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- TL;DR
- Understanding the definition
I note the general Taylor series for ##a(t)## as:
\begin{equation}
\begin{split}
a(t)&\approx a(t_0) + a'(t_0) (t-t_0) + \frac{1}{2!} a''(t_0) (t-t_0)^2 ...
\end{split}
\end{equation}
which I rewrite as:
\begin{equation}
\begin{split}
a(t)&\approx a(t_0)\left(1 + \frac{a'(t_0)}{a(t_0)} (t-t_0) + \frac{1}{2} \frac{a''(t_0)}{a(t_0)} (t-t_0)^2 + ... \right)
\end{split}
\end{equation}
In this context, the Hubble constant ##H(t_0)## is defined as follows:
\begin{equation}
\begin{split}
H(t_0) &\doteq \frac{a'(t_0)}{a(t_0)}
\end{split}
\end{equation}
and the deceleration parameter ##q(t_0)## is defined as follows:
\begin{equation}
\begin{split}
q(t_0)&\doteq -\frac{a''(t_0)}{a(t_0) H^2(t_0)}
\end{split}
\end{equation}
such that
\begin{equation}
\begin{split}
a(t)&\approx a(t_0)\left(1 + H(t_0) (t-t_0) - \frac{1}{2} q(t_0) H^2(t_0) (t-t_0)^2 + ... \right)
\end{split}
\end{equation}
My question: Does a negative value for ##q(t_0)##, which is what is observed in the data, therefore imply that the term ##\left[- \frac{1}{2} q(t_0) H^2(t_0) (t-t_0)^2\right]## is positive?
\begin{equation}
\begin{split}
a(t)&\approx a(t_0) + a'(t_0) (t-t_0) + \frac{1}{2!} a''(t_0) (t-t_0)^2 ...
\end{split}
\end{equation}
which I rewrite as:
\begin{equation}
\begin{split}
a(t)&\approx a(t_0)\left(1 + \frac{a'(t_0)}{a(t_0)} (t-t_0) + \frac{1}{2} \frac{a''(t_0)}{a(t_0)} (t-t_0)^2 + ... \right)
\end{split}
\end{equation}
In this context, the Hubble constant ##H(t_0)## is defined as follows:
\begin{equation}
\begin{split}
H(t_0) &\doteq \frac{a'(t_0)}{a(t_0)}
\end{split}
\end{equation}
and the deceleration parameter ##q(t_0)## is defined as follows:
\begin{equation}
\begin{split}
q(t_0)&\doteq -\frac{a''(t_0)}{a(t_0) H^2(t_0)}
\end{split}
\end{equation}
such that
\begin{equation}
\begin{split}
a(t)&\approx a(t_0)\left(1 + H(t_0) (t-t_0) - \frac{1}{2} q(t_0) H^2(t_0) (t-t_0)^2 + ... \right)
\end{split}
\end{equation}
My question: Does a negative value for ##q(t_0)##, which is what is observed in the data, therefore imply that the term ##\left[- \frac{1}{2} q(t_0) H^2(t_0) (t-t_0)^2\right]## is positive?