Enhanced sensitivity to time-variation of m_p/m_e in the inversion spectrum of ammonia
aa r X i v : . [ a s t r o - ph ] A p r Enhanced sensitivity to time-variation of m p /m e in the inversion spectrum of ammonia V. V. Flambaum , and M. G. Kozlov , School of Physics, University of New South Wales, Sydney, 2052 Australia Institute for Advanced Study, Massey University (Albany Campus),Private Bag 102904, North Shore MSC Auckland, New Zealand and Petersburg Nuclear Physics Institute, Gatchina, 188300, Russia (Dated: November 1, 2018)We calculate the sensitivity of the inversion spectrum of ammonia to possible time-variationof the ratio of the proton mass to the electron mass, µ = m p /m e . For the inversion transition( λ ≈ .
25 cm − ) the relative frequency shift is significantly enhanced: δω/ω = − . δµ/µ . Thisenhancement allows one to increase sensitivity to the time-variation of µ using NH spectra for highredshift objects. We use published data on microwave spectra of the object B0218+357 to placethe limit δµ/µ = (0 . ± . × − at redshift z = 0 . µ/µ = ( − ± × − yr − . PACS numbers: 06.20.Jr, 06.30.Ft
INTRODUCTION
The possible time-variation of the fundamental con-stants has been discussed for a long time. The interestin this discussion has grown considerably after the re-cent discovery of the acceleration of the universe. Thelatter is usually regarded as evidence for the existenceof dark energy. Cosmological evolution of dark energymay cause variations in fundamental constants, such asthe fine-structure constant α and the proton to electronmass ratio, µ ≡ m p /m e . The electron mass is one ofthe parameters of the Standard Model, it is proportionalto the vacuum expectation value of the Higgs field (theweak scale). The proton mass is proportional to anotherfundamental parameter, the quantum chromodynamicsscale Λ QCD ( m p ≈ QCD ). The proportionality coef-ficients cancel out in the relative variation. Therefore,we are speaking about the relative variation of a veryimportant dimensionless fundamental parameter of theStandard Model, the ratio of the strong to weak scale,defined as δ (Λ QCD /m e ) / (Λ QCD /m e ) = δµ/µ .It is known that µ defines the scales of electronic, vi-brational, and rotational intervals in molecular spectra, E el : E vib : E rot ∼ µ − / : µ − . Similarly, the ratio ofelectronic and hyperfine intervals in atoms and moleculesalso depend on µ , E el : E hfs ∼ α g p µ − , where g p is theproton g -factor. These scalings are used to look for thetime-variation of µ by comparing electronic, vibrational,rotational, and hyperfine spectra of atoms and molecules[1, 2]. In the most recent astrophysical studies [3] a non-zero effect was reported for two quasars at 3 . σ level: δµ/µ = (20 ± × − , (1)at a time scale of approximately 12 Gyr. Assuming lin-ear variation with time this result translates into ˙ µ/µ =( − ± × − yr − . A different method, compari-son of the hyperfine transition in atomic hydrogen with optical transitions in ions, was used in Refs. [4, 5]. Thismethod allows one to study variation of the parameter x = α g p /µ . Analysis of 9 quasar spectra with redshifts0 . ≤ z ≤ .
35 gave δx/x = (6 . ± . × − , (2)˙ x/x = ( − ± × − yr − , (3)which is consistent with zero variation of µ . In Refs. [6,7, 8] the 18 cm λ -doublet lines in OH molecule were stud-ied from objects at the redshifts z ≈ . z ≈ . z ≈ .
765 and no time-variation of the parameter g p ( α µ ) ν was seen, where ν < ∼ µ to placea new limit on the time-variation of µ at the cosmologi-cal timescale. The NH molecule has a pyramidal shapeand the inversion frequency depends on the exponentiallysmall tunneling of three hydrogens through the potentialbarrier [9]. Because of that it is very sensitive to anychanges of the parameters of the system, particularly tothe reduced mass for this vibrational mode [10].We use high-resolution ammonia spectra for gravita-tional lens B0218+357, published by Henkel et al. [11].The redshifts for ammonia lines are compared to the red-shifts for the rotational lines of other molecules measuredin Refs. [11, 12, 13]. The ammonia lines have an orderof magnitude stronger dependence on µ than the usualvibrational lines; this enhancement allows us to place thebest limit on the variation of µ . INVERSION SPECTRUM OF NH The inversion spectrum of NH has been studied for avery long time [9] and is considered a classical exampleof the tunneling phenomenon. The inversion vibrationalmode is described by a double well potential with firsttwo vibrational levels lying below the barrier. Because ofthe tunneling, these two levels are split in inversion dou-blets. The lower doublet corresponds to the wavelength λ ≈ .
