METHANOL AS A TRACER OF FUNDAMENTAL CONSTANTS

The hypothetical variation of the dimensionless physical constant μ—the electron-to-proton mass ratio—can be probed through the spectral observations of certain molecular transitions which are particularly sensitive to changes in μ. The corresponding sensitivity coefficients Q_μ of different molecular transitions relevant to astrophysics were calculated at first for H₂ (Varshalovich & Levshakov 1993) and later on for OH (Darling 2003; Chengalur & Kanekar 2003), ¹⁵ND₃ (van Veldhoven et al. 2004), H₂ (Ubachs et al. 2007), NH₃ (Flambaum & Kozlov 2007), OH and CH (Kozlov 2009), NH₂D and ND₂H (Kozlov et al. 2010), H₃O⁺ (Kozlov & Levshakov 2011), CH₃OH (Jansen et al. 2011), and H₃O⁺, H₂DO⁺, HD₂O⁺, and D₃O⁺ (Kozlov et al. 2011). Among them ammonia, NH₃, is actively used in extragalactic and galactic observations of dense molecular clouds. In extragalactic observations the most stringent limits on the mass ratio, Δμ/μ = (μ_space − μ_lab)/μ_lab, were obtained by Henkel et al. (2009) at redshift z = 0.89 and by Kanekar (2011) at z = 0.69: |Δμ/μ| < 1400 ppb (3σ) and |Δμ/μ| < 400 ppb (3σ), respectively (1ppb = 10⁻⁹). Observations in the Milky Way have shown, however, a tentative signal Δμ/μ = 26 ± 3 ppb (Levshakov et al. 2010a, 2010b), but its nature remains unclear. To distinguish whether this is a real signal or an artefact due to unaccounted systematic effects, additional independent measurements involving other molecules are required.

One of such molecules—CH₃OH—was recently suggested by Jansen et al. (2011, hereafter J11). CH₃OH is a widespread interstellar molecule observed in the Milky Way, external galaxies (Herbst & van Dishoeck 2009; Sjouwerman et al. 2010; Martín et al. 2006) and even in comets (Bockelée-Morvan et al. 1991). The purpose of this paper is to probe the variability of μ in the Milky Way using narrow emission lines of the methanol masers.

The microwave spectrum of CH₃OH is very rich because of the internal rotation of the OH group. The basic theory of the non-rigid tops with internal rotation was established in the 1950s (Lin & Swalen 1959; Herschbach 1959) and the main features of the methanol spectrum were explained. Later on the theory was refined many times and currently there is a very impressive agreement between the theory and experiment (Anderson et al. 1990; Müller et al. 2006; Xu et al. 2008; Kleiner 2010).

In this section the sensitivity coefficients Q_μ of the methanol lines are calculated. Our computational procedure differs from that described in J11. This allows us to check the values of Q_μ and their validity.

To calculate the microwave spectrum of CH₃OH we use an approach from Rabli & Flower (2010, hereafter RF), where a simple and convenient form of the effective Hamiltonian with six spectroscopic constants is suggested. This Hamiltonian is physically transparent and sufficiently accurate for calculations of sensitivity coefficients. All six parameters of the RF model have clear physical meaning and their dependence on μ is easily understood within the Born–Oppenheimer approximation.

To show how rotational parameters scale with μ, we consider an example of a diatomic molecule in its ground vibrational state:

$$\begin{equation} B_0 = \frac{1}{M} \left\langle v=0\left|\frac{1}{R^2}\right|v=0\right\rangle =B_e -\alpha _e+\cdots, \end{equation} \tag{ 1 }$$

where B_e = 1/(MR²₀) corresponds to the equilibrium internuclear distance R₀ and α_e is the vibrational correction. It is clear that B_e ∼ μ, but α_e has an additional dependence on μ via the vibrational wave function and, hence, scales as μ^3/2. Thus, there is no one-to-one correspondence between terms of the effective Hamiltonian and the terms of the Born–Oppenheimer perturbation theory. As a result, B₀ scales as δB₀/B₀ = Q^r_μδμ/μ, with Q^r_μ ≈ 1. Typically, α_e/B₀ is of the order of 10⁻² and Q^r_μ varies from 0.995 for the NO molecule to 0.981 for the H-bearing HF molecule.

