis drives $a_{\rm max}$ upward from cycle to cycle, the anisotropy density $\rho_{\rm anis}\equiv \Sigma/a^6$ decreases correspondingly [1203.0395]. 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 [1603.02805]. Later work extended the construction to $f(R)$ and $f(T)$ gravity, where the loop integral is retained but the background dynamics is modified by curvature or torsion corrections [2506.08109].

| Framework | Representative structure | Reported hysteresis behavior |
|---|---|---|
| EHGB braneworld | $C(\rho+\sigma)^2=(A+H^2)(B+H^2)^2$ | $\delta a_{\min}\propto \oint p\,dV$; sign and model parameters control growth or shrinkage [1603.02805] |
| DGP braneworld | Modified Friedmann equation with crossover scale $r_c$ | Same hysteresis logic with branch parameter $\epsilon=\pm1$ [1603.02805] |
| Quadratic $f(R)$ | $f(R)=R+\alpha R^2$ | $173$ cycles; $\langle W\rangle=-1.39\times10^{-4}$; gradual decrease of $a_{\min}$ and $a_{\max}$ [2506.08109] |
| Reconstructed $f(T)$ | Torsion scalar $T=-6H^2$ and reconstructed $f(T)$ | Non-zero, negative work per cycle and an arrow of time [2602.15924] |

In the Einstein–Gauss–Bonnet case with space-like extra dimension, the modified Friedmann equation can be written as
\[
\frac{\kappa_5^4}{36}(\rho+\sigma)^2
=
\bigl(h(a)/a^2+H^2\bigr)
\Bigl[1+\tfrac{4\alpha}{3}\bigl((3k-h(a))/a^2+2H^2\bigr)\Bigr]^2,
\]
or, in compact form,
\[
C(\rho+\sigma)^2=(A+H^2)(B+H^2)^2.
\]
In the DGP model,
\[
H^2+\frac{k}{a^2}
=
\Bigl[\sqrt{\frac{\kappa^2\rho}{3}+\frac{1}{4r_c^2}+\frac{\epsilon}{2r_c}}\Bigr]^2.
\]
For both classes, the change in the minimum scale factor satisfies $\delta a_{\min}\propto\oint p\,dV$ up to a positive function of model parameters, and the conditions for ever-increasing expansion depend on the signature of $\oint p\,dV$ and the membrane parameters such as $\sigma$, $\alpha$, and $r_c$ [1603.02805].

The quadratic $f(R)$ study sharpens this point by showing that hysteresis does not universally imply growing cycles. Starting from
\[
S=\int d^4x\sqrt{-g}\Bigl[\tfrac1{2\kappa^2}f(R)+\mathcal{L}_m\Bigr],\qquad f(R)=R+\alpha R^2,
\]
the modified Friedmann equations become
\[
3H^2F(R)
=
\tfrac12\bigl[F(R)R-f(R)\bigr]-3H\dot F(R)+\kappa^2\Bigl(\tfrac12\dot\phi^2+V(\phi)\Bigr),
\]
\[
-2\dot HF(R)
=
\ddot F(R)-H\dot F(R)+\kappa^2\dot\phi^2.
\]
The work over one cycle is
\[
W=\oint p_\phi\,dV
=3\int_{\rm cycle}a^3H\Bigl(\tfrac12\dot\phi^2-V(\phi)\Bigr)\,dt.
\]
For the numerical setup
\[
a(0)=1,\;H(0)=0,\;\phi(0)=0.1,\;\dot\phi(0)=0.05,\;
\alpha=0.1,\;m^2=0.04,\;\kappa^2=1,
\]
the system undergoes $173$ complete cycles, with average work $\langle W\rangle=-1.39\times10^{-4}$ and total cumulative work $W_{\rm total}=-2.41\times10^{-2}$. In that realization, successive bounces and turnarounds are accompanied by a slight secular decrease in both $a_{\min}$ and $a_{\max}$, and the loop areas in the $(w_\phi,a)$ plane range from $10^{-5}$ to $10^{-3}$ [2506.08109].

A torsion-based extension reconstructs exact $f(T)$ functions from prescribed nonsingular bounces, again with a minimally coupled scalar field and the same hysteretic work integral,
\[
W=\oint p_\phi\,dV=\int_{\rm cycle}3a^2\dot a\,p_\phi\,dt.
\]
In that setting the loop appears in the $(w_\phi,a)$ or $(p_\phi,a)$ 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 [2602.15924].

