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3 Benefits to Automating Your Manufacturing Business Processes
Here's are three ways in which business process automation in manufacturing pays off.
Automation in manufacturing is already commonplace, but as consumers become more demanding and the industry becomes more competitive, more and more companies are turning to technology to help automate and optimize time and labor-intensive processes and workflows.
By automating the way data is being captured, stored, and shared using advances in cloud and mobile-based technologies, manufacturers can optimize their processes and make better business decisions based on access to key operational information.
Here's are three ways in which business process automation in manufacturing pays off:
No. 1 - A More Connected Manufacturing Process
Today's manufacturers are always chasing the big picture. Having the ability to access and interpret business information across the enterprise — from the shop floor to the head office and across the supply chain is no longer a nice to have — it's a must-have. Yet manual workflows and disconnected systems can hinder the right information getting to the right place at the right time.
With business process automation you get visibility into the process. Teams that are geographically dispersed can share work, communicate better, and find the experts they need to keep productivity at 100 percent.
In the past this was only possible with costly bespoke solutions but using the ubiquitous connectivity of mobile applications and the power of cloud workflows, you can skip the paper chase and fast forward to automated business processes. This allows scenarios that were previously reserved for mega corporations.
For example, with easy-to-use cloud-based mobile platforms you can move your most commonly used paper forms and data collection processes to smartphone-based applications, creating a standardized and automated workflow for collecting and sharing important information across the business.
Such applications are easy-to-use, require no coding skills, are available online and offline (they synch automatically with the cloud once you're back in range) and can be customized to any process, whether it's quality assurance, maintenance equipment logs, or inspections checklists.
Taking the latter as an example. Mobile inspection checklist applications streamline and automate the end-to-end process of conducting manufacturing inspection reviews. Pre-loaded or easily customized with all the form fields you need (and none you don't), inspectors can enter the results of the inspection — who conducted it, what they found, and what, if any, anomalies were corrected — from any location. The data is then automatically and instantly uploaded to the cloud, where it's stored securely. This data can then be presented in easy-to-use dashboards, to determine which machines are most likely to fail.
These smart applications not only connect the business processes you already have, they can also be scaled or adapted to other scenarios without the need to build new applications every time.
No. 2 - A More Complete End-to-End Manufacturing Process
The magic of digitized business process automation is that it eliminates silos across your firm.
For example, data captured from systems and personnel into mobile applications can be linked seamlessly with other software (such as a Google Analytics, Salesforce, or legacy CRM tools) and processes using cloud-based integrations and APIs. This eliminates redundant data entry, speeds insights, and fuels course-correcting actions in a way that is impossible when the data stays on the rockface of the business or locked away in a filing cabinet. This automated process is also just one of many opportunities to cut the hours and effort involved in collecting and keying that data in manually.
Mobile applications also allow you to easily build in conditional route linking options for streamlined actions. For example, if an inspections checklist returns an anomaly with a temperature sensor, your quality assurance manager receives an automatic corrective action to fix the issue. He can also dispatch tasks to others and ensure approval sign-offs are captured with digital signatures, all within a single application.
Finally, business process automation can make it easier to search information captured over time. Whether you need to find data generated by a work order or site inspection form, because it's digitally captured and stored in the cloud, rather than rifle through old filing cabinets, the data you need is at your fingertips.
In this way, manufacturers maximize investments in existing technology investments and achieve higher levels of interoperability than is possible with manual processes.
No. 3 - A More Compliant Manufacturing Process
Manufacturers are subject to a range of regulations, many of which are constantly changing. Keeping track of all these regulations can be overwhelming. But compliance isn't just a regulatory burden, in the interests of efficiency, workers and plant managers must also comply with standard operating procedures (SOPs).
We hear repeatedly how companies are faced with the consequences of non-compliance and the potential loss of revenue due to individuals forgetting or unintentionally not filling in items on the forms they were assigned. But mobile applications designed to support business process automation can be customized easily without the help of an application developer so required fields can be locked down.
For example, a safety manager who travels to different locations throughout the day can use an application to easily run through OSHA requirements and evaluate compliance right from within a safety checklist application. And, because the data that powers the application is stored in a centralized cloud repository, mobile form fields and checklists can be updated in minutes by the head office to reflect new regulations or SOPs and deployed immediately across the plant.
This is a huge asset to compliance processes enforcing discipline, simplicity, and efficiency into the process.
Be Connected, Be Complete, Be Compliant
In conclusion, business process automation is about reducing the amount of human intervention as much as possible, bringing uniformity to certain processes, saving time, achieving standardization and compliance, and creating a more connected business.
Take advantage of the tools available to you to automate the way your company captures, stores, and shares its data across people and systems — in a complete end-to-end framework. These tools also open the door for improvement in other areas as well. Having better data visibility and connected workflows allows your teams to see where they can improve and gives you the ability to stay ahead of the competition, while staying compliant.
James Robins is Chief Marketing Officer at GoCanvas.
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American Actress, known for playing Kat Gardener on TV series Eastwick, Tess on Eureka and appearing in film Catch Me if you Can. In an IESB.net interview, she commented on her height, saying "I'm only 5'3". I'm unassuming"
No Aaron, she was taller then you, don't be jealous.
I saw her during filming of "Mind Games" in Chicago. I am 5"1.5 exactly, barely 5"2. I was wearing flats that day. She was shorter than I was so I would say she is 5"2 at the most.


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Human aging stages
When asked about how we age, most will say, "Gradually." And it turns out to be wrong, judging by the latest research by scientists from Stanford University in the United States, as reported by RIA Novosti.
In fact, experts identify certain stages of aging. Researchers announced three age points, refuting the idea of a gradual, daily change in the body, says Focus.
In the course of scientific research, for which they used blood plasma samples taken from 4263 people 18-95 years old, they came to the conclusion that the proportion of about 3000 different proteins does not change over decades. But at the age of 34, 60 and 78, there were significant changes at the protein level. At the same time, according to the study report, these fluctuations may be associated with changes in the body as a whole.
It remains unclear how blood composition and proteins affect aging. But experiments on mice show that blood transfusions from young rodents have a positive effect on older ones. Interesting research, promising important discoveries, continues.
Earlier in February, it was reported that in Japan, scientists have developed a drug that can rejuvenate the body and stop the aging process. According to Makoto Nakanishi, professor at the Research Institute of Medicine at the University of Tokyo, its use may begin very soon – in 5-10 years.
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Online Shopping is rather popular nowadays. We buy every little thing from food to digital devices online Imagine Mattress computer tutorial. When it comes to acquiring a product like a bed mattress that we are as well utilized to going to the store and buy, we often tend to be reluctant Imagine Mattress computer tutorial occasionally. This can be rather justified because a mattress is a quite a substantial buy and returning such an item can be a headache if you happen to make an error with your selection. Right here are some of things that you need to understand prior to you buy mattress online.
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Come Hear Famed Costume Designer Patricia Field Speak at Our New York Conference
Mark your calendars — and get your tickets today.
Patricia Field. Photo: Courtesy
Anyone with a career in the fashion industry today will tell you that it takes hard work to make it. We're holding our annual New York "How to Make It in Fashion" conference on Friday, Nov. 4 to get inside the minds of some of fashion's hardest-working power players, with keynote speakers including esteemed fashion critic Cathy Horyn and Patricia Field, one of the most celebrated costume designers working today.
A native New Yorker, Field got her start in the industry by designing and opening her New York boutique before catapulting into the international spotlight for her Emmy Award-winning costume designs on "Sex and the City." She would go on to work on other on-screen fashion classics like "Ugly Betty" and "The Devil Wears Prada" — for which she earned an Academy Award nomination — and currently designs the costumes for TV Land's "Younger."
If you want to hear Field's advice for making it in fashion; take in panels covering the rise of menswear, how to get a fashion label off the ground and what it means to be an influencer today; and get one-on-one mentoring with someone who's made it in your chosen field — get your tickets today, right here. And do it soon: early birds get $50 off the regular ticket price and our conferences always sell out.
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Our 'How to Make it in Fashion' Conference is Coming to New York on Oct. 23
Get your ticket before they're sold out!
Save the Date: Our 'How to Make It in Fashion' Conference Is Coming Back to New York and Los Angeles!
Mark these dates on your calendar — and secure your ticket now.


