text:
# Simple Rules Don’t Mean a Simple Universe

It’s always fun when nature surprises you.

This week, the Perimeter Colloquium was given by Laura Nuttall, a member of the LIGO collaboration.

In a physics department, the colloquium is a regularly scheduled talk that’s supposed to be of interest to the entire department. Some are better at this than others, but this one was pretty fun. The talk explored the sorts of questions gravitational wave telescopes like LIGO can answer about the world.

At one point during the talk, Nuttall showed a plot of what happens when a star collapses into a supernova. For a range of masses, the supernova leaves behind a neutron star (shown on the plot in purple). For heavier stars, it instead results in a black hole, a big black region of the plot.

What surprised me was that inside the black region, there was an unexpected blob: a band of white in the middle of the black holes. Heavier than that band, the star forms a black hole. Lighter, it also forms a black hole. But inside?

Nothing. The star leaves nothing behind. It just explodes.

The physical laws that govern collapsing stars might not be simple, but at least they sound straightforward. Stars are constantly collapsing under their own weight, held up only by the raging heat of nuclear fire. If that heat isn’t strong enough, the star collapses, and other forces take over, so the star becomes a white dwarf, or a neutron star. And if none of those forces is strong enough, the star collapses completely, forming a black hole.

Too small, neutron star. Big enough, black hole. It seems obvious. But reality makes things more complicated.

It turns out, if a star is both massive and has comparatively little metal in it, the core of the star can get very very hot. That heat powers an explosion more powerful than a typical star, one that tears the star apart completely, leaving nothing behind that could form a black hole. Lighter stars don’t get as hot, so they can still form black holes, and heavier stars are so heavy they form black holes anyway. But for those specific stars, in the middle, nothing gets left behind.

This isn’t due to mysterious unknown physics. It’s actually been understood for quite some time. It’s a consequence of those seemingly straightforward laws, one that isn’t at all obvious until you do the work and run the simulations and observe the universe and figure it out.

Just because our world is governed by simple laws, doesn’t mean the universe itself is simple. Give it a little room (and several stars’ worth of hydrogen) and it can still surprise you.

# What Space Can Tell Us about Fundamental Physics

Back when LIGO announced its detection of gravitational waves, there was one question people kept asking me: “what does this say about quantum gravity?”

The answer, each time, was “nothing”. LIGO’s success told us nothing about quantum gravity, and very likely LIGO will never tell us anything about quantum gravity.

The sheer volume of questions made me think, though. Astronomy, astrophysics, and cosmology fascinate people. They capture the public’s imagination in a way that makes them expect breakthroughs about fundamental questions. Especially now, with the LHC so far seeing nothing new since the Higgs, people are turning to space for answers.

Is that a fair expectation? Well, yes and no.

Most astrophysicists aren’t concerned with finding new fundamental laws of nature. They’re interested in big systems like stars and galaxies, where we know most of the basic rules but can’t possibly calculate all their consequences. Like most physicists, they’re doing the vital work of “physics of decimals”.

At the same time, there’s a decent chunk of astrophysics and cosmology that does matter for fundamental physics. Just not all of it. Here are some of the key areas where space has something important to say about the fundamental rules that govern our world:

1. Dark Matter:

Galaxies rotate at different speeds than their stars would alone. Clusters of galaxies bend light that passes by, and do so more than their visible mass would suggest. And when scientists try to model the evolution of the universe, from early images to its current form, the models require an additional piece: extra matter that cannot interact with light. All of this suggests that there is some extra “dark” matter in the universe, not described by our standard model of particle physics.

If we want to understand this dark matter, we need to know more about its properties, and much of that can be learned from astronomy. If it turns out dark matter isn’t really matter after all, if it can be explained by a modification of gravity or better calculations of gravity’s effects, then it still will have important implications for fundamental physics, and astronomical evidence will still be key to finding those implications.

2. Dark Energy (/Cosmological Constant/Inflation/…):

The universe is expanding, and its expansion appears to be accelerating. It also seems more smooth and uniform than expected, suggesting that it had a period of much greater acceleration early on. Both of these suggest some extra quantity: a changing acceleration, a “dark energy”, the sort of thing that can often be explained by a new scalar field like the Higgs.

Again, the specifics: how (and perhaps if) the universe is expanding now, what kinds of early expansion (if any) the shape of the universe suggests, these will almost certainly have implications for fundamental physics.

3. Limits on stable stuff:

Let’s say you have a new proposal for particle physics. You’ve predicted a new particle, but it can’t interact with anything else, or interacts so weakly we’d never detect it. If your new particle is stable, then you can still say something about it, because its mass would have an effect on the early universe. Too many such particles and they would throw off cosmologists’ models, ruling them out.

Alternatively, you might predict something that could be detected, but hasn’t, like a magnetic monopole. Then cosmologists can tell you how many such particles would have been produced in the early universe, and thus how likely we would be to detect them today. If you predict too many particles and we don’t see them, then that becomes evidence against your proposal.

4. “Cosmological Collider Physics”:

A few years back, Nima Arkani-Hamed and Juan Maldacena suggested that the early universe could be viewed as an extremely high energy particle collider. While this collider performed only one experiment, the results from that experiment are spread across the sky, and observed patterns in the early universe should tell us something about the particles produced by the cosmic collider.

People are still teasing out the implications of this idea, but it looks promising, and could mean we have a lot more to learn from examining the structure of the universe.

5. Big Weird Space Stuff:

If you suspect we live in a multiverse, you might want to look for signs of other universes brushing up against our own. If your model of the early universe predicts vast cosmic strings, maybe a gravitational wave detector like LIGO will be able to see them.

