text:
Q: How do I add transition to styles changed on mouseover and mouseout events I changed styles on anchor tags and icons I want to add a transition to them and I don't know how.

const listItems = document.querySelectorAll("nav ul li");
const icons = document.querySelectorAll("nav ul li i");
const visitButtons = document.querySelectorAll("nav ul li a.visit--link");

listItems.forEach((item, id) => {
  item.addEventListener("mouseover", function() {
    icons[id].style.display = "none";
    visitButtons[id].style.transform = "translateX(0%)";
    visitButtons[id].style.display = "inline-block";
  });
  item.addEventListener("mouseout", function() {
    icons[id].style.display = "inline-block";
    visitButtons[id].style.transform = "translateX(100%)";
    visitButtons[id].style.display = "none";
  });
});
nav {
  width: 30%;
  border: 1px solid #333;
  border-radius: 5px;
  display: flex;
  flex-direction: column;
}

h2 {
  font-size: 36px;
  text-align: center;
  border-bottom: 1px solid #333;
  justify-content: center;
  margin: 0;
  padding: 15px 30px;
}

nav ul {
  padding: 0;
  margin: 0;
}

nav ul li {
  list-style-type: none;
  position: relative;
  display: flex;
  justify-content: space-between;
  align-items: center;
  transition: 2s;
}

nav ul li i {
  margin: 0px 5px;
}

nav ul li a {
  text-decoration: none;
  margin: 0;
  color: #000;
}

nav ul li i,
nav ul li a {
  font-size: 36px;
  padding: 15px;
}

nav ul li a.visit--link {
  background-color: #80d6ff;
  color: #fff;
  transform: translateX(100%);
  display: none;
}
<!DOCTYPE html>

  <meta charset="UTF-8" />
  <meta http-equiv="X-UA-Compatible" content="IE=edge" />
  <meta name="viewport" content="width=device-width, initial-scale=1.0" />
  <title>Sliding Sidebar</title>

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I am adding a Visit link at mouseover on list items and hide the icons. On mouseout I'm hiding the link and showing the icons. I thought that just adding transition 1s to anchor tags would add the transition to all styling changes on anchor tags.

A: Avoid using display: none when you want to transition the element because it will ignore all transitions and animations, instead you can use opacity: 0 and width: 0 or move it outside screen view using transform property:

const listItems = document.querySelectorAll("nav ul li");
const icons = document.querySelectorAll("nav ul li i");
const visitButtons = document.querySelectorAll("nav ul li a.visit--link");

listItems.forEach((item, id) => {
  item.addEventListener("mouseover", function () {
    icons[id].style.transform = "translateX(-100px)"; // to move it outside screen
    icons[id].style.width = 0; // remove width for the text next to it to move in place of it
    visitButtons[id].style.transform = "translateX(0%)";
    visitButtons[id].style.opacity = 1; // fade in
  });
  item.addEventListener("mouseout", function () {
    icons[id].style.transform = "translate(0)"; // return it to it's initial position
    icons[id].style.width = "auto"; // add width back to icon
    visitButtons[id].style.transform = "translateX(100%)";
    visitButtons[id].style.opacity = 0; // unfade, hide
  });
});
<!DOCTYPE html>

    <meta charset="UTF-8" />
    <meta http-equiv="X-UA-Compatible" content="IE=edge" />
    <meta name="viewport" content="width=device-width, initial-scale=1.0" />
    <title>Sliding Sidebar</title>

MOST POPULAR CASINOS

            MUCHO VEGAS

          VISIT

            LOKI

          VISIT

            GOLDEN STAR

          VISIT

            BITSTARY

          VISIT

            GUNSBET

          VISIT


Even as the sport celebrates the 10,000 meter debut of American master phenom Bernard Lagat at last night's Payton Jordan Invitational in Palo Alto, California — a world master's record win in 27:49 — we recall that six years ago today Chris Solinsky ran his own 10,000m debut at Payton Jordan in what turned out to be the run of his career. Here is how I reported it on our old Runnerville.com website, created by by Tracksmith founder Matt Taylor.
Well, they sold it as an American record attempt at 10,000 meters. Just top-billed the wrong guy. In a stunning debut in track's longest event, former University of Wisconsin All-American Chris Solinsky shocked the field, the fans, and even himself with a sensational 26:59.60 win at the Payton Jordan Invitational 10,000, ripping 14 seconds off Meb Keflezighi's 27:13.98 record set on the same track in 2001.
Galen Rupp, the Alberto Salazar-trained Nike star who the record attempt had been built around, ended up doing much of the second half pace work. And though he achieved his goal of dipping under Meb's old record, his 27:10.74 clocking could only garner him fourth place as a slew of records fell in the wake of Solinksky's shocker.
The Payton Jordan meet was selected at the last-minute by Salazar and Rupp after the weather outlook at the previous night's Oregon Relays didn't produce the best opportunity for a record attempt. Salazar had called around for a month hunting for pacers able to run through 7K at sub-27:13 tempo. But pacing became problematic from the gun.
Kenyan Simon Ndirangu took the pack through an uneven first mile in 4:24, but got back on pace, whirring through a 4:17 third mile, passing 5K in 13:34. After Ndirangu fell off at 4K, fellow Kenyan Matthew Kisorio, 20, towed Rupp, Kenyans Sam Chelanga and Daniel Salel, and Solinsky for two more kilometers. When Kisorio gave way, it was left to Rupp to muster the effort. Coach Salazar, himself a 27:25 man in his own heyday, encouraged his prized pupil on each lap. Rupp's former NCAA rival Sam Chelanga, 19 year-old Daniel Sale, Canada's Simon Bairu, and Solinsky remained in the lead group.
Mile four fell in 4:22 as Solinsky suffered through a painful stitch riding the tail of the Rupp's string. But with a series of deep breathing exercises Solinsky managed to stave off the cramp and regather his power.
As the small crowd rose to the record assault, Solinsky took his first steps in front at the top of the stretch with 900 meters to go. Quickly he opened his advantage. He blitzed the final 800 in 1:56, his last lap fell in sixty seconds. His time makes the 25 year-old Stevens Point, Wisconsin native the fastest non-African ever over 10,000 meters, and the first non-African to break 27 minutes. Notwithstanding, he still fancies himself a 5000 meter man, and has no intention of moving up in distance.
Behind the chesty Solinsky, another debutant, Daniel Salel of Kenya, finished second in 27:07.85. Sam Chelanga broke his own collegiate record in 27:08.39 in third, with a conflicted Rupp achieving his goal of breaking the old American record in 27:10.74, but not in the manner he had hoped. But Solinsky was well aware of Rupp's contribution.
Solinsky's Oregon Track Club teammate Simon Bairu broke the Canadian record in 27:23.63 in fifth. England's Chris Thompson, OTC, ran the third fastest 10,000m in British history in 27:29.61. Next came Tim Nelson, another of the Solinsky/Bairu teammates trained by Wisconsin transplant Jerry Schumacher. Nelson clocked in at 27:31.56.
With all eyes and expectations on Galen Rupp, Solinsky may have entered the race without any pressure, but knew he was ready for something.
Brussels in 2011 when he posted a 26:48.00. He then lowered that mark in 2014 at his hometown Prefontaine Classic in Eugene to the still standing 26:44.36 clocking.
But Chris Solinsky, still just 31, had reached the peak of his career in 2010. In 2011 Chris began to be plagued by a string of setbacks following a hamstring injury. He had hoped to mount a comeback in time to compete in this year's Olympic Trials in Eugene, Oregon, but distance running is a cruel taskmaster, a build-upon sport that takes years to reach the heights.
Chris Solinsky knows those heights, and now can carry that knowledge forward to help the next generation discover it as well as coach at William & Mary College.
Funny how fast this sport roars through time, taking no prisoners in its relentless march. All part of what compels our attention, I'd say. Thanks for the memories, Chris!
I clearly remember Rupp crying as he left the track. Crying. Salazar and Michael Johnson wouldn't let anyone near him either.
WHAT MEAN THE STRUGGLES IF THE BATTLES GO UNNAMED?


text:
Cameron Diaz Source
Posted on July 26, 2008 Written by adminNo Comments on Cameron gets mentioned in Forbes 'Ultimate Star Payback List'
Cameron gets mentioned in Forbes 'Ultimate Star Payback List'
Given what his movies cost to make, what he's paid to appear in them, and what they bring in at the box office, Vince Vaughn ranks No. 1 on Forbes magazine's Ultimate Star Payback List. The comedian's films earned $14.73 for every dollar he was paid, the magazine said. Because they took sizable pay cuts to appear together in Ocean's Twelve and Ocean's Thirteen, Matt Damon, George Clooney, Brad Pitt and Julia Roberts also ranked high on the list.
However, Will Ferrell, Jim Carrey and Nicole Kidman landed at the bottom for receiving huge salaries for films that flopped.
But the magazine also noted that some films would never be produced without a top movie star being brought on board with the promise of a huge payday. "When Cameron Diaz attaches herself to a script," the magazine noted "suddenly the producers can hire the hottest director and supporting actors and secure a good date for the film's release. Without a star, the same script can sit in limbo for years."
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Cameron Michelle Diaz (born August 30, 1972) is an American actress, producer, and former fashion model. She rose to stardom with roles in The Mask (1994), My Best Friend's Wedding (1997) and There's Something About Mary (1998), and is also known for voicing the character of Princess Fiona in the Shrek series (2001–10). Other high-profile credits include Charlie's Angels (2000) and its sequel Charlie's Angels: Full Throttle (2003), The Sweetest Thing (2002), In Her Shoes (2005), The Holiday (2006), What Happens In Vegas (2008), My Sister's Keeper (2009), Knight and Day (2010), The Green Hornet (2011), Bad Teacher (2011), What to Expect When You're Expecting (2012), The Counselor (2013), The Other Woman, Sex Tape, and Annie (all 2014).
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redpajama_set_name: RedPajamaCommonCrawl


text:
Q: How to use gdb with LD_PRELOAD I run a program with LD_PRELOADing a specific library. Like this.
LD_PRELOAD=./my.so ./my_program

How do I run this program with gdb?

