Detection prospects for solar and terrestrial chameleons

Dark energy models, such as the chameleon, where the acceleration of the expansion of the Universe results from the dynamics of a scalar field coupled to matter, suffer from the potential existence of a fifth force. Three known mechanisms have been proposed to restore general relativity in the solar system and the laboratory, which are the symmetron/Damour-Polyakov effect, the Vainshtein property and the chameleon screening. Here, we propose to probe the existence of chameleons in the laboratory, considering their particle physics consequences. We envisage the resonant and nonresonant production of chameleons in the sun and their back-conversion into X-ray photons in a solar helioscope pipe such as the one used by CAST. We find that the resonant contribution is only present for a very small range of chameleon couplings to matter and occurs in extremely narrow magnetized regions in the sun. However, in the nonresonant case, chameleons are created in all available solar magnetic regions and for a very wide range of chameleon couplings to photons and matter. Such chameleons could leave the sun and be back-converted into X-ray photons above the photosphere or in the magnetic pipe of a helioscope. A detection of these X-rays would indicate the existence of chameleons. The number of produced chameleons and regenerated X-ray photons varies depending on the strength of the solar magnetic field in the production shell, the matter and photon couplings of chameleons, and the magnetic field length and magnitude inside a dipole magnet. We focus on a template model for the solar magnetic field: a constant magnetic field in a narrow shell surrounding the tachocline. The X-ray photons in a helioscope pipe obtained from back-conversion of the chameleons created inside the sun have a spectrum which is peaked in the sub-keV region, just below the actual sensitivity range of the present axion helioscopes. Nevertheless they are detectable by present-day magnetic helioscopes like CAST and Sumico, which were built originally for solar axions. We also propose a chameleon-through-a-wall experiment whereby X-ray photons from a synchroton radiation source could be converted into chameleons inside a dipole magnet, then pass a wall which is opaque to X-rays before being back-converted into X-ray photons in a second magnet downstream. We show that this could provide a direct signature for the existence of chameleon particles.

Figure 1

The matter density at the resonance in $\mathrm{g}·{\mathrm{cm}}^{-3}$ as a function of the matter coupling constant $\beta$ for $n=1$ and $n=4$. Notice that only a limited range of $\beta$ can lead to a resonance in the sun. Nuclear physics constraints always impose $\beta \lesssim {10}^{11}$ [29].

Figure 2

The matter coupling constant $\beta$ leading to a resonance as a function of the radial distance of the resonance (in units of the solar radius) when $n=1$ and $n=4$. Nuclear physics constraints impose $\beta \lesssim {10}^{11}$ [29].

Figure 3

The width of the resonance region in cm when $n=4$ as a function of $\beta =\frac{m_{\mathrm{Pl}}}{M}$ for ${\beta }_{\gamma }={10}^{10.32}$ and $\omega =7\mathrm{keV}$ when a magnetic field is assumed to exist in a shell of width $0.01R_{\odot }$ around the tachocline. The width scales like ${\beta }_{\gamma }$ (see (24)). Here we have chosen ${\beta }_{\gamma }$ which corresponds to the saturation of the solar energy loss bound by escaping chameleons. Notice that the width is of the order of the photon mean free path $\approx 0.3$ cm here and much shorter than the width of the shell, i.e., the width of the shell does not influence the resonance. For each value of $\beta$ the location of the resonance can be deduced from Fig. 2.

Figure 4

The sensitivity region at the $2\sigma$ level above the red region for a magnetic helioscope like CAST in vacuum at a temperature of 1.8K and for the photon coupling $\log {\beta }_{\gamma }$ as a function of the matter coupling $\log \beta$. We have assumed that there is a constant magnetic field of 30T in a shell of $0.01R_{\odot }$ surrounding the tachocline. The precise form of the diagram depends on $n$ although the overall shape remains the same. Notice that for values of $\beta$ below the resonance values, where the dip occurs, the $2\sigma$ contour is constant. In this work, we have chosen this value as a template for the photon coupling. Larger values of the photon couplings would violate the solar energy loss bound due to escaping chameleons with $\beta \lesssim {10}^{10.32}$. As such, the solar model we have chosen for the magnetized region, where chameleons are produced, is almost optimal. Notice that the value $\beta ={\beta }_m$ would only lead to interesting physics at the $2\sigma$ level for very large values of the coupling to photons, higher than the CHASE bound.

Figure 5

The mass of the chameleon in eV in the helioscope pipe (vacuum) as a function of $\log \beta$. The mass is constant and independent for low values of $\beta \lesssim {10}^9$, and becomes $\beta$-dependent for larger values. The constant value is obtained from (69) and the $\beta$ dependence from (4).

Figure 6

The conversion probability as a function of the photon/chameleon energy in keV with vacuum in the magnetic helioscope pipe. Here we have chosen the matter coupling to be $\beta ={10}^6$ and the photon coupling ${\beta }_{\gamma }={10}^{10.32}$. The magnetized region is such that $B_{\mathrm{helio}}L\approx 90T·m$. Notice the parabolic shape at low energy followed by a sinusoidal trend which would saturate at much higher energies (not shown here).

Figure 7

The resonant spectrum of back-converted photons from solar chameleons giving the number of counts per hour and per keV predicted to be seen in a magnetic helioscope like CAST in vacuum as a function of the reconverted photon energy. Here we have chosen the matter coupling to be $\beta =1.8\times {10}^7$, the photon coupling ${\beta }_{\gamma }={10}^{10.29}$ and a shell magnetic field in the sun of 30T over a width of 0.01 solar radius above the tachocline for the resonant case (left) and for the nonresonant case (right) with $\beta ={10}^6$ and ${\beta }_{\gamma }={10}^{10.29}$.

