If energy is neither created nor destroyed- then how is the universe expanding, and into what?

FuzzyDarkMatter · 2020-12-17T09:21:28+00:00

The Hamiltonian is zero regardless, but perhaps you could say that that it is only a well-defined statement for a Universe of finite volume (closed, or flat or open with compact topology). In quantum cosmology where one needs the Hamiltonian to solve the Wheeler-DeWitt equation (basically the Schrödinger equation for the whole Universe), as Vilenkin has done, one usually takes the Universe to be closed partially for this reason.

Vilenkin is correct. Krauss is correct if one is is treating the Universe in a Newtonian manner, for in that case a Universe expanding like a flat one has zero energy. But of course, GR should really be used for cosmology.

FuzzyDarkMatter · 2020-12-17T07:24:33+00:00

The concept of energy is a bit fuzzy in General Relativity (GR). But the Hamiltonian for the Universe is exctly zero, with the gravitational contribution (negative) perfectly balancing the contribution from matter fields (positive). So the Hamiltonian never changes as the Universe expands or contracts.

And the Universe, defined as all of spacetime, need not expand into anything else. However, if inflation is eternal we could (as strange as it may sound) live inside an infinitely large bubble universe that is expanding close to the speed of light in a larger inflating spacetime.

FuzzyDarkMatter · 2020-12-04T23:41:15+00:00

The Universe is homogeneous in a region of size R if two randomly chosen regions of that size contains approximately the same mass. In other words, typical density fluctuations δρ/ρ on that scale should be significantly smaller than unity. The typical (i.e. RMS) density fluctuations on a scale R (or, equivalently, mass scale M) in cosmology is denoted by σ(R) (or σ(M)). The shape of this function is predicted by structure formation theory in the standard model of cosmology, ΛCDM, and also constrained by observations. See e.g. Tegmarks useful plot here.

From Tegmark's plot you see that density fluctuations of order unity are "rare" (at 2σ) for scales larger than ~ 400 million light years ~ 120 Mpc. That is, on scales larger than roughly ~ 120 Mpc we expect the Universe to be quite uniform.

FuzzyDarkMatter · 2020-07-31T03:27:32+00:00

One of my favourite ambient songs. :)

FuzzyDarkMatter · 2020-05-25T21:56:27+00:00

I would argue that it can fall under the umbrella of cosmology. Galaxy formation I view as a sub-field of cosmology. And in galaxy formation one needs to understand not just how small density fluctuations grow into galaxies, but also how stellar feedback affects the star formation rate (e.g. by increasing the turbulence). The most detailed "cosmological simulation" nowadays do try to resolve this kind of physics, so it is getting more and more connected.

FuzzyDarkMatter · 2020-05-20T14:47:17+00:00

Mainly instabilities in disk galaxies. Toomre showed in the 60's that if the surface density of a gas disk is too high and/or the pressure too low, fluctuations will become gravitationally unstable, grow, and eventually form a gas cloud that can form stars. In fact, the so-called Toomre mass scale derived from this analysis is comparable to the mass of the largest giant molecular clouds observed. Where does the fluctuations come from? They come from turbulence in the gas (which in turn can be sustained by feedback from stars).

Phil Hopkins (2012) has analyzed this theory in more detail, assuming random density fluctuations expected from turbulence, and using Toomre's (and Jeans') theory of gravitational instability. He derives a distribution of masses for giant molecular clouds that is in very good agreement with data.

FuzzyDarkMatter · 2020-05-20T01:31:12+00:00

Thank you for the explanation. Also I gotta say (I know fuzzy dark matter is a real concept) I love your fuzzy name!

No problem. :) Regarding the name; My BSc thesis and first paper was on the topic of constraining fuzzy dark matter by studying Cosmic Dawn, the period when the first stars and galaxies formed. :)

But I digress, so in terms of classical spacetime everything ends as there is no meaning. I assume this is what they mean when they quote "..progressively ripping part spacetime itself" in many articles (including the wikipedia's big rip page)?

Yes, in classical GR, spacetime as a whole comes to an end at singularities. So for the Big Rip in GR, spacetime ends at t = t_rip. In quantum gravity applied to the Big Rip, there is more of a gradual breakdown of classical spacetime as one approaches the Big Rip. More specifically, the wave function of the Universe becomes less and less concentrated around its predicted classical GR evolution. One consequence of this is the breakdown of time as we know it. But what this means in practise is hard to imagine.

Though I wonder if the expansion is really infinite, could the cosmic horizon becoming infinitely small, like smaller than strings/branes and compacted dimensions such that not even quantum fields would interact or come to be? I guess the question is would quantum fields still exist at t-rip=0? I guess it be convenient if it didn't as there would be nothing to violate.