A: Do the following.
gdb your_program

(gdb) set environment LD_PRELOAD ./yourso.so
(gdb) start

A: Posting because we ran into a case where set environment didn't work:
From GDB documentation:

set exec-wrapper wrapper
show exec-wrapper
unset exec-wrapper

When 'exec-wrapper' is set, the specified wrapper is used to launch programs for debugging. gdb starts your program with a shell command of the form exec wrapper program. Quoting is added to program and its arguments, but not to wrapper, so you should add quotes if appropriate for your shell. The wrapper runs until it executes your program, and then gdb takes control.
You can use any program that eventually calls execve with its arguments as a wrapper. Several standard Unix utilities do this, e.g. env and nohup. Any Unix shell script ending with exec "$@" will also work.
For example, you can use env to pass an environment variable to the debugged program, without setting the variable in your shell's environment:
         (gdb) set exec-wrapper env 'LD_PRELOAD=libtest.so'
         (gdb) run

A: A way to set both the environment and arguments in one command:
gdb --args env LD_PRELOAD=/usr/local/lib/libstderred.so /usr/bin/ls -l

This uses env to the same effect as an exec wrapper (like Alexey Romanov's answer), except that GDB doesn't know about it. The side effect is that your session will start in env. Fortunately, GDB will follow the exec into the target program as if nothing happened, and the backtrace is identical.
The convenience of having everything in one command is that your shell history will help you run the exact same thing again.

A: You can supply env as an exec-wrapper on the command line using the -iex flag:
gdb -iex "set exec-wrapper env LD_PRELOAD=./my.so" ./my_program

A: I am using gdbserver with VS Code, the simplest way is launching your program wrapped in a shell:
gdbserver :8888 sh -c 'LD_PRELOAD=/libtest.so your_prog'

A: You can basically do it the same way, just add gdb before the program name:
LD_PRELOAD=./my.so gdb ./my.program

You can check the environment variables using:
(gdb) show environment LD_PRELOAD

In the rare case you actually need to change it inside gdb, e.g. when debugging a dlopen(), you ca do that:
(gdb) set environment LD_PRELOAD ./my.so

Oh, wait, it doesn't work for me with gdb 7.6.2! The library doesn't get loaded, that means none of the answer here are entirely correct, at least with current tools.

redpajama_set_name: RedPajamaStackExchange


Why lithium battery life shortening ?
2. Lithium ion battery active material irreversible consumption.
3. The large lithium-ion battery, the battery operating current is large , cooling is slow, easy to damage the battery structure.
4.If the lithium battery additional safety protection measures, need to be provided by the lithium battery to maintain the power.
1.Lithium battery working environment, such as the impact of a direct impact on the length of life: lithium battery charging temperature of 0 ℃ ~ 45 ℃, lithium battery discharge temperature of -20 ℃ ~ 60 ℃.
2. In the lithium battery charge and discharge process, shouldn't over charge ,over discharge and so on.
3. In the lithium battery work should not appear incorrect operation: short circuit.
4. The use of regular matching lithium battery charger to charge the battery.
5.For the just-filled lithium battery need to hold for half a minute, and then use, otherwise it will affect the battery performance.
1.When the lithium battery matching digital products do not work, it should be said lithium battery equipment from the equipment to remove.
2. For long-term storage of lithium batteries, lithium batteries need to charge to 97% ~ 98% saturation, during storage should be 3 to 6 months on a lithium battery charge and discharge.
3. Lithium battery storage temperature should be controlled at 10 ° ~ 25 ° .
4. Don't knock, needle, trample, refit, or put the battery under the sun. Don't put the batteries under microwave, high pressure or other conditions.


They said Expedia payment didn't go through so I am waiting while the hotel attendant called Orbitz. This is really ridiculous.
Room was clean but small roach in bathtub. Other than that, worked for a quick place to sleep before flight.
Good location close to the airport. Great shuttle service to and from the airport.
The room had a very unclean smell. When I pulled the covers back there was a tiny spider on the bed and the bedding did not smell clean. The carpet also felt dirty as well. Great location to airport but the room and bedding needed a major scrubbing and cleaning!
Very friendly and helpful staff and the room was clean and had everything a traveler needs.
Great hotel for a one night stay if you are passing thru. We had to called for the shuttle to be picked up. The staff was very friendly and accommodating. I really think they need to get rid of the carpet concept and go tile or hardwood floor. The room I was put in was very smelly and I had to request another room. The other room was far better and very clean. The breakfast bar was very clean and a wide selection of breakfast choices. The hotel is in walking distance of a very nice mall which is a bonus for the hotel. It was raining, so the shuttle dropped us off, since this is not a service that they offer. I would recommend to family and friends, but for a short stay.
Check in was great. Jamie & Lee took great care of me at check-in. Had a very early flight the next morning so wake up call was scheduled, taxi was set up for 3:45 and they gave me yogurt, bagels and milk for breakfast! The room was small but clean and the bed was very comfortable. It's an older hotel that clearly needs updating but it's just what I needed at the time! Fairfield by Marriot in Kenner, Louisiana. Oh and let me mention my dinner from the restaurant across the street, the Harbor/Fisherman's Cove, which I highly recommend. Seafood done right with a great price!
Nothing really in the area, shuttle driver took me to McDonald's, which was nice because I was hungry but other than a seafood place no restaurants in the area.No vending machine, so you have to buy stuff at the front desk, which is weird.No mini fridge in the room,again, weird.Nice friendly staff, very helpful but had to call to get a shuttle from the airport, which was frustrating.Stood there for 20 minutes watching all the other hotel shuttles come and go before I figured out they weren't coming.Wanted to change my plans because I was going to a wedding in Mississippi Saturday night and I was going to stay there but I was informed I would be charged for the second day anyway so drove back two hours and stayed again.Bought a bag of chips for dinner when I gassed up the rental car and drank a water with no way to keep the other one cold that I bought for the plane ride home. They took that one at the airport so I bought one on the other side of security at a kiosk for $4.45!Sometimes you just can't win!I'm going into business selling water at the airport!One last thing,they must of had a dance competition in town one night,because they had a prayer meeting going on when I was trying to print out my boarding passes on the lobby computer in the morning.Praise Jesus honey boo boo!Party time in the Big Easy!
The staff was so nice and even let my friend check in for us since my flight didn't get in until after hers. They had an amazing shuttle service snd Sara the driver was so sweet! She told us great spots for restaurants and tours! The breakfast was really good and our room was beautiful! In the morning I had business stuff to attend and they were kind enough to let me work on their computer, fax and even print at no extra charge. Will definitely stay here again!! Amazing staff!
Located in Kenner - Louis Armstrong International Airport within a 5-minute walk of Cafe Du Monde, Fairfield Inn & Suites Kenner New Orleans Airport is within a mile (2 km) of Esplanade Mall. This 106-room hotel has conveniences like free breakfast, an outdoor pool, and a 24-hour fitness center.
An LCD TV comes with cable channels, and guests can stay connected with free WiFi and wired Internet. The pillowtop bed provides a full night's sleep and the bathroom offers hair dryers. Other standard amenities include a sofa bed, a coffee maker, and free newspapers.
Guests of Fairfield Inn & Suites Kenner New Orleans Airport have access to an outdoor pool, a 24-hour fitness center, and a spa tub. You can take advantage of free parking, along with a free airport shuttle (available 24 hours). The 24-hour front desk has multilingual staff ready to assist with dry cleaning/laundry and securing valuables. Additional amenities include free WiFi in public areas, conference space, and a business center.
Thank you for your feedback. We will review with our management team.
Thank you for sharing your experience at our hotel with a mission to provide superior customer service, we take great pride in serving your needs and are delighted to hear that we have met your expectations. As a VALUED CUSTOMER, your feedback will be shared with the team to continue to inspire and motivate them to deliver exceptional customer service.
Thank you for taking the time to review our hotel regarding your recent stay. As a VALUED GUEST, we see that there is room for improvement based on your feedback. We also noticed that there were several missed opportunities. Please accept our apology for not meeting your expectations during your stay. As we strive to deliver a flawless guest experience we are working on addressing the issues you noticed so that we can continue to improve our hotel operations.
The preferred airport is New Orleans, LA (MSY-Louis Armstrong New Orleans Intl.): 10-min drive, 3.3 mi/5.3 km.
The front desk is open 24 hours a day, and checking out is easy with express services. For convenient arrivals and departures, you can take advantage of the free airport shuttle (available 24 hours). Driving? This property offers free parking.


Since the Sun Journal has changed its policy on polling questions, it has opened the door to groups with political agendas. The Obamacare/Congress voting poll is a perfect example. When first polled on Sept. 28, there were 369 participants. 33 percent Obamacare is the law, 51 percent Congress should repeal Obama care and 16 pecent Repeal Obamacare but not at expense of shutting down the government. On Oct. 2 there were 627 votes with similar results (35 percent, 51 percent, 14 percent). Then, one day later, the vote count jumps up to 1,202 with dramatic opinion changes.
Since the Sun Journal has changed its policy on polling questions, it has opened the door to groups with political agendas. The Obamacare/Congress voting poll is a perfect example. When first polled on Sept. 28, there were 369 participants. 33 percent Obamacare is the law, 51 percent Congress should repeal Obama care and 16 pecent Repeal Obamacare but not at expense of shutting down the government. On Oct. 2 there were 627 votes with similar results (35 percent, 51 percent, 14 percent). Then, one day later, the vote count jumps up to 1,202 with dramatic opinion changes. Latest results are 62 percent, 29 percent, 8 percent, respectively. Doing the math, the surge of another 575 voters in one day, would require 526 votes (or 91.5 percent) to change �Obamacare is the Law� from 35 percent to 62 percent. If that isn�t stuffing the ballot box, I don�t know what is. Maybe you need to implement voter ID to avoid skewing results.


