Technical Issues:
How does the cyclic
model evade the Tolman entropy problem that plagues the oscillatory models
of the 1920's?
In the oscillatory models of the 1920's, the entropy created during one expansion phase draws together during the periods of contraction and adds to the entropy density at the beginning of the next cycle and to the total entropy an observer measures within the horizon.
In the new cyclic model, the entropy created during one cycle is diluted during the period of accelerated expansion, but is not draw together (i.e., remains dilute) during the contraction period. The total entropy of the universe as a whole increases steadily from bounce to bounce, as demanded by the second law of thermodynamics. However, the entropy from the previous cycle is spread to regions beyond the horizon during the period of dark energy domination. So, as far as a local observer is concerned, the entropy density and the total entropy within the horizon is driven to zero each cycle and the universe appears to begin afresh..
The brane picture is the best way to explain how this occurs. The entropy
consists of particles and radiation that lie on the brane. During the period
of matter-, radiation- and dark energy-domination, the branes stretch and
the entropy density is steadily reduced. After a trillion years of
dark energy domination,
so much expansion has occurred that
the entropy density is nearly zero. During the contraction phase, the branes
cease stretching, but they do not contract significantly, either. Instead, the
extra dimension contracts to zero. Hence, the entropy density on the branes
remains dilute all the way up to the bounce. At the bounce,
new matter-radiation is created by the collision whose entropy density
exponentially overwhelms any tiny remnant of
older entropy density, making the latter irrelevant
in the next period of cosmic evolution.
Does the cyclic model conserve energy?
Yes. The question is motivated by the fact that some brane kinetic energy is converted into matter-radiation at the bounce, and, yet, the scenario requires that there be enough kinetic energy for the branes to get back to their original maximal separation. This would be a perpetual motion machine of the first kind (i.e., creation of mechanical energy from nothing) if there were only the interbrane potential energy plus kinetic energy driving the branes back and forth. However, there is also gravitational potential energy.
The natural dynamics causes some gravitational potential energy to be converted to brane kinetic energy, adding to the contribution from the interbrane potential. This added kinetic energy makes it possible to produce matter-radiation at the collision and still have enough kinetic energy for the branes to return to their original separation. The important point is that gravity can be an unending source of energy. As we know from Newtonian gravity, gravitational potential energy is negative and it magnitude is unbounded. So, it is possible to increase the positive brane kinetic energy at the expense of decreasing the gravitational potential energy by the same magnitude. Solving the equations that describe the motion of branes in the presence of gravity shows that this occurs naturally during periods of cosmic contraction when all forms of kinetic energy - the kinetic energy of the branes in this case - are blue shifted. (The converse occurs during periods of expansion: the kinetic energy of light or particles is red shifted, converted into gravitational potential energy.)
Note that a similar conversion of gravitational potential energy to matter-radiation occurs in inflationary cosmology. Gravitational potential energy is first converted to vacuum energy (the universe expands exponentially in size but the density remains the same) and then, at reheating, to matter-radiation. In the case of inflation, the conversion occurs all in one brief period of inflation. In the cyclic model, it occurs in repeated steps cycle by cycle during each phase of contraction.
Is the cyclic model a perfectly efficient engine?
No. This question is motivated by the picture in which infinite branes bounce at regular intervals forever, a seemingly dissipationless scenario.
However, the picture is naive. It describes large regions of space for long periods of times, but it does not capture the full global picture. In the cyclic model, there are various sources of dissipation, which lead to flaws in the regular cycles in localized regions. For example, black holes formed during a cycle of structure formation are too massive to evaporate before the next brane collision. A black hole distorts the space-time in its vicinity, effectively traversing the two branes and gluing them together. Close to the black hole, the gravitational field is strong enough to prevent the regular periodic collision of the branes in that region. The build-up in the number of black holes from cycle to cycle is a form of dissipation.