6. Unexpected weirdness:

In all likelihood, nothing visibly “quantum” happens at the event horizons of astrophysical black holes. If you think there’s something to see though, the Event Horizon Telescope might be able to see it. There’s a grab bag of other predictions like this: situations where we probably won’t see anything, but where at least one person thinks there’s a question worth asking.

I’ve probably left something out here, but this should give you a general idea. There is a lot that fundamental physics can learn from astronomy, from the overall structure and origins of the universe to unexplained phenomena like dark matter. But not everything in astronomy has these sorts of implications: for the most part, astronomy is interesting not because it tells us something about the fundamental laws of nature, but because it tells us how the vast space above us actually happens to work.

# Things You Don’t Know about the Power of the Dark Side

Last Wednesday, Katherine Freese gave a Public Lecture at Perimeter on the topic of Dark Matter and Dark Energy. The talk should be on Perimeter’s YouTube page by the time this post is up.

Answering twitter questions during the talk made me realize that there’s a lot the average person finds confusing about Dark Matter and Dark Energy. Freese addressed much of this pretty well in her talk, but I felt like there was room for improvement. Rather than try to tackle it myself, I decided to interview an expert on the Dark Side of the universe.

Twitter doesn’t know the power of the dark side!

Lord Vader, some people have a hard time distinguishing Dark Matter and Dark Energy. What do you have to say to them?

Fools! Light side astronomers call “dark” that which they cannot observe and cannot understand. “Fear” and “anger” are different heights of emotion, but to the Jedi they are only the path to the Dark Side. Dark Energy and Dark Matter are much the same: both distinct, both essential to the universe, and both “dark” to the telescopes of the light.

You ask an empty question. “Matter” has been defined in many ways. When we on the Dark Side refer to Dark Matter, we merely mean to state that it behaves much like the matter you know: it is drawn to and fro by gravity, sloshing about.

It is distinct from your ordinary matter in that two of the forces of nature, the strong nuclear force and electromagnetism, do not concern it. Ordinary matter is bound together in the nuclei of atoms by the strong force, or woven into atoms and molecules by electromagnetism. This makes it subject to all manner of messy collisions.

Dark Matter, in contrast, is pure, partaking neither of nuclear nor chemical reactions. It passes through each of us with no notice. Only the weak nuclear force and gravity affect it. The latter has brought it slowly into clumps and threads through the universe, each one a vast nest for groupings of stars. Truly, Dark Matter surrounds us, penetrates us, and binds the galaxy together.

Could Dark Matter be something we’re more familiar with, like neutrinos or black holes? What about a modification of gravity?

Many wondered as much, when the study of the Dark Side was young. They were wrong.

The matter you are accustomed to composes merely a twentieth of the universe, while Dark Matter is more than a quarter. There is simply not enough of these minor contributions, neutrinos and black holes, to account for the vast darkness that surrounds the galaxy, and with each astronomer’s investigation we grow more assured.

As for modifying gravity, do you seek to modify a fundamental Force?

If so, you should be wary. Forces, by their nature, are accompanied by particles, and gravity is no exception. Take care that your tinkering does not result in a new sort of particle. If so, you may be unknowingly walking the path of the Dark Side, for your modification may be just another form of Dark Matter.

What sort of things could Dark Matter be? Can Dark Matter decay into ordinary matter? Could there be anti-Dark Matter?

As of yet, your scientists are still baffled by the nature of Dark Matter. Still, there are limits. Since only rare events could produce it from ordinary matter, the universe’s supply of Dark Matter must be ancient, dating back to the dawn of the cosmos. In that case, it must decay only slowly, if at all. Similarly, if Dark Matter had antimatter forms then its interactions must be so weak that it has not simply annihilated with its antimatter half across the universe. So while either is possible, it may be simpler for your theorists if Dark Matter did not decay, and was its own antimatter counterpart. On the other hand, if Dark Matter did undergo such reactions, your kind may one day be able to detect it.

Of course, as a master of the Dark Side I know the true nature of Dark Matter. However, I could only impart it to a loyal apprentice…

Yeah, I think I’ll pass on that. They say you can only get a job in academia when someone dies, but unlike the Sith they don’t mean it literally.

Let’s move on to Dark Energy. What can you tell us about it?

Dark “Energy”, like Dark Matter, is named for what people on your Earth cannot comprehend. Nothing, not even Dark Energy, is “made of energy”. Dark Energy is “energy” merely because it behaves unlike matter.

Matter, even Dark Matter, is drawn together by the force of gravity. Under its yoke, the universe would slow down in its expansion and eventually collapse into a crunch, like the throat of an incompetent officer.

However, the universe is not collapsing, but accelerating, galaxies torn away from each other by a force that must compose more than two thirds of the universe. It is rather like the Yuuzhan Vong, a mysterious force from outside the galaxy that scouts persistently under- or over-estimate.

Umm, I’m pretty sure the Yuuzhan Vong don’t exist anymore, since Disney got rid of the Expanded Universe.

That perfidious Mouse!

Well folks, Vader is now on a rampage of revenge in the Disney offices, so I guess we’ll have to end the interview. Tune in next week, and until then, may the Force be with you!

# You Go, LIGO!

Well folks, they did it. LIGO has detected gravitational waves!

FAQ:

What’s a gravitational wave?

Gravitational waves are ripples in space and time. As Einstein figured out a century ago, masses bend space and time, which causes gravity. Wiggle masses in the right way and you get a gravity wave, like a ripple on a pond.

Ok, but what is actually rippling? It’s some stuff, right? Dust or something?

In a word, no. Not everything has to be “stuff”. Energy isn’t “stuff”, and space-time isn’t either, but space-time is really what vibrates when a gravitational wave passes by. Distances themselves are changing, in a way that is described by the same math and physics as a ripple in a pond.