We conclude that, in general, the rotational parameters A, B, and C scale linearly with μ within the uncertainty interval of 1%–2%. These vibrational corrections are of the same order of magnitude as centrifugal corrections considered in J11. Both types of corrections are included below in the estimate of the error of the calculated sensitivity coefficients.

The rotational part of the Hamiltonian H_rot corresponds to the slightly asymmetric top and includes rotational constants A, B, and C (B ≈ C). Here we use the standard convention A > B > C, while in RF C > B > A. The hindered rotation is described by the Hamiltonian

$$\begin{equation} H_\mathrm{hr}=-F\frac{{d}^2}{{d}\omega ^2} + \frac{V_3}{2}\left(1-\cos \,3\omega \right), \end{equation} \tag{ 2 }$$

where the kinetic coefficient F is proportional to μ and the electronic potential V₃ is independent on μ. This model does not include centrifugal distortions. Interaction of internal rotation with overall rotation is described by a single parameter D, which scales linearly with μ (Lin & Swalen 1959). For the rotational degrees of freedom we use the basis set of the prolate symmetric top and plane waves exp (imω) for the internal rotation. The Hamiltonian (2) mixes waves with m' − m = 3n. All relevant matrix elements are tabulated in RF. The fitted values of the parameters are also given there.

After the effective Hamiltonian is formed it is diagonalized numerically. Due to the C₃ symmetry of the Hamiltonian (2) the final eigenstates can be classified as A-type states and twofold degenerate E-type states. A-type states have definite parity p: for A⁺ states p = (− 1)^J and for A⁻ states p = (− 1)^{J + 1}. Because methanol is close to the symmetric top its states are classified with an approximate quantum number K, which corresponds to the projection of the angular momentum $\bm {J}$ on the axis of the CH₃ group. In this study we are interested only in the lowest states of the internal rotation of A and E symmetry: excitation of higher states requires kinetic temperatures T ≳ 300 K, while warm molecular clouds have typical temperatures T ≲ 100 K.

The CH₃OH transitions with ΔK = 0 can be considered approximately as rotational, where the state of the internal motion does not change. These transitions have "normal" sensitivities to μ-variation with Q_μ ≈ 1. Because our model does not include centrifugal corrections, Q_μ must be exactly equal to 1. We estimate that these corrections do not exceed 3% for J ⩽ 10. The J11 model accounts for these corrections and provides the sensitivity coefficients for the ΔK = 0 transitions between 1.00 and 1.03 in agreement with our estimates. We note that this deviation from unity is of the same order of magnitude as the vibrational correction which is not accounted for in both the RF and J11 effective Hamiltonians. The transitions with ΔK = ±1, on the contrary, lead to the change in the internal motion and have sensitivities to μ-variation which vary in a wide range. Such transitions are of primary interest for our purpose.

In order to determine the sensitivity coefficients Q_μ for the microwave transitions we first find the dependence of the eigenvalues E_i on Δμ/μ:

$$\begin{equation} \Delta E_i = q_i \frac{\Delta \mu }{\mu }, \end{equation} \tag{ 3 }$$

where the coefficient q_i, individual for each level, shows a response of the level E_i to a small change of μ (|Δμ/μ| ≪ 1). This is done by diagonalizing the effective Hamiltonian for three sets of the parameters, which correspond to μ = μ₀ and μ = μ₀(1 ± ε), where ε equals to 0.001 or 0.0001 (in both cases we obtain the same q-factors). The parameters A, B, C, D, and F, all scaling linearly with μ, should then take the values A₀, A₀(1 ± ε), etc., where the first value A₀ corresponds to the fit of the experimental spectrum. The dimensionless sensitivity coefficient Q_μ for the transition ω = E_up − E_low is found through calculation of the corresponding q-factor