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The UK's Grocery Shopping Habits, Revealed
Adele Halsall
Whether you eat a plant-based, flexitarian, pescatarian or omnivorous diet, there are many ways to grab your groceries to fit around your busy lifestyle. But which dietary preferences shop most sustainably, and which habits will help you save money on your grocery bill?
To find out, we've taken a deep dive into the grocery shopping habits of Brits, looking at where they buy their groceries from, how much they spend, what motivates their buying choices and what we're doing to save money on food.
Here's what we found …
Pescatarians and vegans spend the least on groceries — an average of £203.02 and £203.51 per month
Those with omnivorous diets spend the most on groceries — an average of £238.29 per month
Since the rise in the cost of living, almost one fifth (19%) of respondents have switched to a more plant-based diet or incorporated more plant-based meals
For the same reason, almost half (49%) of all respondents have found it more difficult to shop sustainably for groceries, with flexitarians finding it most difficult of all (58%).
Where does the UK buy their groceries?
Unsurprisingly, major supermarkets are the most common place to shop (84%), followed by mini-marts or convenience stores such as Tesco Express or Co-Op (50%). Meanwhile, 29% shop via online supermarkets.
More niche places to pick up groceries include speciality shops/delis (22%), health food shops (20%), and farmers markets (16%), with 20% also getting a fruit and veg delivery each week.
Vegan diets seem to coincide with more local or health-focused shopping choices, with over half (58%) of vegans shopping at health food shops, 52% shopping at speciality shops/delis and 50% choosing to shop at farmers markets at least once a week.
While major supermarkets are still the main place for vegans to buy their groceries, a smaller percentage of vegans visit them than followers of other diets. Pescatarians are nearly as eco and health-conscious, with 43% shopping at health food shops and 40% choosing to visit farmers markets.
Vegans and vegetarians are also the most likely to shop with convenience in mind, with 69% and 62% respectively shopping at convenience shops, and 60% and 52% shopping at online supermarkets. Vegans and pescatarians are the most likely to have a weekly fruit and veg delivery (62% for vegans and 43% for pescatarians), which possibly ties into their concerns around the environment and their preference for locally sourced or organic food.
How much do Brits spend on groceries?
When looking at how much Brits spend on groceries each month, the average household was found to spend £222 — £51.23 per week.
Pescatarians and vegans buck this trend, spending considerably lower than their peers (£203.02 and £203.51 per month respectively — an average of £46.85 and £46.96 per week). Followers of omnivorous and flexitarian diets — which both include meat — have a higher average spend of £238.29 and £229.66 per month respectively, working out at £54.99 and £52.99 per week.
Vegans most likely to shop with eco-credentials in mind
Even before the cost-of-living increase, 89% of the UK public already had price topmost in mind when choosing where to shop for groceries. This is followed by quality of items, which 88% of Brits rate as very or somewhat influential, and location, which 84% agree is also important. Variety and availability of products is also important to Brits, as is the option to buy locally, with two thirds (66%) rating them both as very or somewhat influential.
Sustainability factors such as brand ethics and availability of refillable options were lower down on the list, though still important, with 53% and 52% of Brits respectively claiming them to be influential. Whether a shop sells organic food is less of a concern, though, with just 36% saying this was influential.
Vegans seem to be the most eco-conscious group when choosing where to shop. Brand ethics and responsibility and sustainable packaging options rank joint second after location (which 87% ranked as most important), with 85% of vegans agreeing these factors are very or somewhat influential. These factors are just as important as price and good variety/availability of products, which shows the lengths vegans will go to ensure their weekly shop is as ethical as possible.
Vegans are also the most likely to shop places where they can find organic produce, with 67% rating this as important compared with just 36% of the nation overall.
When it comes to what we put in our trolleys, quality of items is the most important factor Brits look at, with 90% agreeing this is very or somewhat influential. This is followed by price (89%), how healthy an item is (71%), and whether a product is locally produced (60%). Other factors, such as the product's packaging, brand ethics, whether it's refillable and whether it's organic, are less important.
Vegans again seem the most eco-conscious in this area, with brand ethics and social responsibility being the top factor that influences their choice of purchases (92%), followed by whether a product has plastic-free, recyclable or compostable packaging (85%). 81% say that price and quality are equally important, while 79% say health.
Pescatarians and flexitarians seem to be the next most eco-conscious group, with 70% of pescatarians ranking packaging of a product as very or somewhat influential, along with 73% of flexitarians.
Vegetarians seem to be the most health-focused when it comes to their choice of grocery items, with 83% ranking health credentials as very or somewhat influential.
What impact has the cost-of-living had on our grocery buying habits?
Since the sharp increase in the cost-of-living, almost a fifth (19%) of Brits say they have incorporated more plant-based meals and/or meat substitutes into their diet, with 4% adapting a fully vegan or veggie lifestyle and 6% adapting a pescatarian lifestyle.
Almost half (49%) of the population say they have found it noticeably more difficult to shop sustainably for groceries, with flexitarians finding it most difficult of all (58%), followed by vegans (54%), omnivores (49%) and vegetarians and pescatarians (43%). 17% say that shopping sustainably was not a concern before, so had not noticed any difference.
When it comes to choosing where to shop, it's obvious the cost-of-living squeeze is directing our buying choices more than ever. 90% of Brits name price as the most important factor (compared to 89% before the cost-of-living increase), while quality is now seen as less important (85% vs 88% before). Location is also less important, suggesting people are willing to travel further in order to get the best deal on price (82% vs 84%).
Other factors now deemed less important are variety and availability of products (60% vs 66%), the option to buy products with sustainable packaging (54% vs 58%), brand ethics and social responsibility (50% vs 53%) and whether there are refillable product options available (47% vs 52%).
When choosing grocery items, quality now takes a backseat to price, with 90% of consumers saying price is most influential compared with 86% for quality. Health credentials are also seen as influential by 68% of people, compared with 71% before cost-of-living increases, while eco-factors such as sustainable packaging and brand ethics have also fallen down the scale of importance with 53% and 49% of people rating them as important respectively, down respectively from 58% and 55%.
How is the UK saving money on groceries?
Since the rise in the cost-of-living, 51% of Brits say they have switched to cheaper brands and items, while 46% say they look for discounts and deals wherever possible. 42% have restricted themselves to buying fewer items, 40% plan their meals to avoid food waste and 26% buy food in bulk to cut down on costs.
Almost a quarter (24%) have switched supermarkets in a bid to get their groceries cheaper, while 11% have made an effort to buy refillable products. Another 11% say they try to shop from local or independent shops and markets. Only 12% of respondents said they haven't done anything differently.
Vegans seem to be making the biggest efforts to cut down on food costs, with just 2% claiming to have not changed any habits. Omnivores seem to be the least affected group, with 17%.
Planning meals to cut down on food waste seems to be the most popular eco-conscious money-saving habit, with consumers of all dietary preferences making it a priority. This is followed by buying items in bulk, which was also high on everyone's list.
Take Home Message
It's clear the UK's grocery buying habits are being significantly affected by the rise in living costs, leading to more choices based around price and convenience rather than sustainability.
The good news is that there are ways to both save money and be environmentally conscious, simply by taking things back to basics. Buying in bulk and planning meals to save on food waste can help your budget go that bit further, while refillable products can also help spread the pennies (and save on packaging use).
Finally, the research is clear: buying less meat and incorporating more plant-based meals into your weekly rota may also help to cut your grocery bill. Check out our range of vegan recipes and plant-based protein snacks to give you plenty of inspiration for your next meal plan.
We surveyed over 2,000 UK adults on their grocery shopping habits, including how much they spend, where they shop and what motivates their shopping choices. We also asked questions around how the cost of living has changed (if at all) their grocery spending habits.
Questions were mostly multiple choice with the option to add additional qualitative data where relevant. Results were split by age, gender, region/city and dietary preferences.
Adele is a keen blogger and yogi with a passion for the vegan lifestyle. She loves exploring new (sometimes weird) foods, cooking & testing new recipes, and always appreciates a good sourdough.
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• By Lauren Dawes


For those following a healthy eating plan, staying "on the wagon" can be a minefield, from the time Thanksgiving dinner hits the table through New Year's Day.
In this season of excess, keeping tabs on sugar intake isn't always easy, and yes, for a lot of people all that vigilance is a bit of a downer. However, there's an easy workaround for those who struggle with sugar: L-glutamine. It's an excellent supplement that helps cut cravings for sweets and helps stabilize blood sugar – two things sugar-lovers need most. It's also helpful for elevating mood, which comes in handy during the short, dark days of December.
· Stick to lower-in-sugar dry red wine or tequila.
· Steer clear of sugar-drenched dessert wines.
· Take beer off the table and forget about sugary cocktails like Cosmos and Moscow Mules.
· Skip drinks involving juices or sweet mixers like tonic water, grenadine, simple syrup, etc.
· When it comes to spirits, drink it straight (and slowly!), or diluted with water or seltzer.
Following a specific dietary plan through the holidays isn't always easy but it's also not fair to put the 'what-do-you-have-that's-on-my-diet on us' on those kind enough to host you. If possible, ask the host if it you can contribute a healthy dish to to ensure you'll have something that works for your eating program – and doesn't make more work for the host. If bringing a dish isn't feasible or gets the thumbs-down, then eat a full, healthy meal before heading to Cousin Larry's for his annual all-carb, all-sugar buffet. When you're already full, unhealthy buffet items will have considerably less appeal.
– Pass on pasta, potatoes, and grains.
– Order organic, pasture raised or grass fed animals.
– Double up on greens and veggies.
– Skip gravies, sauces and most condiments, which are often laced with flour and sugar.
– Trade sweet desserts for a cappuccino with organic whole milk or unsweetened almond milk, topped with cinnamon and whipped cream.
Let's face it – all of us will slip up from time to time, but it's how you respond to the slip-up that counts. Minor bumps in the road are to be expected, so drive right on over them, veer around them and get back on the path as soon as you can.


No matter what type of event you're planning, Green Light Booking offers country bands for hire that will perfectly fit the occasion. Our country music bands for hire are award-winning groups who know how to put on a tight and energetic show.
If you're looking to please a country dance crowd, Jagertown, Artie Hemphill, and Charley Jenkins are fast-pickin' country music bands that really know how to wow crowds. Due West, Gwen Sebastian, and Nathan Osmond are some of the most exciting country artists and best live performance bands in America when it comes to live concert performances. Our top country bands and artists are all seasoned performers who know how to turn any event a spectacular one.
Jagertown is one hot, rockin' band, touring the country and performing their original hits to rave reviews at many of the biggest country music festivals in the nation. But performing at prestigious venues is not all they do. Invite them to your corporate event, country-themed wedding reception, or private party, and they'll pump out a constant stream of the best country cover dance hits all night long, treating your guests to a rip-roaring, fun time on the dance floor. Whether it's a live concert or private event, this band always brings the heat.
Artie Hemphill and the Iron Horse Band are one of the most popular country bands for hire in the entire Intermountain West. Artie spent over ten years in Nashville, honing his remarkable vocal and performance skills and is one of the best country singers in the state. Artie and his band have shared the stage with Little Texas, Miranda Lambert, Sawyer Brown, and The Little River Band. This popular group plays classic, contemporary, and original tunes which appeal to both country and rock fans.
Country singer Gwen Sebastian, a popular finalist on the hit TV show The Voice, is starting to get noticed in a big way. Her hit debut single in 2009, Hard Rain, announced the arrival of one of the purest and most magnetic country voices in a long time. Gwen is certainly one of the top female country artists to hit the scene in quite awhile, and there's no question she'll be around for a long time to come.
If you're searching for country bands to hire, the Due West trio should be at the top of your list. Due West is fast making a name for themselves as their music and videos climb the charts. When the Smoke Clears, off their album Forget the Miles, catapulted to #15 on the GAC Top 20 Country Countdown thanks to fan voting. With major radio airplay, Due West is quickly becoming the talk of the country music industry.
If you want to hire a country cover band that boasts one of the best Nashville-quality country singers around, you'll need to take a close look at the Chad Truman Country Cover Band. Smooth, creamy vocals like Chad's are extremely rare and in hot demand, so book him early before somebody else does.
Charley Jenkins and his high-powered band have opened for the biggest names in country – Taylor Swift, LeAnn Rimes, and over thirty others. If you're looking for high-energy country entertainment, invite Charley to your party.
Country artist Nathan Osmond has certainly gained most of his fame by being a member of the world-famous Osmond family, but Nathan is not just one of the Osmond brothers, he's one of the standouts. He already has multiple number one hits on the Indie Country Charts. With Nathan's larger-than-life personality and seasoned vocal chops, it's easy to see why he is one of the most popular country bands for hire.
The Bootleg Blvd country cover band is one of the most fun and exciting country bands for hire in Utah. With quality musicians and vocalists, this band pumps out irresistible country dance music that everyone enjoys. Playing from a long list of hit cover tunes, Bootleg Blvd will have everyone at your party in a tizzy.
Country star Mark Owens is larger than life on stage. He is a natural performer who routinely makes a lasting impression on audiences. This is one of the top country bands for hire in Utah.
Country dance band Mister Pickins is a real crowd-pleaser. Once you hear them play, you'll understand why they're invited back again by nearly everyone who hires them. Mister Pickins' players have shared the stage with Miranda Lambert, Diamond Rio, Little Texas, Kenny Chesney, and many others. This is one fun dance band that makes it impossible for anyone to sit still.
When it comes to high energy dance music for the country crowd, nobody does it better than Green Light Booking's hot country bands for hire. If you want to put on a party that will wow your crowd, you'll want to hire one of our hot live bands, no doubt about it.


At Keng Soon Auto, we are fully committed to the development and well-being of our employees as they are our greatest assets. Our success was largely attributed through the efforts of our employees, dedicated to ensuring customer satisfaction as their main priority.
If you're an independent, highly-motivated and hardworking individual, we want to hear from you!
Respond to routine inquiries from internal or external sources, for example customer enquiry on office's location, hours of operation technical enquiries.


An orchestral work featuring immensely gifted individual and collective voices from around the world (see above!), meshing as one to manifest a pair of deep double-drummer-and-bass driven mantras; this is serious (other) World teleportation music that achieves psychedelic proportions of sonic interplay.
"Double Sunrise was borne of a concept I call universal tonality, which is based off of the idea that all sounds like human beings come from the same place. All sound has a heartbeat and breathes the same as each human being. Some sounds are born in Africa; others are born in Asia, Europe, Australia or America. These sounds pass through certain human beings. We don't invent sounds, we are allowed to encounter them; we don't own them, they existed before we were born and will be here after we are gone.
This project is supported by the New York State Music Fund, established by the New York State Attorney General at Rockefeller Philanthropy Advisors, and is released in cooperation with Arts for Art, Inc.


