Designing dark energy afterglow experiments

Chameleon fields, which are scalar field dark energy candidates, can evade fifth force constraints by becoming massive in high-density regions. However, this property allows chameleon particles to be trapped inside a vacuum chamber with dense walls. Afterglow experiments constrain photon-coupled chameleon fields by attempting to produce and trap chameleon particles inside such a vacuum chamber, from which they will emit an afterglow as they regenerate photons. Here we discuss several theoretical and systematic effects underlying the design and analysis of the GammeV and CHASE afterglow experiments. We consider chameleon particle interactions with photons, Fermions, and other chameleon particles, as well as with macroscopic magnetic fields and matter. The afterglow signal in each experiment is predicted, and its sensitivity to various properties of the experimental apparatus is studied. Finally, we use CHASE data to exclude a wide range of photon-coupled chameleon dark energy models.

Figure 1

An idealized afterglow experiment. (a): Production phase. Photons stream through chamber via entrance and exit windows, occasionally oscillating into chameleon particles which are trapped inside. (a): Afterglow phase. The photon source is turned off and a detector is uncovered. Chameleon particles oscillate back into photons, which emerge from the chamber and reach the detector.

Figure 2

Minimum value of the matter coupling ${\beta }_{\mathrm{m}}$ required for the electron cloud and the proton, respectively, to have thin shells, in the case of a chameleon dark energy $V(\varphi )=M_{\Lambda }^4+M_{\Lambda }^{4-n}|\varphi {|}^n$. (Top): $n>2$. (Bottom): $n<0$.

Figure 3

Chameleon field and effective mass for a proton, approximated as a uniform-density ball of mass $m_{\mathrm{prot}}=938\mathrm{MeV}/{\mathrm{c}}^2$ and radius $r_{\mathrm{P}}=0.83\mathrm{fm}$. The potential $V(\varphi )$ is that of a chameleon dark energy with $n=-1$ and a matter coupling ${\beta }_{\mathrm{m}}={10}^{12}$. A vertical dotted line shows the surface of the proton.

Figure 4

Fragmentation processes in ${\varphi }^4$ chameleon models.

Figure 5

Chameleon mass in matter, modelled as a lattice of spherical nuclei in a homogeneous gas of electrons. In each plot, the horizontal dotted line shows $m=\omega =2.33\mathrm{eV}$, the chameleon energy in CHASE. (Top) Approximate chameleon mass. (Bottom) Critical mass of Refs. 21, 22, the mass above which the inhomogeneous nature of the matter becomes apparent to the chameleon field.

Figure 6

Chameleon reflection phase ${\xi }_V$ vs distance from wall, with ${\beta }_{\mathrm{m}}={10}^{12}$ and ${\rho }_{\mathrm{mat}}=1\mathrm{g}/{\mathrm{cm}}^3$. (Top) $n=-1$ chameleon dark energy (9), $V(\varphi )=M_{\Lambda }^4(1+M_{\Lambda }/\varphi )$. (Bottom) potential $V(\varphi )=M_{\Lambda }^4\exp (M_{\Lambda }/\varphi )$. In both cases, the dotted line shows ${\xi }_V=\pi /3$, the expected phase in the $x\to \infty$ limit for $n=-1$ chameleon dark energy.

Figure 7

Norms and arguments of the complex amplitude reflectivities $A_{\mathrm{ref},S}$ and $A_{\mathrm{ref},P}$. Colored lines assume the parameters $n_1=1.6$, ${\tilde{n}}_1=3.2$, and $f_{\mathrm{vis}}=0.82$ appropriate to the CHASE chamber walls. Dotted lines show the reflectivities in the thin-skin case; the two amplitudes are equal and real. See the text for a discussion of the conventions used here. In brief, our $S$ and $P$ directions differ from the standard definition, and our phases are normalized so that ${\xi }_S={\xi }_P$ at normal incidence.

Figure 8

Particle in an afterglow experiment. The magnetic field region is shaded. Inside this region, where chameleon-photon oscillation takes place, the particle trajectory is drawn as a dashed path. Next to each dashed segment is shown the contribution of that segment to the total photon amplitude. From 34.

Figure 9

(Top) GammeV apparatus and (bottom) CHASE apparatus (side views, not to scale). Note that CHASE has no section 2 (${\ell }_3=0$) and has glass windows dividing the $B_0$ region into three partitions. Values for these parameters are given in Table .

