# Why does space expansion not expand matter?

I have looked at other questions on this site (e.g. "why does space expansion affect matter") but can't find the answer I am looking for.

So here is my question: One often hears talk of space expanding when we talk about the speed of galaxies relative to ours. Why, if space is expanding, does matter not also expand? If a circle is drawn on balloon (2d plane), and the balloon expands, then the circle also expands. If matter is an object with 3 spatial dimensions, then when those 3 dimensions expand, so should the object.

If that was the case, we wouldn't see the universe as expanding at all, because we would be expanding (spatially) with it.

I have a few potential answers for this, which raise their own problems:

1. Fundamental particles are 'point sized' objects. They cannot expand because they do not have spatial dimension to begin with. The problem with this is that while the particles would not expand, the space between them would, leading to a point where the 3 non-gravity forces would no longer hold matter together due to distance

2. Fundamental particles are curled up in additional dimensions a la string theory. These dimensions are not expanding. Same problems as 1, with the added problem of being a bit unsatisfying.

3. The answer seems to be (from Marek in the previous question) that the gravitational force is so much weaker than the other forces that large (macro) objects move apart, but small (micro) objects stay together. However, this simple explanation seems to imply that expansion of space is a 'force' that can be overcome by a greater one. That doesn't sound right to me.

• Because space expansion means field expansion, field due to matter is much more expanded than matter itself. – Anubhav Goel Feb 11 '16 at 16:41
• Landau Lifshitz showed how energy+gravitational energy is conserved. Increasing say the size of an atom would actually raise it's energy whereas the expanding universe seems to take energy from the stress energy tensor. Thus if anything, one might expect an atom to shrink in radius as the universe grows! haha – R. Rankin Aug 9 '16 at 6:44
• I'm not sure, it would be nice if someone could correct me, but when people say the universe is expanding, aren't they saying the boundaries of space is increasing? – Adamawesome4 Aug 18 '16 at 0:48
• @Adamawesome4 maybe you've learned since you posted comment above, but my understanding, and most will agree, is the universe is unbounded and that the expansion of space is occurring everywhere. Others might argue differently. It's only the observable universe that's bounded. Bounded by the speed of light and our ability to observe the earliest luminous objects. – docscience Mar 9 '17 at 15:14
• As stated, the question asserts a falsehood and then asks why it's true. It is not true that cosmological expansion produces no strain on matter. It is true that the strain is much too small to measure. – Ben Crowell Jun 6 '17 at 23:35

Let's talk about the balloon first because it provides a pretty good model for the expanding universe.

It's true that if you draw a big circle then it will quickly expand as you blow into the balloon. Actually, the apparent speed with which two of the points on the circle in a distance $$D$$ of each other would move relative to each other will be $$v = H_0 D$$ where $$H_0$$ is the speed the balloon itself is expanding. This simple relation is known as Hubble's law and $$H_0$$ is the famous Hubble constant. The moral of this story is that the expansion effect is dependent on the distance between objects and really only apparent for the space-time on the biggest scales.

Still, this is only part of the full picture because even on small distances objects should expand (just slower). Let us consider galaxies for the moment. According to wikipedia, $$H_0 \approx 70\, {\rm km \cdot s^{-1} \cdot {Mpc}^{-1}}$$ so for Milky way which has a diameter of $$D \approx 30\, {\rm kPc}$$ this would give $$v \approx 2\,{\rm km \cdot s^{-1}}$$. You can see that the effect is not terribly big but the given enough time, our galaxy should grow. But it doesn't.

To understand why, we have to remember that space expansion isn't the only important thing that happens in our universe. There are other forces like electromagnetism. But most importantly, we have forgotten about good old Newtonian gravity that holds big massive objects together.