How we conceptualize our body, our joints and tissues, affects how we react to situations that involve those tissues. Our joints and tissues represent, in the minds of most laypersons, a static entity. That is, "they are what they are". They can often be viewed as the sum of all stresses, where stresses are responsible for "wearing out" the tissues. This analysis can be misleading. How we conceptualize our tissues dictates, or at least influences, how we behave. In essence, we act out in the real world what we believe to be true about our bodies.
The "wear and tear" model of joint structure, while intuitive and rational in terms of what we know about the world (we all are well aware discontinuity of all things that exist, aside change), has some misconceptions when applied to the living adaptable system that is our body. The main misconception involves the belief that tissue wear itself is irriversable tissue degeneration. In a world where tissue wear = degeneration = pain, the body can be viewed much like a car, in that based on amount of use (ie at 200k miles) it requires a replacement.
I can't remember the reference, but I once read of a tribe whom held the belief that a human has a finite number of heart beats before death. The tribe would act in accordance to this belief, and move very slowly, avoiding any increase in heart rate. They became fragile, soft and sloth like sedentary citizens, and their king existed in a near immobile state, spoon fed and manually transported. Yes, their hearts beat slowly. But at what cost was this belief? Was what they were acting out advancing the intentions of their belief, or to their detriment?
There is mounting evidence and exposure on the tenuous link between tissue and damage, and the more we can re-conceptualize our bodies as adaptive, the less likely we are to fall in line with the aforementioned tribe, mistaking cure for disease.
Bringing this to light can be of therapeutic value in many ways. If we can change what someone believes about their condition, then we can change what they act out, which has incremental returns. The job doesn't end with just stating information about the body being adaptable. In order for a person to make an informed decision to change how they react to their pain, they need to find this new conceptualization of an adaptable system interesting. If it isn't interesting, then there is likely no cognitive shift in understanding, and behavior would be unlikely to change.
The idiom "Wear and Tear" (a phrase also labeled by Butler/Moseley to be a particularly "sticky" neuro-tag in is roll-off-the-tongue ease) is usually somewhere in the mind of a patient. In reality, there is tear. There is breakdown. It is completely unavoidable. What can be missed though in the patient-physio interaction is the education that tissue "learns", changes and responds. This is part and parcel of what Butler and Moseley are bringing to the rehab table. There are too many real interactions between the immune system, thoughts and beliefs, stress and sleep, behavior etc… to view pain in a purely isolated biological tissue mechanistic sense. That is, it isn't exactly the painful joint (or the muscle or body part) but how that body part is acting in its environment. The "how it is acting" has implications both at the tissue state, and at the individual tissue owner's state.
I am reminded of a recent PT visit of a woman in her 80's that had severe scoliosis and it was affecting her gait. She wasn't coming in because of pain, as she didn't have any. She was just concerned about the way her walk looked, and her balance. Here is a great example of wear and tear, structural changes and tissue "dysfunction" over the course of an entire life without manifesting any mentionable pain level. But when we are under the influence of pain, it is much more difficult to zoom out and look at the big picture: there exists a present reminder of pain that can consistently confirm our beliefs that something is wrong, that we are at 200k on that tissue. It is difficult to imagine the educational messages of adaptation outweighing innate messages of pain (yes pain is output, but its output circles back around to a cognitive psychological input). But part of that depends on how relieved the patient is to find out the truth about their condition (this goes back to the interest). If for months you suspected a sneaking terminal cancer, awaking in the middle of the night dreading it and imagining and feeling symptoms that confirmed this daily, and finally getting to a specialist only to find that your symptoms were, all from say a gluten allergy, imagine your relief. Its not terminal cancer! Its just an allergy! A weight is lifted…this can change.
Can we shift our views, in a similar sense, when we think about low back pain, neck pain or knee pain? Are our pains conformation of a static and irreversibly damaged tissue? Or could they be a feared cancer without any relevant diagnosis? What would lift the weight of our concerns? Part of this is education. The other part is learned through experience, when the person starts to behave differently and treat the tissue based on what they may have learned through the education. We act out what we believe (we hobble on a joint we believe to be fragile, we move carefully and slow when expecting pain, we change what we do if we are afraid that activity will cause damage, we tense up with pain, and we avoid positions and loads that our tissues have disliked in the past). These behaviors can have their place, but their protection can often quickly become unwarranted.
Back to wear and tear, lets take a metaphor. Imagine a small cabin in the forest of Alaska. In that cabin lives a particularly industrious jack-of-all trades woodsman. His goal is survival, and to survive he keeps his cabin fortified. He learns how to properly fortify his cabin based on what he experiences over the years; wet falls, heavy winters, bears in the summer clawing their way in…you might liken that cabin to your body, where its under constant surveillance of your nervouse system and brain. The tissues are constantly being monitored, and changing in small ways based on what they experience. In a simple mechanical sense, these could be stress, strain, and all things detected by "mechanoreceptors". In an immunological sense, these could be chemical changes, cellular response, and all things "chemoreceptor". The cabin does eventually break down over 100 years. Tissue will always eventually degenerate. But, it also gets fixed up along the way, making areas of weakness more robust, and adding support where it is most needed. In a living system, there is adaptation.
Disk degeneration, or "DDD" is a common term brought up by patients with back pain. One argument against some of the medical overemphasis on disc loading strategy is disk research is conducted in-vitro to "non-living" tissue, and doesn't respond the same to actual living tissue. But even then, disc degeneration doesn't necessarily have anything to do with pain (back to the woman with severe scoliosis, and all sorts of disc degeneration). There are many angles we can help address where the body is working for the bothersome tissue, not against it. This may come in the form of load management, sleep education, graded exposure to reduce sensitivity, getting stronger globally to support the painful area, increasing cardio for improved blood flow (this could be an arm bike for a person with knee pain, or a leg bike for a person with neck pain… just something to globally improve vascularity).
Not only is the term "wear and tear" misleading (as it should be more something like "wear/load, and create adaptations to best/better tolerate future wear/load"), it also isn't strongly correlated with pain, which is almost always the primary complaint. Take knee OA for example. Researchers are now referring to this condition as "symptomatic knee OA" (and more and more conditions are allowing this discriminating preamble) because radiographic knee OA does not equate pain.
The consensus is clear that as age and OA findings have a linear relationship, which does not come as a surprise. But remember, OA does not equal pain. The Framingham Osteoarthritis Study found that 10% of people aged 63 years and over had symptomatic knee OA (pain) in the presence of radiographic changes (1). In people with severe OA, it seems about half report pain (2). Although some studies find the link of reported pain and OA severity less clear, some large scale studies (4000 subjects) finds a clear relations between the roughly 40% of persons with radiographic OA and pain (3). Even in these studies, 60% of the persons with radiographic OA did not have pain. Further, can the 40% improve? If so, their "radiographic" OA isn't changing….
Back to the education component. A study I like to reference is a recent "JOSPT perspective for patients." (4) The important part to get across is, the more you load the tissue doesn't necessarily mean the worse off the tissue is. Again, the structural life of tissue is not a ticking clock that only that only counts down to zero. People that don't jump around, load and move (sedentary individuals) have a higher chance of radiographic knee OA than recreational runners. And people that are obsessive about running (over 57 miles/week runners) only have slight more signs of usage than sedentary people! The take home point is that, to a large level, tissue adapts and becomes more robust, depending what it is introduced to and how it is introduced (ie graded VS overworked). I also like to bring up the fact that people can train themselves to run 26 miles, without their body reacting to that load with severe pain and inflammation. If a runner runs 15 miles in the first month, they may be hobbling in pain for a few weeks. If they train, running 3, 2, 5, 4, 8, 3, 3, 8, 4, 10 miles and so on, they can undergo the same "wear and tear" load, with a totally different response. The response is different not because the person is just stronger or more fit (even though that may be part) but because the body gets familiar with the jolts and pounding of running, and has time to adjust to this and not be alarmed by it. If it (the brain) isn't alarmed, then the alarm signals (pain) are not present. I also like to reference the Moseley / Butler diagram of "twin peaks" (5). This illustrates that the alarm is going to be set at a lower threshold after injury (even though the actual tissue tolerance can be still robust, especially after proper healing time).
I like tangible examples that don't get too deep in the physiology of pain. We don't want to move into a completely mechanistic explanation of neurophysiology to replace a mechanistic explanation of structure. There are a lot of simple examples out there that can be used to open up a conversation about the pain in question and how it relates to structure. There is growing evidence that many things influence pain, and how we understand the issue may dictate how we then interact with the world. Not only do we need to change beliefs through our rhetorical explanations, we need to disconfirm beliefs (a term used by Peter O'sullivan in his CFT workshop) in a physical, real body activity and movement sense.
There will always be cases where structure wins, where structure truly needs to be replaced, or "fixed". Unfortunately, for the same reason we mention tissue damage doesn't equate pain, fixing the structure doesn't promise pain relief, and we can look to lumbar fusion outcomes to see it isn't a cure-all to decompress and immobilize areas reported painful. Some areas respond better to "fixing" than others. We can often mistake symptoms for a disease, and be confused on the relevancy of a patient with one, or both of those variables. Either way, if we are going for conservative treatment, we need to go about it right, with the right outlook and information. This conceptualization should act as a critical foundational understanding to which the rest of the rehab experience can fall under. Following a narrative of adaptation, strength, desensitization and graded exposure, patients will have a better understanding of what physical therapy means, and their individual role in rehab from the specific irritable tissue to the global influencing factors in their lives.
-Tal Blair, DPT - via Motion Guidance blog - check out the MG Clinicians' Kit below!


text:
\section{Introduction}

\dlabel{intro}

The observed primordial curvature perturbation $\zeta(\bfx)$ presumably originates in the
early universe, and  a central task of theoretical cosmology
is to determine its origin and the  evolution $\zeta(\bfx,t)$.
It is usually  supposed to be generated
from the perturbation of one or more scalar fields, making it
 statistically isotropic
which is consistent with observation.   But it
may also receive a contribution from one or more vector fields, making it statistically
anisotropic. Observation does allow significant anisotropy and in particular a spectrum
  of the form\footnote
{We use hats to indicate unit vectors}
\be
{\calp_\zeta}(\bfk) = {\calp_\zeta}(k) \[ 1 + g_* (\hat\bfw \cdot \hat \bfk)^2 \]
 \dlabel{gstardef} \end{equation}
is allowed with  $|g_*|\lsim 10\mone$. The upcoming PLANCK results will
give $|g_*|\lsim 10\mtwo$ barring a detection \cite{anis}.

 Most schemes  invoke a  $U(1)$ gauge field $B_\mu$.
 If the field has the canonical kinetic
term and no coupling to gravity, its
spectrum increases with wavenumber like $k^2$,
 and its contribution to the
 spectrum of $\zeta$ has the same behaviour making it  almost certainly
  negligible on cosmological
scales \cite{ouranis}. One way of avoiding this may be to keep the canonical kinetic
term but invoke a coupling to gravity
given by $-R B_\mu B^\mu/6$. This apparently  \cite{nonmin,ouranis}
 generates a flat spectrum
but the model has instability and (possibly) non-linearity \cite{ghost} and it is not known
 \cite{health} whether or not these spoil the prediction.