25 cm and is used in ammonia masers. Molec-ular rotation leads to the centrifugal distortion of thepotential curve. Because of that the inversion splittingdepends on the rotational angular momentum J and itsprojection on the molecular symmetry axis K : ω inv ( J, K ) = ω − c (cid:2) J ( J + 1) − K (cid:3) + c K , (4)where we omitted terms with higher powers of J and K .Numerically, ω ≈ .
787 GHz, c ≈ . c ≈ . N, the hyperfine struc-ture is dominated by the electric quadrupole interaction( ∼ K = 0 the levels with J = K are metastable and inlaboratory experiments the width of the correspondinginversion lines is usually determined by collisional broad-ening. In astrophysics, the hyperfine structure for spectrawith high redshifts is not resolved and we will not discussit here.For our purposes it is important to know how theparameters in (4) depend on fundamental constants.One can measure only dimensionless ratios of frequencieswhich do not depend on the units used. It is convenientto consider all parameters in atomic units. The energyunit Hartree is E H = m e e / ¯ h = e /a B , where a B is theBohr radius ( E H =2 Ry=219475 cm − ). In these unitsall electron energies ( E e /E H ) and electrostatic poten-tials ( U ( r ) /E H ) have no dependence on the fundamentalconstants (here we neglect small relativistic correctionswhich give a weak α dependence), the vibrational inter-vals ∼ µ − / and the rotational intervals ∼ µ − . Theinversion frequency ω /E H and constants c , /E H arealso functions of µ only (see below). Note that the coef-ficients c i depend on µ through the reduced mass of theinversion mode and because they are inversely propor-tional to the molecular moments of inertia. That impliesa different scaling of ω and c i with µ . The magnetichyperfine structure of NH is due to the interaction ofnuclear magnetic moments and proportional to α g p µ − .We see that different frequencies in the inversion spec-trum scale differently with µ and α . In principle, thisallows one to study time-variation of µ and α by com-paring different lines of the inversion spectrum. On theother hand, it may be preferable to use independent ref-erences (see below). INVERSION HAMILTONIAN
The inversion spectrum (4) can be approximately de-scribed by the following Hamiltonian: H inv = − M ∂ x + U ( x ) (5)+ I ( x ) (cid:2) J ( J + 1) − K (cid:3) + I ( x ) K , where x is the distance from N to the H-plane, I , I are moments of inertia perpendicular and parallel to themolecular axis correspondingly and M is the reducedmass for the inversion mode. If we assume that the length d of the N—H bond does not change during inversion,then M = 2 . m p and I ( x ) ≈ m p d (cid:2) . x/d ) (cid:3) , (6) I ( x ) ≈ m p d (cid:2) − ( x/d ) (cid:3) . (7)The dependence of I , on x generates a correction to thepotential energy of the form C ( J, K ) x /µ . This changesthe vibrational frequency and the effective height of thepotential barrier, therefore changing the inversion fre-quency ω inv given by Eq. (4).Following [15] we can write the potential U ( x ) in (5)in the following form: U ( x ) = kx + b exp (cid:0) − cx (cid:1) . (8)Fitting vibrational frequencies for NH and ND gives k ≈ . b ≈ . c ≈ . δω inv ω inv ≈ − . δµµ . (9)It is instructive to reproduce this result from an analyt-ical calculation. In the semiclassical approximation theinversion frequency is estimated as [16]: ω inv = ω v π exp ( − S ) (10a)= ω v π exp (cid:18) − h Z a − a p M ( U ( x ) − E )d x (cid:19) , (10b)where ω v is the vibrational frequency of the inversionmode, S is the action in units of ¯ h , x = ± a are classicalturning points for the energy E . For the lowest vibra-tional state E = U min + ω v . Using the experimentalvalues ω v =950 cm − and ω inv =0.8 cm − , we get S ≈ . ω on the mass ratio µ . Let us present S in the following form: S = Aµ / R a − a p ( U ( x ) − E ) /E H d ( x/a B ), where A is a numerical constant. We see that the dependence of ω on µ appears from the factor µ / in S and fromthe vibrational frequency ω v and E − U min = ω v whichare proportional to µ − / . Below we assume that allenergies are measured in atomic units and omit theatomic energy unit E H . Then we obtaind ω d µ = − ω (cid:18) µ + d S d µ (cid:19) (11a)= − ω (cid:18) µ + ∂S∂µ + ∂S∂E ∂E∂µ (cid:19) , (11b)where we took into account that ∂S/∂a = 0 because theintegrand in (10b) turns to zero at x = ± a .It is easy to see that ∂S/∂µ = S/ µ . The value ofthe third term in Eq. (11b) depends on the form of thepotential barrier: ∂S∂E = − q SU max − E , (12)where for the square barrier q = 1, and for the triangularbarrier q = 3. For a more realistic barrier shape q ≈ U max we get: δω ω ≈ − δµ µ (cid:18) S + S ω v U max − E (cid:19) = − . δµµ . (13)We see that the inversion frequency of NH is an orderof magnitude more sensitive to the change of µ than typ-ical vibrational frequencies. The reason for that is clearfrom Eq. (13): it is the large value of the action S for thetunneling process.Let us also find the dependence of the constants c , on µ in Eq. (4). According to Eqs. (5) – (7) both constantsmust have the same dependence on µ . Below we focuson the constant c , which is linked to the last term inHamiltonian (5). It follows from Eq. (7) that this termgenerates a correction to the potential: δU ( x ) = K m p d x . (14)This correction does not change the height of the barrier,but changes the energy E = U min + ω v in (10b) by rais-ing the potential minimum and increasing the vibrationalfrequency: U min → U min + K m p d x , (15) ω v → ω v (cid:18) K m p d k (cid:19) . (16)With the help of Eq. (12) with q = 2 we can find theconstant c : c = ω m p d k (cid:18) kx + ω v U max − E S (cid:19) . (17)We can differentiate Eq. (17) to estimate how c dependson µ . This leads to: δc /c = − . δµ/µ , while the nu-merical solution with Hamiltonian (5) gives: δc , c , = − . δµµ . (18) It is clear that NH is not the only molecule with en-hanced sensitivity to variation of µ . Similar enhancementshould take place for all tunneling transitions in molecu-lar spectra. For example, the inversion frequency for ND molecule is 15 times smaller than for NH and Eq. (10a)leads to S ≈ .
4, compared to S ≈ . . Accord-ing to Eq. (13) that leads to a slightly higher sensitivityof the inversion frequency to µ [10]:ND : δω inv ω inv ≈ − . δµµ , δc c ≈ − . δµµ . (19) REDSHIFTS FOR MOLECULAR LINES IN THEMICROWAVE SPECTRA OF B0218+357
In the previous section we saw that the inversion fre-quency ω and the rotational intervals ω inv ( J , K ) − ω inv ( J , K ) have different dependencies on the constant µ . In principle, that allows one to study time-variation of µ by comparing different intervals in the inversion spec-trum of ammonia. For example, if we compare the rota-tional interval to the inversion frequency, then Eqs. (9)and (18) give: δ { [ ω inv ( J , K ) − ω inv ( J , K )] /ω } [ ω inv ( J , K ) − ω inv ( J , K )] /ω = − . δµµ . (20)The relative effects are substantially larger if we comparethe inversion transitions with the transitions between thequadrupole and magnetic hyperfine components. How-ever, in practice this method will not work because of thesmallness of the hyperfine structure compared to typicalline widths in astrophysics.It is more promising to compare the inversion spectrumof NH with rotational spectra of other molecules, where δω rot ω rot = − δµµ . (21)In astrophysics any frequency shift is related to a corre-sponding redshift: δωω = − δz z . (22)According to Eqs. (9) and (21), for a given astrophysicalobject with z = z variation of µ will lead to a change ofthe redshifts of all rotational lines δz rot = (1 + z ) δµ/µ and corresponding shifts of all inversion lines of ammo-nia δz inv = 4 .
