Kurzgesagt – In a Nutshell
We thank following experts for their critical reading and input.
Prof. Matt Caplan
Illinois State University
– Today we know that the universe had a beginning 14 billion years ago and that it has been expanding ever since.
Unfortunately, our universe has no circles like trees nor can we count the wrinkles on it. So calculating the age of the universe is a tad bit complicated with deciding on various cosmological parameters and repeated measurements. The details of the calculation are beyond the scope of this document. The commonly accepted age of the universe, ~13.8 billion years, which we rounded up to 14 billion years for the sake of simplicity.
#Planck Collaboration. Planck 2018 results - VI. Cosmological parameters. 2020.
https://www.aanda.org/articles/aa/full_html/2020/09/aa33910-18/aa33910-18.html
Quote: “We could, therefore, disregard these tensions and conclude that the 6-parameter ΛCDM model provides an astonishingly accurate description of the Universe from times prior to 380 000 years after the Big Bang, defining the last-scattering surface observed via the CMB, to the present day at an age of 13.8 billion years.”
– Information can’t travel faster than light. That means that we can only see parts of the universe whose light has had time to reach us in the last 14 billion years. When we look outwards, what we see is a sphere centered on us, the observable universe. But it gets a tad more complicated – because the universe has been expanding, we know that the most far away things whose light we can see are actually 45 billion lightyears from us right now. So the observable universe is a sphere with a radius of 45 billion light years.
The following infographic nicely explains how our 14 billion years old universe can have a radius of 45 billion light years. If you need to brush over some cosmological misconceptions, the following article is a good place to start.
#Charles H. Lineweaver and Tamara M. Davis. Misconceptions about the Big Bang. 2005.
https://www.mso.anu.edu.au/~charley/papers/LineweaverDavisSciAm.pdf
So the radius of the ball that is the observable universe is set by the distance that light has traveled in 14 billion years. But since the universe is expanding, that distance is more than 14 billion light-years. How much more than 14 billion light-years it is therefore depends on the speed of the expansion. When both taken together, we reach a value of about 45 billion light-years.
If you want to notch the technical level up, the same author has a scientific publication on this topic.
#Tamara M. Davis, Charles H. Lineweaver. Expanding Confusion: common misconceptions of cosmological horizons and the superluminal expansion of the Universe. 2003.
– It contains around 200 billion galaxies, each with hundreds of billions of stars.
#ESA. The Color Of The Stars. Retrieved Feb 2024.
https://www.cosmos.esa.int/web/cesar/the-color-of-the-stars
Quote: “The Universe has about 200 billion galaxies. Our Galaxy, the Milky Way is a very normal spiral galaxy.”
Though 200 billion is the most commonly cited number, there are some recent studies suggesting that there are ten times more galaxies than that. It is important to note that the number is still unclear, but most estimates are between a few hundred billion and about a trillion galaxies. So we stick to the number 200 billion in the script.
#NASA Hubble Mission Team. Hubble Reveals Observable Universe Contains 10 Times More Galaxies Than Previously Thought. 2016.
Quote: “One of the most fundamental questions in astronomy is that of just how many galaxies the universe contains. The landmark Hubble Deep Field, taken in the mid-1990s, gave the first real insight into the universe's galaxy population. Subsequent sensitive observations such as Hubble's Ultra Deep Field revealed a myriad of faint galaxies. This led to an estimate that the observable universe contained about 200 billion galaxies.
The new research shows that this estimate is at least 10 times too low.
Conselice and his team reached this conclusion using deep-space images from Hubble and the already published data from other teams. They painstakingly converted the images into 3-D, in order to make accurate measurements of the number of galaxies at different epochs in the universe's history. In addition, they used new mathematical models, which allowed them to infer the existence of galaxies that the current generation of telescopes cannot observe. This led to the surprising conclusion that in order for the numbers of galaxies we now see and their masses to add up, there must be a further 90 percent of galaxies in the observable universe that are too faint and too far away to be seen with present-day telescopes. These myriad small faint galaxies from the early universe merged over time into the larger galaxies we can now observe."
– So for us, there is an edge: We are looking at the past until there is just no past left. This edge is really more like an edge in time, and in a sense meaningless.
The outer boundary of the observable universe is “made” by the cosmic microwave background, or the “afterglow of the Big Bang”, which looks like the following image: the cosmic microwave background divided into two hemispheres. This is basically a picture of the plasma that emerged from the Big Bang. The points that emitted this light are today 45 billion light-years from us although today they don’t look like that anymore. It’s only that we cannot see those points as they are now but only as they were 14 billion years ago.
#NASA. Astronomy Picture of the Day. 2003.
#ESA. Planck and the cosmic microwave background. Retrieved Jan 2024.
Quote: “The cosmic microwave background (or CMB) fills the entire Universe and is leftover radiation from the Big Bang. When the Universe was born, nearly 14 billion years ago, it was filled with hot plasma of particles (mostly protons, neutrons, and electrons) and photons (light). In particular, for roughly the first 380,000 years, the photons were constantly interacting with free electrons, meaning that they could not travel long distances. That means that the early Universe was opaque, like being in fog.
