Cosmological hysteresis and the cyclic universe

A universe filled with a homogeneous scalar field exhibits “cosmological hysteresis.” Cosmological hysteresis is caused by the asymmetry in the equation of state during expansion and contraction. This asymmetry results in the formation of a hysteresis loop: $\oint pdV$, whose value can be nonvanishing during each oscillatory cycle. For flat potentials, a negative value of $\oint pdV$ leads to the increase in amplitude of consecutive cycles and to a universe with older and larger successive cycles. Such a universe appears to possess an arrow of time even though entropy production is absent and all of the equations respect time-reversal symmetry. Cosmological hysteresis appears to be widespread and exists for a large class of scalar-field potentials and mechanisms for making the universe bounce. For steep potentials, the value of $\oint pdV$ can be positive as well as negative. The expansion factor in this case displays quasiperiodic behavior in which successive cycles can be both larger as well as smaller than previous ones. This quasiregular pattern resembles the phenomenon of beats displayed by acoustic systems. Remarkably, the expression relating the increase or decrease in oscillatory cycles to the quantum of hysteresis appears to be model independent. The cyclic scenario is extended to spatially anisotropic models and it is shown that the anisotropy density decreases during successive cycles if $\oint pdV$ is negative.

Figure 1

The expansion factor of a spatially closed matter dominated universe is described by the cycloid (2.5).

Figure 2

An idealized illustration of cosmological hysteresis. The hysteresis loop shown above has $\oint pdV<0$ evaluated over a single expansion-contraction cycle. Rapid oscillations of the scalar field will modify this picture, since the equation of state of the scalar field varies between $-1$ and $+1$ during oscillations. Indeed, as discussed later in the text, scalar-field oscillations are indispensible for providing a nonvanishing value to the hysteresis loop. Oscillations of the scalar during expansion scramble its phase making all values of $\dot{\varphi }$ equally likely at turnaround and leading to $p_{\mathrm{expansion}}\ne p_{\mathrm{contraction}}$ and $\oint pdV\ne 0$. By contrast only a unique value of $\dot{\varphi }$, namely, $\dot{\varphi }=0$ at turnaround, enables the scalar field to follow its original trajectory in reverse during contraction, resulting in $p_{\mathrm{expansion}}=p_{\mathrm{contraction}}$ and $\oint pdV=0$. Consequently only a very small set of initial conditions does not give rise to hysteresis, the latter appearing to be ubiquitous for scalar-field models which can oscillate.

Figure 3

The change in amplitude of successive expansion maxima is linked, via (2.15, 2.16), to the hysteresis loop defined by $\oint pdV≔{\int }_b^{d}pdV$. By contrast, the change in amplitude of successive expansion minima is related, via (2.20, 2.21), to the hysteresis loop defined by $\oint pdV≔{\int }_a^{c}pdV$.

Figure 4

The presence of hysteresis can increase the amplitude of successive expansion maxima in a cyclic universe sourced by the potential $V(\varphi )=\frac{1}{2}{m}^2{\varphi }^2$. In the left panel, cosmic turnaround is caused by the presence of a negative cosmological constant, so that $n=0$ in (2.11). In the right panel, turnaround is caused by a (positive) curvature term, so that $n=2$ in (2.11). In both cases the increased amplitude of successive cycles is described by the hysteresis equations (2.15, 2.20). The successively increasing expansion cycles appear to endow the universe with an arrow of time even though the equations governing cosmological evolution are formally time reversible and there is no entropy production.

Figure 5

A cyclic universe sourced by the potential $V=V_0(\cosh \lambda \varphi /{\tilde{m}}_P-1)$. Increasing the value of the control parameter $\lambda$ causes cyclic behavior to change dramatically, as illustrated in the four panels. In all cases turnaround is caused by a negative cosmological constant: $n=0$ in (2.11), while the control (steepness) parameter has values $\lambda =2$, 2.5 (upper left and right) and $\lambda =4,6$ (lower left and right). As the control parameter increases the behavior of the universe undergoes a remarkable change. For $\lambda \lesssim 2$ the amplitude of successive cycles increases (top left), with the increase being larger for smaller values of $\lambda$. For moderate values of $\lambda$ (top right and bottom left panels) the universe displays a quasiperiodic pattern reminiscent of beats in sound waves. During beats the value of $\oint pdV$ is negative during the first half of the larger (parent) cycle and positive during the second half. Beats gradually disappear as the value of $\lambda$ is increased. The universe now begins to show oscillatory behavior in which all cycles have roughly the same amplitude and duration signifying $\oint pdV\simeq 0$ (bottom right). In all panels the amplitude of successive cycles is governed by the simple formulas (2.17, 2.20).

Figure 6

A cyclic universe sourced by the potential $V=V_0(\cosh \lambda \varphi /{\tilde{m}}_P-1)$. Increasing the value of the control parameter $\lambda$ causes cyclic behavior to change dramatically, as illustrated in the four panels. In all cases turnaround is caused by a positive curvature term: $n=2$ in (2.11), while the control (steepness) parameter has values $\lambda =1.5$, 2 (upper left and right) and $\lambda =3.3,6$ (lower left and right). The top two panels show the expansion factor in logarithmic units. In the top left panel each cyclic epoch is larger in amplitude and duration than its predecessor, indicating $\oint pdV<0$. As the steepness parameter is increased the behavior of successive cycles becomes highly stochastic (top right and bottom left) which is indicative of the fact that the hysteresis loop can be negative ($\oint pdV<0$) as well as positive ($\oint pdV>0$) during cycles. Once more we find larger amplitude cycles to have a longer duration. Further increase of $\lambda$ leads to a stage when hysteresis is virtually absent and all cyclic epochs become similar. In all panels the amplitude of successive cycles is governed by the simple formulas (2.18, 2.20).

Figure 7

The expansion factor of a spatially closed universe dominated by a scalar field with the potential $V\propto {\varphi }^{-\alpha }$, $\alpha >0$. The universe soon settles into an oscillatory mode in which all cycles are of equal amplitude indicating $\oint pdV=0$.

Figure 8

The expansion factor (dotted line) of a spatially anisotropic Bianchi I universe dominated by a massive scalar field. The anisotropy density (solid line), given by ${\rho }_{\mathrm{anis}}=\Sigma /a^6$, first increases and then decreases at the time of each bounce. The hysteresis loop for this universe is negative $\oint pdV<0$, with the result that the amplitude of successive cycles gradually increases. This leads to the waning of anisotropy with each successive cycle with the result that ${\rho }_{\mathrm{anis}}$ becomes vanishingly small at the commencement of the fifth cycle and cannot be resolved on the scale of the figure.