Hysteresis in the Sky

Hysteresis in the Sky

Sayantan Choudhury 111Presently working as a Visiting (Post-Doctoral) fellow at DTP, TIFR, Mumbai,
Alternative E-mail: sayanphysicsisi@gmail.com
.
   Shreya Banerjee sayantan@theory.tifr.res.in, shreya.banerjee@tifr.res.in
Abstract

Hysteresis is a phenomenon occurring naturally in several magnetic and electric materials in condensed matter physics. When applied to cosmology, aka cosmological hysteresis, has interesting and vivid implications in the scenario of a cyclic bouncy universe. Most importantly, this physical prescription can be treated as an alternative proposal to inflationary paradigm. Cosmological hysteresis is caused by the asymmetry in the equation of state parameter during expansion and contraction phase of the universe, due to the presence of a single scalar field. This process is purely thermodynamical in nature, results in a non-vanishing hysteresis loop integral in cosmology. When applied to variants of modified gravity models -1) Dvali-Gabadadze-Porrati (DGP) brane world gravity, 2) Cosmological constant dominated Einstein gravity, 3) Loop Quantum Gravity (LQG), 4) Einstien-Gauss-Bonnet brane world gravity and 5) Randall Sundrum single brane world gravity (RSII), under certain circumstances, this phenomenon leads to the increase in amplitude of the consecutive cycles and to a universe with older and larger successive cycles, provided we have physical mechanisms to make the universe bounce and turnaround. This inculcates an arrow of time in a dissipationless cosmology. Remarkably, this phenomenon appears to be widespread in several cosmological potentials in variants of modified gravity background, which we explicitly study for- i) Hilltop, ii) Natural and iii) Colemann-Weinberg potentials, in this paper. Semi-analytical analysis of these models, for different potentials with minimum/minima, show 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.

Keywords:
Alternatives to inflation, Cosmological hysteresis, Cyclic cosmology, Bouncing cosmology, Cosmology beyond the standard model, Cosmology from effective theory.

TIFR/TH/15-18

Department of Theoretical Physics, Tata Institute of Fundamental Research, Colaba, Mumbai - 400005, India Department of Astronomy and Astrophysics, Tata Institute of Fundamental Research, Colaba, Mumbai - 400005, India

1 Introduction

Hysteresis is a phenomenon that arises in systems with a lag between its input and output. When this lag is dynamic i.e it changes with time, we get hysteresis loops. It can be evaluated by purely thermodynamical expressions where the output depends on the current and past inputs. Such phenomenon naturally occurs in many laboratory systems like ferromagnetic and ferroelectric materials and is often incorporated artificially in several electrical systems.

In analogy with hysteresis, in cosmology, we have the phenomenon of cyclic universe in which the universe re-borns repeatedly after each cycle. Just as in hysteresis, where the material undergoes through the same process over and over again, in a cyclic universe, the universe starts from big bang and ends in big crunch repeatedly. Several models of cyclic universe have been proposed in literature eliade (); jaki (); starobinsky (). Such models also arise naturally as exact solutions of Einstein equations for a closed universe filled with perfect fluid. However in most of these models, all the cycles are identical to one another. Also all these models does not provide any prescription for avoiding singularity. Hence these models remain unsuccessful in solving some of the major problems of big bang model i.e the flatness and horizon problem, and avoidance of singularity. However Tolman tolman () in his paper used a radically different approach by which one could get an oscillating universe with increasing expansion maximum after each cycle 222The same phenomenon occurs naturally in other cyclic universe models as proposed in refs. Steinhardt:2001st (); Lehners:2008vx ().. He postulated the presence of a viscous fluid which gave rise to asymmetric, irreversible equation of motion. This created an inequality between the pressures at the time of expansion and contraction phases which resulted in the growth of both energy and entropy. Thus he showed a novel way of linking thermodynamical principles to the model of cyclic universe. This unusual approach helped in solving the horizon and flatness problem. In later years, theory of inflation Baumann:2014nda (); Baumann:2009ds (); Lyth:1998xn () was developed which addressed both the horizon and flatness problem. But none of these models were able to avoid big bang singularity. Also the model proposed by Tolman led to an inevitable increase in entropy with each cycle.

However, in Kanekar:2001qd (); Sahni:2012er (), the authors have proposed a method of avoiding both the presence of singularity and increasing entropy by using a cosmological analog model of hysteresis. The basic idea of generating the cyclic universe with an increasing maximum remains the same and in the words of Tolman is “if the pressure tends to be greater during a compression than during a previous expansion, as would be expected with a lag behind equilibrium conditions, an element of fluid can return to its original volume with increased energy..". The authors created the asymmetry in pressure using the scalar field dynamics generated during inflationary paradigm, thereby maintaining the symmetric nature of the equation of motion hence avoiding entropy production. Originally proposed in Kanekar:2001qd () and later extended in Sahni:2012er (), the authors demonstrated that “a universe filled with a scalar field possess the intriguing property of ‘hysteresis’." The central idea was to show that the presence of a massive scalar field "under certain reasonable conditions at the bounce Cai:2013kja (); Cai:2013vm (); Cai:2012ag (); Cai:2012va (); Li:2014era (); Brandenberger:2012zb (); Cai:2011zx (); Lilley:2015ksa (); Falciano:2008gt (); Lilley:2010av (); Lilley:2011ag (); Battefeld:2014uga (); Graham:2011nb (); Koehn:2013upa (), gives rise to growing expansion cycles, the increase in expansion amplitude being related to the work done by/on the scalar field during the expansion/contraction of the universe." This leads to the production of hysteresis loop defined as , during each oscillatory cycle. The loop area is largest in case of inflationary potentials since they give rise to largest asymmetry between expansion and contraction pressures. But the phenomenon of hysteresis is generic i.e. independent of the nature of potential. Any potential with a proper minima which randomizes the phase of the scalar field as it oscillates around the minima during expansion, thereby making all possible values of probable at turnaround, is cable of causing the phenomenon of hysteresis. However potentials without any proper minimum will result in a unique value of which will make

