Guest Post: Tom Banks Contra Eternal Inflation

By Sean Carroll | October 24, 2011 8:38 am

Now that we’ve softened you up by explaining a bit about eternal inflation and its puzzles, we’re very happy to host a guest post by Tom Banks in which he really hits on some of these problems hard.

Tom is a professor at Rutgers and UC Santa Cruz, an extremely accomplished researcher in field theory and string theory, and the author of a textbook on quantum field theory. In collaboration with Fischler, Shenker, and Susskind, he proposed the (M)atrix Theory non-perturbative formulation of string theory. Most recently, he (often working with Willy Fischler) has been exploring the connections between holography and cosmology, developing a detailed model of the evolution of the universe that is compatible with the holographic principle. Here is video of a lecture Tom recently gave on holographic cosmology.

This post is at a more technical level than most of our entries here at CV, and we’re going to try to keep the discussion useful for workers in the field. Sincere questions are welcome, but we’ll be deleting any unproductive philosophical gripes or advertisements for anyone’s personal outsider theories.


Why I Don’t Believe in Eternal Inflation

A lot of research in high energy theory has been devoted to the topic of eternal inflation. More and more, over the last few years, I’ve come to regard this as an enormous waste of intellectual resources and I’ve chosen Cosmic Variance as a very public way to make my objections to this theoretical mistake clear. The theory was developed in the 1980s, when it seemed plausible that quantum field theory in curved space-time was a good approximation to a real theory of quantum gravity whenever the energy densities and curvatures of the background geometry were small in Planck units. This idea is simply wrong. The fact that its falsification came through a back door, the rather philosophical discussion of whether black hole evaporation violates the rules of quantum mechanics, has led to a widespread but unfortunate tendency to ignore this FACT.

There are two other psychological reasons for the widespread interest in Eternal Inflation, which I will discuss below. They have led even the inventors of the resolution of the black hole information paradox through the notion of holography, to try to find a sensible holographic theory which incorporates the notion of EI. While this attempt itself is subject to a number of objections, I will not go into them here. Instead, I’ll concentrate on evidence from the seminal Coleman-De Luccia (CDL) theory of tunneling in quantum gravity, which is one of the two biggest clues to what the theory of quantum gravity really is.

There are, in my opinion, two serious conceptual errors behind the theory of EI. The first is the notion that space-time geometry is a fluctuating quantum variable. The second is that de Sitter (dS) space is a system with an ever increasing number of quantum degrees of freedom. The increase is supposed to take place as the global dS time coordinate, or the time coordinate in flat coordinates, goes to future infinity. I’ll end this post with a brief discussion of the formalism of Holographic Space Time (HST), in which both of these ideas are seen to be false, in a very explicit manner. The fact that the HST formalism is able to give an approximate description of particle physics in a curved space-time background is by itself enough to falsify any claim that the semi-classical ideas that lead to EI are inevitable consequences of ANY sensible theory of quantum gravity. For this purpose, it’s not even necessary that HST be right, only that it have a limit in which it reduces to QFT in curved space-time.

There are two flavors of EI. The first comes from applying the theory of vacuum tunneling in QFT to individual horizon volumes at late times in an exponentially expanding space-time. In QFT, tunneling proceeds, like the boiling of water, via the formation of bubbles. It is argued that as long as the bubble radius is much smaller than the horizon scale, then bubbles will form independently in each horizon volume, since these are causally separated regions. This is true in QFT is a fixed background dS space , but the question is which background dS space are we talking about. The whole formalism of QFT in curved space-time is predicated on the notion that fluctuations of space-time geometry are negligible and small, and can be treated as gravitons propagating in a fixed space-time background. This is true even when we “derive” QFT in curved space-time from a formalism in which space-time geometry is assumed to be a fluctuating quantum variable (the Wheeler-DeWitt (WD) equation). In other words, the formalism of QFT in curved space-time is not the proper tool for evaluating the validity of EI, whose fundamental premise is that tunneling events change the asymptotic structure of space-time by changing the cosmological constant (c.c.).