What’s LIGO?

LIGO is the Laser Interferometer Gravitational-Wave Observatory. In simple terms, it’s an observatory (or rather, a pair of observatories in Washington and Louisiana) that can detect gravitational waves. It does this using beams of laser light four kilometers long. Gravitational waves change the length of these beams when they pass through, causing small but measurable changes in the laser light observed.

Are there other gravitational wave observatories?

Not currently in operation. LIGO originally ran from 2002 to 2010, and during that time there were other gravitational wave observatories also in operation (VIRGO in Italy and GEO600 in Germany). All of them (including LIGO) failed to detect anything, and so LIGO and VIRGO were shut down in order for them to be upgraded to more sensitive, advanced versions. Advanced LIGO went into operation first, and made the detection. VIRGO is still under construction, as is KAGRA, a detector in Japan. There are also plans for a detector in India.

Other sorts of experiments can detect gravitational waves on different scales. eLISA is a planned space-based gravitational wave observatory, while Pulsar Timing Arrays could use distant neutron stars as an impromptu detector.

What did they detect? What could they detect?

The gravitational waves that LIGO detected came from a pair of black holes merging. In general, gravitational waves come from a pair of masses, or one mass with an uneven and rapidly changing shape. As such, LIGO and future detectors might be able to observe binary stars, supernovas, weird-shaped neutron stars, colliding galaxies…pretty much any astrophysical event involving large things moving comparatively fast.

What does this say about string theory?

Basically nothing. There are gravity waves in string theory, sure (and they play a fairly important role), but there were gravity waves in Einstein’s general relativity. As far as I’m aware, no-one at this point seriously thought that gravitational waves didn’t exist. Nothing that LIGO observed has any bearing on the quantum properties of gravity.

But what about cosmic strings? They mentioned those in the announcement!

Cosmic strings, despite the name, aren’t a unique prediction of string theory. They’re big, string-shaped wrinkles in space and time, possible results of the rapid expansion of space during cosmic inflation. You can think of them a bit like the cracks that form in an over-inflated balloon right before it bursts.

Cosmic strings, if they exist, should produce gravitational waves. This means that in the future we may have concrete evidence of whether or not they exist. This wouldn’t say all that much about string theory: while string theory does have its own explanations for cosmic strings, it’s unclear whether it actually has unique predictions about them. It would say a lot about cosmic inflation, though, and would presumably help distinguish it from proposed alternatives. So keep your eyes open: in the next few years, gravitational wave observatories may well have something important to say about the overall history of the universe.

Why is this discovery important, though? If we already knew that gravitational waves existed, why does discovering them matter?

LIGO didn’t discover that gravitational waves exist. LIGO discovered that we can detect them.

The existence of gravitational waves is no discovery. But the fact that we now have observatories sensitive enough to detect them is huge. It opens up a whole new type of astronomy: we can now observe the universe not just by the light it sheds (and neutrinos), but through a whole new lens. And every time we get another observational tool like this, we notice new things, things we couldn’t have seen without it. It’s the dawn of a new era in astronomy, and LIGO was right to announce it with all the pomp and circumstance they could muster.

My impressions from the announcement:

Speaking of pomp and circumstance, I was impressed by just how well put-together LIGO’s announcement was.

As the US presidential election heats up, I’ve seen a few articles about the various candidates’ (well, usually Trump’s) use of the language of political propaganda. The idea is that there are certain visual symbols at political events for which people have strong associations, whether with historical events or specific ideas or the like, and that using these symbols makes propaganda more powerful.

What I haven’t seen is much discussion of a language of scientific propaganda. Still, the overwhelming impression I got from LIGO’s announcement is that it was shaped by a master in the use of such a language. They tapped in to a wide variety of powerful images: from the documentary-style interviews at the beginning, to Weiss’s tweed jacket and handmade demos, to the American flag in the background, that tied LIGO’s result to the history of scientific accomplishment.

Perimeter’s presentations tend to have a slicker look, my friends at Stony Brook are probably better at avoiding jargon. But neither is quite as good at propaganda, at saying “we are part of history” and doing so without a hitch, as the folks at LIGO have shown themselves to be with this announcement.

I was also fairly impressed that they kept this under wraps for so long. While there were leaks, I don’t think many people had a complete grasp of what was going to be announced until the week before. Somehow, LIGO made sure a collaboration of thousands was able to (mostly) keep their mouths shut!

Beyond the organizational and stylistic notes, my main thought was “What’s next?” They’ve announced the detection of one event. I’ve heard others rattle off estimates, that they should be detecting anywhere from one black hole merger per year to a few hundred. Are we going to see more events soon, or should we settle into a long wait? Could they already have detected more, with the evidence buried in their data, to be revealed by careful analysis? (The waves from this black hole merger were clear enough for them to detect them in real-time, but more subtle events might not make things so easy!) Should we be seeing more events already, and does not seeing them tell us something important about the universe?

Most of the reason I delayed my post till this week was to see if anyone had an answer to these questions. So far, I haven’t seen one, besides the “one to a few hundred” estimate mentioned. As more people weigh in and more of LIGO’s run is analyzed, it will be interesting to see where that side of the story goes.

# The Universe, Astronomy’s Lab

There’s a theme in a certain kind of science fiction.

Not in the type with laser swords and space elves, and not in cyberpunk dystopias…but when sci-fi tries to explore what humanity might do if it really got a chance to explore its own capabilities. In a word, the theme is scale.

We start out with a Dyson sphere, built around our own sun to trap its energy. As time goes on, the projects get larger and larger, involving multiple stars and, eventually, reshaping the galaxy.