(1)

thereby making:

(2)

Such potentials are not suitable candidates for causing hysteresis. In order to avoid big bang and big crunch, the authors have made used of the presence of existing models like brane world scenario in the early universe, and the presence of negative density or phantom like density in the late universe. These models replaces the big bang singularity by bounce and the big crunch by re-collapse or turnaround.

Our present paper is based on the analysis done by Sahni:2012er (). We have further investigated the phenomenon of hysteresis in different models like the variants of cosmological constant model including CDM, higher dimensional models like Dvali-Gabadadze-Porrati (DGP) brane world gravity model, Loop Quantum Gravity model and Einstein Gauss-Bonnet brane world gravity model in brane world, and in models where the dynamics of the scalar field gets modified which can be achieved by making the cosmological constant field dependent. Our aim is to study not only the phenomenon of hysteresis in different models but also to constrain the parameters of the model using hysteresis. We have mostly studied the models which can give rise to both the phenomena of bounce in the early universe and turnaround in the late universe. We have also investigated the equivalent conditions required to achieve such bouncing and re-collapsing scenarios. We have shown that our analysis holds true for any general form of the potential of the scalar field with a proper minimum. We have also shown that the phenomenon of hysteresis or the asymmetry in pressure can be achieved irrespective of whether the slow roll conditions of inflation are satisfied or not. A notable feature of this analysis is that an increase in expansion maximum after each cycle now depends not only on the sign of but also on the parameters of the models that we have considered. Thus we see that using the remarkable cosmological effect of hysteresis as proposed by Kanekar:2001qd (); Sahni:2012er (), there are numerous methods and models in which a cyclic universe with an ever increasing amplitude maximum can be achieved.

The plan of the paper is as follows:

  • In section 2, we have discussed the mathematical formulation leading to the phenomenon of hysteresis in cosmological scenario.

  • In section 3 we have tried to draw an analogy between the hysteresis in ferromagnetic materials and in a cyclic universe. This simple analysis leads to the conclusion that the equation of state parameter w plays the role of magnetic field in cosmology and the scale factor mimics the behavior of magnetization .

  • In section 4, 5, 6, 7 ,9.1, we have studied the behavior of each of the above mentioned models for flat, closed and open universe and have discussed various unexplored important cosmological features.

  • In section 4, 6, 7 ,9.1, we have explicitly studied cosmological hysteresis in the context of higher dimensional theories, where we have shown the results for both space-like and time-like extra dimensions. For the sake of simplicity we restrict ourselves up to five dimensions (). But one can extend the computation for dimensions, .

In this paper, we have drawn various physical conclusions by explicitly solving the equations governing the dynamics of the system using the semi-analytical techniques. Though the analysis is perfectly true for any kind of cosmological potential with a proper minimum/minima, we have the studied the detailed features for three different potentials - hilltop potential, natural potential and Colemann-Weinberg potential. All these potentials have well defined minimum/minima and have free parameters which can be adjusted to get the required results. Thus this analysis helps us to put stringent constraints on the characteristic parameters of these models in the bouncing scenario along with cosmological hysteresis. Though the analysis that we have performed holds good under certain physically acceptable approximations and limiting cases, but we can at least show mathematically if there are any limiting cases in which these potentials combined with the models can give rise to the phenomenon of cosmological hysteresis i.e. make non-zero. In this paper we have also explicitly derived the expression for work done in one complete cycle of expansion and contraction, and have shown it to be non zero. But the sign of the integral depends on how we have chosen the sign and magnitudes of the parameters of our models. The most interesting result of our analysis is that there are several models which can give rise to a cyclic universe with an increasing amplitude of expansion.

2 Basics of cosmological hysteresis

Before going to the technical details of the “Cosmological hysteresis” let us mention clearly the underlying assumptions. It is important to note that throughout the analysis in this paper we assume that during a specified period in the time line of the universe, it is described by a single massive scalar field which is minimally interacting with the gravity sector. The presence of this scalar field is responsible to generate the required asymmetry in the pressure of the universe and this leads to an overall increase in the energy of the universe and hence an increase in its amplitude of the expansion rate.