What mitigates this effect is the expansion of the universe during the dark energy dominated phase and the contraction of the universe after the dark energy dominated phase ends. The expansion spreads the black holes apart and the contraction shrink the observable universe to a diameter much smaller than the spacing between black holes. Hence, there are more black holes overall after a cycle, but their mean separation is larger than the horizon size and so they have no effect on a typical observer.)
Over time, then, in a comoving picture of the two branes (a map in which you constantly rescale coordinates so that there is no apparent expansion), regions far from black holes are regularly colliding; but, since more and more black holes are produced as the cycles continue, less and less of the comoving volume is cycling. This is steady, long-term dissipation.
A corollary is that, standing at any one location, cycling ends after a finite period due to the tiny but non- zero chance that, after many cycles, a black hole in the vicinity will end the cycling in the particular region. The probability can be estimated based on the fractional volume occupied by black holes today. Assuming 10 billion galaxies with million solar mass black holes at the core we find the fractional volume to be 10^(-30). In other words, cycling ends in any given region after 10^30 cycles, although an exponentially large amount of new volume is created for each black hole, so cycling continues overall.
Other kinds of rarer events, such as quantum fluctuations, are also dissipative processes that can also throw a region off-kilter from the average colliding of branes, but which also have an insignificant effect on the scenario overall.
Is the cyclic model truly cyclic? Does it have an arrow of time?
The cyclic model is difficult to classify because it has aspects of oscillatory, big bang, steady state, and de Sitter models, and yet it is different from all of these at the same time.
A local observer (such as ourselves) confined to measurements within the Hubble horizon certainly views the universe as cyclic. The "intrinsic" or locally measurable quantities such as the temperature, density, and other physical conditions are the same today as they were a cycle ago.
On the other hand, the brane stretches, the volume of space increases, and the gravitational potential energy decreases from cycle to cycle without returning to their original values. These physical characteristics change in the same direction as time evolves, and, hence, serve as an arrow of time (a feature that distinguishes forward from backward in time). Evolution and an arrow of time sounds similar to a big bang model.
Yet, one can equally argue that, averaging over many cycles, the cyclic picture is more like a modified steady state model. In the steady state model, all physical characteristics remain the same over time. For the cyclic model, the temperature and energy density, say, do not remain constant, but their average value over one cycle is the same from cycle to cycle.
Or perhaps the cyclic model is a de Sitter universe since the de Sitter universe has constant energy density. Normally, a de Sitter universe contains only vacuum energy and no matter energy. An exception is the steady state model where it postulated that there is continuous matter creation that keeps the matter density constant. If one wishes, then, the cyclic model can be viewed as a periodic variation of the de Sitter (or steady state) universe in which matter creation occurs at regular intervals (the big bangs) and the matter density changes within a cycle but keeps the same average value.
So, it seems that the cyclic model can be viewed as oscillatory, evolving, steady state or de Sitter depending on what variables are tracked and the period of time over which one averages.
Does the cyclic model require more or less fine-tuning than inflationary models?
The result of careful analysis is that, surprisingly, there is virtually no difference in the fine-tuning requirements. Specifically, both inflationary and cyclic models have the following requirements:
- a minimum of one degree of freedom is necessarily, typically a scalar field rolling down a potential;
-
in order to have a nearly scale invariant spectrum (
,
say), two dimensionless ratios must be tuned:
and 
,
where each is expressed in Planck units and the ratios are evaluated for
values of the scalar field when quantum fluctuation modes leave the horizon.
The deviations of these dimensionless parameters (either 
or 
)
from unity are an unbiased, model-independent measure of the degree of
tuning. In inflationary models, flat potentials with 
and 
of order 
are required; for cyclic models, steep potentials with 
and 
of order
are required. -
in order to obtain the right density perturbation amplitude, the ratio
of the height of the potential (or some similar characteristic scale during
the phase of perturbation generation) to the Planck scale must be fixed
to a value of
(or smaller) in each model; - additional tuning is required to fix the current dark energy density.
This net result is the number of degrees of freedom, the number of tunings, and the quantitative degree of tuning is remarkably similar in the two models; see Khoury et al. [6] and Gratton et al. [7] for more details.