However, the Universe was expanding and as it expanded, it cooled, as the fixed amount of energy within it was able to spread out over larger volumes. After about 380,000 years, it had cooled to around 3000 Kelvin (approximately 2700ºC) and at this point, electrons were able to combine with protons to form hydrogen atoms, and the temperature was too low to separate them again. In the absence of free electrons, the photons were able to move unhindered through the Universe: it became transparent.
Over the intervening billions of years, the Universe has expanded and cooled greatly. Due to the expansion of space, the wavelengths of the photons have grown (they have been ‘redshifted’) to roughly 1 millimetre and thus their effective temperature has decreased to just 2.7 Kelvin, or around -270ºC, just above absolute zero. These photons fill the Universe today (there are roughly 400 in every cubic centimetre of space) and create a background glow that can be detected by far-infrared and radio telescopes.”
Therefore, our boundary is more of a boundary in time, rather than a physical wall in space.
#Dana Berry / SkyWorks Digital, Inc. and SDSS collaboration and WMAP cosmic microwave background image credit: NASA/WMAP Science Team. Retrieved February 2024.
– But that leads to a weird problem. Such a universe should have an edge, a cosmic wall where space ends. And if there is an edge there should be something outside that edge. But the universe by definition is “all there is”, so how could there be stuff outside all there is? Does the idea of something outside of everything even make sense?
The universe is thought to have no boundary (or “no border”) in current cosmology - so no wall at the end of space that we are going to bump into. Properly explaining why would take pages and pages of textbook information however it is neatly summed up in the following.
#Brief Answers to Cosmic Questions. Retrieved February 2024.
https://lweb.cfa.harvard.edu/seuforum/faq.htm
Quote: “Does the Universe have an edge, beyond which there is nothing?
Galaxies extend as far as we can detect... with no sign of diminishing.There is no evidence that the universe has an edge. The part of the universe we can observe from Earth is filled more or less uniformly with galaxies extending in every direction as far as we can see - more than 10 billion light-years, or about 6 billion trillion miles. We know that the galaxies must extend much further than we can see, but we do not know whether the universe is infinite or not. When astronomers sometimes refer (carelessly!) to galaxies "near the edge of the universe," they are referring only to the edge of the OBSERVABLE universe - i.e., the part we can see.”
– The universe is not like the skin of an orange, but it could be very similar – instead of a sphere, it could be a hypersphere, where 3D-space is curled on itself. Which is impossible for your brain to visualize unfortunately. But the point is: no borders, no outside of the hyper sphere – from our human ant perspective, our whole 3D space is like the peel of the orange. If you were aboard a spaceship flying in a straight line, you would eventually come back to Earth.
A hypersphere is a higher dimensional analog of a normal sphere. A circle is a 1-sphere and a normal sphere is a 2-sphere, so a hypersphere is 3-sphere or higher. If we assume that the ant has a circular field of view, then the ant's observable orange-universe will always be a uniform circle as it walks along the surface of the orange. Tying in with the ant example, if we go up a dimension, our observable universe coulda 3-dimensional ball (the inner part of a 2-sphere) within a hypersphere (3-sphere) universe. Unfortunately, it is not possible to draw this shape on a piece of paper. Therefore, we have to use the 2D analogy here for the visuals.
– How does any of this make sense? The actual physics is hard, so we have to simplify and lie a bit here. But in a nutshell it all boils down to gravity: The way it works is that mass creates gravity by bending spacetime. This bending is the strongest where the mass is, but sort of stretches on forever, like a very mild tension in the fabric of spacetime itself. This could bend the whole universe in a way where it bends back on itself, which then makes the hypersphere. If you find this confusing, we are with you.
Before we jump back into the hyperspheres and other unimaginable structures, it might be good to take a step back and take a look at the geometry of the universe, since the question “what is outside the universe?” is essentially related to the geometry of the universe. And there are two levels for that geometry, local vs global. In terms of the ant and orange, you can think of the whole surface of the orange as the global geometry, and the circle that is visible to the ant as the local geometry. Local geometry can have implications on the global one and to our advantage it is a bit more manageable when it comes to making measurements. With our universe, we can also start by figuring out the local geometry and move on from there.
There can be more due to more complicated physics but for now let’s say there are basically three possible shapes for the universe: a flat Universe, a spherical or closed Universe and a hyperbolic or open Universe. One of the main, and locally measurable, differences between these three is the curvature, which would be zero in a flat universe, positive in a spherical universe and negative in a hyperbolic universe. And as you might remember from the trampoline-bowling ball demonstration of General Relativity, matter tells spacetime how to curve. In the following image, these three options are depicted alongside the corresponding density parameter Ω0. Density parameter is in the most simple terms the ratio of the average density of matter to the critical density, which is the density required for the Universe to be spatially flat. If the actual density is bigger, Ω0>1, then the universe is closed. If the actual density is smaller, Ω0<1, then the universe is open.
In the article below you can find a good beginner-level explanation regarding the geometry of the universe.
