For a static observer, the boundary of observed $dS^4$ spacetime locates on $r=\ell$, $\ell$ is the de Sitter radius which is reversely proportional to spacetime curvature or say cosmological constant $\Lambda^{\frac{1}{2}}$ up to a constant factor.

There is seemingly a physical and smooth limit $\ell\rightarrow\inf$ back to flat spacetime, and thus the cosmological boundary becomes the null-infinity.

For a finite $\ell$, the bifurcation 2-sphere $B$ is located on $r=\ell,t=0$ without any singularity or ambiguity. One should notice that this 2-sphere is a common boundary of two horizons $\mathcal{H}^-,\mathcal{H}^+$.

If one takes the flat limit, seemingly the $\mathcal{H}^-,\mathcal{H}^+$ deform to null-infinity $\mathcal{I}^-,\mathcal{I}^+$ defined on Minkowski spacetime, and $B$ becomes the spatial infinity $i^0$. But, as a famous result, $i^0$ in the description of Penrose's diagram is singular and is not a common boundary of $\mathcal{I}^-,\mathcal{I}^+$. Or exactly, $\mathcal{I}^-,\mathcal{I}^+$ have no a common boundary.

Is there something wrong with this limit? or say is this limit smooth?

For a static observer, the boundary of observed $dS^4$ spacetime locates on $r=\ell$, $\ell$ is the de Sitter radius which is reversely proportional to spacetime curvature or say cosmological constant $\Lambda^{\frac{1}{2}}$ up to a constant factor.

There is seemingly a physical and smooth limit $\ell\rightarrow\inf$ back to flat spacetime, and thus the cosmological boundary becomes the null-infinity.

For a finite $\ell$, the bifurcation 2-sphere $B$ is located on $U=V=0$ without any singularity or ambiguity. One should notice that this 2-sphere is a common boundary of two horizons $\mathcal{H}^-,\mathcal{H}^+$.

If one takes the flat limit, seemingly the $\mathcal{H}^-,\mathcal{H}^+$ deform to null-infinity $\mathcal{I}^-,\mathcal{I}^+$ defined on Minkowski spacetime, and $B$ becomes the spatial infinity $i^0$. But, as a famous result, $i^0$ in the description of Penrose's diagram is singular and is not a common boundary of $\mathcal{I}^-,\mathcal{I}^+$. Or exactly, $\mathcal{I}^-,\mathcal{I}^+$ have no a common boundary.

Is there something wrong with this limit? or say is this limit smooth?

For a static observer, the boundary of observed $dS^4$ spacetime locates on $r=\ell$, $\ell$ is the de Sitter radius which is reversely proportional to spacetime curvature or say cosmological constant $\Lambda$ up to a constant factor.

There is seemingly a physical and smooth limit $\ell\rightarrow\inf$ back to flat spacetime, and thus the cosmological boundary becomes the null-infinity.

For a finite $\ell$, the bifurcation 2-sphere $B$ is located on $r=\ell,t=0$ without any singularity or ambiguity. One should notice that this 2-sphere is a common boundary of two horizons $\mathcal{H}^-,\mathcal{H}^+$.

If one takes the flat limit, seemingly the $\mathcal{H}^-,\mathcal{H}^+$ deform to null-infinity $\mathcal{I}^-,\mathcal{I}^+$ defined on Minkowski spacetime, and $B$ becomes the spatial infinity $i^0$. But, as a famous result, $i^0$ in the description of Penrose's diagram is singular and is not a common boundary of $\mathcal{I}^-,\mathcal{I}^+$. Or exactly, $\mathcal{I}^-,\mathcal{I}^+$ have no a common boundary.

Is there something wrong with this limit? or say is this limit smooth?

For a static observer, the boundary of observed $dS^4$ spacetime locates on $r=\ell$, $\ell$ is the de Sitter radius which is reversely proportional to spacetime curvature or say cosmological constant $\Lambda^{\frac{1}{2}}$ up to a constant factor.

There is seemingly a physical and smooth limit $\ell\rightarrow\inf$ back to flat spacetime, and thus the cosmological boundary becomes the null-infinity.

For a finite $\ell$, the bifurcation 2-sphere $B$ is located on $r=\ell,t=0$ without any singularity or ambiguity. One should notice that this 2-sphere is a common boundary of two horizons $\mathcal{H}^-,\mathcal{H}^+$.

If one takes the flat limit, seemingly the $\mathcal{H}^-,\mathcal{H}^+$ deform to null-infinity $\mathcal{I}^-,\mathcal{I}^+$ defined on Minkowski spacetime, and $B$ becomes the spatial infinity $i^0$. But, as a famous result, $i^0$ in the description of Penrose's diagram is singular and is not a common boundary of $\mathcal{I}^-,\mathcal{I}^+$. Or exactly, $\mathcal{I}^-,\mathcal{I}^+$ have no a common boundary.

Is there something wrong with this limit? or say is this limit smooth?