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Eq. (1.14) and (1.15) of David Tong's lectures (p.g. 12) writes:

Consider a galaxy which, in co-moving coordinates, traces a trajectory $\vec{x}(t)$. Then, in physical coordinates, the position is $$\overrightarrow{x}_{\small{\text{phys}}}(t)=a(t)\overrightarrow{x}(t)\tag{1.14}$$ The physical velocity is then $$\overrightarrow{v}_{\small{\text{phys}}}(t)=\frac{d\overrightarrow{x}_{\small{\text{phys}}}}{dt}=H\overrightarrow{x}_{\small{\text{phys}}}+\overrightarrow{v}_{\small{\text{pec}}}\tag{1.15}$$ where $a(t)$ is the scale factor and $H=\dot{a}/a$.

My question is this:

Assuming that $\overrightarrow{v}_{\small{\text{pec}}}$ is constant, can I write $$\overrightarrow{x}_{\small{\text{phys}}}(t)=a(t)(\overrightarrow{v}_{\small{\text{pec}}}t+C)$$ where $C$ is a fixed distance. If so, would that make $C$ the co-moving distance? Thus, $$\overrightarrow{x}_{\small{\text{phys}}}(t)=a(t)\overrightarrow{v}_{\small{\text{pec}}}t+\overrightarrow{x}_{\small{P}}(t),$$ where $\overrightarrow{x}_{\small{P}}(t)=a(t)C$ is the proper distance?

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The terminology "proper coordinates" and "physical coordinates" are interchangeable. Similarly, "proper distance" and "physical distance" mean the same thing. One always has to be careful to be very precise when talking about proper/physical properties of course, especially when dealing with distances between points at different cosmic times.

Consider a stationary point in comoving coordinates (i.e. a comoving observer), call it $\vec{x}_0$. The trajectory in proper coordinates is then $\vec{x}_{phys}(t) =a(t) \vec{x}_0$, which is just your first equation with zero peculiar velocity. So it makes sense to interpret your $C$ as a (fixed) comoving coordinate. The $\vec{v}_{pec}t$ part is problematic, though, because any physical distance traversed due to the peculiar velocity subsequently expands, but only after having been traversed - you can't avoid an integral in this equation.

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