Hello! This is Matt Johnson, and I was recently asked to write a brief guest post for FQXi about some of my recent work with Stephen Feeney, Daniel Mortlock, and Hiranya Peiris on observational tests of eternal inflation. To supplement some popular stories (e.g. Science) and blog posts (cosmic variance, backreaction) that describe this work (as well as the papers themselves), I thought I'd use...

view entire post

arXiv:1012.1995v2

Hello! This is Matt Johnson, and I was recently asked to write a brief guest post for FQXi about some of my recent work with Stephen Feeney, Daniel Mortlock, and Hiranya Peiris on observational tests of eternal inflation. To supplement some popular stories (e.g. Science) and blog posts (cosmic variance, backreaction) that describe this work (as well as the papers themselves), I thought I'd use this space to give some background on why thinking about eternal inflation is an interesting problem, give a realistic picture of what is theoretically and observationally possible to do, and paint a broad picture of our recent work.

I've been fascinated by the idea of eternal inflation (I'll describe what this is fully below, and you can also look at the article "When Universes Collide" for some of the background too) ever since my PhD advisor (and FQXi director) Anthony Aguirre as well as Tom Banks and Michael Dine got me thinking about it. I find it truly amazing that science could provide the tools to predict and search for evidence of such radical phenomena. Ever since, I suppose I've been spending a significant amount of time searching for other universes. Sound crazy? Maybe, but read on and perhaps I can convince you otherwise...

Perhaps one of the most speculative ideas in modern physics is that of the "multiverse." In this scenario, our observable universe is purported to be only a very small pocket embedded in a much larger cosmos. Outside of our pocket, the physical constants we view as fixed, and perhaps even the laws of physics themselves, could be different. In this picture, the pocket containing our observable universe was born from, and is embedded inside, some parent vacuum. What might we learn about these origins, and the possibility that we live in a multiverse?

This question has received a lot of attention in the past few years. Part of the reason is that eternal inflation, which gives rise to the multiverse described above, seems to follow fairly naturally from string theory. (You can read more about this in the article, "Tying Down the Multiverse with String.")

Two things are necessary to obtain eternal inflation from string theory: multiple metastable vacua and an effective four-dimensional positive cosmological constant. The multiple vacua correspond to the many ways that the 6 extra dimensions predicted by string theory can be compactified (and thus hidden from our view). Although they are classically stable, the size and shape of a compactification can be adjusted through quantum mechanical tunneling. In the four-dimensional picture, this corresponds to the formation of an expanding bubble inside of which things like the vacuum energy, masses of particles, strength of gravity, and perhaps other properties, are different. Since an effective four dimensional cosmological constant is necessary to explain the currently observed dark energy, and to source an inflation in the early universe, the second criteria is expected to be accommodated by string theory. When the formation and expansion of bubbles is outpaced by the expansion caused by the cosmological constant, the bubbles do not meet on average, and the original (inflating) spacetime is never completely eaten up: this epoch of inflation is eternal.

Each undisturbed bubble contains a homogenous open universe, which if there is a secondary epoch of inflation, could be consistent with our observable universe. Some of the properties of our parent vacuum could be imprinted in the density fluctuations produced during this secondary epoch of inflation (even allowing us to probe the dimensionality of our parent vacuum, as described in this paper by Graham, Harnik & Rajendran Phys.Rev.D82:063524,2010), providing one observational check on the theory of eternal inflation. However, these effects are typically on very large scales, where there is a large theoretical error (due to cosmic variance), and are therefore difficult to distinguish from the standard picture. Another possibility comes from the fact that although the bubbles do not merge completely, any given bubble will undergo many collisions with other bubbles. Each collision perturbs the bubble interior, giving rise to inhomogeneities. If any of these collisions affected the observable portion of our bubble interior, the resulting inhomogeneities are imprinted on the cosmic microwave background, opening up the exciting possibility of directly observing the dynamics of eternal inflation.

Even this is still quite difficult, since the signal from a bubble collision must make it through the second epoch of inflation I mentioned above. The longer inflation lasts inside our bubble, the weaker the signal. In addition, since we only have causal access to a small portion of our bubble interior, we only see a subset of all the collisions. To expect to have a collision in our past, the (exponentially suppressed!) probability of bubble formation in our parent vacuum must be balanced against the ratio of energy scales in our parent vacuum and the secondary epoch of inflation inside of our bubble. Assessing if these criteria can be met requires a detailed theory (say of a set of string theory compactifications), which at the moment we do not have.