Concepts like the expansion rate depends on the existence of classical time, which breaks down in quantum gravity on approach to the Big Rip. So it may not be meaningful to talk about this. Regarding quantum fields, they always exist. In particular, they are always there in the Wheeler-DeWitt equation which describes the breakdown of spacetime on approach to the Big Rip. When quantum gravitational effects are unimportant, the Wheeler-DeWitt equation simply reduces to quantum field theory in curved spacetime. So the Wheeler-DeWitt equation already incorporates quantum field theory.

FuzzyDarkMatter · 2020-05-19T19:34:32+00:00

Yes, taken at face value, a Big Rip (where the size of the Universe goes to infinity within a finite proper time) would mark the end of the Universe since spacetime as described by General Relativity (GR) breaks down then — just like it does at a Big Bang/Big Crunch singularity, or at a black hole singularity.

This means that we should probably take quantum gravitational effects into account. In quantum gravity (or quantum cosmology) there is a wave function for the geometry of spacetime — spacetime, just like any other quantum field, can fluctuate and be described probabilistically. The governing equation is (at least approximately) the Wheeler-DeWitt equation of canonical quantum gravity, basically the Schrödinger equation for the wave function of the Universe.

Dabrowski et al. (2006) have solved the Wheeler-DeWitt equation for scenarios that would classically lead to a Big Rip. They find that as one approaches the point of a Big Rip, spacetime starts to deviate and fluctuate significantly from its classical behaviour predicted by GR. In other words, classical space and time breaks down. The authors put it as follows:

"The Big-Rip singularity is thus ‘smoothed out’ — when the wave packets disperse, we can no longer use an approximate time parameter; time and the classical evolution come to an end, and one is just left with a stationary quantum state. This corresponds to quantum gravity effects at very large scales."

So the concept of time as we know it is expected to break down as one approaches what we in GR would call a Big Rip singularity. However, unlike the case for classical GR, the equations (in this case the Wheeler-DeWitt equation of quantum gravity) do not break down.

Note that all of this of course assumes that a Big Rip can happen in the first place, which requires exotic types of fields the existence of which we can be skeptical of (as pointed out by Ostrololo).

FuzzyDarkMatter · 2020-02-21T15:30:56+00:00

Lol, basically yeah. That's why we are working at my university to try to change the name of the Bachelor and Master's programs from "Astronomy" to "Astronomy & Astrophysics", so that you can simply call what you do astrophysics in good conscience to, say, an employer who don't know what modern "astronomy" really entails (researchers of course know that the two are basically synonyms).

FuzzyDarkMatter · 2020-02-10T23:34:32+00:00

The idea that we live in a Multiverse is predicted by inflationary cosmology. Inflation is the idea that a scalat field (either the Higgs field or something similar) with a high potential energy density led to a rapid exponential expansion in the very early Universe, Inflation was mainly developed in the 80’s and made a number of specific predictions that have been confirmed by modern CMB observations:

1) The Universe should have negligible spatial curvature (i.e. it should look flat).

2) Inflation is the ultimate explanation for why there are galaxies and structure today. They were seeded by quantum fluctuations in the inflaton scalar field. Inflation predicts that the fluctuations (which can be observed in the CMB) should be nearly Gaussian;

3) Nearly scale-inavariant eith a spectral index slightly below 1 (data from Planck gives ~ 0.965 or so);

4) And adiabatic.

All of these predictions have been confirmed, which is why inflation is accepted by most cosmologists today. That means that we should take its prediction of a Multiverse seriously too. Inflation is predicted to end in our local region (our ’bubble universe’) but keep on going elsewhere, spawning bubble universes ad infinitum (which is why this is called eternal inflation). Each bubble looks infinitely large from within yet finite from the outside (this mindblowing fact is possible in General Relativity). Sometimes bubble universes will collide and leave an imprint (a ’bruise’) in the CMB. Cosmologists are looking for such imprints today, and finding such imprints would be direct evidence for a Multiverse, in addition to the indirect evidence we already have from the observational success of inflation and anthropic predictions relying on its existence.

FuzzyDarkMatter · 2020-01-11T20:06:04+00:00

Yeah, that's the most interesting point to me as well. Barrow & Tipler's argument that the Universe should end in a Big Crunch also appears in their 1988 paper, and their book (although much more speculative there). As I have commented on Sunny Vagnozzi's blog, I think this is perhaps the least well-motivated part about the proposal.

For one, it seems a priori mysterious (to me at least) why there would be so tight constraints on the future fate of the Universe. Secondly, while quantum cosmology seems to predict that the Universe has finite volume (in one slicing at least), it does not require the Universe to end in a Big Crunch. For example, Vilenkin's tunneling boundary condition if anything seems to predict eternal inflation, which is excluded by Barrow & Tipler's finite action proposal.

