Graviton condensate stars and its gravitational echoes

We construct the exact stellar configurations that contain an ordinary perfect-fluid matter that interacts minimally with a condensate of gravitons with distinct pressure conditions on the surface. We propose vanishing transverse pressure on the surface for, namely graviton condensate type 1 and vanishing radial pressure on the surface for type 2. The condition for the radial pressure of type 1 requires the existence of a thin shell that will balance the pressure discontinuity while for type 2, the discontinuity on transverse pressure does not require the additional thin shell. It is found that the Buchdahl inequality of the resulting stellar configurations depends on the parameter related to the graviton condensate, such that we can find the ultra-compact regime of the stellar models. Moreover, the echo time and echo frequency within the ultra-compact regime are computed. At the same compactness, it is found that the presence of the graviton condensate will delay the gravitational echoes for type 2 and will expedite the gravitational echoes for type 1 compared to constant density star, τ echo 2 > τ CDS > τ echo 1 . Furthermore, the gravitational perturbation of a massless scalar wave is also investigated to support these results. These results could open more opportunities for the observational study of graviton in the near future, mostly from the compact astrophysical objects.