However, in the space of possible theories, there are cases where bubble collisions are potentially observable and expected to be there. We can therefore focus on constraining, and possibly verifying, those theories which give rise to signatures that are observable with current and future technology. Although it is certainly possible (and perhaps very likely) that we live in an unfortunate region of theory space where there are no signatures to be found, the significance of a definitive detection of other bubble universes is hard to understate. This outlook shifts the questions being asked from "what is the best fundamental theory and what does it predict?" to things like: Given the set of possibly observable collisions, what type of signatures would they leave? How would we go about detecting the signatures of a potentially observable bubble collision in the CMB?

The latest installment in trying to answer these questions, which I will describe in the remainder of this blog post, can be found in a pair of papers I wrote with Stephen Feeney, Daniel Mortlock, and Hiranya Peiris (arXiv:1012.1995v2 & arXiv:1012.3667v1. Important previous work on these questions includes papers by Chang, Kleban, and Levi as well as Czech, Kleban, Larjo, Levi, and Sigurdson, and some of the work I've done with Aguirre, Shomer, and Tysanner. Building on this previous work, we outlined a generic set of signatures of bubble collisions in the temperature fluctuations of the CMB, and designed an algorithm that targets these signatures, which we then applied to the WMAP 7-year data release.

First, let's talk about the generic signatures we might expect for a bubble collision. The spherical (and Lorentz boost) symmetry of two colliding bubbles implies that a collision will have circular symmetry on the CMB sky. The fact that a collision affects only a portion of the bubble interior, together with this circular symmetry, implies that the temperature pattern will be confined to a disc. Depending on how the collision propagates through our bubble interior, the temperature and its derivatives need not be constant across the boundary of the disc. Finally, the inhomogeneities produced by the collision are stretched by inflation, implying that the temperature modulation will be very broad inside the disc. The signature is therefore a circular hot or cold spot, with a possible temperature discontinuity along its edge, which we specify using a five parameter phenomenological model.

WMAP

Of course, there are lots of hot and cold spots in the CMB, so how does one identify a bubble collision? Finding and explaining "weird" features in the CMB data is in general a dangerous business. For example, do the fluctuations spelling out Stephen Hawking's initials really need an explanation? To avoid such pitfalls, our methods are as close to a blind analysis of the full sky data as possible. An optimal search algorithm would consist of taking the full sky data, and asking if the addition of an arbitrary number of bubble collisions at arbitrary locations on the sky provides a better fit to the data than the standard picture of purely Gaussian fluctuations. Unfortunately, this is computationally impossible, and so we have designed a method to segment the sky into regions which have bubble collision-like fluctuations, which are then analyzed further.

This first step of segmenting the sky is accomplished using wavelet analysis. Wavelets provide information about the position and angular scale of features in the CMB data. Comparing the strength of a signal in the data to the signals obtained from simulations with and without bubble collisions, we can locate the most promising features at a variety of angular scales. Within these "blobs," we then look for a circular temperature discontinuity. Finally, we determine if the addition of a collision is a better fit to the data than the standard hypothesis of Gaussian fluctuations using Bayesian model selection (the degree to which the bubble collision hypothesis is preferred is quantified through the "evidence ratio").

Using simulations of the CMB data with bubble collisions, we segment the parameter space of possible collision strength into regions in which our algorithm would: definitively detect a collision, would have some sensitivity to the presence of a collision, or would not be able to detect a collision. We control for systematics by applying our pipeline to a full simulation of the WMAP experiment, providing a set of realistic data containing no collision. We found that the presence of a detectable circular temperature discontinuity was the most distinguishing evidence for a bubble collision. Applying our algorithm to the WMAP 7 year data release, we find no evidence for circular temperature discontinuities. However, we identify four features which are consistent with being bubble collisions. These features all have best-fit parameters in the region of parameter space we are sensitive to, and yield evidence ratios consistent with similar simulated bubble collisions. However, in this region of parameter space the ability to make a definitive detection is strongly dependent on the level of experimental noise and the particular realization of the background Gaussian CMB fluctuations. We therefore cannot claim that we have detected bubble collisions, but also cannot rule out the possibility that these features are bubble collisions.