#WMAP Special Exhibit. The Shape of the universe. Retrieved Jan 2024.
https://imagine.gsfc.nasa.gov/observatories/satellite/wmap/shape.html
Quote: “For starters, there's the spherical universe, where an airplane sent in one direction ultimately comes back around, as is the case on Earth. A triangle in this universe has angles adding up to more than 180 degrees. Parallel lines ultimately meet, just like two friends walking parallel to each other on their way to the North Pole. Cowboys may fancy the hyperbolic universe, which is saddle shaped. A triangle's angles here add up to less than 180 degrees. Then there's the good, old flat universe, which obeys Euclidean geometry: triangles add up to 180 degrees; parallel lines never meet.
WMAP studied the likely shape of our universe. Prior to WMAP, the data pointed to things being flat. How can we tell this? WMAP will measure a great big "triangle" in the sky and see if it adds up to what a flat, Euclidean-based universe should be.
[...]
So for our triangle, we have one distance across the sky (the sound horizon distance, which is the distance the sound waves travel for 400,000 years), and an angle, which was determined by WMAP. The unknown distance was the distance the microwave background had traveled since the last scattering. If the angle was one degree on the sky, as many scientists predict, then the universe may be flat. If the angle between peaks in temperature differences is, say, two degrees, then sound waves will have had to have covered twice the ground in 400,000 years. This would imply that the universe is open or curved.
The TOCO experiment in Chile made the first detection, and balloon-borne experiments and a South Pole telescope have all found that angle to be about one degree. WMAP found that the universe should obey the rules of Euclidean geometry so the sum of the interior angles of a triangle add to 180 degrees - so confirmed that the universe should be flat.”
#Luminet. Cosmic Hall of Mirrors. 2005.
https://www.researchgate.net/publication/2173949_A_Cosmic_Hall_of_Mirrors
Quote: “The first testable predictions about the size and shape of the universe were made by Einstein in 1916 as part of his general theory of relativity. In general relativity massive bodies such as stars change the shape of space--time around them, much as a bowling ball would change the shape of a trampoline. Indeed, it is this local deformation of space--time that is responsible for gravity in Einstein's theory. The average curvature of space therefore depends on the overall density of matter and energy in the universe. This density is usually expressed in terms of the parameter Ω , which is defined as the ratio of the actual density to the critical density required for space to be flat or Euclidean. Space can therefore have three possible curvatures: zero curvature (Ω = 1), which means that two parallel lines remain a constant distance apart as they do in the familiar Euclidean space; negative curvature (Ω < 1), with parallel lines diverging as they do on the hyperbolic surface of a saddle; or positive curvature (Ω > 1), which means that parallel lines eventually cross one another as they do on the surface of a sphere.”
– If the universe happens to be a hypersphere, how could we find out how big it is? On Earth we can see things disappear below the horizon and that helps us calculate how big the Earth is. Scientists tried to find some sort of “universe horizon” that would reveal the scale of the cosmic sphere – but didn’t see anything. Which means that if the universe is a hypersphere, it needs to be so big that from our perspective it looks like we are living on a flat surface.
In the most simple and rough terms, we can calculate the radius of Earth using the distance to the horizon as demonstrated in the following figure. However, if we imagine that we are standing on a much much larger body than Earth, then the tangent line OH would be flat for an observer.
#Australian Space Academy. Distance to the Horizon. Retrieved February 2024.
Coming back to our hypersphere and the corresponding 2D analogy, it would be similar to that we’ve measured the curvature of the circular patch inside the yellow line and found that it’s really close to a flat patch. This can only mean two things: Either the “bigger” universe is NOT a hypersphere or, if it is one, the hypersphere is so much bigger than our circular patch (aka observable universe) that the curvature in the patch is super small.
#Luminet. A Cosmic Hall of Mirrors. 2005.
https://www.researchgate.net/publication/2173949_A_Cosmic_Hall_of_Mirrors/figures?lo=1
Quote: “Figure 1: How flat is the universe? The curvature of space and our horizon radius is determined by the average density of the universe and its expansion rate. Cosmologists often say that space is nearly flat because the observed value of the density is close to the critical value for a flat universe. However, if the density was 2% more than the critical density, the horizon radius (red line) would be about 46 billion light-years and the radius of curvature of the corresponding hypersphere would be only 2.6 times greater. We would therefore see a modest but non-trivial portion of the hypersphere. If the density is exactly equal to the critical value, space is Euclidean, the radius of curvature is infinite and we can only see an infinitesimal portion of the universe.”
– For this to make sense, a hyperspherical universe should be at least 1,000 times bigger than our observable part. It could be a trillion times bigger for all we know, but not smaller than that.
In the same way that the curvature of an ordinary sphere is related to its radius (the smaller the curvature at the surface of the sphere, the larger the radius of the sphere), in the case of a hypersphere it is also possible to define a curvature radius RC:
#Vardanyan et al. Applications of Bayesian model averaging to the curvature and size of the Universe. 2011.
where H0 is the value of the Hubble constant (the present expansion rate of the universe) and ΩK is a dimensionless number that quantifies the empirically measured bound on the overall curvature of the universe. Recent and high confidence empirical values for H0 and ΩK can be taken from the last (2018) dataset of cosmological parameters from the Planck collaboration:
#Planck Collaboration, N. Aghanim, Y. Akrami, M. Ashdown, J. Aumont, C. Baccigalupi, M. Ballardini, A. J. Banday, R. B. Barreiro, N. Bartolo. Planck 2018 results - VI. Cosmological parameters. 2020.
https://www.aanda.org/articles/aa/full_html/2020/09/aa33910-18/aa33910-18.html#T1
Quote: “Assuming the base-ΛCDM cosmology, the inferred (model-dependent) late-Universe parameters are: Hubble constant H0 = (67.4 ± 0.5) km s−1 Mpc−1 [...] The joint constraint with BAO measurements on spatial curvature is consistent with a flat universe, ΩK = 0.001 ± 0.002.”
