Australia: The Land Where Time Began

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Hysteresis in the Sky

Hysteresis is a phenomenon that occurs naturally in several magnetic and electric materials in condensed matter physics. It has interesting, vivid implications for the scenario of a cyclic bouncy Universe when it is applied to cosmology, i.e. cosmological hysteresis. The most important feature of this physical prescription is that it can be treated as an alternative proposal to an inflationary paradigm. The asymmetry in the equation of state parameter during expansion and contraction phase of the Universe causes cosmological hysteresis, as a consequence of a single scalar field. This process is purely thermodynamical in nature, resulting in a non-vanishing integral in cosmology. When applied to variants of modified gravity models:

1)      Dvali-Gabadadz-Porrati (DGP) brane world gravity,

2)      Einstein gravity dominated by the cosmological constant,

3)      Loop Quantum Gravity (LQG),

4)      Einstein-Gauss-Bonnet brane world gravity,

5)      Randall Sundrum single brane world gravity (RSII), in certain circumstances, this phenomenon leads to the increase of amplitude of the consecutive cycles as well as to a universe with older and larger successive cycles provided we have physical mechanisms to make the universe bounce and turnaround.

An arrow of time is inculcated in a dissipationless cosmology by this. It is said to be remarkable that it appears this phenomenon is widespread in several cosmological potentials in variants of modified gravity background, which Choudhury et al. explicitly study for-

i)                    Hilltop,

ii)                  Natural and

iii)                Coleman-Weinberg potentials, in this paper.

It is shown by semi-analytical analysis of these models, for different potentials with minimum/minima that the conditions which creates a universe with an ever increasing expansion, depend on the signature of the hysteresis loop integral as well as on the variants of model parameters. 

The phenomenon of hysteresis arises in systems where there is a lag between input and output. Hysteresis occurs when the lag is dynamic, i.e. changing over time. It can be evaluated by purely thermodynamical expressions where the output is dependent on the current and past inputs. In many lab systems such as ferromagnetic and ferroelectric materials such a phenomenon occurs naturally and is incorporated artificially in several electrical systems.

In cosmology, analogously with hysteresis, there is a phenomenon of cyclic universes where the Universe repeatedly comes into existence after each cycle. As occurs in hysteresis, where the material goes through the same process repeatedly, in a cyclic universe the Universe starts from big bang and ends in a big crunch repeatedly. There have been several proposals for cyclic universes in the literature (Eliade, 1934; Jaki, 1974; Starobinsky, 1979). In exact solutions of Einstein’s equations for a closed Universe filled with perfect fluid such models of cyclic universe also arise. In most of these models, however, all the cycles are identical to one another. Also, all of these models do not provide any prescription for avoiding a singularity. Therefore these models have failed to be successful for solving some of the major problems with the big bang model, such as the flatness and horizon problem, and avoidance of a singularity. In his paper Tolman (Tolman, 1934) used a radically different approach in which an oscillating universe could be obtained with increasing maximum expansion after each cycle. In other cyclic universe models that have been proposed (Steinhardt & Turok, 2002; Lehners, 2008), the same phenomenon occurs naturally. Tolman has proposed the presence of a viscous fluid which gave rise to an asymmetric, irreversible equation of motion. This generated an inequality between the pressures at the time of expansion and contraction phases which resulted in the growth of both energy and entropy. He therefore demonstrated a novel way for linking thermodynamical principals to the model of a cyclic universe. The solving of the flatness and horizons was helped by this unusual approach. But none of these models avoided a big bang singularity. Also, Tolman’s proposed model led to an increase in entropy with each cycle which was inevitable.