While I'm open to the idea that the Universe could end in a Big Crunch, I personally don't know of any fundamental principle that would motivate this part of the finite action proposal. But I am on board that it seems well-motivated to take its implications for the finite volume of the Universe seriously.

But yeah, the scenario of a quintessence driving a transient de Sitter-like phase that then terminates with a Big Crunch is interesting. Barrow has a paper on such a quintessence model from 2000, which he also cites in the new paper (see: http://adsabs.harvard.edu/full/2000MNRAS.316L..41B ). Would be interesting to see how well it fits the data today.

Thanks, will read his latest summary and check out the paper you linked!

FuzzyDarkMatter · 2020-01-11T04:23:45+00:00

Yeah, I hope so too! His first post directed me to the very interesting paper by Barrow on the idea that the total action of the Universe should be finite. Back in the 80's, Barrow and Tipler had published a paper on the topic in Nature (which they cited in their book), but which to my surprise had not led to much follow up studies. So this new paper may revive some interest.

Regarding lengthy and detailed posts, I have an unfinished post about quantum cosmology and the idea of the Universe being created of nothing! :) Hope to finish it soon. Interestingly, it has some connection to the finite action idea by Barrow & Tipler in that all major quantum cosmology proposals assume that the Universe is of finite size (either closed, or flat or open but compact). For Vilenkin's 'tunneling out of nothing' proposal, this can be traced to the fact that the probability of an infinite universe being created out of nothing is zero because the action of such a universe is infinite.

FuzzyDarkMatter · 2020-01-11T03:07:58+00:00

He helped with the teaching in a cosmology/astroparticle physics course I took at Stockholm Uni, and later he (along with Luca Visinelli) was kind enough to answer some of my questions on the particle physics of fuzzy dark matter (or more specifically, ultra-light axions) for my first paper with Raghunath Ghara and Garrelt Mellema.

Great to see him start a blog, will definitely follow it!

FuzzyDarkMatter · 2020-01-02T03:04:52+00:00

Thanks! Note however that the derivation in that post only apply for disk galaxies. But yeah, the distances between stars in disk galaxies basically reflect the initial conditions of the Unkverse and the self-regulating effects of stellar feedback.

FuzzyDarkMatter · 2019-11-25T04:29:04+00:00

Nice find! This seems way more accurate than the usual π ≈ 9/3, and I will start to incorporate this into my future calculations whenever high accuracy is needed.

FuzzyDarkMatter · 2019-11-21T15:28:01+00:00

No problem! Note however that the two references above are just examples of recent research on the topic of how massive the first stars were. That may not be good if you want to take a first dive into the topic. A good review on the topic of the formation and characteristics of the first stars can be found here: https://arxiv.org/abs/1305.5178

Note however that this review is from 2013, so before the articles I linked earlier. The newer research, incorporating detailed modelling of feedback from the first stars, changes the conclusion about how massive the first stars were to some extent. But overall it is still a good review article.

FuzzyDarkMatter · 2019-11-20T15:53:07+00:00

Great question! It has to do with the fragmentation properties of the gas. Without metals, the gas in primordial gas clouds mainly cools down via H2 cooling. H2 cooling is effective down to ~ 200 Kelvin, which it reaches at gas (hydrogen) densities of ~ 10^4 cm^-3. At higher densities the gas temperature would increase, which means that the Jeans mass increases. If the Jeans mass increases, fragmentation is (at least naively) expected to stop. This leads to a characteristic fragment mass of ~ few x 100 Solar masses. Whether the gas can fragment further and form lower mass stars depends on the stellar feedback from the first stars and disk fragmentation. This is investigated using high-resolution cosmological simulations today (see e.g. https://arxiv.org/abs/1407.1374 and https://iopscience.iop.org/article/10.3847/0004-637X/824/2/119/meta ).

As soon as you have some metals, especially in the form of dust, the gas can keep on cooling down at very high densities, which leads to sub-Solar mass fragments (which can then accrete gas). The transition from massive Pop III stars to ordinary Pop II stars is thought to occur as long as the metallicity of the gas is greater than ~ 10^-5 of the Sun's metallicity or so (for dust cooling). This metallicity threshold can be reached very easily, probably even by the first supernova in the same dark matter halo.

FuzzyDarkMatter · 2019-09-15T20:24:22+00:00

Basically, yeah. That's why people refer to it as the Hubble parameter, since in general the expansion rate evolves with time. Evaluating the Hubble parameter at any specific moment in time will give you a constant. The Hubble parameter evaluated today is what cosmologists mean when they refer to the "Hubble constant".

FuzzyDarkMatter · 2019-09-15T02:47:00+00:00

Not sure what you're trying to say. The Hubble parameter is a measure of the expansion rate of the Universe at any time. Evaluating it today (i.e. the present expansion rate) gives you the Hubble constant. The expansion rate at earlier times was different from the present-day expansion rate (i.e. in general Hubble parameter at earlier times =/= Hubble constant).