There’s an expectation, though, that this sort of thing is far in our future. Treating the galaxy as a resource, as a machine, seems well beyond our present capabilities.

On Wednesday, Victoria Kaspi gave a public lecture at Perimeter about neutron stars. At the very end of the lecture, she talked a bit about something she covered in more detail during her colloquium earlier that day, called a Pulsar Timing Array.

Neutron stars are one of the ways a star can end its life. Too big to burn out quietly and form a white dwarf, and too small to collapse all the way into a black hole, the progenitors of neutron stars have so much gravity that they force protons and electrons to merge, so that the star ends up as a giant ball of neutrons, like an enormous atomic nucleus.

Many of these neutron stars have strong magnetic fields. A good number of them are what are called pulsars: stars that emit powerful pulses of electromagnetic radiation, often at regular intervals. Some of these pulsars are very regular indeed, rivaling atomic clocks in their precision. The idea of a Pulsar Timing Array is to exploit this regularity by using these pulsars as a gravitational wave telescope.

Gravitational waves are ripples in space-time. They were predicted by Einstein’s theory, and we’ve observed their indirect effects, but so far we have yet to detect them directly. Attempts have been made: vast detectors like LIGO have been built that bounce light across long “arms”, trying to detect minute disruptions in space. The problem is, it’s hard to distinguish these disruptions from ordinary vibrations in the area, like minor earthquakes. Size also limits the effectiveness of these detectors, with larger detectors able to see the waves from bigger astronomical events.

Pulsar Timing Arrays sidestep both of those problems. Instead of trying to build a detector on the ground like LIGO (or even in space like LISA), they use the pulsars themselves as the “arms” of a galaxy-sized detector. Because these pulsars emit light so regularly, small disruptions can be a sign that a gravitational wave is passing by the earth and disrupting the signal. Because they are spread roughly evenly across the galaxy, we can correlate signals across multiple pulsars, to make sure we’re really seeing gravitational waves. And because they’re so far apart, we can use them to detect waves from some of the biggest astronomical events, like galaxies colliding.

Earth very much not to scale.

Longtime readers know that I find astronomy really inspiring, but Kaspi’s talk woke me up to a completely different aspect, that of our mastery of scale.

Want to dream of a future where we use the solar system and the galaxy as resources? We’re there, and we’ve been there for a long time. We’re a civilization that used nearby planets to bootstrap up the basic laws of motion before we even had light bulbs. We’ve honed our understanding of space and time using distant stars. And now, we’re using an array of city-sized balls of neutronium, distributed across the galaxy, as a telescope. If that’s not the stuff of science fiction, I don’t know what is.

By the way, speaking of webcast lectures, I’m going to be a guest on the Alda Center’s Science Unplugged show next week. Tune in if you want to hear about the sort of stuff I work on, using string theory as a tool to develop shortcuts for particle physics calculations.

# What’s the Matter with Dark Matter, Matt?

It’s very rare that I disagree with Matt Strassler. That said, I can’t help but think that, when he criticizes the press for focusing their LHC stories on dark matter, he’s missing an important element.

From his perspective, when the media says that the goal of the new run of the LHC is to detect dark matter, they’re just being lazy. People have heard of dark matter. They might have read that it makes up 23% of the universe, more than regular matter at 4%. So when an LHC physicist wants to explain what they’re working on to a journalist, the easiest way is to talk about dark matter. And when the journalist wants to explain the LHC to the public, they do the same thing.

This explanation makes sense, but it’s a little glib. What Matt Strassler is missing is that, from the public’s perspective, dark matter really is a central part of the LHC’s justification.

Now, I’m not saying that the LHC’s main goal is to detect dark matter! Directly detecting dark matter is pretty low on the LHC’s list of priorities. Even if it detects a new particle with the right properties to be dark matter, it still wouldn’t be able to confirm that it really is dark matter without help from another experiment that actually observes some consequence of the new particle among the stars. I agree with Matt when he writes that the LHC’s priorities for the next run are

1. studying the newly discovered Higgs particle in great detail, checking its properties very carefully against the predictions of the “Standard Model” (the equations that describe the known apparently-elementary particles and forces)  to see whether our current understanding of the Higgs field is complete and correct, and

2. trying to find particles or other phenomena that might resolve the naturalness puzzle of the Standard Model, a puzzle which makes many particle physicists suspicious that we are missing an important part of the story, and

3. seeking either dark matter particles or particles that may be shown someday to be “associated” with dark matter.

Here’s the thing, though:

From the public’s perspective, why do we need to study the properties of the Higgs? Because we think it might be different than the Standard Model predicts.

Why do we think it might be different than the Standard Model predicts? More generally, why do we expect the world to be different from the Standard Model at all? Well there are a few reasons, but they generally boil down to two things: the naturalness puzzle, and the fact that the Standard Model doesn’t have anything that could account for dark matter.

Naturalness is a powerful motivation, but it’s hard to sell to the general public. Does the universe appear fine-tuned? Then maybe it just is fine-tuned! Maybe someone fine-tuned it!

These arguments miss the real problem with fine-tuning, but they’re hard to correct in a short article. Getting the public worried about naturalness is tough, tough enough that I don’t think we can demand it of the average journalist, or accuse them of being lazy if they fail to do it.

That leaves dark matter. And for all that naturalness is philosophically murky, dark matter is remarkably clear. We don’t know what 96% of the universe is made of! That’s huge, and not just in a “gee-whiz-cool” way. It shows, directly and intuitively, that physics still has something it needs to solve, that we still have particles to find. Unless you are a fan of (increasingly dubious) modifications to gravity like MOND, dark matter is the strongest possible justification for machines like the LHC.

