Expansion History Calculator Expansion History of the Universe Instructions Ho 72 Sem 0.3 2, 0.2 2 0.7 to 13.6 9.9 Help Data Analysis Now Vacuum Matter Radiation Redshift Scale Factor Relative to Today Fraction Energy Density The simulation shows the plot of Scale Factor Relative to Today versus Time measured in units of billion years. When the Hubble parameter Ho is 72, 12m is 0.3, 12, is 0.2, 12, is 0.7, the value of this 13.6 and the value of to is 9.9. 24 -20 -16 -12 -8 -4 Time (Billion Years) 0 4 8 12 16 Line color - Line Width 0 Line Alpha 100 - 72 2m 0.2 28 Hubble Redshift Densities 0.30 0.70 SAT Plot new Grid Clear Print Using the Expansion History Calculator above set all 2 values to zero using the three sliders and the Hubble parameter, Ho, to 50. Choose a color using the color selection, and click the "Plot" button to generate the corresponding curve. Repeat this for Ho values of 72 and 80, choosing a different color for the curve each time. Toggle the redshift scale on by clicking on the "Redshift" button, then fill out the table below from the information displayed on the plot for each of the values of the Hubble parameter that you plotted. Note that the age of the Universe is the negative of the intercept with the horizontal axis for the corresponding curve of the scale parameter as illustrated in this example: Using the Expansion History Calculator above set all 12 values to zero using the three sliders and the Hubble parameter, Ho, to 50. Choose a color using the color selection, and click the "Plot" button to generate the corresponding curve. Repeat this for Ho values of 72 and 80, choosing a different color for the curve each time. Toggle the redshift scale on by clicking on the "Redshift" button, then fill out the table below from the information displayed on the plot for each of the values of the Hubble parameter that you plotted. Note that the age of the Universe is the negative of the intercept with the horizontal axis for the corresponding curve of the scale parameter as illustrated in this example: Examples: Redshift Age = 10.8 billion years -10.8 -8 0 The age corresponding to a given redshift is determined by finding the redshift on the redshift axis, moving horizontally to the curve for the scale parameter, and then vertically to the horizontal axis. In the example above, a redshift of 1 corresponds to an intercept of -8 on the horizontal axis, implying that a redshift of 1 corresponds to a time 8 billion years ago when the light was emitted. Time of Redshift = 1 Age of Universe HO (billion years) (billion years ago) 50 19.307 0, 9.777 72 13.460 6.790 80 12.125 6.111 Now let's investigate the behavior of universes that consist only of matter (no radiation or vacuum energy density). Using the Expansion History Calculator click "Clear" to remove any previous calculation. Set the 2 values corresponding to the density parameters to zero using the three sliders. Set the Hubble parameter Ho to 72 and choose a color for the curve then click "Plot new" to generate the corresponding curve. Repeat this for values of 2m equal to 0.25, 0.75 and 1, choosing a different color for the curve each time (keep the other 2 values equal to zero for each case). You should now have three curves plotted (plus a straight line), all intersecting at the current time (0 on the horizontal axis and 1 on the vertical axis). Fill in the ages of the Universe corresponding to these four cases in the table below: 2 Age of Universe (billion years) 0.0 19.307 0.25 16.090 0.75 13.767 1.00 13.035 Next we investigate the behavior of universes that consist only of radiation. Using the Expansion History Calculator click "Clear" to remove any previous calculation. Set the 2 values corresponding to the density parameters to zero using the three sliders. Set the Hubble parameter Hoto 72 and choose a color for the curve then click "Plot new" to generate the corresponding curve. Repeat this for values of 2 equal to 0.25, 0.75 and 1, choosing a different color for the curve each time (keep the other 2 values equal to zero for each case). You should now have three curves plotted (plus a straight line), all intersecting at the current time (O on the horizontal axis and 1 on the vertical axis). Fill in the ages of the Universe corresponding to these four cases in the table below: 22 Age of Universe (billion years) o 0.25 Now we investigate the behavior of universes that consist only of vacuum energy. Using the Expansion History Calculator click "Clear" to remove any previous calculation. Set the 2 values corresponding to the density parameters to zero using the three sliders. Set the Hubble parameter Hoto 72 and choose a color for the curve then click "Plot" to generate the corresponding curve. Repeat this for values of 2 equal to 0.25, 0.75 and 0.99, choosing a different color for the curve each time (keep the other 2 values equal to zero for each case). You should now have three curves plotted (plus a straight line), all intersecting at the current time (0 on the horizontal axis and 1 on the vertical axis). Fill in the ages of the Universe corresponding to these four cases in the table below: Age of Universe (billion years) RA 0.0 0.75 Finally, let's investigate the case corresponding to what we think are the best current values for the paramters governing the evolution of the Universe. Using the Expansion History Calculator click "Clear" to remove any previous calculation. Set the S2 values to correspond to radition of O, mass of 0.3 and vacuum energy of 0.7. Set the Hubble parameter Hoto 72 and choose a color for the curve then click "Plot" to generate the corresponding curve. This curve corresponds to a flat Universe (because the sum of density parameters is approximately 1) that is dominated by vacuum energy (70% of energy density) and mass (30% of energy density, most from dark matter), with negligible contribution from radiation. Click the "Densities" button to display the energy densities as a function of time. Assuming the parameters used here are correct, the present Universe is dominated by vacuum energy. From the energy density curves, how long ago was it when the amount of vacuum energy first became larger than that associated with matter? billion years ago. Click "Densities" again to hide the densities and click the "Redshift" button to display the redshift scale. Use this scale and this choice of cosmological parameters to fill out the table below: Time Redshift (billion years) Expansion History Calculator Expansion History of the Universe Instructions Ho 72 Sem 0.3 2, 0.2 2 0.7 to 13.6 9.9 Help Data Analysis Now Vacuum Matter Radiation Redshift Scale Factor Relative to Today Fraction Energy Density The simulation shows the plot of Scale Factor Relative to Today versus Time measured in units of billion years. When the Hubble parameter Ho is 72, 12m is 0.3, 12, is 0.2, 12, is 0.7, the value of this 13.6 and the value of to is 9.9. 