Can you check the jco conncetions for the RFC is working fine or not?
http:<hostname>:<port no>/index.html ->WD tools ->Content administrator->Manage JCOs.
It looks like there is some problem with initilization in the List.
Please have a look at this thread for [initializing AbstractList Model in Custom Controller|initializing AbstractList Model in Custom Controller;. and make changes accordingly.


text:
Sonia Harmand (née en 1974) est une préhistorienne française, spécialisée dans les industries lithiques du Paléolithique inférieur.

Formation
Sonia Harmand a obtenu son doctorat à l'Université de Paris X Nanterre.

Carrière académique
Elle a été associée de recherche au CNRS en France, puis professeure associée à l'Université d'État de New York à Stony Brook.

Découverte
Sonia Harmand a dirigé une campagne de fouilles archéologiques sur le site de Lomekwi 3, à l'ouest du lac Turkana, au Kenya, où des outils lithiques datés de 3,3 millions d'années ont été découverts en 2012. La découverte a été publiée en 2015 dans la revue scientifique Nature. Il s'agit des plus anciens outils lithiques découverts à ce jour. Ils précèdent de  le plus ancien fossile connu attribué au genre Homo. Sonia Harmand a proposé d'attribuer ces outils archaïques à une nouvelle industrie dénommée Lomekwien, qui précèderait ainsi l'Oldowayen.

Distinctions
En 2017, Sonia Harmand a été nommée par l'édition française du magazine Vanity Fair parmi les .

Références

Articles connexes
 Lomekwi 3
 Masol

Liens externes

Préhistorien français
Docteur de l'université Paris-X
Professeur à l'université d'État de New York à Stony Brook
Chercheuse au CNRS
Naissance en 1974

redpajama_set_name: RedPajamaWikipedia


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Unemployed Lyrics: Walking in the street, tonight in a foggy downtown / Thinking about my sorrows, I'm feeling down / I lost my job today and I am losing my. Find a Stand High Patrol, Pupajim - Unemployed first pressing or reissue. Complete your Stand High Patrol, Pupajim collection. Shop Vinyl and. Album · · 2 Songs. Available with an Apple Music subscription. Try it free.
Stand High Patrol "Unemployed" "I lost my job today and I am loosing my mind tonight [ ] How long Am I gonna deal with it How long Am I gonna deal. Chords for STAND HIGH PATROL: "UNEMPLOYED" + Riddim (7inch - promo mix - Stand High Records - SH). Play along with guitar, ukulele, or piano with . Chords for STAND HIGH PATROL - "Unemployed" + Riddim (SH promo mix) - Stand High Patrol. Play along with guitar, ukulele, or piano with interactive.
With it's riddim in a pure channel One style, this new record sounds like a tribute to STAND HIGH PATROL's main influences. Heavy drum and bass, killer. B Bm C#m A D G F#m Em] Chords for STAND HIGH PATROL. Stand High Patrol Unemployed (Single). Label: Stand High Patrol. ref: Une seule copie par personne!! Only one copie per buyer! Tracks.


text:
TWYGs at Tacchi Morris Open Day
The TWYGs performed a series of numbers at the Tacchi Morris Arts Centre Open Day on 19th Sept 2015. Here are some photos showing them warming up and during their performances. Well done to all those who took part. Thanks to Group Leader, Sarah Whittaker, and Choreographer, Sheila Driver, for getting them so well prepared. Oh, and not forgetting Sheila's able assistant Sarah Lock. And thanks to Emma Jones for her interview with Tone FM:
The Cast for Robinson Crusoe and the Pirates
The Auditions are over. Thank you to all those who auditioned and took part in the workshops. There were so many talented people that making the decisions for some of the parts was really quite hard. However, the decisions have been made and our cast is as follows: ROBINSON CRUSOE CAST LIST Chorus Alec Whyborn Matt Batterbsy Tom Kelvey Charlotte Whittaker Cassie Sutton Dancers Eve Whyborn Gabriella-Rose Gonta Kizzy Preece Lara Adkins Helen Coombes Emma Knight Feature Dancers Emma Jones…
Congratulations Emma & Rosie!
Congratulations to Emma Jones and Rosie Taylor who won "Comedy Duo of the Year" at this year's Cinderella Awards for their roles in Beauty and the Beast as Belle's sisters, Ermengarde and Esmeralda. Nominations were also received for: Choreographer of the Year – Sheila Driver Best Chorus Best Special Effects (Roses in the Opening UV) Best Theatre Stage Management – Garry Stacey Best Theatre Design – Garry Stacey Principal of the Year – Kim Barnard Beauty and the Beast came…
Presentation to Somerset Prostate Support Association
Wayfarers' Treasurer, Keith Smith, met the members of the Somerset Prostate Support Association at one of their regular meetings to present a cheque for £751, which was half the proceeds from this year's bucket collection. SPSA provide services associated with identifying prostate cancers and then advice and guidance in that period during and after diagnosis. They require funds in order to run drop in PSA testing sessions, which they run during the year, providing men the opportunity to check for…
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text:
\section{Introduction}
\label{sec:intro}

Several recent studies, concerned primarily  with  anisotropic
astrophysical environments,  have demonstrated that the nonlinear
neutrino self-coupling results in spatial
instabilities~\cite{Duan:2014gfa,Abbar:2015mca,Mirizzi:2015fva,
Chakraborty:2015tfa}.  These works indicate the importance of taking
into account deviations from spherical symmetry in describing the
evolution of, for example, the supernova neutrino flavor field.
Previous studies of the neutrino
evolution~\cite{Dolgov:1997mb,Dolgov:2002ab,Mangano:2005cc,
Grohs:2015eua,Grohs:2015tfy} in the early universe generally assume
homogeneity and isotropy obtains at all times from the epochs of weak
equilibrium through neutrino decoupling and Big Bang nucleosynthesis
(BBN). The objective of the present study is to address the validity
of this assumption in an exploratory calculation.  To this end,  in
this Letter we study the stability of a two-dimensional, two-flavor
dense neutrino gas with respect to the growth of seed spatial
inhomogeneity driven by neutrino self-coupling. In this model, we
also investigate the impact of incoherent elastic collisions on the
stability properties.  We find, perhaps surprisingly, that elastic
scattering-angle dependent collisions are not always effective at
damping flavor instabilities.

In extreme environments with large neutrino number densities, such as
the interior of a core-collapse supernovae (CCSN) and in the early
universe,  the nonlinearity associated with neutrino self-interactions
lead to an array of interesting effects beyond the ``standard''
MSW~\cite{Wolfenstein:1977ue,Mikheev:1986gs}.  These effects, which
have their origin in the non-linear character of the neutrino
evolution equations, include collective oscillations in an idealized
model of supernova~\cite{Pantaleone:1994ns,
Duan:2005cp,Duan:2006an,Duan:2006jv,Duan:2007mv,Duan:2008za,
Raffelt:2007xt,EstebanPretel:2007bz,EstebanPretel:2008ni,
Raffelt:2008hr, Dasgupta:2010ae}, matter-neutrino resonant
behavior~\cite{Malkus:2014iqa,Zhu:2016mwa,Wu:2015fga}, spontaneous
breaking of the axial symmetry~\cite{Raffelt:2013rqa,Mirizzi:2013wda},
and emergence of spatial
inhomogeneity~\cite{Duan:2014gfa,Abbar:2015mca}, to cite a few
examples.

In this context, the early universe is particularly interesting to study since
non-equilibrium and neutrino flavor oscillation effects may be important for
precision cosmological probes,  such as BBN.
In the early universe neutrino oscillations can be  induced by
flavor and/or  lepton asymmetries.

Studies of neutrino weak decoupling in a Boltzmann equation
approach~\cite{Dolgov:1997mb,Mangano:2001iu,Grohs:2015tfy}  suggest
that neutrino spectra  deviate from equilibrium Fermi-Dirac
distributions at the percent level and display flavor dependence (\ensuremath{\nu_e}\xspace
versus  \ensuremath{\nu_\mu}\xspace/\ensuremath{\nu_\tau}\xspace) at the  level of a few
percent~\cite{Dolgov:1997mb,Grohs:2015tfy}.  The flavor asymmetry
triggers flavor oscillations and a full analysis of this problem is
still lacking. Significant progress, however, has been made in
Refs.~\cite{Dolgov:2002ab,Mangano:2005cc,deSalas:2016ztq}.

Additionally, and complementary to the question of neutrino flavor
asymmetry,   neutrino oscillations play a significant role in the
presence of a lepton number asymmetry, namely a difference in  the
spectra for neutrinos and antineutrinos  (e.g. $\nu_{e}$ and
$\bar{\nu}_{e}$).
An electron lepton asymmetry large enough for collective neutrino
oscillations to be important in the  early universe can be generated
in  several well  motivated
models~\cite{Harvey:1981cu,Foot:1995qk,Shi:1996ic,Casas:1997gx,
MarchRussell:1999ig,Kawasaki:2002hq,Yamaguchi:2002vw,Shaposhnikov:2008pf,
Gu:2010dg}.    Moreover, a recent study~\cite{Johns:2016enc},
examining a range of lepton number flavor asymmetries over six orders
of magnitude, identified a variety of qualitatively distinct regimes
and  non-trivial behavior at very small lepton asymmetry.

The departure point for this study, in light of  the complex and
sundry array of behavior driven by the nonlinear neutrino
self-interaction, is the question of spatial inhomogeneity. We ask the
question of whether the assumption of spatial homogeneity and
isotropy, unexamined in previous works on the flavor field evolution
in the early universe~\cite{Dolgov:2002ab,Mangano:2005cc}, is
sustained in the presence of the nonlinear neutrino self-interaction.
Our finding is that, for a two-dimensional, two-flavor
exploratory model of a neutrino gas, the system is indeed unstable
with respect to spatial inhomogeneity. Further, we ask the question of
whether incoherent collisions, with a static background field of
elastic scatterers, can serve to drive the neutrino field back to
homogeneity. We investigate inhomogeneity for both normal and
inverted neutrino mass hierarchies.  We find that for the inverted
neutrino spectrum such collisions  (irrespective of the coupling
strength) are incapable of stabilizing the system for a significant
range of length scales.