What observations can be used to test the cyclic model?
The cyclic model predicts an nearly scale-invariant, adiabatic, nearly gaussian spectrum of density perturbations and a very blue spectrum of gravitional waves whose amplitude on cosmic scales is exponentially smaller than the density perturbation amplitude. The deviation from gaussianity, though small, is typically ten times greater (or more) than for inflation. Compared to the inflationary model, theise cyclic predictions are difficult ot evade. Hence, the cyclic picture can be ruled out if any of the following are observed:
- a nearly scale-invariant spectrum of gravitational waves on cosmic scales;
- primordial density perturbations whose non-gaussian component is less than 0.01 percent of the gaussian component (inflation predicts that the non-gaussian component should be less than 0.001 percent);
- primordial isocurvature perturbations. **
)
smaller during the slow contraction phase of the cyclic model compared
to the rapid expansion phase of inflation. The gravitational wave spectrum
is blue in cyclic models rather than red because the slowly contracting
cosmic background is nearly Minkowski. The scalar spectrum is purely
adiabatic
because only one component has scale invariant curvature
fluctuations.
(In inflation,
all light fields obtain scale-invariant fluctuations. Consequently, it
is possible to have several independent contributions to the large-scale
inhomogeneity and, hence, significant isocurvature perturbations.)
The spectrum has a significantly larger non-gaussian contribution even though both can be described in terms of a scalar field rolling down a potential because, in the cyclic model, the potential is very steep during the phase when the perturbations are generated, whereas the potential is very flat during inflation. A steep potential implies strong non-linear self-interactions that sizeable non-gaussian fluctuations; a flat potential implies weak non-linearity and comparatively tiny non-gaussian fluctuations.
Experiments planned for the coming decade should be able to detect
gravitational wave fluctuations in the amplitude range
expected from
inflation and also the degree of non-gaussianity predicted by the cyclic
model.
**(We note that there are conceivable effects that can occur after the bounce which can add isocurvature or non-gaussian features on top of the primordial spectrum generated before the bounce. A similar statement holds for inflationary models after inflation is complete. The simplest cyclic and inflationary models do not have these additional effects, but they cannot be ruled out entirely. Hence, to be cautious, the term "primordial" is used to above to refer to the properties of the spectrum stemming directly from the cyclic (or inflationary) phase, assuming no such additional effects.)
Has any progress been made in understanding what happens when branes bounce and the cosmic singularity problem?
Yes! Although there remain important unresolved issues (see, for example, [10] and [11]), recent work by Tolley et al. [5] provides a detailed proposal for how branes collide and the universe passes through a bounce from a big crunch to a big bang. Recent work by string theorists, including Craps and Ovrut [12], Berkooz et al. [13], Berkooz et al. [14], entails closely related ideas. So, there is good reason to hope for significant progress on the singularity problem in the near future.
Is the cyclic model simply a variant of inflation?
No! The cyclic model includes a period of dark energy dominated, accelerated expansion, but we have recently come to understand that this does not play a key role in making the universe homogeneous and isotropic after the bounce. The phase that makes the universe homogeneous and isotropic is the contraction epoch after dark energy domination is completed, as discussed here. During the contraction phase, the potential energy density associated with the interbrane force is negative, creating an equation of state with large positive pressure. A contracting phase with positive pressure exceeding the energy density becomes increasingly isotropic and flat as the universe contracts, as explained in the reference above.
This new understanding has led to a revision in our understanding of the role of dark energy in the cyclic picture. We have often described the dark energy dominated phase as lasting trillions of years and transforming the universe into nearly perfect vacuum. In fact, we now appreciate that the dark energy phase could last a must shorter period, perhaps only 10 billion years, and the scenario would not be adversely affected.
Regions where matter is densely packed today might prevent cycling from occuring near those locations, but cycling would continue, for example, in the current voids. During the contraction phase, the horizon shrinks to an exponentially small size, and the new observable universe is born from just a tiny subvolume of the observable universe of the previous cycle. So, a new region like our observable universe could have formed from some volume in the interior of a void in the previous cycle.