You see, when equations of space-time expansion are derived, nothing of the above is taken into account because all of it is negligible on the macroscopic scale. One assumes that universe is a homogenous fluid where microscopic fluid particles are the size of the galaxies (it takes some getting used to to think about galaxies as being microscopic). So it shouldn't be surprising that this model doesn't tell us anything about the stability of galaxies; not to mention planets, houses or tables. And conversely, when investigating stability of objects you don't really need to account for space-time expansion unless you get to the scale of galaxies and even there the effect isn't that big.

• "given enough time, our galaxy should grow. But it doesn't" - source? And after that, your answer is (rephrased): "With the assumptions we made to derive expansion, we assumed that galaxies are points" - So you've only said "our derivation doesn't say anything about finite size galaxies", which is interesting, but the question @SoulmanZ asked is unanswered: Can we explain why galaxies are not expanding? I'd like an answer myself. – doublefelix Nov 3 '15 at 1:20
• @user3141592 From an energy perspective (landau lifshitz gravity pseudotensor plus tensor energy conserved), an expanding universe seems to remove energy from local systems (such as the em wave) from this perspective, one can see that expanding a galaxy increases it's energy, thus one might expect (counterintuitively) rather that a galaxy would experience an inward force of contraction. – R. Rankin Aug 9 '16 at 6:51
• @doublefelix Galaxies aren't expanding because they are held together by local gravitational forces that aren't accounted for in the simple cosmological model. This is, in fact, contained in Marek's answer. – AGML Sep 28 '16 at 19:15
• you are being misquoted in the question. – anna v Oct 13 '17 at 6:05
• @doublefelix For more insight into why galaxies are not expanding, see my answer to this question. – Andrew Steane Nov 17 '18 at 16:53

If the question is interpreted as why don't atoms and other bound systems expand the answer is that the general expansion of space cannot do continuous work against the electromagnetic force that holds an atom together or any other force that holds a bound system together.

However the accelerating expansion of the universe can exert a small "constant" negative force between the electrons and nucleus and make the atom very very slightly bigger than it would have been in a non-accelerating expanding universe. But in the current best theory of dark energy which is that it is a constant vacuum energy, this effect will be constant and the atoms have already expanded as much as they ever would.

There are theoretical speculations of an acceleration of the accelerating expansion of the universe where this effect increases with time such that eventually in an exponential way the universe ends in a big rip where atoms and eventually nuclei would be ripped apart.

On another website, I answered a question about whether we could extract energy from the expanding universe and this is the answer I wrote which I think will be helpful in understanding this issue:

The universe is expanding at 74 km/sec/Mpc (Mpc is a mega parsec which is 3.26 million light years). So let's take two heavy objects and place them far from any galaxy cluster or other influence and space them just one parsec apart (3.26 light years).  Then they will effectively be moving apart at 7.4 cm/sec.  Now imagine that your monomolecular filament rope between the objects puts a force on the objects that will decelerate the objects.  Then during the time that they are decelerating you can extract work from the objects. That work per second comes from the force the rope is exerting being applied over the 7.4 cm/sec that the objects are moving apart. However, once the force causes their relative velocity to drop to 0, you won't be able to get any more energy from the objects since they are no longer moving apart.  There will still be a constant force on your rope but you need to have a force applied over a distance to get work.

Now this is all from just the "Big Bang" expansion of space.  Once the rope's force has gotten their relative velocity to zero, the two objects are like a gravitational bound system and it will stop "expanding".  However, in addition to the "standard" expansion of space, we now know that there is dark energy which is causing an accelerating expansion of the universe.  This means that the two objects are not just "moving" apart at constant 7.4 cm/sec but that this velocity is actually increasing with time.  So if you setup your rope such that the force it is exerting on the objects results in an deceleration that is slightly smaller than this cosmic acceleration, you can extract work continuously and indefinitely.  Unfortunately, I have not been able to convert the dark energy measurements into units of acceleration in this particular case of objects at one parsec.  I suspect it is a small number but current estimates are that it is definitely positive.  Note that if your rope exerts more force that causes a deceleration larger than the cosmic acceleration then the objects will eventually stop moving apart and the work you can extract will drop to zero again.

