In a word, if you are sitting on the Earth, if I'm not mistaken you are experiencing Time Dilation compared to being in deep solar system space. Due to the mass of the Earth.
However. We're all sitting in a galaxy, and it weighs a lot.
(Perhaps 2e40 kg? I have no idea if issues like "dark matter" radically effect this.)
Can anyone quantify,
a) my Time Dilation sitting on Earth – versus in deeper space
b) my Time Dilation sitting in the Milky Way – versus deeper in space
But wait.
We're all in the Universe. It weighs a lot.
In fact ….. are we all experiencing Time Dilation because of this being-in-the-rather-heavy-Universe affair?
c) if so, how much? Thanks.
Important ancillary question a'): I've never quite found the answer to this: we experience Time Dilation due to planet Earth. Now, if you are at the center of the Earth, you experience no gravitational pull, but, do you experience the Time Dilation??
Best Answer
In the weak field limit, which applies to all the cases you've described, the difference between the time rates for two observers with a Newtonian gravitational potential energy difference of $\Delta\Phi$ is given by:
$$ \frac{\Delta t_1}{\Delta t_2} = \sqrt{1 - \frac{2\Delta\Phi}{c^2}} \tag{1} $$
Note that the time dilation is related to the gravitational potential energy not the gravitational force - you'll see why this matters when we come to your ancillary question.
So to answer your questions (a) and (b) just work out what the difference in gravitational potential energy between your two observers and plug it into equation (1). I'll leave this as an exercise for the reader.
The answer to question (c) is a bit subtle, because the key feature of an FLRW universe is that it is homogenous i.e. the gravitational potential is the same everywhere in the universe. That means whatever pair of observers you choose $\Delta\Phi$ is always zero and therefore the time dilation is always zero. You can't ask about the time dilation relative to an observer outside the universe because an FLRW universe has no outside.
Now on to the ancilliary question: at the centre of the Earth the gravitational force is indeed zero, but the gravitational potential is not. As you move from infinity to the surface of the earth $\Phi(r)$ decreases (i.e. gets more negative) as $r^{-1}$, but as you move below the surface to the centre $\Phi(r)$ carries on decreasing only not as fast. So compared to an observer at infinity the time dilation at the centre of the Earth is greater than the time dilation at the surface.
Actually I've just spotted this has already been addressed in the question Gravitational time dilation at the earth's center.