Colloquium: Search for a drifting proton-electron mass ratio from ${\mathsf{H}}_{\mathsf{2}}$

An overview is presented of the ${\mathrm{H}}_2$ quasar absorption method to search for a possible variation of the proton-electron mass ratio $\mu =m_p/m_e$ on a cosmological time scale. The method is based on a comparison between wavelengths of absorption lines in the ${\mathrm{H}}_2$ Lyman and Werner bands as observed at high redshift with wavelengths of the same lines measured at zero redshift in the laboratory. For such comparison sensitivity coefficients to a relative variation of $\mu$ are calculated for all individual lines and included in the fitting routine deriving a value for $\mathrm{\Delta }\mu /\mu$. Details of the analysis of astronomical spectra, obtained with large 8–10 m class optical telescopes, equipped with high-resolution echelle grating based spectrographs, are explained. The methods and results of the laboratory molecular spectroscopy of ${\mathrm{H}}_2$, in particular, the laser-based metrology studies for the determination of rest wavelengths of the Lyman and Werner band absorption lines, are reviewed. Theoretical physics scenarios delivering a rationale for a varying $\mu$ are discussed briefly, as well as alternative spectroscopic approaches to probe variation of $\mu$, other than the ${\mathrm{H}}_2$ method. Also a recent approach to detect a dependence of the proton-to-electron mass ratio on environmental conditions, such as the presence of strong gravitational fields, are highlighted. Currently some 56 ${\mathrm{H}}_2$ absorption systems are known and listed. Their usefulness to detect $\mu$ variation is discussed, in terms of column densities and brightness of background quasar sources, along with future observational strategies. The astronomical observations of ten quasar systems analyzed so far set a constraint on a varying proton-electron mass ratio of $|\mathrm{\Delta }\mu /\mu |<5\times 10^{-6}$ ($3\sigma$), which is a null result, holding for redshifts in the range $z=2.0–4.2$. This corresponds to look-back times of $(10–12.4)\times {10}^9$ years into cosmic history. Attempts to interpret the results from these ten ${\mathrm{H}}_2$ absorbers in terms of a spatial variation of $\mu$ are currently hampered by the small sample size and their coincidental distribution in a relatively narrow band across the sky.

Figure 1

Shifts in the rest wavelengths of Lyman ($L$) and Werner ($W$) lines as functions of the (exaggerated) relative variation in $\mu$. The solid (red) curves indicate transitions that are redshifted with increasing $\mathrm{\Delta }\mu /\mu$, while the dashed (blue) curves are blueshifters. The $L1R0$ transition, indicated by a dot-dashed (black) curve, represents a line with very low sensitivity to $\mu$ variation, i.e., it acts as an anchor line.

Figure 2

Excitation schemes used to determine the Lyman and Werner transition wavelengths. Left panel: A direct excitation (a) scheme from $X$ to $B$, $C$ states using a narrowband XUV-laser system (or, alternatively, detection of XUV emission from excited $B$ and $C$ states). Right panel: An indirect determination scheme is represented comprising two-photon Doppler-free excitation (b) in the $EF-X$ system and Fourier-transform emission measurements (c) of the $EF-B$ and $EF-C$ systems.

Figure 3

Recording of the XUV-absorption spectrum of $L15R2$, i.e., the $R(2)$ line in the $B{{}^1\mathrm{\Sigma }}_u^{+}-X{{}^1\mathrm{\Sigma }}_g^{+}$ (15,0) Lyman band (lower spectrum) with the $I_2$ reference spectrum (middle spectrum) and etalon markers (top spectrum) for the calibration. The line marked with an asterisk is the $a_1$ hyperfine component of the $R(66)$ line in the $B-X(21,1)$ band in $I_2$ at $17718.7123{\mathrm{cm}}^{-1}$. From [152].

Figure 4

(a) Laser spectroscopic recording of the $Q(0)$ two-photon transition in the $EF{{}^1\mathrm{\Sigma }}_g^{+}-X{{}^1\mathrm{\Sigma }}_g^{+}$ (0,0) band. (b) Recording of part of the Fourier-transform spectrum of the $EF{{}^1\mathrm{\Sigma }}_g^{+}-B{{}^1\mathrm{\Sigma }}_u^{+}$ emission.

Figure 5

Sensitivity coefficients $K_i$ of individual lines in the Lyman and Werner bands showing the range of values from around $-0.02$ to 0.06. Solid (blue) symbols are associated with Lyman lines and open (red) symbols to Werner lines. From $L8$, level crossings occur between levels in the $B{{}^1\mathrm{\Sigma }}^{+}$ and $C{{}^1\mathrm{\Pi }}^{+}$ states, leading to irregularities in the $K$ progression of the Lyman $P$, $R$ and Werner $Q$ transitions.

Figure 6

Sensitivity coefficients $K_i$ for $\mu$ variation in ${\mathrm{H}}_2$ as a function of rest-frame wavelengths. Solid (blue) symbols are associated with Lyman lines and open symbols to Werner lines. The size of the data points reflects line oscillator strengths $f_i$ for the spectral lines.