The purpose of the CDL theory is precisely to deal with tunneling events that change the c.c., and the fact that in every case its results are drastically different from those of tunneling in QFT in a fixed space-time background are universally ignored by proponents of EI. Let us begin with the case of a potential on field space, all of whose minima have positive c.c. If there are many minima, there are many instantons (imaginary time classical solutions describing tunneling events from one minimum to another), but they all share several characteristics. They are all compact Euclidean manifolds with negative Euclidean action, I < 0. The compactness means that there are no collective coordinates for translation invariance, and the dilute instanton gas approximation (DGA), which is the mathematical basis for the claim that there are multiple bubbles in disjoint space-like regions of the real space-time does not work. Even when the bubble size is much less than the dS radius, one finds fewer than (RdS/Rbubble)3 bubbles. This estimate is based on sticking instantons into a fixed dS background, but in fact in the CDL instanton, the geometry responds to the instanton and the fixed background approximation breaks down for multiple instantons. Furthermore, there are important interactions between the bubbles, which are neglected in the DGA. I suspect that they correspond to attractive forces (in the DGA sense) between instanton centers, which means that no multi-instanton solutions exist. The standard theory of EI has well known problems with infinities (THE MEASURE PROBLEM) and one might have thought that these observations about CDL instantons would be relevant to regulating the infinite numbers of bubbles encountered in the conventional treatment. In fact, these obvious facts about CDL instantons are mentioned nowhere in that literature. The upshot of this is that there is really no theoretical basis for a picture of bubbles nucleating independently in causally disconnected patches of an exponentially expanding universe.

The problem with negative action is well known, and has a very illuminating solution. As in any tunneling problem, one must subtract the action of the meta-stable configuration. In this case this is dS space, whose imaginary time version is just a sphere. The sphere is a compact Euclidean manifold with negative action, and the difference IinstIF is positive so the the reciprocal of its exponential can be a decay probability. Here we’ve subtracted the action of the higher or false dS minimum, but the action of the lower minimum is even larger so exp(ITIinst) can be interpreted as the probability of transition between the lower and upper state. The thermal nature of dS space leads us to expect such upward fluctuations. The ratio of forward to backward rates is given by exp(ITIF) , and this is rather miraculous, because the absolute value of the Euclidean action for dS space is precisely the Bekenstein- Gibbons-Hawking entropy of the state. This entropy, like that of black holes, was initially rather mysterious, but black hole entropy counting in string theory has given us confidence in the idea that it really does come from a counting of quantum states. In the black hole context we already knew from the thermodynamics of Hawking radiation that black hole entropy was necessary in order to obey the laws of thermodynamics. The CDL instanton formula is the analogous statement for dS space. In the dS case, there is no process which continuously varies the c.c., but the CDL process tells us how to change it in discrete jumps. The formula for the ratio of forward and backward rates is simply the principle of detailed balance for a system at infinite temperature. There is a matrix element for transitions between the states represented by the two dS minima of the potential. Fermi’s golden rule tells us that the ratio of forward to backward transitions is just the ratio of the number of available quantum microstates corresponding to each macrostate. The fact that entropies, rather than free energies, enter the rate equations, tells us that the system is at infinite temperature, and that the total number of quantum states is finite. This bolsters two other arguments that dS space has a finite number of states.

The first is simply that the Covariant Entropy Bound tells us that no observer in dS space can ever probe more than a finite number of states. The Observer Complementarity Principle tells us that different observers in dS space simply use different non-commuting Hamiltonians. Each explores the same finite space of states in a different manner. Taken together these two principles imply that the quantum theory of all possible observers in dS space is finite dimensional. A less abstract argument for the same point comes from examining ”scattering” in dS space. Consider incoming small perturbations of the dS solution of any Lagrangian. In global coordinates this solution looks like a sphere which contracts down from infinite radius to some finite radius and then re-expands to infinite radius. In the remote past or future we can look at small amplitude wave packets. However, as we approach the neck of dS space, the wave packets are pushed together. If we put too much information into the space in the remote past, then the packets will collide and form a black hole whose horizon is larger than the neck. The actual solution is singular and does not resemble dS space in the future. Thus, it is extremely plausible, given the Bekenstein Hawking entropy formula for black holes, that the quantum theory of a space-time , which is dS in both the remote past and remote future, has a finite dimensional Hilbert space.

