A friend believes the universe is very young, less than 10,000 years old. Based on Hubble's law, how would you argue that the universe is older? What would the Hubble constant be for a million-year-old universe?

Estimating the age of the universe is akin to piecing together a cosmic puzzle. To illustrate, imagine taking a video of fireworks exploding and playing it in reverse. As the colorful trails converge, you pinpoint the moment of ignition. Analogously, Hubble's Law allows astronomers to run the clock backwards on the universe's expansion.

When galaxies are observed to be speeding away from us in every direction, Hubble's Law's velocity-distance correlation suggests a common starting point - the Big Bang. By calculating backward using the speeds and distances of galaxies, we can estimate the time required for these galaxies to travel from that singular event to their present locations. This time frame, which far exceeds a mere 10,000 years, provides a lower limit to the universe's age, declaring the cosmos to be ancient beyond ordinary reckoning. To propose a universe only a million years old, you'd need a drastically different Hubble constant, yielding a galactic recessional ballet serenely unobserved in our telescopes.

Diving deeper into Hubble's Law, the velocity-distance relationship is a cornerstone concept in modern cosmology. It's the idea that the farther a galaxy is from us, the faster it appears to be moving away. This relationship is not due to the galaxies themselves speeding through space, but rather the expansion of space itself – as if the galaxies are dots on a balloon that’s being inflated.

To bring this into perspective with real-world imagery, consider a loaf of raisin bread rising in the oven. As the dough expands, the raisins (similar to galaxies) move apart from each other. In spacetime, the 'dough' expands, and thus, galaxies recede. This forms the basis of understanding how the universe swells over time and aids in piecing together our cosmic history. The relationship between velocity (\(v\)) and distance (\(d\)) is linear and proportional, elegantly summarized by the equation \(v = H_0 d\), where \(H_0\) is the Hubble constant – the universe's expansion rate.

Cosmological measurements are the tools by which astronomers measure the vastness of the cosmos and its constituents. Among those tools, the Hubble constant (\(H_0\)) is paramount, serving as a yardstick for the expansion of the universe. Determining its value is crucial for calculating the age and size of our universe.

However, pinpointing the exact figure for the Hubble constant remains a challenge due to the various methods used for measurement, each yielding slightly different outcomes. Techniques involve observations from Type Ia supernovae, the Cosmic Microwave Background (CMB) radiation, and baryon acoustic oscillations, among others. Jewel-like precision in these measurements is not just academic; they have the weight to decipher the universe's dynamics, its past, and even predict its fate. These cosmological endeavors continue to fuel our understanding and ensure that our celestial map stretches accurately across the eons and the light-years.