Using multiwavelength observations of radio afterglows, we confirm the hypothesis that the flux density of gamma-ray bursts (GRBs) at a fixed observing frequency is invariable when the distance of the GRBs increases, which means the detection rate will be approximately independent of redshift. We study this behavior theoretically and find that it can be well explained by the standard forward shock model involving a thin shell expanding in either a homogeneous interstellar medium (ISM) or a wind environment. We also found that short GRBs and supernova-associated GRBs, which are at relatively smaller distances, marginally match the flux-redshift relationship and they could be outliers. We rule out the assumption that the medium density evolves with redshift as $npropto(1+z)^4$ from the current measurements of $n$ and $z$ for short and long GRBs. In addition, the possible dependence of host flux on the redshift is also investigated. We find that a similar redshift independence of the flux exists for host galaxies as well, which implies that the detection rate of radio hosts might also be independent of the redshift. It is also hinted that most radio hosts have the spectral indices ranging from $beta_hsimeq-1$ to 2.5 in statistics. Finally, we predict the detection rates of radio afterglows by the next-generation radio telescopes such as the Five-hundred meter Aperture Spherical Telescope (FAST) and the Square Kilometer Array (SKA).
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