I was reading a book on Topological defects of the very early universe as an example of the fundamental groups, and they say that "in an expanding homogeneous and isotropic universe, the background energy density is a function of time", I do not know what do they mean by "background", are they referring to the spacetime.

Could somebody please explain this to me? I am not into physics but I like general relativity.


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    $\begingroup$ There's no background, there's only background energy (aka. vacuum energy). $\endgroup$ – Hagen von Eitzen Feb 14 '14 at 16:27

It sounds like they are talking about perturbing around a cosmological solution to Einstein's equations. The starting point in cosmology is an assumption that the universe is spatially homogeneous and isotropic, so the metric can depend only on an overall, time dependent scale factor. Then to make things interesting and match what realistically happens, you generally consider small perturbations to this solution. In this case, you call the "background" the solution you are perturbing around, i.e. the exact homogeneous isotropic universe.

So when they say "background energy density", it sounds like they are referring to the background $00$-component of the matter stress tensor.

To be slightly more explicit, you assume a metric of the form $$ds^2 = -dt^2+a^2(t)(\gamma_{ij}dx^i dx^j)$$ where $\gamma_{ij}$ is the metric for a maximally symmetric 3 dimensional space. The Einstein equations are $$R_{ab}-\frac12 R g_{ab} = 8\pi G T_{ab}.$$ The $00$-component of this equation is $$3 H^2 = 8\pi G \rho$$ where $H\equiv\dfrac{\dot{a}}{a}$ is the Hubble parameter, and $\rho\equiv T_{00}$ is the energy density. This $\rho$ is the background energy density they are referring to.

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