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Why does inflation generate gravity waves?

  1. Oct 3, 2014 #1
    I've been trying to understand the whole BICEP story. Here's what I think I understand so far:
    • Inflation produces primordial gravitational waves.
    • During recombination, photons decouple from matter and the CMB is formed.
    • Propagating gravity waves stretch space in one direction and squeeze it in an orthogonal direction.
    • The presence of gravity waves during CMB formation would therefore redshift the wavelength of the photons in one direction and blueshift it in an orthogonal direction.
    • Thus, gravity waves cause quadrupole anisotropies in the CMB temperature.
    • Quadrupole temperature anisotropies lead to differing intensities of photons incident on any given electron from orthogonal directions, which in turn leads to the Thomson-scattered outgoing radiation having a linear polarization.
    • This is one of several mechanisms to produce polarization in the CMB, but only this mechanism produces polarization patterns with nonzero curl ("B-mode" polarization).
    Please feel free to correct, clarify, or expand any of these points, but my primary question is about the very first one. Why do we expect that inflation should generate primordial gravitational waves?
  2. jcsd
  3. Oct 6, 2014 #2


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    The generation of primordial gravity waves by inflation is a quantum process: the inflationary expansion excites the gravitational vacuum. This is a general phenomenon affecting quantum fields in inflating spacetimes: the rapidly expanding space results in the generation of quanta in a manner analogous to the creation of charged particles in a time-dependent electric field.
  4. Oct 6, 2014 #3
    Fair enough. But why is this unique to exponential expansion of spacetime, rather than any dynamical evolution of spacetime?
  5. Oct 6, 2014 #4


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  6. Oct 6, 2014 #5


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    Technically speaking, it is not unique. Particle production (and hence, for certain cases graviton production) is a generic phenomenon in curved space in general. However, the effect is typically tiny and unobservable, unless it is amplified by processes in the early universe. Here the exponential expansion amplifies the wave numbers through many efolds, and once theses modes reenter the horizon, they are visible in CMB searches.
  7. Oct 8, 2014 #6
    Not being a cosmologist but interested, my understanding is based on this paper:

    "... But scalar density perturbations are not the only perturbations that are generated. Gravitational waves are also produced (see SOM for details).

    When some background field gets stuck in a metastable state, energy gets stored in space, acting like a cosmological constant, producing a negative pressure and a positive energy density. Solving Einstein’s equations in such a background results in an accelerated expansion which quickly becomes exponential. The appropriate metric for such a universe is the de Sitter metric.

    In a de Sitter background all massless or light quantum fields will fluctuate with a magnitude which is proportional to the only dimensional parameter governing the expansion, the stored energy density, ρ, which by Einstein’s equations is related to the square of the Hubble expansion parameter, H at the time. This implies that a fluctuating background gravitational waves, which in the quantum theory correspond to massless quantum fields, will also be generated. As inflation stretches distances, their wavelength soon becomes much larger than the Hubble radius – they are driven outside the horizon – after which their amplitude remains fixed since no causal process can act over distances larger than the horizon.

    After inflation ends, these modes return inside the horizon as a stochastic background of gravitational waves with a power that is proportional to H2 (9–12). ... "

    [ http://arxiv.org/pdf/1004.2504.pdf }

    So now we know how primordial gravity waves (PGWs) behave. They are produced during inflation, gets expanded with the universe, eventually stuck at the cosmological horizon (when spacetime expands faster than signals can reach us), and released later.

    But how are PGWs generated? The Supporting Online Material (SOM) gives more detail:

    "It is well known that in a de Sitter background all massless or light quantum fields will fluctuate with a magnitude which is proportional to the only dimensional parameter governing the expansion, the stored energy density, ρ. Einstein’s Equations relate this energy density to the expansion rate H, so that, for every scalar, for modes with wavenumber k ≈ H,

    <(δφi)2> ≈ H2/2π.