The LHC won’t confirm dark matter on its own. It might not directly detect it, that’s still quite up-in-the-air. And even if it finds deviations from the Standard Model, it’s not likely they’ll be directly caused by dark matter, at least not in a simple way.

But the reason that the press is describing the LHC’s mission in terms of dark matter isn’t just laziness. It’s because, from the public’s perspective, dark matter is the only vaguely plausible reason to spend billions of dollars searching for new particles, especially when we’ve already found the Higgs. We’re lucky it’s such a good reason.

# All Is Dust

Joke stolen from some fellow PI postdocs.

The BICEP2 and Planck experiment teams have released a joint analysis of their data, discovering what many had already suspected: that the evidence for primordial gravitational waves found by BICEP2 can be fully explained by interstellar dust.

For those who haven’t been following the story, BICEP2 is a telescope in Antarctica. Last March, they told the press they had found evidence of primordial gravitational waves, ripples in space-time caused by the exponential expansion of the universe shortly after the Big Bang. Soon after, though, doubts were raised. It appeared that the BICEP2 team hadn’t taken proper account of interstellar dust, and in particular had mis-used some data they scraped from a presentation by larger experiment Planck. After Planck released the correct version of their dust data, BICEP2’s predictions were even more evidently premature.

Now, the Planck team has exhaustively gone over their data and BICEP2’s, and done a full analysis. The result is a pretty thorough statement: everything BICEP2 observed can be explained by interstellar dust.

A few news outlets have been describing this as “ruling out inflation” or “ruling out gravitational waves”, both of which are misunderstandings. What Planck has ruled out are inflation (and gravitational waves caused by inflation) powerful enough to have been observed by BICEP2.

To an extent, this was something Planck had already predicted before BICEP2 made their announcement. BICEP2 announced a value for a parameter r, called the tensor-scalar ratio, of 0.2. This parameter r is a way to measure the strength of the gravitational waves (if you want to know what gravitational waves have to do with tensors, this post might help), and thus indirectly the strength of inflation in the early universe.

Trouble is, Planck had already released results arguing that r had to be below 0.11! So a lot of people were already rather skeptical.

With the new evidence, Planck’s bound is relaxed slightly. They now argue that r should be below 0.13, so BICEP2’s evidence was enough to introduce some fuzziness into their measurements when everything was analyzed together.

I’ve complained before about the bad aspects of BICEP2’s announcement, how releasing their data prematurely hurt the public’s trust in science and revealed the nasty side of competition for funding on massive projects. In this post, I’d like to talk a little about the positive side of the publicity around BICEP2.

Lots of theorists care about physics at very very high energies. The scale of string theory, or the Planck mass (no direct connection to the experiment, just the energy where one expects quantum gravity to be relevant), or the energy at which the fundamental forces might unify, are all much higher than any energy we can explore with a particle collider like the LHC. If you had gone out before BICEP2’s announcement and asked physicists whether we would ever see direct evidence for physics at these kinds of scales, they would have given you a resounding no. Maybe we could see indirect evidence, but any direct consequences would be essentially invisible.

All that changed with BICEP2. Their announcement of an r of 0.2 corresponds to very strong inflation, inflation of higher energy than the Planck mass!

Suddenly, there was hope that, even if we could never see such high-energy physics in a collider, we could see it out in the cosmos. This falls into a wider trend. Physicists have increasingly begun to look to the stars as the LHC continues to show nothing new. But the possibility that the cosmos could give us data that not only meets LHC energies, but surpasses them so dramatically, is something that very few people had realized.

The thing is, that hope is still alive and kicking. The new bound, restricting r to less than 0.13, still allows enormously powerful inflation. (If you’d like to work out the math yourself, equation (14) here relates the scale of inflation $\Delta \phi$ to the Planck mass $M_{\textrm{Pl}}$ and the parameter r.)

This isn’t just a “it hasn’t been ruled out yet” claim either. Cosmologists tell me that new experiments coming online in the next decade will have much more precision, and much better ability to take account of dust. These experiments should be sensitive to an r as low as 0.001!

With that kind of sensitivity, and the new mindset that BICEP2 introduced, we have a real chance of seeing evidence of Planck-scale physics within the next ten or twenty years. We just have to wait and see if the stars are right…


text:
Waves on a String Video Lessons

Concept

# Problem: Two Velocities in a Traveling WaveWave motion is characterized by two velocities: the velocity with which the wave moves in the medium (e.g., air or a string) and the velocity of the medium (the air or the string itself).Consider a transverse wave traveling in a string. The mathematical form of the wave isy(x,t) = Asin(Kx - wt) Part AFind the velocity of propagation vp of this wave.Express the velocity of propagation in terms of some or all of the variables A, k,and w.Part BFind the y velocity  vy(x,t)  of a point on the string as a function of x and t.Express the y velocity in terms of w, A,k,x,and t.Part CWhich of the following statements about, the x component of the velocity of the string, is true?a) vx(x;t) = vpb) vx(x;t) = vy(x;t) c) vx(x;t)  has the same mathematical form as vy(x;t) but is 180° out of phase.d) vx(x;t) = 0Part DFind the slope of the string  as a function of position x and time t.Express your answer in terms of A,k,w, x,and t.

###### FREE Expert Solution

Propagation constant k:

$\overline{){\mathbf{k}}{\mathbf{=}}\frac{\mathbf{2}\mathbf{\pi }}{\mathbf{\lambda }}}$

Period, T:

$\overline{){\mathbf{T}}{\mathbf{=}}\frac{\mathbf{2}\mathbf{\pi }}{\mathbf{\omega }}}$

Part A

vp = λ/T = (2π/k)/(2π/ω) = ω/k

89% (350 ratings)
###### Problem Details

Two Velocities in a Traveling Wave

Wave motion is characterized by two velocities: the velocity with which the wave moves in the medium (e.g., air or a string) and the velocity of the medium (the air or the string itself).

