Anomalous phosphine sensitivity coefficients as probes for a possible variation of the proton-to-electron mass ratio

Abstract

1 INTRODUCTION

Recently, the J = 2 − 1 rotational transition of phosphine (PH₃) was detected in the carbon star envelope IRC +10216 (Agúndez et al. 2014), thus confirming the presence of PH₃ in the outflows of evolved stars but more significantly outside of the Solar system. The appearance of PH₃ has been predicted in numerous other astrophysical environments (see the discussion by Sousa-Silva et al. 2015 and references therein), and because of prominent ‘irregularities’ displayed by its rotation–vibration spectrum, it is a promising system for investigating the cosmological variability of the proton-to-electron mass ratio, μ = m_p/m_e. Observing PH₃ outside of our Galaxy is no easy feat; however, nearby Galactic molecular clouds offer a means to constrain μ through the so-called chameleon scenario (Brax et al. 2004; Khoury & Weltman 2004) as evidenced by studies of ammonia (Levshakov et al. 2010b,a) and methanol (Daprà et al. 2017).

At present, the most robust constraint on a temporal variation of μ was determined from methanol absorption spectra observed in the lensing galaxy PKS1830−211 (Kanekar et al. 2015). The three measured transitions possessed sensitivity coefficients, T, ranging from −7.4 to −1.0 and resulted in a constraint of |$\dot{\mu }/\mu < 2\times 10^{-17}\,$|yr⁻¹ assuming a linear rate of change. This translates to no change in μ over the past ≈7.5 billion years and is in agreement with the best laboratory constraint to date, which measured optical transitions in ¹⁷¹Yb⁺ ions to derive |$\dot{\mu }/\mu = (0.2\pm 1.1)\times 10^{-16}\,$|yr⁻¹ (Godun et al. 2014) again assuming a linear rate of change. Whilst the use of methanol has led to several astronomical constraints (Jansen et al. 2011; Levshakov, Kozlov & Reimers 2011; Bagdonaite et al. 2013a,b; Thompson 2013; Kanekar et al. 2015), it is worthwhile identifying other molecular absorbers with notable sensitivities to expand the search for a drifting μ.

Due to the small difference between its rotational constants B and C, and also because of the strong x − y Coriolis interaction between the coinciding ν₂/ν₄, ν₁/ν₃ and |$2\nu _4^{\ell =0}/2\nu _4^{\ell =2}$| states (see Fig. 1), phosphine is a potential candidate system for probing μ. Notably, the spectrum of PH₃, and presumably other molecules of |$\boldsymbol {C}_{3\mathrm{v}}\mathrm{(M)}$| symmetry, is special due to the anomalous behaviour of the A₁ − A₂ splittings (Ulenikov et al. 2002). A large number of spectroscopic studies of PH₃ have been reported in the literature (see Müller 2013 and references therein) and highly accurate data is available for the majority of its states. Furthermore, a robust theoretical description of this molecule, which we utilize for this work, has been developed over the years (Yurchenko et al. 2003, 2005, 2006; Ovsyannikov et al. 2008a,b; Sousa-Silva, Yurchenko & Tennyson 2013; Sousa-Silva et al. 2014, 2015; Sousa-Silva, Tennyson & Yurchenko 2016), culminating in the construction of a comprehensive rotation–vibration line list applicable for elevated temperatures (Sousa-Silva et al. 2015).

Figure 1.

The lowest vibrational energy levels of PH₃.

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Model radiative transfer calculations of phosphine excitation in the envelope of IRC +10216 (Cernicharo et al. 1999; Agúndez et al. 2014) highlighted the importance of infrared pumping from the ground to the first excited vibrational states, helping explain the presence of strong emission bands in the observed spectra. We therefore find it useful to investigate the sensitivity of the ground, fundamental and low-lying combination and overtone vibrational states of PH₃ (see Fig. 1) to a possible space–time variation of μ using a robust variational approach. The paper is structured as follows: in Section 2, we describe the variational approach used to compute sensitivity coefficients. The results for the phosphine molecule are presented and discussed in Section 3. Concluding remarks are given in Section 4.