The good news is that our ability to detect bubble collisions will markedly improve with data from the Planck satellite. Better resolution and lower noise will improve every step of our analysis pipeline, and the much better polarization data can be used to find complementary signatures of bubble collisions (derived by Czech, Kleban, Larjo, Levi, and Sigurdson). We will be analyzing this data as soon as possible, which will hopefully remove the ambiguity in our results.

It’s also important to emphasize that bubble collisions during eternal inflation are not the only way to get circular spots in the CMB. Some other hypotheses include

textures, primordial inhomogeneities, and kinks in the path of the field driving inflation. Our analysis pipeline can be applied to test these theories as well, and used to compare how well each hypothesis fits the data. This analysis will also shed light on the possible observability and distinguishability of bubble collisions. With any luck, in the coming year or two, we will have a more complete picture of what might be out there!

view post as summary

The main question I have with this model is the extent to which the inflation is “eternal.” The dynamics of the space manifold

(a’/a)^2 + 8πGΛ/3 – k/a^2 = 0

is driven by the cosmological constant term Λ that has an energy density ~ 10^{110}GeV^4. For k = 0 the solution to this DE is exponential and the spacetime is de Sitter-like. The cosmological constant...

view entire post

The main question I have with this model is the extent to which the inflation is “eternal.” The dynamics of the space manifold

(a’/a)^2 + 8πGΛ/3 – k/a^2 = 0

is driven by the cosmological constant term Λ that has an energy density ~ 10^{110}GeV^4. For k = 0 the solution to this DE is exponential and the spacetime is de Sitter-like. The cosmological constant Λ ~ ρ = vacuum energy density, and this drops in a bubble nucleation which initiates reheating, a concept advanced by Coleman back around 1983. The vacuum energy level is determined by the scalar inflaton field, which in local regions exhibits a quadratic potential that corresponds to a bubble of nucleation.

The vacuum energy drops enormously in a region so the vacuum is at a much lower level. So with eternal inflation these bubbles will expand at a much slower rate than the embedding spacetime which is exponentially expanding like a mad hatter.

However, there does seem to be a bit of a problem with the eternal aspect of this. The Lagrangian for a scalar field is L = (1/2)∂^aφ∂_aφ – V(φ) and in QFT we work with the Lagrangian density £ = L/vol so that S = ∫d^3xdt£(φ, ∂φ). We run this into the Euler-Lagrange equation ∂_a(∂£/∂(∂_aφ)) - ∂£/∂φ = 0, and keep in mind vol ~ x^3. This gives a dynamical equation

∂^2φ – (3/vol^{4/3})∂_aφ – ∂V(φ)/∂φ = 0.

If we assume the inflaton field is more or less constant on the space for a given time on the Hubble frame this DE may be simplified to

φ’’ – (3/vol^{4/3})φ’ – ∂V(φ)/∂φ = 0

That middle term is interesting for it is a sort of friction. It indicates the inflaton field, the thing which drives the inflationary expansion, is running down or becoming diffused in the space. The potential function here is complicated and not entirely known, but it is approximately constant --- similar to the accelerated expansion of the universe we observe. What then happens, which is not entirely understood, is that the field experiences a phase transition, the potential becomes V(φ) ~ φ^2 with a minimum about 110 orders of magnitude smaller than it was in the unbroken phase. The phase transition has a latent heat of fusion that is released and this is the reheating.

The phase transition is a bubble nucleation from a high energy vacuum to a low energy one, which occurs throughout the spatial manifold of this universe. Our observable universe out to about 10^6 billion light years is one of these bubbles. The space R^3 becomes fractured into these bubble nucleation zones which exponentially expands over the long run according to the residual vacuum energy which is left. It is a sort of soap suds foam that extends throughout the space, and where each of these bubble volumes is expanding exponentially. The universe we can observe may then be about 1 part in 10^{20} of one of those “soap bubbles.”

What is unknown is the nature of this phase transition and the sort of “Higgsing” which goes on with it. There are lots of models out there about this sort of thing. However, it seems likely the embedding spacetime will “poop out” pretty quickly, which does give an enhanced probability for interactions between these bubble nucleation zones, such as the one we observe. In fact it seems then do not even need to “bump” into each other, for they should interact by Casimir effects.