Consider a transverse wave traveling in a string. The mathematical form of the wave is

y(x,t) = Asin(Kx - wt)

Part A

Find the velocity of propagation vp of this wave.
Express the velocity of propagation in terms of some or all of the variables A, k,and w.

Part B

Find the y velocity  vy(x,t)  of a point on the string as a function of x and t.

Express the y velocity in terms of w, A,k,x,and t.

Part C

Which of the following statements about, the x component of the velocity of the string, is true?

a) vx(x;t) = vp

b) vx(x;t) = vy(x;t)

c) vx(x;t)  has the same mathematical form as vy(x;t) but is 180° out of phase.

d) vx(x;t) = 0

Part D

Find the slope of the string  as a function of position x and time t.

format: plain


text:
# 2nd PUC Statistics Previous Year Question Paper March 2017

Students can Download 2nd PUC Statistics Previous Year Question Paper March 2017, Karnataka 2nd PUC Statistics Model Question Papers with Answers helps you to revise the complete Karnataka State Board Syllabus and score more marks in your examinations.

## Karnataka 2nd PUC Statistics Previous Year Question Paper March 2017

Time: 3 Hrs 15 Min
Max. Marks: 100

Instructions:

1. Statistical table and graph sheets will be supplied on request.
2. Scientific calculators may be used.
3. All working steps should be clearly shown.

Section – A

I. Answer any ten of the following questions: ( 10 × 1 = 10 )

Question 1.
Define fertility.
Fertility refers to the births occurring to women of child bearing age.

Question 2.
Does Marshall-Edgeworth’s Index number satisfy factor reversal test?
No

Question 3.
Which price of the commodities is used in the construction of cost of living index number?
Retail/consumer prices.

Question 4.
Give an example for seasonal variation.
Incease in employement during harvest season in agriculture sector.

Question 5.
In which distribution mean and variance are equal?
Poisson distribution.

Question 6.
Write the mean of /-distribution.
Mean = 0.

Question 7.
What is point estimation?
‘While estimating the unknown parameter, if a specific’value is proposed as an estimate, which is called Point estimation’.
‘While estimating the unknown parameter instead of a specific value, an interval is proposed, which is likely to contain the parameter is called Interval estimation’

Question 8.
What is power of a test?
It is the probability of rejecting the null hypothesis when it is not true. (1 – β ),
Here β = P (Type II Error) which is referred as ‘consumers risk

Question 9.
When is the pooling of the frequencies done in testing of goodness of fit?
When expected frequencies are less than 5.

Question 10.
What is defect?

• A defect is a quality characteristic, which does not conform to specifications.
• The presence of one or more defects in a product/article is a defective.

Question 11.
Mention a need of replacement of an item.

• As the equipment grow older, its maintenance cost would be costlier.
• Production cost would be lesser in new equipments.

Question 12.
What is meant by inventory?
An Inventory is a physical stock of goods, which is held for the purpose of future production or sales.

Section – B

II. Answer any ten of the following questions: ( 10 × 2 = 20 )

Question 13.
Mention two uses of life table.

They are used in computation of actuarial of premium, bonus etc, of policies by Insurance Agencies. OR

• Life Tables are used in research activities in Biology, Medicine, Pharmacology, Demography, Psychology, Sociology etc.
• They are used to study population growth and forecast the size and sex distribution of the
Population. OR
• Life Tables give Mortality and Survival ratios at different ages.
• Life tables give the life expectancy at different age. OR
• These are useful in public administration, heath care, planning and population control

Question 14.
Why is Fisher’s index number called as an ‘Indeal Index Number’?
Fisher’s Index Number is called as ideal because, it is free from bias in use of weights ie. it takes both current and base year quantities as weights.

• It is based on geometric mean which is considered as best average
• It satisfies both time reversal tests and factor reversal tests

Question 15.
If Iptq = 600 and Epgq = 500 then, find suitable price index number.
Kelly’s P.I.No.

Question 16.
What are the different phases in a business cycle?
(1) Prosperity (2) Decline (3) Depression and (4) Improvement.

Question 17.
Write down the assumptions of interpolation and extrapolation.

• The assumptions made in interpolation and extrapolations are:
• There are no sudden jumps in the values of dependent variable(Y) from one period to another(X).
• The rate of change of figures (Y) from one period to another(X) is uniform.
• There will be no consecutive missing values in the series.

Question 18.
Write down the p.m.f. of Bernoulli distribution with parameter p = 0.4.
p(x) = px q1 – x; x = 0,1, : p(x) = 0.4x 0.61 – x; x = 0,1
Here = 1 – p = 1 – 0.4 = 0.6

Question 19.
If Z1 and Z2 are two independent SNV’s, then name the distribution of Z12 + Z22 and find its mean.
It is a chi-square distribution with 2 d.f. Mean = n = 2.

Question 20.
Define type – I and type – II errors.
Type I Error is taking a wrong decision to reject the null hypothesis when it is actually true.
Type II Error is taking a wrong decision to accept the null hypothesis when it is actually not true.3

Question 21.
If n = 10, s2 = 20 and σ2 = 25, find chi-square test statistic.

Question 22.
What do you mean by process control and product-control?

• Controlling the quality of the product during the manufacturing process itself is called process control.
• Controlling the quality of the finished products/manufactured products is called product control.

Question 23.
If the depreciation and the cumulative maintenance costs of an equipment for the third year are Rs. 10,000 and Rs. 10,400 respectively, find the annual average cost.