The action governing the dynamics of this scalar field with potential within effective field theory description, is given by

(3)

where the signature of the metric throughout our analysis is (-, +, +, +). The total action can be written as the sum of the standard Einstein Hilbert action () and the action for the scalar field (). The energy-momentum tensor for the scalar field can be computed from the matter part of the action

(4)

as:

(5)

and for a homogeneous and isotropic spatially flat () FLRW cosmological background, the energy density and pressure for a scalar field can be computed from the energy momentum tensor as:

(6)
(7)

and the resulting equation of state parameter can be written as:

(8)

where we assume that the energy momentum tensor for scalar field can be approximated via perfect fluid. Also in the spatially flat FLRW cosmological background the scalar field equation of motion is given by:

(9)

where is the d’Alembertian operator four dimension defined as:

(10)

Here is the Hubble parameter defined as:

(11)

where is the scale factor.
Origin of cosmic hysteresis: If we closely look at Eq (9), then we can conclude that when the universe expands i.e. the second term mimics the role of friction and opposes the motion of the scalar field, thus serves the purpose of damping during its motion. This lowers the kinetic energy of the scalar field compared to its potential energy, giving rise to a soft equation of state ( in case i.e. slow roll regime). By contrast, in a contracting () phase of the universe, the term behaves like anti-friction and favors the motion of the scalar field and hence accelerates it. This makes the kinetic energy of the scalar field much larger than the potential energy, giving rise to stiff equation of state ( in case ). As a result, from second law of thermodynamics, we can convey that a net asymmetry in the pressure (during expansion and contraction cycle) leads to a net non-zero work done by/on the scalar field. In addition, if we now postulate the presence of bouncing and recollapsing mechanisms during contraction and expansion respectively, one can expect that a non-zero work done during a given oscillatory cycle to be converted into expansion energy, resulting in the growth in the maximum amplitude and hence maximum volume of the universe of each successive expansion cycle. Thus producing older and larger cycles. In Sahni:2012er (), using simple thermodynamic arguments, the authors have developed the equations which relate the change in maximum amplitude of the scale factor after successive cycles to the work done. The authors have shown that these equations have a universal form which is independent of the scalar field potential responsible for hysteresis. As has been discussed in Sahni:2012er (), though the process is independent of the potential, “the presence of hysteresis is closely linked to the ability of the field to oscillate”. This urges the presence of potential minimum/minima. This is because only potentials with well defined minimum can make the field oscillate. As a result Oscillations of the scalar during expansion makes its phase arbitrary, thereby making all values of equally likely at turnaround. Thus assuring that the values of and , when the universe turns around and contracts, are nearly uncorrelated with its phase space value when the field began oscillating. As a result, the field almost always rolls up the potential along the different phase space trajectory compared to the one along which it had descended during expansion. This gives rise to unequal pressure during expansion and contraction hence to non-zero work done.

In the present work, we have reconsidered the above phenomenon of hysteresis and applied it to models or physical mechanisms which can make the universe bounce and turnaround in the presence of potentials having well defined minimum/minima. Further solving Eq. (7) and Eq. (9) simultaneously, along with the Friedmann equations derived from various cosmological background model, and applying the proper bounce and turnaround conditions, which will be discussed in the following sections, we get the explicit expressions for the scalar field and scale factor as a function of time.

It has been first pointed out in ref.Kanekar:2001qd () that when we plot the equation of state given by vs the scale factor from a specified cosmological model, we get a hysteresis loop whose area contributes to the work done by/on the scalar field during expansion and contraction of the Universe. The general expression for the work done by/on the scalar field during one cycle is given by

(12)

where the contributions and represent the work done by/on the scalar field during the phase of contraction and expansion of the Universe respectively. The signature of the integral depends on the pressure during the phase of contraction and expansion i.e if the pressure corresponding to the contraction phase is greater than the pressure due to expansion phase i.e.

(13)

then the overall signature of the work done is negative i.e.

(14)

On the other hand, if the pressure corresponding to the contraction phase is smaller than the pressure due to expansion phase i.e.

(15)

then the overall signature of the work done is positive i.e.

(16)

For a universe characterized by a scale factor , the total volume at any given time is given by (neglecting the overall constant factor). Hence using this input, the area of the cosmological hysteresis loop or equivalently the work done is given by:

(17)

Hence we follow the following algorithm:

  • In the context of various cosmological frameworks i.e. DGP branewold gravity, Loop Quantum gravity (LQG), Einsitein-Gauss-Bonnet (EGB) gravity and in presence of pure and field dependent cosmological constant within Einstein gravity we explicitly derive an expression for this characteristic integral in terms of the scale factor and the scalar field degrees of freedom.

  • Then we solve the equations of motions for the scalar field under various limiting approximations and hence elaborately study the physical conditions under which the above integral gives non zero value.

  • Further we repeat this mentioned two step process for three different cosmological potentials i.e. for Hilltop potentials, Natural potential and Coleman-Weinberg potential within the framework of effective field theory prescription, which will be discussed in the next section in detail.

  • We also analyze the whole process graphically to see whether we get a net increase in the amplitude of the scale factor after one cycle of expansion and contraction.

Though the phenomenon of hysteresis is commonly attached to magnetic and electric systems, its appearance for different cosmological models, almost naturally, makes us appreciate and acknowledge its importance in the field of cosmology. The various advantages which a cyclic universe along with the phenomenon of hysteresis, which is our main focus in this paper, have are:

  • The phenomenon of hysteresis is important due to the simplicity with which it can be generated. Only a thermodynamic interplay between the pressure and density, creating an asymmetry during expansion and contraction phase of the universe, succeeds in causing cosmological hysteresis.