Note that from just the normal expansion of the universe you can only extract a finite total amount of energy, but that with the accelerated expansion you can extract a small but positive amount of energy per second forever.  However, your rope needs to get longer and longer with time (at the rate of 7.4 cm/sec, in this example), so, as they say TANSTAFL (there ain't no such thing as a free lunch). The rope needs to get longer because you have to have your very small force applied to continuously moving objects to get work done.  Since it will take continuous energy to make a continuously lengthening rope, and you cannot win this battle by starting with objects that are further apart since then the rope is lengthening at an even faster rate than the 7.4 cm/sec of this example.  You can increase the energy per second you extract by making the objects more massive, but then the force on the rope increases so you need to make a thicker rope.

The bottom line is that I think this free energy project is impractical, even though it is theoretically possible.  The problem that needs to be solved is the energy cost of the continuously lengthening rope.

• If the question is interpreted as why don't atoms and other bound systems expand the answer is that the general expansion of space cannot do continuous work against the electromagnetic force that holds an atom together or any other force that holds a bound system together. This is wrong. For example, the solar system does expand due to cosmological expansion, but the effect is undetectably small. See Cooperstock, Faraoni, and Vollick, "The influence of the cosmological expansion on local systems," arxiv.org/abs/astro-ph/9803097v1 – Ben Crowell Jun 6 '17 at 23:28
• However the accelerating expansion of the universe can exert a small "constant" negative force between the electrons and nucleus and make the atom very very slightly bigger than it would have been in a non-accelerating expanding universe. This is also wrong. The strain on a bound system is proportional to $(d/dt)(\ddot{a}/a)$, where $a(t)$ is the cosmological scale factor. This quantity is not constant in realistic models, and can be nonzero even if the cosmological constant is zero. Also, it vanishes identically in a cosmology that consists only of dark energy (=cosmological constant). – Ben Crowell Jun 6 '17 at 23:31
• @BenCrowell I believe your answer misses the main point. The statement "the solar system does expand ... but the effect is small" seems to suggest that we can apply the cosmological expansion without adjustment to things like solar systems. This, I believe, is wrong (see my answer to this question). This is not to say there is no influence at all on the solar system, but the influence is in competition with the Sun's gravity and the latter dominates. – Andrew Steane Nov 17 '18 at 17:06

This was written for a question that closed during my composition of this. The question is how does the CC effect atomic physics, by Ashton.

Dark energy has the mass-energy equivalent of a proton every 1-10 cubic meters. That is a pretty diffuse energy. An atom is on the scale of $10^{-8}$cm in length or has a volume of about $10^{-30}m^3$. So about that proportion of a proton’s mass-energy worth of dark energy acts on an atom, or perturbs its atomic levels. That is about $10^{-21}$ Gev or $10^{-12}$ev. That is very small.

Now your question is not entirely without merit. Some very sensitive atomic measurements get atomic level splittings to within $10^{-6}$ev. I will not say for certain, but these atomic-quantum optics people can be quite clever on the bench. It is not entirely unimaginable that with squeezed states, entangled squeezed states of photons and electrons and so forth that this might be measured. If there is an EM response due to a level splitting the wave would be around the sub Hertzian range.

The interaction Hamiltonian for the cosmological constant would be an inverted harmonic oscillator potential $H_{cc}~=~\Lambda r^2/3$. Some analysis for avoided crossings of energy levels and states and the rest might not be an unreasonable thing to work on.

• Although Marek's answer is good, it is not answering the question, even if itself poses the same question, arguing: “You can see that the effect is not terribly big but the given enough time, our galaxy should grow. But it doesn't.”. I feel it doesn' t answer that. On the other hand, I feel that this answer gives something: that is that we are still expecting experiments to conclude. Maybe, B. Crowell, if there is such a result you could please post an update. Thank you. – Constantine Black May 28 '16 at 17:33
• @ConstantineBlack fair comment; see my answer. – Andrew Steane Nov 17 '18 at 19:04

Quick fun analogy:

If we think of the expansion of space as a sheet stretching, particles of matter move away from each other. Hooray, as explained several times before.

