## Friday, August 10, 2012

### The Journey of a Gravitational Wave III: GWs Stretch Out

What happens to a gravitational wave between when it is produced and when LIGO can detect it?  It turns out not much, which makes it a key new medium in which to observe the Universe!

Today I will be discussing how a gravitational wave can get stretched during its journey to Earth.  Previously, we have discussed how gravitational waves can be bent away from a straight path and how things cannot absorb or reflect a gravitational wave.

THE DOPPLER EFFECT

If you have ever been passed by an ambulance or police car with its sirens on, then you likely noticed that as the sound from the siren approached you, the tone was higher than when the siren passed you.  This is due to something called the Doppler effect.  As the source of a sound is moving toward toward you, the distance between the sound wave's crests (wavelength) are compressed, resulting in a higher frequency.  We hear higher frequencies as higher tones.  Conversely, when the source of a sound is moving away from you, the sounds wave's crests are stretched apart resulting in a lower frequency and a lower tone.  Consider the following animated example:

 Animated example of the Doppler effect: how motion and cause the wavelength of sound to be affected. [Image from Wikipedia]

In the beginning of this animation, the car is not moving and the sound waves have the same wavelength both in front and behind the car.  Once it starts moving forward (to the left), the sound waves in front of the car are closer together and the sound waves behind the car are farther apart.  This results in us hearing a higher tone when you are in front of the car and a lower tone when you are behind the car.

COSMOLOGICAL REDSHIFT

This Doppler effect can also affect the wavelength of light and of a gravitational wave depending on the motion of the object creating the waves.  First, let's think about what happens to light.  As the source moves toward you, the wavelength of the light will get shorter.  Since we can't hear light, instead of hearing a higher tone, we will see the light to be shifted toward the blue end of the spectrum (where the shortest wavelengths we can see are).  This is called blueshift.  For a source moving away from us, the wavelengths would be longer and we would see the light shifted toward the red end of the spectrum (where the longest wavelengths we can see are).  This is called redshift.

We can measure this cosmological Doppler effect by measuring the spectral signatures of stars.  Below is an example of such a spectral signature.  Stars give off almost every color of light, but there are particular colors that get absorbed by different elements before that light can make it to Earth.  Those absorbed colors show up as dark lines in an otherwise complete spectrum of colors:

 The optical spectrum from our Sun (left) compared to the optical spectrum of a supercluster of galaxies (right).  Note that there are similar dark lines (from absorption of those wavelengths by different chemical elements), but the lines on the right are shifted toward the red end of the spectrum.  This phenomena is called "redshift" because of this.  [Source: Wikipedia]

The spectrum to the left is what we observe from our local star, the Sun.  The spectrum on the right is from a distant cluster of galaxies (containing many stars).  Note that the spacing of the dark lines is essentially the same, just shifted toward the red end of the spectrum (this is highlighted by the arrows).  That means that this cluster of galaxies is moving away from us.  Just about everything (but not quite) in the Universe shows this redshift which implies that the Universe is expanding!  (There is more than one cause of redshift, but we will only be discussing the redshift due to the expanding Universe in this blog post.)

Redshift is measured by the change in wavelength of light (or a gravitational wave) divided by the wavelength of light if the source wasn't moving.  For the spectrum example here, this would be the amount the color of the light changed (the difference between the right and the left side) divided by its original color (the left side).  The redshift is a result of the Doppler effect; it can also be used to measure an object's velocity.  To do that, we need to determine the proportionality constant between redshift and velocity.  This constant is known as the Hubble constant, H (named after Edwin Hubble who first determined this relationship).  The best estimate of this constant is currently:

H = 71.0 ± 2.5 km/s/Mpc
67.2 ± 1.2 km/s/Mpc
(revised on 20 March 2013 with the parameters released by Planck)

This means that for every light-year away an object is, its velocity away from us increases by about 257 feet/hour.  (This may not sound like much, but a light-year is really a very small distance in the Universe; our own galaxy is about 110,000 light-years across!)  This expansion is believed to have originated with the Big Bang.  So, will the Universe expand forever?  Will it slow to a stop?  Will it stop expanding and start shrinking until everything is a compact ball again (this is called the Big Crunch)?  From current observations, the ultimate fate of the Universe appears to be an eternal slowing expansion which will cause all of the energy to be evenly distributed throughout this even larger Universe (compared to now).  That means that there won't be enough energy in any one place to make anything happen (all events in the Universe need an imbalance in energy).  This is called the Big Freeze or the heat death of the Universe.  However, more observations (especially of things like dark matter and dark energy) of our Universe are needed before we can be sure of the Universe's future.

BLUESHIFT

There are a few things in our Universe that are moving towards us and therefore have a blueshift.  Of most note is the Andromeda Galaxy which is moving towards us at about 300,000 meters/s (671,801 mph).  This is because our Milky Way Galaxy and the Andromeda Galaxy are gravitationally bound to each other; this means that the gravitational attractions between these two galaxies are greater than the expansion of the Universe.  In the future, our two galaxies will collide, but not for an estimated 4.5 billion years.  This is also about the time the that our Sun will become a red giant and all life on Earth will be extinct by then anyway!  I wouldn't stress about it if I were you :P

Also, portions of rotating galaxies can be blueshifted.  In this case, the side that is rotating toward us will be blueshifted and the part that is rotating away from us will be redshifted.

GRAVITATIONAL WAVES AND REDSHIFT

Just like redshift stretches out the wavelength of light, the expansion of the Universe will also stretch out the wavelength of a gravitational wave during its journey to Earth.  In a previous post, I discussed what gravitational waves would sound like if you put their signal through a speaker (remember: gravitational waves don't make sound!).  So, I decided I wanted to hear what the change in the sound of a gravitational wave is due to different amounts of redshift.  I did the calculations, generated the sounds, and assembled them into the video below for your viewing pleasure:

So, how can we make use of the redshifting of a gravitational wave to learn more about our Universe?  Well, most astronomers and physicists don't believe that the rate of expansion of the Universe (the Hubble constant, H, above) has been the same since the Big Bang.  Particularly, there is a period believed to have existed in the past history of the Universe called inflation which is a time of rapid expansion.  From a certain kind gravitational wave source, we will be able to measure the Hubble constant around the neighborhood of the source.  This will be a direct probe of how the expansion of the Universe (the Hubble constant) has changed through the history of the Universe because observing objects in the distance is the same as observing them as they were in the past (e.g. observing an object 1 light-year away is the same as observing it as it was a year ago since light and gravitational waves travel at the speed of light - at least gravitational waves are expected to travel at the speed of light!).

1. "This means that for every light-year away an object is, its velocity away from us increases by about 257 feet/hour."

The expansion of the universe sounds kind of unimpressive when you put it that way - roughly the speed of a beetle :)

1. True, but I wanted to express this in terms that people in the US may be able to identify with...

Perhaps I should add that a light year is really not that far in the Universe. After all, the Milky Way galaxy is about 110,000 light-years across!

Thanks for pointing out that this needs some perspective!

2. Well it takes billions of years so I guess it can't be that fast. I suppose you could say the space between us and Betelgeuse is expanding at about 30 mph (and the path to the Galactic centre at Mach 1.7)

2. Can you/LIGO tell the difference between a low-frequency source nearby and a high-frequency one that's been redshifted? (E.g. does a chirp source slowed down still sound like a chirp source?)