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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?

Qmechanic
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teacher
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    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 – ProfRob Aug 01 '22 at 07:45
  • @ProfRob https://i.postimg.cc/hvhzNFVM/600-C9-BDC-68-DC-4-B16-A9-FC-795500-CE312-E.jpg – teacher Aug 01 '22 at 08:11
  • That passage does not say that the Hubble parameter was smaller in the past. – ProfRob Aug 01 '22 at 09:28
  • @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? – teacher Aug 01 '22 at 12:47
  • The fact that the graph is getting steeper indicates a larger Hubble parameter in the past. – ProfRob Aug 01 '22 at 14:12
  • @ProfRob The graph is based on the z observed at the current point in time and the luminosity distance. How can you infer historical Hubble parameters by change in the slope of a graph? I understand that the slope comparison of the graph determines the density distribution of the universe to determine whether it is deceleration expansion or acceleration expansion, but how do you infer the Hubble parameters of the past by the slope itself? Theoretically, I understand how the Hubble parameters vary depending on the density distribution. – teacher Aug 01 '22 at 15:11
  • @ProfRob https://physics.stackexchange.com/questions/68493/according-to-hubbles-law-how-can-the-expansion-of-the-universe-be-accelerating. According to the attached link, the difference in z-values indicates how much expansion has occurred between time intervals. Does the instantaneous slope mean interpreting it in this way? – teacher Aug 01 '22 at 15:21
  • You might be right about that, but you are changing the subject. Where in your text does it say H(t) was smaller in the past? – ProfRob Aug 01 '22 at 17:32

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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.

Fineman
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  • 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? – teacher Aug 01 '22 at 15:26