Readers used to the statement that the inverse dS temperature is the circumference of the sphere may be puzzled by the statement that the temperature is infinite. This is, quite literally, a question of point of view. One can look at the Hamiltonian corresponding to an observer following a trajectory of fixed distance from the origin of the static coordinates, which cover a single horizon volume of dS space. The trajectory at the origin is a geodesic, but the others are all accelerated and see an Unruh temperature different from the conventional dS temperature. As the radius approaches the horizon the temperature goes to infinity. The Hamiltonian P0 describing the geodesic observer approximately decouples the particle degrees of freedom of the system from those on the horizon. The usual dS temperature is a reflection of the degeneracy of P0 eigenstates in the dS vacuum ensemble. This will all be discussed in some detail in a forthcoming paper on the Unruh effect in Holographic Space-time.

This interpretation of the CDL instanton can be generalized to a large class of potentials which have both positive and negative energy densities at their minima. Given such a potential, one can add a negative constant so that the lowest positive c.c. minimum is brought down to zero. We can then ask if there is a positive energy theorem for the resulting Minkowski space solution. There obviously is if all the other minima are still positive. The results of the CDL paper show us that this persists for SOME potentials with negative minima. We say that potentials with a positive energy theorem when the lowest positive c.c. is translated to zero energy are Above the Great Divide. It is easy to see that one can move a potential from below to above the Great Divide by tuning a single parameter. Let f be any smooth non-negative function on the space of fields, which vanishes exponentially rapidly when one moves away from those minima of the potential V, which have negative energy density. Then for sufficiently large ε, V + εf is above the Great Divide. It is easy to show, for potentials above the great divide, that CDL transitions from the lowest dS minimum to the negative region of the potential (which, as shown by CDL always leads to a Big Crunch solution in which the field is driven out of the negative region as one approaches the singularity), are suppressed by an entropy factor. That is to say: just like the upward transitions to high positive c.c., for these potentials the downward jumps to Big Crunches are low entropy transitions of a finite system, analogous to all of the air in a room gathering into one cubic centimeter. In fact, when the covariant entropy bound is applied to the crunching region of space-time, one finds that no observer in that region can observe an entropy comparable to that of the minimal c.c. dS space.

To summarize: for potentials above the great divide, all of the evidence from CDL instantons points to a quantum theory with a finite number of states, most of which resemble the empty dS vacuum with the lowest c.c. All CDL transitions away from this state have an entropic suppression, and the principle of detailed balance assures us that the inverse transition will occur much more rapidly. In the case of upward transitions the CDL formula confirms this, but reverse transitions from the Big Crunch to dS space are not amenable to purely semi-classical analysis. For these potentials, if we accept this evidence, the entire formalism of conventional EI is invalid. Indeed, in that formalism, the difference between a potential above and below the divide is merely quantitative, so potentials above the divide have the same issues with infinite numbers of bubbles as potentials below the divide. The standard discussion of EI uses a treatment appropriate for field theory instantons in a fixed space-time background, to discuss CDL transitions in which the final state has a radically different space-time geometry than the initial state. For potentials above the great divide, EI gets the physics infinitely wrong.

I think that CDL transitions for potentials below the divide are unlikely to have a sensible interpretation in quantum theory. They are like quantum Hamiltonians for systems that do not have energy bounded from below. This leaves the question of whether there is a sensible quantum interpretation of low energy Lagrangians which allow a meta-stable dS space to decay to a region of the potential with vanishing energy density. These transitions are central to the FRW/CFT program of the Stanford group. As I said above, there are numerous issues with the attempts to make a theory of these transitions, but I won’t discuss them here. I should note that for “phenomenological” reasons (see the discussion of Boltzmann Brains below) these authors often insist that dS solutions in their models be below the great divide for transitions to negative c.c. Crunches in addition to being unstable to decay into the vanishing energy density region.