Extending this to 3D, we're basically stretching objects at a very slow rate. 1.62038964 × 10^-17 m/s / meter, to be precise. Thus, a typical person is stretched at about 3x10^-17 meters per second. I couldn't find any good estimates on the ideal spring constant of the human body but you'd probably be stretched by within 10 orders of magnitude of 10^-15 meters.

(Due to the electromagnetic force being so strong, your tensile strength is very high.)

Thus, with Hubble expansion you're about 0.0000000000001% taller. Congrats!

• so density of matter is decreasing? – Muhammad Umer Mar 20 at 16:08
• is there a unit of measurement that determines density of an object independent of space. so meter would still say you are n meters. but as you say person is 0.0000000000001% taller. Any unit that would show that? – Muhammad Umer Mar 20 at 16:12

Let's go back to the expanding balloon, which is a very fine analogy even if it's lacking a dimension. Suppose there is an ant on the balloon's surface. As time goes by it will realise that its six legs are moving further apart. As it begins to feel uncomfortable it will shuffle them in to adjust. If there are two ants talking together they will drift apart, so they will move back towards each other. Similarly electrons in atoms, and atoms in solids, will be expanded apart but move back together.

My understanding of the current theory is that galaxies are moving away from each other at an accelerated rate due to dark energy repulsion--creating an expanding universe.

However, within galaxies, dark matter keeps the galaxies themselves together--so much so that the outer rim of the galaxy spins at the same rate of the inner rim--meaning there must be an awful lot of matter holding the galaxies themselves together.

Well, according to the answer given to this question: At which point of the universe $R_{\mu \nu}=0$ if there is a source of gravitation (point mass), the universe expansion refers actually to matter expansion:

"So Einstein's equations in vacuum mean exactly that: that $G_{\mu \nu} = 8\pi T_{\mu \nu} = 0$ in a region without mass-energy. That is far from saying that there is no gravity, just as it would be silly to say there is no electric field in the exterior of a charged ball."

So if this is true that the Einstein's equation:

$$R_{\mu\nu}-\frac {1}{2}g_{\mu\nu}R+g_{\mu\nu}\Lambda=\frac {8\pi G}{c^4}T_{\mu\nu}$$

(if not null) is restricted to matter only (because if the $T_{\mu\nu}=0$ then the left hand side of the equation also vanishes), and since it contains the cosmological constant $\Lambda$, the logical conclusion is that matter must be expanding.

Summing up: the vacuum Einstein's equation requires that either (1) the universe under consideration must always be all void of matter, or (2) matter is expanding (or (3) Einstein's equation is wrong). Since (1) is considered not true, and (3) is not claimed by mainstream physics, then we have only (2) left - i.e. matter is expanding.

P.S. You can also formulate it differently - gravity ($T_{\mu\nu} \neq0$) is the source of universe expansion within matter.

EDIT: It should be obvious, that $\Lambda$ is not just a force that can be overcome (and therefore not apparent). It is on the side of the equation that shows actual curvature.

EDIT2: As I wrote elsewhere, (possible) expansion would keep relative values (of various "constants, planck's constant, electron mass, speed of light, elementary charge, and permitivity of free space") intact. It's like with the time dilatation and length contraction in SR - for the moving frame nothing changes, and local observer wouldn't notice. The difference being here that the outside observer also wouldn't notice, because the change would be taking place everywhere, at the same time, and with equal acceleration.

Answer to this question is only by understanding and not proof. If the Universe is expanding then I say Yes, to that the Matter is also expanding. Explanation for this is, all the Matter is expanding means, even the scale to measure the Matter is also expanding. Consider a small Example: A rectangular wooden block is expanding, Measure the initial dimensions of the block, say they are x,y,z after expansion of only the wooden block, measure again, then it is x+a,y+b,z+c. But what if the scale is also expanding at the same rate as of the wooden block, then measure it any time you will get its dimensions as x,y,z only and not x+a,y+b,z+c. Coming back to reality, the same thing is happenning in the Universe. i.e. Universe is expanding along with it everything else is also expanding. Only thing is that we cannot notice the small things. We can notice for larger things such as Galaxies, Planets, Stars etc.

