As pointed out by Weinberg in his book Cosmology (note, this is NOT Gravitation and Cosmology. He also has a book of that name), inflation was proposed to explain 3 problems:
1)Horizon problem
2)Flatness problem
3)Monopole problem

1)Horizon problem: The evolution of the scale factor before and after decoupling is $\sqrt{t}$ and $t^{\frac{2}{3}}$. We compute the linear dimension of the forward and backward lightcones at the time of decoupling in the hot big bang model. The radius of this light cone is the physical size of the region on the last scattering surface from which we receive the CMB. The backward lightcone is $l_{B} \approx 3(t_{dec}^{2}t_{0})^{1/3}$ ($t_{0}$ is present time.). The forward lightcone radius is $l_{F} = 2t_{dec}$. The ratio $R \equiv \frac{l_{B}}{l_{F}} \approx 70$. The physical wavelength associated with cosmological perturbations grows faster than the Hubble radius as we go back in time. If, a causal mechanism is responsible for the inhomogenities, then these scales should be inside Hubble scale in very early universe. This is possible if, the perturbation associated wavelength decreases faster than Hubble radius as we go back in time. So, $-\frac{d}{dt}\left( \frac{\lambda}{d_{H}}\right) <0 $ ($d_{H}$ is the Hubble radius). This leads to $\ddot{a} >0$. In most cases, we model it as a single scalar field which causes this inflation in a de Sitter background ($\Lambda$ dominated universe)

2)Flatness problem: A less convincing argument of inflation. Experimentally, we observe a vanishing spatial curvature parameter $\Omega_{K} = -\frac{K}{a^2 H^2} = -\frac{K}{a^2}$. In solving this problem, we assume that nothing much happens to the cosmic scale factor and expansion rate from the end of inflation to the beginning of the radiation dominated era i.e $a_{Inflation}H_{Inflation} \approx a_{rad. domination}H_{rad. domination}$. The small value of $|K|/\dot{a}^2$ could be explained by taking $K=0$ i.e a spatially flat universe. However, inflation opens up the possibility that the universe is not at all homogenous and isotropic and that its apparent flatness of the cosmic metric is just the result of inflation.

3)Monopole problem: Standard Model predicts that in a hot early universe, a large number of monopoles must be produced by symmetry breaking from some single gauge theory since it is at an energy scale of about $M = 10^{16}$ GeV. Those monopoles should have persisted even to the present days. However, that is not the case.

Amongst all the above problems, the horizon problem is the most serious one. Since, the other two can be explained by other mechanisms. Also, any number of $e$-foldings not only solves the horizon problem but also the flatness problem and the monopole problem.

This post imported from StackExchange Physics at 2014-07-28 11:18 (UCT), posted by SE-user Debangshu