Before going on to the second class of EI models I want to mention the only response I got from EI theorists to my complaint that the multi-instanton solutions they are positing simply don’t exist in the CDL formalism. This response was that the same issue exists for quantum field theory in a fixed dS background, where we ”know” the EI discussion is correct. This is simply not correct. QFT on a large sphere has the symmetries of the sphere. Every point-like instanton configuration will have rotational collective coordinates, which approach the translational collective coordinates of flat Euclidean space in the large radius limit. If the size of the instanton is much smaller than the radius of the sphere then there are approximate solutions which are just superpositions of solutions with a rotated center and large distances between the centers. That is to say, the DGA configurations are approximately correct. By contrast, the CDL instanton for decay of dS space has no exact collective coordinates. When the instanton size is much smaller than the dS radius, then the single instanton solution looks like a small cap cut out of the dS sphere. But when we put in another instanton, the geometry changes in response to its matter content. Once one starts thinking about solutions with many instantons one loses all control over the shape of the solution or whether it exists. Above I gave the bound (RdS/Rinst)3 as an upper bound on the number of instantons, based on the field theory in a fixed dS background. My guess is that the actual number is much smaller than this, and that inter-instanton forces may preclude the existence of any multi-instanton CDL solutions at all. At any rate, without a thorough investigation of this question it seems to me that the standard discussion of EI is simply wild conjecture. The alternative interpretation of CDL transitions above the great divide, in terms of a system with a finite number of states, suggests strongly that EI is a wrong conjecture for those potentials.

The second flavor of EI is called chaotic eternal inflation or the self-reproducing inflationary universe. In these models, the formalism for calculating small fluctuations around a slow roll inflation model is extended to the regime of large fluctuations. In particular, Starobinsky’s proposal that one view the fluctuations in each independent horizon volume as a stochastic force added to the classical slow roll equations leads, for a certain range of values of the parameters in the potential, to a regime in which the stochastic kicks up the potential are more important than the force coming from the slope of the potential. Any given point will eventually roll down and inflation will end, but there are “always” (The Starobinsky equations have a global time for all points. One has to ask how this time is defined in a global picture of the space-time geometry. This leads to one aspect of the dreaded Measure Problem) points which are still inflating. This formalism takes for granted that the global picture of the inflationary space-time, with independent degrees of freedom in an infinite number of horizon volumes in the flat slicing of dS space, is valid. It also ignores the ultimate fate of the post-inflationary regimes. Indeed, in the eternal inflation regime of parameters, it is guaranteed that fluctuations on large enough scales become of order one, and there is probability one that every post-inflationary regime will be crunched inside a black hole. It is argued that those crunches occur on a time scale much longer than the age of our universe, so we can ignore them. However, if one is looking for a mathematical definition of the theory one cannot be so cavalier.

Advocates of this point of view will counter that the formalism has been validated by its application to the calculation of observable fluctuations in the Cosmic Microwave Background. I don’t think this argument is correct. The CMB fluctuations can be equally well accounted for by slow roll inflation models in the self reproducing regime, and by models without self reproduction. The CMB fluctuations are very small and extend over a range of scales that is at most 105 in size (if one includes the fluctuations that form galaxies). The fact that a set of equations works over this limited regime says nothing about the validity of its use in a much more ambitious context. There have, for example, been suggestions that “string theory may never give rise to potentials in the self-reproducing regime”. I no longer think that that is the correct way to think about the problem, but it seems to me to be an indicator that this version of EI is on no firmer footing than that based on tunneling.

The Psychology of EI

I believe that there are two rather different psychological arguments that are driving the renewed interest in EI and the rejection of a more holographic point of view (or the attempt to shoehorn EI into a holographic theory via FRW/CFT). These are the success of the calculation of inflationary fluctuations, and that of Weinberg’s anthropic estimate of the c.c. Andrei Linde and I independently invented the first inflationary models which led to a mechanism for anthropic determination of the c.c. Subsequently, Weinberg estimated an upper bound on the c.c. from the requirement that galaxies form, assuming the size of primordial fluctuations and the dark matter density at the beginning of matter domination were fixed. Many of the proponents of the String Landscape and EI view it as the only framework which will give rise to a distribution of universes, with varying c.c., on which one can do anthropic selection. This is incorrect and the HST formalism I will describe below is an explicit counterexample. The string landscape version of anthropic selection seems to point to the likelihood that all of the parameters in low energy effective field theory are random variables constrained only by anthropic selection. Such a proposal has enormous phenomenological problems, which to my knowledge have only been addressed by recourse to our ignorance about the nature of the (so far entirely hypothetical) String Landscape. I think it is likely that only a very small number of parameters in low energy physics are random numbers fixed by selection effects.