  • As the phenomena of cosmological hysteresis deals with the bouncing as well as the re-collapsing phase of the universe, one can avoid the appearance of Big Bang Singularity as well as the Big Crunch at early and late times.

  • Hysteresis can be generated by the presence of a single massive scalar field, which has already been studied for a wide variety of physical situations. The most exciting issue is, it do not require any other fields for its occurrence. Hence it is very easy to handle and its properties can be studied extensively in cosmological literature.

  • A cyclic universe having the conditions to cause hysteresis, can solve all the Big Bang puzzles, hence acts as an alternative proposal to inflation Choudhury:2014hja (); Choudhury:2014sxa (); Choudhury:2013woa (); Choudhury:2013zna (); Choudhury:2011jt ().

  • In this scenario, we can always start with a closed or open universe and after allowing the universe go through a number of cycles, we get the present observable flat universe. In this paper, we will extensively deal with several models that can give rise to such cyclic universe. Through analytical calculations, we will show that irrespective of the nature of the universe, we can get a cyclic model with increasing amplitude of the scale factor for a wide variety of models.

  • It results in dissipative cosmology Kanekar:2001qd (); Sahni:2012er (); Vilenkin:2013rza (), which makes the whole process irreversible, which finally causes an arrow of time according to widespread belief. But recently in the ref. Sahni:2015kga () the authors have explicitly shown that for cosmological hysteresis phenomena such an arrow of time can appear even if equations describing cosmological evolution are dissipationless, which makes the cumulative process reversible provided they possess cosmological attractors in the expanding and contracting phase of the universe. Additionaly, it is important to note that for flat cosmological potential the increament in the expansion cycles are routinely observed. But for the steep potentials two fold cyclic pattern with lesser expansion cycles is observed, which being nested in the larger expansion cylces. This phenomena is exactly analogous to the ‘beat’ formation in acoustics systems. In ref. Kanekar:2001qd (); Sahni:2012er () the authors have studied the cosmological beat formation in the context of chaotic potential .

  • Cosmology in this scenario has not been explored in a very wide sense. Earlier it has been studied by the authors of refs. Kanekar:2001qd (); Sahni:2012er (); Sahni:2015kga (). In this paper, we have further explored this phenomenon for several other models which have not been discussed earlier in refs. Kanekar:2001qd (); Sahni:2012er (); Sahni:2015kga ().

  • In future we plan to connect these analyses with CMB observations, by rigorous study of the cosmological perturbation theory Biswas:2015kha (); Biswas:2012bp (); Battarra:2014tga () in various orders of metric fluctuations and computation of two point correlations to get the expressions for scalar and tensor power spectrum in this context. Hence we extend the study of this paper to compute the primordial non-Gaussianity in CMB from three and four point correlations Choudhury:2015yna (); Choudhury:2014uxa (); Gao:2014hea (); Gao:2014eaa (); Maldacena:2002vr (); Maldacena:2011nz (); Arkani-Hamed:2015bza (); Mata:2012bx (); Ghosh:2014kba (); Kundu:2014gxa (). We also plan to derive the explicit expression for various modified consistency relations between the non-Gaussian as well as other cosmological parameters in the present context.

  • Also we plan to setup the formalism of magnetogenesis from the present setup which can be treated as the alternative version of the inflationary magnetogenesis Choudhury:2014hua (); Choudhury:2015jaa (); Subramanian:2015lua ().

  • We also carry forward our analysis in the development of density inhomogeneities, which is the prime component to form large scale structures at late times. Also the specific role of cosmological hysteresis in the study of cosmological perturbations i.e. for interacting/decoupled dark matter and dark energy have not been explored at all earlier. We have some future plan to do some computations from this setup.

  • Further using the reconstruction techniques Choudhury:2014kma (); Choudhury:2013iaa (); Choudhury:2014sua (); Lidsey:1995np () we want to study the generic features of scalar field potentials in the framework of cosmological hysteresis.

  • Last but not the least, we also plan to further study this outstanding cosmological phenomenon for different modified gravity pictures i.e. for variants of gravity Sotiriou:2008rp (); DeFelice:2010aj (); Faraoni:2008mf (); Capozziello:2011et (), two brane-world model in presence of the Einsitein-Hilbert term and the Einstein-Hilbert-Gauss-Bonnet gravity setup Choudhury:2012yh (); Choudhury:2013yg (); Choudhury:2013eoa (); Choudhury:2013qza (); Choudhury:2013aqa (); Choudhury:2014hna (); Choudhury:2015wfa ().

