Vector dark matter from inflationary fluctuations

Peter W. Graham, Jeremy Mardon, and Surjeet Rajendran

Phys. Rev. D 93, 103520 – Published 18 May 2016

Abstract

We calculate the production of a massive vector boson by quantum fluctuations during inflation. This gives a novel dark-matter production mechanism quite distinct from misalignment or thermal production. While scalars and tensors are typically produced with a nearly scale-invariant spectrum, surprisingly the vector is produced with a power spectrum peaked at intermediate wavelengths. Thus dangerous, long-wavelength, isocurvature perturbations are suppressed. Further, at long wavelengths the vector inherits the usual adiabatic, nearly scale-invariant perturbations of the inflaton, allowing it to be a good dark-matter candidate. The final abundance can be calculated precisely from the mass and the Hubble scale of inflation, $H_{I}$ . Saturating the dark-matter abundance we find a prediction for the mass $m \approx 10^{- 5} eV \times (10^{14} GeV / H_{I})^{4}$ . High-scale inflation, potentially observable in the cosmic microwave background, motivates an exciting mass range for recently proposed direct-detection experiments for hidden photon dark matter. Such experiments may be able to reconstruct the distinctive, peaked power spectrum, verifying that the dark matter was produced by quantum fluctuations during inflation and providing a direct measurement of the scale of inflation. Thus a detection would not only be the discovery of dark matter, it would also provide an unexpected probe of inflation itself.

Received 27 January 2016

DOI:https://doi.org/10.1103/PhysRevD.93.103520

Physics Subject Headings (PhySH)

Cosmology Dark matter Inflation Quantum fields in curved spacetime

Gravitation, Cosmology & Astrophysics

Authors & Affiliations

Peter W. Graham¹, Jeremy Mardon¹, and Surjeet Rajendran²

¹Stanford Institute for Theoretical Physics, Department of Physics, Stanford University, Stanford, California 94305, USA
²Berkeley Center for Theoretical Physics, Department of Physics, University of California, Berkeley, California 94720, USA

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Issue

Vol. 93, Iss. 10 — 15 May 2016

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Images

Figure 1
Cosmological evolution of length scales, and nature of the longitudinal mode in different regimes. The line labeled “horizon” shows the comoving horizon size $1 / a H$ , which shrinks during inflation and grows after reheating. The line labeled “Compton wavelength” shows the comoving Compton wavelength of the vector, $1 / a m$ . Modes of the vector field maintain fixed comoving wave vector $k$ , evolving along straight lines from left to right. In the pale-blue-shaded region, where modes are relativistic ( $m ≪ k / a$ ), the longitudinal mode behaves identically to a massless Nambu-Goldstone boson. In the pale-brown-shaded region, where Hubble damping is not important ( $H ≪ m$ ), the longitudinal mode behaves identically to a free massive scalar. In the red triangle between these regions, the longitudinal mode has a new behavior unlike any scalar. Modes crossing the tip of this region reenter the horizon just as they become nonrelativistic—their wave vector defines the special scale $k_{*}$ .
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Figure 2
Evolution of the energy density in longitudinal modes, from inflationary production through to matter radiation equality. As expected from Fig. 1, the evolution is the same as it would be for a massive scalar in all regions except the red triangle labeled “Vector regime.” In that regime, the energy stored in a scalar would be constant, whereas for the vector it damps as $a^{- 2}$ . This damping suppresses large-scale isocurvature modes, allowing the produced vector abundance to make up the dark matter. This abundance is dominated by modes of comoving size $1 / k_{*}$ indicated by the dashed line. The details of reheating do not affect these modes, as long at it occurs before they reenter the horizon.
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Figure 3
Primordial power spectrum of the amplitude of a massive vector field’s longitudinal modes, produced purely by inflationary fluctuations. The spectrum is shown at a time just three $e$ -folds after $H = m$ , and shorter-wavelength modes (on the right of the plot) are still relativistic. At later times the $k^{- 1}$ scaling will continue all the way to the right of the plot.
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Figure 4
(Lower plot) Primordial density power spectrum of a massive vector produced by inflationary fluctuations. The spectrum is shown at a time when all modes covered are nonrelativistic, but before self-gravitation of the modes is important (this should be approximately valid until matter-radiation equality). The peaked part of the spectrum at large $k$ is the isocurvature power produced by the inflationary fluctuations of the field itself. The also flat part of the spectrum at small $k$ corresponds to the usual adiabatic fluctuations, which are imprinted onto the field from inflaton fluctuations. The values $m = 10^{- 5} eV$ and $H_{I} \approx 10^{14} GeV$ were used here, corresponding to $k_{*} \approx 1400 {pc}^{- 1}$ . Because the density fluctuations are $O (1)$ for $k \sim k_{*}$ , fluctuations on shorter scales (in the gray region) are not well described by the power spectrum alone (see Fig. 5). These higher $k$ modes are also expected to be affected by quantum pressure later in their evolution. For comparison, the top plot shows the power spectrum of the field amplitude (Fig. 3).
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Figure 5
Typical variation of the field and energy density along a line. This is a random example, generated according to the power spectra shown in Fig. 4, i.e. before structure formation occurs. (Note the $y$ and $z$ components of the field are not shown, and so the lower plot is not exactly the square of the top plot.) As can be seen, the density is dominated by “lumps” of size $L \sim π / k_{*}$ . On longer scales, overdensities are caused by random clustering of these lumps. On smaller scales, there is complicated substructure both within and between the main lumps. If the vector is discovered in direct-detection experiments, its present-day field profile will be mapped out as the experiments sweep though the dark-matter halo.
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Figure 6
Prospects for direct detection of a cosmological vector abundance through kinetic mixing with the photon. Direct-detection experiments are sensitive to the combination $ϵ \sqrt{ρ_{γ^{'}}}$ shown on the $y$ axis ( $m_{γ^{'}}$ is the mass of the new vector, and $ρ_{γ^{'}}$ its cosmic abundance—this plot does not assume the vector makes up all the dark matter). In the vertical pink band, high-scale inflation ( $10^{13} GeV ≲ H_{I} ≲ 10^{14} GeV$ ) generates the full dark-matter abundance for the vector. Masses to the right of this band require lower values of $H_{I}$ , while to the left of this band the vector can constitute an interesting dark-matter subcomponent. The blue dashed line shows the projected reach of a recently proposed search with resonant $L C$ oscillators [19], with the blue dotted line showing the potential improvement by multiplexing in the high-frequency regime. Tan-shaded regions are excluded due to cosmological disturbances caused by the relic vectors, while green regions are excluded by null results from the Xenon10/100 and ADMX dark-matter searches. Gray-blue-shaded regions indicate constraints on the existence of the vector independent of its cosmological abundance (these are truly constraints on $ϵ$ , so to plot these here it was assumed that the vector makes up the largest possible fraction of the dark matter that is consistent with inflationary production). See Sec. 5 for more details.
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