Enhanced sensitivity to a possible variation of the proton-to-electron mass ratio in ammonia
Alec Owens, Sergei N. Yurchenko, Walter Thiel, Vladimir Špirko
EEnhanced sensitivity to a possible variation of theproton-to-electron mass ratio in ammonia
A. Owens , , ∗ S. N. Yurchenko , W. Thiel , and V. ˇSpirko , Department of Physics and Astronomy,University College London, Gower Street,WC1E 6BT London, United Kingdom Max-Planck-Institut f¨ur Kohlenforschung,Kaiser-Wilhelm-Platz 1, 45470 M¨ulheim an der Ruhr, Germany Academy of Sciences of the Czech Republic,Institute of Organic Chemistry and Biochemistry,Flemingovo n´am. 2, 166 10 Prague 6, Czech Republic and Department of Chemical Physics and Optics,Faculty of Mathematics and Physics,Charles University in Prague, Ke Karlovu 3,CZ-12116 Prague 2, Czech Republic (Dated: August 17, 2018) a r X i v : . [ phy s i c s . c h e m - ph ] A ug bstract Numerous accidental near-degeneracies exist between the 2 ν and ν rotation-vibration energylevels of ammonia. Transitions between these two states possess significantly enhanced sensitivityto a possible variation of the proton-to-electron mass ratio µ . Using a robust variational approachto determine the mass sensitivity of the energy levels along with accurate experimental values forthe energies, sensitivity coefficients have been calculated for over 350 microwave, submillimetre andfar infrared transitions up to J = 15 for NH . The sensitivities are the largest found in ammoniato date. One particular transition, although extremely weak, has a sensitivity of T = − ,
738 andillustrates the huge enhancement that can occur between close-lying energy levels. More promisinghowever are a set of previously measured transitions with T = −
32 to 28. Given the astrophysicalimportance of ammonia, the sensitivities presented here confirm that NH can be used exclusivelyto constrain a spatial or temporal variation of µ . Thus, certain systematic errors which affect theammonia method can be eliminated. For all transitions analyzed we provide frequency data andEinstein A coefficients to guide future laboratory and astronomical observations. ∗ [email protected] . INTRODUCTION Molecules are an attractive testing ground for probing two particular dimensionless fun-damental constants. Electronic transitions are sensitive to the fine structure constant α ,whilst vibration, rotation and inversion transitions are sensitive to the proton-to-electronmass ratio µ = m p /m e . If any variation did exist it would manifest as observable shiftsin the transition frequencies of certain molecular species. Such shifts can be detected byhigh-precision laboratory experiments over short time scales (years), or from astronomicalobservation of spectral lines at high redshift. The idea that the fundamental constants ofnature may be understood within the framework of a deeper cosmological theory dates backto Dirac [1]. As of yet there is no theoretical justification for the values they assume, oreven if they have always had the same values that we measure today.Research in the field has become more active after claims of a temporal variation in thefine structure constant where observations of atomic absorption spectra of distant quasarssuggested that α was smaller in the past [2]. A few years later, measurements of H spectraindicated that the proton-to-electron mass ratio was larger by 0 . T u,l = µE u − E l (cid:18) d E u d µ − d E l d µ (cid:19) (1)where E u and E l is the energy of the upper and lower state, respectively, quantifies the effectthat a possible variation of µ would have for a given transition. It is related to the frequencyshift of the probed transition through the expression,∆ νν = T u,l ∆ µµ (2)where ∆ ν = ν obs − ν is the change in the frequency and ∆ µ = µ obs − µ is the change in µ ,both with respect to their present day values ν and µ .3he ammonia method [8] (adapted from van Veldhoven et al. [9]) compares inversiontransitions in the vibrational ground state of NH (henceforth referred to as NH ) withrotational lines from other molecular species. By employing this approach several con-straints on a temporal variation of µ have been established from measurements of the objectB0218+357 at redshift z ∼ .
685 [8, 10, 11], and from the lensing galaxy PKS1830 − z ∼ .
886 [12]. However, the reliance on other reference molecules, particularly thosewhich are non-nitrogen bearing, is a major source of systematic error (see Refs. [10–12] fora detailed discussion).Methanol is now preferred because not only is it astronomically abundant, but it can beused exclusively to place limits on a drifting µ [13–18], circumventing the errors which arisefrom comparing different molecular species. The most robust constraint to date measuredCH OH absorption lines in the system PKS1830 −
211 [18]. The three observed transitionspossessed sensitivities ranging from T = − . − . NH , NH , ND and ND [7] offered perspectives for the development of the ammonia method. Inversiontransitions in the ν vibrational state had sensitivities from T = − .