$$\begin{equation} \Delta \omega = q \frac{\Delta \mu }{\mu }, \end{equation} \tag{ 4 }$$

where q = q_up − q_low, and

$$\begin{equation} Q_\mu = \frac{q}{\omega }. \end{equation} \tag{ 5 }$$

The results of calculations for transitions with ΔK = ±1 are listed in Table 1, where we also give experimental and calculated transition frequencies. One can see that the accuracy of our model is about 1 GHz for all considered transitions. Consequently, the relative error for the high-frequency transitions is only a fraction of a percent, but for the lowest frequency 6.7 GHz it is 15%. In order to improve the accuracy of the Q_μ-values we use experimental frequencies in Equation (5). To check the computational procedure we also determined Q_μ for a few transitions with ΔK = 0 (0₀ − 1₀A⁺ 48.4 GHz, 1₀ − 2₀A⁺ 96.9 GHz, 1₁ − 2₁A⁺ 96.1 GHz, 1₁ − 2₁A⁻ 97.7 GHz, and 4₂ − 5₂A⁻ 242.2 GHz) and for all of them found Q_μ = 1.00.

Table 1. Numerical Calculation of the Q-factors for the Low-frequency Mixed Rotation-tunneling Transitions (ΔK = ±1) in Methanol

Transition	ω (MHz)		Qμ		Transition	ω (MHz)		Qμ
${J_l}_{K_l}\hbox{--}{J_u}_{K_u}$	Experiment	Theory			${J_l}_{K_l}\hbox{--}{J_u}_{K_u}$	Experiment	Theory
60–51 A+	6668.5192(4)	5777.8	+43(3)		40–5−1 E	84521.169(10)	85413.3	−3.5(4)
8−2–9−1 E	9936.202(2)	11281.9	−14(1)		63–72 A−	86615.600(5)	86033.5	+5.9(3)
3−1–20 E	12178.597(2)	11487.4	+32(2)		63–72 A+	86902.949(5)	86315.2	+5.9(3)
30–21 E	19967.3961(2)	20012.6	+6.3(3)		71–80 A+	95169.463(10)	96244.8	−1.9(3)
31–32 E	24928.707(7)	25565.7	−17(1)		10−2−11−1 E	104300.414(7)	105949.0	−0.45(16)
41–42 E	24933.468(2)	25597.1	−17(1)		40−31 A+	107013.803(5)	106283.3	+3.6(2)
21–22 E	24934.382(5)	25551.1	−17(1)		1−1–00 E	108893.963(7)	108370.8	+4.5(2)
51–52 E	24959.0789(4)	25655.3	−17(1)		50–6−1 E	132890.692(10)	133898.0	−1.9(3)
61–62 E	25018.1225(4)	25752.6	−16(1)		81–90 A+	146618.794(50)	147799.4	−0.9(2)
71–72 E	25124.8719(4)	25902.7	−16(1)		8−1–80 E	156488.868(10)	155888.7	+3.4(2)
81–82 E	25294.4165(2)	26120.3	−16(1)		30–21 A+	156602.413(10)	155946.7	+2.8(1)
91–92 E	25541.3979(4)	26419.4	−16(1)		7−1–70 E	156828.533(10)	156258.3	+3.4(2)
31–40 E	28316.031(8)*	28351.5	−2.8(3)		6−1–60 E	157048.625(10)	156506.3	+3.4(2)
91–82 A−	28969.942(50)	29091.1	+11.1(6)		5−1–50 E	157179.017(10)	156661.9	+3.4(2)
30–4−1 E	36169.265(30)	36956.3	−9.6(9)		4−1–40 E	157246.056(10)	156750.9	+3.4(2)
8−1–7−2 E	37703.700(30)	36486.0	+5.1(3)		1−1–10 E	157270.851(10)	156817.2	+3.4(2)
53–62 A−	38293.268(50)	37660.7	+12.1(7)		3−1–30 E	157272.369(10)	156795.3	+3.4(2)
53–62 A+	38452.677(50)	37817.3	+12.1(7)		20–21 A+	304208.324(13)*	303751.3	+1.91(7)
61–70 A+	44069.410(10)	45048.7	−5.3(6)		40–41 A+	307165.911(13)*	306658.4	+1.89(7)

