GR predicts accelerated expansion of spacetime should create gravitational wave, how?
How are gravitational waves generated?
Gravitational waves are disturbances in the curvature of spacetime, generated by accelerated masses, that propagate as waves outward from their source at the speed of light.
Gravitational waves transport energy as ...
The flatness problem
To understand this we need to use the Friedmann equation$$
Where $\Omega_M$ and $\Omega_\Lambda$ correspond to the fraction of matter and dark energy in the universe, k the curvature of the universe($k=-1,0,1$ corresponds to hyperbolic, flat, spherical geometry), $a$ the scale factor and $H^...
Quantum fields can exchange energy. So for example when an electron and a positron annihilate the energy that was in the electron/positron field is transferred to the photon field. The result is that one electron and one positron disappear and two photons are created. The fields themselves are still both present - only the energy in the fields has changed.
Within the Standard model of particle physics, the most general experimentally verified model of fundamental physics (excluding gravity), there are quantum fields, such as the electron-positron field, that are truly fundamental. Other fundamental fields within the Standard model include the tau, muon, muon- tau- and electron-neutrino fields, quark fields, ...
Tong says he is using $k=0$, so one can easily use
\mathrm d\Sigma^2=\mathrm dx^2+\mathrm dy^2+\mathrm dz^2$$
instead of using radial coordinates. And hence your FLRW metric is of the form,
This clearly gives the desired form of