In this paper we consider a different scheme, which  invokes a
gauge kinetic function $f^2(\phi)$ that depends on the slowly
 rolling inflaton field $\phi$. To allow a significant contribution on cosmological scales
one needs  (at least approximately)
 $f\propto a\mtwo$ where $a(t)$ is the scale factor, which may be reasonable in the context
of string theory \cite{p12jcap}.
A well-defined contribution to $\zeta$ is generated during slow-roll inflation
\cite{beforemarco,marco} and additional
contributions may be generated afterwards \cite{jiro,p12jcap,ouranis}.
In this paper we give for the first
time a master equation that includes all contributions, but our main focus is on the
contribution generated during slow-roll inflation. Under the assumption
that the homogeneous part of the
gauge field dominates the perturbation, it has been calculated
\cite{beforemarco,marco}
using the in-in formalism of quantum field theory. Using instead
the classical formula for $\zeta$ in terms of the energy
density perturbation, we reproduce that calculation and extend it to the case where
the  homogeneous part is  sub-dominant or even negligible. We also identify the
 assumptions made by the calculation, and show how to perform a more complete
calculation with weaker assumptions.
We end by discussing the significance of our result.

While this paper was in its final stages of preparation ref. \cite{hassanNew}
appeared. The authors of this work uses the so called $\delta N$ formalism to
calculate $g_*$ and non-gaussianity in the same model. Our results agree with their
findings where there is an overlap.

\section{Action and field equations}

\dlabel{saction}

During inflation we are taking the action to be
\be
S = \int d^4x \sqrt{-g} \[ \frac12 \mpl^2 R
- \frac12 \pa_\mu \phi \pa^\mu \phi - V(\phi)
 - \frac14 f^2(\phi) F_\mn F^\mn \]
, \dlabel{action} \end{equation}
where $F_\mn\equiv \pa_\mu B_\nu-\pa_\nu B_\mu$  with  $B_\mu$  a
 gauge field, and $f$ is some function of the inflaton field $\phi$.

Extremizing the action in \eq{action} with respect to fields $\phi$, $B_\mu$ and their derivatives we obtain field equations
\begin{eqnarray}
\left[\partial_{\mu}+\partial_{\mu}\ln\sqrt{-g}\right]\partial^{\mu}\phi+V'+
\frac{1}{2}ff'F_{\mu\nu}F^{\mu\nu}&=&0; \dlabel{basic1}  \\
\left[\partial_{\mu}+\partial_{\mu}\ln\sqrt{-g}\right]fF^{\mu\nu}&=&0,
\dlabel{basic2}
\end{eqnarray}
where $g\equiv \mathrm{det} (g_{\mu\nu})$, $V'\equiv\partial V/\partial \phi $ and
$f'\equiv\partial f/\partial\phi$. We make the gauge choice $B_0=0$,
which fixes the spatial components $B_i(\bfx,t)$ up to a constant.

We are going to assume that the inflationary expansion is nearly
isotropic, checking later that
this is  justified. Writing
\be
ds^2 = -dt^2 + a^2(t) \delta_{ij} dx^i dx^j
, \end{equation}
we then have
\begin{eqnarray}
\ddot\phi+3H\dot\phi-a^{-2}\nabla^{2}\phi+V' & = & -\frac{1}{2}f f' F_{\mu\nu}F^{\mu\nu},
\dlabel{EoM-phi} \\
\ddot B_i+\left(H+2\frac{\dot f}{f}\right)\dot{B}_{i}-a^{-2}\nabla^{2}B_{i} & = & a^{-2}2\frac{\partial_{j}f}{f}\partial_{j}B_{i}, \dlabel{EoM-W}
\end{eqnarray}
where $H\equiv \dot a/a$ and $\nabla^2 \equiv \delta_{ij}\pa^2/\pa x^i\pa x^j$.

\section{Pure slow-roll inflation and the curvature perturbation}

\dlabel{pure}

If the effect of the gauge field is completely negligible
 we have pure slow-roll inflation, described for instance
in \cite{review,book}.
 The last term of \eq{basic1} is absent, and
for   the  unperturbed inflaton $\phi(t)$ we have
\be
\ddot\phi+ 3H\dot\phi + V'(\phi) = 0
. \end{equation}
The potential is supposed to satisfy  the
 flatness conditions  $\epsilon\ll 1$ and $|\eta|\ll 1$ where
$\epsilon \equiv \mpl^2(V'/V)^2/2$ and $\eta\equiv \mpl^2 V''/V$. Then, more or less
independently of the initial condition
\be
3H\dot\phi \simeq -V'
. \dlabel{phidot} \end{equation}
With the flatness conditions this is called the slow-roll approximation, which
we use throughout.
It  gives $|\dot H|/H\ll H$ and $|\dot\epsilon|/H\ll \epsilon$. Except where stated
we take  $H$ and $\epsilon$ to be constant.

We    write $\phi(\bfx,t)=\phi(t) +\delta\phi(\bfx,t)$ and define
\be
\delta\phi_\bfk(t) \equiv \int d^3x e^{-i\bfk\cdot\bfx} \delta \phi(\bfx,t)
, \end{equation}
and similarly for other quantities.

Let $N(k)$ be the number of $e$-folds of slow-roll inflation after
the epoch of horizon exit $k=aH$. The `cosmological scales' on which $\zeta_\bfk$
is observed have $N(k_0)-15\lsim N(k) \lsim N(k_0)$, where
 $k_0=a_0H_0$ and the subscript on the right hand side denotes the present epoch.
Also,
\be
15 \lsim  N(k_0) \lsim  70
, \dlabel{nbound} \end{equation}
where the upper bound assumes $P\leq \rho$ after inflation, were $P$ is the pressure and $\rho$
the energy density. The lower bound is
needed         so that cosmological scales leave the horizon.
Typical cosmologies give a value in the upper third of the range.

Perturbing \eq{basic1} to first order, and ignoring
 the metric perturbation
(back-reaction) we have
\be
\delta\ddot\phi_\bfk + 3H\delta\dot\phi_\bfk + (k/a)^2 \delta\phi_\bfk=
-V'' \delta\phi_\bfk
. \dlabel{phiddot} \end{equation}
Evaluating the metric perturbation to first order, on
 the flat slicing of spacetime
(such that the 3-curvature scalar vanishes) one finds \cite{sasmuk,book}
that it is significant only
after horizon exit, when \eq{phiddot} becomes
\be
 \delta\ddot\phi + 3H\delta\dot\phi_\bfk \simeq
-  V'' \delta\phi_\bfk - 6H^2\epsilon  \delta\phi_\bfk
. \dlabel{phiddot3} \end{equation}
The last  term  represents  effect of the
metric perturbation (back-reaction).
{}From \eq{basic1} this  term can be written
\be
a\mthree \pa_0 \delta(\ln\sqrt{-g}) a^3 \dot\phi \simeq
3H \delta(\ln\sqrt{-g}) \dot\phi
. \dlabel{backreaction} \end{equation}
Comparing \eqs{phiddot3}{backreaction}
we find\footnote{We assume $\dot\phi<0$ as is the case for hybrid inflation.}
\be
\delta(\ln\sqrt{-g}) \simeq \frac{-\sqrt{2\epsilon}}\mpl \delta\phi_\bfk
. \dlabel{deltaln} \end{equation}

Regarding $\delta\phi_\bfk(t)$ as an operator its mode function
$\phi(k,t)$ satisfies  \eq{phiddot}. Well before horizon
exit at $k=aH$  \eqs{action}{phidot} describe a free field in flat spacetime.
We  choose
\be
a(t)\phi(k,t) = e^{-ik/aH}/\sqrt{2k}
, \dlabel{phiinit} \end{equation}
and the vacuum state corresponding to the absence of $\phi$ particles. Then
we have
\begin{eqnarray}
\vev{ \delta\phi_\bfk \delta\phi_\bfkp } &=&
 (2\pi)^3\delta^3(\bfk+\bfkp)(2\pi^2/k^3) {\cal P}_\phi(k,t), \dlabel{vev} \\
(2\pi^2/k^3){\cal P}_\phi(k,t) &=&  \phi^2(k,t).
\end{eqnarray}
Also,
\begin{eqnarray}
\vev{
\delta\phi_{\bfk_1} \delta\phi_{\bfk_2} \delta\phi_{\bfk_3} } &=& 0,
\dlabel{phi3} \\
\vev{\delta\phi_{\bfk_1} \delta\phi_{\bfk_2} \delta\phi_{\bfk_3} \delta\phi_{\bfk_4} }
&=& \vev{\delta\phi_{\bfk_1} \delta\phi_{\bfk_2} }
\vev{\delta\phi_{\bfk_3} \delta\phi_{\bfk_4} } + \nonumber \\
&+& \vev{\delta\phi_{\bfk_1} \delta\phi_{\bfk_3} }
\vev{\delta\phi_{\bfk_4} \delta\phi_{\bfk_2} } +
\vev{\delta\phi_{\bfk_1} \delta\phi_{\bfk_4} }
\vev{\delta\phi_{\bfk_3} \delta\phi_{\bfk_2} }
 \dlabel{phicorr} , \end{eqnarray}
and similarly for higher products.

Well after  horizon exit ($k\ll aH\ll 1$)
the phase of $\phi(k,t)$ becomes constant which
means that $\delta\phi$ can be regarded as a classical perturbation
with the above  correlators (gaussian perturbation).
In the initial regime
\be
|\eta|,\epsilon\ll k/aH \ll 1
, \end{equation}
the left hand  side of \eq{phiddot} is still close to zero, and the mode function
is
\be
\phi(k,t)\simeq \phi_0(k,t) \equiv \frac{e ^{-ik/aH}}{a\sqrt{2k}}
 \( 1-iaH/k \) \simeq \frac{iH}{k\sqrt{2k}}
. \dlabel{phi0} \end{equation}
This gives a slowly varying perturbation, $H\mone |\delta\dot\phi_\bfk|
\ll |\delta\phi_\bfk|$ with
\be
{\cal P}_\phi(k,t) \simeq  {\cal P}_\phi\su 0\equiv  \( \frac H{2\pi} \)^2
. \dlabel{pphi0} \end{equation}
The subsequent evolution is given by \eq{phiddot3}. It has two independent solutions
but  the initial slow variation will pick out the solution
\be
 3H\delta\dot\phi_\bfk \simeq
- \[  V''+ 6H^2\epsilon \] \delta\phi_\bfk
. \dlabel{phiddot2} \end{equation}

Going to second order in the perturbations of the field and metric one finds
additional correlators of $\delta\phi$ (non-gaussianity) \cite{seery},
starting with the three-point correlator which defines the bispectrum
$B_\phi$:
\be
\vev{\delta\phi_{\bfk_1} \delta\phi_{\bfk_2} \delta\phi_{\bfk_3} }
=(2\pi)^3 \delta^3(\bfk_1+\bfk_2+\bfk_3) B_\phi(\bfk_1,\bfk_2,\bfk_3)
. \dlabel{bphi} \end{equation}

