# Is there a distance from a gravitational source where the influence of gravity and dark energy are balanced out?

While gravity is a force that attracts objects with mass, dark energy (or, alternatively, the accelerated expansion of the universe) is not.

However, I have found numerous articles, forums, questions in the stack exchange network... where people seem to say that if you get further from a gravitational source (e.g. a galaxy) there would be a point where the influence of gravity (attractive) and dark energy (repulsive) would be balanced out (sometimes even using the term "force").

But again, dark energy is not a force (as said in here and also here). So what is happening here? Is there such a point? If not, then, why are there so many people saying that there is? This is confusing...

PS: Examples of people mentioning a point where there is a balance between Dark Energy and gravity:

https://imgur.io/IMUhq

https://arxiv.org/abs/1206.1433

https://astronomy.stackexchange.com/questions/54826/dark-energy-affecting-the-ejection-and-infall-of-material-in-galaxies

### Dark energy exerts a repulsive gravitational influence.

Dark energy is not normally considered to be itself a force. Instead it is a substance or form of energy that is uniformly distributed throughout the universe, and (like everything else) it exerts a gravitational influence, but its gravitational influence is repulsive. This verbage matches how dark energy is treated mathematically in general relativity. Whereas matter is modeled as a (nearly) pressureless perfect fluid and radiation is modeled as a fluid with pressure $$p=\rho/3$$ (where $$\rho$$ is the energy density and I take $$c=1$$), dark energy is a fluid with $$p=-\rho$$. Due to how both pressure and energy density contribute to gravity, a fluid with $$p<-\rho/3$$ exerts gravitational repulsion inside it.

### At what distance does dark energy's gravitational repulsion balance a mass's gravitational attraction?

For a mass $$M$$ and uniformly distributed dark energy with density $$\rho_\Lambda$$, the gravitational attraction and repulsion balance when the distance $$r$$ from the mass is such that the average enclosed mass density is twice the dark energy density, i.e. $$M/(4\pi r^3/3)=2\rho_\Lambda$$. This leads to $$r = \left(\frac{3}{8\pi}\frac{M}{\rho_\Lambda}\right)^{1/3}. \tag{1}$$ The density comparison comes from the second Friedmann equation, according to which the gravitational acceleration of a spherical shell is proportional to $$\rho+3p$$ (taking $$c=1$$), where $$\rho$$ and $$p$$ are the enclosed density and pressure, respectively. Matter has zero pressure while dark energy has $$p=-\rho_\Lambda$$, so the gravitational acceleration is proportional to $$\rho_m-2\rho_\Lambda$$, where $$\rho_m$$ is the matter density.

If dark energy is a cosmological constant, its value is $$\Lambda=8\pi G\rho_\Lambda$$, so an alternative expression is $$r = \left(\frac{3GM}{\Lambda}\right)^{1/3}.\tag{2}$$ This can alternatively be seen to follow directly from the Newtonian limit of de Sitter space. In the Newtonian limit, the gravitational acceleration induced by dark energy at position $$\vec r$$ with respect to any freely falling observer (at $$\vec r=0$$) is $$\ddot{\vec{r}}=\frac{\Lambda}{3}\vec r=\frac{8\pi G}{3}\rho_\Lambda \vec r.$$

### In our universe

The density of dark energy is about $$9\times 10^{10}$$ M$$_\odot$$ Mpc$$^{-3}$$, so from equation (1), the forces of the matter and dark energy balance when $$r\simeq 1.1~\mathrm{Mpc}\left(\frac{M}{10^{12}~\mathrm{M}_\odot}\right)^{1/3}.\tag{3}$$ Notice that $$10^{12}~\mathrm{M}_\odot$$ is about the mass of the Milky Way, and it's within a factor of a few of the mass of the Local Group. Thus, in the asymptotic future (assuming dark energy is a cosmological constant), only a sphere about 1 Mpc in radius is expected to remain bound to us. For reference, that's comparable to the present distance to the Andromeda galaxy (which is about 0.75 Mpc away).

• With verbalization substituted for the math, Brian Greene reaches a comparably dismal conclusion in the 'Duration and Impermanence' chapter of his 2020 book titled "Until the End of Time", so I'm afraid Sten is right. The dreariness of it seems to have swung Greene, more recently, toward cosmological models of a multiverse of black holes each causally separated from the others, perhaps because black holes containing black holes ad infinitum might allow an infinitely larger amount of lebensraum. Oct 2 at 18:52
• You said: "Dark energy is not normally considered to be itself a force." It should be noted that in general relativity gravity isn't a force either. Oct 3 at 7:06