What Shape Is the Universe, Closed or Flat?
quantamagazine.org
quantamagazine.org
If I remember right, there was some analogy likening us to ants crawling on the surface of the Earth, but I could be conflating it with something else.
I read this more than 20 years ago, and I'm no cosmologist, so forgive me if I butchered that summary.
If you have a baloon with a band attached to two opposite points on it's surface and blow the ballon, the distance between points (the band length) on the surface will change 2 times as the distance from the middle. Moreover, if you place some markers on the band the distance between them will also strech. You get that effect in plain Euclidean 3d space. It's not that surprising even for nice orthogonal geometry.
Einstein suggested we could directly measure the curvature of the universe by comparing distances between probes located at angles to each other.
They would need to be pretty far apart from each other for this to work, but they can be very small: just an RTG and an antenna.
So hopeful light enough to build up an enormous velocity.
The receiver can be a mylar reflecting dish, which would be lightweight.
Use a gyroscope to steer the entire probe (both for transmitting and receiving).
They only need to communicate very very sporadically, and you only need a single bit of data transmitted (amongst each other).
Make some kind of schedule where the probes face each other, measure distance (i.e. time between send and reply), then steer toward the next probe.
To earth you would basically use Hubble to read the data (and let each probe send duplicate data back to earth).
Source: https://arxiv.org/abs/1502.01589
The CMB results assume dark matter and dark energy are real, it assumes a specific type of big bang theory. But none of those things have direct evidence.
We absolutely do need direct measurement of this.
And this is an especially bad type of modeling because we only have a single example. (In contrast to say, weather, where we can keep refining the model and comparing to the real world.)
Dark matter and dark energy are not assumptions. The assumptions are the general relativity and the FLRW metric which is the solution of Friedmann equations. The assumptions of Friedmann equations are that the universe is homogeneous, isotropic, and a perfect fluid. A perfect fluid means the universe has only a local density and pressure. There are no other assumptions in deriving the FLRW metric.
Even if were to verify these result by direct measurements, we wouldn't be able to send a probe across the universe. Please note that local curvature in the Milky Way or even the Virgo cluster wouldn't say anything about the curvature of the universe at very large scales. We are talking about curvature at scales larger than 100Mpc (more than 300 million light years).
The CMB is the evidence for a hot big bang (note, this is not the same as a singularity) although of course observational cosmologists want to trace back from the first starlight to the CMB and have various plans to do so. Surprises in that "dark age" are more than welcome because they are bound to reveal things about low-density momentum-energy.
> assume dark matter and dark energy are real
Nobody has put together a self-consistent physical cosmology model that reasonably matches actual observations of the shape, scatter, and shift of galaxies across our sky, without the theory containing some additional nonluminous tension and pressure terms that take effect earlier than the surface of last scattering and persist to the present day. The self-consistency problems are expected to blow apart consistency with observation. (Details in e.g. Famey & McGaugh 2011 §9).
The need for a tension term to match observations arose even in 1917 [1], and the more detailed 1929 Hubble observations supported the idea of a constant tension independent of visible sources; that is what modern observations still support. The need for an additional pressure term became obvious from the 1960s Rubin observations; the term was pretty simple initially and that simplicity was well-supported through the 1980s and 1990s. One could through that period support theories wherein the pressure term appeared everywhere luminous sources appeared in some straightforward function (e.g. Milgrom 1983). Modern observations have several advantages in the observable details, and those support a rather more complicated interplay between luminous sources and the pressure term, particularly at scales much larger than kiloparsecs.
If you can find a theoretical mechanism that produces the observed small anisotropies of the CMB without introducing some tension and pressure acting on an almost entirely homogeneous photon gas with an almost perfect (albeit highly redshifted) blackbody spectrum, well, that'd be amazing. It should of course also be compatible with the actual observations of less-highly-redshifted molecular gas spectra (and the Lyman-alpha forest), and how those lense around rather than through radio-clumpy foreground galaxy clusters.
There are a lot of galaxies in the sky, so many that we've discovered that their shapes are pretty regular across an enormous redshift-range. That's additional observational evidence for the spatial flatness of the universe. Even fairly gentle spatial curvature would distort the shapes of face-on spirals at high redshifts, and that is not what we see.