Notes. The rest frequencies are taken from Müller et al. (2004; the uncertainties of the first three transitions at 6.668, 9.936, and 12.178 GHz given in Müller et al. 2004 correspond to the 2σ errors in accord with Breckenridge & Kukolich 1995) except for those marked by the asterisk which are from Lovas (2004). The 1σ uncertainties of the least significant figure of the rest frequencies and Q_μ-values are given in parenthesis.

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Table 1 also lists the errors of the sensitivity coefficients in their last digits which are given in parenthesis. These errors were estimated in the following way. The effective Hamiltonian from RF includes the rotational and tunneling parts as well as the interaction between them. The transition frequencies and the sensitivity coefficients are found from the numerical diagonalization of this Hamiltonian followed by the numerical differentiation of the results in respect to μ.

Only the low-frequency transitions can have an enhanced sensitivity to the μ-variation because they are of a mixed character with the rotational and tunneling contributions to the transition energy:

$$\begin{equation} \omega = \omega _r - \omega _t. \end{equation} \tag{ 6 }$$

These two contributions have the following dependences on μ:

$$\begin{equation} \frac{\Delta \omega _r}{\omega _r} = Q^r_\mu \frac{\Delta \mu }{\mu }, \quad \frac{\Delta \omega _t}{\omega _t} = Q^t_\mu \frac{\Delta \mu }{\mu }. \end{equation} \tag{ 7 }$$

Then, the resultant sensitivity of the transition ω is given by

$$\begin{equation} \frac{\Delta \omega }{\omega } =\bigg(Q^r_\mu \frac{\omega _r}{\omega } -Q^t_\mu \frac{\omega _t}{\omega } \bigg)\frac{\Delta \mu }{\mu } \equiv Q_\mu \frac{\Delta \mu }{\mu }. \end{equation} \tag{ 8 }$$

Let us suppose that the sensitivities Q^r_μ and Q^t_μ are known with a relative error ε. Then, the absolute error of Q_μ is equal to

$$\begin{equation} \Delta Q_\mu =\varepsilon \times \frac{Q^r_\mu \omega _r +Q^t_\mu \omega _t}{\omega }. \end{equation} \tag{ 9 }$$

In order to use this expression to estimate the errors ΔQ_μ we need to know the decomposition (6) and the sensitivity coefficients Q^r_μ and Q^t_μ. As we discussed earlier, Q^r_μ ≈ 1. We can estimate the tunneling sensitivity to be Q^t_μ ≈ 2.6 from the semi-classical (Wentzel-Kramers-Brillouin (WKB)) approximation (e.g., Kozlov et al. 2010). The tunneling part of the transition energy ω_t can be estimated from the model of Hecht & Dennison (1957). The rotational energy ω_r is then given by the experimental frequency ω and Equation (6).

We conservatively estimate the relative error of our calculations of Q^r_μ and Q^t_μ to be of about 3%, i.e., ε = 0.03. This error is associated with the missing centrifugal distortion and vibration correction which changes the rotational and tunneling frequencies by a few percent at the most. The resulting errors ΔQ_μ are estimated from Equation (9).