Instability does not imply, of course, the persistence of
inhomogeneity for all times. Eventually, incoherent collisions may in
fact drive the system toward homogeneity. In this case, the associated
transient inhomogeneity is associated with entropy generation. And, if
sufficiently large, this added entropy may effect the evolution of the
light element nucleosynthesis in early universe. This is a question
that can only be addressed with detailed numerical simulations; such a
code is currently under
development~\cite{Grohs:2015tfy,Grohs:2016vef,Grohs:2016cuu}.

Our consideration of a simplified model of the early universe is
driven mainly by pragmatic concerns.  The rich treasure of effects
that are closely related to nonlinearity in the collective neutrino
oscillations makes prediction difficult for realistic physical
systems. Any such prediction involving collective neutrino
oscillations in, for example, the interior of the proto-neutron star
formed in CCSN requires the solution of the neutrino-flavor density
matrix in a large dimensional space,\footnote{The general case of
inhomogeneous and anisotropic environments, relevant for supernovae,
corresponds to an equation of motion for the neutrino density matrix
in a space of seven dimensions corresponding to three spatial, three
momentum coordinates and time.} which is beyond the capability of even
modern large-scale, multiprocessor computational platforms. However,
it is possible to solve these equations for simplified models that
assume deviations from highly symmetrical spatial geometries may be
neglected thereby reducing the dimensionality of the
solution space. It is hoped that, with this approach, insight may be
gained into the development of methods that will allow the solution of
the full problem; or, at least, to obtain results of phenomenological
relevance.

This Letter is organized as follows.
In Sec.~\ref{formalism}, we detail the model of neutrino oscillations
employed in this exploratory study of neutrino flavor stability in the
early universe. Section \ref{collision} discusses the neutrino-nucleus
model of the elastic, angle-dependent collision term adopted for the
model. In Sec.~\ref{linearstability}, we perform linear stability
analysis of the model and present results in Sec.~\ref{sec:results}.
Finally, in Sec.~\ref{sec:concl}, we explore the implications and
limitations of the present study in the context of more realistic
models. In particular, we comment on effects that inelastic
contributions to the collisions may have on collective neutrino
oscillations.

\section{Model setup}
\label{formalism}

The simplification of the spacetime and four-momentum dependence of
the equations of motion of a dense neutrino gas in conditions similar
to that of the early universe is effected in three stages:
simplification of the geometry; restriction to two neutrino flavors;
and approximation of the collision terms. The equations of motion that
govern the evolution of the neutrino density matrix in the early
universe depend on spacetime position and
the neutrino three-momentum and energy. The equations are simplified
by reducing the  spatial dimensions of the universe from three to two.
Since we are performing the linearized
stability analysis at a given instant in the evolution of the system,
we may further neglect the spacetime curvature, without loss of
generality.   Our assumption of elastic collisions with a background static array of
nucleon-like scatterers allows the further assumption that neutrino
energies are decoupled; energy transport is neglected. Thus we
consider the neutrinos to be mono-energetic. This reduces the momentum
dependence to that of a single, angular coordinate $\theta$. The
resulting equations of motion are  four-dimensional -- two spatial
dimensions, one momentum direction and time.
We work with two active neutrino flavors, instead of three.

The purpose of restricting the dimensionality of spacetime to a planar
spatial surface is to limit the size of the space of Fourier modes
(see Eq.~\eqref{mom:exp} below) that must be considered in the
stability analysis. In fact, we expect that this approximation is not
too severe in the sense that, all other approximations taken equally,
a treatment considering the universe as three spatial dimensions would
simply result in a larger number of Fourier modes. The stability
behaviors  of at least some of the modes in this case would be
similar to those of the two-dimensional case since the spatial mode
wave number couples explicitly only to the local velocity of the
neutrino field.  The two-flavor approximation also reduces the size of
the space in which the linearized stability analysis is performed.
The purpose of restricting the collision term to have only angular
dependence in elastic scatterings of the type $\nu_\alpha N \to
\nu_\alpha N$, where the field $N$ is represented by an infinitely
massive, immobile object, is to allow consideration of just a single
neutrino energy, which is conserved by this process.  With these
simplifications the universe can be viewed as a square of length $L$
on each side; we may take periodic boundary conditions without loss of
generality.

The dense neutrino gas is described in terms of the density matrix in
flavor space with elements $f_{\alpha \beta}(x,p)$, where $x$ and $p$
are position and momentum four-vectors, formally related to the Wigner
transform of the neutrino correlation function  $\langle \nu_\alpha
(x) \bar{\nu}_\beta (y)\rangle$ in
medium~\cite{Cirigliano:2014aoa,Vlasenko:2013fja,Blaschke:2016xxt}.  Factorization of the total
population (flux) $n(\vec{p})$ in momentum bin $\vec{p}$, a scalar
quantity, from the correlation function yields the $2\times 2$ density
matrices $\rho^{\theta}(\vec{x},t)$ and
$\bar{\rho}^{\theta}(\vec{x},t)$ for neutrinos and anti-neutrinos
respectively, which describe the flavor content of the system; they
satisfy ${\rm Tr } \rho^\theta  = 1 = {\rm Tr } \bar \rho^\theta$. Here
$\vec{x}$ is a two-dimensional vector specifying the position,
$\theta$ is the polar angle characterizing the direction of the
neutrino momentum $\vec{p}$, with respect to a given, arbitrary
direction (taken as the $y$-axis);  $t$ is the time. The diagonal
components of the density matrix represent the relative populations
for the two neutrino flavors, denoted \ensuremath{\nu_e}\xspace and \ensuremath{\nu_\mu}\xspace. The off-diagonal
terms express correlations between the two flavors.
The spacetime evolution equations of the density matrices are
the quantum kinetic equations (QKEs)~\cite{Sigl:1992fn,Raffelt:1992uj,McKellar:1992ja,
Enqvist:1990ad,Strack:2005ux,Volpe:2013jgr,Volpe:2015rla,Vlasenko:2013fja,
Zhang:2013lka,Cirigliano:2014aoa,Serreau:2014cfa,Blaschke:2016xxt}, which are given, in the limit that we neglect incoherent
scattering, by:
\begin{align}
\label{eqn:dm_eom}
i\left(\frac{\partial}{\partial t}
+\vec{v}\cdot\vec{\nabla}
+\vec{F}\cdot\nabla_{p}\right)\rho^{\theta}(\vec{x},t)
&= [\mathsf{H},\rho^{\theta}(\vec{x},t)], \cr
i\left(\frac{\partial}{\partial t}
+\vec{v}\cdot\vec{\nabla}
+\vec{F}\cdot\nabla_{p}\right)\bar{\rho}^{\theta}(\vec{x},t)
&= [\bar{\mathsf{H}},\bar{\rho}^{\theta}(\vec{x},t)],
\end{align}
where $\vec{v} = \dot{\vec{x}}$, $\vec{F} = \dot{\vec{p}}$.
Here the Hamiltonian $\mathsf{H}$ for neutrino evolution and that for
antineutrino evolution $\bar{\mathsf{H}}$ is written as a sum of terms
\begin{align}
\mathsf{H} &=\mathsf{H}_{0} + \mathsf{H}_{\nu\nu}, \nonumber \\
\bar{\mathsf{H}} &=-\mathsf{H}_{0} + \mathsf{H}_{\nu\nu},
\label{eom}
\end{align}
where $\mathsf{H}_{0}$ is that part of the time-evolution operator
independent of the density matrix, which is comprised of the vacuum and
matter parts of the Hamiltonian. We ignore the term
$\vec{F}\cdot\nabla_{p}$, which contributes when external forces, such
as those induced by the Hubble expansion, are present.  It does not
contribute significantly in the early universe to the linearized
stability analysis.  The neutrino-neutrino evolution part of the
Hamiltonian, which depends linearly on the density matrix, is denoted by
$\mathsf{H}_{\nu\nu}$. These components are given explicitly as:
\begin{align}
\mathsf{H}_{0} &= \frac{1}{2}
\begin{pmatrix} -\omega & 0 \cr 0 & \omega \end{pmatrix} \\
\label{eqn:Hnn}
\mathsf{H}_{\nu\nu} &= \mu
\int_{0}^{2\pi} d\theta^{\prime}
\left( \rho^{\theta^{\prime}}(t,\vec{x})
      -\alpha\bar{\rho}^{\theta^{\prime}}(t,\vec{x})\right)
      (1-\cos(\theta-\theta^{\prime})).
\end{align}
Here  $\mu  \propto  G_F n_{\nu} $  denotes the effective
self-coupling strength, proportional to the neutrino number density,
and $\alpha$  denotes the ratio $n_{\bar{\nu}}/n_{\nu}$.  The
linear, vacuum/matter term $\mathsf{H}_0$ is written for zero vacuum
mixing angle  ($\theta_V=0$) since $\theta_V$  does not alter the linearized stability analysis
we perform in Sec.~\ref{linearstability}.\footnote{Including  $\theta_V \neq 0$  would  have two effects:
(i) it would rescale the vacuum/matter frequency $\omega \to  \omega \times \cos 2 \theta_V$; (ii) it would
introduce a source term $\sim  \omega \times \sin 2 \theta_V$ in the evolution of the homogeneous perturbations,
which would lead to a linear growth with time, thus not impacting the stability analysis concerned with exponential growth.
Physically, the effect (ii)   corresponds to the onset of vacuum oscillations.}
We have carried out the integration over energy assuming a $\delta$-function distribution for energy.
Unlike the case of supernova there is no matter suppression of flavor
oscillations in a homogeneous and isotropic gas.
We note, for later
discussion, that the normal hierarchy case corresponds to $\omega>0$
and $\omega<0$ is the inverted case for this equivalent two-flavor
system; we may refer to $\omega = \Delta m^2 / (2 E)$ as the `scaled' vacuum frequency.