Question 24.
Given R = 3600 units/year, C3 = Rs. 50 and C1 = Rs. 4/unit/year, find Q0.

Section – C

III. Answer any eight of the following questions: ( 8 × 5 = 40 )

Question 25.
Find standardized death rate for the follwoing data:

Let A be ASDRS and Ps be the standard population.

Question 26.
Mention any five steps involved in the construction of Index number.
Any two of the following :

1. Purpose and the scope
2. Selection of base period or
3. Selection of commodities or items
4. Obtaining price quotations
5. Choice of an average
6. 6. Selection of weights
7. Selection of suitable formula

Question 27.
Find cost of living index number for the following data.

Weights – W are given, use family budget method.

Question 28.
Find five yearly moving average for the following data.

Question 29.
Following is the data regarding annual nt life insurance premium. Using binomial expansion method estimate the premium at the age 30 and 45.

Let X and Y be the age and premium. The number of known values of y : n = 4, the Binomial exponsion (y – 1)4 = 0. .

y4 – 4y3 + 6y2 – 4y1+ y0 = 0 ……………(1)
By increasing sufficiences of y by T we get
y5 – 4y4 + 6y3 – 4y2 + y1 = 0 …………. (2)
From (1) 2256 – 4(1996) + 6y2 – 4(1581) +1426 = 0.
2256 – 7984 + 6y2 – 6324 + 1426 = 0 ,
6y2 – 10626 = 0
∴ y2 = Rs.1771
From (2) y5 – 4(2256) + 6(1996) -4(1771)+ 1581 =0
y5-9024 + 11976 – 7084 + 1581 =0
∴ y5 = Rs. 2551

Question 30.
On an average a typist makes 2 mistakes per page. Find the probability that a page types by him has (i) 2 mistakes (ii) at most 1 mistake.
Let X be the number of typing mistakes is a poisson variable with the parameter λ = 2.

(i) P(2 mistakes) =p(x = 2) = $$\frac{e^{-2} 2^{x}}{2 !}=\frac{0.1353 \times 4}{2}$$ = 0.2706
(ii) P(at most 1 mistake) =p(x ≤ 1) – p(x = 0) + p(x = 1)

Question 31.
A box contains 8 red and 4 white balls. From this box 5 balls are drawn at random What is the probability that the sampel contains 3 red balls?
Let X be the number of red balls is a Hypergeometric variate then : a(x) = 8, b = 4 and n = 5.

Question 32.
Following is the data regarding mean weights of randomly selected boys and girls of PUC students. Test whether, mean weight of boys is less than mean weight of girs. [Use 5% I.o.S]

Given: n1 = 64, n2 = 48, x̄1 = 50, x̄2 = 54, S1 = 8, s2 = 12.
H0 : Mean weight of Boys and Girls is same (µ1 = µ2)
H1 : Mean weight of Boys is less than Girls (µ1 < µ2) {Lower fail test -k}
Under H0 the test statistic is:

At a = 5% the lower critical value -k = – 1.65
Here Zcal lies in rejection region (RR).
∴ H0 is rejected and H1 accepted.
Conclusion: Mean weight boys is less than girsl.

Question 33.
A specified brand of automobile tyre is known to have average life of 40000 km. A random sample of 10 types of this brand, when tested resulted in the average life 39000 km and S.D 1200 km. At 1% I.o.s what is your conclusion regarding average life of tyres?
Given : µ = 40,000, n = 10, x̄ = 39,000, s = 1200, α = 1%
H0: Mean life of tyres is 40,000 kms(µ = 40,000)
H1: Mean life of tyres differs from 40,000 kms/(µ ≠ 40,000) (two tailed test)
Under H0 the t-test statistics is :-

At α = 1.1 the two tail critical values for (n – 1) = 10 – 1 = 9 d.f are ±k = +3.25
Here tcal lies in Acceptance region (A.R).
∴ H0 is accepted
Conclusion: Mean life of tyres is 40,000 kms

Question 34.
For the following data find control limits of X̄ chart. (Given A2 = 0.729]

Standards are not known, for x̄ – chart

The control limits are:
C.L = X̿ = 50,
L.C.L= X̿ – A2 R̄ = 50 – (0.729 × 5) = 46.355
U.C.L= X̿ + A2 R =50+ (0.729 × 5) = 53.645

Question 35.
Solve the following LPP graphically.
Maximize Z = 100X + 170Y ‘
s.t 2X + 3Y ≤ 36
X + 2T ≤ 20 and X,Y ≥ 0.
(For visually challenged students only)
A tailor gets a profit of Rs. 100 from a shirt and Rs. 170 from a pant. In a week of 56 hours, he uses 36 hours for cutting and 20 hours for stitching. For cutting he requires tw0 hours for a shirt and three hours for a pant. He, requires one hour to stitch a shirt and two hours to stitch a pant, formualte L.P.P.
Consider the constraints as equalities (i.e., as equaion).
2x + 3y = 36 Co-ordinates (x, y)
put x = 0: we get y = $$\frac{36}{3}$$ = 12: (0,12) .
put y = 0; we get x = $$\frac{36}{2}$$ = 18; (18 , 0)
and x + 2y = 20
put x = 0 ; y = $$\frac{20}{0}$$ =10; (0, 10)
put y = 0; x = 20 (20 , 0)
Also x = 0, y = 0

From the graph teasible region exhists at the corner points 0(0,0), A(0, 10), B( 12,4), C( 18,0) optimum value of the objective function is:

From the above table the objectivity of an L.P.P maximises at the corner point B(12,4). The optimum solution is: x= 12, y = 4 and the optimum value ZB= 1880.
For visually challenged students.
(OR)

The L.P.P is : objective function : Max z = 100x + 170y
Subject to constraints : Cutting: 2x + 3y ≤ 36
Stiching: x + 2y ≤ 20
and x ≥ 0, y ≥ 0

Question 36.
Solve the following game by dominnance rule.