27 to 4 .
67, whilst the NH astronomically observed 2 +1 ← − and 0 − ← +0 transitions in the ν state [20, 21]possessed values of T = 17 .
24 and T = − .
59, respectively. Here states are labelled as J ± K where J is the rotational quantum number, K is the projection onto the molecule-fixed z axis, and ± denotes the parity of the state. Because of the abundance of NH throughout theUniverse and the ease with which its spectrum can be observed, identifying more transitionswith large sensitivities in the microwave, submillimetre or far infrared regions could lead toa much tighter constraint on µ .A recent analysis of 56 sources of high-resolution NH spectra utilizing the MARVELprocedure determined 4961 rovibrational energy levels of experimental quality, all labelledusing a consistent set of quantum numbers [22]. This has allowed us to investigate newtransitions of NH and accurately calculate their sensitivity to a possible variation of µ . Asshown in Fig. 1, numerous accidental near-degeneracies occur between the the 2 ν and ν rovibrational energy levels of ammonia. The strong Coriolis interaction between these twostates [23] can give rise to highly anomalous sensitivities. Furthermore, a large number of4 +2 ← +1 +2 → +3 ν ν E n e r g y ( c m − ) FIG. 1. Accidental near-degeneracies between the 2 ν and ν rotation-vibration energy levels ofammonia. Energy levels are labelled as J ± K . For illustrative purposes only part of the rovibrationalmanifold is shown. transitions between these levels have been measured experimentally [24, 25]. II. VARIATIONAL APPROACH
To compute sensitivity coefficients the dependence on µ of the energy levels is required,i.e. the derivatives in Eq. (1). Under the assumption that all baryonic matter may be treatedequally [26], µ is assumed to be proportional to the molecular mass and it is sufficient tosimply compute the mass dependence of the desired energy levels. The variational approachwe employ here is identical to our previous study of ammonia and we refer the reader toRef. [7] (and references therein) for a detailed description. In short, a series of calculationsare performed employing a scaled value for the mass of NH , from which numerical valuesof the derivatives d E/ d µ are obtained by finite differences. After matching the derivativeswith the experimentally determined energy levels from the MARVEL analysis, sensitivitiesare calculated through Eq. (1). We also compute Einstein A coefficients to determine whichtransitions could realistically be detected. All calculations were carried out with the nuclearmotion program TROVE [27]. Note that sensitivities have been computed for H O + andD O + [28] using exactly the same approach.5 II. RESULTS AND DISCUSSION
In Fig. 2 we have simulated the intensities at room temperature for 38 previously observedtransitions from Ref. [24, 25] and plotted their corresponding sensitivity coefficients. Thelargest difference in sensitivity is ∆ T = 59 .
6, which is over nine times more sensitive than the∆ T of the methanol lines used to establish the most robust constraint to date [18], and overseventeen times larger than the ∆ T of the transitions utilized in the ammonia method [8].As well as being consistently large, the mixture of positive and negative sensitivities is highlybeneficial for detecting a change in µ as transitions are shifted in opposing directions. FromFig. 2, one could imagine scanning this frequency window at two separate instances in timeto produce a displaced spectrum if any variation of µ had occurred. In addition to thefrequencies of Ref. [24, 25], there are 153 transitions with similar Einstein A coefficients andsensitivities from T = − .
40 to 17 .
27 in the frequency range 100 to 900 GHz. We providecomprehensive tables of all investigated transitions as supplementary material.The accuracy of the calculated sensitivity coefficients depends on the MARVEL energylevels and the computed TROVE numerical derivatives. The MARVEL analysis offers a rig-orous evaluation of high-resolution NH spectra. The 2 ν energy levels have an average errorof 0 . − for the 251 levels up to J = 15, whilst a similar uncertainty of 0 . − isgiven for the 495 ν energies. As such, the error on the predicted sensitivities is significantlyreduced by replacing the computed TROVE energy levels with the corresponding MARVELvalues in Eq. (1). This is not to say that the TROVE frequencies are unreliable. As part ofthe MARVEL procedure the derived experimental energy levels are checked against theoret-ical predictions using the same potential energy surface (PES) and computational setup [29]as utilized for the present study. This PES is based on extensive high-level ab initio calcu-lations [30] and has subsequently been refined to experimental data up to J ≤ .