We are now in a position to investigate the role of collective neutrino
oscillations in the early universe and whether the equations of motion
of the neutrino density matrix are stable with respect to seed
inhomogeneous perturbations. The analysis is carried out through the
decomposition of the equations of motion of the density matrices
[Eq.~\eqref{eqn:dm_eom}] in terms of their Fourier modes,
\begin{align}
\rho^{\theta}(t,\vec{x})
&= \sum_{\vec{k}}\rho^{\theta}_{\vec{k}}(t) \exp(i \vec{k}\cdot \vec{x}).
\label{mom:exp}
\end{align}
Here the sum is over $k_i = 2\pi n_i / L$ for $n_i \in \mathbb{Z}$ and
$i=x,y$. We note that previous studies have exclusively considered
only the homogeneous mode $\vec{k}=(0,0)$ (in two spatial
dimensions); all other modes $\vec{k}\ne\vec{0}$ are inhomogeneous. In
the remainder of the present work we write the equations of motion
[Eq.~\eqref{eqn:dm_eom}] in terms of the Fourier modes
$\rho^{\theta}_{\vec{k}}$. The objective of the linearized stability
analysis is to identify those modes that grow exponentially with time
and are therefore unstable to seed perturbations. Substituting
Eq.~\eqref{mom:exp} in Eq.~\eqref{eom} and taking the projection to
the mode $\vec{k}$ gives
\begin{align}
i\frac{\partial}{\partial t} \rho^{\theta}_{\vec{k}}(t)
&= \vec{v}\cdot \vec{k} \rho^{\theta}_{\vec{k}}(t)
+ \left[\mathsf{H}_{0},\rho^{\theta}_{\vec{k}}(t)\right] \nonumber \\
&+ \mu \sum_{k^{\prime}}\int_{0}^{2\pi} d\theta^{\prime}
  \left[(\rho^{\theta^{\prime}}_{\vec{k}-\vec{k^{\prime}}}(t)
  -\alpha\bar{\rho}^{\theta^{\prime}}_{\vec{k}-\vec{k^{\prime}}}(t)),
  \rho^{\theta}_{\vec{k^{\prime}}}(t) \right] \nonumber \\
  &\times (1-\cos(\theta-\theta^{\prime})).
\label{eom:mom2}
\end{align}
Pairs of modes that appear in the last term of Eq.~\eqref{eom:mom2}
sum to $\vec{k}$.
Before considering the stability properties of the above collisionless
equations in Sec.~\ref{linearstability} we first discuss, in the
following section, the general form of the
collision term and the elastic, angular dependent approximation we
make for the purposes of the present study.

\section{The collision term}
\label{collision}

The  QKEs presented  in the previous section assumed that the
neutrinos experience only the effects of forward scattering from other
neutrinos. We must, however, take into account the effect of
incoherent, direction changing
scattering off other neutrinos, electrons and nucleons present in the
early
universe~\cite{Sigl:1992fn,Dolgov:1997mb,Grohs:2015tfy,Grohs:2016vef,Grohs:2016cuu}.
In this section, we
briefly review the incorporation of momentum changing scattering to
the neutrino flavor evolution. We continue to   assume the
reduced two-dimensional configuration space with  monoenergetic
neutrinos,   which greatly simplifies the scattering matrices.
The stipulation that neutrinos remain monoenergetic
throughout the scattering requires that the scattering
kernel only changes the direction of  the momentum and not magnitude.
We note that energy transport among neutrinos in  scattering events
is expected to contribute to  additional suppression of the collective effects (see discussion in Sec.~\ref{sec:concl}).

Previous studies have included the effect of incoherent scattering
on neutrino oscillations in an approximate   manner. Some references have
included an off-diagonal damping term with Boltzmann collision integrals
in the diagonal terms~\cite{Mangano:2011ip,Castorina:2012md,Pastor:2008ti,
Mangano:2010ei}. (A recent study~\cite{deSalas:2016ztq} has considered
collision terms off-diagonal in flavor but neglected terms where
neutrinos incoherently scatter from neutrinos and antineutrinos.) These
treatments, which assume homogeneity and isotropy, have not considered
the dependence of their results on the neutrino mass hierarchy. The
mass hierarchy, however, plays a role when inhomogeneity and anisotropy
are present.

In order to calculate the effect of collisions on collective neutrino
oscillations we use the formalism developed
in~\cite{Blaschke:2016xxt}, albeit with a modified notation. The
general form for the rate of change of the density matrix due to
incoherent collisions is written as:
\begin{align}
\label{coll:nuc}
\frac{d}{dt}\rho^{\theta}(\vec{x},t)
\Big\vert_{\rm coll}
= &-\frac{1}{2}\left\{\Pi^{\textrm{loss}},\rho^{\theta}(\vec{x},t)\right\}
\nonumber \\
& +\frac{1}{2}\left\{\Pi^{\textrm{gain}},\left(1-\rho^{\theta}(\vec{x},t)\right)\right\}
.
\end{align}
The first term on the right-hand side encodes the flux loss for
neutrinos traveling in the $\theta$ direction and the second term
encodes the flux gain for a neutrino being scattered into the
direction $\theta$. We note that both $\rho$ and $\Pi^{\rm gain/loss} [\rho]$ are $2\times 2$
matrices and gain and loss terms are expressed in term of
anti-commutators denoted by $\left\{\ ,\ \right\}$.  The factor
$\left(1-\rho^{\theta}\right)$ in the second term is due to
Pauli-blocking, a consequence of the fermionic nature of the
neutrinos. Qualitatively speaking, this factor arises due to the
degeneracy pressure of filled fermionic levels that disfavor
scattering of neutrinos to the direction $\theta$ when $\rho^{\theta}$
is sufficiently close to 1.

The general form of the collision rate [Eq.~\eqref{coll:nuc}] is too
complex to consider in the linearized stability analysis we pursue
here. In order to make the problem tractable and to include the effect
of angle dependent scattering in a basic way, we approximate the
collision integral in two ways. First, we ignore the effect of Pauli
blocking in the second term of Eq.~\eqref{coll:nuc}.  This will be
problematic in regimes of high neutrino degeneracy but is not too
drastic an approximation during the epoch of
weak-decoupling/nucleosynthesis that we're ultimately interested in
understanding. As a further simplifying approximation, we consider
only the neutrino-nucleon-like scattering processes  $\nu_\alpha N \to
\nu_\alpha N$, which is flavor $\alpha$ independent, neutral current
scattering.  Incorporating the aforementioned assumptions allows the
reduction of the general form of scattering matrices given in
Ref.~\cite{Blaschke:2016xxt}.
Using Eqs.~(3.21)-(3.22)   (Eqs. (60)-(61))
in the e-print archive (published) version of Ref.~\cite{Blaschke:2016xxt}, the
form of the collision integral can be calculated by explicitly taking the inner
product in 2+1 dimensional space.
We find that the matrix-valued
gain and loss collision contributions are proportional to the
flavor-space identity matrix
\begin{align}
\label{coll:form}
\Pi^{\textrm{loss/gain}}
& \sim \Big[ c_{1} - c_{2} \cos(\theta-\theta^{\prime}) \Big]
\times \mathds{1}_{2\times2}
\end{align}
where, $c_{1}$ and $c_{2}$ are constants. In three dimensional space
one has  $c_{1}  \propto G_{F}^{2}n_{\textrm{nuc}}p^{2}$ with
proportionality  constant of order unity. For equal number of protons
and neutrons in the early Universe the ratio $c_{1}/c_{2}$ is fixed
and is equal   to 4.83, a value which we use for rest of the paper.

We note that the above form for the approximate collision rate in
Eq.~\eqref{coll:form} is energy-local; that is, there is no energy
transport. And it is angle dependent, being a function of the angle of
scattering $\theta'$ relative to the incoming neutrino angle $\theta$.

Combining Eqs.~\eqref{eom} and \eqref{coll:form} we obtain the
complete equation describing the evolution of flavor density matrices
in presence of collective effects and collisions.

\section{Linear stability analysis}
\label{linearstability}

Nonlinear neutrino self-interaction in the equations of motion for the
density matrix [Eqs.~\eqref{eqn:dm_eom}] presents a computationally
challenging scenario for the general case, particularly when $\vec{k}\ne 0$.
In the presence of seed spatial inhomogeneity perturbations,
the reduction of the number of
independent variables by symmetry is no longer possible.  It is,
however, possible to gain insight into the stability properties
of Eqs.~\eqref{eqn:dm_eom}  through a linear stability
analysis~\cite{Banerjee:2011fj}. In this section we describe the
formalism and later,  in Sec.~\ref{sec:concl}, we
discuss the reliability and limitations of linear stability analyses
in the context of collective neutrino oscillations.

The linear stability analysis of collective neutrino oscillations is
predicated on the basis that, for significant neutrino flavor
oscillations to occur, the off-diagonal elements of the Hamiltonian
should be comparable in magnitude to the diagonal elements. Since
$\mathsf{H}_{0}$ is diagonal (neglecting vacuum mixing),
this can occur if the off-diagonal component of density matrices
increase rapidly with time.