    Remarkably, as first demonstrated by Grishchuk [1], when one linearizes Einstein’s equation for weak fields, then each of the two helicity states of a gravitational wave hi obeys precisely the equation for a massless, minimally coupled real scalar field, with a normalization factor of √16πG. Thus when the wavelength of a gravitational wave is of order k-1 = H-1, the Hubble radius, it fluctuates with a variance

    <(δhi)2> ≈ 8GH2.

    As inflation stretches distances, this wavelength soon becomes much larger than the Hubble radius – it is driven outside the horizon – after which its amplitude remains fixed since no causal process can act over distances larger than the horizon. After inflation ends, these modes return inside the horizon as a stochastic background of gravitational waves and begin to oscillate and redshift."

    [ http://www.sciencemag.org/content/suppl/2010/05/18/328.5981.989.DC1/Krauss.SOM.pdf ]

    So PGWs are generated because de Sitter spaces gets massless or light quantum fields fluctuating.

    As you can understand, the reference [1] becomes somewhat obscure for someone like me, not well versed in either general relativity or quantum field theory. I found teaching material that facilely describes the physics:

    "...Recall that perfect de Sitter space is defined by constant H, while inflation (or quasi-de Sitter space) is characterized by a small time-evolution of H, ...

    As we have just seen, the inflaton evolution φ(t) governs the energy density of the early universe ρ(t) and hence the end of inflation. Essentially, φ ̄(t) + δφ(t, x) plays the role of a local clock reading off the amount of inflationary expansion remaining. The space-dependent fluctuations δφ imply that different regions of space inflate by different amounts. Intuitively, microscopic clocks are quantum mechanical objects with necessarily some variance (by the uncertainty principle). In quantum theory, local fluctuations in ρ and hence ultimately in the CMB temperature T(t, x) are therefore unavoidable.

    [Discusses inflation as an action including general relativity, and the resulting fluctuation modes.]

    A priori, we have 5 scalar modes: 4 metric perturbations – δg00, δgii, δg0i ∼ ∂iB and δgij ∼ ∂ijH – and 1 scalar field perturbation δφ."

    [ http://www.damtp.cam.ac.uk/user/db275/TEACHING/ICTP/QFTinDS.pdf ]

    So inflation is a quasi-de Sitter space, which when expanding will get light quantum fields fluctuating, inclusive the spacetime metric, because fluctuations in the inflation field gives fluctuations in the inflation rate.

    Finally, in this context I recently learned that the quantum fluctuations that pertain to cosmological horizons (i.e. whether or not quantum measurements are valid) approach exotic physics:

    "Although it is possible to use the term “vacuum fluctuations” in a consistent manner, referring to well-defined phenomena, people are usually way too sloppy. Most physicists never think clearly about quantum measurements, so the term is widely misunderstood and should be avoided if possible. Maybe the most dangerous result of this is the confident, unexplained use of this term by experienced physicists talking to students; it has the awful effect of giving these student the impression that their inevitable confusion is normal and not indicative of deep misunderstanding[1].

    Here is everything you need to know:

    1. A measurement is best specified by a basis, not by an observable.[2]

    2. Real-life processes amplify microscopic phenomena to macroscopic scales all the time, thereby effectively performing a quantum measurement. (This includes inducing the implied wave-function collapse). These do not need to involve a physicist in a lab, but the basis being measured must be an orthogonal one.[3]

    3. “Quantum fluctuations” are when any measurement (whether involving a human or not) is made in a basis which doesn’t commute with the initial state of the system.

    4. A “vacuum fluctuation” is when the ground state of a system is measured in a basis that does not include the ground state; it’s merely a special case of a quantum fluctuation.

    [Goes on to introduce the model used to derive quantum fluctuations as measured, which is based on considering amplification.]

    [ http://blog.jessriedel.com/2014/08/...quantum-mechanics-part-2-vacuum-fluctuations/ ]

    Of quantum fluctuations that pertain to the particle physics vacuum, specifically the ones called vacuum fluctuations (the ground state fluctuations) can only be agreed on by going back to the minimal "shut up and calculate" quantum mechanics we learn at the university. See the link for further links to physics research on how the quantum physics can play out. (But that quantum fluctuations in inflation is safe.)
    Last edited: Oct 8, 2014
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