3 Analogy with magnetic hysteresis

The phenomenon of lagging of magnetic induction B or magnetization M behind the magnetic field H when a specimen of a magnetic material (such as iron) is subjected to a cycle of magnetization is called “magnetic hysteresis”. The closed loop that is traced by the material in the or plane is known as the hysteresis loop. It is related to the change in the alignment of the magnetic dipoles, as one varies the magnetic field which leads to magnetization and demagnetization of the material. It is a beautiful way of depicting the effect of varying H on the system. Fig. 1(a) is a representative schematic diagram in which we have explicitly shown one such hysteresis loop in ferromagnetic material. Since we want to draw an analogy between the phenomenon of hysteresis in magnetism and cosmology, we have also shown Fig. 1(b) to illustrate how an ideal hysteresis loop looks in the case of a cyclic universe. In a cyclic universe, the hysteresis loop is traced by the universe in the plane where a is the scale factor of the universe and is the equation of state parameter. Closely following the two figures we can say that just as in a magnetic material, we also vary the magnetic field and try to find the behavior of the magnetization of the material, in this cyclic model, we vary the pressure or the equation of state in order to find the variation of the scale factor or expansion of the universe. This motivates us to consider that the parameter playing the role of H in cosmological hysteresis is and that of is . Just as in magnetic material, where the magnetization oscillates within a specified maximum and minimum values, analogously in cyclic model, the scale factor goes through maximum and minimum values which has been clearly shown in Kanekar:2001qd (); Sahni:2012er () for few cosmological models and will again be shown in this paper for some other cosmological models. While the primary cause of magnetic hysteresis is the asymmetry in the behavior of the alignment of the magnetic dipoles with increase and decrease of , the cause for hysteresis in cosmology is the asymmetry in pressure during expansion and contraction phases of the universe. In magnetic hysteresis loop, the minimum and maximum corresponds to the state when all the magnetic dipoles are aligned in reverse and along the direction of respectively. In cosmological hysteresis, the maximum is reached when the scale factor reaches its maximum value and the density of the scalar field reached at its minimum i.e when the condition for re-collapse is generated, and the corresponding minimum is reached at bounce when scale factor becomes minimum and density of the scalar field reached at its maximum value. But unlike in magnetic hysteresis where the parameters can take all kind of values i.e positive, negative and zero, in cyclic universe, one of our primary goal is to avoid singularity, i.e. 0, so that singularity is replaced by bounce. While repeated cycles of magnetic hysteresis loop results in loss of finite amount of energy, hence decrease in the area of the loop, in cosmological hysteresis, repeated cycles may be either larger or smaller in area compared to the previous one. Thus, we find that a closer look at these two phenomena draws lots of similarities in their behavior. In table. 1 we have depicted the analogy and the parallelism between the different features of these two kinds of hysteresis.

(a) An illustration of hysteresis in ferromagnetic materials. The loop has been plotted in the plane.
(b) An idealised illustration of cosmological hysteresis. The loop has been plotted in the plane.
Figure 1: Analogy between magnetic hysteresis in ferromagnetic materials and cosmological hysteresis.
Characteristics Magnetic Hysteresis Cosmological Hysteresis
Loop parameters
Hysteresis loop
Largest value of Given by hard Cyclic Universe
hysteresis loop ferromagnetic materials with inflationary conditions
Upper and lower No limit on maximum
limits of the loop and minimum values of
Characteristic equation
for the loop
Nature of Soft ferromagnetic materials Universe with softer equation
the loop have smaller loop area of state have smaller loop area
Table 1: Table showing the analogy between magnetic hysteresis and cosmological hysteresis.

In this section we have shown the complete analysis of different models and studied the conditions under which we get an increase in the amplitude of the scale factor after each cycle. We have studied open closed and flat universe. We have also explicitly calculated the work done in one cycle for three different potentials.

4 Hysteresis from Dvali-Gabadadze-Porrati (DGP) brane world gravity model

The DGP model is a model of modified gravity theory proposed by Gia Dvali, Gregory Gabadadze, and Massimo Porrati. The model consists of a 4D Minkowski brane embedded in a 5D Minkowski bulk like Randall Sundrum (RS) II model. But unlike in RSII model Randall:1999vf (); Maartens:2010ar () 333See also the details of Randall Sundrum (RS) I model studied in ref. Randall:1999ee () for completeness., here the infinitely large 5th extra dimension is flat. The Newton’s law can be recovered by adding a 4D Einstein–Hilbert action sourced by the brane curvature to the 5D action. While the DGP model recovers the standard 4D gravity for small distances, the effect from the 5D gravity manifests itself for large distances.

The DGP model is described by the following action Dvali:2000hr ():

(18)

where is the metric in the 5D bulk and

(19)

is the induced metric on the brane with being the coordinates of an event on the brane labeled by . The first and second terms in Eq. (426) correspond to Einstein–Hilbert actions in the 5D bulk and on the brane, respectively. Note that and are 5D and 4D gravitational constants, respectively, which are related with 5D and 4D Planck masses, and , via

(20)
(21)

The Lagrangian describes matter localized on the 3-brane.

The equations of motion read

(22)

where is the 5D Einstein tensor. The Israel junction conditions on the brane, under which a symmetry is imposed:

(23)

where is the extrinsic curvature calculated on the brane, is the energy-momentum tensor of localized matter and is the crossover length scale,

(24)

because it sets the scale above which the effect of extra dimension becomes important.

The modified Friedmann equations in this model are given by Copeland:2006wr ():

(25)
(26)

In the present analysis, we have set the extra dimension as time-like which is necessary for getting late time acceleration in DGP model. but in generalized prescription one can comsider both space and time like extra dimensions. Also it is important to note that, in Eq. (26) the spatial curvature can take values or . Now following the proposal of Kanekar:2001qd (); Sahni:2012er (), we know that in order to get a cyclic universe, the condition for bounce and turn around are given by:

  • Bounce:

    (27)

    and

    (28)
  • Turn around:

    (29)

    and

    (30)

In the following subsections we will discuss all of these possibilities in detail for DGP brane world gravity framework.

