**CYCLIC MODEL FAQs**

The following are brief qualitative answers to FAQs (and criticisms) concerning the Cyclic Universe Model. Wherever possible, the answer is given in plain language, although some questions require technical answers and reference to journal articles for a complete answer. Further questions, comments, corrections and criticisms are always welcome. Send to steinh@princeton.edu. We will update the site when as more questions arise and more progress is made.

- How does the cyclic model evade the Tolman entropy problem that plagues the oscillatory models of the 1920's?
- Does the cyclic model conserve energy?
- Is the cyclic model a perfectly efficient engine?
- Is the cyclic model truly cyclic? Does it have an arrow of time?
- Does the cyclic model require more or less fine-tuning than inflationary models?
- What observations can be used to test the cyclic model?
- Has any progress been made on understanding what happens when branes bounce and the cosmic singularity problem?
- Is the cyclic model simply a variant of inflation?
- Does radiation produced at the bounce necessarily prevent the universe from cycling?
- Does the cyclic model violate the equivalence principle at a level inconsistent with current constraints?
- Are inflationary and ekpyrotic models the only possibilities for having nearly scale-invariant, adiabatic, nearly gaussian spectra of perturbations?
- Why have different theorists obtained different answers regarding the passage of perturbations through the bounce?
- What is causal structure of cyclic models?
- Has the cyclic model been cycling forever?
- Is the cyclic model consistent with the holographic principle?
- Can a single particle in the bulk cause the entire universe to collapse into a black hole?
- Bibliography
- 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.
- 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. **
- 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.

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:

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:

**(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:

**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?**

It is possible that the universe has been cycling forever into the past and will cycle forever into the future without violating any known laws of physics. On the other hand, it is also conceivable that the universe did have a beginning and settled into regular cycling behavior. In the latter case, there would only be a finite number of cycles in our past. Both ideas are currently being explored.

The issue is more of a theoretical issue than a practical one because the basic predictions of the cyclic model are insensitive to whether there was a beginning or not. In either case, 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 a hypothetical pre-cyclic phase would be annihilated or thermalized by the present epoch. Hence, there would be no imprint to inform the observer whether the number of previous cycles has been few or infinite. (Some argue that, as a matter of principle, even if there were an infinite number of cycles in the past, there must have been something that preceded the cycling. The argument is based on analogy with inflation, for which it is known that the expansion phase is "geodesically incomplete," a technical way of saying space-time must include something in the past other than inflationary expansion: there must be some beginning or pre-inflation phase or cosmic singularity. However, because of the bounces, it is not clear whether the same arguments apply to cycling.)

**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].