To define the primordial curvature perturbation $\zeta(\bfx,t)$, one
 smoothes  the metric and the energy-momentum tensor
on a super-horizon scale and chooses the comoving
threading and the uniform energy-density slicing of spacetime.
Then
\be
\zeta(\bfx,t) = \delta \ln(a(\bfx,t) =\delta\[ \ln a(\bfx,t)/a(t_1) \]
\equiv \delta [N(\bfx,t,t_1)] \dlabel{deltaN}
, \end{equation}
where $a(\bfx,t)$ is the local scale factor such that a comoving volume element
is proportional to $a^3$, and  $a(t_1)$ is its unperturbed
value which can be evaluated at any epoch. As we discuss in Section \ref{just}
it is usually enough to work to first
order in $\zeta$; then one can choose $t_1=t$ to get
\be
\zeta(\bfx,t) = H\delta t(\bfx,t) = -H\delta\rho(\bfx,t)/\dot\rho(t)
, \dlabel{zeta} \end{equation}
where $\delta t$ is the displacement of the uniform-$\rho$ slice from the
flat slice and $\delta\rho$ is evaluated on the flat slice and
\be
\dot\rho = - 3H(\rho+P)
. \dlabel{econ} \end{equation}

Since $\rho$ is smoothed on a super-horizon scale, \eq{econ} applies at each location,
which implies  that  $\zeta(\bfx,t)$ is constant during any era when $P(\bfx,t)$
is a unique function of $\rho(\bfx,t)$.
Galaxy surveys and observation of the
CMB anisotropy detect the value of $\zeta$ at the epoch with temperature
$T\sim 10\mone$\,MeV; the universe is then radiation-dominated ($P=\rho/3$)
giving $\zeta$ a constant value that we  denote   simply by $\zeta(\bfx)$. Its
 spectrum is nearly scale-independent with \cite{wmap9}\footnote
{The spectrum and bispectrum of any perturbation are defined as in \eqs{vev}{bphi}.}
\begin{eqnarray}
{\calp_\zeta}(k) &\simeq&   (5\times 10\mfive)^2   \dlabel{calpzobs} \\
n(k)-1 &\equiv& d\ln{\calp_\zeta}(k)/d\ln k  \simeq
-0.029\pm 0.010
. \dlabel{nobs} \end{eqnarray}
The result for $n(k)$ assumes that it has negligible scale dependence. It also
 assumes a tensor fraction $r\ll 10\mone$, which will soon be tested by PLANCK
\cite{planck}.)

For the reduced bispectrum defined by
\begin{eqnarray}
f\sub{NL}(\bfk_1,\bfk_2,\bfk_3) &\equiv& \frac56 \frac {B_\zeta}
{P_\zeta(k_1)P_\zeta(k_2) +P_\zeta(k_2)P_\zeta(k_3) +P_\zeta(k_3)P_\zeta(k_4)}
\dlabel{fnldef}
 \end{eqnarray}
where $P_\zeta(k) \equiv (2\pi^2/k^3) {\calp_\zeta}$,
current observation \cite{komatsu}
give $|f\sub{NL}|\lsim 100$ and barring a
detection PLANCK will give $|f\sub{NL}|\lsim 10$.
For $f\sub{NL}$  to ever be observable we need
$|f\sub{NL}|\gsim 1$.

During pure slow-roll inflation,
  $\zeta$ has a  time-independent value that we denote
by $\zeta_\phi$. It is well-approximated by \eq{zeta} with
\be
\rho_\phi  \equiv V(\phi) + \frac12 \dot\phi^2 \simeq V(\phi)
, \end{equation}
leading to the  gaussian perturbation $\zeta=\zeta_\phi\equiv
-H\delta\phi/\dot\phi$ and
\be
{\cal P}_{\zeta_\phi}(k) =  \frac1{2\epsilon \mpl^2} \( \frac H{2\pi} \)^2
, \dlabel{calpzphidef} \end{equation}
where  $\epsilon$ and $H$ are evaluated at horizon exit.

Going to higher order in the perturbation of the metric and of $\delta\phi$,
one finds \cite{seery,lms} a non-zero
$B_\phi$.
Then, using the $\delta N$ formula to go beyond the first-order formula
\eqreff{zeta}, one can use \eqss{vev}{phi3}{phicorr} to find
  $|f\sub{NL}|\sim ({\cal P}_{\zeta_\phi}/{\calp_\zeta})^2 10\mtwo$ which
 is too small to observe    \cite{seery,lms}.\footnote
{This result was first found in \cite{maldacena}
by a different method.}

\section{Background (unperturbed) universe}

\dlabel{background}

Recasting \eqs{EoM-phi}{EoM-W}
 in terms of $\mathbf W$ with components
$W_i \equiv f  B_i /a$ and dropping gradient terms, one arrives at equations of motion
for homogeneous fields $\phi(t)$ and $\bfw (t)$
\begin{equation}
\ddot\phi+3H\dot\phi+V'=\frac{f'}{f}\left|
\dot{\mathbf{W}}+\left(H-\frac{\dot{f}}{f}\right)\mathbf{W}\right|^{2}, \dlabel{hom-phi}
\end{equation}
\be
\ddot{\mathbf{W}}+3H\dot{\mathbf{W}}+\left(2H^{2}-H\frac{\dot{f}}{f}-
\frac{\ddot{f}}{f}\right)\mathbf{W}=0.  \dlabel{hom-W}
\end{equation}

 We  assume   $f\propto a\mtwo$ so that the last
 term of \eq{hom-W} vanishes and  there is a solution with
$\dot\bfw=0$. The other solution is decaying, $\bfw\propto a\mthree$, and
 corresponds to
$B_i=$\,constant. With the action \eqreff{action} this
can be set to zero
by a  gauge transformation and has no  physical effect. We therefore take
$\bfw(t)$ to be constant.
(For hybrid inflation with the waterfall coupling to $\bfw$ both solutions are
physical \cite{p12jcap} but we  assume that the decaying solution is anyhow negligible.)
It would make no essential difference if  we allow $f\propto a^\alpha$
with $\alpha$ slightly different from $-2$, or more generally if we just assume
\be
H\dot W \simeq - \left(2H^{2}-H\frac{\dot{f}}{f}-\frac{\ddot{f}}{f}
\right)\mathbf{W},\qquad H\mone|\dot\bfw| \ll W\equiv |\bfw|
. \end{equation}

We assume that the right hand side of \eq{hom-phi} is small enough that the approximation
$3H\dot\phi\simeq -V'$ still applies. Then
$f'/f \simeq  2/\sqrt{2\epsilon} \mpl$ and treating the right hand side as a first
order perturbation we have
\be
3H\dot \phi \simeq - V' +  \frac{18 H^2 W^2}{\sqrt{2\epsilon} \mpl}\simeq -V'
. \dlabel{homphi2} \end{equation}

The energy density of the gauge field  is \cite{fF2curv}
$\rho_{W}(\bfx,t)=-f^{2}F_{\mu\nu}F^{\mu\nu}/4$. Smoothed on a super-horizon scale
this gives
\be
\rho_{W}=\frac92 H^2 W^2
\dlabel{rhow}. \end{equation}
Using this result for the unperturbed energy density, we see that  our
assumption that the right hand side of
 \eq{homphi2} is dominated by the first term is consistent if
\be
\frac{2\rho_W}{\epsilon\rho} \equiv \frac{3 W^2}{\epsilon\mpl^2} \ll 1 ,
\dlabel{rhowcon}\end{equation}
which we assume.

\section{Perturbed universe}

\dlabel{perturbed}

Evaluating  \eqs{EoM-phi}{EoM-W} to first order in $\delta\phi$ and
dropping the right hand side of \eq{EoM-W} we get
\begin{eqnarray}
\delta\ddot\phi + 3H\delta\dot\phi - \nabla^2\delta \phi
&=& -V'' \delta \phi +  \frac2{\sqrt{2\epsilon}\mpl} \delta \[ \left|
\dot{\mathbf{W}}_\bfx+3H \mathbf{W}_\bfx \right|^{2} \] - \nonumber \\
& - &\frac{8H\mathbf{W}_\bfx}{\sqrt{2\epsilon}\mpl}  \left(
\dot{\mathbf{W}}_\bfx+3H\mathbf{W}_\bfx\right)
\frac{\delta\dot\phi}{\dot\phi}  \dlabel{a9} \\
\delta\ddot \bfw+3H \delta\dot\bfw - \nabla^2 \delta \bfw
&=& - \frac{ 2\bfw_\bfx }{ \sqrt{2\epsilon}\mpl } \[ \delta\ddot\phi
-3H\delta\dot\phi - \nabla^2 \delta\phi \]
 \dlabel{a10} \\
\bfw_\bfx &\equiv& \bfw(\bfx,t) \equiv \bfw + \delta\bfw(\bfx,t) ,
\end{eqnarray}
where $\bfw$ is the unperturbed value.

This ignores the metric perturbation (back-reaction).
Since slow-roll inflation is supposed to be a good approximation, the effect of the
 metric perturbation in  \eqs{basic1}{basic2} is significant
only well after horizon exit, and
is then given by  \eq{deltaln}. Including it, \eqs{a9}{a10} become well after
horizon exit
\begin{eqnarray}
\delta\ddot\phi + 3H\delta\dot\phi
&=& -\(V''+6H^2\epsilon \) \delta \phi \nonumber + \\
&+&  \frac2{\sqrt{2\epsilon}\mpl} \delta \[ \left|
\dot{\mathbf{W}}_\bfx+3H \mathbf{W}_\bfx\right|^{2} \]
-\frac{8H\mathbf{W}_\bfx}{\sqrt{2\epsilon}\mpl}  \left(
\dot{\mathbf{W}}_\bfx+3H\mathbf{W}_\bfx\right)
\frac{\delta\dot\phi}{\dot\phi}  \dlabel{a93} \\
\delta\ddot \bfw_\bfx+3H \delta\dot\bfw_\bfx
&=& - \frac{ 2\bfw_\bfx }{ \sqrt{2\epsilon}\mpl } \[ \delta\ddot\phi
-3H\delta\dot\phi + \frac32
H  \dot\epsilon \delta\phi \]
, \dlabel{a103}
\end{eqnarray}
where in \eq{a93} the  back-reaction $-6H^2\epsilon\delta\phi$ is the same as
in \eq{phiddot2}.

Let us first set the  right hand sides of \eqs{a9}{a10} to zero.
We saw in Section \ref{homphi2} how the vacuum fluctuation of $\delta\phi$
is then converted at horizon exit
to a nearly gaussian classical perturbation, and the same thing happens
to the  vacuum fluctuation of $\bfw_\bfk$  \cite{ouranis}.\footnote
{In contrast with the case for scalar field perturbations \cite{seery},
the non-gaussianity of $\delta\bfw$ has yet to be evaluated, but we will assume
that it still has a negligible effect.}
Its  left- and right-handed components have the same
mode function $W(k,t)=\phi_0(k,t)$ in \eq{phi0}, which gives
well  after horizon exit
\be
 \vev{ \delta W_\bfk^i \delta W_\bfkp^j }
= (2\pi)^3 \(\delta^{ij} - \hat k^i \hat k^j \)
\delta^3(\bfk+\bfkp) (2\pi^2/k^3)(H/2\pi)^2   \dlabel{wspec}
. \end{equation}

Now we consider the effect of the right hand sides of \eqs{a9}{a10}.
In the regime  $k\gg aH$, spacetime curvature is negligible and we deal with field
theory in flat spacetime.
The theory involves massless gauge bosons and the nearly massless
inflaton particles ($|V''|\ll (k/a)^2$
which propagate as  nearly free particles (perturbative regime), justifying
the initial condition   \eqreff{phiinit}.