A method has been proposed (Kanekar, Sahni & Shtanov, 2001; Sahni & Toporensky, 2012) of both avoiding the presence of a singularity and increasing entropy by the use of a cosmological analogue model of hysteresis. The basic idea of generating the cyclical universe with an increasing maximum remains unchanged. According to Tolman if there is a tendency for the pressure to be greater during a compression than during a previous expansion, as he suggests would be expected with a lag following equilibrium conditions, an element of fluid can return to the volume it had originally with energy that had increased. Chaudhary et al. used the scalar field dynamics that were generated during the inflationary paradigm to generate the pressure asymmetry in order to maintain the symmetric nature of the equation of motion and thereby avoiding the production of entropy. It was originally proposed (Kanekar, Sahni & Shtanov, 2001) and extended later (Sahni & Toporensky, 2012) Chaudhary et al. demonstrated that “a universe filled with a scalar field possesses the intriguing property of ‘hysteresis’.” According to Chaudhary et al. the central idea was to show that in the presence of a massive scalar field under certain reasonable conditions when the bounce occurred (Cai et al., 2013; Cai, Brandenberger & Peter, 2013; Cai, Gao & Saridakis, 2012; Cai. Easson & Brandenberger, 2012; Li, Brandenberger & Cheung, 2014; Brandenberger, ????; Cai, Brandenberger & Zhang, 2011; Lilley & Peter, ????; Falciano, Lilley & Peter, ????; Lilley, Di Marco, Martin & Peter, 2010; Lilley, Lorenz & Clesse, ????; Battefeld & Peter, 2015; Graham et al., 2014; Koehn, Lehners & Ovrut, 2014), growing expansion cycles arise, the increase in amplitude of the expansion being related to the work done by/on the scalar field during the expansion/contraction of the universe. This leads to the production of a hysteresis loop defined to as ∮ pdV=0 during each oscillatory cycle. But the phenomenon of hysteresis is independent of the nature of potential, i.e. generic. Any potential which has a proper minima that randomises the phase of the scaler field as it oscillates around the minima during expansion, and thereby makes all possible values of ϕ probable at turnaround, is capable of causing the phenomenon of hysteresis. Potentials that have no proper minimum, however, will result in a unique value of ϕ which will make

                                                            Ƥexp = Ƥcont,

Thereby making:

                                                            ∮ pdV = 0

According to Choudhury et al. such potentials are not suitable candidates for causing hysteresis. The authors have used existing models such as brane world scenario in the early universe so as to avoid a big bang and big crunch, as well as the presence of negative density or phantom-like density in the early universe. In these models a bounce replaces the big bang singularity, and re-collapse or turnaround replaces the big crunch.

This paper is based on an analysis carried out (Sahni & Toporensky, 2001). Choudhury et al. have investigated further the phenomenon of hysteresis in different models such as variants of a cosmological constant model that includes ACDM, higher dimensional models such as Dvali-Gabadadz-Porrati (DGP) brane world gravity model in brane world. Models in which the dynamics of the scalar field are modified were used, which can be achieved by making the cosmological constant field dependent. The aim of the study was to study the phenomenon of hysteresis in different models as well as well as to constrain the models by using hysteresis. Models which can give rise to bounce in the early universe and turnaround in the late universe were studied. Also, Choudhury et al. investigated the equivalent conditions required to achieve such bounding and re-collapsing scenarios. It was shown that their analysis holds true for any form of the potential of the scalar field with a proper minimum. They have also shown that the phenomenon of hysteresis or the asymmetry in pressure can be achieved whether or not the slow roll conditions of inflation are satisfied. An increase in expansion maximum following each cycle depends on the sign of ∮ pdV as well as the parameters of the models that have been considered is a notable feature of this analysis. It is therefore can be seen that by using the remarkable cosmological effect of hysteresis as proposed (Kanekar, Sahni & Shtanov, 2001; Sahni & Toporensky, 2001), there are many methods and models in which a cyclical universe in which there is an ever increasing amplitude maximum can be achieved.

 In this paper Choudhury et al. have drawn various physical conclusions by explicitly solving the equations that govern the dynamics of the system by the use of semi-analytical techniques. They have studied the detailed features for 3 different potentials – hilltop potential, natural potential and Colemann-Weinberg potential; though the analysis is perfectly true for any kind of cosmological potential that has a proper minimum/minima. All of these potentials have minimum/minima that are well defined and have free parameters which can be adjusted to obtain the required results. This analysis therefore helps to apply stringent constraints to the characteristic parameters of these models in the bounding scenario along with cosmological hysteresis. According to Choudhury et al. they can at least show mathematically if there are any limiting cases in which these potentials combined with the models that can give rise to the phenomenon of cosmological hysteresis, i.e. Make ∮ pdV non-zero, though the analysis they carried out holds good under certain approximations that are physically acceptable and limiting cases. In this study we have explicitly derived the expression for work done in 1 complete cycle of expansion and contraction, and have shown that it is non-zero. But the sign of the integral depends on how the sign and magnitudes of the parameters of the models used in the study have been chosen.  That there are several models capable of giving rise to a cyclical universe with an increasing expansion magnitude is the most interesting result of their analysis.  

Sources & Further reading

1. Choudhury, S. and S. Banerjee (2016). "Hysteresis in the Sky." Astroparticle Physics 80: 34-89.