4.1 Condition for bounce

In the early universe, at high energy or equivalently in the high density regime of the brane world, , one can expand Eq. (26) as Gumjudpai:2003vv ()

(31)

At first order approximation, Eq. 31 reduces to the following expression:

(32)

This is obviously a valid assumption because at the time of bounce, the density of the matter content of the Universe (in our case scalar field) is at its maximum, hence we can neglect the contribution from the other higher order terms in Eq.  (31). Now applying the at bounce, and setting in Eq.  (2(c)), we get:

(33)

where and are the density and scale factor at bounce, respectively. Similarly the mass content at bounce (neglecting the constant factor) is given by,

(34)

Hence applying the energy conservation in the present context, we get:

(35)

where is the work done during each expansion-contraction cycle which is given by which includes contribution from the area of the hysteresis loop. By setting,

(36)

we get the expression for change in amplitude of the scale factor at each successive cycle as

(37)

Therefore we clearly observe that the change in amplitude of the scale factor after each cycle depends on the sign of the integral, the curvature parameter and the cross over length scale for DGP brane world. It is independent of the density of the matter content of the universe. During our computation we also observe that the if we fix the spatial curvature parameter , this makes the energy density of the scalar field imaginary, which is not at all physically possible. Hence, for the cosmological bounce from DGP brane world model is not at all possible. But if we neglect all the higher order terms, Eq. (31) reduces to the standard Friedmann equation for which the condition for bounce becomes

(38)

which is possible for both but not for . Hence we see that the bouncing condition depends largely on the order of terms which we are including because our analysis is possible only under approximations. But since our present observed universe is flat (), we will be interested to study the first case in more detail. Hence in our further analysis for DGP model, we will show the results for Eq. (2(c)) only i.e exclude the possibility. Here we have the following expression for :

(39)

Let us now briefly mention the characteristic feature of the results for cosmological bounce for DGP brane world model in the following:

  1. For a closed universe, depending on whether the quantity appearing in the denominator for is positive or negative, an increase in the scale factor after each cycle is possible if:

    • The hysteresis loop integral

      (40)

      or equivalently

      (41)
    • The hysteresis loop integral

      (42)

      or equivalently

      (43)
  2. On the other hand, for the case , the quantity in the denominator is always positive and finally we get an increase in the scale factor only if the hysteresis loop integral

    (44)

4.2 Condition for acceleration

From Eq.  (26), at bounce or at high energy, contribution from term is small compared to the density of the scalar field and consequently one finally gets the standard cosmology from the condition for acceleration as:

(45)

which clearly implies that the cosmological bounce can be obtained by violating the strong energy condition.

Substituting the expression for for , we get the conditions for acceleration at bounce as:

(46)

This implies that the pressure of the matter content of the Universe at the time of cosmological bounce is related to the scale factor and the cross over length scale . Also we observe that the condition for acceleration depends on whether we have considered a possibility of a closed or flat universe. For flat universe, we see that the condition for acceleration is independent of the scale factor at cosmological bounce.

Finally substituting the expressions for the energy density and pressure from Eq. (7) into Eq. (533), we get the condition for the expansion of the Universe in terms of the scalar field degrees of freedom as:

(47)

which is same as that we get for the standard cosmological inflationary scenario. Therefore, we can conclude that the conditions on the dynamics of scalar which is needed for causing acceleration in standard case remains unchanged for the DGP model at the time of cosmological bounce. This implies that the cosmological potentials in the standard cosmological scenario, capable of satisfying the above condition, will also be able to cause acceleration in a universe described by the DGP brane world model.

(a) An illustration of the bouncing condition for a universe with an equation of state .
(b) An illustration of the bouncing condition for a universe with an equation of state .
(c) An illustration of the acceleration condition at the time of bounce for a universe with an equation of state .
(d) An illustration of the acceleration condition at the time of bounce for a universe with an equation of state .
Figure 2: Graphical representation of the phenomena of bounce and acceleration for DGP model.

In Fig. 2, we have shown the phenomena of bounce and acceleration in the DGP model. We can draw the following conclusions from the above figures:

  • In Fig. 2(a) and Fig. 2(b), we have plotted the r.h.s of Eq. (26) using the relation . The graphs have been plotted only for the cases when . This is because, Eq. (33) shows that density becomes imaginary for the case when , which is unphysical.

  • We have used , since we require a soft equation of state for causing the acceleration and expansion. The case made only approximately zero. The value of has been chosen such that we get proper bounce and acceleration. Thus we find that we require a larger value of for causing bounce in a closed universe as compared to a flat universe.

  • From Fig. 2(a), we get the bounce at . Similarly from Fig. 2(b), we get .

  • In Fig. 2(a) and Fig. 2(b), going to negative values may be interpreted as the universe changing its direction of motion at bounce. As had been discussed in Kanekar:2001qd (), the condition of bounce/turnaround can be imposed by either the condition of making the scale factor changing sign with other quantities remaining same, or, going to negative values with other quantities remaining same.

  • Figs. 2(c) and 48 show the necessary condition of acceleration () at the time of bounce. Here we have plotted the r.h.s of Eq. (26) and Eq. (26). These plots have also been obtained for the same parameter values as the earlier graphs.

  • Thus Fig. 2 shows graphically that the phenomenon of bounce is possible for dgp model having .