To discuss the subsequent evolution, let us  take $\delta\phi(\bfx,t)$
and $\delta\bfw(\bfx,t)$ to include only  modes with $k$ in some small interval,
so that there is a well-defined
epoch of horizon exit.
 In the (still quantum) regime $k\sim aH$,
 $H$ is the  only relevant scale and  we have typical magnitudes
\begin{eqnarray}
\delta\phi \sim\delta \bfw\sim H,\qquad
\delta\dot\phi \sim\delta \dot\bfw\sim H^2,\\
\delta\ddot\phi \sim\delta \ddot\bfw\sim \nabla^2 \delta\phi
\sim \nabla^2 \delta\bfw \sim H^3
. \end{eqnarray}
Using these with \eq{rhowcon}, we see that
 the right hand sides of
\eqs{a9}{a10} are much smaller than $H^3$,
and hence have only a small effect.\footnote
{The findings of this  and the previous paragraph remain valid when the last term of
\eq{a10} is included, and that term is negligible in the super-horizon regime
that we are about to discuss.}
 It is therefore reasonable to  assume that the perturbations become
 classical  soon after  horizon
exit,  with \eqs{pphi0}{wspec} initially a good approximation giving
typical values $|\delta\phi| \sim|\delta \bfw|\sim H$.

The evolution of classical perturbations is described by
\eqs{a93}{a103} with $k=0$.
One of their solutions is
\begin{eqnarray}
3H\delta\dot\phi &\simeq&  - \( V'' + 6H^2\epsilon \) \delta\phi+
\frac{18 H^2}{\sqrt{2\epsilon}\mpl}   \delta\( W^2(\bfx,t) \) \\
&\equiv& - \( V'' + 6H^2\epsilon \) \delta\phi+
\frac{18 H^2}{\sqrt{2\epsilon}\mpl}
\( 2\bfw\cdot\delta\bfw + \delta\bfw\cdot\delta\bfw \)
\dlabel{a92}  \\
\delta\dot\bfw  &\simeq& -\frac{2\bfw}{\sqrt{2\epsilon}\mpl}
 \delta\dot\phi , \dlabel{a102}
\end{eqnarray}
which implies $|\delta\dot\phi|\ll H|\delta\phi|$
and $|\delta\dot\bfw|\ll H|\delta\bfw|$.
The  self-consistency of this solution can be
checked by inserting it into the right hand sides of \eqs{a92}{a102}.
It is presumably  picked out by the initial condition,
just as in the case of slow-roll inflation.

Since all of the quantities appearing in the second term of \eq{a92}
 vary slowly on the Hubble timescale,
we will  take them all to be constant, giving
\begin{eqnarray}
\delta\phi_\bfk(t\sub{end}) &=& \delta\phi_\bfk\su 0(t\sub{end}) +
\frac{6N(k)}{\sqrt{2\epsilon}\mpl}   \( \delta(W^2)\)_\bfk
, \end{eqnarray}
where $\delta\phi_0(t\sub{end})$ is the slow-roll result.

\section{Contribution of $\delta\bfw$ to the  curvature perturbation}

\dlabel{contribution}

Now we calculate $\zeta$
at the end of slow-roll inflation, using \eq{zeta}. Smoothed on a super-horizon
scale the energy density is $\rho=\rho_\phi+\rho_W$
\begin{eqnarray}
\delta\rho(\bfx,t\sub{end}) &\simeq& V'\delta\phi(\bfx,t\sub{end})  + \frac92 H^2\delta(W^2) \\
&\simeq& V' \delta\phi_0(\bfx,t\sub{end}) + 18N(k)H^2 \delta(W^2)
+ \frac92 H^2\delta(W^2) \\
&\simeq& V' \delta\phi_0(\bfx,t\sub{end}) + 18N(k)H^2 \delta(W^2)
. \end{eqnarray}
This gives
 $\zeta_\bfk(t\sub{end})=\zeta^\phi_\bfk +\zeta\su W_\bfk$,  where
\begin{eqnarray}
\zeta^W_\bfk &=& \frac12 C(k) \delta(W^2)_\bfk \nonumber \\
&=& C(k) \[ \bfw\cdot\delta\bfw_\bfk + \frac12 \[ \(\delta\bfw\)^2\]_\bfk \]
\dlabel{zetawofc},
\end{eqnarray}
with
\be
C(k)  =     \frac{6N(k)}{\epsilon\mpl^2} \dlabel{cofk}
. \end{equation}

We can now  calculate the spectrum and bispectrum of
$\zeta_W$ using \eq{wspec} and the analogues of \eqs{phi3}{phicorr}.
The result depends on the value of the  unperturbed field $\bfw$,
and in particular on its magnitude $W$.

\subsection{Working in a finite box}

An  unperturbed quantity  is the zero mode of its Fourier expansion, and at this
point we need to remember that within the cosmological context that
expansion has to be done within
 some   box of finite coordinate size $L$  \cite{mybox}.
Assuming nearly exponential inflation, the box
 leaves the horizon $N_L(k)\equiv
\ln(kL)$ $e$-folds
before the scale $k$.
To    avoid making unverifiable assumptions about
an era that will never be constrained by observation (and, as we will see,
also in to simplify the calculation) one should  take  $k_0L$ to be
as small as is consistent with the requirement that the
periodic boundary condition implied
by the use of the Fourier expansion have  a negligible effect  (minimal box).
 Demanding say $1\%$ accuracy in the
calculation, it should be enough to choose
$k_0L\sim 100$ corresponding to $\ln(k_0L) \sim 5$.
After choosing $L$ one writes  for a given quantity
$g(\bfx,t)= g(t) + \delta g(\bfx,t)$, where the `unperturbed' value
$g(t)$ is the average of $g$ within the box.

Expectation values like \eqreff{vev}  are in general  defined
with respect to an ensemble of universes,
one of which is the observed universe.
But  under the usual assumption that perturbations originate
as a vacuum fluctuation
the translation invariance of the vacuum makes them translation invariant.
As a result the expectation values can be
 defined as spatial averages within a single realisation of the ensemble.
Thus the $\vev{(\delta g)^2}$
can be defined as the spatial average within the box,
 $\vev{\delta g(\bfx+{\bf X}) \delta g(\bfx) }$ can be defined as the average
with respect to ${\bf X}$ and so on \cite{book}.

Keeping only classical modes, \eqs{pphi0}{wspec} give
\begin{eqnarray}
\vev{(\delta\phi)^2)} &=& \int^k_L {\cal P}_\phi(k) dk/k =\ln(k_0L) (H/2\pi)^2  \\
\vev{ |\delta \bfw|^2 } &\simeq& 2 \ln(k_0L)(H/2\pi)^2 \dlabel{deltawsq1}
. \end{eqnarray}
For the minimal box size this corresponds to typical values
$|\delta\phi|\sim |\delta\bfw|\sim H$.

\subsection{The value of  $W$}

As was discussed in \cite{mybox}, there
 are two possible viewpoints about the  magnitude of an  unperturbed
field like $\bfw$.\footnote
{These  apply to
 any non-inflaton field that acquires a perturbation from its vacuum
fluctuation.
The inflaton field is an exception because the inflation model and the cosmology
determines its value $N(k_0)$ before the end of inflation.
We ignore the issue of anthropic selection, assuming that statistical anisotropy
(like non-gaussianity \cite{book})
is neither favoured or disfavoured in that respect.}
One is to regard the magnitude
 $W$ as  a free parameter, that one gets to choose just like
the masses and couplings appearing in the action.
The other viewpoint  is to
 estimate the likely value of $W$,  assuming that we are at a typical
location within  a box whose   size $M$ is very much bigger than the size $L$ of the
minimal box within which the calculations are done.

Adopting the second  viewpoint one has to make an assumption about the evolution
of the universe long before the observable universe leaves the horizon.
The usual assumption is that there is almost
exponential  inflation,  beginning
$N_M$ $e$-folds before the observable
universe leaves the horizon with $N_M$ fairly large. One can then estimate
the likely value with suitable assumptions about the relevant physics during
those $e$-folds. In our case, let us assume that
the dependence $f\propto a\mtwo$
continues to (at least approximately) hold during those $e$-folds.
Then,
after smoothing $\bfw$ on the scale $L$ \eq{wspec} gives for the average within
the exponentially inflated patch
\be
\vev{ |\delta \bfw|^2 } \simeq 2 \ln(M/L)(H/2\pi)^2 \simeq 2 N_L^2(k) (H/2\pi)^2
, \dlabel{deltawsq} \end{equation}
The expected value of $W^2$ if our location is typical and therefore
\be
W^2 = W_M^2 + \ln(M/L) (H/2\pi)^2 > \ln(M/L)(H/2\pi)^2
= N_M(k_0) (H/2\pi)^2
, \dlabel{nmeq} \end{equation}
where $W_M$ is the average within the inflated patch.
We conclude that  if we occupy a typical location within an
inflated patch that left the horizon many $e$-folds before the observable
universe, then  $W\gg H$.

\subsection{The case $W\gg H$}

If $W\gg H$, the second term in the square bracket of \eq{zetawofc}
can be treated as a first-order perturbation. Then,
assuming that $\zeta_W$ gives the only contribution to the
anisotropy of ${\calp_\zeta}$,  Eq.~(39) of \cite{ouranis}  gives
\be
{\calp_\zeta}(\bfk) = {\calp_\zeta}(k)  \[ 1 + g_*(k) \(\hat\bfw\cdot \hat \bfk\)^2 \] \dlabel{ourcalpz}
, \end{equation}
with
\begin{eqnarray}
g_*(k) &=&  - {\cal P}_{\zeta_W}(k)/{\calp_\zeta}(k)  \dlabel{gofc} \\
 &=& -\frac{ C^2(k) W^2 }{ {\calp_\zeta}(k)} \( \frac H{2\pi} \)^2 \dlabel{calpzw2} \\
&=& -48N^2(k) \frac{\rho_W}{\epsilon\rho} \frac{{\cal P}_{\zeta_\phi}}{{\calp_\zeta}(k)}. \dlabel{ourg}
\end{eqnarray}
Here, ${\cal P}_{\zeta_\phi}$ is given by \eq{calpzphidef}, and is independent of $k$ since we
are taking $H$ and $\epsilon$ to have negligible time-dependence.

The anisotropy
 is of the form \eqreff{gstardef} but with a strongly scale-dependent $g_*(k)\propto N(k)$.
This  has not been compared with observation but the constraint is
presumably similar to the scale-independent case,
 currently  $|g_*|\lsim 10\mone$.
Another constraint  comes from the strong scale-dependence of
 ${\cal P}_{\zeta_W}(k)$, giving a contribution
$n(k)\supset  - g_*(k)$ to the spectral index. Using \eq{nobs},
 this  requires barring a cancellation
$|g_*(k_0)|\lsim 0.03$.
In any case, one certainly needs $|g_*(k_0)|\ll 1$, which with
 \eq{ourg} is  stronger than \eq{rhowcon}, justifying the latter.
Our assumption that the anisotropy of the expansion has a negligible effect
is also justified, because that anisotropy is
 presumably only of  order $(\rho_W/\rho)$, which presumably gives
a contribution $(\rho_W/\rho)({\cal P}_{\zeta_\phi}/{\calp_\zeta}(k))$ to $g_*$ which is
 smaller than the one that we have calculated.