If we take the upper bound ΩK = 0.003 (i.e. 0.3%), from the above formula we get:
RC = 265 ·109 light-years = 265 Gly
So IF the universe is a hypersphere, this is the smallest curvature radius compatible with the empirical data. As we see, this is much larger than the radius of the observable universe (about 45 billion light-years), which means that, if the universe is curved, its curvature will only be noticeable at distances much larger than the observable universe.
To quantify such distances we can do the following. The volume of a hypersphere of radius RC is given by:
S3 = 2π2RC3
(this formula can be found in textbooks and it’s the generalization to three dimensions of the more familiar formula 4πR2 for the surface of an ordinary sphere of radius R). Plugging the value we’ve just found for the smallest possible curvature radius of a hyperspherical universe (265 Gly) into this formula, we get:
S3 = 367·106 Gly3
In other words, this would be the minimal volume of a hyperspherical universe. It could be bigger, but no smaller than that.
On the other hand, the volume VO of the observable universe (neglecting the curvature induced by the embedding of the 3-dim ball into the 3-sphere, since the ratio between radii is small enough) is given by the volume comprised within a ball of radius RO ~ 45 Gly. This gives:
VO = (4/3)πRO3 = 380·103 Gly3
Which means that inside a hyperspherical universe with volume S3, we could pack
N = S3/VO = 966 ~ 1000
observable patches. That is: if the universe is a hypersphere, it has to be at least about 1000 times bigger than the observable universe.
This bound is obtained from the empirical bound on ΩK reported by the Planck dataset, which for most uses is considered as the current golden standard regarding the fundamental cosmological parameters. However, It has to be noted that other analyses have yielded different bounds for ΩK, for example, the following paper.
#Liu et al. Model-independent Constraints on Cosmic Curvature: Implication from Updated Hubble Diagram of High-redshift Standard Candles. 2020.
https://iopscience.iop.org/article/10.3847/1538-4357/abb0e4
Quote: “Meanwhile, we find that, for the Hubble diagram of 1598 quasars as a new type of standard candle, the spatial curvature is constrained to be ΩK = 0.08 ± 0.31. For the latest Pantheon sample of SNe Ia observations, we obtain ΩK = − 0.02 ± 0.14.”
obtains values for ΩK of the order of 1% (instead of 0.1%). We will omit the exact calculations here (they are identical to the ones we’ve made above) but, in order of magnitude, this would imply that such a hyperspherical universe would be about ~100 times bigger (instead of 1000 times bigger) than the observable universe.
At any rate, the conclusion is clear: the universe COULD be a hypesphere, at least in principle, But we’ve measured the curvature of our observable patch and have found it so small that, IF the universe is indeed a hypersphere, it has to be many times bigger than the observable universe..
– Some scientists thought all of this is way too straight forward and came up with a wilder option: The universe could be like the frosting of a donut. A hyperdonut – also impossible to visualize for your brain. This, too, means that if you travel in a straight line, you would get back to where you started. But with fun complications.
The curvature is a local parameter and it does not necessarily tell us the overall topology of the whole universe, even though it implies certain restrictions on it. As in the ant and orange analogy, the ant would not know the overall shape of the orange just by its view from the patch it is sitting on. It would need to know how its patch is connected to the other patches on the orange to know the overall topology. How the different regions in the universe “are connected” to make up the whole universe is what determines the shape or topology of it, as well as it being finite or infinite.
Similarly, the seemingly flat appearance of our local universe can give rise to different topologies. One option is the so called "simply connected" one, like a plane. In this case, the universe would be infinite. We would need to go into mathematics' area of topology to explain it properly, but the full account is beyond the scope of this document. However, simply put, "simple connected" domain means without holes. However, it is possible to construct "multiply connected" topologies from 2D planes. In the image below, construction of one such "multiply connected" topology is depicted: torus, or more properly Euclidean 2-torus, which is, simply put, the outer surface of a doughnut. We can have a very interesting universe shape if we go up a dimension: a hyper-doughnut, i.e. Euclidean 3-torus. Here the relation between 2-torus and 3-torus is analogues to the relation between 2-sphere and 3-sphere. Therefore, instead of starting with a square as in the case of Euclidean 2-torus, we need to start with a cube to construct the Euclidean 3-torus. Again, it is impossible for our brains to visualize this, so we stick to lower dimension analogies and go on with the torus.
#Jean-Pierre Luminet, Glenn D. Starkman and Jeffrey R. Weeks. Is Space Finite?. 1999.