2 VARIATIONAL APPROACH

Sensitivity coefficients for PH₃ have been computed with the same variational approach as was previously employed for ammonia (Owens et al. 2015a, 2016) and the hydronium cation (Owens et al. 2015b). Calculations were carried out with the nuclear motion program trove (Yurchenko, Thiel & Jensen 2007; Yachmenev & Yurchenko 2015; Yurchenko, Yachmenev & Ovsyannikov 2017) and utilized the potential energy surface (PES), dipole moment surface (DMS) and computational set-up of Sousa-Silva et al. (2015), which have all undergone rigorous testing and are known to be reliable. We refer the reader to Sousa-Silva et al. (2015) for further details of the nuclear motion computations. All sensitivity coefficients, equation (1), have been determined with calculated frequencies, E_u − E_l, as oppose to experimental values when available. This was done for consistency and to verify the trend in sensitivities displayed by PH₃, which we will discuss further in Section 3.

3 RESULTS AND DISCUSSION

In general, as shown in Table 1, Figs 2 and 3, the majority of the calculated sensitivity coefficients for the low-lying vibrational states acquire the expected values of T ≈ −1 and T ≈ −1/2 for rotational and ro-vibrational transitions, respectively. Notably, this is the case for the J = 2 − 1 and J = 1 − 0 rotational transitions observed in the carbon star envelope IRC +10216 (Agúndez et al. 2008, 2014). For a small fraction of the probed transitions, the sensitivities deviate from the usual values. Accidental coincidences between ro-vibrational states can cause the magnitude of these ‘irregularities’ to strongly increase with vibrational excitation, as illustrated in Fig. 4.

Figure 2.

Sensitivity coefficients T for pure rotational transitions in the ground, ν₂, and ν₄ vibrational states of PH₃. Here, n is a running number which counts the number of transitions.

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Figure 3.

Sensitivity coefficients T for ro-vibrational transitions from the ground to the lowest vibrational states of PH₃. Here, n is a running number which counts the number of transitions.

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Figure 4.

The wavenumbers ν (in cm⁻¹) and sensitivity coefficients T of the ν₄ ← ν₂ ro-vibrational transitions of PH₃. Here, n is a running number which counts the number of transitions.

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The most striking sensitivities are displayed by the A₁ − A₂ doublets of PH₃. As is well known for a molecule with |$\boldsymbol {C}_{3\mathrm{v}}\mathrm{(M)}$| symmetry, all rotation–vibration energy levels corresponding to the same K ≡ |k| ≠ 0 rotational quantum number and having overall A₁, A₂ symmetry are split into doublets due to different ro-vibrational interactions (see, for example, Chen & Oka 1989). For the non-degenerate vibrational states, the A₁ − A₂ splittings occur for rotational levels with K = 3n (n = 1, 2, …). For the doubly degenerate fundamental vibrational states characterized by the vibrational angular momentum quantum number ℓ ≠ 0, the splittings occur for the K = 1, 2, 4, 5, 7, 8… levels.

In Tables 2–9, we have computed sensitivity coefficients for a large number of the A₁ − A₂ doublets for low-lying vibrational states. The results suggest that sensitivities of the A₁ − A₂ splittings for non-coinciding ro-vibrational states possess values dependent on the rotational quantum number J. For example, T ≈ −1.5, −2, −3 for k = 1, 2, 3, respectively (see Tables 2–5). It would be interesting to see if this trend is present in other molecules of |$\boldsymbol {C}_{3\mathrm{v}}\mathrm{(M)}$| symmetry. For the sensitivities corresponding to coinciding states, there is a strong and irregular dependence on the x − y Coriolis interaction that can produce values at least one order of magnitude larger than the respective Coriolis-free predictions. This behaviour is similar to that of NH₃ (Špirko 2014; Owens et al. 2015a, 2016) and H₃O⁺ (Owens et al. 2015b) .