The success of the inflationary predictions for CMB fluctuations points to the reality of independent degrees of freedom in different inflationary horizon volumes. However, as I emphasized above, there is nothing in the data that attests to more than about 25 e-foldings of inflation and a finite number of horizon volumes. It’s been evident since Bousso’s first papers on the covariant entropy bound that inflationary cosmology is not dS space. Indeed, it’s precisely the observability of the CMB fluctuations that shows us the difference. The maximal causal diamond of an observer in an inflationary universe is determined not be the inflationary c.c. but by the value of the c.c., which dominates the asymptotic future. Using the covariant entropy bound to estimate the number of allowed degrees of freedom we find that about 85 e-folds of GUT scale inflation are compatible with the covariant entropy bound and the value of the c.c. suggested by cosmological data. Thus, the success of the CMB calculations gives us no reason to believe the hyperbolic claims of EI theorists.

Holographic Space Time

I want to outline the theory of Holographic Space Time (HST) and how it addresses the problems of dS space and inflation. HST is an infinite collection of quantum systems, each of which describes the entire universe as seen from the perspective of a different time-like trajectory in space-time. A segment of such a trajectory, with finite proper time defines a causal diamond – the region in space-time that can be probed by a detector following that trajectory. The diamond is the intersection of the interior of the backward light cone emanating from the future end of the trajectory and that of the forward light cone of its past endpoint. Alternatively, we can think of the trajectory as being defined by a nested sequence of causal diamonds, each one larger than the former. The Holographic Principle tells us that the Hilbert space corresponding to all observations in the diamond is finite dimensional. When the dimension is large, it defines the largest area 2-surface on the boundary of the diamond (this surface is called the holographic screen of the diamond), via the asymptotic formula D → eA/4, where the area is measured in Planck units (10−66cm2). Using this formula we say that a time-like trajectory is equivalent to a sequence of nested Hilbert spaces, each containing the previous one as a tensor factor. The maximal dimension Hilbert space, for infinite proper time along the trajectory, has dimension Dmax, which might be infinite. The time evolution operator must be time dependent, since causality tells us that for proper time T it must factor into an operator acting only on the Hilbert space of measurements that could have been done during that time, and one which acts on degrees of freedom that could not yet have been measured at that time, because they are at space-like separation. Mathematically, the latter space is the tensor complement of the Hilbert space accessible at time T in the Hilbert space at infinite proper time.

Now consider another time-like trajectory, which doesn’t intersect the first. It consists of a sequence of Hilbert spaces and evolution operators, as above. HST also specifies, at each time, an overlap Hilbert space, which corresponds geometrically to the Hilbert space associated with the maximal area causal diamond in the intersection between the causal diamonds of the individual trajectories. Thus, for any pair of trajectories there is a sequence of overlap Hilbert spaces. The dynamics associated with each individual trajectory, plus a choice of initial state, determines a sequence of density matrices in this sequence of overlap spaces. The basic dynamical constraint of the theory is that the two different sequences of density matrices, be unitarily equivalent to each other. Finally, we put a topology on the space of trajectories, which we can think about as defining the topology of a Cauchy surface in space-time. In the limit of large areas, each HST quantum system defines a Lorentzian metric. The space-time metric is not a fluctuating quantum variable, but is implicit in the dynamics of the individual quantum systems and their relations. We have found several solutions of the consistency conditions, and each corresponds to a simple space-time geometry, satisfying Einstein’s equations with a simple stress tensor.

