# Distance as a function of time in the case of an increasing Hubble value

I am trying to deduce the traveled distance of, e.g. a galaxy, as a function over time $$D(t)$$ in scenarios in which the Hubble value stays fixed or increases proportionally with time.

For the fixed Hubble scenario, I did this by deducing the time needed to travel an infinitesimally small distance $$dD$$ in which the speed doesn't increase (caused by of the fixed Hubble value). Starting from an initial distance $$D_0$$ and adding all these time intervals until a certain distance $$D$$ has been reached gives: $$t(D) = \int_{D_0}^{D} \frac{1}{H\cdot D}\cdot dD = \ln\bigg(\frac{D}{D_0}\bigg)\cdot \frac{1}{H}$$ $$D(t) = e^{Ht}\cdot D_0$$ I am struggling with the scenario in which the Hubble value increases proportionally with time according to $$H(t)=H_0+C\cdot t$$, where $$H_0$$ is the value at $$t=0$$. I know that in this case $$t(D) = \int_{D_0}^{D} \frac{1}{(H_0+Ct)\cdot D}\cdot dD$$ The $$t$$ within the integrand must be expressed in terms of $$D$$ to do the integration but then I'd have to know the relationship between $$D$$ and $$t$$ which I am trying to deduce in the first place. Rewriting the integration over $$dD$$, $$dt$$ or $$dH$$ always leaves me with an unknown function within the integrand. For example, knowing that $$dt=\frac{dH}{C}$$ and wanting to integrate over $$dH$$ until $$H_t$$ has been reached: $$D(H)=D_0+\int_0^{\frac{H_t-H_0}{C}}H\cdot D\cdot \frac{dH}{C}$$ Here, $$D$$ should increase with $$H$$ during the integration but merely knowing that $$D(H)=\frac{HD}{H}$$ doesn't help me.

Perhaps I'm missing something obvious but is there a way to deduce a definite formula for $$D(t)$$ when $$H$$ increases proportionally with time?

I suggest that you differentiate both sides of your 2nd integral equation, and reposition some terms to get: $$(H_0 + Ct) dt = \frac{dD}{D}.$$ If you then integrate both sides, you get the result for $$D(t)$$.
• This slipped me! Thanks a lot, I was able to deduce that $$D(t)=D_0\cdot e^{\frac{1}{2}t(2H_0+Ct)}$$