Is the accelerated expansion of the universe consistent with conservation of energy?

Question

If energy can neither be created nor destroyed, how come the universe is expanding at an accelerating rate? Where does this energy come from?

Answer 1

Quoting Sean Carrol's article linked by Симон Тыран, which makes the case for energy not being conserved:

Having said all that, it would be irresponsible of me not to mention that plenty of experts in cosmology or GR would not put it in these terms. We all agree on the science; there are just divergent views on what words to attach to the science. In particular, a lot of folks would want to say “energy is conserved in general relativity, it’s just that you have to include the energy of the gravitational field along with the energy of matter and radiation and so on.” Which seems pretty sensible at face value.

Furthermore,

there’s energy in the gravitational field, but it’s negative, so it exactly cancels the energy you think is being gained in the matter fields

However,

there is no such thing as the density of gravitational energy

General relativity makes no distinction between gravity and inertia, and gravitational energy is like the centrifugal potential: It's necessary for energy conservation to hold, but it can be made to vanish (at least pointwise) in particular frames of reference and thus does not come with an associated energy density.

For Friedmann cosmology, the energy budget is basically given by the first Friedmann equation $$ \left( \frac {\dot a}a \right)^2 = \frac 13 \left( 8\pi\rho + \Lambda \right) $$ For an expanding volume of lateral length $d = d_0 a$ containing dust of mass $M$ and a photon gas of energy $U_0$ at $t = t_0$, we arrive at the following total energy by re-arraging the terms after multiplying the equation by $\frac 3{8\pi} d^3$: $$ M + \frac {U_0}a + \frac 1{8\pi} \Lambda d^3 - \frac 3{8\pi} d \dot d^2 = 0 $$ The last term is the contribution of the gravitational field that becomes more and more negative to cancel the increasing amount of dark energy given by the third term.

Answer 2

Long story short: conservation of energy only holds locally where you can assume a static spacetime. On large scales the expansion of the universe gets relevant, so energy is said not to be conserved universally since the amount of dark energy per volume stays the same while the volume increases, see Sean Carroll's article, from which I quote:

The famous Sean Carroll wrote:

"Energy is not conserved in general relativity. The point is pretty simple: back when you thought energy was conserved, there was a reason why you thought that, namely time-translation invariance. A fancy way of saying “the background on which particles and forces evolve, as well as the dynamical rules governing their motions, are fixed, not changing with time.” But in general relativity that’s simply no longer true.

Einstein tells us that space and time are dynamical, and in particular that they can evolve with time. When the space through which particles move is changing, the total energy of those particles is not conserved."

Answer 3

Whether energy is or isn't conserved in an expanding universe is a somewhat vexed issue. On the one hand you have an experienced physicist claiming that energy is conserved, and on the other hand you have an experienced physicist claiming that energy is not conserved.

The problem is that accounting for energy in general relativity is a complicated business. Philip Gibbs and Luboš Motl are accounting for the energy in different ways, and reaching different conclusions as a result. Who, if either, of them is correct is not clear - possibly both are.

Conservation of energy is derived from a symmetry called time shift symmetry. See more about this in my answer to On what basis do we trust Conservation of Energy?. Annoyingly Googling has failed to find a good article on time shift symmetry - maybe I should write one. What is clear is that in an expanding universe time shift symmetry does not apply so conservation of energy does not necessarily apply either.