[Inverted Passion] Notes from the book ‘First Three Minutes’ by Steven Weinberg

Notes from the book ‘First Three Minutes’ by Steven Weinberg

By Paras Chopra on Oct 13, 2021 04:31 am

It’s mind-blowing that we humans are able to talk about what happened in the first 3 minutes of The Big Bang. This book was written in 1976 which was quite a while back but it’s interesting to note that while there have been extensions in the ideas presented, I’m not aware of any idea being rejected or overturned yet. This should perhaps be unsurprising because most scientific ideas that are accepted as truth are consilient, i.e. they’re supported by multiple lines of evidence.

This means that when we talk about what happened in the first 3 minutes of The Big Bang, we’re confident about those events because only those descriptions make sense when we account for what we observe in the universe today.

I’m writing these notes primarily to solidify what I understood from the book. I’d love to get corrected if I’m wrong somewhere and to learn from people who are informed a lot more about cosmology than me. As you read, feel free to email your feedback to me.

How do we know that the Big Bang really happened?

The evidence for Big Bang essentially comes from the observation that we see different galaxies moving away from us and their speed of movement is proportional to how far they’re from us. This speed — called the Hubble Constant — is an empirical measurement (i.e. cannot be derived from first principles yet). Currently, it’s measured to be 70 (km/s)/Mpc. The unit Mpc is mega parsec where 1 parsec is approximately equal to 3.26 light years.

If we roll back this expansion, we’d naturally find that all these galaxies once were at the same place. To understand this, notice that a galaxy twice as far from us as another galaxy moves at twice the velocity. So if you roll back time, you’ll find that all galaxies (no matter how far from us currently) were once coincident in space.

Of course, this doesn’t mean that our location on Earth is special and the Big Bang started here. The expansion of universe can be observed from any location in the universe (which is the basis of cosmological principle). No location is privileged. Big Bang happened everywhere in the universe.

We can do a rough estimate of when did Big Bang happen by simply inverting the Hubble Constant. Which means if we know that currently a galaxy 1 mega parsec away is moving from us at 70 km/s, we need to ask how much time it would have taken for the galaxy to reach 1 megaparsec from us at that speed.

So, Time since Big Bang = 10^6 * 3.26\,\textup{light years} / 70 \,\textup{km/s}

Converting 70 km/s into light years / years (natural units).

70\,\textup{km/s} = 0.000233495\,\textup{light years/year}

Plugging in:

Time since Big Bang = 10^6 * 3.26 \,\textup{light years} / 0.000233495\,\textup{light years/year}

= 10^6 * 13961\,\textup{years}

= 13.9\,\textup{billion years}

The correct calculation would include the changing velocity of expansion with time (which is primarily due to initial slowing down due to gravitation but current acceleration due to dark energy / cosmological constant).

How do we measure the velocity of galaxies?

We don’t see galaxies moving away from us when we observe them in the telescope because such movement is too imperceptible to human eyes. But we have another way of measuring movement.

This method of measuring how far away something is moving is called redshifting of atomic spectra.

Stars usually emit light at a spectrum of multiple wavelengths. As the light comes out of a star, some of the wavelengths get absorbed by the elements present on its surface. The light gets absorbed because photons at those specific wavelengths cause jumping of electrons in these elements from one energetic state to another. The result of such absorption is that when we pass the light through a sensitive prism, we see dark bands at some locations. These dark bands correspond to the elements in the star. This method is, in fact, how we study the composition of stars.

For example, here is the spectrograph of sun’s radiation (called Fraunhofer lines).

The dark bands correspond to various elements present in the Sun (such as Fe, Na, etc.)

As light from distant galaxies comes towards us, the intervening space itself expands, stretching the wavelength of the emitted light. As I said, this stretching is known as redshifting.

So when we measure spectrum of a distant galaxy, we can compare how much the known absorption lines (as compared to a closer object like Sun) have moved. The difference in wavelengths of what we expect and what we see indicates the velocity of the object.

(Note that measuring Hubble’s constant requires us to know the distance as well (apart from speed). Measuring distances of distant objects is tricky and astronomers use multiple methods. One method is to rely on stars that are called Cephid variables whose absolute luminosity can be calculated from its periodicity of light. We can easily measure periodicity of light which gives us how luminous we expect the star to be. Now we can compare this absolute luminosity to apparent luminosity that we observe in our telescopes to estimate distance needed to dilute absolute luminosity enough to give rise to what we see in our telescope.)

By the way, we can measure the redshift of light from objects at different distances in the universe to figure out the speed of expansion of space during different times in the history of the universe. It’s crazy but in our universe, looking far in space = looking far in time.

