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We study linear cosmological perturbations in a previously introduced familyof deformations of general relativity characterized by the absence of newdegrees of freedom. The homogeneous and isotropic background in this class oftheories is unmodified and is described by the usual Friedmann equations. Thetheory of cosmological perturbations is modified and the relevant deformationparameter has the dimension of length. Gravitational perturbations of thescalar type can be described by a certain relativistic potential related to thematter perturbations just as in general relativity. A system of differentialequations describing the evolution of this potential and of the stress-energydensity perturbations is obtained. We find that the evolution of scalarperturbations proceeds with a modified effective time-dependent speed of sound,which, contrary to the case of general relativity, does not vanish even at thematter-dominated stage. In a broad range of values of the length parametercontrolling the deformation, a specific transition from the regime of modifiedgravity to the regime of general relativity in the evolution of scalarperturbations takes place during the radiation domination. In this case, theresulting power spectrum of perturbations in radiation and dark matter issuppressed on the comoving spatial scales that enter the Hubble radius beforethis transition. We estimate the bounds on the deformation parameter for whichthis suppression does not lead to observable consequences. Evolution of scalarperturbations at the inflationary stage is modified but very slightly and theprimordial spectrum generated during inflation is not noticeably different fromthe one obtained in general relativity.

Cosmological perturbations in a family of deformations of general relativity