  The direct measurements of cosmic rays (CRs), after correction for the
propagation effects in the interstellar medium, indicate that their source
spectra are likely to be significantly steeper than the canonical $E^{-2}$
spectrum predicted by the standard Diffusive Shock Acceleration (DSA)
mechanism. The DSA has long been held responsible for the production of
galactic CRs in supernova remnant (SNR) shocks. The $\gamma$-ray 'probes' of
the acceleration spectra of CRs on-the-spot, inside of the SNRs, lead to the
same conclusion. We show that the steep acceleration spectrum can be attributed
to the $combination$ of (i) spherical expansion, (ii) tilting of the magnetic
field along the shock surface and (iii) shock deceleration. Because of (i) and
(ii), the DSA is efficient only on two ``polar caps'' of a spherical shock
where the local magnetic field is within $\simeq45^{\circ}$ to its normal. The
shock-produced spectrum observed edge-on steepens with the particle energy
because the number of freshly accelerated particles with lower energies
continually adds up to a growing acceleration region. We demonstrate the
steepening effect by obtaining an exact self-similar solution for the particle
acceleration at expanding shock surface with an arbitrary energy dependence of
particle diffusivity $\kappa$. We show that its increase toward higher energy
steepens the spectrum, which deeply contrasts with the standard DSA spectrum
where $\kappa$ cancels out.