46 (1 + z ) δµ/µ . Therefore, comparing theredshift for NH with the redshifts for rotational lines wecan find δµ/µ : δµµ = 0 . z inv − z rot z . (23) TABLE I: Redshifts for molecular rotational lines, ammoniainversion lines, and hydrogen hyperfine line in the spectrumof B0218+357. Rotational linesCO J = 1 → + , HCN average 0.68466(1) [17]Inversion lines of NH NH ( J, K ) = (1 ,
1) red-shifted 0.684679(3) [11]blue-shifted 0.684649(15) [11]= (2 ,
2) red-shifted 0.684677(3) [11]blue-shifted 0.684650(17) [11]= (3 ,
3) red-shifted 0.684673(3) [11]blue-shifted 0.684627(33) [11]average red-shifted 0.684676(3)average blue-shifted 0.684647(11)H λ = 21 cm average 0.68466(4) [18] In Table I we list the redshifts for microwave lines inthe spectrum of the object B0218+357. Three inver-sion lines (
J, K ) = (1 , , (2 , , and (3 ,
3) are reported inRef. [11]. Each of them consists of a narrow red-shiftedand a wide blue-shifted component. The splitting be-tween the red-shifted and blue-shifted components, whichis about 5 km/s, is ascribed to the complicated struc-ture of the molecular cloud [11]. Using average red-shifts of these inversion components (0.684676(3) and0.684647(11)) from Table I we can calculate the aver-age deviation of the inversion redshift in respect to theaverage molecular redshift (0.68466(1)):∆ z unweightedav = (0 . ± . × − , (24)∆ z weightedav = (0 . ± . × − . (25)Eq. (23) gives the following estimate for variation of µ : δµµ = 10 − × (cid:26) . ± . , . ± . . (26)As a final result we present a conservative limit withlarger error bars to cover the total interval between theminimal and maximal values for both estimates: δµµ = (0 . ± . × − . (27)We can also compare averaged redshift for ammoniawith that of hydrogen to get a restriction on the variationof the parameter y = α g p µ . : δyy = z inv − z hfs z = (1 ± × − . (28)The estimates (26–28) can be further improved bydedicated analysis of the molecular spectra published inRefs. [11, 12, 13]. As it was mentioned in [12], the major-ity of molecular lines from B0218+357 have two velocity components. The same applies to the hydrogenic 21 cmline [19]. Instead of taking an average, as we have done in(26–28), all red-shifted and all blue-shifted componentsshould be analyzed independently. That may allow oneto reduce the error bars significantly.We thank M. Kuchiev for helpful discussions andJ. Ginges for reading the manuscript. This work is sup-ported by the Australian Research Council, Godfrey fundand Russian foundation for Basic Research, grant No.05-02-16914. [1] D. A. Varshalovich and A. Y. Potekhin, Astronomy Let-ters , 1 (1996).[2] M. J. Drinkwater, J. K. Webb, J. D. Barrow, and V. V.Flambaum, Mon. Not. R. Astron. Soc. , 457 (1998).[3] E. Reinhold, R. Buning, U. Hollenstein, A. Ivanchik,P. Petitjean, and W. Ubachs, Phys. Rev. Lett. , 151101(2006).[4] P. Tzanavaris, J. K. Webb, M. T. Murphy, V. V. Flam-baum, and S. J. Curran, Phys. Rev. Lett. , 041301(2005), astro-ph/0412649.[5] P. Tzanavaris, J. K. Webb, M. T. Murphy, V. V. Flam-baum, and S. J. Curran, Mon. Not. R. Astron. Soc. ,634 (2007).[6] N. Kanekar, J. N. Chengalur, and T. Ghosh, Phys. Rev.Lett. , 051302 (2004), astro-ph/0406121.[7] N. Kanekar, C. L. Carilli, G. I. Langston, et al., Phys.Rev. Lett. , 261301 (2005).[8] J. N. Chengalur and N. Kanekar, Phys. Rev. Lett. ,241302 (2003), astro-ph/0310764.[9] C. Townes and A. Schawlow, Microwave Spectroscopy (McGraw-Hill, New York, 1955).[10] J. van Veldhoven, J. K¨uper, H. L. Bethlem, B. Starkov,A. J. A. van Roij, and G. Meijer, Eur. Phys. J. D , 337(2004).[11] C. Henkel, N. Jethava, A. Kraus, K. M. Menten, C. L.Carilli, M. Grasshoff, D. Lubowich, and M. J. Reid, As-tronomy and Astrophysics , 893 (2005).[12] T. Wiklind and F. Combes, Astronomy and Astrophysics , 382 (1995).[13] F. Combes and T. Wiklind, Astronomy and Astrophysics , L61 (1995).[14] P. T. P. Ho and C. H. Townes, Ann. Rev. Astron. Astro-phys. , 239 (1983).[15] J. D. Swalen and J. A. Ibers, J. Chem. Phys. , 1914(1962).[16] L. D. Landau and E. M. Lifshitz, Quantum mechanics (Pergamon, Oxford, 1977), 3rd ed.[17] F. Combes and T. Wiklind, Astrophysical Journal ,L79 (1997).[18] C. L. Carilli, M. P. Rupen, and B. Yanny, AstrophysicalJournal , L59 (1993).[19] M. T. Murphy, J. K. Webb, V. V. Flambaum, C. W.Churchill, and J. X. Prochaska, Mon. Not. R. Astron.Soc.327