In order to investigate the conditions (parameter ranges) under which
exponential growth of the Fourier modes may occur, we assume that the
density matrices have a form describing the presence of seed
inhomogeneity perturbations ($\Delta^\theta(t,\vec{x})$  and  $\epsilon^{\theta}(t,\vec{x})$  for neutrinos
and $\bar \Delta^\theta(t,\vec{x})$  and  $\bar \epsilon^{\theta}(t,\vec{x})$  for antineutrinos):
\begin{align}
& \rho^{\theta}(t,\vec{x}) =
\begin{pmatrix}
1-\Delta^\theta(t,\vec{x}) & \epsilon^{\theta}(t,\vec{x}) \\
 \epsilon^{\theta*}(t,\vec{x}) & \Delta^\theta(t,\vec{x})
\end{pmatrix}
& |\epsilon^{\theta}(t,\vec{x})|,|\Delta^\theta(t,\vec{x})| \ll 1
\nonumber \\
&\bar{\rho}^{\theta}(t,\vec{x}) =
\begin{pmatrix}
1-\bar{\Delta}^\theta (t,\vec{x}) & \bar{\epsilon}^{\theta}(t,\vec{x}) \\
\bar{\epsilon}^{\theta*}(t,\vec{x}) & \bar{\Delta}^\theta (t,\vec{x})
\end{pmatrix}
& |\bar{\epsilon}^{\theta}(t,\vec{x})|,|\bar{\Delta}^\theta(t,\vec{x})| \ll 1.
\label{rho:lin}
\end{align}
We leave the origin of these seed inhomogeneity perturbations
unspecified. They will be present in generic theories of the early
universe unless expressly forbidden by some exact symmetry.

The perturbations in the diagonal terms, $\Delta^\theta(t,\vec{x})$ and
$\bar{\Delta}^\theta (t,\vec{x})$ can be ignored upon linearization of the
equations of motion, because they contribute only at second order.  In terms of
the Fourier modes, Eq.~\eqref{rho:lin} reads
\begin{eqnarray}
\rho^{\theta}_{\vec{k}}(t)=
\begin{pmatrix}
\delta_{\vec{k},0} & \epsilon^{\theta}_{\vec{k}}(t) \\
 \epsilon^{\theta*}_{-\vec{k}}(t) & 0
\end{pmatrix} &
\bar{\rho}^{\theta}_{\vec{k}}(t)=
\begin{pmatrix}
\delta_{\vec{k},0} & \bar{\epsilon}^{\theta}_{\vec{k}}(t) \\
\bar{\epsilon}^{\theta*}_{-\vec{k}}(t) & 0
\end{pmatrix}~.
\end{eqnarray}
The equations for $\epsilon_{\vec{k}}$ for the various Fourier modes
$\vec{k}$ are decoupled at linear order in the perturbations
$\ensuremath{\epsilon}^\theta_{\vec{k}}$, $\bar\ensuremath{\epsilon}^\theta_{\vec{k}}$:
\begin{align}
&i \frac{\partial}{\partial t} \epsilon^{\theta}_{\vec{k}}(t)
= \vec{v}\cdot \vec{k} \epsilon^{\theta}_{\vec{k}}(t)
- \omega \epsilon^{\theta}_{\vec{k}}(t) \nonumber \\
&+ 2 \pi \mu (1-\alpha) \epsilon^{\theta}_{k}
- \mu \int_{0}^{2\pi} d\theta^{\prime}
(\epsilon^{\theta^{\prime}}_{k}-\alpha
\bar{\epsilon}^{\theta^{\prime}}_{k})
(1-\cos(\theta-\theta^{\prime})) \nonumber \\
&-i2\pi c_{1}\epsilon^{\theta}_{\vec{k}}(t)
+ i\int_{0}^{2\pi}   d\theta^{\prime}
(c_{1}-c_{2}\cos(\theta-\theta^{\prime}))
\epsilon^{\theta^{\prime}}_{\vec{k}}(t),\nonumber \\
\label{off:eom}
&i \frac{\partial}{\partial t} \bar{\epsilon}^{\theta}_{\vec{k}}(t)
= \vec{v}\cdot \vec{k} \bar{\epsilon}^{\theta}_{\vec{k}}(t)
+ \omega \bar{\epsilon}^{\theta}_{\vec{k}}(t) \nonumber \\
&+ 2 \pi \mu (1-\alpha) \bar{\epsilon}^{\theta}_{k}
- \mu \int_{0}^{2\pi} d\theta^{\prime}
(\epsilon^{\theta^{\prime}}_{k}-\alpha
\bar{\epsilon}^{\theta^{\prime}}_{k})
(1-\cos(\theta-\theta^{\prime}))\nonumber \\
&-i2\pi c_{1}\bar{\epsilon}^{\theta}_{\vec{k}}(t)
+ i\int_{0}^{2\pi}  d\theta^{\prime}
(c_{1}-c_{2}\cos(\theta-\theta^{\prime}))
\bar{\epsilon}^{\theta^{\prime}}_{\vec{k}}(t) .
\end{align}
For a given set of parameters, exponential growth of $\epsilon$ and
$\bar{\epsilon}$ implies flavor instability.
The dependence of collective neutrino oscillations on hierarchy can be
viewed as being a result of the relative sign between the second term
on right-hand side of Eq.~\eqref{off:eom}, which
is proportional to the vacuum term, and the third and forth terms, which
is proportional to the self-interaction strength $\mu$. However, it should
be noted that the relative sign between the vacuum and self-interaction
term can also be changed by going from $\alpha<1$ to $\alpha>1$.
All the results therefore for $1-\alpha>0$ with normal hierarchy
are identical to those for $1-\alpha<0$ with inverted hierarchy.
There are two different
ways in which we can investigate the flavor instability of the system.
One is the brute-force method in which we discretize the angular
dependence of $\epsilon$ and $\bar{\epsilon}$ and solve the resulting
system of coupled differential equations.  A major drawback of
this approach is that sometimes we get `spurious' flavor instabilities
as a result of discretization which reduce as we increase the number
of angle-bins\cite{Abbar:2015mca}.

The second method is a bit more formal but does not have the problem
of spurious instabilities. Based on the fact that we are investigating
collective modes, we seek solutions to Eqs.~(\ref{off:eom}) of the form
\begin{align}
&\epsilon^{\theta}_{\vec{k}}(t) = Q^{\theta}_{\vec{k}} \exp(-i\Omega_{\vec{k}} t) \nonumber\\
&\bar{\epsilon}^{\theta}_{\vec{k}}(t) =
\overline{Q}^{\theta}_{\vec{k}} \exp(-i\Omega_{\vec{k}} t).
\label{Q:form}
\end{align}
Existence of a solution with $\kappa > 0$, where
\begin{align}
\label{eqn:def_kappa}
\kappa\equiv\text{Im }(\Omega_{\vec{k}}),
\end{align}
implies the existence of flavor instabilities.  Combing
Eq.~\eqref{off:eom} and Eq.~\eqref{Q:form} we obtain
\begin{align}
\label{Q:wcoll}
&\left[\Omega_{\vec{k}} - \vec{v}\cdot \vec{k} + \omega -2\pi\mu(1-\alpha)
       + i2\pi c_{1} \right]Q^{\theta}_{\vec{k}} \nonumber \\
&= -\mu \int_{0}^{2\pi} d\theta^{\prime}
(Q^{\theta^{\prime}}_{\vec{k}}-
  \alpha \overline{Q}^{\theta^{\prime}}_{\vec{k}})
  (1-\cos(\theta-\theta^{\prime})) \nonumber \\
& +i \int_{0}^{2\pi}d\theta^{\prime}
  Q^{\theta^{\prime}}_{\vec{k}}
  (c_{1}-c_{2}\cos(\theta-\theta^{\prime})),
  \end{align}
for the neutrino modes, and
\begin{align}
\label{barQ:wcoll}
&\left[\Omega_{\vec{k}} - \vec{v}\cdot \vec{k} - \omega -2\pi\mu(1-\alpha)
        + i2\pi c_{1} \right]\overline{Q}^{\theta}_{\vec{k}} \nonumber \\
&= - \mu \int_{0}^{2\pi} d\theta^{\prime}
(Q^{\theta^{\prime}}_{\vec{k}}
-\alpha \overline{Q}^{\theta^{\prime}}_{\vec{k}})
(1-\cos(\theta-\theta^{\prime})) \nonumber \\
&+i \int_{0}^{2\pi}d\theta^{\prime}
\overline{Q}^{\theta^{\prime}}_{\vec{k}}
(c_{1}-c_{2}\cos(\theta-\theta^{\prime})),
\end{align}
for the antineutrino modes.

\begin{figure*}[!ht]
\includegraphics[width=0.49\textwidth]{nocollnor.pdf}
\includegraphics[width=0.49\textwidth]{nocollinv.pdf}
\caption{\label{nocoll}Imaginary parts, $\kappa$ of the unstable mode
frequency $\Omega_{\vec{k}}$ as a function of the ratio of the
strength of the effective neutrino coupling $\mu$ [Eq.~\eqref{eqn:Hnn}]
to that of the scaled vacuum frequency $\omega$ for the collisionless
system. The normal hierarchy is displayed in the left panel and the
inverted in the right panel.  The red curve is the homogeneous mode
$\vec{k}=0$ and the green curve is a high wave number, small scale
inhomogeneity with $|\vec{k}|=1000 \omega$ and $\alpha=0.99$.}
\end{figure*}