Table 1 shows that the sensitivity coefficients of the mixed rotation-tunneling transitions at low frequencies lie in the range between −17(1) and +43(3). For comparison, the H₂ Lyman and Werner transitions have Q_μ ∼ 0.03 (Varshalovich & Levshakov 1993), and the ammonia inversion transition has Q_μ = 4.46 (Flambaum & Kozlov 2007). This means that methanol is almost 1000 times more sensitive to the change in μ than molecular hydrogen and about 10 times more sensitive than ammonia. Besides, CH₃OH has many lines with different sensitivities of both signs which allows us to estimate Δμ/μ from observations of only this one molecule. This is the advantage over, e.g., ammonia NH₃ which has the same sensitivity coefficients for the inversion transitions (1,1), (2,2), etc., and, hence, requires some rotational transitions of other molecule as a reference in order to trace Δμ/μ.

Now we can compare our Q_μ-values with the sensitivity coefficients K_μ derived in J11. We note that μ was defined in J11 as m_p/m_e and, hence, one expects Q_μ = −K_μ. This comparison is shown graphically in Figure 1. The error bars in this figure mark the 2σ uncertainties. In total we have 35 common computations of the sensitivity coefficients. The theoretical calculations of their values show a good concordance except for the six points at 9.936, 37.703, 38.293, 38.453, 86.615, and 86.902 GHz where the discrepancy |Q_μ + K_μ| exceeds the 2σ level (in Figure 1, the corresponding Q_μ values are −14, 5.1, 12.1, 12.1, 5.9, and 5.9). The reason for such deviations is not clear. The highest offsets of 2.8σ and 2.7σ are found for the J = 5–6 transitions at 38.293 and 38.452 GHz, respectively, where the RF model should have a sufficiently accurate result. On the other hand, the errors of the K_μ values were taken in J11 to be 5% if |K_μ| ⩾ 1 (relative error) or 0.05 if |K_μ| < 1 (absolute error). However, analysis based on Equation (9) shows that the relative errors noticeably differ from line to line.

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The agreement between the values determined with independent methods gives us confidence that, given the high Q_μ values with different signs, methanol is excellent for testing the electron-to-proton mass ratio. Below we consider such a test based on high angular and high spectral resolution observations of CH₃OH maser lines.

Equations (4) and (5) show that for a given transition from Table 1, ω_i, with the sensitivity coefficient Q_i, the expected frequency shift, Δω_i/ω_i, due to a change in μ is given by

$$\begin{equation} \frac{\Delta \omega _i}{\omega _i} = Q_i\frac{\Delta \mu }{\mu }. \end{equation} \tag{ 10 }$$

Then the value of Δμ/μ can be estimated from two transitions with different sensitivity coefficients Q_i and Q_j:

$$\begin{equation} \frac{\Delta \mu }{\mu } = \frac{V_j - V_i}{c(Q_i - Q_j)}, \end{equation} \tag{ 11 }$$

where V_j and V_i are the apparent radial velocities of the corresponding CH₃OH transitions and c is the speed of light.

Interstellar CH₃OH lines were widely observed in the last two decades and we can obtain some preliminary estimates of Δμ/μ using the published data. In particular, the maser CH₃OH emission is of a special interest here since maser lines are narrow and the error of the line center measurement is correspondingly low.

The CH₃OH molecules originate in star-forming regions and are observed as maser emission in two types of sources: class I and class II (e.g., Menten 1991). Class II methanol masers are radiatively pumped and located in the vicinity of young stellar objects (YSOs), whereas class I methanol masers are believed to trace distant parts of the outflows from YSOs and are collisionally pumped. The most accurate Δμ/μ values can be estimated from narrow and symmetric line profiles with similar shapes. Such profiles are provided in interferometric observations of the class I masers located in the vicinity of IRAS 16547–4247 (G343.12–0.06) which is a luminous YSO with a radio jet (Voronkov et al. 2006, hereafter V06). This methanol maser emission consists of a cluster of six spots spread over an area of 30 arcsec in extent. One spot (called B in V06) shows activity in 12 CH₃OH transitions (Figure 3 of V06). Among them, the 9.9 GHz and 104 GHz lines have a narrow spike (FWHM < 30 m s⁻¹) on top of a broader (FWHM ≈ 300 m s⁻¹) symmetric line. The widths of these spikes do not exceed the channel spacing which is 29 m s⁻¹ and 22 m s⁻¹ at 9.9 GHz and 104 GHz, respectively. According to V06, these are the narrowest spectral features ever found. Their line widths imply that the masers are unsaturated and that the turbulent motion in the gas is strongly suppressed. The 9.9 GHz and 104 GHz masers show similar angular sizes, θ_9.9 = 0.10 ± 0.09 arcsec, and θ₁₀₄ = 0.2 ± 0.1 arcsec, and almost the same coordinates, |Δα| = 0.03 ± 0.04 arcsec, and |Δδ| = 0.09 ± 0.04 arcsec. The lines were observed in a month interval on 2005 June 16 (9.9 GHz) and on 2005 August 18 (104 GHz).