Even the Hubble Ultra Deep Field image imposes pretty significant constraints. It's littered with galaxies and with goodies like the Gravitational Lens Galaxy Cluster CL0024+1654, which have been imaged in various wavelengths by both ground and space telescopes.
The match between the small anisotropies in the CMB and galaxies at increasing redshift strongly ties the hands of speculative theorists. Those are the people who do the "modelling and simulation". They are stuck having to match actual observation, assuming their goal is to theorize about the single example we inhabit. Bolshoi and MultiDark had to match Sloan, not the other way around.
> bad type of modelling
Whose word are you taking for this?
> we only have a single example. In contrast to ... weather .. where we can keep ... comparing to the real world
As said at the top, we've kept comparing the scatter and shape of galaxies and the scatter and shape of fluctuations in the CMB. How is dealing with evidence questioning the standard value of the H_0 parameter not "refining the model"?
Also, today we only have a single example of an Earthlike planet's weather data. And until the late 1960s at the earliest we had weather data of any sort for only one planet. So was the numerical weather modelling work done in the 1950s (e.g. by Charney, von Neumann et al. on ENIAC) also "an especially bad type of modelling"?
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[1] E Schrödinger, “Über ein Lösungssystem der allgemein kovarianten Gravitationsgleichungen”, Physikalische Zeitschrift,19, 20-22 (1918) cf. https://arxiv.org/abs/1211.6338
Yeah, like light-years apart. Many light-years, too.
If you have a topological space, the entire space is always both open and closed. https://en.wikipedia.org/wiki/Topological_space#Definition_v...
In relativity, the idea is all time-like (basically path of ordinary matter) curves will all converge, while the other case, time-like curves always diverge (so open).
The Poincaré conjecture states that closed simply connected 3-manifolds are all diffeomorphic to the 3-sphere. Even stronger, every closed 3-manifold whose fundamental group is finite is a quotient of the 3-sphere by a discrete subgroup of its group of isometries, SO(4) (called the elliptization theorem, which is what Perelman proved). I've been told some astronomers once looked into whether the cosmic background radiation suggested that we lived in Poincaré dodecahedral space.
Thurston's geometrization conjecture (all proved as of 2012) is that all closed 3-manifolds can be built out of certain 3-manifolds with standard Riemannian geometries by gluing them together along their torus boundaries and by introducing wormholes, essentially. Some interesting cases are the spherical geometries (constant positive curvature, classified above) and hyperbolic geometries (constant negative curvature, also classified by Perelman). The only actually flat closed 3-manifolds are the 10 finite-order mapping tori of the torus -- one example is S^1 x S^1 x S^1, where the 3-dimensional version of the game Asteroids would be played.
There are infinitely many closed hyperbolic 3-manifolds. I don't understand why space can't be negatively curved.
Jeff Weeks has a cool program for flying through different spaces: http://geometrygames.org/CurvedSpaces/index.html
4-manifolds are definitely more complicated. Lots of techniques that work for 3-manifolds and (5+)-manifolds don't work.
This statement is meaningless. It makes no sense to speak of whether a set is open or closed outside the context of a topology.
Also, like... in any topological space of the sort we usually deal with (specifically: T_1 spaces), any open set must be infinite, as any finite set is closed? So I'm really not sure what you're thinking of here. (And in something like Euclidean space -- or a manifold -- any open set must be uncountably infinite...)
(Also, as others have already mentioned, "closed" here doesn't mean "closed" in the sense of point-set topology; "closed manifold" means a compact manifold [without boundary, if manifolds with boundary are being considered]. Yes, this is a bit confusing at first.)
Some models suggest our universe is just one "foam-bubble" in a big ocean of universes. Hell, the mole on your face may contain its own universes. It really could be "turtles all the way down", or Pikachus all the way down.
https://www.vice.com/en_us/article/j5yngp/the-universe-is-ma...
Because of our Earthly experience, we tend to assume there's an absolute limit to the quantity of "stuff", but perhaps that's not true: there is infinite stuff, we just can't touch most of it.
“Worship not God but Regigigas, on whom the world rests.”
— Henry David Thoreau