The radio signal used for communicating with Pioneer 10 and 11 (and indeed, used routinely with other spacecraft) is circularly polarized. Therefore, the question naturally arises as to whether the coupling between the helicity of the radio signal and the rotation of either the transmitter or the receiver could contribute to the observed Doppler anomaly in the Pioneer radio signal . To first order, this coupling can increase or decrease the frequency of a radio signal by the rotational frequency of the transmitter (or receiver):
For the Pioneer spacecraft, (Pioneer 10) and (Pioneer 11). However, Anderson and Mashhoon  note that this effect cannot account for the Pioneer anomaly. The effect of rotation on the radio signal is already phenomenologically incorporated into the Doppler data analysis (see Section 4.5.4).
The fact that the anomaly was discovered using Doppler techniques leaves duality in the nature of the detected signal – it is either true physical acceleration or a time acceleration that is connected with the former by the relationship . This fact motivated Anderson et al.  to try to look for purely phenomenological “time” distortions that might fit the Pioneer data. The question was “is there any evidence that some kind of ‘time acceleration’ is being seen?”.
A number of models were investigated and discarded for various reasons (see  for discussion), but there was one model that was especially interesting. This model adds a term that is quadratic in time to the light time, as seen by the DSN station as . In particular, let any labeled time be given as
Expression (6.29) mimics a line of sight acceleration of the spacecraft, and could be thought of as an expanding space model. Note that affects only the data, not the trajectory. It was pointed out by Anderson et al. that this model fits both Doppler and range very well for several spacecraft used in their study . This fact motivated the discussion on the nature of the implied numerical relationship between the Hubble constant and (Section 6.6.2). To investigate further the nature of this relation one would need to check the data of other spacecraft, compare modern clocks with accuracy much higher then that used in the navigation of the Pioneers, as well as the data on millisecond binary pulsars. Presently, not all of these venues are yet properly explored.
Rañada  investigated the effect of a background gravitational potential that pervades the universe and is increasing because of the expansion, provoking a drift of clocks ; however, such an effect should also be observed in the radio signals from pulsars [198, 410], which is not the case. Further refining their argument, Rañada and and Tiemblo  investigated the nonequivalence of atomic and astronomical times and concluded that these times could be accelerating adiabatically with respect to one another.
Ostvang  proposes that cosmic expansion applies directly to gravitationally bound systems according to the quasi-metric framework. According to [315, 316], the scale factor of the spacetime background would cause an anomaly in the frequency. The cosmological constant has also been invoked to produce acceleration  or a gravitational frequency shift [199, 200].
Lämmerzahl  considered the possibility that an expanding universe may have an effect on the Doppler microwave signals traveling in the solar system. The basic question is whether or not, if it exists, the coupling of the expansion of the universe to light has an observable effect. It was shown that for a spacecraft moving with velocity , the cosmologically-induced acceleration would have the following form:[73, 176].
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