By dominance rule, all the pay-off of B1 less than all the pay-offs of B1 B3 [B1 < B2, B3]

B1 dominates B2, B3, So delete B2, B3

Pay-ofFof A2 is more than, A1, A3, A4, A2 dominates A1, A3. So, delte A1, A3
A2[7] Saddle point occurs at (2, 1)
The suggested best strategies are : for player A: A2: for player B : B1
The value of the game : V = 7

Section – D

IV. Answer any two of the foilwing questions: ( 2 × 10 = 20 )

Question 37.
Calculate GRR and NRR for the following data and comment on the result.

G. R.R = (i) $$\sum_{i=15}^{49}$$ W.S.F.R = 5 × 220 = 1100 female
Children born per 1000 women
N.R.R = (i) $$\sum_{i=15}^{49}$$ (W.S.F.R × S) = 5 × 194.6 = 973
Female children bom per 1000 women.

Question 38.
For the following data find Laspeyer’s, Paasche’s and Dorbish-Bowley’s price index numbers.

Question 39.
For the following data fit a straight line trend and find trend values also estimate the production of 2017.

Let x and y be the year and production

Here x̄ = 2013
From the normal equaiton na + bΣx = Σy
From Σa + bΣx2 = Σxy b = $$\frac{\Sigma x y}{\Sigma x^{2}}=\frac{84}{28}=3$$
Since Σx $$\frac{\Sigma y}{n}=\frac{630}{7}$$ = 90 and
The fitted straight line trend equation is
y = a + bx : ŷ = 90 + 3x
Trend values : (2010) = 90 + 3(-3) = 81
ŷ(2011) = 90 + 3(-2) = 84.
Similarly all remaining trend values can be compated.
Estimation of production for 2017; x = 4
ŷ (2017) = 90 + 3(4) = 102 Tons.

Question 40.
(a) Five unbiased coins are tossed 128 times. Calculate the expected freuencies for the number of heads obtained. .
Let x be the number of heads obtained is a Binomial variate is with the parameters n = 5
and p = 1/2 = 0.5 [coin is unbiased] N = 128. Then the p.m.f is :
p(x) = ncxpxqn – x; x = 0,1,2, …… n
p{x) =5cx 0.5x 0.55 – x; x = 0,1,2, …… 5
Theoretical/Expected frequency : Tx = p(x) N
T0 = p(x = 0) N = 5c00.500.55 – 0 × 128 = 4

The expected frequencies of number of heads is:

(b) Seventy accidents that have occurred in a state in a week are tabulated as follows:

Test whether accident occurs uniformly throughtout the week.
[Given K2 = 16.81].
H0 : Accidents occur uniformly throughout the week
H1 : Accidents does not occur uniformly throughout the week…(upper tail test)
The test statistic is :

Let ‘O’ and ‘E’ be the observed and expected number of accidents.
Since accidents occur uniformly throughout the week.
Expected frequencies each is $$\frac { 70 }{ 7 }$$ = 10

∴ χ2cal= 6.8
For (n – 1) = (7 – 1) = 6 degrees of freedom at 1% level of significance, the upper tail critical value given k2= 16.81 (χ2tab)
Here χ2cal < k22cal lies on (AR-) acceptance region.
∴ H0 is accepted.
Conclusion: Accidents occur uniformly throughout the week.

Section – E

V. Answer any two of the following questions: ( 2 × 5=10 )

Question 41.
X is a normal variate with mean 100 and standard deviation 5. Find the probability
that (i) X ≥ 105 (ii) 90 ≤ X ≤ 110.
X is a normal variate with the parameters

µ = 100 and σ = 5. Then S.N.V.Z = $$\frac{x-\mu}{\sigma}$$ N(0,1);
$$\mathrm{z}=\frac{\mathrm{X}-100}{5}$$
(i)

= p(z > 1) = Area from 1 to ∞ = 0.1587

(ii)

= p(-2 < Z < 2)
= Area from (-2) to 2 = Area from (-2) to
= Area from (-2) to ∞ = 0.9772 – 0.0028 = 0.9544.

Question 42.
In a sample of 400 people ¡n Kerala 220 are tea drinkers and the rest are coffee drinkers. Can we assume that both tea and coffee are equally popular in this state?
Use 5% level of significance.
Given : n 400, x – Tea drinkers 220, po = 0.5 [equally popular].
Sample proportion p = $$\frac{x}{n}=\frac{220}{400}$$ = 0.55
H0: Tea and coffee drinkers are equally popular [Po = 0.5]
H1 : Tea and coffee drinkers are not equally poular [Po ≠ 0.5] {Two tailed test)

At α = 5% the two tail critical values are ±k = ±1.96
Here zcallies in Rejection Region (R.R). Therefore-H0: is rejected and H1: is Accepted. Conclusion: Tea and coffee drinkers are not equally popular (p0: ≠0.5)

Question 43.
For the following data test whether the attributes ‘smoking’ and literacy’ are independent. Use 1% level of significance.

H0: Literacy and smoking are independent.
H1 : Literacy and smoking are dependent {upper tail test)
The 2 × 2 contingency table is:

Under H0 the χ2 – test statistic is

At α = 1% for 1 .d.f. the uppertail critical value k2 = 6.65
Here χ2 lies in acceptance Region (A.R)
.’. H0 is accepted
Conclusion: literacy and smoking are independent.

Question 44.
For the following transportation problem find initial basic feasible solution by NWCR method. Compute transportation cost. Is the solution degenerate?