For example, the redshift of relatively closer objects is higher than much further objects, indicating that the universe now is expanding faster than it was doing in the past. In other words, it appears our universe’s expansion is accelerating. (Light from the objects nearer would have spent a larger proportion of time traveling through higher rate of space expansion than light from the objects further away, hence we should expect higher relative redshift for closer objects than objects further away)

Estimating what happened in the first three minutes

We have essentially two signals from the very first moments of The Big Bang:

  1. The radiation signal – the leftover light from the Big Bang should be observable today, albeit its wavelength would have been stretched enormously with the expansion of the universe. We detect this radiation as Cosmic Microwave Background, a faint electromagnetic radiation coming from all directions.
  2. The matter signal (Big Bang nucleosynthesis) – all hydrogen that we observe is from the Big Bang (since hydrogen cannot be made in stars as it is their fuel to make higher elements). Also, even though stars produce helium, it turns out most of the helium that exists in the universe is from the Big Bang (we know this because abundance of Helium is estimated to be ~25% in the universe; this amount of Helium if produced in stars would have created intense radiation, something that we would have observed. Moreover, this abundance of helium is pretty constant across stars and galaxies while other heavy elements’ abundance varies which indicates its primordial nature). Since hydrogen and helium were created at the Big Bang, our current ratio of their relative abundance (which is 3:1) should have been the same at the Big Bang and that’s something we can use in our deduction of what happened then.

So, we can put together the radiation signal and the matter signal to roughly reconstruct what must have happened at the Big Bang.

Let’s start with finding more about the temperature of the universe during its first moments.

All objects at thermal equilibrium emit electromagnetic radiation (called blackbody radiation) at a specific spectrum of wavelengths called Plank distribution. This distribution of wavelengths is not as important. What matters is that every object in the universe emits radiation with a specific mean temperature. The higher the temperature of a body, the lower the mean wavelength of emitted radiation. The black body radiation is why infrared cameras work. Animals like humans who maintain a temperature of their bodies emit radiation that peaks in the infrared spectrum, so even in dark when there is no visible light, an infrared camera can pick up this radiation.

Just like all bodies emit a radiation that depends on temperature, we can consider the leftover radiation signal (the cosmic microwave background) to also correspond to a temperature of the universe today. The CMB today is measured to have a characteristic temperature of 2.5 degree kelvin. This temperature corresponds to the microwave length radiation that we observe coming from all directions in the universe.

Since the Big Bang, the photons universe started with have neither been created nor destroyed, so net number of photons today should be similar to what we had during universe’ initial moments. However, wavelength of these photos has been stretched by the expansion of the universe which means volume filled by the same photons grew as a cube. As wavelength increased by the cube, frequency of photons decreased by the cubed. Since temperature of an object depends on energy and energy depends on frequency, so energy per unit volume also got diluted as universe expanded.

If we run this logic backwards in time, this means that per unit volume, as the universe shrunk by a factor of X, there would be a factor of X more energy in that volume and hence the temperature will be higher than a factor of X.

The upshot of this insight is that when the universe was 1000 times smaller in radius than the current size, the background radiation energy of the universe was 1000 times more (since total background radiation is neither created or destroyed since the Big Bang).

This enables us to link the background temperature of the universe with the size and hence age of the universe since the Big Bang.

Why is knowing temperature of the universe during its history important? It’s because at different threshold temperatures, different transitions happen (such as the formation of first atoms, collisions between particles, etc.).

The first moments of the Big Bang

We measure our current observable universe to be roughly a sphere with 45 billion light years radius, which means when the universe was roughly 45 billion times smaller, the background radiation temperature would have been 2.5 degree kelvin * 45 billion = ~100 billion Kelvin in the first instance. This is intensely hot (as a comparison, Sun’s core is 15 million Kelvin). At this high temperature, most interactions and dynamics are at an equilibrium (i.e. the number of particles of matter and antimatter created by pure energy is equal to energy released by annihilation of matter and antimatter).

Because the early universe is super-hot and most interactions are at equilibrium, we cannot say much about the universe other than properties that are conserved (that is, those that are fixed and don’t change with time or size of the universe). Three such conserved quantities describe the first moments:

  1. Net charge: this is the total charge (positive or negative) in the universe. We know it to be zero from modern observations and also from the fact that if there was a net charge, universe would not have been the way as we observe it today.
  2. Baryon number: baryons are large nuclear particles such as neutrons and protons (and their anti partners). Baryon number is the total number of baryons per photon. We know that that this number is not zero because baryons exist (since we’re made from protons and neutrons). This number is experimentally observed to be 1 per 1000 million photons. Note that in early universe, baryons and anti-baryons were constantly created and destroyed. As temperature dropped, this creation stopped and most baryons and anti-baryons annihilated into radiation leaving us with the extra remaining baryons that make up stars, planets and living beings like us. Nobody knows why there are extra baryons or first principles basis of knowing the exact Baryonic number. Perhaps, anthropic principle can explain – if this number would have been different, we would have not existed. It’s speculated that outside our observable universe, in different pockets of the entire universe, this number could be different.
  3. Lepton number: leptons are particles such as electrons, positrons, neutrinos and anti-neutrinos. This number is similar to baryon number and hence represents extra number of leptons per photon. The exact number is hard to estimate but is believed to be smaller than 1.