Cheers LC

view post as summary

show all replies (1 not shown)

hide replies

show all replies (8 not shown)

hide replies

show all replies (7 not shown)

hide replies

show all replies (3 not shown)

hide replies

show all replies (38 not shown)

hide replies

Here is the DEFINITIVE and FUNDAMENTAL mathematical AND physical proof of fundamentally unified physics:

F=ma fundamentally and ultimately represents the totality of space equivalently with force/distance -- as inertia is ultimately equivalent with gravity/acceleration (so 1/2 x 1/2 equals 1/4). This concurs with the fourth dimension of space (also one fourth) that unifies Maxwell's and...

view entire post

Here is the DEFINITIVE and FUNDAMENTAL mathematical AND physical proof of fundamentally unified physics:

F=ma fundamentally and ultimately represents the totality of space equivalently with force/distance -- as inertia is ultimately equivalent with gravity/acceleration (so 1/2 x 1/2 equals 1/4). This concurs with the fourth dimension of space (also one fourth) that unifies Maxwell's and Einsteins's equations, as force/energy is equated with distance in/of space ultimately.

Indeed, force/energy IS ultimately reflected consistent with/distance in/of space as follows below -- Half of one half (m x a) = one fourth of the totality of space as it is reflected by fundamental inertial/gravitational/electromagnetic equivalency:

Space manifests as inertial/gravitational/electromagnetic energy in/as dream experience. Full gravity involves full mobility in relation to, and in conjunction with, distance in/of space. Full gravity involves full distance in/of space. The experience of space in/as dream experience is that of the middle distance in/of space in keeping with half gravity and half inertia. Gravity, inertia, and electromagnetism are key to distance in/of space. Note that vision begins invisibly inside the eye/body. Gravity, invisible and visible, is key to distance in/of space. Note the experience of colors in dreams.

Half distance = one fourth distance OVERALL, as follows:

Since one fourth of the TOTALITY of space is [effectively] VISIBLE (one half of one half) in/as dream experience, here we have a union of force/energy as it is fundamentally and equivalently expressed (and manifested) as distance in/of space. A beautiful match of theoretical and actual/observed, as 1/2 x 1/2 equals one fourth = 4th dimension AND also equals F=ma as it is fundamentally expressed as force/energy consistent with distance in/of space.

F=ma and the fourth dimension of space ultimately BOTH reflect "half of a half of space" as a fundamental unification of force/energy (with inertia and gravity both equivalent at half force/strength) as this is equivalently/consistently expressed with/as distance in/of space.

Half inertia and half gravity is necessarily how they are equivalent. One half multiplied times one half (m x a) equals one fourth. Here we unify equivalently with the fourth dimension of space -- space manifesting as inertial/gravitational/elecromagnetic energy.

While waking, the eye [typically] sees, basically and importantly, half of space (in/as the visual field of experience). So, in dreams, half distance then equals one fourth distance in/of space OVERALL.

In my prior post, THE PROOF of how F=ma (1/2 x 1/2) fundamentally and necessarily equals, represents, and concurs with the [fourth] dimensional unification of gravity and electromagnetism BY ITSELF is a huge breakthrough in physics. I have shown force/energy to be representationally and fundamentally equivalent with distance in/of space.

One fourth of the totality of space is thus shown to be [representationally] visible in/as dream experience.

view post as summary

As I said, conservation of energy is interesting in other settings. The case with general relativity is such a case, though not for the reasons you cite.

Conservation laws are established in general relativity if there is a Killing vector K_a, where for some values of the index a there may be zero entries, so that for a momentum vector p^a = (p_t, p) the inner product p^aK_a =constant. ...

view entire post

As I said, conservation of energy is interesting in other settings. The case with general relativity is such a case, though not for the reasons you cite.

Conservation laws are established in general relativity if there is a Killing vector K_a, where for some values of the index a there may be zero entries, so that for a momentum vector p^a = (p_t, p) the inner product p^aK_a =constant. The Killing vector is then an isometry such that a vector along a parallel translation defines a conserved quantity relative to the Killing vector. That conserved quantities are variables conjugate to the components of the Killing vector. How to find Killing vectors is somewhat involved, but as a rule, if a metric coefficient g_{ab} = g_{ab}(t) then there is no Killing vector with a component along that coordinate direction. The general line element for a cosmology involves a scale factor a = a(t),

ds^2 = -dt^2 + a^2(t)(dr^2 + r^2d\Omega^2

which is a pretty good clue that this spacetime has not fundamental conservation of energy. There is no timelike directed part of a Killing vector, therefore conservation of the energy conjugate variable can’t be established fundamentally.