24 -20 -16 -12 -8 -4 Time (Billion Years) 0 4 8 12 16 Line color - Line Width 0 Line Alpha 100 - 72 2m 0.2 28 Hubble Redshift Densities 0.30 0.70 SAT Plot new Grid Clear Print Using the Expansion History Calculator above set all 2 values to zero using the three sliders and the Hubble parameter, Ho, to 50. Choose a color using the color selection, and click the "Plot" button to generate the corresponding curve. Repeat this for Ho values of 72 and 80, choosing a different color for the curve each time. Toggle the redshift scale on by clicking on the "Redshift" button, then fill out the table below from the information displayed on the plot for each of the values of the Hubble parameter that you plotted. Note that the age of the Universe is the negative of the intercept with the horizontal axis for the corresponding curve of the scale parameter as illustrated in this example: Using the Expansion History Calculator above set all 12 values to zero using the three sliders and the Hubble parameter, Ho, to 50. Choose a color using the color selection, and click the "Plot" button to generate the corresponding curve. Repeat this for Ho values of 72 and 80, choosing a different color for the curve each time. Toggle the redshift scale on by clicking on the "Redshift" button, then fill out the table below from the information displayed on the plot for each of the values of the Hubble parameter that you plotted. Note that the age of the Universe is the negative of the intercept with the horizontal axis for the corresponding curve of the scale parameter as illustrated in this example: Examples: Redshift Age = 10.8 billion years -10.8 -8 0 The age corresponding to a given redshift is determined by finding the redshift on the redshift axis, moving horizontally to the curve for the scale parameter, and then vertically to the horizontal axis. In the example above, a redshift of 1 corresponds to an intercept of -8 on the horizontal axis, implying that a redshift of 1 corresponds to a time 8 billion years ago when the light was emitted. Time of Redshift = 1 Age of Universe HO (billion years) (billion years ago) 50 19.307 0, 9.777 72 13.460 6.790 80 12.125 6.111 Now let's investigate the behavior of universes that consist only of matter (no radiation or vacuum energy density). Using the Expansion History Calculator click "Clear" to remove any previous calculation. Set the 2 values corresponding to the density parameters to zero using the three sliders. Set the Hubble parameter Ho to 72 and choose a color for the curve then click "Plot new" to generate the corresponding curve. Repeat this for values of 2m equal to 0.25, 0.75 and 1, choosing a different color for the curve each time (keep the other 2 values equal to zero for each case). You should now have three curves plotted (plus a straight line), all intersecting at the current time (0 on the horizontal axis and 1 on the vertical axis). Fill in the ages of the Universe corresponding to these four cases in the table below: 2 Age of Universe (billion years) 0.0 19.307 0.25 16.090 0.75 13.767 1.00 13.035 Next we investigate the behavior of universes that consist only of radiation. Using the Expansion History Calculator click "Clear" to remove any previous calculation. Set the 2 values corresponding to the density parameters to zero using the three sliders. Set the Hubble parameter Hoto 72 and choose a color for the curve then click "Plot new" to generate the corresponding curve. Repeat this for values of 2 equal to 0.25, 0.75 and 1, choosing a different color for the curve each time (keep the other 2 values equal to zero for each case). You should now have three curves plotted (plus a straight line), all intersecting at the current time (O on the horizontal axis and 1 on the vertical axis). Fill in the ages of the Universe corresponding to these four cases in the table below: 22 Age of Universe (billion years) o 0.25 Now we investigate the behavior of universes that consist only of vacuum energy. Using the Expansion History Calculator click "Clear" to remove any previous calculation. Set the 2 values corresponding to the density parameters to zero using the three sliders. Set the Hubble parameter Hoto 72 and choose a color for the curve then click "Plot" to generate the corresponding curve. Repeat this for values of 2 equal to 0.25, 0.75 and 0.99, choosing a different color for the curve each time (keep the other 2 values equal to zero for each case). You should now have three curves plotted (plus a straight line), all intersecting at the current time (0 on the horizontal axis and 1 on the vertical axis). Fill in the ages of the Universe corresponding to these four cases in the table below: Age of Universe (billion years) RA 0.0 0.75 Finally, let's investigate the case corresponding to what we think are the best current values for the paramters governing the evolution of the Universe. Using the Expansion History Calculator click "Clear" to remove any previous calculation. Set the S2 values to correspond to radition of O, mass of 0.3 and vacuum energy of 0.7. Set the Hubble parameter Hoto 72 and choose a color for the curve then click "Plot" to generate the corresponding curve. This curve corresponds to a flat Universe (because the sum of density parameters is approximately 1) that is dominated by vacuum energy (70% of energy density) and mass (30% of energy density, most from dark matter), with negligible contribution from radiation. Click the "Densities" button to display the energy densities as a function of time. Assuming the parameters used here are correct, the present Universe is dominated by vacuum energy. From the energy density curves, how long ago was it when the amount of vacuum energy first became larger than that associated with matter? billion years ago. Click "Densities" again to hide the densities and click the "Redshift" button to display the redshift scale. Use this scale and this choice of cosmological parameters to fill out the table below: Time Redshift (billion years)