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Quantum Fluctuations as Seeds of Large Scale Structure

December 16, 2013
By Ward Struyve, Rutgers.

On very large scales (over hundreds of megaparsecs) the universe appears to be homogeneous. This fact formed one of the original motivations for inflation theory. According to inflation theory, the very early universe went through a phase of accelerated expansion that stretched a tiny portion of space to the size of our entire observable universe, stretching initial inhomogeneities over unobservable distances. On smaller scales, the universe is far from homogeneous; one can identify all kinds of structures, such as stars, galaxies, clusters of galaxies etc. According to inflation theory, these structures are considered to originate from quantum vacuum fluctuations. The usual story is that during the inflationary period these quantum fluctuations grew to become classical inhomogeneities of the matter density. These inhomogeneities then gave rise to structures through gravitational clumping. The primordial quantum fluctuations are also the source of the temperature fluctuations of the microwave background radiation.

This explanation of the origin of structures is regarded as part of the success story of inflation theory. One aspect of the explanation is, however, problematical, namely how the quantum fluctuations became classical. The quantum fluctuations are described by a state (a wave function) which is homogeneous and the Schrödinger dynamics does not spoil this homogeneity. So how can we end up with classical fluctuations which are no longer homogeneous? According to standard quantum theory this could only happen through wave function collapse. Such a collapse is supposed to happen upon measurement. But the notion of measurement is rather vague and therefore it is ambiguous when exactly collapses happen. This is the notorious measurement problem. This problem is especially severe in the current context. Namely, in the early universe there are no measurement devices or observers which could cause a collapse. Moreover, structures such as measurement devices or observers (which are obviously inhomogeneous) are themselves supposed to be generated from the quantum fluctuations. In order to deal with the measurement problem and with the quantum-to-classical transition, we need to consider an alternative to standard quantum theory that is free of this problem. A number of such alternatives exist: Bohmian mechanics, many worlds, and collapse theories.

According to many worlds theories, the wave function always evolves according to Schrödinger’s equation and never collapses. In this case, the quantum-to-classical transition could be explained through decoherence. Decoherence could lead to the identification of worlds which are themselves not homogeneous. Various possible sources for decoherence have been considered. However, it seems fair to say that no conclusive results have been obtained yet, neither on the source of the decoherence, nor on the time-scales over which this could happen.

According to collapse theories, collapses happen objectively, at random times. A number of research groups have explored possible collapse models that could account for the quantum-to-classical transition in inflation theory, see e.g. [1-4]. Especially Daniel Sudarsky and collaborators have been very active on this topic, see e.g. [2] and references therein. (Sudarsky also wrote a detailed exposition of the problem of the quantum-to-classical transition [5].)

Bohmian mechanics provides a different possible explanation of the quantum-to-classical transition. According to Bohmian mechanics, the fluctuations are described by an actual field configuration. There is also a wave function, the same as in standard quantum theory, but it never collapses, i.e., it always evolves according to Schrödinger’s equation. Its role is to guide the actual field in its motion. So the equation of motion of the actual field depends on the wave function. Now, the actual field is typically inhomogeneous, even though the wave function may be homogeneous. This is how Bohmian mechanics explains the inhomogeneity of the primordial vacuum fluctuations. Furthermore, Bohmian mechanics allows for a simple analysis of the quantum-to-classical transition. The classical limit is obtained whenever the actual field approximately evolves according to the usual classical field equations. Together with Nelson Pinto-Neto and Grasiele Santos, I have looked into this and we found that the actual field starts behaving classically during the inflationary period, exactly at the expected time (see [6] and my talk at the Santa Cruz summer school, see also [7] for a similar discussion for bouncing models).


  1. A. Perez, H. Sahlmann, D. Sudarsky, On the quantum origin of the seeds of cosmic structure, Class.Quant.Grav. (2006); arXiv:gr-qc/0508100
  2. P. Cañate, P. Pearle and D. Sudarsky, CSL Wave Function Collapse Model as a Mechanism for the Emergence of Cosmological Asymmetries in Inflation , Phys. Rev. D (2013); arXiv:1211.3463
  3. J. Martin, V. Vennin and P. Peter, Cosmological Inflation and the Quantum Measurement Problem, Phys. Rev. D (2012); arXiv:1207.2086
  4. S. Das, K. Lochan, S. Sahu, T.P. Singh, Quantum to Classical Transition of Inflationary Perturbations – Continuous Spontaneous Localization as a Possible Mechanism, Phys. Rev. D (2013); arXiv:1304.5094
  5. D. Sudarsky, Shortcomings in the Understanding of Why Cosmological Perturbations Look Classical, Int. J. Mod. Phys. D (2011); arXiv:0906.0315
  6. N. Pinto-Neto, G.B. Santos and W. Struyve, Quantum-to-classical transition of primordial cosmological perturbations in de Broglie-Bohm quantum theory, Phys. Rev. D 85, 083506 (2012); arXiv:1110.1339
  7. N. Pinto-Neto, G.B. Santos and W. Struyve, Quantum-to-classical transition of primordial cosmological perturbations in de Broglie-Bohm quantum theory: the bouncing scenario, Phys. Rev. D (2013);arXiv:1309.2670
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