One might wonder if we need a dark energy dominated phase at all. There are two roles that dark energy still plays in the scenario. First, dark energy plays a role in stabilizing the cycling behavior. If the branes are given a "kick" and bounce apart too far compared to the regular cycling solution, the interbrane potential energy is greater and the dark energy period lasts longer. Dark energy dissipates the kinetic energy of particles and branes, so the energy associated with the extra "kick" is red shifted away as the branes contract. Hence, by the next bounce, the branes return to regular cycling.
The second role of dark energy is indirectly insuring that the strength of the gravitational force (and other forces) remain constant today. As branes move, couplings constants change their values. During dark energy domination, the rapid expansion helps by red shifting away the kinetic energy of the branes so that the motion is too slow to cause an observable effect on coupings.
Other differences between acceleration in the cyclic model and the (big bang-)inflationary scenario include:
- The accelerated expansion occurs 10 billion years before the next bang (instead of 10^(-35) seconds after the last bang).
- The acceleration begins when the universe is cold (3K) and dilute, rather than hot and dense.
- The acceleration phase does not determine the spectrum of density perturbations. The observed perturbations of the CMB are generated during the contraction phase, and have nothing to do with the details of the accelerated expansion phase. The amplitude and tilt only depend on the conditions during the contraction phase and the bounce.
- The acceleration is 10^50 times slower than in usual inflation
- The energy density during acceleration is 10^100 smaller
- The acceleration is followed by contraction rather than further expansion.
- The acceleration repeats periodically
- The acceleration is due to the same energy source that causes the currently observed acceleration.
Does radiation produced at the bounce necessarily prevent the universe from cycling?
No. At the bounce, radiation is produced but its energy is
negligible compared to the brane kinetic energy. The branes
fly apart and approach their original maximally separated positions.
During this period, the brane kinetic energy density decreases
more rapidly than the radiation density; so eventually
the radiation density catches up and dominates. But,
this can be
arranged to occur near the time that the branes come to a halt.
All this has considered in the original proposal
and taken into account in the
constraints on the cyclic
model; see discussion in [2].
When this and all other constraints on the model are taken into account
(see [6]), one finds that the temperature
at radiation domination must be less than 
GeV, a physically plausible value.
Does the cyclic model necessarily violate the equivalence principle at a level inconsistent with current constraints?
No. The concern arises because the cyclic model assumes
that the universe can be described in terms
of two orbifolds (branes) in 5d whose separation
varies with time and that
matter lies on one of these orbifolds.
This picture is well-approximated by
an effective 4d field theory in which matter couples to a
massless scalar field known as the
radion field 
, where the radion is related to the separation between
branes. The coupling between matter and a massless field generally produces
interactions that violate the equivalence principle of general
relativity.
All this has considered in the original proposal
and taken into account in the
constraints on the cyclic
model; see discussion in [2].
Namely, If the coupling between the radion and matter is given by
,
then the violation of the equivalence principle is
small enough that all current constraints are satisfied if 
.
Improved tests of the equivalence principle may reduce this bound by three
or more orders of magnitude in the next few years (or detect a violation).
It also may be possible to have 
and still evade test of general relativity, as recently suggested by Khoury
and Weltman [9],
or by giving the radion a finite mass.
In sum, there are a number of
ways to satisfy the equivalence principle constraint in the cyclic model.
Are inflationary and ekpyrotic/cyclic models the only possibilities for having nearly scale-invariant, adiabatic, nearly gaussian spectra of perturbations?
Yes, as demonstrated
by Gratton et al. [7], which
is important because current measurements of CMB fluctuations suggest
primordial perturbations with these properties. Gratton et al.
show that
the only possible equations of state which are attractor solutions to the
cosmic evolution equations (essential for producing a spectrum of perturbations
with an exponentially broad range of wavelengths) and which produce nearly
scale invariant perturbation spectra are 
and 
.
The first corresponds to
an expanding inflationary phase and the second is for a slowly
contracting (ekpyrotic/cyclic) phase.
