# Gravitational effect of the distant universe

When we observe at close to the cosmological horizon (as close as we currently can) we see the universe as a much younger, denser place. As we feel the gravitational forces from these objects distant, will we measure a tidal force increasing the distance between an object less distant than the cosmological horizon and ourselves? Or for that matter between any two objects at different distances from earth? If so is it great enough to account for the accelerating expansion of the universe?

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Sorry, but I can't manage to figure out what you're getting at. The gravitational effect on us from very distant objects is very weak -- inverse square law! Certainly there's no sense in which the gravitational pull of distant objects causes the expansion to accelerate. On the contrary, ordinary matter (as opposed to things like vacuum energy) cause the expansion to decelerate. – Ted Bunn Feb 22 '11 at 21:39
If we look to the distant universe like it is a shell of dense material surrounding us- the gravitational field will be zero at the center (earth) but pointing away from the center elsewhere and greater the further you get from the center right (objects distant from earth)? While I suspect that it is weak- I just wonder if anyone has done the calculation- or if I am somehow very mistaken somewhere along the line... – user1567 Feb 22 '11 at 21:55
I feel this is related to: physics.stackexchange.com/questions/3061/… – user1567 Feb 22 '11 at 22:01
@jaskey13: note that the universe is not just a shell. If anything it's actually more like a uniform density sphere. – David Z Feb 22 '11 at 22:11
@jaskey13 : Are you familiar with Mach's principle? It was a discussion pre-General Relativity about whether the distant Universe could have any gravitational or Inertial effect on isolated matter. It didnt have anything to do with accelerating Universes though. – Roy Simpson Feb 22 '11 at 22:30

I think the question here asks what the role of distant gravity fields are. This can be addressed with Newtonian gravity. The motion of a body in Newtonian gravity has the total energy $$E~=~\frac{1}{2}mv^2~-~\frac{GMm}{r}.$$ We may use this in cosmology for the scale factor $a$. For a coordinate system with coordinates $x_i$, the distance between objects with these coordinates expands by a scale factor. The relevant coordinate is the radius, and for $r~=~0$ our origin any radial distance then scales according to $r~=~ar_0$. The radial velocity of objects away from the origin is then $dr/dt~=~(da/dt)r_0$. We may then write this equation with $r_0~=~1$ (normalized) as $$E~=~\frac{1}{2}\Big(\frac{da}{dt}\Big)^2~-~\frac{GM}{a}$$ So far this has assumed a homogeneous distribution of matter, and the motion of this mass appears isotropic. So this mass $M$ corresponds to everything “out there” out to some radius given by the scale factor a. So this mass is determined by a constant density $\rho$, and $M~=~\rho\times vol$ for the volume $vol~=~(4\pi/3)r^3$ so that $$E~=~\frac{1}{2}{\dot a}^2~-~\frac{4\pi G\rho a^2}{3}~\rightarrow~\frac{\dot a}{a}~=~\frac{8\pi G\rho}{3}~+~\frac{2E}{a^2}.$$ We may then simplify all of this. The energy $E$ is set to zero, where we assume the space if flat. This is the $k~=~0$ case in the Friedman-Lemaitre-Robertson-Walker (FLRW) spacetime. We have derived a central part of it without using general relativity!
So this equation describes the motion of particles, or galaxies, for a coordinate origin. That origin is chosen to be where we are, and we do indeed observe galaxies moving outwards. Hubble empirically “derived” or demonstrate an equation. The equation says that a galaxy at a distance $d$ from the Earth is moving outwards with a velocity $v~=~Hd$. The distance out was measured with Cepheid variable luminosity and the velocity measured by Doppler shifting. The Hubble factor (constant on a spatial surface) is then $H~=~{\dot a}/a$, which is our scale factor “velocity.”
Now let us consider matter that is moving or expanding. The density will scale as $a^3$ and we assume that the scale factor evolves with some power of time, $a(t)~=~bt^n$, for $b$ a constant. We put this into both sides of the FLRW equation and we find that the scale factor evolves as $a(t)~=~bt^{-2/3}$. The calculation is a bit more complicated, but a universe with radiation, such as the time prior to the CMB limit, the universe evolved as $a(t)~=~bt^{-1/2}$. This is the primary role of matter in the universe upon the structure and dynamics of the universe.