Recognizing that the solutions to the above  equations for the collective mode
amplitudes $Q^{\theta}_{\vec{k}}$ and $\overline{Q}^{\theta}_{\vec{k}}$ are
spanned by the functions $\{1, \cos\theta, \sin\theta\}$
leads to the  ansatz
\begin{align}
\label{eqn:Q}
Q^{\theta}_{\vec{k}} &= -\frac{\mu}{D_{\vec{k}}}
\left[(a_{\vec{k}}-\alpha\bar{a}_{\vec{k}})
-\sin\theta(s_{\vec{k}}-\alpha\bar{s}_{\vec{k}})
-\cos\theta(c_{\vec{k}}-\alpha\bar{c}_{\vec{k}})\right]\nonumber\\
& + \frac{i}{D_{\vec{k}}}
\left[c_{1}a_{\vec{k}}-c_{2}\sin\theta s_{\vec{k}}
-c_{2}\cos\theta c_{\vec{k}}\right]\\
\label{eqn:barQ}
\overline{Q}^{\theta}_{\vec{k}} &= -\frac{\mu}{\overline{D}_{\vec{k}}}
\left[(a_{\vec{k}}-\alpha\bar{a}_{\vec{k}})
-\sin\theta(s_{\vec{k}}-\alpha\bar{s}_{\vec{k}})
-\cos\theta(c_{\vec{k}}-\alpha\bar{c}_{\vec{k}})\right]
\nonumber\\
& + \frac{i}{\overline{D}_{\vec{k}}}
\left[c_{1}\bar{a}_{\vec{k}}-c_{2}\sin\theta \bar{s}_{\vec{k}}
-c_{2}\cos\theta \bar{c}_{\vec{k}}\right],
\end{align}
where we have defined the collective mode functions
$a_{\vec{k}}$, corresponding to modes independent of the angle
$\theta$ and the angular-dependent modes
$s_{\vec{k}}$ and $c_{\vec{k}}$ (and their barred counterparts).
We have abbreviated the denominators as
\begin{align}
\label{eqn:DbarD}
D_{\vec{k}} &= D(\omega,\Omega_{\vec{k}})
= {\left[\Omega_{\vec{k}} - \vec{v}\cdot \vec{k}
+ \omega -2\pi\mu(1-\alpha)\right] + i2\pi c_{1}}, \nonumber \\
\overline{D}_{\vec{k}} &= D(-\omega,\Omega_{\vec{k}}).
\end{align}

The collective mode functions are projected from Eqs.~\eqref{eqn:Q} and \eqref{eqn:barQ} as
\begin{align}
&a_{\vec{k}} = \int_{0}^{2\pi} d\theta\, Q^{\theta}_{\vec{k}} &
&\bar{a}_{\vec{k}} = \int_{0}^{2\pi} d\theta\, \overline{Q}^{\theta}_{\vec{k}} \cr
&s_{\vec{k}} = \int_{0}^{2\pi} d\theta \sin\theta Q^{\theta}_{\vec{k}} &
&\bar{s}_{\vec{k}} = \int_{0}^{2\pi} d\theta \sin\theta \overline{Q}^{\theta}_{\vec{k}} \cr
&c_{\vec{k}} = \int_{0}^{2\pi} d\theta \cos\theta Q^{\theta}_{\vec{k}} &
&\bar{c}_{\vec{k}} = \int_{0}^{2\pi} d\theta \cos\theta \overline{Q}^{\theta}_{\vec{k}}
\label{asc:newdef}
\end{align}
Consistency among these relations yields linear equations for the
collective modes, given by

\begin{widetext}
\begin{align}
\begin{pmatrix}
-J_{1}[1](c_{1})-1         & \alpha I_{1}[1]           & J_{1}[\sin\theta](c_{2})           & -\alpha I_{1}[\sin\theta]           & J_{1}[\cos\theta](c_{2})           & -\alpha I_{1}[\cos\theta]            \\
-I_{2}[1]                  & -J_{2}[1](c_{1})-1        & I_{2}[\sin\theta]                  & J_{2}[\sin\theta](c_{2})            & I_{2}[\cos\theta]                  & J_{2}[\cos\theta](c_{2})             \\
-J_{1}[\sin\theta](c_{1})  & \alpha I_{1}[\sin\theta]  & J_{1}[\sin^{2}\theta](c_{2})-1     & -\alpha I_{1}[\sin^{2}\theta]       & J_{1}[\sin\theta\cos\theta](c_{2}) & -\alpha I_{1} [\sin\theta\cos\theta] \\
 -I_{2}[\sin\theta]        & -J_{2}[\sin\theta](c_{1}) & I_{2}[\sin^{2}\theta]              & J_{2}[\sin^{2}\theta](c_{2})-1      & I_{2}[\sin\theta\cos\theta]        & J_{2}[\sin\theta\cos\theta](c_{2})   \\
 -J_{1}[\cos\theta](c_{1}) & \alpha I_{1}[\cos\theta]  & J_{1}[\sin\theta\cos\theta](c_{2}) & -\alpha I_{1}[\sin\theta\cos\theta] & J_{1}[\cos^{2}\theta](c_{2})-1     & -\alpha I_{1}[\cos^{2}\theta]        \\
 -I_{2}[\cos\theta]        & -J_{2}[\cos\theta](c_{1}) & I_{2}[\sin\theta\cos\theta]        & J_{2}[\sin\theta\cos\theta](c_{2})  & I_{2}[\cos^{2}\theta]              & J_{2}[\cos^{2}\theta](c_{2})-1
\end{pmatrix}
\begin{pmatrix}
a_{\vec{k}} \\ \bar{a}_{\vec{k}} \\ s_{\vec{k}} \\ \bar{s}_{\vec{k}} \\ {c}_{\vec{k}} \\ \bar{c}_{\vec{k}}
\end{pmatrix}
&=0,
\label{uglymatrix}
\end{align}
\end{widetext}
\noindent in terms of the functionals
\begin{align}
I_{1}[f] &= \int_{0}^{2\pi}d\theta\,
  \frac{\mu f}{D_{\vec{k}}}, &
I_{2}[f] &= \int_{0}^{2\pi}d\theta\,
  \frac{\mu f}{\overline{D}_{\vec{k}}},\\
I_{3}[f] &= \int_{0}^{2\pi}d\theta\,
  \frac{if}{D_{\vec{k}}}, &
I_{4}[f] &= \int_{0}^{2\pi}d\theta\,
  \frac{if}{\overline{D}_{\vec{k}}},
\end{align}
with $J_{1,2}$ defined as
\begin{align}
J_{1}[f](c) &= I_{1}[f]-c I_{3}[f], \nonumber \\
J_{2}[f](c) &= -\alpha I_{2}[f] - c I_{4}[f]~.
\end{align}

\begin{figure*}
\includegraphics[width=0.49\textwidth]{k1e4nor.pdf}
\includegraphics[width=0.49\textwidth]{k1e4inv.pdf}
\caption{\label{fig:k1e4}The quantity $\kappa/\omega$ of small-scale
$|\vec{k}|=1000 \omega $ inhomogeneity as a function of
$\mu/\omega$ [Eq.~\eqref{eqn:Hnn}] for various values of the strength
of the incoherent collision term, Eq.~\eqref{coll:form} as measured by
the constant $c_1$. The normal hierarchy is displayed in the left
panel and the inverted in the right panel. In each panel, the red
curve is the collisionless case ($c_1 = 0$), the green curve is the
intermediate value of collision strength ($c_1= 1.0\, \omega$), and
blue is strong coupling ($c_1 = 10.0 \, \omega$). The ratio
$\bar{\nu}_{e}/\nu_{e}$ defined as $\alpha$ is fixed at 0.99.
}
\end{figure*}

We determine the values of $\Omega_{\vec{k}}$ for which
Eq.~\eqref{uglymatrix} is satisfied by requiring the determinant of the
$6\times 6$ matrix vanishes. As the determinant of the matrix is a
$6^{\textrm{th}}$-order polynomial we obtain up to six different
values of $\Omega_{\vec{k}}$ for each value of $\vec{k}$. While
multiple solutions are possible, the instability is dominated by the
$\Omega_{\vec{k}}$ with largest positive imaginary part. We have
numerically solved for $\Omega_{\vec{k}}$ by using standard root
finding numerical algorithms. For the results presented in the next
section we use the open source GNU Scientific Library
(GSL)~\cite{Gough:2009:GSL:1538674}.

Note that for $\vec{k}=0$ (homogeneous) modes  the matrix in
Eq.~\eqref{uglymatrix} has a block diagonal form with three $2\times 2$
blocks corresponding to subspaces spanned by the pairs of mode
functions $\{a_{\vec{k}}$, $\bar{a}_{\vec{k}}\}$, $\{s_{\vec{k}}$,
$\bar{s}_{\vec{k}}\}$, and $\{c_{\vec{k}}$, $\bar{c}_{\vec{k}}\}$;
these subspaces are decoupled.  The latter two subspaces are identical
(those for the $s_{\vec{k}}$, $\bar{s}_{\vec{k}}$ and $c_{\vec{k}}$,
$\bar{c}_{\vec{k}}$ blocks).  In this case, the determinant can be
calculated analytically and the solutions for
$\Omega_0=\Omega_{\vec{k}=0}$ are given by
\begin{equation}
\label{eq:Oma}
\Omega_{0,\pm}^{(a)} = \pi \mu (1 - \alpha)
\pm \sqrt{ \omega^2  + 2 \pi  \mu (1 + \alpha) \omega
   + \pi^2 \mu^2 (1 - \alpha)^2 },
\end{equation}
for  the $a_{\vec{k}=0}=a_0,\bar{a}_{\vec{k}=0}=\bar{a}_0$ block, and
\begin{align}
\label{eq:Omsc}
\Omega_{0,\pm}^{(s,c)} &= \frac{5}{2}
   \pi \mu (1 - \alpha) - i \pi (2 c_1 + c_2) \nonumber \\
&\pm   \sqrt{ \omega^2  -   \pi  \mu (1 + \alpha) \omega
   + \frac{1}{4}   \pi^2 \mu^2 (1 - \alpha)^2},
\end{align}
for the $s_{0}$, $\bar{s}_{0}$ and $c_{0}$, $\bar{c}_{0}$ blocks.
Despite the fact that the $\vec{k}=0$ modes are homogeneous, it is
instructive to study these results for this $\vec{k}=0$ mode since
similar behavior is observed for the inhomogeneous $\vec{k}\ne 0$
modes.