• In Woody Allen's "Annie Hall" this question comes up; the answer? "Brooklyn is not expanding!" The reason is that it is spacetime that is expanding; objects that have any kind of binding energy are being held together by forces. Photons expand, so would cosmic sound waves. But Brooklyn is held together by motherly love. See the scene at youtube.com/watch?v=5U1-OmAICpU – Peter Diehr Feb 13 '16 at 20:40
• Hi. @PeterDiehr and Sushant23 . But why Brooklyn doesn't? If on the small scale we agree on the posted answer, that we cannot see the expansion since everything is expanding, then why see it on the big scale? Is it because it expands faster on the big scales but on the small the speed is the same for all objects? Thanks. – Constantine Black May 28 '16 at 17:37
• @ConstantineBlack: the expansion is equivalent to a very weak force - local binding forces always overwhelm it: atoms, molecules, people (eg, not a valid excuse for the waistline!), planets, solar systems, and galaxies. But you can see it over very great distances - hence the red shift due to cosmic expansion is a good proxy for distance, though other proxies are used to set the distance scale. See Hubble's Law – Peter Diehr May 28 '16 at 17:46
• @PeterDiehr Thanks for the fast response. I find it conceptually wrong to admit that expansion is a force such that you can use an equation like Newton's or any argument at least saying that: the total force on the object is expansion + other_forces so that the result in small scales is not-expansion. It' s more reasonable to either say that experiments say this or that or that in small scales, the expansion rate is the same for all objects( even inside the galaxy??) so that we don't observe it. Am I losing something here? Thanks. – Constantine Black May 28 '16 at 18:01
• @ConstantineBlack: make this a new question, and we'll provide an answer that is hopefully correct at all levels. – Peter Diehr May 28 '16 at 18:24

I am a non-physicist who uses cognitive reasoning not mathematical, so I like non-physicist answers because they make more sense.

You have two boats, one 20 feet off shore and one 100 feet off shore. The boat closest to the shore is getting pulled into shore by large waves. The boat 100 feet out is getting pulled away from the shore because of the ocean's current.

Think of the ocean as space, and the boats as Galaxies. Because the two are free-floating, over time, their distance will grow further apart.

Now, tie the two boats together via a rope. Think of the rope as Gravity.

Now, unless the current and the waves get strong enough to snap the rope, they will never drift apart.

Since Space is not just expanding, but expanding at an ever-accelerating rate; eventually, in trillions of years, it will accelerate faster than the weak, strong, electromagnetic, and gravitational forces can counter, and they will actually fly apart.

• This is kinda insulting to physicists, isn't it? – sammy gerbil Jan 31 '18 at 3:56

The answer to your question, in the case of the size of things like solar systems and galaxies, is contained in a simple observation in General Relativity called Birkoff's theorem which I will explain below. The main result is that, to first approximation, things like solar systems and galaxies do not participate in the cosmological expansion. Galaxies get further apart from one another while each staying roughly of fixed size. And furthermore, the reason for this, and the way it comes about, is because the local spacetime around isolated objects such as stars is static, i.e. not expanding, and therefore there is no reason why physical objects located in that spacetime, such as planet Earth, and tables, chairs, atoms and so on, should have any inclination to expand at all. In the rubber balloon analogy, the situation is that the rubber of the balloon is not stretching at all in the vicinity of any given star, but in between these patches the rubber is stretching.

It is not too hard to sketch the underlying maths for the above, which the rest of this answer briefly presents.

