I was reading an interesting book from cosmomogist Viatcheslav Mukhanov Physical Foundations of Cosmology and I had a specific question about it:

It is usually said that energy conservation is difficult to define in cosmological scales since, for example, dark energy density appears to be constant in each point of space, so its total energy increases as the universe expands.

In section problem 8.10, Mukhanov mentions that cosmological perturbations can "violate" energy conservation and be excited (therefore, gaining energy) from the Hubble flow. Also,in this article Mukhanov says:

Since the primordial fluctuations were obtained as a result of the amplification of initially Gaussian quantum fluctuations by the external classical source (they acquired energy from the Hubble expansion), the resulting gravitational potential must be described by a Gaussian random field up to the second order corrections due to the nonlinearity of the Einstein equations

I had a question about this technical aspect:

Has this phenomenon been observed or experimentally verified? Can any types of perturbations (or anything else) actually gain energy fron the Hubble expansion?

  • $\begingroup$ It hasn't been directly experimentally verified. But, the predictions of inflation are reasonably consistent with observations. $\endgroup$
    – Virgo
    Nov 21, 2022 at 0:33

1 Answer 1


You can make a formal analogy between the perturbation equations during inflation, and a harmonic oscillator with a time dependent spring constant. For example, see Section 9 of https://arxiv.org/abs/gr-qc/9909001. In the latter case, there is a well known method (the "Bogoliubov transformation") that lets you compute particle creation due to the time dependent Hamiltonian. Then, the energy in the perturbations are extracted from the external driving force causing the spring constant to change with time.

The concept of energy in GR is murky though, and I would treat Mukhanov's statements as an analogy that works at the level of the equations for the linearized perturbations. In particular, there is no external driving force acting on an expanding Universe, or explicit time dependence in the field equations.

  • $\begingroup$ but has there been any observations or experiments that verified these kind of phenomena? Have we seen any system that actually gains energy and "gets excited" from expansion? @Andrew $\endgroup$
    – vengaq
    Nov 21, 2022 at 22:51
  • $\begingroup$ @vengaq Depends on what you mean. Given that inflation happened, the equations for the perturbations take the form of a harmonic oscillator with a time dependent spring constant. This is also called the Mathieu equation and is well known and studied. Among other applications, there are analog systems that implement particle creation type phenomena. If your question is whether or not inflation happened, that's up in the air. The observational evidence is consistent with inflation, but not conclusive. $\endgroup$
    – Andrew
    Nov 21, 2022 at 23:09
  • $\begingroup$ For what it's worth, I don't think the question of whether there are observations or experiments is really the right question here, actually. If inflation is right and the equations in the review I linked are correct or approximately correct, then particle creation occurred. The question is whether inflation really happened. Directly looking evidence of this specific effect might not be the most effective way to establish whether inflation happened observationally. $\endgroup$
    – Andrew
    Nov 21, 2022 at 23:11

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