Why have different theorists obtained different answers regarding the passage of perturbations through the bounce?
The reason is that the propagation of perturbations through the bounce
depends sensitively on conditions near the bounce and different authors
are considering different kinds of bounces. First, there are non-singular
vs. singular bounces. In non-singular bounces, the scale-factor
contracts to a small but finite value, and then increases. Equivalently,
the branes rush toward one another but stop and reverse before colliding.
In either case, this requires stress-energy that violates the null energy
condition (equation of state 
).
The singular bounce is one in which the scale factor reaches zero, or,
equivalently, the branes collide.
The bounce relevant for cyclic models is singular.
Another difference is that some authors imagine a bounce without radiation production (to simplify the analysis). Yet, the cyclic model requires radiation production.
These differences -- the singular bounce and radiation -- are important because they change the conditions as the bounce approaches and thereby modify the matching of perturbations.
In non-singular bounces, the curvature on comoving hypersurfaces is constant and conserved for long wavelength perturbations; in these cases, there are generically no scale-invariant perturbations after the bounce.
But, as shown by Tolley et al. [5], the curvature on comoving hypersurfaces undergoes a jump when branes collide, and, consequently, scale-invariant perturbations produced earlier in the contracting phase transform into a scale-invariant spectrum of curvature perturbations after the bounce. The curvature jump depends on the rapidity of the collision which, in turn, determines the magnitude of the density perturbation amplitude.
A key feature of the Tolley et al. analysis is that the conserved curvature perturbation in the 5d brane picture only carries information about the curvature of the branes themselves, and does not include perturbations of the bulk. At the bounce, both branes and bulk meld and the bulk perturbations are converted, in part, to brane curvature perturbations. This effect is inherently related to the 5d nature of the theory and the presence of branes (orbifold fixed planes) vs. bulk. The resulting amplitude depends on 5d physical parameters (e.g., the rapidity of the brane collision) that have no analogue in the 4d effective theory. Hence, the result cannot be recovered by the inherently 4d analysis employed by most authors.
What is the causal structure of cyclic models?
The cyclic model has the same average positive energy density from cycle to cycle (see above). Consequently, the average expansion rate and the space-time structure of cyclic models is de Sitter (or the same as an infinitely inflating universe). In particular, this means that the scale factor increases by an exponential factor from cycle to cycle. This may seem counterintuitive for a cyclic model. Bear in mind, though, that the temperature and density depend on the Hubble parameter, the ratio of the time derivative of the scale factor to the scale factor itself. By the end of a cycle, both the numerator and denominator change by the same constant factor, so the Hubble parameter returns to its value a cycle earlier. The temperature and density, therefore, do the same.
Has the cyclic model been cycling forever?
In principle, it is possible that the universe has undergone a semi-infinite number of cycles in its past during which the volume increases from cycle to cycle. Even though this would take an infinite time according to ordinary clocks, this cannot be the full story. This cycling regime would not cover all space-time. Something must have preceded the cycles.
Recall that the cyclic model is approximately de Sitter when averaged over many cycles. The cyclic model in which the volume increases from cycle to cycle is analogous to the de Sitter phase in which the volume expands. But, this represents only half of the de Sitter space-time. It is preceded by another half of de Sitter space which is contracting.
Could there be another half of the cyclic model in which the universe is contracting on average from cycle to cycle? That is, the dark energy dominated phase would be like the de Sitter contraction phase instead of the de Sitter expansion phase. The answer appears to be no. A contracting phase with dark energy does not smooth out and dilute the matter and radiation from the previous cycle. It does just the opposite -- it amplifies inhomogeneities and concentrates the matter and radiation. This makes cycling unstable. It is implausible to suppose that either model underwent an infinite contraction phase prior to expansion given the instabilities. So, what did precede the expanding phase?
A similar issues arises in inflationary cosmology. In both cases, this is an open question. The issue is referred to as the problem of geodesic incompleteness referring to the fact that a purely expanding phase does not span the entire space-time and one has to consider what happened before.