So is energy absolutely not conserved in our universe? The answer to this depends upon upon some other conditions; for it does turn out that our universe may have a unique condition which recovers energy conservation.. The FLRW equation for the scale parameter a = a(t) is

(a’/a)^2 = 8π\pi Gρ/3 – k/a^2

where (a’/a)^2 = H, the Hubble parameter, and flatness means k = 0, spherical geometry is k = 1 and hyperbolic geometry is k = -1. There is an equation of state for the mass-energy and pressure in the spacetime

d(ρa^3)/dt}+ pda^3/dt = 0

I will consider an approximate de Sitter spacetime, which has ρ = constant, and is identified in ways not entirely understood with the quantum field vacuum. The FLRW equation for k = 0 is then

da/dt = sqrt{8π\pi Gρ/3}a

which has the solution a = \sqrt{3/8ρ\pi G\rho}~ exp(\sqrt{8πGρ/3}t). Using the Einstein field equation we then have that the stress energy is T^{00} = 8π\pi Gρ = Λ, which is the cosmological constant. Returning the first equation, the FLRW equation, we then see that H^2 = Λ/3. The dynamical equation for the dS spacetime with ρ = const gives

ρd(a^3)/dt + pda^3/dt = 0

or p = -ρ. This is the equation of state for p = wρ, and w = -1. This corresponds to a case where the total energy is zero and the first law of thermodynamics is dF = dE – pdV = 0 means the energy that is increased in a unit volume of the universe under expansion is compensated for by a negative pressure which removes work from the system. Further pdV = d(NkT), and for a constant thermal energy for the vacuum and Nk = S the entropy of the universe.

For this particular special case we do have an equation of state which gives a conservation of energy. Another way of seeing this is this spacetime has a time dependent conformal factor a(t) , and this metric is conformally equivalent to a flat spacetime, where one can define an ADM mass that is conserved.

Of course the question might be raised whether this pertains to our physical universe. The inflationary period had a huge exponential acceleration, or equivalently a scale factor which grew extremely rapidly. This period should then have had conservation of energy. As for the time period before then, who knows? After the reheating period the universe became radiation dominated and energy conservation is not immediately apparent. Energy conservation may only then be established in our universe in the very distant future as it approaches an empty de Sitter vacuum state.

Cheers LC

view post as summary

Very recently there have been unexpected advances in understanding dark energy. In fact if the claim of the Egyptian Scientist M. S. El Naschie is correct, then there is no more a mystery regarding dark energy. El Naschie’s solution is disarmingly simple and was presented at two conferences which were almost entirely devoted to his work. The first was held in Bibliotheca Alexandrina early...

view entire post

Very recently there have been unexpected advances in understanding dark energy. In fact if the claim of the Egyptian Scientist M. S. El Naschie is correct, then there is no more a mystery regarding dark energy. El Naschie’s solution is disarmingly simple and was presented at two conferences which were almost entirely devoted to his work. The first was held in Bibliotheca Alexandrina early October 2012 and the second was in Shanghai a week or so ago. On both occasions El Naschie presented a revision of Einstein’s theory leading to an equation very similar to that of Einstein’s namely Energy equals mass x the square of the speed of the light. However unlike Einstein’s equation, the result is divided by 22. His explanation of 22 is as follows: As in the old string theory of strong interaction, space time of relativity should have been considered 26 dimensional. Taking 4 only is what Einstein did and that is how he got his famous result. Nevertheless Einstein ignored 22 dimensions. This is a scaling factor following Nottale’s theory as argued by El Naschie. Even in simpler terms, he reasons that Einstein knew only one elementary messenger particle namely the photon. He knew nothing about the other 11 messenger particles of the standard model which were not known in 1905. Adding 11 super partners it turned out that Einstein did not know about an additional 22 elementary particles. These are the particles needed to explain the missing dark energy. In this way El Naschie was able to show that 95.5% of the energy of the Universe is missing. Alternatively this energy was never there to start with because space time is a fractal and although it looks puffed up it boils down to very little similar to cotton candy. In addition the compactified 22 dimensions are the cause for the negative pressure which increases the acceleration of the Universe’s expansion. He claims to have tested his theory using 25 different methods including Witten’s M-Theory and reached the same result. Even more importantly this result agrees completely with observation. In other words mathematics and physics have been substantiated by measurement which led last year to the award of the Nobel Prize to the 3 team who obtained this incredible measurement and data. Click on this link to get more info re the above (under news) http://www.msel-naschie.com/ and also http://mohamed-elnaschie.blogspot.com/.

view post as summary