We consider a simple case to get the main idea. The simple case is the vacuum around a spherically symmetric distribution of matter. This gives the Schwarzschild solution. This Schwarzschild metric is the formula for the way spacetime behaves; it in turn tells us the possible orbits of planets and things like that. The important point is that this solution is unique, meaning that the spherical symmetry is enough to fix the solution for the spacetime metric sufficiently that the only parameter remaining is the mass of the central body. This important observation is called Birkoff's theorem. It has some simple consequences. One consequence is that if the body at the centre oscillates in the purely radial direction while maintaining its spherical shape, no change at all happens to the spacetime outside the body (no gravitational waves). Another consequence is that if the spherically symmetric vacuum region is itself a hole in a larger spherically symmetric fluid, then again the spacetime in the region is unchanged. And this remains true even if that surrounding fluid is expanding outwards. This is the crucial observation for us.

Now consider the solar system. The local spacetime and its curvature is dominated by the effect of the Sun. The universe at the largest scale is like a continuous fluid, equally dense in all directions. As that fluid expands, the spacetime around the Sun remains unaffected, so the planets follow the same orbits at the same radii, and furthermore each planet is situated in a static spacetime, one which is not expanding locally.

Of course the spherical symmetry will not be perfect, but this argument gives the main story for gravitationally bound systems at scales up to those where you can no longer approximate the dominant contribution to the local gravity as coming from a central body with vacuum around it. It also ignores the cosmological constant.

The expansion of space is not an inexorable fact which nothing can resist or oppose, it is more like the net result of gravitation and initial conditions at the largest scales. In any given local region, pretty much any force can prevent the expansion locally.

I should add that I am not an expert on cosmology; I am merely a physics professor reporting what he has picked up from books such as "Relativity: Special, General, and Cosmological" by Wolfgang Rindler, and the famous "Gravitation" by Misner, Thorne, Wheeler. I would be happy to be corrected if I have misremembered this argument.

• I don't think you are correct. For example, the Schwarzschild solution to GR is only a solution to the GR equation with cosmological constant = 0. On the other hand, the solution to GR for an empty universe with a non-zero cosmological constant has a metric with a scale factor a(t) multiplying the spatial part which is an exponential function of time. The correct Schwarzschild solution with a non-zero cosmological constant would have to exhibit this behavior at large distances from the black hole, and thus it will also be different at all distances from the BH. – FrankH Nov 18 '18 at 20:07
• Thanks; I acknowledge that my answer left out the cosmological constant. I still find it useful as an example of the fact that local gravitation can 'win' in a competition with cosmological effects, which are the net result of initial conditions and the distribution on the largest scale. – Andrew Steane Nov 20 '18 at 10:00

To accept that the space is expanding you have to admit that the ruler, made of atoms, is invariant, i.e. it has always the same length, and no one has provided a convincing argument of this. The space expantion relies in the belief that this is a fact. If atoms are expanding at the same rate we were not able to measure any expantion. If, on the contrary, the atoms are shrinking thru time we can measure a space expantion without any de facto happening to the space. I dont know why the space is expanding except that we measure it. The matter may be contracting because the gravitoelectric fields have energy that is expanding and are sourced by the particles since matter is born and, obviously, we are not able to measure this fact in the lab. Un discussed model, out of academia, un-peer-reviewed, is 'A self-similar model of the Universe unveils the nature of dark energy' that does not need any Dark Energy, Inflation, etc.

Concluding: to the question 'Why does space expansion not expand matter?': if matter expanded at the same rate no one would be able to measure any variation. The measuring act is to obtain a ratio between two quantities and both the numerator or the denominator (the standard) can change to obtain a specific measured value. But the standard is based in the 'atom' properties (in the first link and in this recent one The physical basis of natural units and truly fundamental constants) that we presume invariables.

Both links provide an inside about units, but the first link is much more interesting because it provides an insight on the rationale of the measuring act.

