# Why is acceleration of expansion of space unexplained?

I didn't know how to better phrase the question so here we go.

It says that the farther we see from our galaxy the faster the other object seems to be going away (accelerating).

My common sense tells me to expect this due to following reason.

If space is expanding at every point equally then farther objects would naturally move away faster as there are more points of space between you and a far object than points of space between you and near object.

So as we move farther from an object the amount of space (say no of space points of length dx) increases hence there is more space expanding away from each other than for a near object and that is why far away object would appear to be accelerating.

But obviously its not that simple as i said or i may be completely wrong as scientists seem to consider this acceleration to non understood phenomenon like dark energy

So What am i missing here?

• Are you confusing the expansion of the universe with the acceleration of the expansion rate? The speed by which distant galaxies move away from us is very nearly linear in their distance (Hubble's law), with the ratio of speed over distance the Hubble constant $H$. There is no acceleration in that yet. However, $H$ is not exactly constant, but increasing, i.e. the expansion of the universe is accelerating. – Walter Jun 7 '16 at 8:34

The FLRW energy equation for the motion of test masses in the universe is $$\left(\frac{\dot a}{a}\right)^2 = \frac{8\pi G\rho}{3}.$$ the scale factor for space is $a$ and its time derivative is $\dot a$. I derived this from Newtonian dynamics. The density of mass $\rho$ for the case of a quantum vacuum energy level is constant. I now replace this with energy density with $\rho\rightarrow~\rho/c^2$. This leads to the dynamical equation $$a(t) = a_0e^{t\sqrt{\frac{8\pi G\rho}{3c^2}}}$$ which is an exponential expansion. The cosmological constant is then $\Lambda~=\frac{8\pi G\rho}{3c^2}$, which is determined by the quantum vacuum energy.

The question is then what is the density of mass-energy, or more exactly what is the nature of the quantum vacuum. The vacuum is filled with virtual quanta. A pendulum sitting vertically will be a motionless plumb in classical mechanics. However, quantum mechanics informs us there is an uncertainty in its position and momentum $\Delta x\Delta p~\simeq~\hbar$. This means it can fluctuate about its vertical plumb position. The Hamiltonian for the harmonic oscillator transitions from the classical to quantum form as $$H = \frac{1}{2m}p^2 + \frac{k}{2}x^2~\rightarrow~\frac{\omega}{2}(a^\dagger a + aa^\dagger)$$ This can by put in the more standard form with the quantum commutator $[a,~a^\dagger]~=~1$, and so $aa^\dagger = a^\dagger a + \frac{1}{2}$. The quantum Hamiltonian is then $$H = \omega a^\dagger a + \frac{1}{2}\omega,$$ where this last term is a zero point energy for the fluctuation of the harmonic oscillator. For a quantum field there is a big summation over Hamiltonians for each frequency $$H = \sum_n\omega_n \left(a_n^\dagger a_n + \frac{1}{2}\right)$$ The sum over this residual energy or zero point energy is summed up to the Planck energy. This leads to a huge vacuum energy. For most quantum work this term is eliminated, often by something called normal ordering. However this leads to a huge energy that can't be ignored in cosmology. The cosmological constant is $\Lambda \simeq 10^{-53}cm^{-2}$. The sum over these zero point energy leads to and expected $\Lambda \simeq 10^{67}cm^{-2}$. The difference is $120$ orders of magnitude off.

There is a lot of work on this subject. Much of it focuses on gauge fluxes through wrapped D-branes. Some progress has been made in reducing the expected vacuum energy. However, so far it has not been possible to show the very small vacuum energy we know from cosmology exists. It also can't be zero! It is my thinking that this reflects our lack of understanding in quantum gravity. We know some things about quantum gravity, but we really do not have a complete theory of it. This vacuum energy that propels cosmological expansion, called dark energy, is then not fully understood.

• This is a most informative answer. May I take the courage an ask you if you would like to read my post below. I would really like your thought and help if you find I am wrong. Feel free of-course to downvote if you thing what I write is beyond any reason. Thank you. – Constantine Black Jun 7 '16 at 13:10
• Uhh. Compared to the level of math in your answer i am an illiterate person. But i get your point the rate of expansion of every single point is accelerating right? Then why do scientists use red shift as its evidence? For all we know the red shift can simply due to acceleration of galaxies moving away from us instead of acceleration of expansion of space – user1062760 Jun 8 '16 at 6:30

I would argue that the expanding of space cannot and should not be understood adding space into space, nothingness into nothingness. We have no way of observing the space itself as a reason like the one you presented likes. Things seem to get away from us through and the observed mechanism is called redshift, which, in close distances(inside let's say the local cluster) can be explain equally through Doppler and Einstein(Granitational) redshifts. To large distances, the observed homogeneity and isotropy of space can be modeled by the FRW solution of General Relativity. Even so, choosing different reference frames may give you some differences on the way you will interpret your results. Also, the way to measure the speed of expansion(note that speed is not a well defined thing in GR because our modeling manifold is curved, so vector cannot be compared when they don't belong in a flat part of spacetime, such as a Minkowski frame) is via the Hubble law. But, theoretically the Hubble law should be valid for the above FRW situation and not in general, so we should take that speed as a way of picturing the so called expansion with more familiarity in contrast with the redshift sense which is what we measure in part. Also, understanding that this speed is referring to object and not space itself is useful since, the way to objectify the space expansion itself is via the Hubble constant which has units of $1 \over t$ , simply no unit of our well known speed.