With these conserved quantities, we can now talk about two big events that happened soon after the Big Bang: nucleosynthesis and recombination.

Nucleosynthesis event

When temperature was high enough, there were similar number of neutrons and protons (both matter and anti-matter) that were constantly getting created and destroyed. As temperature dropped, there was not enough energy in the radiation for this creation, hence it stopped. At that point, matter and anti-matter for baryons were still annihilating into radiation. (The temperature was still high enough for electrons and positrons to still constantly get created and destroyed).

The remaining baryons after annihilation consisted of matter neutrons and protons (almost in equal proportions). Where did antimatter go? It got annihilated. Why was there leftover matter? We don’t know for sure.

At high temperatures, neutrons can be converted into protons and vice versa. However, as temperature drops, the likelihood of a neutron converting into proton is much higher than the other way around. This means with expanding universe (and correspondingly decreasing temperature), more and more neutrons converted into protons (but the reverse didn’t happen much).

Eventually, at ~13 seconds since the Big Bang, temperature dropped to 3 billion degrees and at this temperature, strong nuclear force was able to bind protons and neutrons into stable nucleus such as hydrogen and helium. We can calculate that we should expect the ratio of neutrons to protons at this temperature to be ~15 percent (neutrons) and ~85 percent (protons). Since Helium contains equal neutrons and protons while hydrogen only contains protons, we should estimate Helium to Hydrogen proportion in the current universe to be double proportion as the proportion of neutrons. That is, we should expect our universe to have ~30% (helium) to ~70% (hydrogen). In fact, that is what we observe today and it’s a fantastic confirmation that we’ve got the story of the Big Bang roughly right.

Recombination event

As universe expanded and cooled further, leptons (electrons, neutrinos and their anti-partners) stopped getting created. Neutrinos and anti-neutrinos do not interact much with anything else, so they remain as free particles (which in principle, we should be able to observe today but they don’t interact much). However, electrons and its anti-partner positron interacts so, as temperature dropped, most of them get annihilated away into radiation, leaving a leftover number of electrons which is exactly equal to to number of protons (so that net charge is zero in the universe).

Universe was still too hot (radiation is still too energetic) for electrons to bind to protons to make stable atoms. Light was constantly getting scattered by the unbounded electrons. It’s only after ~350k years that the temperature drops enough for free electrons to be captured by hydrogen and helium nucleus to give rise to hydrogen and helium atoms. These atoms eventually coalesced into galaxies and stars, giving rise to living beings like us.

This is incidentally when the universe became transparent as light didn’t have any free electrons to bump into / scatter from. With no more free electrons to bump into, the radiation started to travel freely and that’s what we see today in CMB (cosmic wave background).

But the first stars didn’t form until 200-500 million years. So until then, the era is called the Dark Ages.

Future of the universe

What happens in the future with our universe depends on whether there is enough mass in the universe for gravity to start dominating the movement of galaxies, so that at some point in future instead of racing away from each other, they start racing towards each other.

Hence, the two scenarios of the future would be:

  • Open: there is not enough matter density. The universe keeps on expanding forever, which means any two distant galaxies (that are not locally gravitationally bound) keep getting further and further away, eventually getting so far at such speeds that even light wouldn’t be able to travel between them.
  • Closed: there is enough matter density that after a point, galaxies stop moving away from each other and start moving towards each other. This would look like running the Big Bang in reverse, eventually compressing everything into a small space. It’s called the Big Crunch.

We don’t know about the future of the universe with certainty, but all current evidence indicates that there isn’t enough matter and hence the universe will keep on expanding forever.

Closing lines

To close my notes, I’ll quote the lines from the last paragraph from the book because they’re so beautiful:

The more universe seems comprehensible, the more it seems pointless.
..
But if there’s no solace in the fruits of our research, there is at least some consolation in the research itself. .. The effort to understand the universe is one of the very few things that lifts human life a little above the level of farce, and gives it some of the grace of tragedy.

This essay started out its life as a raw note in my knowledge garden.


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