De Sitter spacetime affine parameter I am reading Chapter 8 in Carroll's "Spacetime and geometry " textbook and I was lead to exercise 8.2, given as: 

Consider de Sitter space in coordinates where the metric takes the form
  $$ds^{2} = -dt^{2} + e^{Ht}[dx^{2} + dy^{2} + dz^{2}]. $$ 
  Solve the geodesic equation for co-moving observers ($x^{i} = constant$) to find the affine parameter as a function of $t$. Show that the geodesics reach $t = -\infty $ in a finite affine parameter, demonstrating that these coordinates fail to cover the entire manifold.  

By solving the geodesic equation I got $\lambda = \frac{1}{H}\exp^{Ht}$. Though I am not sure whether its correct as I am having trouble solving the last part. 
 A: The exercise is wrong (the errata says so), but it's easy to see why. The geodesic equations can be derived from the Lagrangian
$$L = \frac12 g_{\mu\nu}(x) \dot{x}^\mu \dot{x}^\nu,$$
where a dot is a derivative with respect to the affine parameter. For our metric (which should say $e^{2Ht}$ instead of $e^{Ht}$ but let's leave it like it is), the Lagrangian is
$$L = -\frac12 \dot{t}^2 + \frac12 e^{Ht} (\underbrace{\dot{x}^2 + \dot{y}^2 + \dot{z}^2}_{\equiv v^2})$$
and we have
$$\frac{\partial L}{\partial t} = \frac{H}{2} e^{Ht} v^2, \qquad \frac{\partial L}{\partial \dot{t}} = -\dot{t},$$
leading to the Euler-Lagrange equation
$$\ddot{t} = - \frac{H}{2} e^{Ht} v^2$$
(and more for the spatial coordinates). Now, comoving observers are precisely those that don't move with respect to the comoving coordinates, so $v^2 = 0$ and $t$ is just a linear function of the affine parameter. To determine the constant of proportionality, we require that $g_{\mu\nu} \dot{x}^\mu \dot{x}^\nu = -1$, which is the condition that our wordline be parametrized by proper time. For our simple case with $\dot{x}^i = 0$, this gives that $\dot{t}^2 = 1$, so $t = \lambda + \text{const}$.
These coordinates do in fact only cover a portion of de Sitter space, but you can't get that from comoving geodesics.