In the end, one has to be strict with the model he uses, so that he/she must always understand the principles of this particular model. General Relativity doesn't allow us of talking about the spacetime field, that is the metric, disconnected from energy and matter. Such an approach isn't valid. The system spacetime and energy is a dynamical one and one can only discuss the expansion properly by understanding that each such behavior is the dynamical result or effect of the two interacting constantly, and so the expansion is the finale of an ever-going process which depends not only on how the universe looks now but as it was before a time interval allowing interactions.

If someone is to talk about spacetime as an independent object he should explain how he does it and with what reason.

Hope this helps.

• Sorry, but I have to downvote this. Things don't "get away from us through a mechanism called redshift". Things get away due to expansion, and as a consequence, light from these things are redshifted. Also, the expansion rate is well-defined despite your discussion of GR. Saying that "you have to be careful which model you use" doesn't really have anything to do with this. – pela Jun 7 '16 at 10:56
• @pela There is no problem from me in downvoting. If nothing else, maybe there is something for me here to learn. 1) As for your first objection, it is my fault for how I have written my answer. I don't mean that redshift is something more that an observation, I mean by mechanism the mathematical procedure for deducing this experimental result. 2) As for your second statement, can you prove/explain what you mean by "Things get away due to expansion, .. the expansion rate is well-defined.. " and "you have to be careful which model you use" doesn't really have anything to do with this." Thanks. – Constantine Black Jun 7 '16 at 12:43
• arXiv:astro-ph/0310808v2 and arxiv.org/abs/0707.0380v1, @pela Also, an example of an alternative theory for expansion, for why the model matters: nbcnews.com/id/44690771/ns/technology_and_science-science/… , if you are interested. – Constantine Black Jun 7 '16 at 12:47
• By my second statement, I mean: the Universe expands. This pulls "things", or galaxies, away from each other. Light traveling through an expanding space is redshifted. We observe this redshift, and deduce the expansion. You edited, but your edit seems a bit confusing; there are some words that should've been deleted, I think. As for my other comments, I see now what you mean. I'll remove my downvote, although I disagree in the alternative explanations of the expansion. However, I still think the question refers to the "maintream" interpretation of expansion. – pela Jun 7 '16 at 13:41
• The first paper you link to (Davis 04) is a great paper. The others… I don't know… – pela Jun 7 '16 at 13:41

The comment by Walter is on the right track: The "acceleration" does not refer to the fact that recession speed increases with distance, because this is just a consequence of space expanding everywhere. This is why we measure the expansion in km/s per megaparsec. Today, the expansion rate (the Hubble constant) is $H_0 \simeq 70\,\mathrm{km}\,\mathrm{s}^{-1}\,\mathrm{Mpc}^{-1}$.

But there's a small mistake in Walter's comment: The Hubble doesn't increase in value, but decreases. This can be seen from the Friedmann equation, which governs the expansion: $$\frac{H^2}{H_0^2} = \frac{\Omega_\mathrm{r}}{a^4} + \frac{\Omega_\mathrm{M}}{a^3} + \frac{\Omega_k}{a^2} + \Omega_\mathrm{\Lambda}.$$ Here, $a$ is the scale factor of the Universe, which gives the relative distances between objects (galaxies), and the $\Omega$'s give the relative densities of the constituents of the Universe. As the Universe increases (i.e. $a$ increases), radiation, matter, and curvature (subscripts r, M, and k, respectively) are "diluted", but dark energy ($\Lambda$) is constant and will eventually dominate, such that $H$ asymptotically reaches $H(a) = H_0\sqrt{\Omega_\Lambda}$ $\simeq 60\,\mathrm{km}\,\mathrm{s}^{-1}\,\mathrm{Mpc}^{-1}$. This means that, at any time you measure the recession velocity of a galaxy 1 Mpc away, it will recede at $60\,\mathrm{km}\,\mathrm{s}^{-1}\,\mathrm{Mpc}^{-1}$. But the velocity of a given galaxy (e.g. one that is 1 Mpc away right now) will always increase, since in the future it will be farther away than 1 Mpc. If it weren't for dark energy, the expansion velocity would go toward zero (an eventually be negative if $\Omega_\mathrm{M}\gt1$, i.e. the Universe would collapse).

The scale factor as a function of the age of the Universe can be obtained by integrating the Friedmann equation. Now if the Universe contained only regular matter, the scale factor would increase in time according to $a(t) \propto t^{2/3}$, i.e. proportionally to time to some factor less than unity. But in a $\Lambda$-dominated Universe — which is where we are going — the solution turns out to be $a(t) \propto e^{H_0 t}$, i.e. an exponential growth.

This is what is meant by "an accelerating Universe".

We have to be careful what we mean when we say "moving away". Imagine a grid in which we are at the origin and there are light sources located at each of the grid intersections. If the grid stays as it is while the light sources accelerate away from us, their light will appear to be redshifted. If you set up the accelerations such that everything moves away from everything else, though, you will end up with a situation that is isotropic from our point of view (i.e. looks the same in all directions), but is not homogeneous because we are at a "special" location.

If, on the other hand, you let each of the light sources stay at its current coordinate location and instead stretch the grid in all directions, then the expansion will be both homogeneous and isotropic. In this scenario, you could almost say that the light sources are not "moving", but the distance between them is still increasing.

Cosmologists generally tend to assume homogeneity and isotropy, which leads to the conclusion that space itself is expanding rather than all objects "moving" away from one another. But expanding space needs an explanation, since normal matter doesn't do that. That is where dark energy comes in; it is more or less defined as the thing that causes space itself to expand.