on cosmological observables, namely the redshift and the luminosity distance. Finally, the consequences on the determination of the cosmological parameters are presented in Sec. \[sec:cosmological\_consequences\].

Description of the Swiss-cheese cosmological model {#sec:Swiss-cheese}
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The construction of Swiss-cheese models is based on the Einstein-Straus method [@EinStr45] for embedding a point-mass within a homogeneous spacetime (the “cheese”). It consists in cutting off a spherical domain of the cheese and concentrating the matter it contained at the center of the hole. This section presents the spacetime geometries inside and outside a hole (Subsec.\[subsec:spacetime\_patches\]), and how they are glued together (Subsec. \[subsec:junction\_conditions\]).

Spacetime patches {#subsec:spacetime_patches}
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### The “cheese”—Friedmann-Lemaître geometry

Outside the hole, the geometry is described by the standard Friedmann-Lemaître (FL) metric $$\label{eqext}
{{\rm d}}s^2 =-{{\rm d}}T^2 + a^2(T)\left[{{\rm d}}\chi^2 + f_K^2(\chi)\,{{\rm d}}\Omega^2\right],$$ where $a$ is the scale factor and $T$ is the cosmic time. The function $f_K(\chi)$ depends on the sign of $K$ and thus of the spatial geometry (spherical, Euclidean or hyperbolic), $$f_K(\chi) = \frac{\sin\sqrt{K}\chi}{\sqrt{K}}, \quad \chi \quad \text{or} \quad \frac{\sinh\sqrt{-K}\chi}{\sqrt{-K}}$$ respectively for $K>0$, $K=0$ or $K<0$. The Einstein field equations imply that the scale factor $a(T)$ satisfies the Friedmann equation $$\label{e.Friedmann}
H^2 = \frac{8\pi G}{3}\rho -\frac{K}{a^2} + \frac{\Lambda}{3},
\quad \text{with} 