Figure 10

Two-point afterglow and decay rates for $B_0=5\mathrm{Tesla}$, ${\beta }_{\gamma }={10}^{12}$, and ${\xi }_V=0$, for the CHASE afterglow experiment (unpartitioned).

Figure 11

Photon amplitude vs $z$ for a magnetic field $B_0=5\mathrm{Tesla}$ with falloff length $\Delta z_{\mathrm{f}}=5\mathrm{cm}$. A coupling ${\beta }_{\gamma }={10}^{12}$ is assumed. Arrows at the right show expected oscillation probabilities for a top hat magnetic field. Low-mass chameleons effectively see a top hat field while chameleons with $m_{\mathrm{eff}}\gtrsim 0.01\mathrm{eV}$ see a nearly adiabatic evolution of the field.

Figure 12

Oscillation probability vs chameleon mass for the real magnetic field $B_{\tanh }(z)$ and the top hat approximation. ${\beta }_{\gamma }={10}^{12}$ is assumed.

Figure 13

Photon amplitude vs $z$ for a magnetic field $B_0=5\mathrm{Tesla}$ with falloff length $\Delta z_{\mathrm{f}}=5\mathrm{cm}$, assuming photon coupling ${\beta }_{\gamma }={10}^{12}$. Vertical dotted lines show the locations of the window and the beginning of the $B_0$ region. Arrows at the right show expected oscillation probabilities for a top hat magnetic field, assuming that a measurement is made at the window. (Top) Oscillation probability for a particle beginning at the entrance window and measured at the interior window. (Bottom) oscillation probability for a second particle beginning on the interior window is added to the probability due to the first particle.

Figure 14

(Top) oscillation probability vs chameleon mass for top hat and $\tanh $ magnetic fields, assuming two interior windows and ${\beta }_{\gamma }={10}^{12}$. (Bottom) ratio of $\tanh $ and top hat oscillation probabilities.

Figure 15

Afterglow signal vs time for various magnetic fields (labelled in plots) and photon couplings (upper right corners of plots). Approximate observation windows for the short data runs (thin, light blue shaded regions) and long data runs (both shaded regions) are shown in each plot; the long runs are used only for $B_0=5\mathrm{Tesla}$. In all cases, $m_{\mathrm{eff}}={10}^{-4}\mathrm{eV}$ is assumed.

Figure 16

Production probability for a chameleon with ${\beta }_{\gamma }={10}^{12}$ in a magnetic field $B_0=5\mathrm{Tesla}$, with and without the CHASE partitions.

Figure 17

Typical particle paths used in the Monte Carlo simulation of CHASE. (Top) Paths used to compute ${\Gamma }_{\mathrm{aft}}$. The red path (with “$+$” symbols showing bounces from chamber walls) begins on an interior window and ends on the exit window. The green path (with “$\times$” symbols denoting bounces) begins on the entrance window and ends on the exit window. The blue path (with “$*$” symbols denoting bounces) begins on the entrance window and ends on an interior window; the subsequent dotted line denotes the photon which may result from the quantum measurement at the interior window. (Bottom) Paths used to compute ${\Gamma }_{\mathrm{dec}}$. The red path (“$+$”) begins on an internal window and ends on the entrance window, while the green path (“$\times$”) begins on the exit window and ends on an interior window. Since a larger range of directions is allowed in the decay rate computation, the typical path bounces from chamber walls much more frequently than those used in the afterglow computation.

Figure 18

Contributions to (top) afterglow and to (bottom) decay rates from paths vs number of wall bounces per path. The afterglow rate receives no contributions from paths bouncing more than 9 times, since these emerge from the chamber at too great an angle to reach the CHASE detector.

Figure 19

Convergence of the Monte Carlo simulated rates as the number of particles simulated is increased. For each particle number $N_{\mathrm{particles}}$, each point represents a different random number seed used in the Monte Carlo simulation. ${\beta }_{\gamma }={10}^{12}$, $m_{\mathrm{eff}}={10}^{-4}\mathrm{eV}$, and $B_0=5\mathrm{Tesla}$ are assumed.