Quote: "Schwarzschild’s example illustrates how one can mentally construct a torus from Euclidean space. In two dimensions, begin with a square and identify opposite sides as the same—as is done in many video games, such as the venerable Asteroids, in which a spaceship going off the right side of the screen reappears on the left side. Apart from the interconnections between sides, the space is as it was before. Triangles span 180 degrees, parallel laser beams never meet and so on—all the familiar rules of Euclidean geometry hold. At first glance, the space looks infinite to those who live within it, because there is no limit to how far they can see. Without traveling around the universe and reencountering the same objects, the ship could not tell that it is in a torus [see illustration below]. In three dimensions, one begins with a cubical block of space and glues together opposite faces to produce a 3-torus. The Euclidean 2-torus, apart from some sugar glazing, is topologically equivalent to the surface of a doughnut. Unfortunately, the Euclidean torus is food only for the mind. It cannot sit in our three-dimensional Euclidean space. Doughnuts may do so because they have been bent into a spherical geometry around the outside and a hyperbolic geometry around the hole."
#James Schombert. Geometry of the Universe. Retrieved February 2024.
https://pages.uoregon.edu/jschombe/cosmo/lectures/lec15.html
Quote: “Standard cosmological observations do not say anything about how those volumes fit together to give the universe its overall shape--its topology. The three plausible cosmic geometries are consistent with many different topologies. For example, relativity would describe both a torus (a doughnut-like shape) and a plane with the same equations, even though the torus is finite and the plane is infinite. Determining the topology requires some physical understanding beyond relativity.
(...)
The usual assumption is that the universe is, like a plane, "simply connected," which means there is only one direct path for light to travel from a source to an observer. A simply connected Euclidean or hyperbolic universe would indeed be infinite. But the universe might instead be "multiply connected," like a torus, in which case there are many different such paths. An observer would see multiple images of each galaxy and could easily misinterpret them as distinct galaxies in an endless space, much as a visitor to a mirrored room has the illusion of seeing a huge crowd.
(...)
One possible finite geometry is donutspace or more properly known as the Euclidean 2-torus, is a flat square whose opposite sides are connected. Anything crossing one edge reenters from the opposite edge (like a video game see 1 above). Although this surface cannot exist within our three-dimensional space, a distorted version can be built by taping together top and bottom (see 2 above) and scrunching the resulting cylinder into a ring (see 3 above). For observers in the pictured red galaxy, space seems infinite because their line of sight never ends (below). Light from the yellow galaxy can reach them along several different paths, so they see more than one image of it. A Euclidean 3-torus is built from a cube rather than a square.”
A simple definition of the simple connected domains can be found in the following article.
#Nykamp DQ, “Simply connected definition.” Retrieved February 2024.
https://mathinsight.org/definition/simply_connected
Quote: "For three-dimensional domains, the concept of simply connected is more subtle. A simply connected domain is one without holes going all the way through it. However, a domain with just a hole in the middle (like a ball whose center is hollow) is still simply connected, as we can continuously shrink any closed curve to a point by going around the hole and remaining in the domain. On the other hand, a ball with a hole drilled all the way through it, or a spool with a hollow central axis, is not simply connected. A closed curve that went around the hole could not be shrunk to a point while remaining in the domain. There is no way for the curve to bypass the the hole so it remains stuck around it."
Following image might also help with understanding different topologies.
#Luminet. Time, Topology and the Twin Paradox. 2009.
https://www.researchgate.net/publication/45881492_Time_Topology_and_the_Twin_Paradox
Quote: (Figure Caption) "Fig. 7. The four multiply connected topologies of the 2dimensional Euclidean plane. They are constructed from a rectangle or an infinite band (the fundamental domain) by identification of opposite edges according to allowable transformations. We indicate their overall shape, compactness and orientability properties"
– In a hyperdonut universe there is not the same amount of stuff in every direction. If two spaceships fly in different directions, one could get back to the start way earlier. This also means that light from faraway galaxies would do fun and confusing stuff, in a sort of cosmic hall-of-mirrors effect. We could see far away things in two places – but not just that, but we would see it in different moments in time! Because its light would have taken much longer to travel in one direction than the other! You could see a star being born in front of you and see that same star die on the opposite side of the sky.
The topology of the universe has implications on our observations of it. For example, in a plane, there would be a single straight path that the light could travel. In the hyperdoughnut universe, there would be multiple paths that light can travel to reach to the same point.
The following image nicely demonstrate this on the 2D version, i.e. on a torus. If, for example, we take the red galaxy to be Earth and the yellow one to be Andromeda, the light from Andromeda can reach to us both by the path depicted by the white arrow and the red arrow (Let's omit the white one for the sake of simplicity.). White path is shorter than the red one. One important implication of this is that we can see a younger Andromeda on the right and a older one on the left.
#Jean-Pierre Luminet, Glenn D. Starkman and Jeffrey R. Weeks. Is Space Finite?. 1999.
This could lead to a hall of mirrors effect as the light rays wraps around the universe multiple times over different paths. Like a ball put in a infinity box which is covered with mirrors inside, a single object can create multiple images.
#Luminet. A Cosmic Hall of Mirrors. 2005.
https://www.researchgate.net/publication/2173949_A_Cosmic_Hall_of_Mirrors
Quote: “Cosmologists usually assume that the universe is simply connected like a plane, which means there is only one direct path for light to travel from a source to an observer. A simply connected Euclidean or hyperbolic universe would indeed be infinite, but if the universe is multiply-connected like a torus there would be many different possible paths. This means that an observer would see multiple images of each galaxy and could easily misinterpret them as distinct galaxies in an endless space, much as a visitor to a mirrored room has the illusion of seeing a crowd. Could we, in fact, be living in such a cosmic hall of mirrors?”