The fluctuating quantum variables of HST are related to the holographic screen of the diamond they describe. They define a non-commutative approximation to the algebra of functions on the screen, which becomes commutative as the screen gets large. If we insist on Lorentz invariance for large spherical screens (the Lorentz group is the conformal group of a sphere, and we need some kind of conformal invariance to control the large sphere limit), then we can show that the degrees of freedom describe (in the limit) an infinite number of massless supersymmetric particles, plus a set of “horizon degrees of freedom”. The horizon degrees of freedom saturate the covariant entropy bound, but to get a sensible large time (which means large area causal diamond) limit, the particles must decouple from the horizon variables. This leads to a scattering matrix for particle states. The particle interactions are mediated by the much more numerous horizon degrees of freedom.

The HST formalism is also capable of describing situations in which particles do not decouple from the horizon, which occur in the presence of black holes and in the very early universe. Fischler and I are in the midst of working out a model for inflationary dynamics in the HST formalism, and we can already see how to get small (approximately) Gaussian fluctuations from a manifestly finite system with none of the paradoxes of EI. The dS invariance of the fluctuation spectrum, which gives the correct answer for the CMB, can be at best approximate. We do not yet understand how to parametrize the deviations from dS invariance and whether our models can reproduce the “red tilted, approximately scale invariant” spectrum, which is seen in the data.

The HST formalism has much to say about numerous other problems in string theory and particle physics. Here I will only outline it’s version of a “multiverse”. The original solution of the HST constraints has a coarse grained description as a homogeneous isotropic spatially flat cosmology, with equation of state that pressure equals energy density. This is what one would expect heuristically for a universe in which every horizon volume is filled with a maximal size black hole, and these black holes merge as the horizon expands, so that the horizon filling black hole condition is satisfied at every moment. We called this solution the D(ense) B(lack) H(ole) F(luid). We also discussed a solution to Einstein’s equations which was a black hole with de Sitter interior embedded in this homogeneous isotropic cosmology. In the paper referred to above, we have found an exact quantum model, satisfying all the consistency conditions of HST, which corresponds to that solution. There is a one parameter family of models corresponding to the choice of dS c.c. We can also find approximate solutions of the consistency conditions corresponding to two or more such black holes, separated by a large distance. Using the Einstein equations as a guide, we surmise that these multi black hole solutions will evolve in a way that depends on the initial positions and velocities of the black holes in the underlying cosmological space-time. The black holes will be stable, except for collisions, when we expect them to merge to form a black hole with larger area. So we can construct models in which there are many values of the c.c. depending on which black hole interior one resides in. Each mini dS universe will be stable, unless it collides with another. Such a model is ripe for anthropic selection arguments, but the parameters of low energy physics that get anthropically selected may be few. In the paper referred to above, Fischler and I conjecture that only the inflationary and asymptotic values of the c.c. vary among these models, with all other parameters determined in terms of them. The basis of this conjecture is that the limiting theory for small asymptotic c.c. is a super-Poincare invariant theory of gravity, with no moduli, and a discrete R symmetry. There are no known theories of this type coming from string theory, and the conditions that define them in low energy supergravity are non-generic (N+1 equations for N unknowns). The fact that the theory becomes supersymmetric for vanishing c.c. puts a strong lower bound, and probably also an upper bound on the c.c. The scales of SUSY breaking and electroweak physics depend strongly on the c.c., and the QCD scale does also, but in a different way. So having stable atoms and stars requires a special value of the c.c. . The inflationary c.c. determines the size of primordial fluctuations in the universe, and for fixed asymptotic c.c., Weinberg’s galaxy formation bound constrains this to be roughly its observed value.

This version of the multi-verse also resolves the Boltzmann Brain problem, which has been claimed to invalidate any theory of stable dS space. In fact the argument is that Boltzmann’s Brains rule out any theory in which the universe eventually comes to thermodynamic equilibrium. This is an extreme version of anthropic reasoning. Once one has bought into trying to explain a lot of the physical world by anthropic arguments, one tends to be forced to start trying to estimate the number of observers that can exist under various hypotheses and saying that, since we could be any one of those observers, we must be a typical one. In a universe that comes to thermodynamic equilbrium, with finite entropy, the most typical observer is a brain with all of YOUR memories, which spontaneously pops from the vacuum by a thermal fluctuation. This happens an infinite number of times, and with much higher probability than a rerun of the history of the universe as it really happened. The BB quickly dies, and according to this way of thinking, we’re constantly doing the experiment that shows we’re not Boltzmann Brains, and so falsifying a theory which predicts that typical observers are BBs.