The cyclic model is relatively insensitive to what preceded the expanding phase since every observer today has undergone an enormous number of cycles of acceleration and dilution of matter since that earlier era. Particles or radiation emitted in that earlier period would be annihilated or thermalized by the present epoch. Hence, there is no imprint to inform the observer whether the number of previous cycles has been few or infinite.
Is the cyclic model consistent with the holographic principle?
The cyclic model and the holographic principle are compatible. Contrary to some suggestions, the holographic principle does not impose a quantitative constraint on the number of cycles, as described here. In de Sitter space, there is a well-known bound that the total entropy in an observer's past light cone must be less than the area of the Hubble horizon. Since the cyclic model has a causal structure similar to a de Sitter universe (see previous two questions) and since each bounce produces a finite entropy density, one might suppose that there is an entropic bound on the number of cycles. However, as described in the reference above, a key difference between the cyclic model and a true de Sitter phase is the bounce. This represents a period when the pressure and density of the universe naturally violate the energy conditions found in a true de Sitter universe and assumed in deriving entropic bounds. Hence, there is no known holographic bound on the number of cycles.
The holographic picture does suggest that any observer experiences only a finite number of bounces, but we have have already explained above that the cyclic model satisfies this condition.
The holographic principle may, however come into play in explaining the entropy produced in a single bounce. According to the principle, the maximal entropy at a bounce would be obtained by producing one black hole per Hubble volume, where here the relevant Hubble volume is approximately the Planck scale. This volume is microscopic compared to today's horizon. Summing up the entropy from all the black holes, one obtains a net entropy of nearly 10^90, comparable to today's observed entropy. Hence, observations suggest that the bounce nearly saturates the holographic bound and that the bounce may be characterized by the production of many small black holes. (These black holes would rapidly evaporate and would not affect the large scale structure of the universe.) See related discussion below .
Can a single particle in the bulk cause the entire universe to collapse into a black hole?
Horowitz and Polchinski suggested that a single particle during the
contracting phase of the cyclic can cause the entire universe (arbitrarily
far away and at any finite time 
if 5d collapses to 4d) to form a giant black hole [10].
As the orbifold fixed planes (or, more loosely speaking, branes) head toward
collision, the universe approaches a compactified Milne space-time. The
orbifold planes have 
symmetry. The gravitational potential of the particle, they argue, adds
to that of its infinite number of boosted images to produce a divergent
result. The sum diverges (in collisions going from 5d to 4d) for all times
and at arbitrarily large distances. This has been viewed by some
as an indication that a single particle can disrupt the behavior
at the bounce and, hence, as a no-go theorem for ekpyrotic/cyclic models.
This argument is too naive. The divergence at arbitrary times is a gauge artifact. (A footnote to this effect can be found in the published paper by Horowitz and Polchinski.)
There is a singularity at the bounce itself where the extra dimensions momentarily shrinks to zero. This singularity is the usual one assumed in ekpyrotic/cyclic models and its properties are not affected by the addition of a particles. Both Horowitz and Polchinski and Liu, Moore, and Seiberg [11] explore this singular point.
The potential disaster is that the entire universe collapses to a giant black hole at the bounce, as posed by Horowitz and Polchinski. These seems unlikely because of two factors. First, the contraction is not perfectly homogeneous due to perturbations produce before the bounce. The approximation of a perfect Milne universe is invalid at large distances along the branes. Instead, the bounce is uniform and coherent on the scale of a few Planck lengths. Second, the time available to form the bounce is limited; one can track the perturbations up to the point where the brane are a small distance apart and they remain linear, so non-linearities have only a few Planck times to form.
Together, these factors suggest a different possibility for the bounce. Namely, if non-linearities grow to form black holes, as Horowitz and Polchinski suggest, they could only be small black holes with a radius of a few Planck lengths. Yet, small black holes would have an insignificant effect on the cyclic scenario. They would evaporate rapidly into radiation and they would not alter the perturbations at much large wavelengths
A detailed proposal for treating the bounce between branes
is given Tolley et al. [5].