Cosmology studies the universe as a whole and how it develops over time: The scale factor [tex]a(t)[/tex] describes the change of distance in the universe. The Hubble parameter [tex]H(t)=\dot{a}(t) / a(t)[/tex] (where the dot represents the rate of change with respect to time) describes the rate at which the universe expands, and was measured to be around [tex]72.6 km / s / Mpc[/tex] using the James Webb Space Telescope in 2024. The deceleration parameter [tex]q[/tex] describes the acceleration of the expansion:
[tex]q=-\left(1+\frac{\dot{H}}{H^2}\right)[/tex]
Assume a model with [tex]a(t)=\lambda \cdot t^\beta[/tex], where [tex] \lambda[/tex] and [tex]\beta[/tex] are real numbers. Knowing that the universe is around 13.7 billion years old, determine if the expansion is accelerating or decelerating.
Answer (1)
Calculate the Hubble parameter: H ( t ) = t β .
Calculate the derivative of the Hubble parameter: H ˙ ( t ) = − t 2 β .
Calculate the deceleration parameter: q = − 1 + β 1 .
Determine the condition for acceleration or deceleration based on β : If 1"> β > 1 , the expansion is accelerating; if β < 1 , it is decelerating; if β = 1 , it is neither. The deceleration parameter is q = − 1 + β 1 .
Explanation
Problem Setup We are given the scale factor a ( t ) = λ t β , where λ and β are real numbers. We need to determine if the expansion of the universe is accelerating or decelerating based on this model. The Hubble parameter is defined as H ( t ) = a ( t ) a ˙ ( t ) , and the deceleration parameter is q = − ( 1 + H 2 H ˙ ) .
Calculating the Hubble Parameter First, we calculate the Hubble parameter H ( t ) .
a ˙ ( t ) = d t d ( λ t β ) = λ β t β − 1
H ( t ) = a ( t ) a ˙ ( t ) = λ t β λ β t β − 1 = t β
Calculating the Time Derivative of the Hubble Parameter Next, we calculate the time derivative of the Hubble parameter, H ˙ ( t ) .
H ˙ ( t ) = d t d ( t β ) = − t 2 β
Calculating the Deceleration Parameter Now, we calculate the deceleration parameter q .
q = − ( 1 + H 2 H ˙ ) = − ( 1 + ( t β ) 2 − t 2 β ) = − ( 1 + t 2 β 2 − t 2 β ) = − ( 1 − β 1 ) = − 1 + β 1
Analyzing the Sign of q To determine if the expansion is accelerating or decelerating, we need to analyze the sign of q .
If q < 0 , the expansion is accelerating. If 0"> q > 0 , the expansion is decelerating. If q = 0 , the expansion is neither accelerating nor decelerating.
q = − 1 + β 1
Analyzing Different Values of Beta We can analyze the behavior of q for different values of β :
If 1"> β > 1 , then β 1 < 1 , so q = − 1 + β 1 < 0 . The expansion is accelerating.
If β < 1 , then 1"> β 1 > 1 , so 0"> q = − 1 + β 1 > 0 . The expansion is decelerating.
If β = 1 , then β 1 = 1 , so q = − 1 + 1 = 0 . The expansion is neither accelerating nor decelerating.
Conclusion Since we don't have a specific value for β , we cannot definitively say whether the expansion is accelerating or decelerating. However, we can express the condition for acceleration or deceleration in terms of β .
If 1"> β > 1 , the expansion is accelerating. If β < 1 , the expansion is decelerating. If β = 1 , the expansion is neither accelerating nor decelerating.
Final Answer Without knowing the value of β , we cannot determine whether the expansion is accelerating or decelerating. The deceleration parameter is q = − 1 + β 1 .
Examples
Understanding the expansion rate of the universe is crucial in cosmology. For example, consider a scenario where scientists are trying to predict the future distribution of galaxies. If the universe's expansion is accelerating ( 1"> β > 1 in our model), galaxies will drift further apart over time, leading to a more dispersed distribution. Conversely, if the expansion is decelerating ( β < 1 ), galaxies might cluster more closely due to gravity overcoming the expansion. This understanding helps refine cosmological models and predict the universe's long-term evolution.