Figure 7

Typical spectrum of a quasar displaying the observed flux (in arbitrary units), in this case for $\mathrm{B}0642-5038$ emitting at redshift $z=3.09$ and registered by the ultraviolet and visual echelle spectrograph (UVES) mounted at the ESO VLT. The quasar spectrum displays the characteristic broad emission line profiles produced by, e.g., Lyman $\alpha$ of H i, and C iv. The Lyman-$\alpha$ forest arises as the light from the quasar crosses multiple neutral hydrogen clouds lying in the line of sight. On the left (blue) side of the emission peak the damped Lyman-$\alpha$ system at $z_{\mathrm{DLA}}=2.66$ causes a series of very strong absorption lines starting at $\lambda =(z_{\mathrm{DLA}}+1)\times 1216\AA =4450\AA$ and absorbs all the flux at $\lambda <(z_{\mathrm{DLA}}+1)\times 912\AA =3340\AA$ producing a so-called Lyman break or Lyman-limit cutoff. Since the rest wavelengths of ${\mathrm{H}}_2$ are in the UV ($<1140\AA$) they are found in the Lyman-$\alpha$ forest. On the right (red) side of the Lyman-$\alpha$ emission peak the sharp absorption lines are due to metallic ions (C iv, Fe ii, Si iv, etc.) at various redshifts including that of the DLA. The absorptions at 6800 and 7600 Å are related to the Fraunhofer $B$ and $A$ bands (molecular oxygen) absorbing in the Earth’s atmosphere.

Figure 8

The ${\mathrm{H}}_2$ transitions $L0R0$ indicated with solid (blue) markers and $L0R1$ indicated with dashed (green) markers seen at different redshifts toward ten quasars. Transitions are imprinted in a number of velocity features as indicated. For further details see the main text.

Figure 9

Values for $\mathrm{\Delta }\mu /\mu$ obtained for nine ${\mathrm{H}}_2$ absorption systems analyzed as a function of redshift and look-back time plotted alongside with the evolution of cosmological parameters for ${\mathrm{\Omega }}_m$ (dashed line) and ${\mathrm{\Omega }}_{\mathrm{\Lambda }}$ (solid line), referred to the right-hand vertical axis. These are the dark-matter and dark energy densities, respectively, relative to the critical density. Open circles refer to data where a correction for long-range wavelength distortions is included; for the filled (red) circles such correction has not been applied. Uncertainties for individual points are indicated at the $1\sigma$ level by vertical line extensions to the points. The result for system $\mathrm{Q}1232+082$ is not plotted in view of its large uncertainty. The gray bar represents the average for $\mathrm{\Delta }\mu /\mu$ with $\pm 1\sigma$ uncertainty limits from the 10 ${\mathrm{H}}_2$ systems analyzed. For comparison results from radio-astronomical observations at lower redshift ($z<1$) are plotted as well (blue diamonds).

Figure 10

Sky map in equatorial coordinates (J2000) showing currently known quasar sight lines containing molecular absorbers at intermediate-to-high redshifts. The open (red) points correspond to the ten ${\mathrm{H}}_2$ absorbers that have been analyzed for a variation of $\mu$, while the dark (blue) points mark the 23 ${\mathrm{H}}_2$ targets that might be used in future analyses as listed in Table 1. The gray (light blue) points correspond to the additional sample of 23 ${\mathrm{H}}_2$ absorption systems found through the Sloan Digital Sky Survey ([16]). The stars (green) indicate the sight lines toward $\mathrm{PKS}1830-211$ and $\mathrm{B}0218+357$, where $\mu$ variation was measured from ${\mathrm{CH}}_3\mathrm{OH}$ ([7]; [8]) and ${\mathrm{NH}}_3$ ([55]), or only from ${\mathrm{NH}}_3$ ([101]; [70]), respectively. The $+$ sign represents the angle ${\mathrm{\Theta }}_{\alpha }$ of the $\alpha$-dipole phenomenon ([162]). The gray line indicates the galactic plane.

Figure 11

Values of $\mathrm{\Delta }\mu /\mu$ of the results for ${\mathrm{H}}_2$ absorbers obtained so far plotted with their angular coordinates in terms of a projection angle $\mathrm{\Theta }$ onto the dipole axis ${\mathrm{\Theta }}_{\alpha }$ ($RA=17.5\mathrm{h}$ and $\text{dec}=-58°$) associated with the dipole found for $\mathrm{\Delta }\alpha /\alpha$ ([162]). Open circles represent values of $\mathrm{\Delta }\mu /\mu$ that were corrected for possible long-range wavelength distortions, while solid (red) circles refer to uncorrected data points. The solid line represents the result from a fit to Eq. (12) with the slope equaling $A_{\mathrm{\Theta }}$. See the main text for several caveats in interpreting the significance of this fit.

Figure 12

Comparison of a part of the spectrum of the absorption system toward $\mathrm{J}2123-005$ at $z=2.05$ by the Keck telescope equipped with the HIRES spectrograph, indicated by a gray (green) line ([93]), recorded at a resolving power of 110 000  and by the very large telescope, equipped with the UVES spectrograph indicated by a dark (black) line ([156]) at a resolving power of 53 000. Note the absence of improved effective resolution in the Keck spectrum and the higher SNR in the UVES spectrum.

Figure 13

Part of the absorption spectrum of the white dwarf G29-38 as obtained with the COS-HST in the range 1340–1370 Å. The solid (red) line is the result of a fit with optimized parameters for $T$, $b$, $N$, $z$, and $\mathrm{\Delta }\mu /\mu$. The tick (blue) marks represent the positions of ${\mathrm{H}}_2$ absorption lines contributing to the spectrum.