4.3 Condition for turnaround

To establish the condition for turnaround we can again rewrite Eq. (26) as Gumjudpai:2003vv ():

(48)

In late time universe, , hence keeping terms upto the first order we get:

(49)

This is also a valid assumption because at the time of re-collapse, the density of the Universe is low or reaches at its minimum but not negligible. Hence we need to keep terms upto the first order in the binomial series expansion. Now by setting , we get:

(50)

where and are the density and scale factor at turnaround respectively.

Similar to the Cosmological bounce case, here we also can equate the change in energy or mass content of the Universe to the work done after each expansion-contraction cycle and finally we get:

(51)

Therefore the change in the scale factor after each successive cycle is given by the following expression:

(52)

Therefore, just like in the case of bounce, where the change in the amplitude is dependent on the parameters of the cosmological model, in such a physical prescrption the change in the amplitude of the scale factor at turnaround also depends not only on the work done, but also on the cross over length scale of the DGP brane world model.

(53)

Let us now briefly mention the characteristic feature of the results for turnaround for DGP brane world model in the following:

  1. For we clearly observe that for an open universe, an increase in amplitude of the scale factor after each successive cycle is possible if the hysteresis loop integral

    (54)
  2. For we observe that the increase in magnitude of the scale factor depends not only on work done, but also on the cross over length scale. We get a positive change in the scale factor with each successive cycle if the hysteresis loop integral

    (55)

    Hence, this is a case where increase in expansion maximum is possible only if the work done is positive.

  3. For if the denominator

    (56)

    or equivalently

    (57)

    we get positive if the hysteresis loop integral

    (58)

    and vice versa. Thus depending on the relative magnitude of the scale factor and cross over length scale at turnaround, we can get an increase in expansion for both positive and negative signature of the work done.

4.4 Condition for deceleration

To establish the condition for deceleration we first take the time derivative of Eq.  (3(a)), and using the energy conservation or equivalently the continuity equation,

(59)

we can write:

(60)

From the above equation we get the condition for deceleration as:

(61)

Therefore, turnaround can be obtained without violating the energy condition. And just like for acceleration, here we see that the condition for deceleration depends on , which was expected because in the late universe, the effect of will become more important.

In place of Eq.  (46), we get the conditions for deceleration at turnaround as:

(62)

Finally substituting the expressions for the density and pressure from Eq. (7) into Eq. (61), we get the following condition for expansion of the Universe in terms of the scalar field as:

(63)

This clearly implies that the lesser the contribution from the potential energy as compared to the standard cosmological inflationary case or equivalently for the acceleration case as mentioned earlier, easier will be to achieve contraction phase of the Universe. We also clearly observe that, even if the potential satisfies the deceleration condition, just like as for standard cosmological scenario, it is not necessary that it will also cause deceleration in DGP brane world model at turnaround as of now the deceleration is also the cross over length scale dependent in the present context.

(a) An illustration of the turnaround condition for a universe with an equation of state .
(b) An illustration of the turnaround condition for a universe with an equation of state .
(c) An illustration of the deceleration condition at turnaround for a universe with an equation of state .
(d) An illustration of the deceleration condition at turnaround for a universe with an equation of state .
Figure 3: Graphical representation of the phenomena of turnaround and deceleration for DGP model.

In Fig. 3, we have shown the phenomena of turnaround and deceleration in dgp model. We can draw the following conclusions from the above figures:

  • In Figs. 3(a) and 3(b), we have plotted the r.h.s of Eqns. (48) and (26) respectively, using the relation . But for this case we have used , since we require a stiff equation of state for causing contraction and deceleration.

  • From Figs. 3(a) and 3(b), we get the turnaround at and at respectively.

  • Figs. 3(c) and 3(d) show the necessary condition of deceleration () at the time of turnaround. Here we have plotted the r.h.s of Eqns. (26)and (48), (26) respectively. This plot has also been obtained for the same parameter values as the earlier graph.

  • Thus Fig. 3 shows graphically that the phenomenon of turnaround is possible for the DGP model having .

4.5 Evaluation of the work done for one cycle

Let us now compute the explicit contribution from the work done for aone complete cycle for DGP brane world cosmological setup. To serve this purpose we start with the following closed loop integral :

(64)

where and refer to the two successive cycle i.e. th and th cycle of expansion and contraction phase of the Universe. In the present context the work done corresponding to the expanding phase and the contracting phase can be expressed in terms of the work done between two successive cycle as:

(65)
(66)

where are the maximum magnitude of the scale factor for th and th cycle of the Universe. Similarly, represents the the minimum magnitude of the scale factor for th cycle of the Universe for DGP brane world model.

Now the volume of the Universe can be written as,

(67)

and an infinitesimal change in the volume can be written as,

(68)

which is frequently used for further computation of the work done. And we also know from Eq.  (7), in the presence of scalar field, the pressure for the scalar field can be written as: . Substituting these in Eq.  (64), we get

Further using the solution of scalar field and the scale factor , one can solve the above equation to get the estimation of this integral. Also it is important to mention here that the above integral can also be expressed in terms of scale factor only using Eq (26), Eq (26) and Eq (7). But since the Friedmann equations in case of DGP brane world model are highly complicated, we can use the late time and early time approximations of the Friedmann equations in order to get an physically relevant approximate analytical expression for the integral for the work done. Therefore, the work done can be decomposed into four parts as follows:

(70)

where the first two terms corresponds to late and early times during the period of contraction respectively and the last two terms corresponds to early and late times during the period of expansion respectively. Here corresponds to the scale factor at the time of transition from early to late time or vice-versa, and corresponds to the values of the scale factor at the time of turnaround and bounce respectively.