## 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 $(V,\beta)$ with $\{\beta,V\}=4\pi G\gamma$, $V=2\pi^2a^3$, and
\[
\rho_{\max}\equiv \frac{3}{8\pi G\gamma^2\lambda^2}.
\]
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 [1912.11490].

In the holonomy-based case, the effective Friedmann equation is
\[
H^2=\frac{8\pi G}{3}(\rho-\rho_1)\Bigl[1-\frac{\rho-\rho_1}{\rho_{\max}}\Bigr],
\]
and each cycle contains a single non-singular bounce at $\rho=\rho_{\max}}$. In the connection-operator quantization,
\[
H^2=\frac{8\pi G}{3}(\rho-\rho_3)\Bigl[1-\frac{\rho-\rho_4}{\rho_{\max}}\Bigr],
\]
with two distinct bounce densities
\[
\rho^{(\pm)}=\rho_{\max}\bigl[(D\mp1)^2+\gamma^2D^2\bigr].
\]
The connection formulation therefore yields two alternating quantum bounces per cycle rather than one [1912.11490].

Despite these differences, the hysteresis phenomenon is reported to be robust for the quadratic potential. With $U(\phi)=\tfrac12 m^2\phi^2$, both quantizations show $P_{\rm expansion}\neq P_{\rm contraction}$, $\oint P\,dV\neq0$, and a secular increase of the maximum scale factor. The explicit relation
\[
\oint P\,dV=-\,\frac{3\pi}{4G}\,\Delta a_{\max}
\]
connects the loop area to the change in successive recollapse points [1912.11490]. For the $\phi^2$ potential with $\Lambda=0$, both holonomy and connection models exhibit $O(10)$ hysteresis cycles before standard inflation sets in, and each cycle increases $a_{\max}$ by $\Delta a_{\max}/a_{\max}\approx10^{-3}$–$10^{-2}$.

The same paper also identifies a genuinely non-monotonic regime. For the cosh-like potential
\[
U(\phi)=m^2\bigl[\cosh(q\,\phi/\tilde m_P)-1\bigr],
\]
the steepness parameter $q$ can be tuned so that some cycles have $\oint P\,dV>0$ and others $\oint P\,dV<0$. The result is quasi-periodic beats, with small-scale oscillations of period $T_{\rm small}\approx O(10^3\,t_P)$ and an envelope varying on $T_{\rm beat}\approx O(10^4\,t_P)$ 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 [1912.11490].

## 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 [1512.08360]. The quadratic $f(R)$ 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 [2506.08109]. 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 $\oint p\,dV$ 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 [1203.0395].

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 $\sim20\%$ at $\sim2$ GV; helium/proton $\sim4\%$ at $\sim2$ GV; 22-year and Forbush-event loops [2306.11026] |
| Solar wind cycle | Permutation entropy $S$ versus sunspot number | Hysteresis over Solar Cycle 23, indicating multistability [1112.5236] |
| Astrophysical dynamos | Magnetic-energy or field-amplitude bifurcation loops | Bistability, subcritical branches, and chaotic transients in $\alpha^2$, Babcock–Leighton, and $\alpha\Omega$ dynamos [2012.02064]; [2107.01546]; [2407.09042] |
| Curved-spacetime kinetic theory | Entropy-rate loop $\oint \dot\varsigma\,dt$ | Curvature-induced memory described as gravitational hysteresis [2410.04537] |

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 [2306.11026]. In the solar-wind study, hysteresis is seen in permutation entropy as the system traverses the ascending and descending phases of Solar Cycle 23 [1112.5236]. In dynamo theory, hysteresis denotes bistability between decaying and strong-field attractors, as in the $\alpha^2$ dynamo with thresholds $\sigma_c\approx0.2138$ and $\sigma_{bc}\approx0.198$, or in Babcock–Leighton and large-scale $\alpha\Omega$ models with subcritical branches [2012.02064]; [2107.01546]; [2407.09042]. 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
\[
\oint \dot\varsigma\,dt\neq 0
\]
set by integrated spacetime curvature [2410.04537].

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 $\oint p\,dV$.

Source: https://www.emergentmind.com/topics/cosmic-hysteresis