For the inverted hierarchy ($\omega<0$),  in absence of collisions the angle-dependent
collective frequencies $\Omega_{0,\pm}^{(s,c)}$ are real and therefore
do not give rise to exponential growth.
On the other hand, Eqs.\eqref{eq:Oma} indicate that for the inverted
hierarchy ($\omega < 0$), the argument of the square-root
$\kappa^2=\omega^2-2\pi\mu(1+\alpha)|\omega|+\pi^2\mu^2(1-\alpha)^2$
becomes negative for values
\begin{align}
\label{eqn:mu_unstable_limits}
\frac{1}{4 \pi} \lesssim \frac{\mu}{|\omega|} \lesssim  \frac{4}{\pi\delta^2},
\end{align}
where we have  defined $\alpha = 1 - \delta$ and expanded for $\delta \ll 1$
(in our numerical studies we  take $\delta = 10^{-2}$).
Figure \ref{nocoll} shows $\kappa/|\omega|$ plotted against the neutrino effective
self-coupling $\mu/|\omega|$; the left panel shows two $\vec{k}$ modes (red and
green curves) for the normal (left panel) and inverted (right panel)
hierarchies.  The effective coupling inequalities for the inverted
hierarchy described by Eqs.~\eqref{eqn:mu_unstable_limits} are clearly
exhibited in the right panel of Fig.~\ref{nocoll} by the curve
corresponding to $\vec{k}=0$ (the red curve).  The left-most
inequality in Eqs.~\eqref{eqn:mu_unstable_limits} corresponds to the
region $\mu/|\omega| \approx 0$;   the right-most inequality corresponds to where
the $|\vec{k}|=0$ intersects the abscissa near $\mu/|\omega| \simeq 1.27 \times 10^{4}$.

Equation \eqref{eq:Oma} demonstrates that for inverted hierarchy ($\omega < 0$) only
$\theta$-independent instabilities can develop (proportional to $a_{\vec{k}}$ and $\bar
a_{\vec{k}}$). Moreover, and perhaps not too  surprisingly, the
direction-changing form for the elastic collision rate we employ
[Eq.~\eqref{coll:form}] in this exploratory calculation has no effect
on such instabilities: the strength of the collisions, as measured by
the quantities $c_{1,2}$, do not feature in the expression for
$\Omega_{0,\pm}^{(a)}$. This is potentially an important observation
since, if this behavior obtains for the full, unapproximated collision
rate then nonlinear mode coupling effects, present in the full QKEs,
could propagate the instability to other, inhomogeneous modes. This
effect could result in a continuous ``feeding'' of inhomogeneity that
may persist for times long compared to other dynamical time scales,
despite collisional damping in the inhomogeneous modes.

For the normal hierarchy ($\omega>0$), the angle-independent
collective frequencies $\Omega_{0,\pm}^{(a)}$ are real and therefore
do not give rise to exponential growth.  On the other hand,
$\theta$-dependent instabilities (proportional to $s_0,c_0$ and $\bar
s_0, \bar c_0$) described by Eq.~\eqref{eq:Omsc} may develop.
In this case, the growth rate due to the
imaginary part of $\Omega^{(s,c)}_{0,\pm}$ is indeed damped by the
direction-changing collisions, as shown by the term proportional to $2
c_1 + c_2$ in Eq.~\eqref{eq:Omsc}. The numerical results discussed in
the next section match the analytic results at $\vec{k}=0$ and show
that a similar behavior persists for $\vec{k} \neq 0$: for normal
hierarchy $\theta$-dependent instabilities ($s_{\vec{k}}$, $\bar
s_{\vec{k}}$ and  $c_{\vec{k}}$, $\bar c_{\vec{k}}$) can develop,
while for the inverted hierarchy only the $\theta$-independent modes
$a_{\vec{k}}$ and $\bar a_{\vec{k}}$ are subject to unstable,
exponential growth.

\begin{figure*}
\includegraphics[width=0.49\textwidth]{k0nor.pdf}
\includegraphics[width=0.49\textwidth]{k0inv.pdf}
\caption{\label{fig:k0}The quantity $\kappa/\omega$ for large-scale
$|\vec{k}|\approx 0$ inhomogeneity as a function of
$\mu/\omega$ [Eq.~\eqref{eqn:Hnn}] for various values of the strength
of the incoherent collision term, Eq.~\eqref{coll:form} as measured by
the constant $c_1$; see Fig.~\ref{fig:k1e4} for the description.}
\end{figure*}

\section{Results}
\label{sec:results}

In this section we present numerical results for the stability analysis of
collective modes with $\vec{k}\ne 0$ corresponding to inhomogeneous
flavor distributions in configuration space.
We continue to take the ratio of electron anti-neutrinos to electron
neutrinos, $\alpha = 0.99$, as in the previous section. As shown in
Fig.~\ref{fig:k1e4}, for the collisionless case ($c_1=0=c_2$), the
inhomogeneous modes with $|\vec{k}|=1000 \omega$
(we use $n=10^{4}$ and $2\pi/L=0.1 \omega$)
are unstable in both the normal (left panel of the figure) and inverted
(right panel) mass hierarchies. The effective neutrino self-coupling,
$\mu$ over which instabilities exist, we note, is different for normal
and inverted hierarchies.  However, for both mass hierarchies the
small scale inhomogeneous modes are unstable over a larger range of
$\mu$ than for the homogeneous modes.

In the presence of collisions we would generally expect a suppression
of instabilities for sufficiently strong interaction rates. This
expectation is driven principally by the observation that incoherent
scattering of the form of Eq.~\eqref{coll:form}, due to its angular
dependence, will tend to redistribute regions of neutrino density
toward the mean.  The expectation of the suppression is supported by
the form of Eq.~\eqref{eq:Omsc}, specifically the term $-i \pi (2c_1 +
c_2)$.  Since $c_{1,2} >0$,   this term contributes an
exponentially damped factor to the collective mode frequency
$\Omega^{(s,c)}_{0,\pm}$.

Indeed, for large values of $k \sim 1000 \omega$ we find suppression of
instabilities for both mass hierarchies, as shown in
Fig.~\ref{fig:k1e4}. As in Fig.~\ref{nocoll}, the left and right panels
give results for the values of $\kappa/\omega$ for the normal and
inverted hierarchies, respectively. In each hierarchy, the effect of
varying the strength of the collision term, measured by the magnitude
of $c_1>0$, is shown by the various curves, as detailed in the figure
caption.  In each panel, we observe a reduction in the width of the
region of instability with increasing $c_1$.

However, as previously indicated by Eq.~\eqref{eq:Oma}, for smaller
values of $k$ ($k \sim $ few $\times 100\omega$) including zero, we find a suppression
due to collisions  only in the case of normal mass hierarchy,
not in the inverted mass hierarchy. Results for the $\vec{k}=0$
homogeneous modes have been obtained both analytically, through
Eq.~\eqref{eq:Oma} and numerically via diagonalization of the $6\times
6$ matrix appearing in Eq.~\eqref{uglymatrix}.  Figure \ref{fig:k0} shows the
numerical results for the homogeneous mode for various values of the
incoherent scattering strength parameter $c_1$; it agrees with
Eq.~\eqref{eq:Oma} at high precision for all values of $c_1$ considered
in this work.

\section{Conclusion}
\label{sec:concl}

We have studied the stability of a two-dimensional dense neutrino gas with respect to spatial inhomogeneity.
In the absence of incoherent collisional effects  we have   found that  the  system exhibits growth of seed inhomogeneity
due to nonlinear coherent neutrino self-interactions and that the effect depends on the  neutrino mass spectrum.
For  normal hierarchy   spatial instability  exists over a larger range
of neutrino number density compared to the inverted case.

We have further considered  the effect of  elastic  (i.e. energy conserving) incoherent (direction-changing)
collisions of the neutrinos with a  static background of heavy, nucleon-like scatterers,
in order to validate  the intuitive expectation  that incoherent
collisions  tend to drive the neutrino flavor field of the early universe
toward homogeneous isotropic distributions.
Our results suggest that  this heuristic picture may be  oversimplified.

At  small  scales,  corresponding to  Fourier modes $|\vec{k}| \gtrsim
100 \omega$, the growth of flavor instability can be  suppressed by
collisions, irrespective of the neutrino mass spectrum.  At large
length scales ($|\vec{k}| \lesssim 100 \omega$)    we find,  perhaps
surprisingly, that for inverted neutrino mass hierarchy
direction-changing collisions fail to  suppress flavor instabilities,
irrespective of the coupling strength.  In the extreme long-wavelength
case ($\vec{k}=0$) our numerical results are substantiated by an
analytic understanding of the problem (see
Section~\ref{linearstability}).  The key to understanding the puzzling
behavior is that for the simplified two-dimensional model, at large
scale ($|\vec{k}| \sim 0$) only $\theta$-independent modes grow
unstable for inverted hierarchy, while only $\theta$-dependent modes
grow unstable for normal hierarchy.

The appearance of inhomogeneity due to collective neutrino
oscillations in the early universe could have implications for the
entropy evolution in the neutrino sector, which in turn could affect
Big Bang nucleosynthesis.
Moreover,  the  non-trivial dependence of the instability
(and its collisional suppression) on the neutrino mass hierarchy
is  very intriguing and deserves attention in future  studies involving more
realistic models.

One obvious step towards a more realistic setup  would involve
generalizing the collision kernels in order to account for
energy-changing collisions in our formalism.  This would allow one to
model neutrino interactions with electrons /  positrons in the early universe,
which is beyond the scope of this Letter.
However, even in absence of a complete calculation,
since the collisional suppression of neutrino flavor oscillations
 gets  contributions from direction change as well as  energy change,
we speculate that the total effect of collisions on  the suppression of neutrino  inhomogeneities
should  still be dependent on the neutrino mass hierarchy.

\section*{Acknowledgments}
The authors would like to thank George Fuller for offering valuable insights.
The authors would also like to thank Sajad Abbar, Daniel Blaschke,
Joe Carlson, Huaiyu Duan, Evan Grohs, Luke Johns, Lei Ma and Joshua Martin
for numerous discussions.
The authors acknowledge support by the LDRD program at Los Alamos National Laboratory.

\section*{References}
\bibliographystyle{elsarticle-num}

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