Using  ${\calp_\zeta}(k_0)=(5\times 10\mfive)^2$,
\be
g_*(k_0) \simeq
 -1.3\times 10\mthree \( \frac{N(k)}{60} \)^2 \( \frac{{\cal P}_{\zeta_\phi}}{{\calp_\zeta}(k_0)}
\)^2 \( \frac WH \)^2  \dlabel{geqn}
. \end{equation}
If we were to assume ${\cal P}_{\zeta_\phi}={\calp_\zeta}$ as in \cite{marco},
the  observational bound $|g_*|\lsim 10\mone$ leads to two conclusions as
those authors notice. First,  our assumption
$W\gg H$ would be only marginally allowed. Also, from
 \eq{nmeq}, one sees that $N_M$ to be very large if we occupy a typical
location.

Now we calculate the contribution of
$\zeta_W$ to  the bispectrum $B_\zeta$ on cosmological scales.
To simplify the calculation we  set $N(k)=N(k_0)$.
 Taking ${\calp_\zeta}(k)$ to be scale-independent and defining
\be
\frac65f\sub{NL}(\bfk_1,\bfk_2,\bfk_3) \equiv \frac1{4\pi^4}\frac {B_\zeta}
{{\calp_\zeta}^2}\frac{\prod k_i^3}{
\sum  k_i^3}
, \end{equation}
Eq.~(41)  of  \cite{ouranis} gives
\be
f\sub{NL}(\bfk_1,\bfk_2,\bfk_3) = f\sub{NL}  \( 1 +
f\sub{ani}(\bfk_1,\bfk_2,\bfk_3) \)
, \dlabel{fnlanis} \end{equation}
where the constant prefactor is given by
\begin{eqnarray}
f\sub{NL}&=&  \frac56 \frac{g_*^2(k_0)}{C(k_0)W^2} \dlabel{fnl1x} \\
&=& -10 N(k_0)  g_*(k_0) ({\cal P}_{\zeta_\phi}/{\calp_\zeta}), \dlabel{fnl2x}
\end{eqnarray}
and
\be
f\su{ani} = \frac{
 - (\hat \bfa \cdot \hat \bfk_1)^2
-(\hat \bfa \cdot \hat \bfk_2)^2 + (\hat \bfk_1 \cdot \hat \bfk_2)
(\hat \bfa \cdot \hat \bfk_1)(\hat \bfa \cdot \hat \bfk_2)  }{
\sum k_i^3/ k_3^3  }
+ \mbox{ 2 perms.}
. \dlabel{fnlanis2} \end{equation}
This  expression has not yet been confronted with data, but
the bound is presumably similar to the one that  takes $g_*(k)$ to be
constant,  currently $|f\sub{NL}|\lsim 10^2$.

\subsection{The case $W \lsim  H$}

Consider now the case $W=0$. Since there is no preferred direction,
$\zeta_W$ is statistical isotropic. Using Eq.~(40)  of \cite{ouranis} we have
\begin{eqnarray}
\frac{{\cal P}_{\zeta_W}}{{\calp_\zeta}}  & =& \frac{{\cal P}_{\zeta_W}\su{loop} }{{\calp_\zeta}}
\equiv \frac43 C^2(k) \( \frac H{2\pi} \)^4 \ln(kL)/{\calp_\zeta}
\dlabel{calpz1}  \\
&=& 3N^2(k){\calp_\zeta}  \ln(kL) \(\frac{ {\cal P}_{\zeta_\phi}(k)}{{\calp_\zeta}} \)^2 \ll 1
. \dlabel{calpz2} \end{eqnarray}
Using \eq{wspec} and the analogues of \eqs{phi3}{phicorr} we find
\begin{eqnarray}
f\sub{NL}&=&f_{NL}\su{loop} \equiv \frac23 C^3 \( \frac H{2\pi} \)^6
\frac{ \ln(kL) }{ ({\calp_\zeta})^2 }
\[
  1 +
\( \frac{ \(\hat \bfk_1\cdot \hat \bfk_2\)^2 }{
\sum_i k_i^3/k_3^3 } + \mbox{c.p.}
\) \]  \dlabel{fnl1} \\
&=&  \frac{3\half}4   \(\frac{{\cal P}_{\zeta_W}}{{\calp_\zeta}} \)^{3/2}
 \ln\mtwo(kL) {\calp_\zeta}\mhalf
\[
  1 +
\( \frac{ \(\hat \bfk_1\cdot \hat \bfk_2\)^2 }{
\sum_i k_i^3/k_3^3 } + \mbox{c.p.}
\) \]
\dlabel{fnl2}  \\
&=& \frac94 N^3 {\calp_\zeta} \( \frac {{\cal P}_{\zeta_\phi}}{{\calp_\zeta}} \)^3 \ln(kL)
\[
  1 +
\( \frac{ \(\hat \bfk_1\cdot \hat \bfk_2\)^2 }{
\sum_i k_i^3/k_3^3 } + \mbox{c.p.}
\)\] \ll 1
\end{eqnarray}
In the general case, ${\cal P}_{\zeta_W}$ and $f\sub{NL}$ are the sum of the `tree' and
`loop' contributions.

\subsection{How to make the calculation more accurate}

\dlabel{just}

The calculation that we have presented is sufficiently accurate, unless and
until  statistical anisotropy is observed. If that does happen a more accurate
calculation may be required.

Such a calculation should in principle
begin with the generation of the classical
perturbations $\delta\phi$ and $\delta\bfw$ from the vacuum, along the lines
of \cite{lms}.
 It may in principle generate significant time-dependence and
correlation for these quantities, when they first become classical.
But  the analysis of Section \ref{pure} suggests
that it will instead confirm  the  result obtained there;
nearly time-independent and uncorrelated perturbations with the spectra
$(H/2\pi)^2$ defined in \eqss{vev}{pphi0}{wspec}.

After the perturbations become classical, their evolution
is given by  \eqs{a93}{a103}, which can
be solved numerical to determine the correlators of the perturbations
at the end of inflation. That allows one to determine $\delta\rho$,
and working to first order in  $\zeta$ we can use \eq{zeta}
to determine ${\cal P}_{\zeta_W}$ and $f\sub{NL}$.

Instead of working with the perturbations,
one can use the $\delta N$ formula \cite{ouranis}
\begin{eqnarray}
\zeta(\bfx,t) &=& \zeta_\phi(\bfx,t) + \zeta_W(\bfx,t) _+
+ \sum_i \frac12 N_{\phi\phi}(t) [ \delta\phi_*(\bfx) ]^2
+ \frac12 N_{\phi i} \delta\phi_*(\bfx) \delta W_i^*(\bfx)  \dlabel{deltanours}\
 \\
\zeta_\phi(\bfx,t) &\equiv&   N_\phi(t) \delta\phi_*(\bfx), \\
\zeta_W(\bfx,t) &\equiv&  \sum_i N_i(t) \delta W_i^*(\bfx) + \frac12 \sum_{ij}
N_{ij}(t) \delta W_i^*(\bfx) \delta W_j^*(\bfx)
. \end{eqnarray}
Here $N$ is defined by \eq{deltaN} with $t_*$ the epoch of horizon exit for
the scale of interest, and the
subscripts on $N$ denote partial derivatives evaluated at the unperturbed
point  in the field space.
But as we now argue,  the perturbative approach of the previous
paragraph is expected to be adequate, i.e.\ the first-order
\eq{zeta} is expected to be adequate.

The  validity of \eq{zeta} for $\zeta_\phi$
is a standard result but we need to justify its use for $\zeta_W$.
The second order correction is presumably of order  $\zeta_W^2$, corresponding to
a tiny fractional correction of order $\zeta_W$ that can certainly be ignored
for the evaluation of ${\cal P}_{\zeta_W}$. To see whether it can be
ignored for $f\sub{NL}$, let us first pretend that $\bfw$ is a scalar field.
Then, setting $C(k)$ to a constant and assuming that the first term
of \eq{zetawofc} dominates, \eqs{zetawofc}{fnl1x} give\footnote
{If instead the second term dominates, it is solely responsible for
$f\sub{NL}$ and  the first-order formula will certainly be adequate.}
\be
\zeta_\sigma(\bfx) = \zeta\sub g(\bfx)
+ \frac35 f\sub{NL} \( \frac{{\calp_\zeta}}{{\cal P}_{\zeta_\sigma}} \)^2
\zeta\sub g^2(\bfx)
 ,\end{equation}
where $\zeta\sub g\equiv C W \delta W$.
The first-order formula will therefore be adequate unless
\be
f\sub{NL} \lsim \( \frac{{\cal P}_{\zeta_\sigma}}{{\calp_\zeta}} \)^2
. \end{equation}
Since we need $|f\sub{NL}|\gsim 1$ for it to be observable,
we conclude that
the first-order formula will be adequate for the evaluation
of $f\sub{NL}$ unless $|f\sub{NL}|\sim 1$ {\em and} $\zeta_W$
is the dominant contribution to $\zeta$. Except for the second proviso
this is a standard result, that was first recognised in the context of
the curvaton scenario \cite{beforelr,lr}.

Keeping the vector nature of $\bfw$, \eqs{zetawofc}{fnl1x} give
\begin{eqnarray}
\zeta_\sigma(\bfx) &=& \zeta\sub g(\bfx)
+ \frac35 f\sub{NL} \( \frac{{\calp_\zeta}}{{\cal P}_{\zeta_\sigma}} \)^2
\tilde\zeta\sub g^2(\bfx) \\
\zeta\sub g(\bfx) &=& C\bfw\cdot \delta\bfw(\bfx)) \\
\tilde \zeta\sub g(\bfx)  &=& C W |\delta \bfw(\bfk)|
 \end{eqnarray}
At a typical location, $\tilde \zeta\sub g\sim \zeta\sub g$, and the
previous conclusion about the validity of the first-order formula still
applies.

\section{Conclusion}

\dlabel{conc}

Working exclusive with the classical perturbation of $\bfw$, we  have
presented a
rather complete   calculation of the contribution to $\zeta_W$ that is
generated during slow-roll inflation. In the regime $W\gg H$ we
 have reproduced the result of
\cite{beforemarco,marco} for the spectrum (\eq{ourg}) and of \cite{marco}
for the bispectrum (\eqs{fnlanis}{fnl2x}, discussing for the first time
the assumptions that are needed to obtain it.

The field $\bfw(\bfx,t)$  may survive after the end of
slow-roll inflation, and generate further contributions to $\zeta$.
The effect of these is to change $C(k,t)$, so that
 \eqs{gofc}{fnl1x} as well as \eqss{calpz1}{fnl1}{fnl2}
still hold but with $C$ the final value.