I think this argument is silly, and mistakes our theory of the universe for the universe itself. Given the dS temperature corresponding to the observed c.c., the probability of fluctuating even the most modest sized BB is of order exp(−1066). This number is so small that the time we would have to wait to see it happen is so long that it’s essentially the same number measured in Planck times as it is in units of the age of the universe. Thus, this theoretical event is not a part of real physics. We can modify our theory in an infinite number of ways (with a time dependent Hamiltonian) which will make identical predictions for everything we or anyone else will measure over the entire history of the universe (including the events which have not yet happened when our local group of galaxies collapses into a black hole, which then evaporates back to the dS vacuum), but change what happens over the time scales on which BB’s are nucleated, and makes the probability of a BB exactly zero. The HST multiverse has an explicit mechanism for making this happen. In order to make a model which explains the history of the universe up to some point in time, it is only necessary that our particular dS black hole interior last for that period of time. Depending on the initial conditions for black hole positions and velocities defining the HST model, we can make one in which our black hole suffers a collision after that point in time. The result of that collision will be catastrophic, and will result in a new equilibrium state with a substantially smaller value of the c.c.. Given the connection between the c.c. and particle physics, it is likely that atoms are no longer meta-stable excitations of this system and so there are no BB’s after that point. The proponents of EI, fixated on CDL instantons as the only mechanism for making a dS space decay, instead argue that the low energy effective field theory MUST be below the Great Divide, in order to eliminate BBs. There are more things in heaven and earth than are dreamed of in their philosophy.

The Takeway

The brevity of a blog post prevents me from expanding on the manifold virtues of the HST approach to quantum gravity. No matter. You can read about them in hep-th 1109.2435 and references therein. Within the next year, I’ll also put out a long review article on HST. What you should take away from this post is the following message: the very popular idea of Eternal Inflation is built on a foundation of sand. One version of it applies intuition about field theory instantons in a fixed space-time to transitions in which the shape of space-time changes dramatically. The actual theory of such transitions, adumbrated by Coleman and De Lucia in 1981, does not support that intuition. The analysis of CDL instantons for roughly half the potentials one can write on the space of scalar fields instead supports a picture of dS space as a quantum system with a finite dimensional Hilbert space, most of whose states resemble the dS vacuum. All CDL transitions are short lived excursions into a low entropy state of the system. Most other CDL transitions are likely to be fictitious. There is no real theory of quantum gravity in which these transitions occur (for example, almost all instabilities of Anti-de Sitter space are of this type). Transitions to a state with zero c.c. are still not well understood.

The other flavor of eternal inflation is based on extrapolating a theory of small fluctuations, without modification, to a regime where the fluctuations are large. It’s really rather hard to check its consistency since the evident disasters caused by the large fluctuations are always ignored by arguing that they will only occur much later in the history of the universe. This sort of phenomenological attitude prevents one from discussing the mathematical meaning of the formalism.

Both types of EI lead to infinities, and I have argued that there is nothing in either theory or observation which should lead us to believe that these infinities are part of a well defined model. I gave the example of HST as a formalism that can probably explain the data within the context of a manifestly finite model of the entire universe we can observe. HST also provides a new venue for applying anthropic reasoning to the determination of the c.c. and many ways to resolve the putative Boltzmann Brain problem of an asymptotically dS universe.

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About Sean Carroll

Sean Carroll is a Senior Research Associate in the Department of Physics at the California Institute of Technology. His research interests include theoretical aspects of cosmology, field theory, and gravitation. His most recent book is The Particle at the End of the Universe, about the Large Hadron Collider and the search for the Higgs boson. Here are some of his favorite blog posts, home page, and email: carroll [at] .


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