For further reading, you can refer to the following paper:
#Lachieze-Rey and Luminet. Cosmic Topology. 2003.
https://arxiv.org/abs/gr-qc/9605010
– How big would such a hyper donut universe be? Well, because of its strange geometry, actually this is kind of the smallest possible universe – potentially just a few times bigger than the observable universe. But it could also be way, way bigger. We don’t know.
Scientists actually looked for clues of this hall of mirrors effect. Some look into the arrangement of galaxies, some into quasars and some use the fluctuations in the cosmic microwave background.
#Jean-Pierre Luminet, Glenn D. Starkman and Jeffrey R. Weeks. Is Space Finite?. 1999.
Quote: "These fluctuations are the key to resolving a variety of cosmological issues, and topology is one of them. Microwave photons arriving at any given moment began their journeys at approximately the same time and distance from the earth. So their starting points form a sphere, called the last scattering surface, with the earth at the center. Just as a sufficiently large paper disk overlaps itself when wrapped around a broom handle, the last scattering surface will intersect itself if it is big enough to wrap all the way around the universe. The intersection of a sphere with itself is simply a circle of points in space.
Looking at this circle from the earth, astronomers would see two circles in the sky that share the same pattern of temperature variations. Those two circles are really the same circle in space seen from two perspectives [see illustration below]. They are analogous to the multiple images of a candle in a mirrored room, each of which shows the candle from a different angle."
In the following paper, the authors searched for circles with matched temperature variations on the cosmic microwave background map. By this, they seeked to put some constraints on the possible topologies and consequently whether the universe is finite or not. So they reasoned that the light from a distant object will be able to reach us along more than one path if the Universe is finite - as in the case of the hyper-doughnut for example. For us to have these multiple observations, the light must have sufficient time to reach us from multiple directions, implying that the Universe should be small enough. However, their calculations found a constraint on the size.
#Cornish et al. Constraining the Topology of the Universe. 2003.
https://arxiv.org/pdf/astro-ph/0310233.pdf
Quote: “The first year data from the Wilkinson Microwave Anisotropy Probe are used to place stringent constraints on the topology of the Universe. We search for pairs of circles on the sky with similar temperature patterns along each circle. We restrict the search to back-to-back circle pairs, and to nearly back-to-back circle pairs, as this covers the majority of the topologies that one might hope to detect in a nearly flat universe. We do not find any matched circles with radius greater than 25deg. For a wide class of models, the non-detection rules out the possibility that we live in a universe with topology scale smaller than 24 Gpc.”
But even if a hyper torus with a length scale smaller than 24 Gpc is ruled out by data, the constraint on the overall size of the universe is still much less stringent than in the case of a hyperspherical universe (which, as we saw above, should have a curvature scale no smaller than 265 Gly ~ 80 Gpc).
Other researchers though are not dismissing the possibility just yet. They also investigated the temperature variation in the cosmic microwave background for a multiply connected topology.
#Aurich et al. The variance of the CMB temperature gradient: a new signature of a multiply connected Universe. 2021.
https://iopscience.iop.org/article/10.1088/1361-6382/ac27f0
Quote: “To date, the Universe models with non-trivial topology such as the toroidal space are the only models that possess a two-point correlation function showing a similar behaviour as the one derived from the observed Planck CMB maps. In this work it is shown that the normalized standard deviation of the CMB temperature gradient field does hierarchically detect the change in size of the cubic three-torus, if the volume of the Universe is smaller than ≃2.5 × 10^3 Gpc^3. It is also shown that the variance of the temperature gradient of the Planck maps is consistent with the median value of simulations within the standard cosmological model. All flat tori are globally homogeneous, but are globally anisotropic. However, this study also presents a test showing a level of homogeneity and isotropy of all the CMB map ensembles for the different torus sizes considered that are nearly at the same weak level of anisotropy revealed by the CMB in the standard cosmological model.”
– Actually, the cosmological model used by most scientists, describes an infinite universe. We mostly use it to calculate what happens inside our observable chunk, but if taken literally, it predicts an infinite universe.
Currently, the best cosmological model we have is “Flat Lambda-CDM” (Flat ΛCDM). It is known as the standard model. The two important features of the Flat ΛCDM model are the cosmological constant Λ which is associated with dark energy and the cold dark matter. A full account of the model is beyond the scope of this document but in the following, one can find a simple explanation.
#ΛCDM Model of Cosmology. Retrieved February 2024.
https://lambda.gsfc.nasa.gov/education/graphic_history/univ_evol.html
Quote: “In this picture, the infant universe is an extremely hot, dense, nearly homogeneous mixture of photons and matter, tightly coupled together as a plasma. An approximate graphical timeline of its theoretical evolution is shown in the figure above, with numbers keyed to the explanatory text below.