Figure 20

(Top) ${\Gamma }_{\mathrm{aft}}$ and ${\Gamma }_{\mathrm{dec}}$ for the 2-point (2pt) calculation of Sec. 5 as well as the Monte Carlo (MC) calculation with all particles required to start at the center of the entrance window. In both calculations, the chamber has no interior windows. ${\beta }_{\gamma }={10}^{12}$ and $B_0=5\mathrm{Tesla}$ are assumed. (Bottom) Fractional difference between the 2-point and Monte Carlo calculations.

Figure 21

Afterglow and decay rates computed using the Monte Carlo simulation. Particles are allowed to begin at any position on a window surface. ${\xi }_V=0$, ${\beta }_{\gamma }={10}^{12}$, and $B_0=5\mathrm{Tesla}$ are assumed.

Figure 22

Fractional change in afterglow and decay rates vs fraction $f_{\mathrm{diff}}$ of diffuse reflection. Rates for six different random number seeds are shown.

Figure 23

(Top) afterglow rates (thin lines) and decay rates (thick lines) as the lengths ${\ell }_1$, $L$, and ${\ell }_2$ are varied relative to their CHASE values in Table . (Bottom) Afterglow and decay rates vs the mean wall reflectivity ${\overline{f}}_{\mathrm{ref}}$ in the thin-skin limit ${\tilde{n}}_1\to \infty$. In both plots, $m_{\mathrm{eff}}={10}^{-4}\mathrm{eV}$ and ${\xi }_V=0$ have been assumed.

Figure 24

Analysis of simulated CHASE data. Yellow regions show previous constraints; blue regions show models excluded relative to the null (no-chameleon) model by $\Delta {\chi }^2=6.0$, corresponding to a 95% confidence level for a Gaussian probability distribution; green regions show chameleon models preferred relative to the null model by $\Delta {\chi }^2=9.2$, corresponding to a 99% confidence level; pink regions within the green regions show chameleons within $\Delta {\chi }^2=6.0$ of the best-fit chameleon model. Black “$+$” signs show the fiducial chameleon model. (Left) Simulated chameleon near the center of the CHASE sensitivity region, $m_{\mathrm{eff}}={10}^{-4}\mathrm{eV}$ and ${\beta }_{\gamma }={10}^{14}$. The inset shows $0.99\times {10}^{-4}\mathrm{eV}$ $<m_{\mathrm{eff}}<1.02\times {10}^{-4}\mathrm{eV}$ on the horizontal axis and $0.9999\times {10}^{14}<{\beta }_{\gamma }<1.0001\times {10}^{14}$ on the vertical axis; the black ellipse shows the $\Delta {\chi }^2=2.3$ contour, corresponding to the 68% confidence level. (Middle) Simulated chameleon near the edge of the CHASE sensitivity region, $m_{\mathrm{eff}}=2.4\times {10}^{-3}\mathrm{eV}$ and ${\beta }_{\gamma }={10}^{12}$. (Right) Simulated data with no chameleon. In all of the analyses, ${\xi }_V=0$ has been assumed.

Figure 25

Model-independent chameleon constraints for ${\xi }_V=0$, from 32. The solid blue region is excluded at the $\Delta {\chi }^2=6.0$ level (95% C.L. for Gaussian probability) for scalar chameleons, and the interior of the green curve is excluded at that level for pseudoscalar chameleons.

Figure 26

Model-independent scalar chameleon constraints for several reflection phases ${\xi }_V$.

Figure 27

Model-independent scalar chameleon constraints for each of the magnetic fields used in CHASE, assuming ${\xi }_V=0$.

Figure 28

CHASE constraints on ${\varphi }^4$ chameleons. Chameleon fragmentation becomes important for $\lambda \gtrsim 0.01$ and low ${\beta }_{\gamma }$. Note that the entire parameter space shown has already been excluded by the Casimir force constraints of 42.

Figure 29

Constraints on chameleon dark energy models (9). (Top) assumes $\kappa =1$ and varies $n$; (bottom) fixes $n=-1$ and varies $\kappa$.

Figure 30

Constraints on exponential dark energy models (77). Models with $\kappa \gtrsim {10}^4$ have particles too massive to be probed by CHASE.

Figure 31

Combined constraints on $n=-1$, $\kappa =1$ chameleon dark energy (9). Solid regions show current constraints from analyses of experimental data, while lines represent forecasts and preliminary analyses.