5% to the MARVEL substitutedsensitivities, the largest difference being 5 . . ν and ν rovibrational manifold. A striking example of this is for the extremely weak6 requency (MHz) A b s o l u t e i n t e n s i t y ( c m / m o l ec u l e )
375 000325 000275 000225 000175 000125 0002.5 × − × − × − × − × − Frequency (MHz) S e n s i t i v i t y c o e ffi c i e n t
375 000325 000275 000225 000175 000125 0003020100-10-20-30 FIG. 2. Observed frequencies [24, 25] and simulated intensities at temperature T = 296 K (toppanel) with the corresponding sensitivities (bottom panel) for transitions between the 2 ν and ν vibrational states of NH . +3 ( ν ) ← +2 (2 ν ) transition. A computed frequency of ν calc = 3540 . T calc = − .
25, already the largest known sensitivity coefficient for ammonia. Replacingwith MARVEL energy levels gives ν exp = 389 . T exp = − , .
52. The dramaticincrease in magnitude occurs because of the inverse dependence on transition frequency(see Eq. (1)) and illustrates the huge enhancement that can happen between close-lyingenergy levels. Given the difference in predicted sensitivities one could question whetherthe computed numerical derivatives are still reliable. The change in frequency is just over3000 MHz ( ≈ . − ) so one would expect that they are reasonable. The difficulty is thatquantifying the uncertainty of the numerical derivatives is not as straightforward becausethere are no analogous highly accurate experimental quantities.To investigate the error of the computed derivatives, new sensitivity coefficients were7alculated using a purely ab initio PES [30]. One can hope to establish a relationshipbetween the difference in ν = E u − E l , with the difference in the quantity d E u / d µ − d E l / d µ ,by comparing values computed using this and the empirically refined PESs. Whilst no cleargeneral correspondence between the uncertainty on these two quantities emerges, for near-coinciding energy levels separated by 1 cm − or less, the percentage difference in d E u / d µ − d E l / d µ is always smaller than the percentage difference in ν . This ranges from 3-4 timessmaller to several orders of magnitude smaller and suggests that for extremely close-lyingenergy levels, the underlying numerical derivatives are relatively stable. Thus, the hugeamplification in sensitivity we predict is a result of replacing the theoretical frequencieswith experimental values.For the transitions shown in Fig. 2 and those with similar Einstein A coefficients, thereis consistent agreement between the TROVE and MARVEL substituted sensitivities anderrors in the computed derivatives will be negligible. When the two predictions differ signif-icantly, which occurs for a number of weaker transitions with incredibly large sensitivitiesranging from T = − .
84 to 509 .
21 (see supplementary material), we are confident thatthe MARVEL substituted sensitivity coefficients are reliable. In all instances the residualbetween experiment and computed transition frequency never exceeds 1 cm − (regarded asspectroscopic accuracy). IV. OUTLOOK
Finally, we briefly comment on possible experimental tests of our predictions. There arenow novel techniques to produce ultracold polyatomic molecules [32], which have rich spectrawell suited for testing fundamental physics. Already experiments which decelerate, cool andtrap ammonia molecules are being developed to probe a temporal variation of µ [9, 33–36].In Table I we list several highly sensitive transitions, which despite being around two ordersof magnitude weaker than the lowest intensity lines displayed in Fig. 2, could possibly bedetected in such high-precision studies.If the transitions in Table I are too weak to be detected directly, the use of combinationdifferences involving infrared transitions from the ground vibrational state to the 2 ν and ν vibrational states should be considered. This technique would apply to any two levelsprovided transitions from a common ground state level can be identified, or for a situation8 ABLE I. Highly sensitive weak transitions between the 2 ν and ν vibrational states of NH . ν (cid:48) ← ν (cid:48)(cid:48) J ± K (cid:48) ← J ± K (cid:48)(cid:48) ν exp /MHz A /s − T ν ← ν +2 ← +1
61 712.7 1.042 × − ν ← ν − ← +1
110 957.2 9.461 × − -54.08 ν ← ν +2 ← +1
123 427.8 4.745 × − -44.11 ν ← ν − ← +1
169 341.3 2.539 × − -37.57 All transitions are of symmetry E (cid:48) ← E (cid:48)(cid:48) . Experimental frequencies have been obtained using energy levelsfrom the MARVEL analysis [22] . T = − . ν =0 .
79 cm − T = − . ν =1596 .
06 cm − T = − . ν =1594 .
79 cm − T =107 . ν =2 .
06 cm − +2 − ground2 +2 (2 ν ) 1 +1 ( ν ) FIG. 3. Use of combination differences involving infrared transitions from the ground vibrationalstate to the 2 ν and ν vibrational states of ammonia. Energy levels are labelled as J ± K . such as that depicted in Fig. 3. Infrared transitions to the respective levels of the 2 +2 (2 ν ) ← +1 ( ν ) transition (sensitivity of T = 107 .
