As soon as any author proposes alternative interpretation of the cosmological red-shift, it is opposed saying that tired-light explanations are not compatible with the observations of time-dilation of super novae light-curves. So this author had shown [1] that any mechanism which can cause ‘cosmological red-shift’ will also cause ‘time-dilation of super-novae light-curves’. Still the alternative interpretations are not taken seriously. So this author is constrained to show some limitations of the current big-bang cosmology, like: (i) the general relativity theory predicts ‘expansion of space’ between the galaxies; but the space within the galaxy is not expanding, because galaxy is a gravitationally-bound-structure. The question raised here is: If so, then what happens at the edge of a galaxy whose external space is expanding but the space within is not expanding? Is there a smooth transition from expanding to non-expanding space? If expanding-space can stretch the wavelength of a cosmologically red-shifting photon, then less and less expanding space, at the boundary of the galaxy, should shrink the wavelength back to its original length, isn’t it? (ii) According to general relativity the planets, like the earth, orbit around the Sun, because the space around the Sun has got curved; and the planets are in inertial-motion traveling along the geodesic path. Now the question raised here is: Inertial-motion of a body can be at any speed. Can the planets travel along the geodesic-path at any speed they like? Can they take a coffee-brake and then proceed further? (iii) According to general relativity there is a radial-distance at which rate of expansion of space is equal to the speed of light, so the wave-front of light beyond this radius is not able to enter the sphere of observable universe. This means that the speed of light is the same, 3 x 108 meters per second. The question raised here is: Since the speed of light is the same in expanding as well as non-expanding space; and f . λ = c , i.e. the product of frequency (f) and wavelength (λ) is always equal to the speed of light (c); then the wavelength (λ) can increase only when frequency (f) gets reduced; and not because of expansion of space. Then a new ‘self-gravitational-acceleration based explanation for the cosmological red-shift is proposed. And it is shown that reduction in energy of ‘cosmologically red-shifting photons’ matches strikingly with this new explanation.