As noted in V06, the 9.9 GHz and 104 GHz transitions belong to the same J₋₂ − (J + 1)₋₁ E transition series with J = 8 and 10, respectively, and are expected to show a similar behavior, i.e., both maser lines should originate in the same volume and, hence, have equal radial velocities. The radial velocities of these spikes, V_9.9 = −31.554 km s⁻¹ and V₁₀₄ = −31.594 km s⁻¹, are measured with the uncertainty of a few m s⁻¹, but, unfortunately, their difference, ΔV = V_9.9 − V₁₀₄ = 40 m s⁻¹, is less certain because of the errors in the rest frequencies, ε_9.9 = 60 m s⁻¹  and ε₁₀₄ = 20 m s⁻¹ (see Table 1). Since at these two frequencies the sensitivity coefficients are slightly different for two calculations [Q_{μ, 9.9} = −14(1), −K_{μ, 9.9} = −11.5(6), and Q_{μ, 104} = −0.45(16), −K_{μ, 104} = −0.18(5)], we use their average values $\bar{Q} = (Q_\mu - K_\mu)/2$ to estimate Δμ/μ from Equation (11). Thus, with $\bar{Q}_{\mu, 9.9} = -12.75(58)$, $\bar{Q}_{\mu, 104} = -0.32(8),$ and ΔV = 40(63) m s⁻¹ we find Δμ/μ = 11 ± 17 ppb (1σ, c.l.) or the upper limit |Δμ/μ| < 28 ppb. This single point estimate is in line with a limit derived from the sample mean of ammonia observations in the Milky Way, Δμ/μ = 26 ± 3 ppb (1σ, c.l.), or |Δμ/μ| < 29 ppb (Levshakov et al. 2010a).

The reliability of the present estimate of Δμ/μ can be further improved if new laboratory frequencies of the methanol 9.9 GHz and 104 GHz transitions will be determined with a higher accuracy. Since maser sources are in general variable in time, additional gains can be obtained from simultaneous observations of several methanol lines in a way described, e.g., in Voronkov et al. (2011) where up to eight methanol maser transitions were observed simultaneously with the Australia Telescope Compact Array. Unfortunately, the observed methanol profiles at 24, 25 GHz and 9.9 GHz have close sensitivity coefficients with ΔQ ≈ 3 which is comparable to the ammonia method. This does not allow us to improve the aforementioned upper limit on Δμ/μ. Without new laboratory studies and improvements in astronomical observations, which require substantial care to determine frequencies with an accuracy better than 10⁻⁸, any further advances in exploring Δμ/μ from methanol maser spectra will be impossible.

We thank Maxim Voronkov for details on CH₃OH maser emission from the spot B toward G343.12–0.06, Irina Agafonova for her comments on the manuscript, and Christian Henkel for many helpful discussions. The work has been supported by the grant no. "SFB 676 Teilprojekt C4," the RFBR grant nos. 09-02-00352 and 11-02-00943, and by the State Program "Leading Scientific Schools of Russian Federation" (grant NSh-3769.2010.2).