95) have been measured experimentally [25], whilstthe corresponding ground state pure inversion frequency is well known [37]. Combinationdifferences could be utilized to determine a possible shift in these energy levels provided thesensitivities of the three involved transitions are also known. The large number of potentialcombination differences prohibits us from carrying out a rigorous evaluation of all possibletransitions. However, if particular combination differences could be readily measured in thefuture, it would be straightforward to compute the required sensitivity coefficients.Although laboratory experiments have greater control over systematic effects they provideonly a local constraint on a drifting constant. It could be argued that even a null resultwould be limited to the age of the Solar System (around 4.6 billion years) and that a9ariation of µ could have occurred at earlier stages in the evolution of the Universe. Moredesirable are molecular systems which are astronomically relevant because observation atdifferent redshifts presents the opportunity to look back to much earlier times in the Universe.Detection in a wide variety of cosmological settings also lends itself to searches for possiblespatial variations of µ , for which a number of studies using ammonia have been reported [38–41].The transitions presented in this study are perhaps more likely to be detected in terres-trial studies given that the rovibrational states involved lie above 1600 cm − . Astronomicaldetection is not impossible however. The energy levels of the ( J, K ) = (18 ,
18) inversiontransition in the ground vibrational state of NH reside at 2176 .
93 and 2178 .
47 cm − , re-spectively, and this line was observed towards the galactic center star forming region SgrB2 [42]. A number of 2 ν ↔ ν transitions which possess sizeable Einstein A coefficientsand involve energy levels lower than the ( J, K ) = (18 ,
18) energies are listed in Table II.Such highly excited states could effectively be populated by exoergic chemical formation pro-cesses [43]. It is hoped that future astronomical observations search for these particularlysensitive transitions.
ACKNOWLEDGMENTS
S.Y. thanks ERC Advanced Investigator Project 267219. V.S. acknowledges researchproject RVO:61388963 (IOCB) and support from the Czech Science Foundation (grantP209/15-10267S). 10
ABLE II. Astronomically relevant transitions between the 2 ν and ν vibrational states of NH . ν (cid:48) ← ν (cid:48)(cid:48) J ± K (cid:48) ← J ± K (cid:48)(cid:48) ν exp /MHz A /s − Tν ← ν +0 ← +1
379 596.5 4.703 × − -18.70 ν ← ν +1 ← +0
824 624.2 6.427 × − -9.132 ν ← ν +1 ← +0
231 528.2 1.180 × − ν ← ν +2 ← +1
687 852.5 6.318 × − -10.702 ν ← ν +3 ← +2
489 672.2 4.360 × − ν ← ν +3 ← +2
557 275.3 7.623 × − -12.872 ν ← ν +2 ← +1
672 644.4 3.223 × − ν ← ν +2 ← +1
679 163.4 6.964 × − -10.79 ν ← ν +1 ← +0
774 889.5 4.660 × − -9.592 ν ← ν +1 ← +0
842 667.6 1.210 × − ν ← ν +4 ← +3
441 874.1 7.796 × − -15.68 ν ← ν +3 ← +2
548 781.8 9.102 × − -12.94 ν ← ν +2 ← +1
657 787.0 6.078 × − -11.07 ν ← ν +5 ← +4
342 797.1 7.054 × − -19.22 ν ← ν +4 ← +3
434 941.1 9.782 × − -15.59 ν ← ν +3 ← +2
527 333.3 8.219 × − -13.31 ν ← ν +2 ← +1
618 776.8 4.583 × − -11.66 ν ← ν +1 ← +0
672 376.5 2.542 × − -10.86 ν ← ν +6 ← +5
261 535.4 5.745 × − -23.29 ν ← ν +5 ← +4
340 322.9 9.137 × − -18.52 ν ← ν +4 ← +3
413 748.1 9.006 × − -15.98 ν ← ν +3 ← +2
488 661.3 6.308 × − -14.12 ν ← ν +2 ← +1
559 214.0 3.027 × − -12.73 ν ← ν +7 ← +6
198 997.4 4.284 × − -27.24 ν ← ν +6 ← +5
266 541.0 7.700 × − -21.20 ν ← ν +5 ← +4
321 935.0 8.437 × − -18.62 ν ← ν +4 ← +3
375 174.5 6.887 × − -16.99 ν ← ν +3 ← +2
430 468.6 4.113 × − -15.29 ν ← ν +8 ← +7
154 415.5 3.036 × − -30.19 For symmetry of transitions see supplementary material. Experimental frequencies from Ref. [24, 25] orobtained using energy levels from the MARVEL analysis [22].
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