A contribution $\zeta\sub{end}$ may be generated during the
waterfall that ends hybrid
inflation. It can be calculated from the `end-of-inflation' formula,
provided that the waterfall is sufficiently brief which requires
$H\lsim 10^{-9}\mpl ({\cal P}_{\zeta\sub{end}} /{\calp_\zeta} )\half$.\footnote
{This result is an obvious extension of the one that was derived in
  \cite{p11} with the assumption
${\cal P}_{\zeta\sub{end}} ={\calp_\zeta} $. It follows from the fact that the
the duration of the waterfall cannot be much less than $1/m$.} The result is
\be
C\sub{end} = -\frac{h^2}{\sqrt{2\epsilon}\mpl mg}
\end{equation}

If the gauge symmetry is spontaneously broken after inflation, the
curvaton mechanism may generate another contribution after inflation.
With the simplest  assumption that $\bfw$ is nearly time-independent until
it begins to oscillate
\be
C\sub{curv} = \frac{\rho_W\su{dec}}{\rho\su{dec}}
\frac1{3 W^2}
, \end{equation}
where `dec' denotes the epoch just before the decay of $\bfw$.

Including both these contributions we have
\be
C = C\sub{sr} + C\sub{end} + C\sub{curv}
, \end{equation}
where $C\sub{sr}=6 N(k)/\epsilon \mpl^2$ is the contribution generated during
slow-roll inflation.
The extra contributions allow  the non-gaussianity to  be observable
even if $W\ll H$.

We close with an important comment, stemming  from \eq{geqn}.
On the reasonable assumption that there were some large number
$N_M$ of $e$-folds of inflation before the observable Universe left the
horizon, one expects $W/H\sim N_M\gg 1$. But then \eq{geqn} is compatible
with the observational bound on $g_*$ only if ${\cal P}_{\zeta_\phi}\ll{\calp_\zeta}$;
in other words, if the the observed curvature perturbation is mostly
generated {\em after} slow-roll inflation by for instance the
end-of-inflation or curvaton mechanism.
Our results therefore   suggest a dichotomy
regarding the generation of  observable statistical anisotropy of the curvature
perturbation.
Either the entire curvature perturbation $\zeta$ (both the isotropic and the
statistically anisotropic part) is likely to be generated during slow-roll inflation,
or it is likely to be generated afterwards.

\section{Acknowledgments}
DHL
 acknowledges support from the Lancaster-Manchester-Sheffield Consortium for
Fundamental Physics under STFC grant ST/J00418/1, and from
 UNILHC23792, European Research and Training Network (RTN) grant.
MK is supported by the grants CPAN (CSD2007-00042) and MICINN (FIS2010-17395)
also by the Academy of Finland grant 131454.

redpajama_set_name: RedPajamaArXiv


text:
Home - Chapters - Pre-employment screening - When is it unlawful to refuse to employ a candidate based on their criminal record?
Pre-employment screeningMay 01, 2019
When is it unlawful to refuse to employ a candidate based on their criminal record?
Discrimination on the basis of a candidate's criminal record may be unlawful in two situations.
Step-by-Step: How to request a criminal record check from a job candidate
Why should you undertake pre-employment screening? How to determine which pre-employment screening methods to use Background and identification checks Medical examinations Checklist: How to conduct a pre-employment medical examination legally Immigration status checks What happens if you allow an illegal worker to work for you? Checklist: How to manage the risk of contravening migration laws Pre-employment assessment and profiling How to ensure your pre-employment assessment and profiling tools are reliable Criminal record checks Step-by-Step: How to request a criminal record check from a job candidate When is it unlawful to refuse to employ a candidate based on their criminal record? Checklist: How to determine whether a candidate's criminal record is relevant What to do if a candidate refuses to undertake your pre-employment screening How to meet your privacy obligations during pre-employment screening
Checklist: How to determine whether a candidate's criminal record is relevant


text:
2013 Search Engine Ranking Factors
Every couple of years, Moz.com does a study about Ranking Factors meant to determine which attributes of pages and sites have the strongest connection with high Google ranking. There are two parts of the survey. The first part consists of a survey of professional SEOs, and the second is a large correlation study.
Some of the key findings of this study are as follows:
Page Authority correlates higher than any other metric we measured.
Social signals, especially Google +1s and Facebook shares are highly correlated.
Despite Penguin, anchor text correlations remain as strong as ever.
New correlations were measured for schema.org and structured data usage.
More data was collected on external links, keywords, and exact match domains.
During the survey, those who took part in it were asked to rate several factors on a scale from 1 to 10, according to how important Google's ranking algorithm was to them. The highest rated factors in the survey had average scores about 7 to 8, while less important factors ranged from 4 to 6.
To compute the correlations, a large set of keywords from Google Adwords that spanned a large range of search volumes across all topic categories were collected. Then, the top 50 organic search results from Google-US were collected in a depersonalized way. Keep in mind that all SERPs were collected after the Penguin 2.0 update, in early June.
All the factors were extracted from each search result that was going to be analyzed, and from that, the mean Spearman correlation across the entire data set was going to be computed. Basically, this is the same general process that Searchmetrics and Netmark used recently in their studies.
Now, To The Data!
Here's the first set of Mozescape link correlations:
Correlations: Page Level
Correlations: Domain Level
Page Authority is a Mozescape machine learning model that predicts ranking ability from links. It's Moz's highest correlation factor in their study.
In the survey, SEOs felt that links were very important:
In the last couple years, Google has put the smack down on over optimized anchor text. Even then, anchor text correlations for both partial and exact match were still pretty large and relevant in the data set:
SEOs that were surveyed though that an organic anchor text distribution is more important than the number of links:
Because the anchor text correlations were quite different between the Moz results and the Searchmetrics study, it was believed that Moz included navigational gueries while Searchmetrics removed them from its data.
On-page
There was a fairly high correlation between the keyword and the document, both with the TF-IDF score and the language model score. This high correlation was found with the title tag, the body of the HTML, the meta description and the H1 tags:
INSERT CORRELATION ON PAGE
SEOs also agreed t hat the keyword in the title were the most important factors on the page:
There was some additional on-page correlations computed to see where structured markup had any relationship to rankings. At this point, because the correlations were close to zero, it could be concluded that they are not used, at least at this point, as ranking signals.
Exact\Partial Match Domain
Recently, the ranking ability of exact and partial match domains (EMD/PMD) have been heavily debated by SEOs. It seems that Google is still adjusting their ranking ability. Before the June 25 update, in the data that was collected, it was found that EMD correlations to be quite high at 0.17.
This information is surprising, since the data given shows that EMD is decreasing. Because of this, after digging in, we see that the EMD percent has decreased over the last year (the blue line):
In the last few months, we've seen a see-saw pattern in the EMD correlations (red line) where they decreased, then rose back up again. This could be attributed to the decrease last fall to Google's EMD update. This increase in correlation says that the EMDs are still there and are still ranking higher overall in the SERPS, despite the fact they are less prevalent.
In 2011, the social signals were some Moz's highest correlating factors:
SEOs believe that social signals aren't a very important factor in the overall algorithm:
In this situation, this could be a place where the correlation is explainable by other factors, like links, and there may not be direct causation.
Takeaway And The Future of Search
Based on Moz's survey respondents, this is how Google's overall all algorithm breaks down:
According to Moz, this is what they see:
Links are still believed to be the most important part of the algorithm (approximately 40%).
Keyword usage on the page is still fundamental, and other than links is thought to be the most important type of factor.
SEOs do not think social factors are important in the 2013 algorithm (only 7%), in contrast to the high correlations.
It's believed by SEOs that there will be a shift away from traditional ranking factors (such as anchor text, exact match domains, etc) to deeper analysis of a site's "value" to users, authorship, social signals, and structured data:


text:
Trending: Email Marketing Customer Satisfaction Small Biz in 15 Online Marketing
How Mansa Systems Crossed $2M Leveraging Salesforce AppExchange
Published: May 21, 2013 Last Updated: Oct 31, 2022 by Sramana Mitra In Startup Advice 3
In 2006, the concept of cloud computing was just beginning to gain ground. Siva Devaki saw the Cloud as a means to achieve his longtime dream of building a full-service IT company. He decided to leave his secure career working for large companies such as Oracle and VeriSign. Drawing upon his past experience with Enterprise Resource Planning (ERP) and Customer Relationship Management (CRM) applications, Siva launched his venture that same year.
Initially, Siva focused his work on the Cloud computing niche. In this high growth market, Siva noted that Salesforce CRM was already quite strong, but lacked the same focus in other related areas. Here, Siva took advantage of AppExchange, a Salesforce-run marketplace of business applications.
Through AppExchange, partners can create specific apps to enhance Salesforce for businesses. Users can shop for available apps, search partners to develop a custom solution or list an app of their own. Siva quickly began work with Salesforce to build apps for enterprise and small and medium sized enterprise (SME) customers.
Today, San Francisco-based Mansa Systems is a provider of Cloud, mobile and social enterprise solutions. Mansa focuses on the creation of apps to fulfill the specific needs of both customers and developers. By leveraging partner solutions like Watchdox, SendGrid, Slideshare, Citrix, and Amazon AWS, each app is guaranteed to meet a certain proven user need.
In recent months, Mansa Systems has become an official Salesforce ISV partner. The company continues to work with Salesforce in the enterprise app space, creating, updating and launching through the Force.com platform. They have successfully launched several new apps at affordable price points.
Among the company's releases are two apps built to deal with Salesforce's data limitations. The Cloud Drop app expands file storage space and file size limits for teams, as well as enables folder hierarchies to better manage storage. Cloud Drop also provides service at the considerably lower price of 10 cents per GB, compared to $5 with Salesforce. MassMailer eliminates Salesforce email limitations for bulk and transactional emails. The app offers white label emailing for consistent branding.
Mansa offers additional apps to manage business data with Salesforce. Webinar2Lead saves time by automatically syncing all GoToWebinar data in order to align marketing efforts with sales for only $14.99 per month. Similarly, Slide2Lead offers auto sync for SlideShare leads, campaigns, documents and statistics, eliminating manual imports and exports.
A final recent release called 37 Seconds increases success with inbound leads by calling new leads as they are received. Calls can be made from any location and routed to a mobile or office line and are automatically recorded for ease tracking. This increased speed of response is a proven means to achieving sales, for only $4.99 a month.
Each app is priced according to its own specific attributes, using a set bracket for number of users or charging a percentage of the partner's original price. As a general formula, service costs are either project-based or calculated according to time and materials spent – both varying from project to project.
In addition to their relationship with Salesforce, Mansa targets mid-size companies in the financial services and public sectors. Their current verticals include education, non-profits and health care and life sciences. Additional leads are generated via company apps in AppExchange and cross-selling current services.
Mansa provides the high-level services of a large IT firm as well as off-the-shelf apps as solutions. This dual specialization sets Mansa apart from competitors such as Appiro, Bluewolf and Astadia. Mansa continues to develop a personalized business by constructing integration templates or connectors for specific areas, which will provide new leads to specialized customers and projects. Through a global delivery framework, Mansa expands customer options by providing expertise on a worldwide scale.
Mansa already lists prominent enterprise-level names Paypal/eBay, Logitech and Topcon among their customers. Wholly self-funded, the company has already passed the $2 million revenue mark by marketing primarily to Salesforce customers through the AppExchange.
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Sramana Mitra
Sramana Mitra is the founder of One Million by One Million (1M/1M) initiative, a virtual incubator that aims to help one million entrepreneurs globally to reach $1 million in revenue and beyond. She is a Silicon Valley entrepreneur and strategist who writes the blog Sramana Mitra On Strategy.
manish goyal
I run a small paints manufacturing business in Kolkata with about 20 field sales people. Is there any app which can help me keep track of them and also orders and collections
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