At early time, from Eq. (31), keeping terms upto first order and neglecting the contribution from terms in the acceleration equation Gumjudpai:2003vv (), we get:

(71)
(72)

Hence using Eq. (71), Eq. (72) and Eq. (7) for the pressure content of the Universe , we get:

(73)

Let this equation be valid upto to a cut-off time scale where the value of the scale factor is . Therefore using Eq. (73) into Eq. (LABEL:hyst1), the second and third integral as appearing in Eq. (70) can be expressed as:

At late time, using Eq. (3(a)) and Eq. (61), we get:

(76)
(77)

Further using Eq. (76), Eq. (77) and Eq. (7) for the pressure content of the Universe , we get:

(78)

This equation is also valid upto to a cut-off time scale where the value of the scale factor is . Therefore using Eq. (78) into Eq. (LABEL:hyst1), the first and last integral as appearing in Eq. (70) can be expressed as:

(79)
(80)

Therefore by substituting Eq. (LABEL:hyst3.1) and Eq. (80) into Eq. (70) the complete expression for the work done in a single expansion-contraction cycle is governed by the following expression:

(81)

In the present context, we clearly visualize from our analysis that, by knowing the solution of the scale factor in the early Universe and as well as in the late Universe, we can get an idea of the nature of the hysteresis loop. Additionally it is important to note that while the evaluation of work done is independent of the parameters of this model at early times, parameter dependence enters through late time evaluation of the integral for work done.

4.6 Semi-analytical analysis for cosmological potentials

In this section, we try to find simple analytical expressions for the work done during one cycle of expansion and contraction from various cosmological models in order to get some idea of the behavior of the cosmological hysteresis loop. Though the analysis is independent of any particular functional form of potential, but in order to study the physical significance as well as the nature of the derived results in the previous section, we need to specify the functional form of the cosmological potential. While doing the analysis for DGP brane world model and for all the other subsequent cosmological models, we will consider three different potentials i.e. Hilltop, Natural and Coleman-Weinberg potential, which will be discussed in the next sections in detail.

Since from Eq. (81), we observe that the cosmological work done can be evaluated if we know the explicit form of the variation of the scale factor with time, our main motivation is to find an explicit expression for the scale factor in terms of time . For this we need to solve Eq. (26), Eq. (7) and Eq. (9) consistently. Therefore, we will basically substitute the expression for the Hubble parameter from the Friedmann equation into Eq. (9) and replace by Eq. (7) with a specified form of the cosmological potential. Then we can solve for the scalar field and substitute it back to the expression as appearing in the Friedmann equation to get an expression for scale factor in terms of . For the sake of simplicity henceforth we will only concentrate for the case of , which is also a valid and natural assumption in the present context, since we know that our present observations predict a nearly flat universe. In order to simplify the analysis further, we will consider the case when the contribution from the kinetic term is lesser than the potential energy i.e.

(82)

during the expansion and similarly in the physical situation where the kinetic term is larger than the potential energy i.e.

(83)

during the contraction phase of the Universe. Hence the approximate forms of Eq. (7) and Eq. (9) during expansion and contraction phase of the Universe will be given by:

(84)
(85)

Next step is to specify the specific form of the potential, in order to get further informations and the constraints from the equations. In the next subsections we have explicitly shown the analysis for three different potentials i.e. Hilltop, Natural and origanted Coleman-Weinberg potential.

4.6.1 Case I: Hilltop potential

In case of hilltop models the potential can be represented by the following functional form Choudhury:2015jaa ():

(86)

where is the tunable energy scale and is the index which characterizes the feature of the potential. In principle can be both positive and negative. Additionally it is important to note that, in the present context, mimics the role of vacuum energy. Since our present job is to substitute the expression for the scale factor into Eq.  (81), we need to find separate expressions for the scale factor for both early and late times during the expansion and contraction phases of the Universe.

A. Expansion

i) Early time:-

When the scale factor is lying within the window, or equivalently when , we can use the approximated version of the Friedmann equation as given by Eq.  (31), with , where the energy density of the scalar field is for now described by the hilltop potentials. But instead of considering only the zeroth order term (as we had done in order to find the condition for bounce), here we will consider upto the first order terms (as was done for finding an expression for cosmological work done). Then substituting the resulting expression for the Hubble parameter into Eq. (84), we get the following integral equation in DGP brane world as:

(87)

The exact solutions of the above integral equation is given in the Appendix, which we see has a very complicated form for Randall Sundrum (RSII) limiting situation. Hence to simplify the integrals, we use the following redefinition of the field variables:

(88)

where now we will solve for . In order to further simplify the expressions, we solve for two limiting cases:

a) :

For this case we can expand the exponentials upto linear order and then using the result the integral on the left hand side of Eq. (87) becomes:

(89)

which is true under the assumption that the quantity

(90)

is small. It is important to note that in the present context is an arbitrary integration constant.

Hence solving the above integral on the left hand side, we get the expression for , hence for as: