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“Their observations measured the evolution of the Hubble parameter over time by constructing a Hubble diagram: a plot of distance modulus versus redshift (Figure 11.33). For small $z$, the Hubble diagram tells the value of Ho. At large $z$, the diagram shows evidence for changes in the expansion rate. At a given distance, for example, if the universe is deceler ating, the expansion rate will have been higher in the past than predicted by $H(t) = Ho$. Therefore, if gravity slows the expansion (as was expected) at large $μ$ (i.e. the universe in the past), the observed redshift should be higher than predicted. This deviation will increase with increasing $µ$.”

I quoted a book that I found on Google. The book says that supernova studies have shown that the value of Hubble parameters has been smaller in the past. However, it is known that H(t) always decreases with time. I think what this book says is a(t)/dt. What does H(t) from the book mean?

And I don't think the meaning of H(t) is the same in many references. What exactly is H(t) used in supernova cosmology?

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    $\begingroup$ The quote does not say that the Hubble parameter was smaller in the past. Can you provide a section of the book that does and give a link to the book $\endgroup$
    – ProfRob
    Commented Aug 1, 2022 at 7:45
  • $\begingroup$ @ProfRob i.postimg.cc/hvhzNFVM/… $\endgroup$
    – teacher
    Commented Aug 1, 2022 at 8:11
  • $\begingroup$ That passage does not say that the Hubble parameter was smaller in the past. $\endgroup$
    – ProfRob
    Commented Aug 1, 2022 at 9:28
  • $\begingroup$ @ProfRob “the evolution of the Hubble parameter over time by constructing a Hubble diagram: a plot of distance modulus versus redshift“ I understand that this sentence means interpreting the graph as a change in Hubble's parameters. Or is this book describing accelerated expansion with a(t)/dt? $\endgroup$
    – teacher
    Commented Aug 1, 2022 at 12:47
  • $\begingroup$ The fact that the graph is getting steeper indicates a larger Hubble parameter in the past. $\endgroup$
    – ProfRob
    Commented Aug 1, 2022 at 14:12

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As far as I'm aware of, the Hubble parameter is usually defined by $H(t)\equiv \frac{1}{a(t)}\frac{da(t)}{dt}$, where $a(t)$ is the scale factor in the FLRW metric.
According to Friedmann equations, one can show the time derivative of the Hubble parameter as:
$\frac{dH(t)}{dt}=-4\pi G_{N}\rho(1+w)+\frac{kc^2}{a(t)^2}$,
where $G_N$ is the gravitational constant, $\rho$ the energy density, $w=\frac{p}{\rho c^2}$ the equation of state, $p$ the pressure, $c$ the speed of light, and $k$ the spatial curvature. Then, for $\rho>0$, $w>-1$, and $k=0$, $\frac{d H(t)}{dt}<0$, i.e., the Hubble parameter decreases in time.

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  • $\begingroup$ Can you infer how the Hubble parameter changes over time just by the slope of the graph itself? Since the graph was created in the present, I think it is impossible to infer the change in Hubble parameters by slope itself. Do you infer the change in Hubble parameters by considering the theoretical relationship and the slope together? $\endgroup$
    – teacher
    Commented Aug 1, 2022 at 15:26

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