At what redshift did the Universe beging accelerating?

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SUMMARY

The discussion centers on deriving an expression for the redshift at which the Universe transitioned from deceleration to acceleration, specifically using the Friedmann acceleration equation. Participants emphasize the need to express this transition in terms of fundamental cosmological parameters, namely \(\Omega_{M0}\) and \(\Omega_{\Lambda0}\). The equation \(a''=0\) is critical, indicating the point of change in the Universe's expansion dynamics. Attempts to manipulate the equations have focused on expressing the scale factor and redshift relationship, but challenges remain in eliminating the \(a^2\) term.

PREREQUISITES
  • Understanding of Friedmann equations in cosmology
  • Familiarity with the concepts of redshift and scale factor
  • Knowledge of cosmological parameters \(\Omega_{M0}\) and \(\Omega_{\Lambda0}\)
  • Basic calculus for manipulating differential equations
NEXT STEPS
  • Study the derivation of the Friedmann acceleration equation in detail
  • Learn about the relationship between redshift and scale factor in cosmology
  • Research methods for manipulating differential equations in cosmological contexts
  • Explore the implications of \(\Omega_{M0}\) and \(\Omega_{\Lambda0}\) on cosmic expansion
USEFUL FOR

Astronomers, cosmologists, and physics students interested in understanding the dynamics of the Universe's expansion and the transition from deceleration to acceleration.

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Homework Statement


Starting with the equation for a'', derive an expression for the redshift at which the deceleration of the Universe turned into acceleration. Your expression should contain only fundamental cosmological parameters at the present day such as [tex]\Omega[/tex]M0 and [tex]\Omega[/tex][tex]\Lambda[/tex]0.

Homework Equations


Freidmann Acceleration equation:
gif.latex?\frac{\ddot{a}}{a}=-4\pi%20G(\rho%20+\frac{3p}{c^2}).gif


Redshift-Scale Factor:

1+z=1/a


The Attempt at a Solution


I've literally been racking my brain for hours, trying to come up with every possible manipulation of this equation. I know a''=0 at the point where the universe stops decelerating and begins to accelerate, but the acceleration equation does seem to make any sense at this point, I tried expressing the Freidmann Equation in terms of [tex]\Omega[/tex]M0 and [tex]\Omega[/tex][tex]\Lambda[/tex]0, namely,

ex?\frac{H^2}{H^2_0}=(\frac{\dot{a}}{a})\frac{1}{H^2_0}=\Omega_{M_0}a^{-3}+\Omega%20_{\Lambda_0}.gif

But I wasn't able to figure out any satisfactory way to manipulate this one.
 
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The other thing I tried was to express a'' in terms of 1+z and the scale factor, namely,Which is really close to what I need, but I still can't figure out how to get rid of the a^2 term. Any help would be appreciated.
 

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