Context :

Aims : We probe the dependence of \alpha ^ { 2 } / \mu on the ambient matter density by means of submm- and mm-wave bands spectral observations in the Milky Way .

Methods : A procedure is suggested to explore the value of F = \alpha ^ { 2 } / \mu , where \mu = m _ { e } / m _ { p } is the electron-to-proton mass ratio , and \alpha = e ^ { 2 } / ( \hbar c ) is the fine-structure constant .
The fundamental physical constants , which are measured in different physical environments of high ( terrestrial ) and low ( interstellar ) densities of baryonic matter are supposed to vary in chameleon-like scalar field models , which predict that both masses and coupling constant may depend on the local matter density .
The parameter \Delta F / F = ( F _ { obs } - F _ { lab } ) / F _ { lab } can be estimated from the radial velocity offset , \Delta V = V _ { rot } - V _ { fs } , between the low-laying rotational transitions in carbon monoxide ^ { 13 } CO and the fine-structure transitions in atomic carbon [ C i ] .
A model-dependent constraint on \Delta \alpha / \alpha can be obtained from \Delta F / F using \Delta \mu / \mu independently measured from the ammonia method .

Results : Currently available radio astronomical datasets provide an upper limit on | \Delta V| < 110 m s ^ { -1 } ( 1 \sigma ) .
When interpreted in terms of the spatial variation of F , this gives | \Delta F / F| < 3.7 \times 10 ^ { -7 } .
An order of magnitude improvement of this limit will allow us to test independently a non-zero value of \Delta \mu / \mu = ( 2.2 \pm 0.4 _ { stat } \pm 0.3 _ { sys } ) \times 10 ^ { -8 } recently found with the ammonia method .
Taking into account that the ammonia method restricts the spatial variation of \mu at the level of | \Delta \mu / \mu| \leq 3 \times 10 ^ { -8 } and assuming that \Delta F / F is the same in the entire interstellar medium , one obtains that the spatial variation of \alpha does not exceed the value | \Delta \alpha / \alpha| < 2 \times 10 ^ { -7 } .
Since extragalactic gas clouds have densities similar to those in the interstellar medium , the bound on \Delta \alpha / \alpha is also expected to be less than 2 \times 10 ^ { -7 } at high redshift if no significant temporal dependence of \alpha is present .

Conclusions :