1- The initial conditions of this early plasma are currently thought to be established during a period of rapid expansion known as inflation. Density fluctuations in the primordial plasma are seeded by quantum fluctuations in the field driving inflation. The amplitude of the primordial gravitational potential fluctuations is nearly the same on all spatial scales (see e.g. reviews by Tsujikawa 2003 and Baumann 2009).
The small perturbations propagate through the plasma collisionally as a sound wave, producing under- and overdensities in the plasma with simultaneous changes in density of matter and radiation. CDM doesn't share in these pressure-induced oscillations, but does act gravitationally, either enhancing or negating the acoustic pattern for the photons and baryons (Hu & White 2004).
2- Eventually physical conditions in the expanding, cooling plasma reach the point where electrons and baryons are able to stably recombine, forming atoms, mostly in the form of neutral hydrogen. The photons decouple from the baryons as the plasma becomes neutral, and perturbations no longer propagate as acoustic waves: the existing density pattern becomes "frozen". This snapshot of the density fluctuations is preserved in the CMB anisotropies and the imprint of baryon acoustic oscillations (BAO) observable today in large scale structure (Eisenstein & Hu 1998).
3- Recombination produces a largely neutral universe which is unobservable throughout most of the electromagnetic spectrum, an era sometimes referred to as the "Dark Ages". During this era, CDM begins gravitational collapse in overdense regions. Baryonic matter gravitationally collapses into these CDM halos, and "Cosmic Dawn" begins with the formation of the first radiation sources such as stars. Radiation from these objects reionizes the intergalactic medium.
4- Structure continues to grow and merge under the influence of gravity, forming a vast cosmic web of dark matter density. The abundance of luminous galaxies traces the statistics of the underlying matter density. Clusters of galaxies are the largest bound objects. Despite this reorganization, galaxies retain the BAO correlation length that was established in the era of the CMB.
5- As the universe continues to expand over time, the negative pressure associated with the cosmological constant (the form of dark energy in ΛCDM) increasingly dominates over opposing gravitational forces, and the expansion of the universe accelerates.”
The most commonly accepted version of the ΛCDM model is the “flat” one:
#Anselmi et al. What is flat ΛCDM, and may we choose it?. 2022.
https://arxiv.org/pdf/2207.06547.pdf
Quote: “The Universe is neither homogeneous nor isotropic, but it is close enough that we can reasonably approximate it as such on suitably large scales. The inflationary-Λ-Cold Dark Matter (ΛCDM) concordance cosmology builds on these assumptions to describe the origin and evolution of fluctuations. With standard assumptions about stress-energy sources, this system is specified by just seven phenomenological parameters, whose precise relations to underlying fundamental theories are complicated and may depend on details of those fields. Nevertheless, it is common practice to set the parameter that characterizes the spatial curvature, ΩK, exactly to zero. This parameter-fixed ΛCDM is awarded distinguished status as separate model, “flat ΛCDM.””
i.e. the one in which the spatial curvature is exactly zero. As explained above, an isotropic universe with zero curvature implies an infinite size. So if taken literally, the most common interpretation of the standard model of cosmology implies a strictly infinite universe.
– An infinite universe goes on, well, forever, with no border anywhere – also impossible to visualize. Wherever you look you will find more and more stuff in every possible direction. This kind of breaks our brain a bit for a few reasons: First of all, if the universe is infinite, is it also eternal and has been there forever? Was there a time where there was nothing everywhere and then there was something everywhere? Well we don’t know – but we have a lot of evidence for the big bang, so it really seems like the universe started at some point in the past. But wait – since the big bang the universe has been expanding. How can an infinite thing that is everywhere become even bigger? Cosmic expansion just means that the distance between galaxies is growing with time. Even an infinite space can become bigger. Welcome to the paradoxes of infinity.
You might wonder how the Universe can be infinite if it all started with the Big Bang. This underlies a common misconception about the Big Bang that it was an explosion which expanded into empty space. In the following you can find a nice explanation showing how the Big Bang can still be compatible with the universe being infinite.
#Edward L. Wright. How can the Universe be infinite if it was all concentrated into a point at the Big Bang? 1998
https://www.astro.ucla.edu/~wright/infpoint.html
Quote: “The Universe was not concentrated into a point at the time of the Big Bang. But the observable Universe was concentrated into a point. The distinction between the whole Universe and the part of it that we can see is important. In the figure below, two views of the Universe are shown: on the left for 1 Gyr after the Big Bang, and on the right the current Universe 13 Gyr after the Big Bang (assuming that the Hubble constant is Ho = 50 km/sec/Mpc and the Universe has the critical density.)
The size of the box in each view is 78 billion light years. The green circle on the the right is the part of the Universe that we can currently see. In the view on the left, this same part of the Universe is shown by the green circle, but now the green circle is a tiny fraction of the 78 billion light year box, and the box is an infinitesimal fraction of the whole Universe. If we go to smaller and smaller times since the Big Bang, the green circle shrinks to a point, but the 78 billion light year box is always full, and it is always an infinitesimal fraction of the infinite Universe.
Note that the black dots represent galaxies, and the galaxies do not expand even though the separation between galaxies grows with time.”