Usually it is accepted that there are no effects on Solar system scale of the space expansion, but recently it was reported, i.e. measured, that the SUN-Earth distance is increasing much more than expected:
Experimental measurement of growth patterns on fossil corals: Secular variation in ancient Earth-Sun distances by Weijia Zhang , 2010 (behind paywall)

Experimental results indicate a special expansion with an average expansion coefficient of $0.57H_0$

Secular increase of astronomical unit from analysis of the major planet motions, and its interpretation by G. A. Krasinsky, 2004 (behind paywall)

measured \$\fraq{dAU,dt}=15 \pm 4 m/cy .. at present there is no satisfactory explanation of the detected secular increase of AU

The author reviewed all developments in lunar system research and paleontology since 1963, found three contradictions between different methods: ... This means that the ancient Earth is closer to the Sun. .. The revolution period of Earth is increasing, recorded by NASA. The semimajor axis of Earth is increasing, recorded by JPL.

in the page 13 we find a table with the measured values of length of a sidereal year (increasing) after 1900.

The increasing distance is deduced in the presented model, as seen at eq. 35), pag 10 of the preliminary paper of 2002 (arxiv) by Alfredo Oliveira
A relativistic time variation of matter/space fits both local and cosmic data

So, to the question 'Why does space expansion not expand matter?'
the answer is because 'the space expansion is the result of the evanescence of matter' i.e. matter is shrinking.

As an exercise: imagine you are siting in the midlle of a room and you start to see the walls moving away of you. When wake of that dream, or hallucination if you are doped, how would you describe it? :
I was sh-shr-shri-shrinking, as Alice in the Wonderland naturally did, or that the house is getting bigger-BIGGER ?

• I will upvote since this post tries to answer the question, in contrast with other answers here. Also, I will upvote because no one is explaining the downvotes. – Constantine Black May 28 '16 at 17:48

Short Answer: At the current time, there is not enough dark energy in the universe to affect the distance between the particles. This may not always be the case though...

An interesting paper was submitted in 2003 by Robert Caldwell of Dartmouth College and his team, on the Big Rip theory.

A universe dominated by phantom energy is an accelerating universe, expanding at an ever-increasing rate. However, this implies that the size of the observable universe is continually shrinking; the distance to the edge of the observable universe which is moving away at the speed of light from any point moves ever closer. When the size of the observable universe becomes smaller than any particular structure, no interaction by any of the fundamental forces can occur between the most remote parts of the structure. When these interactions become impossible, the structure is "ripped apart". The model implies that after a finite time there will be a final singularity, called the "Big Rip", in which all distances diverge to infinite values.

$$t_{rip}-t_0\approx{2\over3|1+w|H_0\sqrt{1-\Omega_m}}$$

In their paper, the authors consider a hypothetical example with $$w$$ = −1.5, $$H_0$$ (Hubble's Constant) = 70 km/s/Mpc, and $$\Omega_m$$ (density of matter in the universe) = 0.3

Using these figures in their calculations they estimate that in 21.94B years from now, the force of gravity will cease holding the galaxies together.

59.1 years later solar systems would cease to be held together by gravity.

Then perhaps days or hours before the big rip, planets and matter would break down and finally at the big rip, subatomic particles would be ripped apart.

There is debate about what the value of w is. Evidence indicates w to be very close to −1 in our universe, which makes w the dominating term in the equation. The closer that w is to −1, the closer the denominator is to zero and the further the Big Rip is in the future. If w were exactly equal to −1, the Big Rip could not happen, regardless of the values of $$H_0$$ or $$\Omega_m$$.

According to the latest cosmological data available, the uncertainties are still too large to discriminate among the three cases w < −1, w = −1, and w > −1. ("WMAP 9 Year Mission Results". wmap.gsfc.nasa.gov.) & (Allen, S. W.; Rapetti, D. A.; Schmidt, R. W.; Ebeling, H.; Morris, R. G.; Fabian, A. C. (2008). "Improved constraints on dark energy from Chandra X-ray observations of the largest relaxed galaxy clusters". Monthly Notices of the Royal Astronomical Society.)

If the paper holds out as true, then the universe will only last another 22B years or so... We will have to wait to see.

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