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1.3.6 Trouble with Hubble

In 1929, Hubble presented evidence that the recessional velocities of distant galaxies were proportional to their distance from the solar system. He used apparent brightness of stars of known absolute brightness (standard candles) to calculate distance, and red shift to calculate recession velocity assuming that the shift was due to Doppler effect. Hubble's Law states that the recession velocity of a distant star or galaxy is proportional to its distance from the solar system. That is, \(v_r = Hr\), where, \(v_r\), is recession velocity, \(r\) is distance and \(H\) is the Hubble constant.

The theory of an expanding energy continuum assumes Hubble's law is correct even though the value of \(H\) calculated from a Hubble graph may be in error if the measurements of recession velocity and distance are in error. We have already shown that measurement of distance using relative brightness of standard candle stars does not follow a pure inverse square law.

The inaccuracies in the measurement of distance and recession velocity mean that the Hubble graph is not a straight line. However, in the range of distances measured by Hubble these inaccuracies are negligible, so that the Hubble graph was straight enough to provide an estimate of the Hubble constant. It turned out that the distances used by Hubble were too small by a factor of about ten. Though modern distance measuring techniques are more accurate, the Hubble graph still deviates from a straight line at very large distances. This is due to the incorrect assumptions that the speed of light is constant, and that of red shift is Doppler Effect.

Light emitted by a star travels through the continuum which itself is continuously expanding. So the expansion of wavelength is governed by the same expansion ratio as radius, distance, and the velocity of light. Hence,

\begin{align} expansion\ ratio =\frac{\lambda_o}{\lambda_e} = \frac{R_o}{R_e} = \frac{r_o}{r_e} =\frac{c_o}{c_e} = e^{H \Delta t} = e^{\theta} \nonumber \end{align}

where \(\lambda_e\) is the wavelength of emitted photons, and \(\lambda_o\) is the wavelength of detected photons. This assumes that emission spectra of atomic elements are constant throughout time and in every part of the universe.

True recession velocity, \(v_r\), is simply the rate of expansion of arc distance, \(r_o\). This can be derived very easily from the expression \(r_o = R_o \theta\). We have that
\begin{align} v_r =\frac{\partial r_o}{\partial t} =\frac{\partial }{\partial t} (R_o \theta)=c_o \theta \nonumber \end{align}

The value of \(\theta\) may be calculated from the expression for expansion ratio,
\begin{align} expansion\ ratio\ e^{\theta} = \frac{\lambda_o}{\lambda_e} \ \ hence\ \theta =ln \frac{\lambda_o}{\lambda_e} \nonumber \end{align}
and consequently,
\begin{align} v_r =c_o ln \frac{\lambda_o}{\lambda_e} \nonumber \end{align}

In his calculations, Hubble assumed that red shift was due to the Doppler Effect so that the relationship between recession velocity, \(v_{dop}\), of the star and the wavelength change by \(\Delta \lambda\) of the spectral lines is given by
\begin{align} v_{dop} = c_o \frac{\Delta \lambda}{\lambda_e}=c_o \frac{(\lambda_o - \lambda_e)}{\lambda_e} = c_o (\frac{\lambda_o}{\lambda_e}-1) \nonumber \end{align}

For convenience, Hubble introduced the parameter \(Z = (\frac{\lambda_o}{\lambda_e}-1)\) and calculated Doppler recessional velocity as \(v_{dop} = c_o Z\).

In terms of the \(Z\) parameter, a little algebra shows that the correct recessional velocity, is given by \begin{align} v_r = c_o\ ln \frac{\lambda_o}{\lambda_e} = c_o\ ln(1+Z) \nonumber \end{align}

Using the Doppler shift formula over-estimates recession velocity, but for low values of \(Z\), the error is small. However, the error rises rapidly as \(Z\) increases. For example, at a distance, of 30Mpc, \(Z\) = 0.01, and the error is about 1 percent. However, at 3000Mpc, \(Z\)=1, and the error is about 30 percent.

The result of this analysis is that a Hubble graph based on \(v_{dop} = c_o Z\) will not be a straight line, especially at large distances. On the other hand, a Hubble graph based on \(v_r = c_o\ ln(1 + Z)\), will be a straight line over the whole distance range.

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