– Infinity gets much weirder though. As you travel with your spaceship in a straight line, you find new galaxies, stars and planets, new wonders, new weird stuff, probably new aliens and new lifeforms stranger than you could ever imagine. But after a long time, you might find the most special thing in the universe: Yourself. An exact copy of you watching this video right now.
This idea is largely based on the following publication by Max Tegmark:
#Max Tegmark. Parallel Universes. 2003.
https://arxiv.org/pdf/astro-ph/0302131.pdf
Quote: “Is there another copy of you reading this article, deciding to put it aside without finishing this sentence while you are reading on? A person living on a planet called Earth, with misty mountains, fertile fields and sprawling cities, in a solar system with eight other planets. The life of this person has been identical to yours in every respect – until now, that is, when your decision to read on signals that your two lives are diverging. You probably find this idea strange and implausible, and I must confess that this is my gut reaction too. Yet it looks like we will just have to live with it, since the simplest and most popular cosmological model today predicts that this person actually exists in a Galaxy about 10^10^29 meters from here. This does not even assume speculative modern physics, merely that space is infinite and rather uniformly filled with matter as indicated by recent astronomical observations. Your alter ego is simply a prediction of the so-called concordance model of cosmology, which agrees with all current observational evidence and is used as the basis for most calculations and simulations presented at cosmology conferences. In contrast, alternatives such as a fractal universe, a closed universe and a multiply connected universe have been seriously challenged by observations.”
The idea is also outlined in a popular science version by the author of the above publication:
#Max Tegmark. Parallel Universes. 2003.
https://space.mit.edu/home/tegmark/PDF/multiverse_sciam.pdf
– How can that be? Well, everything in existence is made of a finite amount of different particles. And a finite number of different particles can only be combined in a finite number of ways. That number may be so large that it feels like infinity to our brains – but it is not really. If you have finite options to build things, but infinite space that is full of things in all directions forever, then it makes sense that by pure chance, there will likely be repetition.
#Max Tegmark. Parallel Universes. 2003.
https://space.mit.edu/home/tegmark/PDF/multiverse_sciam.pdf
Quote: “Observers living in Level I parallel universes experience the same laws of physics as we do but with different initial conditions. According to current theories, processes early in the big bang spread matter around with a degree of randomness, generating all possible arrangements with nonzero probability. Cosmologists assume that our universe, with an almost uniform distribution of matter and initial density fluctuations of one part in 100,000, is a fairly typical one (at least among those that contain observers). ”
The following infographic from the article above nicely explains the concept.
– Unfortunately you will never meet. Because almost zero still means the chance is incredibly small. Earth as it exists right now is so unlikely, you’d have to travel incredibly far to find a second identical Earth: Some 10 to the 10 to the 29 – a 1 followed by 100 octillion zeros – times the size of the observable universe. So far that it kind of means forever far away.
#Max Tegmark. Parallel Universes. 2003.
https://arxiv.org/pdf/astro-ph/0302131.pdf
Quote: “Inflation in fact generates all possible initial conditions with non-zero probability, the most likely ones being almost uniform with fluctuations at the 10−5 level that are amplified by gravitational clustering to form galaxies, stars, planets and other structures. This means both that pretty much all imaginable matter configurations occur in some Hubble volume far away, and also that we should expect our own Hubble volume to be a fairly typical one — at least typical among those that contain observers. A crude estimate suggests that the closest identical copy of you is about ∼ 10^10^29m away. About ∼ 10^10^91 m away, there should be a sphere of radius 100 light-years identical to the one centered here, so all perceptions that we have during the next century will be identical to those of our counterparts over there. About ∼ 10^10^115 m away, there should be an entire Hubble volume identical to ours.∗∗
[...]
∗∗This is an extremely conservative estimate, simply counting all possible quantum states that a Hubble volume can have that are no hotter than 108K. 10^115 is roughly the number of protons that the Pauli exclusion principle would allow you to pack into a Hubble volume at this temperature (our own Hubble volume contains only about 1080 protons). Each of these 10°115 slots can be either occupied or unoccupied, giving N = 2^10^115 ∼ 10^10^115 possibilities, so the expected distance to the nearest identical Hubble volume is N 1/3 ∼ 10^10^115 Hubble radii ∼ 10^10^115 meters. Your nearest copy is likely to be much closer than 10^10^29 meters, since the planet formation and evolutionary processes that have tipped the odds in your favor are at work everywhere. There are probably at least 10^20 habitable planets in our own Hubble volume alone.”
The source above says that the closest copy of you would be at 10^10^29 meters, while we are saying that you would have to travel 10^10^29 times “the size of the observable universe”. Although both claims look very different, actually they are not that different in practice. The reason is that 10^10^29 is so amazingly huge that the units we use to measure it, be it meters, light-years, or the radius of the observable universe, become largely irrelevant. In other words: 10^10^29 meters is, to a very good degree of accuracy, the same distance as 10^10^29 times the size of the observable universe:
10100,000,000,000,000,000,000,000,000,000 ~ 10100,000,000,000,000,000,000,000,000,000 x 1026 m
= 10100,000,000,000,000,000,000,000,000,000+ 26 m
~ 10100,000,000,000,000,000,000,000,000,000 m
This is so because the difference between a meter and the size of the observable universe (a factor of 10^26) is extremely tiny compared to the prefactor 10^10^29.