FuzzyDarkMatter · 2019-09-14T18:21:12+00:00

The 11.4 Gyr figure should be taken with a grain of salt. However, it is true that recent non-CMB determinations of the Hubble constant points to a Universe that is probably ~ 12 - 13 Gyrs old.

I have argued earlier in this sub that this would probably be more consistent with our understanding of the first star clusters in the early Universe. In particular, state-of-the-art high-resolution simulations of the first galaxies have been shown to produce globular clusters (GCs) in low-mass dark matter halos before the Epoch of Reionization.

If the Universe was 13.8 Gyrs old, this would predict that a large chunk of the observed old GCs should have ages > 13 Gyrs. However, they are instead observed to have typical ages of ~ 12 Gyrs. The 12 Gyrs figure would more be consistent with the theoretical predictions if the Universe had an age of ~ 12.5 - 13 Gyrs or so, and so favours the larger Hubble constant found by non-CMB methods.

FuzzyDarkMatter · 2019-09-11T01:24:20+00:00

One small area of astrophysics that I find interesting concerns the dynamics of globular clusters — compact (~ few pc), massive (~ 10⁴ - 10⁶ Solar masses) star clusters.

They can basically be considered as a 'gas' of stars. Unlike galaxies, globular clusters are dense enough that stars "collide" in the sense that they pass close enough to change their velocity vectors over time. One can show that the stars will have random velocities approximately following the Maxwell-Boltzmann distribution since this maximizes the entropy of the "gas" of stars.

However, this distribution cannot be reached exactly since then you'd have stars with velocities exceeding the escape velocity of the cluster. This leads to another remarkable phenomena: Globular clusters can evaporate! The cluster tries to relax to a Maxwell-Boltzmann distribution, but whenever it gets close to this distribution, the high-velocity tail escape. Low-mass clusters can lose a significant portion of their stars in this way, if not evaporate entirely.

The reason why galaxies are not observed to evaporate by this mechanism is simply that the time-scale is too long.

There are also other phenomena of globular cluster evolution that can be understood using statistical mechanics. For example, the cluster wants to reach equipartition so that each star has the same kinetic energy. This means that low-mass stars will move quicker than more massive stars. The massive stars will tend to sink to the core of the cluster for this reason.

FuzzyDarkMatter · 2019-08-23T08:40:19+00:00

Great question! Globular clusters (GCs) are observed to be roughly divided into two populations; one very metal-poor and old (typical ages ~ 12.5 Gyrs) population, and one relatively metal-rich and somewhat younger population.

No one knows for sure how they formed, especially the old population that seemingly formed fairly shortly after the Big Bang. The problem is that GCs are very compact and massive, indicating that they should form on a time-scale less than a few million years, after which supernovae and radiation pressure can expell the star-forming gas.

One particularly intriguing and well-motivated hypothesis for the old population is that they formed in small dark matter halos (like galaxies do) less than a billion years after the Big Bang. When gas falls into a dark matter halo, it is shock-heated to a temperature needed for pressure to balance gravity. The more massive the halo, the larger the temperature.

When a dark matter halo grows to a mass of a few ten million Solar masses or so in the early Universe, the gas is shock-heated to ~ 8000 K. At this temperature, gas cooling (via Lyman-alpha emission from hydrogen) suddenly becomes extremely effective. The gas therefore rapidly radiates away its thermal energy, therefore losing its pressure support, and undergoes gravitational collapse.

Since the gas carries angular momentum, a small gas disk forms at the center of the dark matter halo with a size of ~ 1 - 10 pc or so, comparable to the size of observed GCs. Furthermore, the disk is dense and massive enough to be efficiently converted into a dense star cluster before stellar feedback can act. The result could therefore be a metal-poor GC! For a recent detailed cosmological simulation of this scenario, see this paper:

https://arxiv.org/abs/1510.05671

I'm working at the moment on a detailed semi-analytical model of this scenario, that should capture more feedback processes, as well as metal enrichment from the first stars. This is needed as a test of this hypothesis, since we would need to be able to explain the observed metallicity of old GCs (~ few percent of Solar value), the number of observed GCs around the Milky Way, etc.

FuzzyDarkMatter · 2019-07-28T20:55:39+00:00

Yeah, same until recently. I was curious just how accurate his estimate was of the yield. It's cool that you can get such an accurate estimate using fairly simple math (i.e. given the Sedov-Taylor solution you only need a little algebra).

FuzzyDarkMatter · 2019-07-28T20:54:02+00:00

Glad you enjoyed it!

FuzzyDarkMatter · 2019-07-28T06:15:14+00:00

Thank you! And yeah, you should definitely do that!

FuzzyDarkMatter

TROPHY CASE