Figure 8

The spectrum of regenerated photons giving the number of counts per hour and per keV as predicted to be seen by a helioscope like CAST as a function of the photon energy in keV. Here we have chosen the matter coupling to be $\beta ={10}^6$, the photon coupling ${\beta }_{\gamma }={10}^{10.29}$ and a shell magnetic field in the sun of 30T over a width of 0.01 solar radius near the tachocline.

Figure 9

On the left, the solar chameleon flux in ${\mathrm{keV}}^{-1}\times {\mathrm{s}}^{-1}\times {\mathrm{cm}}^{-2}$ at 1 AU as a function of the chameleon energy in keV. Here we have chosen the matter coupling to be $\beta ={10}^6$, the photon coupling ${\beta }_{\gamma }={10}^{10.29}$ and a shell magnetic field in the sun of 30 T over a width of 0.01 solar radius above the tachocline. On the right, the chameleon brightness in $\mathrm{erg}\times {\mathrm{keV}}^{-1}\times {\mathrm{s}}^{-1}\times {\mathrm{cm}}^{-2}$ at the surface of the sun as a function of the energy in keV. The estimated total energy loss in chameleons is $6·{10}^9$ $\mathrm{erg}\times {\mathrm{s}}^{-1}\times {\mathrm{cm}}^{-2}$, i.e., around 10% of the photon solar luminosity.

Figure 10

The X-ray luminosity spectrum of back-converted chameleons in keV (in $\mathrm{erg}\times {\mathrm{keV}}^{-1}\times {\mathrm{s}}^{-1}\times {\mathrm{cm}}^{-2}$) leaving the sun as a function of the photon energy. The photons are back-converted at $-500\mathrm{km}$, $-100\mathrm{km}$, 0 km and $+500\mathrm{km}$ with magnetic fields 0.2 T in the first three cases and 0.1 T in the last case. In the first three cases, the conversion takes place over one mean free path of order 15 km, 40 km and 50 km while in the last case the conversion takes place over the size of the magnetic region around 100 km. In each case, the coherence length is much smaller than the size of the chameleon-to-photon interaction region and the spectrum has been averaged out over many scatterings. Here we have chosen the matter and photon coupling to be $\beta ={10}^6$ and ${\beta }_{\gamma }={10}^{10.29}$ respectively.

Figure 11

The photon luminosity spectrum from back-converted chameleons ($\mathrm{erg}\times {\mathrm{keV}}^{-1}\times {\mathrm{s}}^{-1}\times {\mathrm{cm}}^{-2}$) in solar flares leaving the sun as a function of the photon energy in keV. The photons are back-converted at $+1000\mathrm{km}$, $+2000\mathrm{km}$, $+10000\mathrm{km}$ and $+80000\mathrm{km}$ with magnetic fields 100 Gauss, 10 Gauss, 10 Gauss and 5 Gauss, respectively. At the altitude of 1000 km, the spectrum has been averaged out over many oscillations. In the other two cases, we have represented the spectrum itself. In each case we have used $B_{\mathrm{out}}{L}_{\mathrm{out}}\approx {10}^4\mathrm{T}.\mathrm{m}$ which determines the length of the magnetic region. Here we have chosen the matter coupling to be $\beta ={10}^6$ and the photon coupling is ${\beta }_{\gamma }={10}^{10.29}$.

Figure 12

The X-ray luminosity spectrum from back-converted chameleons leaving the quiet sun as a function of the photon energy in keV in $\mathrm{erg}\times {\mathrm{keV}}^{-1}\times {\mathrm{s}}^{-1}\times {\mathrm{cm}}^{-2}$ (linear scale on the left and logarithmic scale on the right). The photons are back-converted at altitudes larger than 5000 km with a magnetic field of 1 Gauss and a density of order ${10}^{-16}\mathrm{g}·{\mathrm{cm}}^{-3}$. We have used $B_{\mathrm{out}}{L}_{\mathrm{out}}\approx {10}^4\mathrm{T}.\mathrm{m}$ to determine the length of the magnetic region. Here we have chosen the matter coupling to be $\beta ={10}^6$ and the photon coupling is ${\beta }_{\gamma }={10}^{10.29}$.

Figure 13

The photon flux of the X-ray source for the chameleon-through-a-barrier experiment in ${\mathrm{cm}}^{-2}\times {\mathrm{s}}^{-1}\times {\mathrm{keV}}^{-1}$ as a function of the photon energy in keV (left) and the regenerated spectrum on the other side of the barrier (right). The integrated flux is ${10}^{19}{\mathrm{s}}^{-1}\times {\mathrm{cm}}^{-2}$ for the source spectrum. The spectrum of regenerated photons presented here is the one predicted to be seen by a chameleon through a wall experiment using two magnetic pipes on both sides of a thick barrier. The coupling to matter is $\beta ={10}^6$ while the coupling to photons ${\beta }_{\gamma }={10}^{10.32}$ implies a corresponding $2\sigma$ result obtained in 300 days with a detection sensitivity of ${10}^{-6}{\mathrm{s}}^{-1}\times {\mathrm{cm}}^{-2}$. For magnetic pipes of 3 times the CAST pipe length, a $5\sigma$ result would only take 6 days.